## Mathematics (math) updates on the arXiv.org e-print archive



Mathematical Selection is a method in which we select a particular choice from a set of such. It have always been an interesting field of study for mathematicians. Combinatorial optimisation is the practice of selecting the best constituent from a collection of prospective possibilities according to some particular characterization. In simple cases, an optimal process problem encompasses identifying components out of a finite arrangement and establishing the function's significance in possible to lessen or achieve maximum with a functional purpose. To extrapolate optimisation theory, it employs a wide range of mathematical concepts. Optimisation, when applied to a variety of different types of optimization algorithms, necessitates determining the best consequences of the specific predetermined characteristic in a particular circumstance. In this work, we will be working on one similar problem - The Maximal Stretch Problem with computational rigour. Beginning with the Problem Statement itself, we will be developing numerous step - by - step algorithms to solve the problem, and will finally pose a comparison between them on the basis of their Computational Complexity. The article entails around the Brute Force Solution, A Recursive Approach to deal with the problem, and finally a Dynamically Programmed Approach for the same.

In this article, a novel barrier function is introduced to convert the box-constrained convex optimization problem to an unconstrained problem. For each double-sided bounded variable, a single monomial function is added as a barrier function to the objective function. This function has the properties of being positive, approaching zero for the interior/boundary points and becomes very large for the exterior points as the penalty parameter approaches zero. The unconstrained problem can be solved efficiently using Newton's method with a backtracking line search. Experiments were conducted using the proposed method, the interior-point for the logarithmic barrier (IP), the trust-region reflective (TR) and the limited-memory Broyden, Fletcher, Goldfarb, and Shanno for bound constrained problems (LBFGSB) methods on the convex quadratic problems of the CUTEst collection. Although the proposed method was implemented in MATLAB, the results showed that it outperformed IP and TR for all problems. The results also showed that despite LBFGSB was the fastest method for many problems, it failed to converge to the optimal solution for some problems and took a very long time to terminate. On the other hand, the proposed method was the fastest method for such problems. Moreover, the proposed method has other advantages, such as: it is very simple and can be easily implemented and its performance is expected to be improved if it is implemented using a low-level language, such as C++ or FORTRAN on a GPU.

Accurately charting the progress of oil production is a problem of great current interest. Oil production is widely known to be cyclical: in any given system, after it reaches its peak, a decline will begin. With this in mind, Marion King Hubbert developed his peak theory in 1956 based on the bell-shaped curve that bears his name. In the present work, we consider a stochasticmodel based on the theory of diffusion processes and associated with the Hubbert curve. The problem of the maximum likelihood estimation of the parameters for this process is also considered. Since a complex system of equations appears, with a solution that cannot be guaranteed by classical numerical procedures, we suggest the use of metaheuristic optimization algorithms such as simulated annealing and variable neighborhood search. Some strategies are suggested for bounding the space of solutions, and a description is provided for the application of the algorithms selected. In the case of the variable neighborhood search algorithm, a hybrid method is proposed in which it is combined with simulated annealing. In order to validate the theory developed here, we also carry out some studies based on simulated data and consider 2 real crude oil production scenarios from Norway and Kazakhstan.

Let $\xi$ be a real analytic vector field with an elementary isolated singularity at $0\in \mathbb{R}^3$ and eigenvalues $\pm bi,c$ with $b,c\in \mathbb{R}$ and $b\neq 0$. We prove that all cycles of $\xi$ in a sufficiently small neighborhood of $0$, if they exist, are contained in a finite number of subanalytic invariant surfaces entirely composed by a continuum of cycles. In particular, we solve Dulac's problem, i.e. finiteness of limit cycles, for such vector fields.

We prove that for every countable ordinal $\xi$, the Tsirelson's space $T_\xi$ of order $\xi$, is naturally, i.e., via the identity, $3$-isomorphc to its modified version. For the first step, we prove that the Schreier family $\mathcal{S}_\xi$ is the same as its modified version $\mathcal{S}^M_\xi$, thus answering a question by Argyros and Tolias. As an application, we show that the algebra of linear bounded operators on $T_\xi$ has $2^{\mathfrak c}$ closed ideals.

Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series, $$C(T):=\sum_{n=0}^\infty C_nT^n.$$ Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this paper we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We obtain new formulae which involve Catalan triangle numbers; the spectrum of $c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ in terms of Catalan polynomials ($\ast$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future.

We establish simple formulae for computing Finkelstein-Rubinstein signs for Skyrme fields generated in two ways: from instanton ADHM data, and from rational maps. This may be used to compute homotopy classes of general loops in the configuration spaces of skyrmions, and as a result provide a useful tool for a quantum treatment beyond rigid-body quantisation of skyrmions.

The present paper considers the model of a homogeneous bubble inside an unbounded isentropic compressible inviscid liquid. The exterior liquid is governed by the Euler equation while the free bubble surface is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. We first proved the local existence and uniqueness of the complete nonlinear system using energy methods under an iteration scheme. Then we proved the almost global existence of the solution and the radiative decay of bubble oscillation through a bootstrap argument. Except for the energy estimate, this bootstrap argument encompasses a generalized KSS (Keel-Smith-Sogge) estimate and the analysis of backward pressure wave using the method of characteristics, which are the novelty of the present paper.

We developed a generalized weighted $L^2_tH^j_x$-estimate, or the so-called KSS estimate, which extends the KSS estimate \cite{MR2015331} to nonlinear wave equations in exterior domains regardless of the boundary conditions, at the cost of only the appearance of a $L_t^2$ norm of the boundary value. To handle this boundary value, we establish a method of characteristics to study the backward pressure wave, which is then used to decouple the ODE of the boundary value from the hyperbolic system of backward and forward pressure wave. The analysis of backward pressure wave takes advantage of a change of variable between the backward and forward characteristics generated by the sound speed field in a geometric way. These two methods can not only be used for the bubble-liquid model studied in this paper, but are expected to be applied on other questions regarding nonlinear wave equations with complex boundary conditions.

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating $\mathbf{F}_p$- or BP-homology nor the associated Ext groups.

In this paper we show that the billiard ball map of the Liouville billiard tables of classical type on the ellipsoid is non-degenerate at the elliptic fixed point. As a corollary we obtain a spectral rigidity result.

We construct a new graph on 120 vertices whose quantum and classical independence numbers are different. At the same time, we construct an infinite family of graphs whose quantum chromatic numbers are smaller than the classical chromatic numbers. Furthermore, we discover the relation to Kochen-Specker sets that characterizes quantum cocliques that are strictly bigger than classical ones. Finally, we prove that for graphs with independence number is two, quantum and classical independence numbers coincide.

In this paper we provide an asymptotic theory for the symmetric version of the Kullback--Leibler (KL) divergence. We define a estimator for this divergence and study its asymptotic properties. In particular, we prove Law of Large Numbers (LLN) and the convergence to the normal law in the Central Limit Theorem (CLT) using this estimator.

Checking whether two quantum circuits are approximately equivalent is a common task in quantum computing. We consider a closely related identity check problem: given a quantum circuit $U$, one has to estimate the diamond-norm distance between $U$ and the identity channel. We present a classical algorithm approximating the distance to the identity within a factor $\alpha=D+1$ for shallow geometrically local $D$-dimensional circuits provided that the circuit is sufficiently close to the identity. The runtime of the algorithm scales linearly with the number of qubits for any constant circuit depth and spatial dimension. We also show that the operator-norm distance to the identity $\|U-I\|$ can be efficiently approximated within a factor $\alpha=5$ for shallow 1D circuits and, under a certain technical condition, within a factor $\alpha=2D+3$ for shallow $D$-dimensional circuits. A numerical implementation of the identity check algorithm is reported for 1D Trotter circuits with up to 100 qubits.

We show that cube-connected cycles graphs $CCC_n$ are distance-balanced, and nicely distance-balanced if and only if $n$ is even.

Selection of hyperparameters in deep neural networks is a challenging problem due to the wide search space and emergence of various layers with specific hyperparameters. There exists an absence of consideration for the neural architecture selection of convolutional neural networks (CNNs) for spectrum sensing. Here, we develop a method using reinforcement learning and Q-learning to systematically search and evaluate various architectures for generated datasets including different signals and channels in the spectrum sensing problem. We show by extensive simulations that CNN-based detectors proposed by our developed method outperform several detectors in the literature. For the most complex dataset, the proposed approach provides 9% enhancement in accuracy at the cost of higher computational complexity. Furthermore, a novel method using multi-armed bandit model for selection of the sensing time is proposed to achieve higher throughput and accuracy while minimizing the consumed energy. The method dynamically adjusts the sensing time under the time-varying condition of the channel without prior information. We demonstrate through a simulated scenario that the proposed method improves the achieved reward by about 20% compared to the conventional policies. Consequently, this study effectively manages the selection of important hyperparameters for CNN-based detectors offering superior performance of cognitive radio network.

We study the symmetric facilitated exclusion process (FEP) on the finite one-dimensional lattice $\lbrace 1,\dots ,N-1\rbrace$ when put in contact with boundary reservoirs, whose action is subject to an additional kinetic constraint in order to enforce ergodicity. We study in details its stationary states in various settings, and use them in order to derive its hydrodynamic limit as $N\to\infty$, in the diffusive space-time scaling, when the initial density profile is supercritical. More precisely, the macroscopic density of particles evolves in the bulk according to a fast diffusion equation as in the periodic case, and besides, we show that the boundary-driven FEP exhibits a very peculiar behaviour: unlike for the classical SSEP, and due to the two-phased nature of the FEP, the reservoirs impose Dirichlet boundary conditions which do not coincide with their equilibrium densities. The proof is based on the classical entropy method, but requires significant adaptations to account for the FEP's non-product stationary states and to deal with the non-equilibrium setting.

Fusion frames are a convenient tool in applications where we deal with a large amount of data or when a combination of local data is needed. Oblique dual fusion frames are suitable in situations where the analysis for the data and its subsequent synthesis have to be implemented in different subspaces of a Hilbert space. These procedures of analysis and synthesis are in general not exact, and also there are circumstances where the exact dual is not available or it is necessary to improve its properties. To resolve these questions we introduce the concept of approximate oblique dual fusion frame, and in particular of approximate oblique dual fusion frame system. We study their properties. We give the relation to approximate oblique dual frames. We provide methods for obtaining them. We show how to construct other duals from a given one that give reconstructions errors as small as we want.

This work focuses on non-adaptive group testing, with a primary goal of efficiently identifying a set of at most $d$ defective elements among a given set of elements using the fewest possible number of tests. Non-adaptive combinatorial group testing often employs disjunctive codes and union-free codes. This paper discusses union-free codes with fast decoding (UFFD codes), a recently introduced class of union-free codes that combine the best of both worlds -- the linear complexity decoding of disjunctive codes and the fewest number of tests of union-free codes. In our study, we distinguish two subclasses of these codes -- one subclass, denoted as $(=d)$-UFFD codes, can be used when the number of defectives $d$ is a priori known, whereas $(\le d)$-UFFD codes works for any subset of at most $d$ defectives. Previous studies have established a lower bound on the rate of these codes for $d=2$. Our contribution lies in deriving new lower bounds on the rate for both $(=d)$- and $(\le d)$-UFFD codes for an arbitrary number $d \ge 2$ of defectives. Our results show that for $d\to\infty$, the rate of $(=d)$-UFFD codes is twice as large as the best-known lower bound on the rate of $d$-disjunctive codes. In addition, the rate of $(\le d)$-UFFD code is shown to be better than the known lower bound on the rate of $d$-disjunctive codes for small values of $d$.

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Non-oscillatory reconstruction is achieved through an adaptive order weighted essentially non-oscillatory (WENO-AO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions (RBF) and Gaussian process (GP) modeling, which we call KFVM-WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple-yet-effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message passing interface (MPI) is also provided.

We consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss on the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one end point of the spatial interval. In particular, we establish uniqueness results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.

This paper investigates new fractional energy methods for variables coupling the Navier-Stokes equations. Micropolar fluids starting from an initial angular velocity with Sobolev regularity close to $-1/2$ are constructed.

This paper studies distributionally robust optimization (DRO) in a dynamic context. We consider a general penalized DRO problem with a causal transport-type penalization. Such a penalization naturally captures the information flow generated by the dynamic model. We derive a tractable dynamic duality formula under mild conditions. Furthermore, we apply this duality formula to address distributionally robust version of average value-at-risk, stochastic control, and optimal stopping.

Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations requires reliable kernel density estimates computed from an ensemble of system realizations. However, in several real world signals such as Big Data, only a single system realization is available -- making it impossible to estimate a reliable kernel density. This study presents an approach for detecting P-type bifurcations using unreliable density estimates. The approach creates an ensemble of objects from Topological Data Analysis (TDA) called persistence diagrams from the system's sole realization and statistically analyzes the resulting set. We compare several methods for replicating the original persistence diagram including Gibbs point process modelling, Pairwise Interaction Point Modelling, and subsampling. We show that for the purpose of predicting a bifurcation, the simple method of subsampling exceeds the other two methods of point process modelling in performance.

In this work we study the one-dimensional stochastic Kimura equation $\partial_{t}u\left(z,t\right)=z\partial_{z}^{2}u\left(z,t\right)+u\left(z,t\right)\dot{W}\left(z,t\right)$ for $z,t>0$ equipped with a Dirichlet boundary condition at $0$, with $\dot{W}$ being a Gaussian space-time noise. This equation can be seen as a degenerate analog of the parabolic Anderson model. We combine the Wiener chaos theory from Malliavin calculus, the Duhamel perturbation technique from PDEs, and the kernel analysis of (deterministic) degenerate diffusion equations to develop a solution theory for the stochastic Kimura equation. We establish results on existence, uniqueness, moments, and continuity for the solution $u\left(z,t\right)$. In particular, we investigate how the stochastic potential and the degeneracy in the diffusion operator jointly affect the properties of $u\left(z,t\right)$ near the boundary. We also derive explicit estimates on the comparison under the $L^{2}-$ norm between $u\left(z,t\right)$ and its deterministic counterpart for $\left(z,t\right)$ within a proper range.

The motivation of the note is to obtain a H\"{o}rmander-type $L^2$ estimate for $\bar\partial$ equation. The feature of the new estimate is that the constant is independent of the weight function. Moreover, our estimate can be used for non-plurisubharmonic weight function.

We study asymptotic behavior of the bottom point of the spectrum of convolution type operators in environments with locally periodic microstructure. We show that its limit is described by an additive eigenvalue problem for Hamilton-Jacobi equation. In the periodic case we establish a more accurate two-term asymptotic formula.

One-shot channel simulation has recently emerged as a promising alternative to quantization and entropy coding in machine-learning-based lossy data compression schemes. However, while there are several potential applications of channel simulation - lossy compression with realism constraints or differential privacy, to name a few - little is known about its fundamental limitations. In this paper, we restrict our attention to a subclass of channel simulation protocols called causal rejection samplers (CRS), establish new, tighter lower bounds on their expected runtime and codelength, and demonstrate the bounds' achievability. Concretely, for an arbitrary CRS, let $Q$ and $P$ denote a target and proposal distribution supplied as input, and let $K$ be the number of samples examined by the algorithm. We show that the expected runtime $\mathbb{E}[K]$ of any CRS scales at least as $\exp_2(D_\infty[Q || P])$, where $D_\infty[Q || P]$ is the R\'enyi $\infty$-divergence. Regarding the codelength, we show that $D_{KL}[Q || P] \leq D_{CS}[Q || P] \leq \mathbb{H}[K]$, where $D_{CS}[Q || P]$ is a new quantity we call the channel simulation divergence. Furthermore, we prove that our new lower bound, unlike the $D_{KL}[Q || P]$ lower bound, is achievable tightly, i.e. there is a CRS such that $\mathbb{H}[K] \leq D_{CS}[Q || P] + \log_2 (e + 1)$. Finally, we conduct numerical studies of the asymptotic scaling of the codelength of Gaussian and Laplace channel simulation algorithms.

A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it does not contain a CM-subfield. For quartic totally imaginary number fields when ordered by discriminant, we show that about 69.95% are of TR-type and about 33.05% are of CM-type. For a sextic totally imaginary number field we classify its type in terms of its Galois group and possibly some additional information about the location of complex conjugation in the Galois group.

This paper proposes a fully distributed termination method for distributed optimization algorithms solved by multiple agents. The proposed method guarantees terminating a distributed optimization algorithm after satisfying the global termination criterion using information from local computations and neighboring agents. The proposed method requires additional iterations after satisfying the global terminating criterion to communicate the termination status. The number of additional iterations is bounded by the diameter of the communication network. This paper also proposes a fault-tolerant extension of this termination method that prevents early termination due to faulty agents or communication errors. We provide a proof of the method's correctness and demonstrate the proposed method by solving the optimal power flow problem for electric power grids using the alternating direction method of multipliers.

The Ising model, originally developed as a spin-glass model for ferromagnetic elements, has gained popularity as a network-based model for capturing dependencies in agents' outputs. Its increasing adoption in healthcare and the social sciences has raised privacy concerns regarding the confidentiality of agents' responses. In this paper, we present a novel $(\varepsilon,\delta)$-differentially private algorithm specifically designed to protect the privacy of individual agents' outcomes. Our algorithm allows for precise estimation of the natural parameter using a single network through an objective perturbation technique. Furthermore, we establish regret bounds for this algorithm and assess its performance on synthetic datasets and two real-world networks: one involving HIV status in a social network and the other concerning the political leaning of online blogs.

We use the Witt index to define and study a refined notion of the local-global principle for isotropy of quadratic forms over a field $k$ and to define and study refined versions of the $m$-invariant of $k$. We also explore connections between these refinements.

We determine the sharp mass threshold for Sobolev norm growth for the focusing continuum Calogero--Moser model. It is known that below the mass of $2\pi$, solutions to this completely integrable model enjoy uniform-in-time $H^s$ bounds for all $s \geq 0$. In contrast, we show that for arbitrarily small $\varepsilon > 0$ there exists initial data $u_0 \in H^\infty_+$ of mass $2\pi + \varepsilon$ such that the corresponding maximal lifespan solution $u : (T_-, T_+) \times \mathbb{R} \to \mathbb{C}$ satisfies $\lim_{t \to T_\pm} \|u(t)\|_{H^s} = \infty$ for all $s > 0$. As part of our proof, we demonstrate an orbital stability statement for the soliton and a dispersive decay bound for solutions with suitable initial data.

In this paper, we study linear convolutional networks with one-dimensional filters and arbitrary strides. The neuromanifold of such a network is a semialgebraic set, represented by a space of polynomials admitting specific factorizations. Introducing a recursive algorithm, we generate polynomial equations whose common zero locus corresponds to the Zariski closure of the corresponding neuromanifold. Furthermore, we explore the algebraic complexity of training these networks employing tools from metric algebraic geometry. Our findings reveal that the number of all complex critical points in the optimization of such a network is equal to the generic Euclidean distance degree of a Segre variety. Notably, this count significantly surpasses the number of critical points encountered in the training of a fully connected linear network with the same number of parameters.

In this paper, we explore the concept of total bondage in finite graphs without isolated vertices. A vertex set $D$ is considered a total dominating set if every vertex $v$ in the graph $G$ has a neighbor in $D$. The minimum cardinality of all total dominating sets in $G$ is denoted as $\gamma_t(G)$. A total bondage edge set $B$ is a subset of the edges of $G$ such that the removal of $B$ from $G$ does not create isolated vertices, and the total dominating number of the resulting graph $G-B$ is strictly greater than $\gamma_t(G)$. The total bondage number of $G$, denoted $b_t(G)$, is defined as the minimum cardinality of such total bondage edge sets. Our paper establishes upper bounds on $b_t(G)$ based on the maximum degree of a graph. Notably, for planar graphs with minimum degree $\delta(G) \geq 3$, we prove $b_t(G) \leq \Delta + 8$ or $b_t(G) \leq 10$. Additionally, for a connected planar graph with $\delta(G) \geq 3$ and $g(G) \geq 4$, we show that $b_t(G) \leq \Delta + 3$ if $G$ does not contain an edge with degree sum at most 7. We also improve some upper bounds of the total bondage number for trees, enhance existing lemmas, and find upper bounds for total bondage in specific graph classes.

In this paper, we establish propagation of chaos (POC) for doubly mean reflected backward stochastic differential equations (MRBSDEs). MRBSDEs differentiate the typical RBSDEs in that the constraint is not on the paths of the solution but on its law. This unique property has garnered significant attention since the inception of MRBSDEs. Rather than directly investigating these equations, we focus on approximating them by interacting particle systems (IPS). We propose two sets of IPS having mean-field Skorokhod problems, capturing the dynamics of IPS reflected in a mean-field way. As the dimension of the IPS tends to infinity, the POC phenomenon emerges, indicating that the system converges to a limit with independent particles, where each solves the MRBSDE. Beyond establishing the first POC result for doubly MRBSDEs, we achieve distinct convergence speeds under different scenarios.

In past work (Onokpasa, Wild, Wong, DCC 2023), we showed that (a) for joint compression of RNA sequence and structure, stochastic context-free grammars are the best known compressors and (b) that grammars which have better compression ability also show better performance in ab initio structure prediction. Previous grammars were manually curated by human experts. In this work, we develop a framework for automatic and systematic search algorithms for stochastic grammars with better compression (and prediction) ability for RNA. We perform an exhaustive search of small grammars and identify grammars that surpass the performance of human-expert grammars.

This paper presents new solutions for Private Information Retrieval (PIR) with side information. This problem is motivated by PIR settings in which a client has side information about the data held by the servers and would like to leverage this information in order to improve the download rate. The problem of PIR with side information has been the subject of several recent studies that presented achievability schemes as well as converses for both multi-server and single-server settings. However, the solutions for the multi-server settings adapted from the solutions for the single-server setting in a rather straightforward manner, relying on the concept of super-messages. Such solutions require an exponential degree of sub-packetization (in terms of the number of messages).

This paper makes the following contributions. First, we revisit the PIR problem with side information and present a new approach to leverage side information in the context of PIR. The key idea of our approach is a randomized algorithm to determine the linear combinations of the sub-packets that need to be recovered from each server. In addition, our approach takes advantage of the fact that the identity of the side information messages does not need to be kept private, and, as a result, the information retrieval scheme does not need to be symmetric. Second, we present schemes for PIR with side information that achieve a higher rate than previously proposed solutions and require a significantly lower degree of sub-packetization (linear in the number of servers). Our scheme not only achieves the highest known download rate for the problem at hand but also invalidates a previously claimed converse bound on the maximum achievable download rate.

This paper revisits the problem of multi-server Private Information Retrieval with Private Side Information (PIR-PSI). In this problem, $N$ non-colluding servers store identical copies of $K$ messages, each comprising $L$ symbols from $\mathbb{F}_q$, and a user, who knows $M$ of these messages, wants to retrieve one of the remaining $K-M$ messages. The user's goal is to retrieve the desired message by downloading the minimum amount of information from the servers while revealing no information about the identities of the desired message and side information messages to any server. The capacity of PIR-PSI, defined as the maximum achievable download rate, was previously characterized for all $N$, $K$, and $M$ when $L$ and $q$ are sufficiently large -- specifically, growing exponentially with $K$, to ensure the divisibility of each message into $N^K$ sub-packets and to guarantee the existence of an MDS code with its length and dimension being exponential in $K$. In this work, we propose a new capacity-achieving PIR-PSI scheme that is applicable to all $N$, $K$, $M$, $L$, and $q$ where $N\geq M+1$ and $N-1\mid L$. The proposed scheme operates with a sub-packetization level of $N-1$, independent of $K$, and works over any finite field without requiring an MDS code.

For turbulent problems of industrial scale, computational cost may become prohibitive due to the stability constraints associated with explicit time discretization of the underlying conservation laws. On the other hand, implicit methods allow for larger time-step sizes but require exorbitant computational resources. Implicit-explicit (IMEX) formulations combine both temporal approaches, using an explicit method in nonstiff portions of the domain and implicit in stiff portions. While these methods can be shown to be orders of magnitude faster than typical explicit discretizations, they are still limited by their implicit discretization in terms of cost. Hybridization reduces the scaling of these systems to an effective lower dimension, which allows the system to be solved at significant speedup factors compared to standard implicit methods. This work proposes an IMEX scheme that combines hybridized and standard flux reconstriction (FR) methods to tackle geometry-induced stiffness. By using the so-called transmission conditions, an overall conservative formulation can be obtained after combining both explicit FR and hybridized implicit FR methods. We verify and apply our approach to a series of numerical examples, including a multi-element airfoil at Reynolds number 1.7 million. Results demonstrate speedup factors of four against standard IMEX formulations and at least 15 against standard explicit formulations for the same problem.

A Thurston map $f\colon (S^2, A) \to (S^2, A)$ with marking set $A$ induces a pullback relation on isotopy classes of Jordan curves in $(S^2, A)$. If every curve lands in a finite list of possible curve classes after iterating this pullback relation, then the pair $(f,A)$ is said to have a finite global curve attractor. It is conjectured by Pilgrim that all rational Thurston maps that are not flexible Latt\{e}s maps have a finite global curve attractor. We present partial progress on this problem. Specifically, we prove that if $A$ has four points and the postcritical set (which is a subset of $A$) has two or three points, then $(f,A)$ has a finite global curve attractor.

We also discuss extensions of the main result to certain special cases where $f$ has four postcritical points and $A=P_f$. Additionally, we speculate on how some of these ideas might be used in the more general case.

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals $\lfloor{\frac{n-k+1}{2}\rfloor}+\ell$, the maximum possible, where $n$ is the vertex number of G. Answering and question of Dong and Wu, we show that every tight $(2,0)$-stable graph with odd vertex number must be an odd cycle. Moreover, we show that for all $k\geq 3$, every tight $(k,0)$-stable graph has at most $k+6$ vertices.

Coding theory revolves around the incorporation of redundancy into transmitted symbols, computation tasks, and stored data to guard against adversarial manipulation. However, error correction in coding theory is contingent upon a strict trust assumption. In the context of computation and storage, it is required that honest nodes outnumber adversarial ones by a certain margin. However, in several emerging real-world cases, particularly, in decentralized blockchain-oriented applications, such assumptions are often unrealistic. Consequently, despite the important role of coding in addressing significant challenges within decentralized systems, its applications become constrained. Still, in decentralized platforms, a distinctive characteristic emerges, offering new avenues for secure coding beyond the constraints of conventional methods. In these scenarios, the adversary benefits when the legitimate decoder recovers the data, and preferably with a high estimation error. This incentive motivates them to act rationally, trying to maximize their gains. In this paper, we propose a game theoretic formulation, called game of coding, that captures this unique dynamic where each of the adversary and the data collector (decoder) have a utility function to optimize. The utility functions reflect the fact that both the data collector and the adversary are interested to increase the chance of data being recoverable at the data collector. Moreover, the utility functions express the interest of the data collector to estimate the input with lower estimation error, but the opposite interest of the adversary. As a first, still highly non-trivial step, we characterize the equilibrium of the game for the repetition code with repetition factor of 2, for a wide class of utility functions with minimal assumptions.

We present two new algorithms for solving norm equations over global function fields with at least one infinite place of degree 1 and no wild ramification. The first of these is a substantial improvement of a method due to Ga\'{a}l and Pohst, while the second approach uses index calculus techniques and is significantly faster asymptotically and in practice. Both algorithms incorporate compact representations of field elements which results in a significant gain in performance compared to the Ga\'{a}l-Pohst approach. We provide Magma implementations, analyze the complexity of all three algorithms under varying asymptotics on the field parameters, and provide empirical data on their performance.

In this paper, we focus on the design of binary constant-weight codes that admit low-complexity encoding and decoding algorithms, and that have size as a power of $2$. We construct a family of $(n=2^\ell, M=2^k, d=2)$ constant-weight codes ${\cal C}[\ell, r]$ parameterized by integers $\ell \geq 3$ and $1 \leq r \leq \lfloor \frac{\ell+3}{4} \rfloor$, by encoding information in the gaps between successive $1$'s of a vector. The code has weight $w = \ell$ and combinatorial dimension $k$ that scales quadratically with $\ell$. The encoding time is linear in the input size $k$, and the decoding time is poly-logarithmic in the input size $n$, discounting the linear time spent on parsing the input. Encoding and decoding algorithms of similar codes known in either information-theoretic or combinatorial literature require computation of large number of binomial coefficients. Our algorithms fully eliminate the need to evaluate binomial coefficients. While the code has a natural price to pay in $k$, it performs fairly well against the information-theoretic upper bound $\lfloor \log_2 {n \choose w} \rfloor$. When $\ell =3$, the code is optimal achieving the upper bound; when $\ell=4$, it is one bit away from the upper bound, and as $\ell$ grows it is order-optimal in the sense that the ratio of $k$ with its upper bound becomes a constant $\frac{11}{16}$ when $r=\lfloor \frac{\ell+3}{4} \rfloor$. With the same or even lower complexity, we derive new codes permitting a wider range of parameters by modifying ${\cal C}[\ell, r]$ in two different ways. The code derived using the first approach has the same blocklength $n=2^\ell$, but weight $w$ is allowed to vary from $\ell-1$ to $1$. In the second approach, the weight remains fixed as $w = \ell$, but the blocklength is reduced to $n=2^\ell - 2^r +1$. For certain selected values of parameters, these modified codes have an optimal $k$.

Many modern applications require the use of data to both select the statistical tasks and make valid inference after selection. In this article, we provide a unifying approach to control for a class of selective risks. Our method is motivated by a reformulation of the celebrated Benjamini-Hochberg (BH) procedure for multiple hypothesis testing as the iterative limit of the Benjamini-Yekutieli (BY) procedure for constructing post-selection confidence intervals. Although several earlier authors have made noteworthy observations related to this, our discussion highlights that (1) the BH procedure is precisely the fixed-point iteration of the BY procedure; (2) the fact that the BH procedure controls the false discovery rate is almost an immediate corollary of the fact that the BY procedure controls the false coverage-statement rate. Building on this observation, we propose a constructive approach to control extra-selection risk (selection made after decision) by iterating decision strategies that control the post-selection risk (decision made after selection), and show that many previous methods and results are special cases of this general framework. We further extend this approach to problems with multiple selective risks and demonstrate how new methods can be developed. Our development leads to two surprising results about the BH procedure: (1) in the context of one-sided location testing, the BH procedure not only controls the false discovery rate at the null but also at other locations for free; (2) in the context of permutation tests, the BH procedure with exact permutation p-values can be well approximated by a procedure which only requires a total number of permutations that is almost linear in the total number of hypotheses.

We show how continuous-depth neural ODE models can be framed as single-layer, infinite-width nets using the Chen--Fliess series expansion for nonlinear ODEs. In this net, the output ''weights'' are taken from the signature of the control input -- a tool used to represent infinite-dimensional paths as a sequence of tensors -- which comprises iterated integrals of the control input over a simplex. The ''features'' are taken to be iterated Lie derivatives of the output function with respect to the vector fields in the controlled ODE model. The main result of this work applies this framework to derive compact expressions for the Rademacher complexity of ODE models that map an initial condition to a scalar output at some terminal time. The result leverages the straightforward analysis afforded by single-layer architectures. We conclude with some examples instantiating the bound for some specific systems and discuss potential follow-up work.

LiNGAM determines the variable order from cause to effect using additive noise models, but it faces challenges with confounding. Previous methods maintained LiNGAM's fundamental structure while trying to identify and address variables affected by confounding. As a result, these methods required significant computational resources regardless of the presence of confounding, and they did not ensure the detection of all confounding types. In contrast, this paper enhances LiNGAM by introducing LiNGAM-MMI, a method that quantifies the magnitude of confounding using KL divergence and arranges the variables to minimize its impact. This method efficiently achieves a globally optimal variable order through the shortest path problem formulation. LiNGAM-MMI processes data as efficiently as traditional LiNGAM in scenarios without confounding while effectively addressing confounding situations. Our experimental results suggest that LiNGAM-MMI more accurately determines the correct variable order, both in the presence and absence of confounding.

The finite-population asymptotic theory provides a normal approximation for the sampling distribution of the average treatment effect estimator in stratified randomized experiments. The asymptotic variance is often estimated by a Neyman-type conservative variance estimator. However, the variance estimator can be overly conservative, and the asymptotic theory may fail in small samples. To solve these issues, we propose a sharp variance estimator for the difference-in-means estimator weighted by the proportion of stratum sizes in stratified randomized experiments. Furthermore, we propose two causal bootstrap procedures to more accurately approximate the sampling distribution of the weighted difference-in-means estimator. The first causal bootstrap procedure is based on rank-preserving imputation and we show that it has second-order refinement over normal approximation. The second causal bootstrap procedure is based on sharp null imputation and is applicable in paired experiments. Our analysis is randomization-based or design-based by conditioning on the potential outcomes, with treatment assignment being the sole source of randomness. Numerical studies and real data analyses demonstrate advantages of our proposed methods in finite samples.

In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a game of Chomp with any starting condition. We also derive and prove formulas for the win probabilities for any game of Chomp with at most two rows. Additionally, we completely analyze the game of nim under random play by finding the expected number of turns and win probabilities from any starting position.

We study the resurgence properties of the coefficients $C_n(\tau)$ appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of $C_n(\tau)$ as $n\to +\infty$ depends on the parity of $n$. Both $C_{2n-1}(\tau)$ and $C_{2n}(\tau)$ exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again $C_{2k-1}(\tau)$ and $C_{2k}(\tau)$, respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, prior to this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.

This paper presents a state-of-the-art algorithm for the vertex enumeration problem of arrangements, which is based on the proposed new pivot rule, called the Zero rule. The Zero rule possesses several desirable properties: i) It gets rid of the objective function; ii) Its terminal satisfies uniqueness; iii) We establish the if-and-only if condition between the Zero rule and its valid reverse, which is not enjoyed by earlier rules; iv) Applying the Zero rule recursively definitely terminates in $d$ steps, where $d$ is the dimension of input variables. Because of so, given an arbitrary arrangement with $v$ vertices of $n$ hyperplanes in $\mathbb{R}^d$, the algorithm's complexity is at most $\mathcal{O}(n^2d^2v)$ and can be as low as $\mathcal{O}(nd^4v)$ if it is a simple arrangement, while Moss' algorithm takes $\mathcal{O}(nd^2v^2)$, and Avis and Fukuda's algorithm goes into a loop or skips vertices because the if-and-only-if condition between the rule they chose and its valid reverse is not fulfilled. Systematic and comprehensive experiments confirm that the Zero rule not only does not fail but also is the most efficient.

In this paper, we investigate the quasi-neutral limit of Nernst-Planck-Navier-Stokes system in a smooth bounded domain $\Omega$ of $\mathbb{R}^d$ for $d=2,3,$ with electroneutral boundary conditions" and well-prepared data. We first prove by using modulated energy estimate that the solution sequence converges to the limit system in the norm of $L^\infty((0,T);L^2(\Omega))$ for some positive time $T.$ In order to justify the limit in a stronger norm, we need to construct both the initial layers and weak boundary layers in the approximate solutions.

In this work we introduce a manifold learning-based surrogate modeling framework for uncertainty quantification in high-dimensional stochastic systems. Our first goal is to perform data mining on the available simulation data to identify a set of low-dimensional (latent) descriptors that efficiently parameterize the response of the high-dimensional computational model. To this end, we employ Principal Geodesic Analysis on the Grassmann manifold of the response to identify a set of disjoint principal geodesic submanifolds, of possibly different dimension, that captures the variation in the data. Since operations on the Grassmann require the data to be concentrated, we propose an adaptive algorithm based on Riemanniann K-means and the minimization of the sample Frechet variance on the Grassmann manifold to identify "local" principal geodesic submanifolds that represent different system behavior across the parameter space. Polynomial chaos expansion is then used to construct a mapping between the random input parameters and the projection of the response on these local principal geodesic submanifolds. The method is demonstrated on four test cases, a toy-example that involves points on a hypersphere, a Lotka-Volterra dynamical system, a continuous-flow stirred-tank chemical reactor system, and a two-dimensional Rayleigh-Benard convection problem

We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to give the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at $O(k^{-1/2})$, independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence and does not use error bounds.

Using elementary methods of algebraic geometry, we present constructions of hyperelliptically fibred surfaces containing nodal fibres.

Consider the task of estimating a random vector $X$ from noisy observations $Y = X + Z$, where $Z$ is a standard normal vector, under the $L^p$ fidelity criterion. This work establishes that, for $1 \leq p \leq 2$, the optimal Bayesian estimator is linear and positive definite if and only if the prior distribution on $X$ is a (non-degenerate) multivariate Gaussian. Furthermore, for $p > 2$, it is demonstrated that there are infinitely many priors that can induce such an estimator.

We consider the redundancy of the exact channel synthesis problem under an i.i.d. assumption. Existing results provide an upper bound on the unnormalized redundancy that is logarithmic in the block length. We show, via an improved scheme, that the logarithmic term can be halved for most channels and eliminated for all others. For full-support discrete memoryless channels, we show that this is the best possible.

This paper is concerned with the ordered statistic decoding with local constraints (LC-OSD) of binary linear block codes, which is a near maximum-likelihood decoding algorithm. Compared with the conventional OSD, the LC-OSD significantly reduces both the maximum and the average number of searches. The former is achieved by performing the serial list Viterbi algorithm (SLVA) or a two-way flipping pattern tree (FPT) algorithm with local constraints on the test error patterns, while the latter is achieved by incorporating tailored early termination criteria. The main objective of this paper is to explore the relationship between the performance of the LC-OSD and decoding parameters, such as the constraint degree and the maximum list size. To this end, we approximate the local parity-check matrix as a totally random matrix and then estimate the performance of the LC-OSD by analyzing with a saddlepoint approach the performance of random codes over the channels associated with the most reliable bits (MRBs). The random coding approach enables us to derive an upper bound on the performance and predict the average rank of the transmitted codeword in the list delivered by the LC-OSD. This allows us to balance the constraint degree and the maximum list size for the average (or maximum) time complexity reduction. Simulation results show that the approximation by random coding approach is numerically effective and powerful. Simulation results also show that the RS codes decoded by the LC-OSD can approach the random coding union (RCU) bounds, verifying the efficiency and universality of the LC-OSD.

We demonstrate that large language models can produce reasonable numerical ratings of the logical consistency of claims. We also outline a mathematical approach based on sheaf theory for lifting such ratings to hypertexts such as laws, jurisprudence, and social media and evaluating their consistency globally. This approach is a promising avenue to increasing consistency in and of government, as well as to combating mis- and disinformation and related ills.

In this paper, we study a class of nonsmooth fractional programs {\rm (FP, for short)} with SOS-convex semi-algebraic functions. Under suitable assumptions, we derive a strong duality result between the problem (FP) and its semidefinite programming (SDP) relaxations. Remarkably, we extract an optimal solution of the problem (FP) by solving one and only one associated SDP problem. Numerical examples are also given.

We consider the dispersion managed nonlinear Schr\"dinger equations with quintic and cubic nonlinearities in one and two dimensions, respectively. We prove the global well-posedness and scattering in $L_x^2$ for small initial data employing the $U^p$ and $V^p$ spaces.

In this paper, we introduce the pluricomplex Green function of the Monge-Amp\{e}re equation for $(n-1)$-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge-Amp\{e}re and Hessian equations on a punctured domain. We prove the pluricomplex Green function is $C^{1,\alpha}$ by constructing approximating solutions and establishing uniform a priori estimates for the gradient and the complex Hessian. The singular solutions turn out to be smooth for the $k$-Hessian equations for $(n-1)$-$k$-admissible functions.

We study variable-length feedback (VLF) codes with noiseless feedback for discrete memoryless channels. We present a novel non-asymptotic bound, which analyzes the average error probability and average decoding time of our modified Yamamoto--Itoh scheme. We then optimize the parameters of our code in the asymptotic regime where the average error probability $\epsilon$ remains a constant as the average decoding time $N$ approaches infinity. Our second-order achievability bound is an improvement of Polyanskiy et al.'s (2011) achievability bound. We also universalize our code by employing the empirical mutual information in our decoding metric and derive a second-order achievability bound for universal VLF codes. Our results for both VLF and universal VLF codes are extended to the additive white Gaussian noise channel with an average power constraint. The former yields an improvement over Truong and Tan's (2017) achievability bound. The proof of our results for universal VLF codes uses a refined version of the method of types and an asymptotic expansion from the nonlinear renewal theory literature.

We introduce and describe relations between Sobolev, Besov and Paley-Wiener spaces associated with three representations of the Lie group of affine transformations of the line. These representations are left and right regular representations and a representation in a space of functions defined on the half-line. The Besov spaces are described as interpolation spaces between respective Sobolev spaces in terms of the Petree's real interpolation method and in terms of a relevant moduli of continuity. By using a Laplace operators associated with these representations a scales of relevant Paley-Wiener spaces are developed and a corresponding $L_{2}$-approximation theory is constructed in which our Besov spaces appear as approximation spaces. Another description of our Besov spaces is given in terms of a frequency-localized Hilbert frames. A Jackson-type inequalities are also proven.

FI-graphs were introduced by the second author and White to capture the idea of a family of nested graphs, each member of which is acted on by a progressively larger symmetric group. That work was built on the newly minted foundations of representation stability theory and FI-modules. Examples of such families include the complete graphs and the Kneser and Johnson graphs, among many others. While it was shown in the originating work how various counting invariants in these families behave very regularly, not much has thus far been proven about the behaviors of the typical extremal graph theoretic invariants such as their independence and clique numbers. In this paper we provide a conjecture on the growth of the independence and clique numbers in these families, and prove this conjecture in one case. We also provide computer code that generates experimental evidence in many other cases. All of this work falls into a growing trend in representation stability theory that displays the regular behaviors of a number of extremal invariants that arise when one looks at FI-algebras and modules.

In this paper, practical utilization of multiple distributed reconfigurable intelligent surfaces (RISs), which are able to conduct group-specific operations, for multi-group multicasting systems is investigated. To tackle the inter-group interference issue in the multi-group multicasting systems, the block diagonalization (BD)-based beamforming is considered first. Without any inter-group interference after the BD operation, the multiple distributed RISs are operated to maximize the minimum rate for each group. Since the computational complexity of the BD-based beamforming can be too high, a multicasting tailored zero-forcing (MTZF) beamforming technique is proposed to efficiently suppress the inter-group interference, and the novel design for the multiple RISs that makes up for the inevitable loss of MTZF beamforming is also described. Effective closed-form solutions for the loss minimizing RIS operations are obtained with basic linear operations, making the proposed MTZF beamforming-based RIS design highly practical. Numerical results show that the BD-based approach has ability to achieve high sum-rate, but it is useful only when the base station deploys large antenna arrays. Even with the small number of antennas, the MTZF beamforming-based approach outperforms the other schemes in terms of the sum-rate while the technique requires low computational complexity. The results also prove that the proposed techniques can work with the minimum rate requirement for each group.

We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $\Psi$-factorizability, the constructions of unipotent nearby cycles over an affine space can be described using the theory of nearby cycles over general bases via the vanishing topos. In particular, this applies to nearby cycles of Satake sheaves on Beilinson-Drinfeld Grassmannians with parahoric ramification.

We study communication over a Gaussian multiple-access channel (MAC) with two types of transmitters: Digital transmitters hold a message from a discrete set that needs to be communicated to the receiver. Analog transmitters hold sequences of analog values, and some function of these distributed values (but not the values themselves) need to be conveyed to the receiver. For the digital messages, it is required that they can be decoded error free at the receiver with high probability while the recovered analog function values have to satisfy a fidelity criterion such as an upper bound on mean squared error (MSE) or a certain maximum error with a given confidence. For the case in which the computed function for the analog transmitters is a sum of values in [-1,1], we derive inner and outer bounds for the tradeoff of digital and analog rates of communication under peak and average power constraints for digital transmitters and a peak power constraint for analog transmitters. We then extend the achievability part of our result to a larger class of functions that includes all linear, but also some non-linear functions.

We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply K-convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect.

Two players take it turn to claim empty cells from an $n\times n$ grid. The first player (if any) to occupy a transversal (a set of $n$ cells having no two cells in the same row or column) is the winner. What is the outcome of the game given optimal play? Our aim in this paper is to show that for $n\ge 4$ the first player has a winning strategy. This answers a question of Erickson.

We introduce the notion of a contractible subshift. This is a strengthening of the notion of strong irreducibility, where we require that the gluings are given by a block map. We show that a subshift is a retract of a full shift if and only if it is a contractible SFT with a fixed point. For virtually polycyclic groups, contractibility implies dense periodic points. We introduce a homotopy theory'' framework for working with this notion, and contractibility'' is in fact simply an analog of the usual contractibility in algebraic topology. We also explore the symbolic dynamical analogs of homotopy equivalence and equiconnectedness of subshifts. Contractibility is implied by the map extension property of Meyerovitch, and among SFTs, it implies the finite extension property of Brice\~no, McGoff and Pavlov. We include thorough comparisons with these classes. We also encounter some new group-geometric notions, in particular a periodic variant of Gromov's asymptotic dimension of a group.

In this paper, we present a signaling design for secure integrated sensing and communication (ISAC) systems comprising a dual-functional multi-input multi-output (MIMO) base station (BS) that simultaneously communicates with multiple users while detecting targets present in their vicinity, which are regarded as potential eavesdroppers. In particular, assuming that the distribution of each parameter to be estimated is known \textit{a priori}, we focus on optimizing the targets' sensing performance. To this end, we derive and minimize the Bayesian Cram\'er-Rao bound (BCRB), while ensuring certain communication quality of service (QoS) by exploiting constructive interference (CI). The latter scheme enforces that the received signals at the eavesdropping targets fall into the destructive region of the signal constellation, to deteriorate their decoding probability, thus enhancing the ISAC's system physical-layer security (PLS) capability. To tackle the nonconvexity of the formulated problem, a tailored successive convex approximation method is proposed for its efficient solution. Our extensive numerical results verify the effectiveness of the proposed secure ISAC design showing that the proposed algorithm outperforms block-level precoding techniques.

For a general time-inhomogenous diffusion process $X$ and a boundary function $g,$ we analyse the probability $F(g)$ that $X$ stays beneath the boundary $g$ during a given finite time interval. We prove that, for $g \in C^2,$ the functional $F(g)$ is G\^ateaux-differentiable in directions $h \in H \cup C^2,$ where $H$ is the Cameron--Martin space, and derive a compact representation for the derivative of $F$. It is shown that, in the time-homogenous case, this representation coincides with the representation in terms of Brownian meander functionals obtained in this special case by Borovkov & Downes (2010). Our results can be useful when approximating $F(g)$ with computable values $F(\widehat{g})$ for special boundaries $\widehat{g}$ that are close to $g.$ We also obtain auxiliary results that can be of independent interest as well: (i) a probabilistic counterpart to the jump relation'' for single layer potentials for backward Kolmogorov equations on time-dependent domains, and (ii) a martingale representation for the indicator of the boundary non-crossing event for time-dependent boundaries.

The paper proves two results involving a pair (A,B) of P-biisometric or (m,P)-biisometric Hilbert-space operators for arbitrary positive integer m and positive operator P. It is shown that if A and B are power bounded and the pair (A,B) is (m,P)-biisometric for some m, then it is a P-biisometric pair. The important case when P is invertible is treated in detail. It is also shown that if (A,B) is P-biisometric, then there are biorthogonal sequences with respect to the inner product <.;.>_P=<P.;.> that have a shift-like behaviour with respect to this inner product.

In this paper, we distinguish two guessing algorithms for decoding binary linear codes. One is the guessing noise decoding (GND) algorithm, and the other is the guessing codeword decoding (GCD) algorithm. We prove that the GCD is a maximum likelihood (ML) decoding algorithm and that the GCD is more efficient than GND for most practical applications. We also introduce several variants of ordered statistic decoding (OSD) to trade off the complexity of the Gaussian elimination (GE) and that of the guessing, which may find applications in decoding short block codes in the high signal-to-noise ratio (SNR) region.

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t, u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t, u_{t}\right)dW_{t}$, where $\left(L_{t}\right)_{t \in \left[0, T\right]}$ is a family of unbounded operators acting on a monotone family of interpolation Hilbert spaces, $\mathbf{X}$ is a two-step $\alpha$-H\"older rough path with $\alpha \in \left(1/3, 1/2\right]$ and $W$ is a Brownian motion. Existence and uniqueness of the mild solution are given through the stochastic controlled rough path approach and fixed-point argument. As a technical tool to define rough stochastic convolutions, we also develop a general mild stochastic sewing lemma, which is applicable for processes according to a monotone family.

We are concerned with the sharp interface limit for the Beris-Edward system in a bounded domain $\Omega \subset \mathbb{R}^3$ in this paper. The system can be described as the incompressible Navier-Stokes equations coupled with an evolution equation for the Q-tensor. We prove that the solutions to the Beris-Edward system converge to the corresponding solutions of a sharp interface model under well-prepared initial data, as the thickness of the diffuse interfacial zone tends to zero. Moreover, we give not only the spatial decay estimates of the velocity vector field in the $H^1$ sense but also the error estimates of the phase field. The analysis relies on the relative entropy method and elaborated energy estimates.

This work focuses on distributed linear precoding when users transmit correlated information over a fading Multiple-Input and Multiple-Output Multiple Access Channel. Precoders are optimized in order to minimize the sum-Mean Square Error (MSE) between the source and the estimated symbols. When sources are correlated, minimizing the sum-MSE results in a non-convex optimization problem. Precoders for an arbitrary number of users and transmit and receive antennas are thus obtained via a projected steepest-descent algorithm and a low-complexity heuristic approach. For the more restrictive case of two single-antenna users, a closed-form expression for the minimum sum-MSE precoders is derived. Moreover, for the scenario with a single receive antenna and any number of users, a solution is obtained by means of a semidefinite relaxation. Finally, we also consider precoding schemes where the precoders are decomposed into complex scalars and unit norm vectors. Simulation results show a significant improvement when source correlation is exploited at precoding, especially for low SNRs and when the number of receive antennas is lower than the number of transmitting nodes.

A maximal planar graph is a graph which can be embedded in the plane such that every face of the graph is a triangle. The center of a graph is the subgraph induced by the vertices of minimum eccentricity. We introduce the notion of quasi-eccentric vertices, and use this to characterize maximal planar graphs that are the center of some planar graph. We also present some easier to check only necessary / only sufficient conditions for planar and maximal planar graphs to be the center of a planar graph. Finally, we use the aforementioned characterization to prove that all maximal planar graphs of order at most 8 are the center of some planar graph -- and this bound is sharp.

Polar codes were originally specified for codelengths that are powers of two. In many applications, it is desired to have a code that is not restricted to such lengths. Two common strategies of modifying the length of a code are shortening and puncturing. Simple and explicit schemes for shortening and puncturing were introduced by Wang and Liu, and by Niu, Chen, and Lin, respectively. In this paper, we prove that both schemes yield polar codes that are capacity achieving. Moreover, the probability of error for both the shortened and the punctured polar codes decreases to zero at the same exponential rate as seminal polar codes. These claims hold for \emph{all} codelengths large enough.

In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process.This generalizes the Generalized Central Limit Theorem for stable random variables infinite dimension. We show that provided we have a control between the randomwalk or the limiting stable process and their respective affine interpolation, we canlift the rate of convergence obtained for multivariate distributions to a rateof convergence in some functional spaces.

We consider a non-conservative nonlinear Schrodinger equation (NCNLS) with time-dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic NLS. Both schemes are second oder accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour-Fortin-Payre scheme and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly non-conservative problems. We finally compare the two numerical schemes and discuss their performance.

Nonnegative Matrix Factorization (NMF) is an important unsupervised learning method to extract meaningful features from data. To address the NMF problem within a polynomial time framework, researchers have introduced a separability assumption, which has recently evolved into the concept of coseparability. This advancement offers a more efficient core representation for the original data. However, in the real world, the data is more natural to be represented as a multi-dimensional array, such as images or videos. The NMF's application to high-dimensional data involves vectorization, which risks losing essential multi-dimensional correlations. To retain these inherent correlations in the data, we turn to tensors (multidimensional arrays) and leverage the tensor t-product. This approach extends the coseparable NMF to the tensor setting, creating what we term coseparable Nonnegative Tensor Factorization (NTF). In this work, we provide an alternating index selection method to select the coseparable core. Furthermore, we validate the t-CUR sampling theory and integrate it with the tensor Discrete Empirical Interpolation Method (t-DEIM) to introduce an alternative, randomized index selection process. These methods have been tested on both synthetic and facial analysis datasets. The results demonstrate the efficiency of coseparable NTF when compared to coseparable NMF.

Oscillations of free intracellular calcium concentration are thought to be important in the control of a wide variety of physiological phenomena, and there is long-standing interest in understanding these oscillations via the investigation of suitable mathematical models. Many of these models have the feature that different variables or terms in the model evolve on very different time-scales, which often results in the accompanying oscillations being temporally complex. Cloete et al [5] constructed an ordinary differential equation model of calcium oscillations in hepatocytes in an attempt to understand the origin of two distinct types of oscillation observed in experiments: narrow-spike oscillations in which rapid spikes of calcium concentration alternate with relatively long periods of quiescence, and broad-spike oscillations in which there is a fast rise in calcium levels followed by a slower decline then a period of quiescence. These two types of oscillation can be observed in the model if a single system parameter is varied but the mathematical mechanisms underlying the different types of oscillations were not explored in detail in [5]. We use ideas from geometric singular perturbation theory to investigate the origin of broad-spike solutions in this model. We find that the analysis is intractable in the full model, but are able to uncover structure in particular singular limits of a related model that point to the origin of the broad-spike solutions.

In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.

Kolmogorov decomposition for a given completely positive definite kernel is a generalization of Paschke's GNS construction for the completely positive map. Using Kolmogorov decomposition, to every quantum dynamical semigroup (QDS) for completely positive definite kernels over a set $S$ on given $C^*$-algebra $\mathcal{A},$ we shall assign an inclusion system $F = (F_s)_{s\ge 0}$ of Hilbert bimodules over $\mathcal{A}$ with a generating unit $\xi^{\sigma}=(\xi^{\sigma}_s)_{s\ge 0}.$ Consider a von Neumann algebra $\mathcal{B}$, and let $\mathfrak{T}=(\mathfrak{T}_s)_{s\ge 0}$ be a QDS over a set $S$ on the algebra $M_2(\mathcal{B})$ with $\mathfrak{T}_s=\begin{pmatrix}\mathfrak{K}_{s,1} & \mathfrak{L}_s\\\mathfrak{L}_s^*& \mathfrak{K}_{s,2} \end{pmatrix}$ which acts block-wise. Further, suppose that $(F^i_s )_{s\ge 0}$ is the inclusion system affiliated to the diagonal QDS $(\mathfrak{K}_{s,i})_{s\ge 0}$ along with the generating unit $(\xi^{\sigma}_{s,i} )_{s\ge 0},$ $\sigma\in S,i\in \{1,2\}$, then we prove that there exists a unique contractive (weak) morphism $V = (V_s)_{s\ge 0}:F^2_s \to F^1_s$ such that $\mathfrak{L}_s^{\sigma,\sigma'}(b)=\langle \xi_{s,1}^{\sigma},V_s b\xi_{s,2}^{\sigma'}\rangle$ for every $\sigma',\sigma\in S$ and $b\in \mathcal{B}.$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families.

Data-driven methods for the identification of the governing equations of dynamical systems or the computation of reduced surrogate models play an increasingly important role in many application areas such as physics, chemistry, biology, and engineering. Given only measurement or observation data, data-driven modeling techniques allow us to gain important insights into the characteristic properties of a system, without requiring detailed mechanistic models. However, most methods assume that we have access to the full state of the system, which might be too restrictive. We show that it is possible to learn certain global dynamical features from local observations using delay embedding techniques, provided that the system satisfies a localizability condition -- a property that is closely related to the observability and controllability of linear time-invariant systems.

The design of zero-delay Joint Source-Channel Coding (JSCC) schemes for the transmission of correlated information over fading Multiple Access Channels (MACs) is an interesting problem for many communication scenarios like Wireless Sensor Networks (WSNs). Among the different JSCC schemes so far proposed for this scenario, Distributed Quantizer Linear Coding (DQLC) represents an appealing solution since it is able to outperform uncoded transmissions for any correlation level at high Signal-to-Noise Ratios (SNRs) with a low computational cost. In this paper, we extend the design of DQLC-based schemes for fading MACs considering sphere decoding to make the optimal Minimum Mean Squared Error (MMSE) estimation computationally affordable for an arbitrary number of transmit users. The use of sphere decoding also allows to formulate a practical algorithm for the optimization of DQLC-based systems. Finally, non-linear Kalman Filtering for the DQLC is considered to jointly exploit the temporal and spatial correlation of the source symbols. The results of computer experiments show that the proposed DQLC scheme with the Kalman Filter decoding approach clearly outperforms uncoded transmissions for medium and high SNRs.

We show that the classification diagram of a relative $\infty$-category arising from a relative simplicial category is equivalent to the levelwise nerve. Applications include the comparison of the diagonal of the levelwise nerve and the homotopy coherent nerve, and a result on the levelwise localizations of simplicial categories.

Wasserstein distortion is a one-parameter family of distortion measures that was recently proposed to unify fidelity and realism constraints. After establishing continuity results for Wasserstein in the extreme cases of pure fidelity and pure realism, we prove the first coding theorems for compression under Wasserstein distortion focusing on the regime in which both the rate and the distortion are small.

Centralized repair refers to repairing $h\geq 2$ node failures using $d$ helper nodes in a centralized way, where the repair bandwidth is counted by the total amount of data downloaded from the helper nodes. A centralized MSR code is an MDS array code with $(h,d)$-optimal repair for some $h$ and $d$. In this paper, we present several classes of centralized MSR codes with small sub-packetization. At first, we construct an alternative MSR code with $(1,d_i)$-optimal repair for multiple repair degrees $d_i$ simultaneously. Based on the code structure, we are able to construct a centralized MSR code with $(h_i,d_i)$-optimal repair property for all possible $(h_i,d_i)$ with $h_i\mid (d_i-k)$ simultaneously. The sub-packetization is no more than ${\rm lcm}(1,2,\ldots,n-k)(n-k)^n$, which is much smaller than a previous work given by Ye and Barg ($({\rm lcm}(1,2,\ldots,n-k))^n$). Moreover, for general parameters $2\leq h\leq n-k$ and $k\leq d\leq n-h$, we further give a centralized MSR code enabling $(h,d)$-optimal repair with sub-packetization smaller than all previous works.

This paper is concerned with the study of a family of fixed point iterations combining relaxation with different inertial (acceleration) principles. We provide a systematic, unified and insightful analysis of the hypotheses that ensure their weak, strong and linear convergence, either matching or improving previous results obtained by analysing particular cases separately. We also show that these methods are robust with respect to different kinds of perturbations--which may come from computational errors, intentional deviations, as well as regularisation or approximation schemes--under surprisingly weak assumptions. Although we mostly focus on theoretical aspects, numerical illustrations in image inpainting and electricity production markets reveal possible trends in the behaviour of these types of methods.

The considered optimal control problem of a stochastic power system, is to select the set of power supply vectors which infimizes the probability that the phase-angle differences of any power flow of the network, endangers the transient stability of the power system by leaving a critical subset. The set of control laws is restricted to be a periodically recomputed set of fixed power supply vectors based on predictions of power demand for the next short horizon. Neither state feedback nor output feedback is used. The associated control objective function is Lipschitz continuous, nondifferentiable, and nonconvex. The results of the paper include that a minimum exists in the value range of the control objective function. Furthermore, it includes a two-step procedure to compute an approximate minimizer based on two key methods: (1) a projected generalized subgradient method for computing an initial vector, and (2) a steepest descent method for approximating a local minimizer. Finally, it includes two convergence theorems that an approximation sequence converges to a local minimum.

Consider a 2-dimensional smooth Riemannian manifold $M$, and let $P(h)$ be a semiclassical pseudodifferential operator on $M$. Assume that $f = f(h)$ is an $O(h)$ quasimode of $P(h)$ localized in phase space. In this work, we establish sharp $L^p$ restriction estimates for quasimodes for all smooth curves in two dimensions. As an application, we address $L^p$ restriction eigenfunction estimates for Laplace eigenfunctions on compact Riemannian manifolds and Hermite functions on $\mathbb R^2$. Our method involves a geometric analysis of the contact order between the curve and the bicharacteristic flow of $P(h)$.

We propose a local modification of the standard subdiffusion model by introducing the initial Fickian diffusion, which results in a multiscale diffusion model. The developed model resolves the incompatibility between the nonlocal operators in subdiffusion and the local initial conditions and thus eliminates the initial singularity of the solutions of the subdiffusion, while retaining its heavy tail behavior away from the initial time. The well-posedness of the model and high-order regularity estimates of its solutions are analyzed by resolvent estimates, based on which the numerical discretization and analysis are performed. Numerical experiments are carried out to substantiate the theoretical findings.

Firewalls in black holes are easiest to understand by imposing time reversal invariance, together with a unitary evolution law. The best approach seems to be to split up the time span of a black hole into short periods, during which no firewalls can be detected by any observer. Then, gluing together subsequent time periods, firewalls seem to appear, but they can always be transformed away. At all times we need a Hilbert space of a finite dimension, as long as particles far separated from the black hole are ignored. Our conclusion contradicts other findings, particularly a recent paper by Strauss and Whiting. Indeed, the firewall transformation removes the entanglement between very early and very late in- and out-particles, in a far-from-trivial way.

We consider relatively prime integer numbers $m$ and $n$ such that each group of order $mn$ has a normal subgroup of order $m$. We prove that each brace of size $mn$ is a semidirect product of a brace of size $m$ and a brace of size $n$. We further give a method to classify braces of size $mn$ from the classification of braces of sizes $m$ and $n$. We apply this result to determine all braces of size $p^2q^2$, for $p$ an odd Germain prime and $q=2p+1$.

Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere $\mathbb{S}^{d-1}$ and on the rotation group SO(3). We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For SO(3), we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on SO(3).

As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior both the free and fixed support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically; besides, the underlying St\"ackel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field.

We expand the theory of 2-classifiers, that are a 2-categorical generalization of subobject classifiers introduced by Weber. The idea is to upgrade monomorphisms to discrete opfibrations. We prove that the conditions of 2-classifier can be checked just on a dense generator. The study of what is classified by a 2-classifier is similarly reduced to a study over the objects that form a dense generator. We then apply our results to the cases of prestacks and stacks, where we can thus look just at the representables. We produce a 2-classifier in prestacks that classifies all discrete opfibrations with small fibres. Finally, we restrict such 2-classifier to a 2-classifier in stacks. This is the main ingredient of a proof that Grothendieck 2-topoi are elementary 2-topoi. Our results also solve a problem posed by Hofmann and Streicher when attempting to lift Grothendieck universes to sheaves.

The combination of multiple-input multiple-output (MIMO) systems and intelligent reflecting surfaces (IRSs) is foreseen as a critical enabler of beyond 5G (B5G) and 6G. In this work, two different approaches are considered for the joint optimization of the IRS phase-shift matrix and MIMO precoders of an IRS-assisted multi-stream (MS) multi-user MIMO (MU-MIMO) system. Both approaches aim to maximize the system sum-rate for every channel realization. The first proposed solution is a novel contextual bandit (CB) framework with continuous state and action spaces called deep contextual bandit-oriented deep deterministic policy gradient (DCB-DDPG). The second is an innovative deep reinforcement learning (DRL) formulation where the states, actions, and rewards are selected such that the Markov decision process (MDP) property of reinforcement learning (RL) is appropriately met. Both proposals perform remarkably better than state-of-the-art heuristic methods in scenarios with high multi-user interference.

We study the stable dynamics of non-polynomial automorphisms of $\mathbb{C}^2$ of the form $F(z,w)=(e^{-z^m}+ \delta e^{\frac{2 \pi}{m}i}\, w\,,\,z)$, with $m\ge 2$ a natural number and $\mathbb{R}\ni\delta>2$.

If $m$ is even, there are $\frac{m}{2}$ cycles of escaping Fatou components, all of period $2m$. If $m$ is odd there are $\frac{m-1}{2}$ cycles of escaping Fatou components of period $2m$ and just one cycle of escaping Fatou components of period $m$.

These maps have two distinct limit functions on each cycle, both of which have generic rank 1. Each Fatou component in each cycle has two disjoint and hyperbolic limit sets on the line at infinity, except for the Fatou components that belong to the unique cycle of period $m$: the latter in fact have the same hyperbolic limit set on the line at infinity.

An innovative approach to hybrid analog-digital precoding for the downlink of wideband massive MIMO systems is developed. The proposed solution, termed Rank-Constrained Coordinate Ascent (RCCA), starts seeking the full-digital precoder that maximizes the achievable sum-rate over all the frequency subcarriers while constraining the rank of the overall transmit covariance matrix. The frequency-flat constraint on the analog part of the hybrid precoder and the non-convex nature of the rank constraint are circumvented by transforming the original problem into a more suitable one, where a convenient structure for the transmit covariance matrix is imposed. Such structure makes the resulting full-digital precoder particularly adequate for its posterior analog-digital factorization. An additional problem formulation to determine an appropriate power allocation policy according to the rank constraint is also provided. The numerical results show that the proposed method outperforms baseline solutions even for practical scenarios with high spatial diversity.

In [2] we show how to construct information sets for Reed-Muller codes only in terms of their basic parameters. In this work we deal with the corresponding problem for q-ary Generalized Reed-Muller codes of first and second order. We see that for first-order codes the result for binary Reed-Muller codes is also valid, while for second-order codes, with q > 2, we have to manage more complex defining sets and we show that we get different information sets. We also present some examples and associated open problems.

In this article we obtain new rigidity results for spacelike submanifolds of arbitrary codimension in Generalized Robertson-Walker spacetimes. Namely, under appropriate assumptions such as parabolicity we prove by means of some maximum principles that they must be contained in a spacelike slice. This enables us to characterize extremal and weakly trapped submanifolds in these ambient spacetimes.

We tackle the problem of Byzantine errors in distributed gradient descent within the Byzantine-resilient gradient coding framework. Our proposed solution can recover the exact full gradient in the presence of $s$ malicious workers with a data replication factor of only $s+1$. It generalizes previous solutions to any data assignment scheme that has a regular replication over all data samples. The scheme detects malicious workers through additional interactive communication and a small number of local computations at the main node, leveraging group-wise comparisons between workers with a provably optimal grouping strategy. The scheme requires at most $s$ interactive rounds that incur a total communication cost logarithmic in the number of data samples.

In this paper we determine a sufficient condition for the quasinilpotency of a commutator of compact operators via block-tridiagonal matrix form associated with a compact operator. We also prove that every compact operator is unitarily equivalent to the sum of a compact quasinilpotent operator and a triangularizable compact operator.

We show that (central) Cowling-Haagerup constant of discrete quantum groups is multiplicative, which extends the result of Freslon to general (not necesarilly unimodular) discrete quantum groups. The crucial feature of our approach is considering algebras $\mathrm{C}(\mathbb{G}), \operatorname{L}^{\infty}(\mathbb{G})$ as operator modules over $\operatorname{L}^1(\mathbb{G})$.

Providing closed form estimates of the decoding failure rate of iterative decoder for low- and moderate-density parity check codes has attracted significant interest in the research community over the years. This interest has raised recently due to the use of iterative decoders in post-quantum cryptosystems, where the desired decoding failure rates are impossible to estimate via Monte Carlo simulations. In this work, we propose a new technique to provide accurate estimates of the DFR of a two-iterations (parallel) bit flipping decoder, which is also employable for cryptographic purposes. In doing so, we successfully tackle the estimation of the bit flipping probabilities at the second decoder iteration, and provide a fitting estimate for the syndrome weight distribution at the first iteration. We numerically validate our results, providing comparisons of the modeled and simulated weight of the syndrome, incorrectly-guessed error bit distribution at the end of the first iteration, and two-iteration Decoding Failure Rates (DFR), both in the floor and waterfall regime for simulatable codes. Finally, we apply our method to estimate the DFR of LEDAcrypt parameters, showing improvements by factors larger than $2^{70}$ (for NIST category $1$) with respect to the previous estimation techniques. This allows for a $\approx 20$% shortening in public key and ciphertext sizes, at no security loss, making the smallest ciphertext for NIST category $1$ only $6$% larger than the one of BIKE. We note that the analyzed two-iterations decoder is applicable in BIKE, where swapping it with the current black-gray decoder (and adjusting the parameters) would provide strong IND-CCA$2$ guarantees.

We develop a framework for learning properties of quantum states beyond the assumption of independent and identically distributed (i.i.d.) input states. We prove that, given any learning problem (under reasonable assumptions), an algorithm designed for i.i.d. input states can be adapted to handle input states of any nature, albeit at the expense of a polynomial increase in copy complexity. Furthermore, we establish that algorithms which perform non-adaptive incoherent measurements can be extended to encompass non-i.i.d. input states while maintaining comparable error probabilities. This allows us, among others applications, to generalize the classical shadows of Huang, Kueng, and Preskill to the non-i.i.d. setting at the cost of a small loss in efficiency. Additionally, we can efficiently verify any pure state using Clifford measurements, in a way that is independent of the ideal state. Our main techniques are based on de Finetti-style theorems supported by tools from information theory. In particular, we prove a new randomized local de Finetti theorem that can be of independent interest.

The controllability issue of control-affine systems on smooth manifolds is one of the main problems in the theory, and it is recently known [Jouan P. Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM: Control Optim. Calc. Var. 2010, 16, 956-973] that it might be connected to that of a particular class of systems called linear control systems on (a homogeneous manifold of) a Lie group. Note that it may become very complicated to establish the controllability property of systems evolving on homogeneous spaces of Lie groups whose dynamics are induced by those of systems in the Lie group under consideration. In fact, even in low-dimensional certain homogeneous spaces, this is quite a challenging task, and for this reason, we have classified in [Da Silva, A., Kizil, E., Duman, O. Linear Control Systems on Homogeneous Spaces of the Heisenberg Group. J. Dyn. Control Syst. 2023, 29, 2065-2086] as a first goal all linear control systems on the homogeneous spaces of the 3-dimensional Heisenberg group $\mathbb{H}$ through its closed subgroups $L$ and, in particular, the controllability and the control sets have been studied for one of the homogeneous spaces $L\setminus \mathbb{H}$.

In this paper, we study the controllability and control sets of the induced linear control systems in the homogeneous spaces left. In particular, we focus on the singularity of the induced drift vector fields that results in many cases and subcases to reveal control sets after quite a technical analysis. We give some nice illustrations to better understand what is going on geometrically.

We investigate the Witsenhausen counterexample in a continuous vector-valued context with a causal encoder and noncausal decoder. Our main result is the optimal single-letter condition that characterizes the set of achievable Witsenhausen power costs and estimation costs, leveraging a modified weak typicality approach. In particular, we accommodate our power analysis to the causal encoder constraint, and provide an improved distortion error analysis for the challenging estimation of the interim state. Interestingly, the idea of dual role of control is explicitly captured by the two auxiliary random variables.

Let $H \subseteq G$ be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic $p> 0$. In our first principal theorem we show that if a closed subgroup $K$ of $H$ is $H$-completely reducible, then it is also $G$-completely reducible in the sense of Serre, under some restrictions on $p$, generalising the known case for $G = GL(V)$. Our second main theorem shows that if $K$ is $H$-completely reducible, then the saturation of $K$ in $G$ is completely reducible in the saturation of $H$ in $G$ (which is again a connected reductive subgroup of $G$), under suitable restrictions on $p$, again generalising the known instance for $G = GL(V)$. We also study saturation of finite subgroups of Lie type in $G$. Here we generalise a result due to Nori from 1987 in case $G = GL(V)$.

In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We establish the possible values for the constant scalar curvature of a compact quasi-Einstein manifold with boundary. Moreover, we show that a $3$-dimensional simply connected compact $m$-quasi-Einstein manifold with boundary and constant scalar curvature must be isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{3}_{+}$, or the cylinder $\left[0,\frac{\sqrt{m}}{\sqrt{\lambda}}\,\pi\right]\times\mathbb{S}^2$ with the product metric. For dimension $n=4,$ we prove that a $4$-dimensional simply connected compact $m$-quasi-Einstein manifold $M^4$ with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{4}_{+},$ or the cylinder $\left[0,\frac{\sqrt{m}}{\sqrt{\lambda}}\,\pi\right]\times\mathbb{S}^3$ with the product metric, or the product space $\mathbb{S}^{2}_{+}\times\mathbb{S}^2$ with the doubly warped product metric. Other related results for arbitrary dimensions are also discussed.

Let $F$ be a non-archimedean local field. For the symplectic group $Sp_{4}(F),$ let $P$ and $Q$ denote respectively its Siegel and Klingen parabolic subgroups with respective Levi decompositions $P=MN$ and $Q=LU.$ For a non-trivial character $\psi$ of the unipotent radical $N$ of $P,$ let $M_{\psi}$ denote the stabilizer of the character $\psi$ in $M$ under the conjugation action of $M$ on characters of $N.$ For an irreducible representation of the Levi subgroups $M$ or $L,$ let $\pi$ denote the respective representation of $Sp_{4}(F)$ parabolically induced either from $P$ or from $Q.$ Let $\psi$ be a character of the group $N$ given by a rank one quadratic form. In this article, we determine the structure of the twisted Jacquet module $r_{N,\psi}(\pi)$ as an $M_{\psi}$-module. We also deduce the analogous results in the case where $F$ is a finite field of order $q.$

Here, we investigate the linear spatial stability of a parallel two-dimensional compressible boundary layer on an adiabatic plate by considering 2D and 3D disturbances. We employ the Compound Matrix Method for the first time for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the original equation. Our study explores flow Mach numbers ranging from low subsonic to supersonic cases, to investigate the effects of flow compressibility and spanwise variation of disturbances. We get some interesting results depending on the flow Mach number. Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for Mach number greater than 3 from viscous calculations (the so-called second mode) that subsequently fuse to create only one unstable zone when Mach number increases. Our calculations show a series of unstable modes for a Mach number greater than 3. The number of such modes is much more than two (unlike what Mack reports). The number and the frequency extent of the corresponding unstable zones increase with an increase in M, which is significantly higher than subsonic or low-supersonic cases. While the shape of the neutral curves for the second unstable mode for a Mach number greater than 4 is similar to the fused neutral curve shown by Mack for a Mach number of 4.8, the characteristics of higher-order spatially unstable modes considering the viscous stability of supersonic boundary layers remain unreported to the best of our knowledge. The last one is the most novel element in the reported results.

Let $L((T^{-1}))$ be the space of (inverse) Laurent serieswith coefficients in some field $L$. It has a standard degree map and the induced topology. With its usual addition and a new product on this space which is continuous and preserves the standard degree map, it will be a complete topological division ring, and called a deformed Laurent series ring. Under mild restrictions, we give the necessary and sufficient conditions for a product on $L((T^{-1}))$ to make it a deformed Laurent series ring. Then we apply the above theory to construct the completions of the Weyl division ring $D_1$, over some field of characteristic 0, with respect to a class of discrete valuations on it. Such completions are topological division rings with nice properties. For instance, their valuation rings are non-commutative Henselian rings; the centralizer of each element not in the center is commutative.

We propose a novel algorithm for online resource allocation with non-stationary customer arrivals and unknown click-through rates. We assume multiple types of customers arrive in a nonstationary stochastic fashion, with unknown arrival rates in each period, and that customers' click-through rates are unknown and can only be learned online. By leveraging results from the stochastic contextual bandit with knapsack and online matching with adversarial arrivals, we develop an online scheme to allocate the resources to nonstationary customers. We prove that under mild conditions, our scheme achieves a best-of-both-world'' result: the scheme has a sublinear regret when the customer arrivals are near-stationary, and enjoys an optimal competitive ratio under general (non-stationary) customer arrival distributions. Finally, we conduct extensive numerical experiments to show our approach generates near-optimal revenues for all different customer scenarios.

We consider the transmission of spatially correlated analog information in a wireless sensor network (WSN) through fading single-input and multiple-output (SIMO) multiple access channels (MACs) with low-latency requirements. A lattice-based analog joint source-channel coding (JSCC) approach is considered where vectors of consecutive source symbols are encoded at each sensor using an n-dimensional lattice and then transmitted to a multiantenna central node. We derive a minimum mean square error (MMSE) decoder that accounts for both the multidimensional structure of the encoding lattices and the spatial correlation. In addition, a sphere decoder is considered to simplify the required searches over the multidimensional lattices. Different lattice-based mappings are approached and the impact of their size and density on performance and latency is analyzed. Results show that, while meeting low-latency constraints, lattice-based analog JSCC provides performance gains and higher reliability with respect to the state-of-the-art JSCC schemes.

For $N \ge 1, s\in (0,1)$, and $p \in (1, N/s)$ we find a positive solution to the following class of semipositone problems associated with the fractional $p$-Laplace operator: $$\tag{SP}

(-\Delta)_{p}^{s}u = g(x)f_a(u) \text{ in } \mathbb{R}^N,$$ where $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is a positive function, $a>0$ is a parameter and $f_a \in \mathcal{C}(\mathbb{R})$ is defined as $f_a(t) = f(t)-a$ for $t \ge 0$, $f_a(t) = -a(t+1)$ for $t \in [-1, 0]$, and $f_a(t) = 0$ for $t \le -1$, where $f \in \mathcal{C}(\mathbb{R}^+)$ satisfies $f(0)=0$ with subcritical and Ambrosetti-Rabinowitz type growth. Depending on the range of $a$, we obtain the existence of a mountain pass solution to (SP) in $\mathcal{D}^{s,p}(\mathbb{R}^N)$. Then, we prove mountain pass solutions are uniformly bounded with respect to $a$, over $L^r(\mathbb{R}^N)$ for every $r \in [Np/N-sp, \infty]$. In addition, if $p>2N/N+2s$, we establish that (SP) admits a non-negative mountain pass solution for each $a$ near zero. Finally, under the assumption $g(x) \leq B/|x|^{\beta(p-1)+sp}$ for $B>0, x \neq 0$, and $\beta \in (N-sp/p-1, N/p-1)$, we derive an explicit positive subsolution to (SP) and show that the non-negative solution is positive a.e. in $\mathbb{R}^N$.

We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the radii of the spheres are restricted to a compact subset of $(0,\infty)$. These bounds extend to general hypersurfaces with non-vanishing Gaussian curvature, and to geodesic spheres on compact manifolds. We also obtain improved maximal function bounds, and pointwise convergence statements, for wave propagators.

This paper answers a fundamental question about the exact distribution of the signal-to-interference-plus-noise ratio (SINR) under matched-filter (MF) precoding. Specifically, we derive the exact expressions for the cumulative distribution function (CDF) and the probability density function (PDF) of SINR under MF precoding over Rayleigh fading channels. Based on the exact analysis, we then rigorously prove that the SINR converges to some specific distributions separately in high SNR and in massive MIMO. To simplify the exact result in general cases, we develop a good approximation by modelling the interference as a Beta distribution. We then shift to the exact analysis of the transmit rate, and answer the fundamental question: How does the exact rate converge to the well-known asymptotic rate in massive MIMO? After that, we propose a novel approximation for the ergodic rate, which performs better than various existing approximations. Finally, we present some numerical results to demonstrate the accuracy of the derived analytical models.

In this paper we introduce a notion of Poincar\'e exponent for isometric representations of discrete groups on Hilbert spaces. Similarly as growth exponents control the geometry this exponent is shown to control the size of spectral gaps. Following similar ideas as Patterson and Sullivan it is used in the case of negatively curved groups to construct weakly contained boundary representations reflecting the spectral properties of the original representation analogously as complementary series representations in the case of semi-simple Lie groups. This is exploited to deduced sharp estimates on spectral invariants. A quantitive property (T) \'a la Cowling is also established proving uniform bound on the mixing rate of representations of hyperbolic groups with property (T). Along the way some properties of boundary representations are discussed. A original characterisation of the positivity of the so-called Knapp-Stein operators and certain fusion rules on the boundary complementary series representations are established.

The Finite Fourier Series (FFS) Shape-Based (SB) trajectory approximation method has been used to rapidly generate initial trajectories that satisfy the dynamics, trajectory boundary conditions, and limitation on maximum thrust acceleration. The FFS SB approach solves a nonlinear programming problem (NLP) in searching for feasible trajectories. This paper extends the development of the FFS SB approach to generate sub optimal solutions. Specifically, the objective function of the NLP problem is modified to include also a measure for the time of flight. Numerical results presented in this paper show several solutions that differ from those of the original FFS SB ones. The sub-optimal trajectories generated using a time of flight minimization are shown to be physically feasible trajectories and potential candidates for direct solvers.

The aim of this paper is to develop hybrid non-orthogonal multiple access (NOMA) assisted downlink transmission. First, for the single-input single-output (SISO) scenario, i.e., each node is equipped with a single antenna, a novel hybrid NOMA scheme is introduced, where NOMA is implemented as an add-on of a legacy time division multiple access (TDMA) network. Because of the simplicity of the SISO scenario, analytical results can be developed to reveal important properties of downlink hybrid NOMA. For example, in the case that the users' channel gains are ordered and the durations of their time slots are the same, downlink hybrid NOMA is shown to always outperform TDMA, which is different from the existing conclusion for uplink hybrid NOMA. Second, the proposed downlink SISO hybrid NOMA scheme is extended to the multiple-input single-output (MISO) scenario, i.e., the base station has multiple antennas. For the MISO scenario, near-field communication is considered to illustrate how NOMA can be used as an add-on in legacy networks based on space division multiple access and TDMA. Simulation results verify the developed analytical results and demonstrate the superior performance of downlink hybrid NOMA compared to conventional orthogonal multiple access.

In this paper, a direct finite element method is proposed for solving interface problems on simple unfitted meshes. The fact that the two interface conditions form a $H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$ pair leads to a simple and direct weak formulation with an integral term for the mutual interaction over the interface, and the well-posedness of this weak formulation is proved. Based on this formulation, a direct finite element method is proposed to solve the problem on two adjacent subdomains separated by the interface by conforming finite element and conforming mixed finite element, respectively. The well-posedness and an optimal a priori analysis are proved for this direct finite element method under some reasonable assumptions. A simple lowest order direct finite element method by using the linear element method and the lowest order Raviart-Thomas element method is proposed and analyzed to admit the optimal a priori error estimate by verifying the aforementioned assumptions. Numerical tests are also conducted to verify the theoretical results and the effectiveness of the direct finite element method.

In this paper, we establish the partial correlation graph for multivariate continuous-time stochastic processes, assuming only that the underlying process is stationary and mean-square continuous with expectation zero and spectral density function. In the partial correlation graph, the vertices are the components of the process and the undirected edges represent partial correlations between the vertices. To define this graph, we therefore first introduce the partial correlation relation for continuous-time processes and provide several equivalent characterisations. In particular, we establish that the partial correlation relation defines a graphoid. The partial correlation graph additionally satisfies the usual Markov properties and the edges can be determined very easily via the inverse of the spectral density function. Throughout the paper, we compare and relate the partial correlation graph to the mixed (local) causality graph of Fasen-Hartmann and Schenk (2023a). Finally, as an example, we explicitly characterise and interpret the edges in the partial correlation graph for the popular multivariate continuous-time AR (MCAR) processes.

This paper investigates the theoretical analysis of intrinsic message passing decoding for generalized product codes (GPCs) with irregular degree distributions, a generalization of product codes that allows every code bit to be protected by a minimum of two and potentially more component codes. We derive a random hypergraph-based asymptotic performance analysis for GPCs, extending previous work that considered the case where every bit is protected by exactly two component codes. The analysis offers a new tool to guide the code design of GPCs by providing insights into the influence of degree distributions on the performance of GPCs.

We prove that the conjugacy relation of transitive homeomorphisms on the Hilbert cube (resp. on the Cantor space) is Borel bireducible with the universal orbit relation induced by a Polish group (resp. by the group $S_\infty$). We also prove that conjugacy relation of minimal homeomorphisms on tame cantoroids (excluding the Cantor space) is Borel and thus this class of spaces is not suitable for identifying the complexity of conjugacy of minimal systems.

We establish the Fa\a di Bruno formula, in the sense of almost everywhere equality, for derivatives of the composed function $f \circ g$, for all function $f : R \rightarrow R$ such that $f$ acts on $W^m_p(R^n)$ by composition, and all $g \in W^m_p(R^n)$, possibly modified on a set of measure 0.

In this paper we deal with a weighted eigenvalue problem for the anisotropic $(p,q)$-Laplacian with Dirichlet boundary conditions. We study the main properties of the first eigenvalue and prove a reverse H\"older type inequality for the corresponding eigenfunctions.

This article bridges the gap between two topics used in sharing an encryption key: (i) Key Consolidation, i.e., extracting two identical strings of bits from two information sources with similarities (common randomness). (ii) Quantum-safe Key Encapsulation by incorporating randomness in Public/Private Key pairs. In the context of Key Consolidation, the proposed scheme adds to the complexity Eve faces in extracting useful data from leaked information. In this context, it is applied to the method proposed in [1] for establishing common randomness from round-trip travel times in a packet data network. The proposed method allows adapting the secrecy level to the amount of similarity in common randomness. It can even encapsulate a Quantum-safe encryption key in the extreme case that no common randomness is available. In the latter case, it is shown that the proposed scheme offers improvements with respect to the McEliece cryptosystem which currently forms the foundation for Quantum safe key encapsulation.

[1] A. K. Khandani, "Looping for Encryption Key Generation Over the Internet: A New Frontier in Physical Layer Security," 2023 Biennial Symposium on Communications (BSC), Montreal, QC, Canada, 2023, pp. 59-64

Let $V$ be quadratic space of even dimension and of signature $(p, q)$ with $p \geq q > 0$. We show that the Kudla-Millson lift of toric cycles - attached to algebraic tori - is a cusp form that is the diagonal restriction of a Hilbert modular form of parallel weight one. We deduce a formula relating the dimension of the span of such diagonal restrictions and the dimension of the span of toric and special cycles.

In this paper, we introduce two metrics, namely, age of actuation (AoA) and age of actuated information (AoAI), within a discrete-time system model that integrates data caching and energy harvesting (EH). AoA evaluates the timeliness of actions irrespective of the age of the information, while AoAI considers the freshness of the utilized data packet. We use Markov Chain analysis to model the system's evolution. Furthermore, we employ three-dimensional Markov Chain analysis to characterize the stationary distributions for AoA and AoAI and calculate their average values. Our findings from the analysis, validated by simulations, show that while AoAI consistently decreases with increased data and energy packet arrival rates, AoA presents a more complex behavior, with potential increases under conditions of limited data or energy resources. These metrics go towards the semantics of information and goal-oriented communications since they consider the timeliness of utilizing the information to perform an action.

This paper belongs to a group of work in the intersection of symbolic computation and group analysis aiming for the symbolic analysis of differential equations. The goal is to extract important properties without finding the explicit general solution. In this contribution, we introduce the algorithmic verification of nonlinear superposition properties and its implementation. More exactly, for a system of nonlinear ordinary differential equations of first order with a polynomial right-hand side, we check if the differential system admits a general solution by means of a superposition rule and a certain number of particular solutions. It is based on the theory of Newton polytopes and associated symbolic computation. The developed method provides the basis for the identification of nonlinear superpositions within a given system and for the construction of numerical methods which preserve important algebraic properties at the numerical level.

Extremely large aperture array (ELAA) is anticipated to serve as a pivotal feature of future multiple-input multiple-output (MIMO) systems in 6G. Near-field (NF) fading channel models are essential for reliable link-level simulation and ELAA system design. In this article, we propose a framework designed to generate NF fading channels for both communication and integrated sensing and communication (ISAC) applications. The framework allows a mixed of line of sight (LoS) and non-LoS (NLoS) links. It also considers spherical wave model and spatially non-stationary shadow fading. Based on this framework, we propose a three-dimensional (3D) fading channel model for ELAA systems deployed with a uniform rectangular array (URA). It can capture the impact of sensing object for ISAC applications. Moreover, all parameters involved in the framework are based on specifications or measurements from the 3rd Generation Partnership Project (3GPP) documents. Therefore, the proposed framework and channel model have the potential to contribute to the standard in various aspects, including ISAC, extra-large (XL-) MIMO, and reconfigurable intelligent surface (RIS) aided MIMO systems. Finally, future directions for ELAA are presented, including not only NF channel modeling but also the design of next-generation transceivers.

We prove a synthetic Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by $K<0$ and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$. In the course of the proof, we show that the space necessarily is a warped product with warping function $\cos:(-\frac{\pi}{2},\frac{\pi}{2})\to\mathbb{R}_+$.

We consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier--Stokes--Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck--Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck--Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.

We consider continuous-time mean-field stochastic games with strategic complementarities. The interaction between the representative productive firm and the population of rivals comes through the price at which the produced good is sold and the intensity of interaction is measured by a so-called "strenght parameter" $\xi$. Via lattice-theoretic arguments we first prove existence of equilibria and provide comparative statics results when varying $\xi$. A careful numerical study based on iterative schemes converging to suitable maximal and minimal equilibria allows then to study in relevant financial examples how the emergence of multiple equilibria is related to the strenght of the strategic interaction.

This paper studies Bayesian optimization with noise-free observations. We introduce new algorithms rooted in scattered data approximation that rely on a random exploration step to ensure that the fill-distance of query points decays at a near-optimal rate. Our algorithms retain the ease of implementation of the classical GP-UCB algorithm and satisfy cumulative regret bounds that nearly match those conjectured in arXiv:2002.05096, hence solving a COLT open problem. Furthermore, the new algorithms outperform GP-UCB and other popular Bayesian optimization strategies in several examples.

Movable antenna (MA) provides an innovative way to arrange antennas that can contribute to improved signal quality and more effective interference management. This method is especially beneficial for full-duplex (FD) wireless, which struggles with self-interference (SI) that usually overpowers the desired incoming signals. By dynamically repositioning transmit/receive antennas, we can mitigate the SI and enhance the reception of incoming signals. Thus, this paper proposes a novel MA-enabled point-to-point FD wireless system and formulates the minimum achievable rate of two FD terminals. To maximize the minimum achievable rate and determine the near-optimal positions of the MAs, we introduce a solution based on projected particle swarm optimization (PPSO), which can circumvent common suboptimal positioning issues. Moreover, numerical results reveal that the PPSO method leads to a better performance compared to the conventional alternating position optimization (APO). The results also demonstrate that an MA-enabled FD system outperforms the one using fixed-position antennas (FPAs).

This paper investigates the link between the null controllability property for some abstract parabolic problems and an inequality that can be seen as a quantified Fattorini-Hautus test. Depending on the hypotheses made on the abstract setting considered we prove that this inequality either gives the exact minimal null control time or at least gives the qualitative property of existence of such a minimal time. We also prove that for many known examples of minimal time in the parabolic setting, this inequality recovers the value of this minimal time.

We leverage the Gibbs inequality and its natural generalization to R\'enyi entropies to derive closed-form parametric expressions of the optimal lower bounds of $\rho$th-order guessing entropy (guessing moment) of a secret taking values on a finite set, in terms of the R\'enyi-Arimoto $\alpha$-entropy. This is carried out in an non-asymptotic regime when side information may be available. The resulting bounds yield a theoretical solution to a fundamental problem in side-channel analysis: Ensure that an adversary will not gain much guessing advantage when the leakage information is sufficiently weakened by proper countermeasures in a given cryptographic implementation. Practical evaluation for classical leakage models show that the proposed bounds greatly improve previous ones for analyzing the capability of an adversary to perform side-channel attacks.

Finite rank perturbations of diagonalizable normal operators acting boundedly on infinite dimensional, separable, complex Hilbert spaces are considered from the standpoint of view of the existence of invariant subspaces. In particular, if $T=D_\Lambda+u\otimes v$ is a rank-one perturbation of a diagonalizable normal operator $D_\Lambda$ with respect to a basis $\mathcal{E}=\{e_n\}_{n\geq 1}$ and the vectors $u$ and $v$ have Fourier coefficients $\{\alpha_n\}_{n\geq 1}$ and $\{\beta_n\}_{n\geq 1}$ with respect to $\mathcal{E}$ respectively, it is shown that $T$ has non trivial closed invariant subspaces provided that either $u$ or $v$ have a Fourier coefficient which is zero or $u$ and $v$ have non zero Fourier coefficients and

$$\sum_{n\geq 1} |\alpha_n|^2 \log \frac{1}{|\alpha_n|} + |\beta_n|^2 \log \frac{1}{|\beta_n|} < \infty.$$

As a consequence, if $(p,q)\in (0,2]\times (0,2]$ are such $\sum_{n\geq 1} (|\alpha_n|^p + |\beta_n|^q )< \infty,$ it is shown the existence of non trivial closed invariant subspaces of $T$ whenever

$$(p,q)\in (0,2]\times (0,2]\setminus \{(2, r), (r, 2):\; r\in(1,2]\}.$$

Moreover, such operators $T$ have non trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity. Likewise, analogous results hold for finite rank perturbations of $D_\Lambda$. This improves considerably previous theorems of Foia\c{s}, Jung, Ko and Pearcy, Fang and Xia and the authors on an open question explicitly posed by Pearcy in the seventies.

In this paper, we present some controllability results for linear and nonlinear phase-field systems of Caginalp type considered in a bounded interval of $\mathbb R$ when the scalar control force acts on the temperature equation of the system by means of the Dirichlet condition on one of the endpoints of the interval. In order to prove the linear result we use the moment method providing an estimate of the cost of fast controls. Using this estimate and following the methodology developed in~\cite{tucsnack}, we prove a local exact boundary controllability result to constant trajectories of the nonlinear phase-field system. To the authors' knowledge, this is the first nonlinear boundary controllability result in the framework of non-scalar parabolic systems, framework in which some hyperbolic'' behaviors could arise.

We propose a class of new cutting planes to strengthen the Lov\'asz theta function, a well-known semidefinite programming (SDP) relaxation for the stable set and the graph coloring problems. For both problems, we introduce two new cutting planes that are derived from odd-cycle constraints and triangle inequalities and are valid for certain subgraphs induced by odd cycles. Computational experiments on graphs from the literature show that for each problem, one of the proposed cutting planes gives very good bounds.

A result of Balogh, Csaba, Jing and Pluh\'ar yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half of its edges in one colour). In this note we prove an analogous result for perfect matchings in $k$-uniform hypergraphs. More precisely, for each $2\leq \ell <k$ and $r\geq 2$ we determine the minimum $\ell$-degree threshold for forcing a perfect matching of significant colour-bias in an $r$-coloured $k$-uniform hypergraph.

Colimits are a fundamental construction in category theory. They provide a way to construct new objects by gluing together existing objects that are related in some way. We introduce a complementary notion of anticolimits, which provide a way to decompose an object into a colimit of other objects. While anticolimits are not unique in general, we establish that in the presence of pullbacks, there is a "canonical" anticolimit which characterises the existence of other anticolimits. We also provide convenient techniques for computing anticolimits, by changing either the shape or ambient category.

The main motivation for this work is the development of a new method, known as anticontraction, for constructing homotopies in the proof assistant homotopy.io for finitely presented $n$-categories. Anticontraction complements the existing contraction method and facilitates the construction of homotopies increasing the complexity of a term, enhancing the usability of the proof assistant. For example, it simplifies the naturality move and third Reidemeister move.

Solar oscillations can be modeled by Galbrun's equation which describes Lagrangian wave displacement in a self-gravitating stratified medium. For spherically symmetric backgrounds, we construct an algorithm to compute efficiently and accurately the coefficients of the Green's tensor of the time-harmonic equation in vector spherical harmonic basis. With only two resolutions, our algorithm provides values of the kernels for all heights of source and receiver, and prescribes analytically the singularities of the kernels. We also derive absorbing boundary conditions (ABC) to model wave propagation in the atmosphere above the cut-off frequency. The construction of ABC, which contains varying gravity terms, is rendered difficult by the complex behavior of the solar potential in low atmosphere and for frequencies below the Lamb frequency. We carry out extensive numerical investigations to compare and evaluate the efficiency of the ABCs in capturing outgoing solutions. Finally, as an application towards helioseismology, we compute synthetic solar power spectra that contain pressure modes as well as internal-gravity (g-) and surface-gravity (f-) ridges which are missing in simpler approximations of the wave equation. For purpose of validation, the location of the ridges in the synthetic power spectra are compared with observed solar modes.

A multi-input multi-output (MIMO) Gaussian channel with two transmit antennas and two receive antennas is studied that is subject to an input peak-power constraint. The capacity and the capacity-achieving input distribution are unknown in general. The problem is shown to be equivalent to a channel with an identity matrix but where the input lies inside and on an ellipse with principal axis length $r_p$ and minor axis length $r_m$. If $r_p \le \sqrt{2}$, then the capacity-achieving input has support on the ellipse. A sufficient condition is derived under which a two-point distribution is optimal. Finally, if $r_m < r_p \le \sqrt{2}$, then the capacity-achieving distribution is discrete.

The present paper is concerned with the well-posedness theory for non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. Differently from previous works, we consider here the full odd viscosity tensor. Similarly to the work of Bresch and Desjardins in compressible fluid mechanics, we identify the presence of an effective velocity in the system, linking the velocity field of the fluid and the gradient of a suitable function of the density. By use of this effective velocity, we propose a new formulation of the original system of equations, thus highlighting a strong similarity with the equations of the ideal magnetohydrodynamics. By taking advantage of the new formulation of the equations, we establish a local in time well-posedness theory in Besov spaces based on $L^\infty$ and prove a lower bound for the lifespan of the solutions implying asymptotically global'' existence: in the regime of small initial density variations, $\rho_0-1= O(\varepsilon)$ for small $\varepsilon>0$, the corresponding solution is defined up to some time $T_\varepsilon>0$ satisfying the property $T_\varepsilon\,\longrightarrow\,+\infty$ when $\varepsilon\to0^+$.

We present a new method to estimate the rate-distortion-perception function in the perfect realism regime (PR-RDPF), for multivariate continuous sources subject to a single-letter average distortion constraint. The proposed approach is not only able to solve the specific problem but also two related problems: the entropic optimal transport (EOT) and the output-constrained rate-distortion function (OC-RDF), of which the PR-RDPF represents a special case. Using copula distributions, we show that the OC-RDF can be cast as an I-projection problem on a convex set, based on which we develop a parametric solution of the optimal projection proving that its parameters can be estimated, up to an arbitrary precision, via the solution of a convex program. Subsequently, we propose an iterative scheme via gradient methods to estimate the convex program. Lastly, we characterize a Shannon lower bound (SLB) for the PR-RDPF under a mean squared error (MSE) distortion constraint. We support our theoretical findings with numerical examples by assessing the estimation performance of our iterative scheme using the PR-RDPF with the obtained SLB for various sources.

Adaptive dynamics describes a deterministic approximation of the evolution of scalar- and function-valued traits. Applying it to the team game developed by Menden-Deuer and Rowlett [Menden-Deuer & Rowlett 2019], we constructed an evolutionary process in the game. We also refined the adaptive dynamics framework itself to a new level of mathamatical rigor. In our analysis, we demonstrated the existence of solutions to the adaptive dynamics for the team game and determined their regularity. Moreover, we identified all stationary solutions and proved that these are precisely the Nash equilibria of the team game. Numerical examples are provided to highlight the main characteristics of the dynamics. The linearity of the team game results in unstable dynamics; non-stationary solutions oscillate and perturbations of the stationary solutions do not shrink. Instead, a linear type of branching may occur. We finally discuss how to experimentally validate these results. Due to the abstract nature of the team game, our results could be applied to derive implications and predictions in several fields including biology, sports, and finance.

In the literature, there are many results about permutation polynomials over finite fields. However, very few permutations of vector spaces are constructed although it has been shown that permutations of vector spaces have many applications in cryptography, especially in constructing permutations with low differential and boomerang uniformities.

In this paper, motivated by the butterfly structure \cite{perrin2016cryptanalysis} and the work of Qu and Li \cite{qu2023}, we investigate rotatable permutations from $\gf_{2^m}^3$ to itself with $d$-homogenous functions.

Based on the theory of equations of low degree, the resultant of polynomials, and some skills of exponential sums, we construct five infinite classes of $3$-homogeneous rotatable permutations from $\gf_{2^m}^3$ to itself, where $m$ is odd. Moreover, we demonstrate that the corresponding permutation polynomials of $\gf_{2^{3m}}$ of our newly constructed permutations of $\gf_{2^m}^3$ are QM-inequivalent to the known ones.

We prove that there exist K\"{a}hler manifolds that are not homotopy equivalent to a quotient of complex hyperbolic space but which admit a Riemannian metric with nonpositive curvature operator. This shows that K\"{a}hler manifolds do not satisfy the same type of rigidity with respect to the curvature operator as quaternionic hyperbolic and Cayley hyperbolic manifolds and are thus more similar to real hyperbolic manifolds in this setting. Along the way we also calculate explicit values for the eigenvalues of the curvature operator with respect to the standard complex hyperbolic metric.

This paper is concerned with the linear stability analysis for the Couette flow of the Euler-Poisson system for both ionic fluid and electronic fluid in the domain $\bb{T}\times\bb{R}$. We establish the upper and lower bounds of the linearized solutions of the Euler-Poisson system near Couette flow. In particular, the inviscid damping for the solenoidal component of the velocity is obtained.

This paper investigates the secure resource allocation for a downlink integrated sensing and communication system with multiple legal users and potential eavesdroppers. In the considered model, the base station (BS) simultaneously transmits sensing and communication signals through beamforming design, where the sensing signals can be viewed as artificial noise to enhance the security of communication signals. To further enhance the security in the semantic layer, the semantic information is extracted from the original information before transmission. The user side can only successfully recover the received information with the help of the knowledge base shared with the BS, which is stored in advance. Our aim is to maximize the sum semantic secrecy rate of all users while maintaining the minimum quality of service for each user and guaranteeing overall sensing performance. To solve this sum semantic secrecy rate maximization problem, an iterative algorithm is proposed using the alternating optimization method. The simulation results demonstrate the superiority of the proposed algorithm in terms of secure semantic communication and reliable detection.

This note presents an upper bound of $1.252 n$ on the size of a set system that satisfies the mod-6 town rules. Under these rules the sizes of the sets are not congruent to $0\bmod 6$ while the sizes of all pairwise intersections are congruent to $0\bmod 6$.

R.Pavlov and S.Schmieding provided recently some synthetic results about generic $\mathbb{Z}$-shifts, which rely mainly on an original theorem stating that isolated points form a residual set in the space of $\mathbb{Z}$-shifts such that all other residual set must contain it. As a direction for further research, they pointed towards genericity in the space of $\mathbb{G}$-shifts, where $\mathbb{G}$ is a finitely generated group. In the present text, we approach this for the case of $\mathbb{Z}^d$-shifts, where $d \ge 2$. As it is usual, multidimensional dynamical systems are much more difficult to understand. Provided the result of R.Pavlov and S.Schmieding, it is natural to begin with a better understanding of isolated points. We prove here a characterization of such points in the space of $\mathbb{Z}^d$-shifts, in terms of the natural notion of maximal subsystems which we also introduce in this article. From this characterization we recover the one of R.Pavlov and S.Schmieding's. We also prove a series of results which exploit this notion. In particular some transitivity-like properties can be related to the number of maximal subsystems. Furthermore, on the contrary of dimension one, the set of isolated shifts is not residual. We also prove that the Cantor-Bendixson rank of the space of $\mathbb{Z}^d$-shifts is infinite when $d > 1$, while it is equal to one when $d=1$.

The general goal of this work is to obtain upper and lower bounds for the $L^2$-norm of biorthogonal families to complex exponential functions associated to sequences $\{ \Lambda_k \}_{k \ge 1} \subset \mathbb C$ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $T > 0$: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $T$ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence.

We indulge in what mathematicians call frivolous activities. In Arithmetic Billiards, a ball is bouncing around in a rectangle. In Parity Checkers we place checkers on a checkerboard under certain parity constraints. Both activities turn out to capture the division of congruence classes modulo a prime into squares and non-squares, allowing fairly simple proofs of the celebrated Law of Quadratic Reciprocity. Since the activities are analyzed somewhat in parallel we don't obtain two independent proofs. But Franz Lemmermeyer's online list of reciprocity proofs already contains well over three hundred items, which seems enough anyway.

Large language models (LLMs) have revolutionized the field of natural language processing, extending their strong capabilities into multi-modal domains. Thus, it is vital to define proper and diversified metrics for the evaluation of LLMs.

In this paper, we introduce matrix entropy, a novel metric rooted in information theory and geometry principles to quantify the data compression proficiency in LLMs. It reflects the model's ability to extract relevant information and eliminate unnecessary elements, thereby providing insight into the language model's intrinsic capability. Specifically, we demonstrate its applicability in both single-modal (language) and multi-modal settings. For language models, our findings reveal that the matrix entropy of representations follows a scaling law type reduction when the model scales up, serving as a complement to the traditional loss scaling law. For the multi-modal setting, we also propose an evaluation method based on matrix entropy for assessing alignment quality and we find that modern large multi-modal models exhibit great alignment performance.

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups $G$ acting on a non-compactly causal symmetric space $M = G/H$, every irreducible unitary representation of $G$ can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over $M$. In the present paper we discuss this procedure for the connected Lorentz group $G = SO_{1,d}(R)_e$ acting on de Sitter space $M = dS^d$. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990's, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain $G_C^{cut}$ in the complexified group $G_C = \SO_{1,d}(C)$, which for $d = 1$ specializes to the complex cut plane $C \setminus (-\infinity, 0]$. A number of special situations is discussed specifically: (a) The case $d = 1$, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for $d > 2$ is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case $d = 3$, where we obtain rather explicit formulas for the matrix-valued spherical functions.

In this paper, we propose a novel approach to test the equality of high-dimensional mean vectors of several populations via the weighted $L_2$-norm. We establish the asymptotic normality of the test statistics under the null hypothesis. We also explain theoretically why our test statistics can be highly useful in weakly dense cases when the nonzero signal in mean vectors is present. Furthermore, we compare the proposed test with existing tests using simulation results, demonstrating that the weighted $L_2$-norm-based test statistic exhibits favorable properties in terms of both size and power.

We study the existence of strong solutions to the initial value problem for the incompressible Navier-Stokes equations in $\mathbb{R}^N, N\geq 3$. Our investigation shows that local in-time classical solutions do not develop singularity as long as the initial velocity lies in $(L^2(\mathbb{R}^N))^N\cap (L^\infty(\mathbb{R}^N))^N$.

We study Markov chains with non-negative sectional curvature on finite metric spaces. Neither reversibility, nor the restriction to a particular combinatorial distance are imposed. In this level of generality, we prove that a 1-step contraction in the Wasserstein distance implies a 1-step contraction in relative entropy, by the same amount. Our result substantially strengthens a recent breakthrough of the second author, and has the advantage of being applicable to arbitrary scales. This leads to a time-varying refinement of the standard Modified Log-Sobolev Inequality (MLSI), which allows us to leverage the well-acknowledged fact that curvature improves at large scales. We illustrate this principle with several applications, including birth and death chains, colored exclusion processes, permutation walks, and attractive zero-range dynamics. In particular, we prove a MLSI with constant equal to the minimal rate increment for the mean-field zero-range process, thereby answering a long-standing question.

We deal with the class of Hausdorff spaces having a $\pi$-base whose elements have an H-closed closure. Carlson proved that $|X|\leq 2^{wL(X)\psi_c(X)t(X)}$ for every quasiregular space $X$ with a $\pi$-base whose elements have an H-closed closure. We provide an example of a space $X$ having a $\pi$-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that $|X|> 2^{wL(X)\chi(X)}$ (then $|X|> 2^{wL(X)\psi_c(X)t(X)}$). Still in the class of spaces with a $\pi$-base whose elements have an H-closed closure, we establish the bound $|X|\leq2^{wL(X)k(X)}$ for Urysohn spaces and we give an example of an Urysohn space $Z$ such that $k(Z)<\chi(Z)$. Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a $\pi$-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a $\pi$-base whose elements have an H-closed closure then such space is Baire.

Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green's function, referred to as BIN-G. We evaluate the Green's function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green's function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green's function for PDEs with variable coefficients. The learned Green's function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process.

We study both a linear and a nonlinear case and describe the concentration profile. In particular, we analyse a hyperbolic (or high frequency) regime that can be interpreted both as a local (microscopic) or as a nonlocal (macroscopic) rescaling. We consider a Hopf-Cole transform and derive a Hamilton-Jacobi equation. The concentrations are then explained as a consequence of the stationary points of the Hamiltonian that is spatially heterogeneous like the velocity-jump process. After revising the classical hydrodynamic limits for the aggregate quantities and the eikonal equation that can be derived from those with a Hopf-Cole transform, we find that the Hamilton-Jacobi equation is a second order approximation of the eikonal equation in the limit of small diffusivity. For nonlinear turning kernels, the Hopf-Cole transform allows to study the stability of the possible homogeneous configurations and of patterns and the results of a linear stability analysis previously obtained are found and extended to a nonlinear regime. In particular, it is shown that instability (pattern formation) occurs when the Hamiltonian is convex-concave.

In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form, and formulate the optimization problem that minimizes the error of PDE solution from data. Using variational calculus, we obtain an evolution equation for the Lagrange multipliers (adjoint equations) allowing us to compute the gradient of the objective function with respect to the parameters of PDEs given data in a straightforward manner. In particular, for a family of parameterized and nonlinear PDEs, we show how the corresponding adjoint equations can be derived. Here, we show that given smooth data set, the proposed adjoint method can recover the true PDE up to machine accuracy. However, in the presence of noise, the accuracy of the adjoint method becomes comparable to the famous PDE Functional Identification of Nonlinear Dynamics method known as PDE-FIND (Rudy et al., 2017). Even though the presented adjoint method relies on forward/backward solvers, it outperforms PDE-FIND for large data sets thanks to the analytic expressions for gradients of the cost function with respect to each PDE parameter.

Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of calibrability, finite time extinction, and some regularity properties of solutions. We focus on the second-order flow (possibly with weights) and the fourth-order flow. We also discuss the fractional cases.

We reiterate the contribution made by Harrow, Hassidim, and Llyod to the quantum matrix equation solver with the emphasis on the algorithm description and the error analysis derivation details. Moreover, the behavior of the amplitudes of the phase register on the completion of the Quantum Phase Estimation is studied. This study is beneficial for the comprehension of the choice of the phase register size and its interrelation with the Hamiltonian simulation duration in the algorithm setup phase.

Logarithmic Number Systems (LNS) hold considerable promise in helping reduce the number of bits needed to represent a high dynamic range of real-numbers with finite precision, and also efficiently support multiplication and division. However, under LNS, addition and subtraction turn into non-linear functions that must be approximated - typically using precomputed table-based functions. Additionally, multiple layers of error correction are typically needed to improve result accuracy. Unfortunately, previous efforts have not characterized the resulting error bound. We provide the first rigorous analysis of LNS, covering detailed techniques such as co-transformation that are crucial to implementing subtraction with reasonable accuracy. We provide theorems capturing the error due to table interpolations, the finite precision of pre-computed values in the tables, and the error introduced by fix-point multiplications involved in LNS implementations. We empirically validate our analysis using a Python implementation, showing that our analytical bounds are tight, and that our testing campaign generates inputs diverse-enough to almost match (but not exceed) the analytical bounds. We close with discussions on how to adapt our analysis to LNS systems with different bases and also discuss many pragmatic ramifications of our work in the broader arena of scientific computing and machine learning.

Tailoring polar code construction for decoding algorithms beyond successive cancellation has remained a topic of significant interest in the field. However, despite the inherent nested structure of polar codes, the use of sequence models in polar code construction is understudied. In this work, we propose using a sequence modeling framework to iteratively construct a polar code for any given length and rate under various channel conditions. Simulations show that polar codes designed via sequential modeling using transformers outperform both 5G-NR sequence and Density Evolution based approaches for both AWGN and Rayleigh fading channels.

Motivated by the structure of the Swanson oscillator, which is a well-known example of a non-hermitian quantum system consisting of a general representation of a quadratic Hamiltonian, we propose a fermionic extension of such a scheme which incorporates two fermionic oscillators, together with bilinear-coupling terms that do not conserve particle number. We determine the eigenvalues and eigenvectors, and expose the appearance of exceptional points where two of the eigenstates coalesce with the corresponding eigenvectors exhibiting the self-orthogonality relation. The model exhibits a quantum phase transition due to the presence of a ground-state crossing. We compute the entanglement spectrum and entanglement entropy of the ground state.

In this paper, we define invariants of links in terms of colorings of link diagrams and prove that these invariants coincide with various notions of widths of links with respect to the standard Morse function. Our formulations are advantageous because they are algorithmic and suitable for program implementations. As an application, we calculate the max-width of over 10000 links up to 14 crossings from the link table.

Given an open, bounded and connected set $\Omega\subset\mathbb{R}^{3}$ and its rescaling $\Omega_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$(\varepsilon^{-2}\chi_{\Omega_{\varepsilon}}+\chi_{\mathbb{R}^{3}\backslash\Omega_{\varepsilon}})\partial_{tt}u=\Delta u+f$$ with initial data and source supported outside $\Omega_{\varepsilon}$; here, $\chi_{S}$ denotes the characteristic function of a set $S$. We provide the first-order $\varepsilon$-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the $L^{\infty}((0,1/\varepsilon^{\tau}),L^{2}(\mathbb{R}^{3}))$-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in $L^{2}(\Omega)$ and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.

We show that transformation formulas of multiple $q$-hypergeometric series agree with wall-crossing formulas of $K$-theoretic vortex partition functions obtained by Hwang, Yi and the author \cite{Hwang:2017kmk}. For the vortex partition function in 3d $\mathcal{N}=2$ gauge theory, we show that the wall-crossing formula agrees with the Kajihara transformation \cite{kajihara2004euler}. For the vortex partition function in 3d $\mathcal{N}=4$ gauge theory, we show that the wall-crossing formula agrees with the transformation formula by Halln\"as, Langmann, Noumi and Rosengren \cite{Halln_s_2022}. Since the $K$-theoretic vortex partition functions are related with indices such as the $\chi_t$-genus of the handsaw quiver variety, we discuss geometric interpretation of Euler transformations in terms of wall-crossing formulas of handsaw quiver variety.

Given a lattice path $\nu$, the alt $\nu$-Tamari lattice is a partial order recently introduced by Ceballos and Chenevi\ere, which generalizes the $\nu$-Tamari lattice and the $\nu$-Dyck lattice. All these posets are defined on the set of lattice paths that lie weakly above $\nu$, and posses a rich combinatorial structure. In this paper, we study the geometric structure of these posets. We show that their Hasse diagram is the edge graph of a polytopal complex induced by a tropical hyperplane arrangement, which we call the alt $\nu$-associahedron. This generalizes the realization of $\nu$-associahedra by Ceballos, Padrol and Sarmiento. Our approach leads to an elegant construction, in terms of areas below lattice paths, which we call the canonical realization. Surprisingly, in the case of the classical associahedron, our canonical realization magically recovers Loday's ubiquitous realization, via a simple affine transformation.

In this work, we study inequalities and enumerative formulas for flags of Pfaff systems on $\mathbb{P}^n_{\mathbb{C}}$. More specifically, we find the number of independent Pfaff systems that leave invariant a one-dimensional holomorphic foliation and deduce inequalities relating the degrees in the flags, which can be interpreted as the Poincar\'e problem for flags. Moreover, restricting to a flag of specific holomorphic foliations/distributions, we obtain inequalities involving the degrees. As a consequence, we prove stability results for the tangent sheaf of some rank two holomorphic foliations/distributions.

The umbral restyling of hypergeometric functions is shown to be a useful and efficient approach in simplifying the associated computational technicalities. In this article, the authors provide a general introduction to the umbral version of Gauss hypergeometric functions and extend the formalism to certain generalized forms of these functions. It is shown that suggested approach is particularly efficient for evaluating integrals involving hypergeometric functions and their combination with other special functions.

A tournament is an orientation of a graph. Vertices are players and edges are games, directed away from the winner. Kannan, Tetali and Vempala and McShine showed that tournaments with given score sequence can be rapidly sampled, via simple random walks on the interchange graphs of Brualdi and Li. These graphs are generated by the cyclically directed triangle, in the sense that traversing an edge corresponds to the reversal of such a triangle in a tournament.

We study Coxeter tournaments on Zaslavsky's signed graphs. These tournaments involve collaborative and solitaire games, as well as the usual competitive games. The interchange graphs are richer in complexity, as a variety of other generators are involved. We prove rapid mixing by an intricate application of Bubley and Dyer's method of path coupling, using a delicate re-weighting of the graph metric. Geometric connections with the Coxeter permutahedra introduced by Ardila, Castillo, Eur and Postnikov are discussed.

We define a (non-decreasing) sequence $\{\text{dTC}_m(X)\}_{m\ge 2}$ of higher versions of distributional topological complexity ($\text{dTC}$) of a space $X$ introduced by Dranishnikov and Jauhari. This sequence generalizes $\text{dTC}(X)$ in the sense that $\text{dTC}_2(X) = \text{dTC}(X)$, and is a direct analog to the classical sequence $\{\text{TC}_m(X)\}_{m\ge 2}$. We show that like $\text{TC}_m$ and $\text{dTC}$, the sequential versions $\text{dTC}_m$ are also homotopy invariants. Also, $\text{dTC}_m(X)$ relates with the distributional LS-category ($d\text{cat}$) of products of $X$ in the same way as $\text{TC}_m(X)$ relates with the classical LS-category ($\text{cat}$) of products of $X$. We show that in general, $\text{dTC}_m$ is a different concept than $\text{TC}_m$ for each $m \ge 2$, but we also provide various examples of spaces $X$ for which the sequences $\{\text{TC}_m(X)\}_{m\ge 2}$ and $\{\text{dTC}_m(X)\}_{m\ge 2}$ coincide.

The classical Andr\'{a}sfai-Erd\H{o}s-S\'{o}s Theorem states that for $\ell\ge 2$, every $n$-vertex $K_{\ell+1}$-free graph with minimum degree greater than $\frac{3\ell-4}{3\ell-1}n$ must be $\ell$-partite. We establish a simple criterion for $r$-graphs, $r \geq 2$, to exhibit an Andr\'{a}sfai-Erd\H{o}s-S\'{o}s-type property (AES), leading to a classification of most previously studied hypergraph families with this property.

For every AES $r$-graph $F$, we present a simple algorithm to decide the $F$-freeness of an $n$-vertex $r$-graph with minimum degree greater than $(\pi(F) - \varepsilon_F)\binom{n}{r-1}$ in time $O(n^r)$, where $\varepsilon_F >0$ is a constant. In particular, for the complete graph $K_{\ell+1}$, we can take $\varepsilon_{K_{\ell+1}} = (3\ell^2-\ell)^{-1}$. Based on a result by Chen-Huang-Kanj-Xia, we show that for every fixed $C > 0$, this problem cannot be solved in time $n^{o(\ell)}$ if we replace $\varepsilon_{K_{\ell+1}}$ with $(C\ell)^{-1}$ unless ETH fails. Furthermore, we establish an algorithm to decide the $K_{\ell+1}$-freeness of an $n$-vertex graph with $\mathrm{ex}(n,K_{\ell+1})-k$ edges in time $(\ell+1)n^2$ for $k \le n/30\ell$ and $\ell \le \sqrt{n/6}$, partially improving upon the recently provided running time of $2.49^k n^{O(1)}$ by Fomin--Golovach--Sagunov--Simonov. Moreover, we show that for every fixed $\delta > 0$, this problem cannot be solved in time $n^{o(\ell)}$ if $k$ is of order $n^{1+\delta}$ unless ETH fails.

As an intermediate step, we show that for a specific class of $r$-graphs $F$, the (surjective) $F$-coloring problem can be solved in time $O(n^r)$, provided the input $r$-graph has $n$ vertices and a large minimum degree, refining several previous results.

It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to $q$-binomial transforms and study the behavior of the leading coefficient in such Hankel determinants. We also investigate the leading coefficient in the Hankel determinants for even-indexed Bernoulli polynomials with recourse to a curious binomial transform. In particular, the degrees of these Hankel determinants share the same nature as those in one of the $q$-binomial cases.

We give a compact tableau formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ in terms of a queue inversion statistic on certain sorted non-attacking tableaux. Our tableaux are in bijection with the multiline queues defined by Martin, from which we obtain an alternative multiline queue formula for $P_{\lambda}$.

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq \mathbb{N}$, such that for all $n\in\Gamma$, there is an explicit set $C_n$ of $(n!)^{\alpha}$ many permutations on $[n]$, such that every pair of permutations in $C_n$ has pairwise Ulam distance at least $(1-\varepsilon)\cdot n$. Moreover, we can compute the $i^{\text{th}}$ permutation in $C_n$ in poly$(n)$ time and can also decode in poly$(n)$ time, a permutation $\pi$ on $[n]$ to its closest permutation $\pi^*$ in $C_n$, if the Ulam distance of $\pi$ and $\pi^*$ is less than $\frac{(1-\varepsilon)\cdot n}{4}$.

Previously, it was implicitly known by combining works of Goldreich and Wigderson [Israel Journal of Mathematics'23] and Farnoud, Skachek, and Milenkovic [IEEE Transactions on Information Theory'13] in a black-box manner, that it is possible to explicitly construct $(n!)^{\Omega(1)}$ many permutations on $[n]$, such that every pair of them have pairwise Ulam distance at least $\frac{n}{6}\cdot (1-\varepsilon)$, for any $\varepsilon>0$, and the bound on the distance can be improved to $\frac{n}{4}\cdot (1-\varepsilon)$ if the construction of Goldreich and Wigderson is directly analyzed in the Ulam metric.

We consider goal-oriented adaptive space-time finite-element discretizations of the parabolic heat equation on completely unstructured simplicial space-time meshes. In some applications, we are interested in an accurate computation of some possibly nonlinear functionals at the solution, so called goal functionals. This motivates the use of adaptive mesh refinements driven by the dual-weighted residual (DWR) method. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal linear problem. The numerical experiment presented demonstrates that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for a model problem with moving domains and a linear goal functional, where we know the exact value.

We consider the Boussinesq-Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well-posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDEconstrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low-dissipation scheme, which contains non-conservative products. The scheme is based on the FORCE-{\alpha} method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.

We prove that for torsion-free amenable ample groupoids, an isomorphism in groupoid homology induced by an \'etale correspondence yields an isomorphism in the K-theory of the associated $\mathrm{C}^\ast$-algebras. We apply this to extend X. Li's K-theory formula for left regular inverse semigroup $\mathrm{C}^\ast$-algebras. These results are obtained by developing the functoriality of the ABC spectral sequence.

Motivated by a balanced ternary representation of the Collatz map we define the map $C_\mathbb{R}$ on the positive real numbers by setting $C_\mathbb{R}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_\mathbb{R}(x)=\frac{3}{2}x$ if $[x]$ is odd, where $[x]$ is defined by $[x]\in\mathbb{Z}$ and $x-[x]\in(-\frac{1}{2},\frac{1}{2}]$. We show that there exists a constant $K>0$ such that the set of $x$ fulfilling $\liminf_{n\in\mathbb{N}}C_\mathbb{R}^n(x)\leq K$ is Lebesgue-co-null. We also show that for any $\epsilon>0$ the set of $x$ for which $(\frac{3^{\frac{1}{2}}}{2})^kx^{1-\epsilon}\leq C_\mathbb{R}^k(x)\leq (\frac{3^{\frac{1}{2}}}{2})^kx^{1+\epsilon}$ for all $0\leq k\leq \frac{1}{1-\frac{\log_23}{2}}\log_2x$ is large for a suitable notion of largeness.

To represent the two-dimensional $N$-body delta-Bose gas for any integer $N\geq 2$, we construct an It\^{o} diffusion satisfying the strong Markov property and establish an associated Feynman-Kac-type formula. Among several properties, the diffusion is singular to the Brownian motion for showing almost sure contacts of particles and has a supercritical drift coefficient in the sense of a Ladyzhenskaya-Prodi-Serrin-type condition. Despite the supercriticality, the construction of the diffusion process and the relevant distributional properties for the Feynman-Kac-type formula are obtained by locally transforming the relative motions in two-body interactions and the necessary free motions. The central mechanism making this reduction to two-body interaction possible is the "no triple contacts" of particles, observed earlier in the functional integrals of the two-dimensional delta-Bose gas and now extended to the stochastic process level.

We construct geometrically a {\bf \em universal ADO link invariant} as a limit of {invariants given by graded intersections in configuration spaces}. The question of providing a link invariant that recovers the coloured Alexander invariants for coloured links (which are non-semisimple invariants) was an open problem. A parallel question about semi-simple invariants is the subject of Habiro's famous universal invariants \cite{H3}.

First, for a fixed level $\mathcal N$, we construct a link invariant globalising topologically all coloured Alexander link invariants at level less than $\mathcal N$ via the {\bf \em set of intersection points between Lagrangian submanifolds} supported on {\bf \em arcs and ovals} in the disc. Then, based on the naturality of these models when changing the colour, we construct the universal ADO invariant. The purely {\bf \em geometrical origin} of this universal invariant provides a {\bf \em new topological perspective} for the study of the asymptotics of these non-semisimple invariants, for which a purely topological $3$-dimensional description is a deep problem in quantum topology.

We finish with a conjecture that our universal invariant has a lift in a module over an extended version of the Habiro ring, which we construct. This paper has a sequel, showing that Witten-Reshetikhin-Turaev and Costantino-Geer-Patureau invariants can both be read off from a fixed set of submanifolds in a configuration space.

We consider cooperative semantic text communications facilitated by a relay node. We propose two types of semantic forwarding: semantic lossy forwarding (SLF) and semantic predict-and-forward (SPF). Both are machine learning aided approaches, and, in particular, utilize attention mechanisms at the relay to establish a dynamic semantic state, updated upon receiving a new source signal. In the SLF model, the semantic state is used to decode the received source signal; whereas in the SPF model, it is used to predict the next source signal, enabling proactive forwarding. Our proposed forwarding schemes do not need any channel state information and exhibit consistent performance regardless of the relay's position. Our results demonstrate that the proposed semantic forwarding techniques outperform conventional semantic-agnostic baselines.

The paper deals with the two-dimensional stochastic incompressible Navier-Stokes equation set in a bounded domain with Dirichlet boundary conditions. We consider an additive noise in the form of a cylindrical Wiener process regularized by a term $A^{-\gamma}$, where $A$ is the Stokes operator, and $\gamma\in(1/4,1/2)$. We prove uniqueness, ergodicity, and a strong mixing property for the invariant measure of the Markov semigroup. While previous results require $\gamma > 3/8$, we uncover the range $\gamma \in (1/4, 3/8]$ by adapting the so called Sobolevskii-Kato-Fujita approach to stochastic Navier-Stokes equations. By means of the mild formulation, this method gives a new \textit{a priori} estimate for the trajectories of the solution, which entails H\"older continuity in time and regularity $D\big(A^{\gamma'}\big)$ in space, where $\gamma'<\gamma$.

Delta lenses are functors equipped with a suitable choice of lifts, generalising the notion of split opfibration. In recent work, delta lenses were characterised as the right class of an algebraic weak factorisation system. In this paper, we show that this algebraic weak factorisation system is cofibrantly generated by a small double category, and characterise the left class as split coreflections with a certain property; we call these twisted coreflections. We demonstrate that every twisted coreflection arises as a pushout of an initial functor from a discrete category along a bijective-on-objects functor. Throughout the article, we take advantage of a reformulation of algebraic weak factorisation systems, due to Bourke, based on double-categorical lifting operations.

We consider the problems arising from the presence of Byzantine servers in a quantum private information retrieval (QPIR) setting. This is the first work to precisely define what the capabilities of Byzantine servers could be in a QPIR context. We show that quantum Byzantine servers have more capabilities than their classical counterparts due to the possibilities created by the quantum encoding procedure. We focus on quantum Byzantine servers that can apply any reversible operations on their individual qudits. In this case, the Byzantine servers can generate any error, i.e., this covers \emph{all} possible single qudit operations that can be done by the Byzantine servers on their qudits. We design a scheme that is resilient to these kinds of manipulations. We show that the scheme designed achieves superdense coding gain in all cases, i.e., $R_Q= \max \left\{0,\min\left\{1,2\left(1-\frac{X+T+2B}{N}\right)\right\}\right\}$.

We study $|A + A|$ as a random variable, where $A \subseteq \{0, \dots, N\}$ is a random subset such that each $0 \le n \le N$ is included with probability $0 < p < 1$, and where $A + A$ is the set of sums $a + b$ for $a,b$ in $A$. Lazarev, Miller, and O'Bryant studied the distribution of $2N + 1 - |A + A|$, the number of summands not represented in $A + A$ when $p = 1/2$. A recent paper by Chu, King, Luntzlara, Martinez, Miller, Shao, Sun, and Xu generalizes this to all $p\in (0,1)$, calculating the first and second moments of the number of missing summands and establishing exponential upper and lower bounds on the probability of missing exactly $n$ summands, mostly working in the limit of large $N$. We provide exponential bounds on the probability of missing at least $n$ summands, find another expression for the second moment of the number of missing summands, extract its leading-order behavior in the limit of small $p$, and show that the variance grows asymptotically slower than the mean, proving that for small $p$, the number of missing summands is very likely to be near its expected value.

Space missions that use low-thrust propulsion technology are becoming increasingly popular since they utilize propellant more efficiently and thus reduce mission costs. However, optimizing continuous-thrust trajectories is complex, time-consuming, and extremely sensitive to initial guesses. Hence, generating approximate trajectories that can be used as reliable initial guesses in trajectory generators is essential. This paper presents a semi-analytic approach for designing planar and three-dimensional trajectories using Hills equations. The spacecraft is assumed to be acted upon by a constant thrust acceleration magnitude. The proposed equations are employed in a Nonlinear Programming Problem (NLP) solver to obtain the thrust directions. Their applicability is tested for various design scenarios like orbit raising, orbit insertion, and rendezvous. The trajectory solutions are then validated as initial guesses in high-fidelity optimal control tools. The usefulness of this method lies in the preliminary stages of low-thrust mission design, where speed and reliability are key.

We give a quantum version of the Danilov-Jurkiewicz presentation of the cohomology of a compact toric orbifold with projective coarse moduli space. More precisely, we construct a canonical isomorphism from a formal version of the Batyrev ring to the quantum orbifold cohomology at a canonical bulk deformation. This isomorphism generalizes results of Givental, Iritani, and Fukaya-Oh-Ohta-Ono for toric manifolds and Coates-Lee-Corti-Tseng for weighted projective spaces. The proof uses a quantum version of Kirwan surjectivity and an equality of dimensions deduced using a toric minimal model program (tmmp). We show that there is a natural decomposition of the quantum cohomology where summands correspond to singularities in the tmmp, each giving rise to a collection of Hamiltonian non-displaceable tori.

We prove that every compact K\"ahler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.

We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict B\'ezout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

We construct extended Weil representations of unitary groups over finite fields geometrically, and show that they are Shintani lifts for Weil representations.

Over $d$-dimensional Cohen-Macaulay rings with a canonical module, $d$-cotilting classes containing the maximal and balanced big Cohen-Macaulay modules are classified. Particular emphasis is paid to the direct limit closure of the balanced big Cohen-Macaulay modules, and the class of modules of depth $d$, which are shown to respectively be the smallest and largest such cotilting classes. Considerations are then given to the interplay between local cohomology, canonical duality and cotilting modules for the class of Gorenstein flat modules over Gorenstein local rings.

In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the $n\times n$ meromorphic linear system of ordinary differential equations with Poncar\'{e} rank $1$. In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary value of the solutions of the isomonodromy equation at a critical point. Motivated by this result, we then describe the regularized limits of Stokes matrices as the irregular data $u={\rm diag}(u_1,...,u_n)$ in the linear system degenerates, i.e., as some $u_i, u_j,...,u_k$ collapse. The prescription of the regularized limit is controlled by the geometry of the De Concini-Procesi wonderful compactification space. As applications, many analysis problems about higher rank Painlev\'e transcendents can be solved.

In the second part of the paper, we show some important applications of the above analysis results in representation theory and Poisson geometry: we obtain the first transcendental realization of crystals in representations of $\frak{gl}_n$ via the Stokes phenomenon in the WKB approximation; we develop a wall-crossing formula that characterizes the discontinuous jump of the regularized limits of Stokes matrices as crossing walls in the compactification space, and interpret the known cactus group actions on crystals arising from representation theory as a wall-crossing phenomenon; and we find the first explicit linearization of the standard dual Poisson Lie group for $U(n)$.

For $n \geq 1$ let $p_n$ denote the $n^{\rm th}$ prime number. Let $$S= \{1,7,11,13,17,19,23,29 \},$$ the set of positive integers which are both less than and relatively prime to $30.$ For $x \geq 0,$ let $T_x := \{ 30x+i \; | \; i \in S\}.$ For each $x,$ $T_x$ contains at most seven primes.

Let $[ \; ]$ denote the floor or greatest integer function.

For each integer $s \geq 30$ let $\pi_7(s)$ denote the number of integers $x, \; 0 \leq x < [\frac {s}{30}]$ for which $T_x$ contains seven primes. In this paper we show that there are infinitely many values of $x$ for which $T_x$ contains seven primes. This in turn proves several cases of Alphonse de Polignac's conjecture that for every even number $k,$ there are infinitely many pairs of prime numbers

$p$ and $p'$ for which $p-p' = k.$

We investigate the algebraicity of compact K\"ahler manifolds admitting a positive rational Hodge class of bidimension $(1,1)$. We prove that if the dual K\"ahler cone of a compact K\"ahler manifold $X$ contains a rational class as an interior point, then its Albanese variety is projective. As a consequence, we answer the Oguiso--Peternell problem for Ricci-flat compact K\"ahler manifolds. We also study related algebraicity problems for threefolds.

We obtain some new results on the unimodal sequences of the real values of rational functions by polynomials with positive integer coefficients. Thus, we introduce the notion of merged-log-concavity of rational functions. Roughly speaking, the notion extends Stanley's $q$-log-concavity of polynomials.

We construct explicit merged-log-concave rational functions by $q$-binomial coefficients, Hadamard products, and convolutions, extending the Cauchy-Binet formula. Then, we obtain the unimodal sequences of rational functions by Young diagrams. Moreover, we consider the variation of unimodal sequences by critical points that separate strictly increasing, strictly decreasing, and hill-shape sequences among almost strictly unimodal sequences. Also, the critical points are zeros of polynomials in a suitable setting.

The study above extends the $t$-power series of $(\pm t;q)_{\infty}^{\mp 1}$ to some extent by polynomials with positive integer coefficients and the variation of unimodal sequences. We then obtain the golden ratio of quantum dilogarithms ($q$-exponentials) as a critical point. Additionally, we consider eta products, generalized Narayana numbers, and weighted $q$-multinomial coefficients, which we introduce.

In statistical mechanics, we discuss the grand canonical partition functions of some ideal boson-fermion gases with or without Casimir energies (Ramanujan summation). The merged-log-concavity gives phase transitions on Helmholtz free energies by critical points of the metallic ratios including the golden ratio. In particular, the phase transitions implies non-zero particle vacua from zero particle vacua as the temperature rises.

Specimens are collected from $N$ different sources. Each specimen has probability $p$ of being contaminated (e.g., in the case of an infectious disease, $p$ is the prevalence rate), independently of the other specimens. In many cases group testing is applicable, namely one can take small portions from several specimens, mix them together and test the mixture for contamination, so that if the test turns positive, then at least one of the samples in the mixture is contaminated.

In this paper we give a detailed probabilistic analysis of a binary search scheme, we propose, for determining all contaminated specimens. More precisely, we study the number $T(N)$ of tests required in order to find all the contaminated specimens, if this search scheme is applied. We derive recursive and, in some cases, explicit formulas for the expectation, the variance, and the characteristic function of $T(N)$. Also, we determine the asymptotic behavior of the moments of $T(N)$ as $N \to \infty$ and from that we obtain the limiting distribution of $T(N)$ (appropriately normalized), which turns out to be normal.

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class of constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation w.r.t. a positive Radon measure supported on a closed subset $K$ of the character space of $A$. We provide a criterion for the existence of such a representation for $L$ when $A$ is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for $A$ not necessarily equipped with a topology. This allows us to extend well-known classical results on truncated moment problems.

In this work, several convergence results are established for nearly critical self-excited systems in which event arrivals are described by multivariate marked Hawkes point processes. Under some mild high-frequency assumptions, the rescaled density process behaves asymptotically like a multi-type continuous-state branching process with immigration, which is the unique solution to a multi-dimensional stochastic differential equation with dynamical mechanism similar to that of multivariate Hawkes processes. To illustrate the strength of these limit results, we further establish diffusion approximations for multi-type Crump-Mode-Jagers branching processes counted with various characteristics by linking them to marked Hawkes shot noise processes. In particular, an interesting phenomenon in queueing theory, well-known as state space collapse, is observed in the behavior of the population structure at a large time scale. This phenomenon reveals that the rescaled complex biological system can be recovered from its population process by a lifting map.

We define a new family of commuting operators $F_k$ in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that $F_2$ satisfies hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen.

We prove that the linear syzygy spaces of a general canonical curve are spanned by syzygies of minimal rank.

This paper is concerned with the evolution dynamics of local times of a spectrally positive stable process in the spatial direction. The main results state that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, the local times at positive half line are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson random measure whose intensity coincides with the L\'evy measure. This helps us to provide not only a simple proof for the H\"older regularity, but also a uniform upper bound for all moments of the H\"older coefficient as well as a maximal inequality for the local times. Moreover, based on this stochastic Volterra equation, we extend the method of duality to establish an exponential-affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra integral equation associated with the Laplace exponent of the stable process.

We prove that a random group, in Gromov's density model with $d<1/16$, satisfies a universal sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group.

We provide a deformation quantization, in the sense of Rieffel, for \textit{all} globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e.\ a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e.\ is \textit{Hadamard}, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).

In the regression framework, the empirical measure based on the responses resulting from the nearest neighbors, among the covariates, to a given point $x$ is introduced and studied as a central statistical quantity. First, the associated empirical process is shown to satisfy a uniform central limit theorem under a local bracketing entropy condition on the underlying class of functions reflecting the localizing nature of the nearest neighbor algorithm. Second a uniform non-asymptotic bound is established under a well-known condition, often referred to as Vapnik-Chervonenkis, on the uniform entropy numbers. The covariance of the Gaussian limit obtained in the uniform central limit theorem is simply equal to the conditional covariance operator given the covariate value. This suggests the possibility of using standard formulas to estimate the variance by using only the nearest neighbors instead of the full data. This is illustrated on two problems: the estimation of the conditional cumulative distribution function and local linear regression.

In large-scale applications including medical imaging, collocation differential equation solvers, and estimation with differential privacy, the underlying linear inverse problem can be reformulated as a streaming problem. In theory, the streaming problem can be effectively solved using memory-efficient, exponentially-converging streaming solvers. In practice, a streaming solver's effectiveness is undermined if it is stopped before, or well-after, the desired accuracy is achieved. In special cases when the underlying linear inverse problem is finite-dimensional, streaming solvers can periodically evaluate the residual norm at a substantial computational cost. When the underlying system is infinite dimensional, streaming solver can only access noisy estimates of the residual. While such noisy estimates are computationally efficient, they are useful only when their accuracy is known. In this work, we rigorously develop a general family of computationally-practical residual estimators and their uncertainty sets for streaming solvers, and we demonstrate the accuracy of our methods on a number of large-scale linear problems. Thus, we further enable the practical use of streaming solvers for important classes of linear inverse problems.

Let $f : X \rightarrow C$ be a genus 1 fibration from a smooth projective surface, i.e. its generic fiber $X_{\eta}$ is a genus 1 curve. Let $j : J \rightarrow C$ be the Jacobian fibration of $f$. The smooth locus of $j$ is the N\'eron model of the Jacobian variety of $X_{\eta}$. In this paper, we prove that the Chow motives of $X$ and $J$ are isomorphic. As an application, we prove Kimura finite-dimensionality for smooth projective surfaces not of general type with geometric genus 0. This can be regarded as a generalization of Bloch-Kas-Lieberman's result to arbitrary characteristic.

Decentralized optimization is gaining increased traction due to its widespread applications in large-scale machine learning and multi-agent systems. The same mechanism that enables its success, i.e., information sharing among participating agents, however, also leads to the disclosure of individual agents' private information, which is unacceptable when sensitive data are involved. As differential privacy is becoming a de facto standard for privacy preservation, recently results have emerged integrating differential privacy with distributed optimization. However, directly incorporating differential privacy design in existing distributed optimization approaches significantly compromises optimization accuracy. In this paper, we propose to redesign and tailor gradient methods for differentially-private distributed optimization, and propose two differential-privacy oriented gradient methods that can ensure both rigorous epsilon-differential privacy and optimality. The first algorithm is based on static-consensus based gradient methods, and the second algorithm is based on dynamic-consensus (gradient-tracking) based distributed optimization methods and, hence, is applicable to general directed interaction graph topologies. Both algorithms can simultaneously ensure almost sure convergence to an optimal solution and a finite privacy budget, even when the number of iterations goes to infinity. To our knowledge, this is the first time that both goals are achieved simultaneously. Numerical simulations using a distributed estimation problem and experimental results on a benchmark dataset confirm the effectiveness of the proposed approaches.

We consider the dynamics of an elastic continuum under large deformation but small strain. Such systems can be described by the equations of geometrically nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material law. The velocity-stress formulation of the problem turns out to have a formal port-Hamiltonian structure. In contrast to the linear case, the operators of the problem are modulated by the displacement field which can be handled as a passive variable and integrated along with the velocities. A weak formulation of the problem is derived and essential boundary conditions are incorporated via Lagrange multipliers. This variational formulation explicitly encodes the transfer between kinetic and potential energy in the interior as well as across the boundary, thus leading to a global power balance and ensuring passivity of the system. The particular geometric structure of the weak formulation can be preserved under Galerkin approximation via appropriate mixed finite elements. In addition, a fully discrete power balance can be obtained by appropriate time discretization. The main properties of the system and its discretization are shown theoretically and demonstrated by numerical tests.

This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface $\mathcal{S}_g$ of genus $g\geq 2$. With each such action of a group $\mathrm{G}$ on $\mathcal{S}_g$ one can associate the fundamental group $\Gamma=\pi(\mathcal{O})$ of the quotient orbifold $\mathcal{O}=\mathcal{S}_g/\mathrm{G}$, isomorphic to a Fuchsian group determined completely by orbifold's signature. The Riemann existence theorem reduces the problem of the existence of an action of $\mathrm{G}$ on $\mathcal{S}_g$ to a purely group-theoretical problem of deciding whether there is an smooth epimorphism mapping the Fuchsian group $\Gamma$ onto the group $\mathrm{G}$. Using computer algebra systems such as \textsc{Magma} or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus $g\geq 2$ becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to topological equivalence. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set of finite group actions on $\mathcal{S}_g$ with the corresponding signature. In this paper we derive several results on the topological equivalence of finite group actions on Riemann surfaces. As an application, we derive complete lists of finite group actions of genus $g\leq 9$ distinguished up to the topological equivalence. A summary of the actions can be found in Appendix, the reader interested in more details is referred to the web page [22]. It is expected that we will be able to extend the list to higher genera, refreshed partial results are available on the web page. The following text is an extended version of the paper [23].

We investigate the modularity constraints on the generating series $h_r(\tau)$ of BPS indices counting D4-D2-D0 bound states with fixed D4-brane charge $r$ in type IIA string theory compactified on complete intersection Calabi-Yau threefolds with $b_2 = 1$. For unit D4-brane, $h_1$ transforms as a (vector-valued) modular form under the action of $SL(2,Z)$ and thus is completely determined by its polar terms. We propose an Ansatz for these terms in terms of rank 1 Donaldson-Thomas invariants, which incorporates contributions from a single D6-anti-D6 pair. Using an explicit overcomplete basis of the relevant space of weakly holomorphic modular forms (valid for any $r$), we find that for 10 of the 13 allowed threefolds, the Ansatz leads to a solution for $h_1$ with integer Fourier coefficients, thereby predicting an infinite series of DT invariants.For $r > 1$, $h_r$ is mock modular and determined by its polar part together with its shadow. Restricting to $r = 2$, we use the generating series of Hurwitz class numbers to construct a series $h^{an}_2$ with exactly the same modular anomaly as $h_2$, so that the difference $h_{2}-h^{an}_2$ is an ordinary modular form fixed by its polar terms. For lack of a satisfactory Ansatz, we leave the determination of these polar terms as an open problem.

We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.

The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge.

We further prove that the limit shape assumption is satisfied for a specific family of distributions.

Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is covered by infinite geodesics starting at the origin is at most an inverse power of $n$. This result is obtained without explicit limit shape assumptions.

This paper explores optimal service resource management strategy, a continuous challenge for health information service to enhance service performance, optimise service resource utilisation and deliver interactive health information service. An adaptive optimal service resource management strategy was developed considering a value co-creation model in health information service with a focus on collaborative and interactive with users. The deep reinforcement learning algorithm was embedded in the Internet of Things (IoT)-based health information service system (I-HISS) to allocate service resources by controlling service provision and service adaptation based on user engagement behaviour. The simulation experiments were conducted to evaluate the significance of the proposed algorithm under different user reactions to the health information service.

We establish the non-commutative analogue of Grothendieck's standard conjecture D for the differential graded category of $G$-equivariant matrix factorizations associated to an isolated hypersurface singularity where $G$ is a finite group.

We investigate the observability of a general class of linear dispersive equations on the torus $\mathbb{T}$. We take one line segment or two line segments in space-time region as the observable set. We give the characteristic on the slopes of the line segments to guarantee the qualitative observability and quantitative observability respectively. The one line segment case, is simple, follows directly from the Ingham's inequality. However, the two line segments case is difficult, the statement of results and the proof rely heavily on the language of graph theory. We also apply our results to (higher order) Schr\"{o}dinger equations and the linear KdV equation.

Given tuples of properly normalized independent $N\times N$ G.U.E. matrices $(X_N^{(1)},\dots,X_N^{(r_1)})$ and $(Y_N^{(1)},\dots,Y_N^{(r_2)})$, we show that the tuple $(X_N^{(1)}\otimes I_N,\dots,X_N^{(r_1)}\otimes I_N,I_N\otimes Y_N^{(1)},\dots,I_N\otimes Y_N^{(r_2)})$ of $N^2\times N^2$ random matrices converges strongly as $N$ tends to infinity. It was shown by Ben Hayes that this result implies that the Peterson-Thom conjecture is true.

Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\bar\rho_0$ be a reducible, odd and semi-simple representation of $G_{\mathbb{Q},Np}$ of dimension $2$ and $\{\ell_1,\cdots,\ell_r\}$ be a set of primes not dividing $Np$. After assuming that a certain Selmer group has dimension at most $1$, we find sufficient conditions for the existence of a cuspidal eigenform $f$ of level $N\prod_{i=1}^{r}\ell_i$ and appropriate weight lifting $\bar\rho_0$ such that $f$ is new at every $\ell_i$. Moreover, suppose $p \mid \ell_{i_0}+1$ for some $1 \leq i_0 \leq r$. Then, after assuming that a certain Selmer group vanishes, we find sufficient conditions for the existence of a cuspidal eigenform of level $N\ell_{i_0}^2 \prod_{j \neq i_0} \ell_j$ and appropriate weight which is new at every $\ell_i$ and which lifts $\bar\rho_0$. As a consequence, we prove a conjecture of Billerey--Menares in many cases.

In this paper, we consider random iterations of polynomial maps $z^2 +c_n$ where $c_n$ are complex-valued independent random variables following the uniform distribution on the closed disk with center $c$ and radius $r$. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters $r$ than expected. We also introduce several open questions worth discussing.

In this paper, we present a review of three widely-used practical square root algorithms. We then describe a unifying framework where each of these well-known algorithms can be seen as a special case of it. The framework with singular curves offers a broad perspective to compare and further improve the existing methods in addition to offering a new avenue for square root computation algorithms in finite fields.

We define essential commutative Cartan pairs of $C^*$-algebras generalising the definition of Renault and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwa\'sniewski and Meyer. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential commutative Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.

We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element.

We study equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials(which might not be orthonormal) in $\mathbb{C}^{m}$. We use certain results obtained in a very recent work of Bayraktar, Bloom and Levenberg and have an equidistribution result in a more general probabilistic setting than what the paper of Bayraktar, Bloom and Levenberg considers even though the basis polynomials they use are more general than $Z$-asymptotically Chebyshev polynomials. Our equidistribution result is based on the expected distribution and the variance estimate of random zero currents corresponding to the zero sets of polynomials. This equidistribution result of general nature shows that equidistribution result turns out to be true without the random coefficients that come from the basis representation being i.i.d. (independent and identically distributed), which also means that there is no need to use any probability distribution function for these random coefficients. In the last section, unlike from the $1$-codimensional case, we study the orthogonal polynomials with respect to the $L^{2}$-inner product defined by the weighted asymptotically Bernstein-Markov measures on a given locally regular compact set, and with a well-studied more general probability distribution including the Gaussian and the Fubini-Study probability distributions as special cases, we have an equidistribution result for codimensions bigger than $1$.

In this article, we introduce a class of invariants of cubic fields termed generalized discriminants. We then obtain asymptotics for the families of cubic fields ordered by these invariants. In addition, we determine which of these families satisfy the Malle--Bhargava heuristic.

Buryak, Feigin and Nakajima computed a generating function for a family of partition statistics by using the geometry of the $Z/cZ$ fixed point sets in the Hilbert scheme of points on $C^2$. Loehr and Warrington had already shown how a similar observation by Haiman using the geometry of the Hilbert scheme of points on $C^2$ could be made purely combinatorial. We extend the techniques of Loehr and Warrington to also account for cores and quotients. In particular, we construct a multigraph $M_{r,s,c}$ that is a direct refinement of Loehr and Warrington's multigraphs $M_{r,s}$, retains the relevant partition data, and is preserved by an involution $I_{r,s,c}$ which we use to prove the equidistribution of a family of partition statistics.

As a consequence, we obtain a purely combinatorial proof of a result of Buryak, Feigin, and Nakajima. More precisely, we define a family of partition statistics $\{h_{x,c}^+, x\in [0,\infty)\}$ and give a combinatorial proof that for all $x$ and all positive integers $c$,

\begin{equation*}

\sum q^{|\lambda|}t^{h_{x,c}^+(\lambda)}=q^{|\mu|}\prod_{i\geq 1}\frac{1}{(1-q^{ic})^{c-1}}\prod_{j\geq 1}\frac{1}{1-q^{jc}t},

\end{equation*}

where the sum ranges over all partitions $\lambda$ with $c$-core $\mu$.

Section 2 recalls background on partitions, cores and quotients and is written with those new to the subject in mind.

The co-optimization of behind-the-meter distributed energy resources is considered for prosumers under the net energy metering tariff. The distributed energy resources considered include renewable generations, flexible demands, and battery energy storage systems. An energy management system co-schedules the consumptions and battery storage based on locally available stochastic renewables by maximizing the expected operation surplus. A stochastic dynamic programming formulation is introduced for which structural properties of the dynamic optimization are derived. A closed-form myopic co-optimization algorithm is proposed, which achieves optimality when the storage capacity constraints are nonbinding. The proposed co-optimization algorithm has linear computation complexity and can be implemented in a decentralized fashion. The myopic co-optimization algorithm's performance and the economic benefits of the co-optimization policy to prosumers and grid operations are evaluated in numerical simulations.

We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite as well as infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this note is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.

We withdraw this note because our calculation of the A(3,3) example, which initially contradicted one of the results of a 2005 paper by Fomin-Fulton-Li-Poon, was incorrect. In the second version of the prepublication arXiv:2303.11653, we explain how the description of the cone A(p,q) obtained by Fomin-Fulton-Li-Poon refines that obtained using the O'Shea-Sjamaar theorem.

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis. This is achieved by revealing some interesting characteristics of the zero-level curves of some functions related to the the alternating zeta function.

We study the extended Bogomolny equations with gauge group $SU(2)$ on $\mathbb {R}^2 \times \mathbb {R}^+$ with generalized Nahm pole boundary conditions and nilpotent Higgs field. We completely classify solutions by relating them to certain holomorphic data through a Kobayashi-Hitchin correspondence.

Inspired by work of Fr\"oberg (1990), and Eagon and Reiner (1998), we define the \emph{total $k$-cut complex} of a graph $G$ to be the simplicial complex whose facets are the complements of independent sets of size $k$ in $G$. We study the homotopy types and combinatorial properties of total cut complexes for various families of graphs, including chordal graphs, cycles, bipartite graphs, the prism $K_n \times K_2$, and grid graphs, using techniques from algebraic topology and discrete Morse theory.

Energy market designs with non-merchant storage have been proposed in recent years, with the aim of achieving optimal market integration of storage. In order to handle the time-linking constraints that are introduced in such markets, existing works commonly make simplifying assumptions about the end-of-horizon storage level, e.g., by imposing an exogenous level for the amount of energy to be left for the next time horizon. This work analyzes market properties under such assumptions, as well as in their absence. We find that, although they ensure cost recovery for all market participants, these assumptions generally lead to market inefficiencies. Therefore we consider the design of markets with non-merchant storage without such simplifying assumptions. Using illustrative examples, as well as detailed proofs, we provide conditions under which market prices in subsequent market horizons fail to reflect the value of stored energy. We show that this problem is essential to address in order to preserve market efficiency and cost recovery. Finally, we propose a method for restoring these market properties in a perfect-foresight setting.

The classical homomorphism preservation theorem, due to {\L}o\'s, Lyndon and Tarski, states that a first-order sentence $\phi$ is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence $\psi$. Given a notion of (syntactic) complexity of sentences, an "equi-resource" homomorphism preservation theorem improves on the classical result by ensuring that $\psi$ can be chosen so that its complexity does not exceed that of $\phi$.

We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first-order, guarded and modal logics.

This note initiates an investigation of packing links into a region of Euclidean space to achieve a maximal density subject to geometric constraints. The upper bounds obtained apply only to the class of homotopically essential links and even there seem extravagantly large, leaving much working room for the interested reader.

We study the endomorphism algebra and automorphism groups of complex tori, whose second rational cohomology group enjoys a certain Hodge property introduced by

F. Campana.

We prove that in the limit of large dimension, the distribution of the logarithm of the characteristic polynomial of a generalized Wigner matrix converges to a log-correlated field. In particular, this shows that the limiting joint fluctuations of the eigenvalues are also log-correlated. Our argument mirrors that of \cite{BouMod2019}, which is in turn based on the three-step argument of \cite{ErdPecRmSchYau2010,ErdSchYau2011Uni}, but applies to a wider class of models, and at the edge of the spectrum. We rely on (i) the results in the Gaussian cases, special cases of the results in \cite{BouModPai2021}, (ii) the local laws of \cite{ErdYauYin2012}(iii) the observable \cite{Bou2020} introduced and its analysis of the stochastic advection equation this observable satisfies, and (iv) the argument for a central limit theorem on mesoscopic scales in \cite{LanLopSos2021}. For the proof, we also establish a Wegner estimate and local law down to the microscopic scale, both at the edge of the spectrum.

Let $A$ be a Noetherian domain and $R$ be a finitely generated $A$-algebra. We study several features regarding the generic freeness over $A$ of an $R$-module. For an ideal $I \subset R$, we show that the local cohomology modules ${\rm H}_I^i(R)$ are generically free over $A$ under certain settings where $R$ is a smooth $A$-algebra. By utilizing the theory of Gr\"obner bases over arbitrary Noetherian rings, we provide an effective method to make explicit the generic freeness over $A$ of a finitely generated $R$-module.

In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to its input. Such a structure is particularly interesting because any traced Cartesian (dataflow) category has an underlying traced comonoid structure. We show that certain subclasses of hypergraphs are fully complete for traced comonoid categories: that is to say, every term in such a category has a unique corresponding hypergraph up to isomorphism, and from every hypergraph with the desired properties, a unique term in the category can be retrieved up to the axioms of traced comonoid categories. We also show how the framework of double pushout rewriting (DPO) can be adapted for traced comonoid categories by characterising the valid pushout complements for rewriting in our setting. We conclude by presenting a case study in the form of recent work on an equational theory for sequential circuits: circuits built from primitive logic gates with delay and feedback. The graph rewriting framework allows for the definition of an operational semantics for sequential circuits.

This paper demonstrates the optimality of an interpolation set employed in derivative-free trust-region methods. This set is optimal in the sense that it minimizes the constant of well-poisedness in a ball centred at the starting point. It is chosen as the default initial interpolation set by many derivative-free trust-region methods based on underdetermined quadratic interpolation, including NEWUOA, BOBYQA, LINCOA, and COBYQA. Our analysis provides a theoretical justification for this choice.

Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the ladders contained in $G$ are reduced to length 4, the treewidth remains unchanged (and that this bound is tight). Our result indicates that, when computing the treewidth of a graph, long ladders can simply be reduced, and that minimal forbidden minors for bounded treewidth graphs cannot contain long ladders. Our result also settles an open problem from algorithmic phylogenetics: the common chain reduction rule, used to simplify the comparison of two evolutionary trees, is treewidth-preserving in the display graph of the two trees.

Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.

We show that the Lipschitz-free space with the Radon--Nikod\'{y}m property and a Daugavet point recently constructed by Veeorg is in fact a dual space isomorphic to $\ell_1$. Furthermore, we answer an open problem from the literature by showing that there exists a superreflexive space, in the form of a renorming of $\ell_2$, with a $\Delta$-point. Building on these two results, we are able to renorm every infinite-dimensional Banach space with a $\Delta$-point.

Next, we establish powerful relations between existence of $\Delta$-points in Banach spaces and their duals. As an application, we obtain sharp results about the influence of $\Delta$-points for the asymptotic geometry of Banach spaces. In addition, we prove that if $X$ is a Banach space with a shrinking $k$-unconditional basis with $k < 2$, or if $X$ is a Hahn--Banach smooth space with a dual satisfying the Kadets--Klee property, then $X$ and its dual $X^*$ fail to contain $\Delta$-points. In particular, we get that no Lipschitz-free space with a Hahn--Banach smooth predual contains $\Delta$-points.

Finally we present a purely metric characterization of the molecules in Lipschitz-free spaces that are $\Delta$-points, and we solve an open problem about representation of finitely supported $\Delta$-points in Lipschitz-free spaces.

We review the status of a program, outlined and motivated in the introduction, for the study of correspondences between spectral invariants of partially hyperbolic flows on locally symmetric spaces and their quantizations. Further we formulate a number of concrete problems which may be viewed as possible further steps to be taken in order to complete the program.

In this paper, we establish novel data-dependent upper bounds on the generalization error through the lens of a "variable-size compressibility" framework that we introduce newly here. In this framework, the generalization error of an algorithm is linked to a variable-size 'compression rate' of its input data. This is shown to yield bounds that depend on the empirical measure of the given input data at hand, rather than its unknown distribution. Our new generalization bounds that we establish are tail bounds, tail bounds on the expectation, and in-expectations bounds. Moreover, it is shown that our framework also allows to derive general bounds on any function of the input data and output hypothesis random variables. In particular, these general bounds are shown to subsume and possibly improve over several existing PAC-Bayes and data-dependent intrinsic dimension-based bounds that are recovered as special cases, thus unveiling a unifying character of our approach. For instance, a new data-dependent intrinsic dimension-based bound is established, which connects the generalization error to the optimization trajectories and reveals various interesting connections with the rate-distortion dimension of a process, the R\'enyi information dimension of a process, and the metric mean dimension.

Tempered stable distributions are frequently used in financial applications (e.g., for option pricing) in which the tails of stable distributions would be too heavy. Given the non-explicit form of the probability density function, estimation relies on numerical algorithms which typically are time-consuming. We compare several parametric estimation methods such as the maximum likelihood method and different generalized method of moment approaches. We study large sample properties and derive consistency, asymptotic normality, and asymptotic efficiency results for our estimators. Additionally, we conduct simulation studies to analyze finite sample properties measured by the empirical bias, precision, and asymptotic confidence interval coverage rates and compare computational costs. We cover relevant subclasses of tempered stable distributions such as the classical tempered stable distribution and the tempered stable subordinator. Moreover, we discuss the normal tempered stable distribution which arises by subordinating a Brownian motion with a tempered stable subordinator. Our financial applications to log returns of asset indices and to energy spot prices illustrate the benefits of tempered stable models.

By providing mathematical estimates, this paper answers a fundamental question -- "what leads to Stokes drift"? Although overwhelmingly understood for water waves, Stokes drift is a generic mechanism that stems from kinematics and occurs in any non-transverse wave in fluids. To showcase its generality, we undertake a comparative study of the pathline equation of sound (1D) and intermediate-depth water (2D) waves. Although we obtain a closed-form solution $\mathbf{x}(t)$ for the specific case of linear sound waves, a more generic and meaningful approach involves the application of asymptotic methods and expressing variables in terms of the Lagrangian phase $\theta$. We show that the latter reduces the 2D pathline equation of water waves to 1D. Using asymptotic methods, we solve the respective pathline equation for sound and water waves, and for each case, we obtain a parametric representation of particle position $\mathbf{x}(\theta)$ and elapsed time $t(\theta)$. Such a parametric description has allowed us to obtain second-order-accurate expressions for the time duration, horizontal displacement, and average horizontal velocity of a particle in the crest and trough phases. All these quantities are of higher magnitude in the crest phase in comparison to the trough, leading to a forward drift, i.e. Stokes drift. We also explore particle trajectory due to second-order Stokes waves and compare it with linear waves. While finite amplitude waves modify the estimates obtained from linear waves, the understanding acquired from linear waves is generally found to be valid.

The main focus of this work is the study of several cones relating the eigenvalues or singular values of a matrix to those of its off-diagonal blocks.

In many occurrences of fluid-structure interaction time-periodic motions are observed. We consider the interaction between a fluid driven by the three dimensional Navier-Stokes equation and a two dimensional linearized elastic Koiter shell situated at the boundary. The fluid-domain is a part of the solution and as such changing in time periodically. On a steady part of the boundary we allow for the physically relevant case of dynamic pressure boundary values, prominent to model inflow/outflow. We provide the existence of at least one weak time-periodic solution for given periodic external forces that are not too large. For that we introduce new approximation techniques and a-priori estimates.

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and $Hom$ quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus $3$ curves, whenever they verify Property $N_2$, using and slightly expanding the theory of integration of ribbons of the authors and E.~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension.

We carry out a similar program for linearly normal hyperelliptic curves of degree $d\geq 2g+3$. We classify surfaces having such a curve $C$ as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the latter case we obtain the existence of a universal extension.

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Non-oscillatory reconstruction is achieved through an adaptive order weighted essentially non-oscillatory (WENO-AO) method cast into a form suited to multidimensional stencils and reconstruction. A kernel-based approach inspired by Gaussian process (GP) modeling is presented here. This approach allows the creation of a scheme of arbitrary order with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple-yet-effective set of reconstruction variables is introduced, as well as an easy-to-implement effective limiter for positivity preservation, both of which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility.

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic-elliptic Keller-Segel system on whole spaces detailized by Euclid space $\mathbb{R}^n\,\,(n \geqslant 4)$ and real hyperbolic space $\mathbb{H}^n\,\, (n \geqslant 2)$. We work in framework of scritical spaces such as on weak-Lorentz space $L^{\frac{n}{2},\infty}(\mathbb{R}^n)$ to obtain the results for Keller-Segel system on $\mathbb{R}^n$ and on $L^{\frac{p}{2}}(\mathbb{H}^n)\, (p>n)$ to obtain the ones on $\mathbb{H}^n$. Our method is based on the dispersive and smoothing estimates of the heat semiroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviours of periodic mild solutions of Keller-Segel system obtained in $\mathbb{R}^n$ and the one in $\mathbb{H}^n$.

In the context of Sobolev spaces with variable exponents, Poincar\'e--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form \begin{equation*}

\int_\Omega \left|f(x)-\langle f\rangle_{\Omega}\right|^{p(x)} \ {\mathrm{d} x}

\leqslant C \int_\Omega|\nabla f(x)|^{p(x)}{\mathrm{d} x},

\end{equation*} are known to be \emph{false}. As a result, all available modular versions of the Poincar\'e- Wirtinger inequality in the variable-exponent setting always contain extra terms that do not disappear in the constant exponent case, preventing such inequalities from reducing to the classical ones in the constant exponent setting. Our contribution is threefold. First, we establish that a modular Poincar\'e--Wirtinger inequality particularizing to the classical one in the constant exponent case is indeed conceivable. We show that if $\Omega\subset \mathbb{R}^n$ is a bounded Lipschitz domain, and if $p\in L^\infty(\Omega)$, $p \geq 1$, then for every $f\in C^\infty(\bar\Omega)$ the following generalized Poincar\'e--Wirtinger inequality holds

\begin{equation*}

\int_\Omega \left|f(x)-\langle f\rangle_{\Omega}\right|^{p(x)} \ {\mathrm{d} x}

\leq C \int_\Omega\int_\Omega \frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\ {\mathrm{d} z}{\mathrm{d} x},

\end{equation*}

where $\langle f\rangle_{\Omega}$ denotes the mean of $f$ over $\Omega$, and $C>0$ is a positive constant depending only on $\Omega$ and $\|p\|_{L^\infty(\Omega)}$. Second, our argument is concise and constructive and does not rely on compactness results. Third, we additionally provide geometric information on the best Poincar\'e--Wirtinger constant on Lipschitz domains.

Dynamical fluctuations or rare events associated with atypical trajectories in chaotic maps due to specific initial conditions can crucially determine their fate, as the may lead to stability islands or regions in phase space otherwise displaying unusual behavior. Yet, finding such initial conditions is a daunting task precisely because of the chaotic nature of the system. In this work, we circumvent this problem by proposing a framework for finding an effective topologically-conjugate map whose typical trajectories correspond to atypical ones of the original map. This is illustrated by means of examples which focus on counterbalancing the instability of fixed points and periodic orbits, as well as on the characterization of a dynamical phase transition involving the finite-time Lyapunov exponent. The procedure parallels that of the application of the generalized Doob transform in the stochastic dynamics of Markov chains, diffusive processes and open quantum systems, which in each case results in a new process having the prescribed statistics in its stationary state. This work thus brings chaotic maps into the growing family of systems whose rare fluctuations -- sustaining prescribed statistics of dynamical observables -- can be characterized and controlled by means of a large-deviation formalism.

Orbit recovery problems are a class of problems that often arise in practice and various forms. In these problems, we aim to estimate an unknown function after being distorted by a group action and observed via a known operator. Typically, the observations are contaminated with a non-trivial level of noise. Two particular orbit recovery problems of interest in this paper are multireference alignment and single-particle cryo-EM modelling. In order to suppress the noise, we suggest using the method of moments approach for both problems while introducing deep neural network priors. In particular, our neural networks should output the signals and the distribution of group elements, with moments being the input. In the multireference alignment case, we demonstrate the advantage of using the NN to accelerate the convergence for the reconstruction of signals from the moments. Finally, we use our method to reconstruct simulated and biological volumes in the cryo-EM setting.

We relate two different proposals to extend the \'etale topology into homotopy theory, namely via the notion of finite cover introduced by Mathew and via the notion of separable commutative algebra introduced by Balmer. We show that finite covers are precisely those separable commutative algebras with underlying dualizable module, which have a locally constant and finite degree function. We then use Galois theory to classify separable commutative algebras in numerous categories of interest. Examples include the category of modules over a connective $\mathbb{E}_\infty$-ring $R$ which is either connective or even periodic with $\pi_0(R)$ regular Noetherian in which $2$ acts invertibly, the stable module category of a finite group of $p$-rank one and the derived category of a qcqs scheme.

This paper proposes a new easy-to-implement parameter-free gradient-based optimizer: DoWG (Distance over Weighted Gradients). We prove that DoWG is efficient -- matching the convergence rate of optimally tuned gradient descent in convex optimization up to a logarithmic factor without tuning any parameters, and universal -- automatically adapting to both smooth and nonsmooth problems. While popular algorithms following the AdaGrad framework compute a running average of the squared gradients to use for normalization, DoWG maintains a new distance-based weighted version of the running average, which is crucial to achieve the desired properties. To complement our theory, we also show empirically that DoWG trains at the edge of stability, and validate its effectiveness on practical machine learning tasks.

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$ is larger than 1. Further, we show that for $\beta$ satisfying a certain condition, the partition function attains all non-negative integers as values.

This work proposes a Momentum-Enabled Kronecker-Factor-Based Optimizer Using Rank-1 updates, called MKOR, that improves the training time and convergence properties of deep neural networks (DNNs). Second-order techniques, while enjoying higher convergence rates vs first-order counterparts, have cubic complexity with respect to either the model size and/or the training batch size. Hence they exhibit poor scalability and performance in transformer models, e.g. large language models (LLMs), because the batch sizes in these models scale by the attention mechanism sequence length, leading to large model size and batch sizes. MKOR's complexity is quadratic with respect to the model size, alleviating the computation bottlenecks in second-order methods. Because of their high computation complexity, state-of-the-art implementations of second-order methods can only afford to update the second order information infrequently, and thus do not fully exploit the promise of better convergence from these updates. By reducing the communication complexity of the second-order updates as well as achieving a linear communication complexity, MKOR increases the frequency of second order updates. We also propose a hybrid version of MKOR (called MKOR-H) that mid-training falls backs to a first order optimizer if the second order updates no longer accelerate convergence. Our experiments show that MKOR outperforms state -of-the-art first order methods, e.g. the LAMB optimizer, and best implementations of second-order methods, i.e. KAISA/KFAC, up to 2.57x and 1.85x respectively on BERT-Large-Uncased on 64 GPUs.

We show that the $p$-adic Siegel Eisenstein series of general degree attached to two kind of number sequences are both linear combinations of genus theta series of level $p$, by applying the theory of mod $p$-power singular forms. As special cases of this result, we derive the result of Nagaoka and Katsurada--Nagaoka.

Assume that $(L,v)$ is a finite Galois extension of a valued field $(K,v)$. We give an explicit construction of the valuation ring $\mathcal O_L$ of $L$ as an $\mathcal O_K$-algebra, and an explicit description of the module of relative K\"ahler differentials $\Omega_{\mathcal O_L|\mathcal O_K}$ when $L|K$ is a Kummer extension of prime degree or an Artin-Schreier extension, in terms of invariants of the valuation and field extension. The case when this extension has nontrivial defect was solved in a recent paper by the authors with Anna Rzepka. The present paper deals with the complementary (defectless) case. The results are known classically for (rank 1) discrete valuations, but our systematic approach to non-discrete valuations (even of rank 1) is new. We also show that the annihilator of $\Omega_{\mathcal O_L|\mathcal O_K}$ can only be equal to the maximal ideal $\mathcal M_L$ of $\mathcal O_L$ if the extension is defectless and $\mathcal M_L$ is principal.

Using our results from the prime degree case, we characterize when $\Omega_{\mathcal O_L|\mathcal O_K}=0$ holds for an arbitrary finite Galois extension of valued fields. As an application of these results, we give a simple proof of a theorem of Gabber and Ramero, which characterizes when a valued field is deeply ramified. We further give a simple characterization of deeply ramified fields with residue fields of characteristic $p>0$ in terms of the K\"ahler differentials of Galois extensions of degree $p$.

The standard methods for calculating Khovanov homology rely either on long exact/spectral sequences or on the algorithmic "divide and conquer" approach developed by Bar-Natan. In this paper, we employ an alternative and arguably simpler tool, discrete Morse theory, which is new in the context of knot homologies. The method is applied for 2- and 3-torus braids in Bar-Natan's dotted cobordism category, where Khovanov complexes of tangles live. This grants a recursive description of the complexes of 2- and 3-torus braids yielding an inductive result on integral Khovanov homology of links containing those braids. The result, accompanied with some computer data, advances the recent progress on a conjecture by Przytycki and Sazdanovi\'c which claims that closures of 3-braids only have 2-torsion in their Khovanov homology.

We study corrections to the scaling limit of subcritical long-range Ising models with (super)-summable interactions on $\mathbb{Z}^d$. For a wide class of models, the scaling limit is known to be white noise, as shown by Newman (1980). In the specific case of couplings $J_{x,y}=|x-y|^{-d-\boldsymbol{\alpha}}$, where $\boldsymbol{\alpha}>0$ and $|\cdot|$ is the Euclidean norm, we find an emergence of fractional Gaussian free field correlations in appropriately renormalised and rescaled observables. The proof exploits the exact asymptotics of the two-point function, first established by Newman and Spohn (1998), together with the rotational symmetry of the interaction.

One of the fundamental results in quantum foundations is the Kochen-Specker (KS) theorem, which states that any theory whose predictions agree with quantum mechanics must be contextual, i.e., a quantum observation cannot be understood as revealing a pre-existing value. The theorem hinges on the existence of a mathematical object called a KS vector system. While many KS vector systems are known, the problem of finding the minimum KS vector system in three dimensions (3D) has remained stubbornly open for over 55 years.

To address the minimum KS problem, we present a new verifiable proof-producing method based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS) that uses an isomorph-free orderly generation technique that is very effective in pruning away large parts of the search space. Our method shows that a KS system in 3D must contain at least 24 vectors. We show that our sequential and parallel Cube-and-Conquer (CnC) SAT+CAS methods are significantly faster than SAT-only, CAS-only, and a prior CAS-based method of Uijlen and Westerbaan. Further, while our parallel pipeline is somewhat slower than the parallel CnC version of the recently introduced Satisfiability Modulo Theories (SMS) method, this is in part due to the overhead of proof generation. Finally, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a size of 42.9 TiB in order 23.

We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include l2 decoupling, small cap decoupling, and estimates of exponential sums.

In a recent work of I. Dynnikov and M. Prasolov a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method we confirm conjectures of Ng and Chongchitmate about Legendrian knots in topological types $7_4$, $9_{48}$ and $10_{136}$. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.

Matroidal entropy functions are entropy functions in the form $\mathbf{h} = \log v \cdot \mathbf{r}_M$ , where $v \ge 2$ is an integer and $\mathbf{r}_M$ is the rank function of a matroid $M$. They can be applied into capacity characterization and code construction of information theory problems such as network coding, secret sharing, index coding and locally repairable code. In this paper, by constructing the variable strength arrays of some matroid operations, we characterized matroidal entropy functions induced by regular matroids and some matroids with the same p-characteristic set as uniform matroid $U_{2,4}$.

Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend potentially nonlinearly on control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

We obtain some results about the spectrum and the upper semi-Fredholm spectrum of weighted composition operators on uniform algebras, assuming that the corresponding map maps the Shilov boundary onto itself. In particular, it follows from our results that in the case of analytic uniform algebras the spectrum is a connected rotation invariant subset of the complex plane, and that the upper semi-Fredholm spectrum is rotation invariant as well.

We provide two families of algorithms to compute characteristic polynomials of endomorphisms and norms of isogenies of Drinfeld modules. Our algorithms work for Drinfeld modules of any rank, defined over any base curve. When the base curve is $\mathbb P^1_{\mathbb F_q}$, we do a thorough study of the complexity, demonstrating that our algorithms are, in many cases, the most asymptotically performant. The first family of algorithms relies on the correspondence between Drinfeld modules and Anderson motives, reducing the computation to linear algebra over a polynomial ring. The second family, available only for the Frobenius endomorphism, is based on a formula expressing the characteristic polynomial of the Frobenius as a reduced norm in a central simple algebra.

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.

The paper deals with three evolution problems arising in the physical modelling of acoustic phenomena of small amplitude in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, which derivation from the physical model is not fully mathematically satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces. The paper also gives several well-posedness and optimal regularity results for the three problems considered, which are new for the Eulerian and Lagrangian models.

In this article, we investigate three classes of equations: the McKean-Vlasov stochastic differential equation (MVSDE), the MVSDE with a subdifferential operator referred to as the McKean-Vlasov stochastic variational inequality (MVSVI), and the coupled forward-backward MVSVI. The latter class encompasses the FBSDE with reflection in a convex domain as a special case. We establish the well-posedness, in terms of the existence and uniqueness of a strong solution, for these three classes in their general forms. Importantly, we consider stochastic coefficients with locally Holder continuity and employ different strategies to achieve that for each class.

Optimal Transport has sparked vivid interest in recent years, in particular thanks to the Wasserstein distance, which provides a geometrically sensible and intuitive way of comparing probability measures. For computational reasons, the Sliced Wasserstein (SW) distance was introduced as an alternative to the Wasserstein distance, and has seen uses for training generative Neural Networks (NNs). While convergence of Stochastic Gradient Descent (SGD) has been observed practically in such a setting, there is to our knowledge no theoretical guarantee for this observation. Leveraging recent works on convergence of SGD on non-smooth and non-convex functions by Bianchi et al. (2022), we aim to bridge that knowledge gap, and provide a realistic context under which fixed-step SGD trajectories for the SW loss on NN parameters converge. More precisely, we show that the trajectories approach the set of (sub)-gradient flow equations as the step decreases. Under stricter assumptions, we show a much stronger convergence result for noised and projected SGD schemes, namely that the long-run limits of the trajectories approach a set of generalised critical points of the loss function.

A geometric perspective of the Higgs Mechanism is presented. Using Thom's Catastrophe Theory, we study the emergence of the Higgs Mechanism as a discontinuous feature in a general family of Lagrangians obtained by varying its parameters. We show that the Lagrangian that exhibits the Higgs Mechanism arises as a first-order phase transition in this general family. We find that the Higgs Mechanism (as well as Spontaneous Symmetry Breaking) need not occur for a different choice of parameters of the Lagrangian, and further analysis of these unconventional parameter choices may yield interesting implications for beyond standard model physics.

We consider locally isotropic Gaussian random fields on the $N$-dimensional Euclidean space for fixed $N$. Using the so called Gaussian Orthogonally Invariant matrices first studied by Mallows in 1961 which include the celebrated Gaussian Orthogonal Ensemble (GOE), we establish the Kac--Rice representation of expected number of critical points of non-isotropic Gaussian fields, complementing the isotropic case obtained by Cheng and Schwartzman in 2018. In the limit $N=\infty$, we show that such a representation can be always given by GOE matrices, as conjectured by Auffinger and Zeng in 2020.

The paper is focused on the four-dimensional visualization of hypersurfaces represented by implicit equations without their parametrization. We describe a general method to find shadow boundaries in an arbitrary dimension and apply it in a three- and four-dimensional space. Furthermore, we design a system of polynomial equations to construct occluding contours of algebraic surfaces in a 4-D perspective. The method is presented on a composed 3-D scene and three 4-D cases with gradual complexity. In general, our goal is to improve the understanding of spatial properties in a four-dimensional space.

Neuron models have attracted a lot of attention recently, both in mathematics and neuroscience. We are interested in studying long-time and large-population emerging properties in a simplified toy model. From a mathematical perspective, this amounts to study the long-time behaviour of a degenerate reflected diffusion process. Using coupling arguments, the flow is proven to be a contraction of the Wasserstein distance for long times, which implies the exponential relaxation toward a (non-explicit) unique globally attractive equilibrium distribution. The result is extended to a McKean-Vlasov type non-linear variation of the model, when the mean-field interaction is sufficiently small. The ergodicity of the process results from a combination of deterministic contraction properties and local diffusion, the noise being sufficient to drive the system away from non-contractive domains.

The notion of twisted sectors play a crucial role in orbifold Gromov-Witten theory. We introduce the notion of dihedral twisted sectors in order to construct Lagrangian Floer theory on symplectic orbifolds and discuss related issues.

Temporal correlation for randomly growing interfaces in the KPZ universality class is a topic of recent interest. Most of the works so far have been concentrated on the zero temperature model of exponential last passage percolation, and three special initial conditions, namely droplet, flat and stationary. We focus on studying the time correlation problem for generic random initial conditions with diffusive growth. We formulate our results in terms of the positive temperature exactly solvable model of the inverse-gamma polymer and obtain up to constant upper and lower bounds for the correlation between the free energy of two polymers whose endpoints are close together or far apart. Our proofs apply almost verbatim to the zero temperature set-up of exponential LPP and are valid for a broad class of initial conditions. Our work complements and completes the partial results obtained in (Ferrari-Occelli'19), following the conjectures of (Ferrari-Spohn'16). Moreover, our arguments rely on the one-point moderate deviation estimates which have recently been obtained using stationary polymer techniques and thus do not depend on complicated exact formulae.

We consider the Macdonald group $\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}\rangle$ and its Sylow 2-subgroup $J=\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}, x^{2^{3m-1}}=y^{2^{3m-1}}=1\rangle$, where $m\geq 1$ and $\ell$ is odd. Then $J$ has order $2^{7m-3}$, and nilpotency class 5 if $m>1$ and 3 if $m=1$. We determine the automorphism group of the 2-groups $J$, $H=J/Z(J)$ and $K=H/Z(H)$, where $|H|=2^{6m-3}$ and $|K|=2^{5m-3}$. Explicit multiplication, power, and commutator formulas for $J$, $H$, and $K$ are given, and used in the calculation of $\mathrm{Aut}(J)$, $\mathrm{Aut}(H)$, and $\mathrm{Aut}(K)$.

We consider 9 infinite families of finite $p$-groups, for $p$ a prime, and we settle the isomorphism problem that arises when the parameters that define these groups are modified.

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case $F(\xi) = \pm \lvert \xi \rvert^2 \xi$ we also examine the large time behaivour and stability of global solutions for arbitrary and small initial data in VMO, respectively.

In this paper we establish the existence and uniqueness of global solutions (in time), as well as the existence, regularity and stability (upper semicontinuity) of the attractor for the semigroup generated by the solutions of a two-dimensional nonlinear hyperbolic-parabolic coupled system with fractional Laplacian. In addition, we also obtain the existence of an exponential attractor and show that this attractor has a finite fractal dimension in a space containing the phase space of the dynamical system.

We study the quasinormal modes (QNM) of the charged C-metric, which physically stands for a charged accelerating black hole, with the help of Nekrasov's partition function of 4d $\mathcal{N}=2$ superconformal field theories (SCFTs). The QNM in the charged C-metric are classified into three types: the photon-surface modes, the accelerating modes and the near-extremal modes, and it is curious how the single quantization condition proposed in arXiv:2006.06111 can reproduce all the different families. We show that the connection formula encoded in terms of Nekrasov's partition function captures all these families of QNM numerically and recovers the asymptotic behavior of the accelerating and the near-extremal modes analytically. Using the connection formulae of different 4d $\mathcal{N}=2$ SCFTs, one can solve both the radial and the angular part of the scalar perturbation equation respectively. The same algorithm can be applied to the de Sitter (dS) black holes to calculate both the dS modes and the photon-sphere modes.

Two CNF formulas are called ucp-equivalent, if they behave in the same way with respect to the unit clause propagation (UCP). A formula is called ucp-irredundant, if removing any clause leads to a formula which is not ucp-equivalent to the original one. As a consequence of known results, the ratio of the size of a ucp-irredundant formula and the size of a smallest ucp-equivalent formula is at most $n^2$, where $n$ is the number of the variables. We demonstrate an example of a ucp-irredundant formula for a symmetric definite Horn function which is larger than a smallest ucp-equivalent formula by a factor $\Omega(n/\ln n)$ and, hence, a general upper bound on the above ratio cannot be smaller than this.

We explore the impact of coarse quantization on low-rank matrix sensing in the extreme scenario of dithered one-bit sampling, where the high-resolution measurements are compared with random time-varying threshold levels. To recover the low-rank matrix of interest from the highly-quantized collected data, we offer an enhanced randomized Kaczmarz algorithm that efficiently solves the emerging highly-overdetermined feasibility problem. Additionally, we provide theoretical guarantees in terms of the convergence and sample size requirements. Our numerical results demonstrate the effectiveness of the proposed methodology.

In this paper, the nonlinear (orbital) stability of static 180^\circ N\'eel walls in ferromagnetic films, under the reduced wave-type dynamics for the in-plane magnetization proposed by Capella, Melcher and Otto [CMO07], is established. It is proved that the spectrum of the linearized operator around the static N\'eel wall lies in the stable complex half plane with non-positive real part. This information is used to show that small perturbations of the static N\'eel wall converge to a translated orbit belonging to the manifold generated by the static wall.

The \emph{Ramsey multiplicity constant} of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labeled copies of $H$ in a $2$-edge colouring of $K_n$. Fox and Wigderson recently identified a large family of graphs whose Ramsey multiplicity constants are attained by sequences of Tur\'an colourings''; i.e. colourings in which one of the colour classes forms the edge set of a balanced complete multipartite graph. Each graph in their family comes from taking a connected non-3-colourable graph with a critical edge and adding many pendant edges. We extend their result to an off-diagonal variant of the Ramsey multiplicity constant which involves minimizing a weighted sum of red copies of one graph and blue copies of another.

Using generalized hydrodynamics (GHD), we exactly evaluate the finite-temperature spin Drude weight at zero magnetic field for the integrable XXZ chain with arbitrary spin and easy-plane anisotropy. First, we construct the fusion hierarchy of the quantum transfer matrices ($T$-functions) and derive functional relations ($T$- and $Y$-systems) satisfied by the $T$-functions and certain combinations of them ($Y$-functions). Through analytical arguments, the $Y$-system is reduced to a set of non-linear integral equations, equivalent to the thermodynamic Bethe ansatz (TBA) equations. Then, employing GHD, we calculate the spin Drude weight at arbitrary finite temperatures. As a result, a characteristic fractal-like structure of the Drude weight is observed at arbitrary spin, similar to the spin-1/2 case. In our approach, the solutions to the TBA equations (i.e., the $Y$-functions) can be explicitly written in terms of the $T$-functions, thus allowing for a systematic calculation of the high-temperature limit of the Drude weight.

In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare, engineering, and beyond. This approach captures the enduring presence or persistence' of signal peaks in time series data arising from Morse functions while preserving their positional information. Through rigorous analysis, we demonstrate that the proposed persistence transformation exhibits stability and outperforms the persistent diagram of Morse functions (with respect to filtration, e.g., the upper levelset filtration). Moreover, we present a modified version of the persistence transformation, termed the reduced persistence transformation, which retains stability while enjoying dimensionality reduction in the data. Consequently, the reduced persistence transformation yields faster computational results for subsequent tasks, such as classification, albeit at the cost of reduced overall accuracy compared to the persistence transformation. However, the reduced persistence transformation finds relevance in specific domains, e.g., MALDI-Imaging, where positional information is of greater significance than the overall signal height. Finally, we provide a conceptual outline for extending the persistence diagram to accommodate higher-dimensional input while assessing its stability under these modifications.

We propose a geometric framework to describe and analyze a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones.

A spectrum-sharing satellite-ground integrated network is conceived, consisting of a pair of non-geostationary orbit (NGSO) constellations and multiple terrestrial base stations, which impose the co-frequency interference (CFI) on each other. The CFI may increase upon increasing the number of satellites. To manage the potentially severe interference, we propose to rely on joint multi-domain resource aided interference management (JMDR-IM). Specifically, the coverage overlap of the constellations considered is analyzed. Then, multi-domain resources - including both the beam-domain and power-domain - are jointly utilized for managing the CFI in an overlapping coverage region. This joint resource utilization is performed by relying on our specifically designed beam-shut-off and switching based beam scheduling, as well as on long short-term memory based joint autoregressive moving average assisted deep Q network aided power scheduling. Moreover, the outage probability (OP) of the proposed JMDR-IM scheme is derived, and the asymptotic analysis of the OP is also provided. Our performance evaluations demonstrate the superiority of the proposed JMDR-IM scheme in terms of its increased throughput and reduced OP.

In this paper, we characterize $\ell$-open and $\ell$-closed $C^*$-algebras and deduce that $\ell$-open $C^*$-algebras are $\ell$-closed, as conjectured by Blackadar. Moreover, we show that a commutative unital $C^*$-algebra is $\ell$-open if and only if it is semiprojective.

Following our reformulation of sheaf-theoretic Virasoro constraints with applications to curves and surfaces joint with Lim-Moreira, I describe in the present work the quiver analog. After phrasing a universal approach to Virasoro constraints for moduli of quiver-representations, I prove them for any finite quiver with relations, with frozen vertices, but without cycles. I use partial flag varieties as a guiding example throughout, but the most exciting upshot is a self-contained proof of Virasoro constraints for Gieseker (semi)stable torsion-free sheaves on $\mathbb{P}^2$ and $\mathbb{P}^1\times \mathbb{P}^1$ relying on derived equivalences to quivers. In the case of Hilbert schemes of points, this can be combined with an existing universality argument to give an independent proof for any surface.

The growing computing power over the years has enabled simulations to become more complex and accurate. While immensely valuable for scientific discovery and problem-solving, however, high-fidelity simulations come with significant computational demands. As a result, it is common to run a low-fidelity model with a subgrid-scale model to reduce the computational cost, but selecting the appropriate subgrid-scale models and tuning them are challenging. We propose a novel method for learning the subgrid-scale model effects when simulating partial differential equations augmented by neural ordinary differential operators in the context of discontinuous Galerkin (DG) spatial discretization. Our approach learns the missing scales of the low-order DG solver at a continuous level and hence improves the accuracy of the low-order DG approximations as well as accelerates the filtered high-order DG simulations with a certain degree of precision. We demonstrate the performance of our approach through multidimensional Taylor-Green vortex examples at different Reynolds numbers and times, which cover laminar, transitional, and turbulent regimes. The proposed method not only reconstructs the subgrid-scale from the low-order (1st-order) approximation but also speeds up the filtered high-order DG (6th-order) simulation by two orders of magnitude.

We consider Prandtl's 1933 model for calculating circulation distribution function $\Gamma$ of a finite wing which minimizes induced drag, under the constraints of prescribed total lift and moment of inertia. We prove existence of a global minimizer of the problem without the restriction of nonnegativity $\Gamma\geq 0$ in an appropriate function space. We also consider an improved model, where the prescribed moment of inertia takes into account the bending moment due to the weight of the wing itself, which leads to a more efficient solution than Prandtl's 1933 result.

We prove that the classification diagram functor from the category of marked simplicial sets to the category of bisimplicial sets carries cartesian equivalences to Rezk equivalences. As a corollary, we obtain Mazel-Gee's theorem on localizations of relative $\infty$-categories.

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee's results on homoclinics and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic-heteroclinic chain in the planar circular restricted three-body problem, using pseudoholomorphic curves.

We propose a novel generalization of the conditional gradient (CG / Frank-Wolfe) algorithm for minimizing a smooth function $f$ under an intersection of compact convex sets, using a first-order oracle for $\nabla f$ and linear minimization oracles (LMOs) for the individual sets. Although this computational framework presents many advantages, there are only a small number of algorithms which require one LMO evaluation per set per iteration; furthermore, these algorithms require $f$ to be convex. Our algorithm appears to be the first in this class which is proven to also converge in the nonconvex setting. Our approach combines a penalty method and a product-space relaxation. We show that one conditional gradient step is a sufficient subroutine for our penalty method to converge, and we provide several analytical results on the product-space relaxation's properties and connections to other problems in optimization. We prove that our average Frank-Wolfe gap converges at a rate of $\mathcal{O}(\ln t/\sqrt{t})$, -- only a log factor worse than the vanilla CG algorithm with one set.

We find polynomial upper bounds on the number of isotopy classes of connected essential surfaces embedded in many cusped 3-manifolds and their Dehn fillings. Our bounds are universal, in the sense that we obtain the same explicit formula for all 3-manifolds that we consider, with the formula dependent on the Euler characteristic of the surface and similar numerical quantities encoding topology of the ambient 3-manifold. Universal and polynomial bounds have been obtained previously for classical alternating links in the 3-sphere and their Dehn fillings, but only for surfaces that are closed or spanning. Here, we consider much broader classes of 3-manifolds and all topological types of surfaces. The 3-manifolds are called weakly generalized alternating links; they include, for example, many links that are not classically alternating and/or do not lie in the 3-sphere, many virtual links and toroidally alternating links.

The random XXZ quantum spin chain manifests localization (in the form of quasi-locality) in any fixed energy interval, as previously proved by the authors. In this article it is shown that this property implies slow propagation of information, one of the putative signatures of many-body localization, in the same energy interval.

In this paper, we explore optimal treatment allocation policies that target distributional welfare. Most literature on treatment choice has considered utilitarian welfare based on the conditional average treatment effect (ATE). While average welfare is intuitive, it may yield undesirable allocations especially when individuals are heterogeneous (e.g., with outliers) - the very reason individualized treatments were introduced in the first place. This observation motivates us to propose an optimal policy that allocates the treatment based on the conditional quantile of individual treatment effects (QoTE). Depending on the choice of the quantile probability, this criterion can accommodate a policymaker who is either prudent or negligent. The challenge of identifying the QoTE lies in its requirement for knowledge of the joint distribution of the counterfactual outcomes, which is generally hard to recover even with experimental data. Therefore, we introduce minimax policies that are robust to model uncertainty. A range of identifying assumptions can be used to yield more informative policies. For both stochastic and deterministic policies, we establish the asymptotic bound on the regret of implementing the proposed policies. In simulations and two empirical applications, we compare optimal decisions based on the QoTE with decisions based on other criteria. The framework can be generalized to any setting where welfare is defined as a functional of the joint distribution of the potential outcomes.

These are notes from a mini-course about the main results of arXiv:2206.03438: I explain how, using suitable valued fields, one obtains a natural notion of canonical stratifications (of e.g. algebraic subsets of $\mathbb{R}^n$). I also explain how the same techniques yield more invariants of singularities, and I present an application to Poincar\'e series. While some rudimentary knowledge of model theory is useful, the notes should also be accessible without such knowledge. In particular, they contain an introduction to the non-standard analysis needed for this approach.

In this paper, we show that the existence of two sequences of Massey iterated product containing zero in the cohomology of a 1-connected CW complex of finite type $X$ directly bears on the unbounded growth of the Betti numbers of the free loop space of $X$.

Optimal resource allocation in wireless systems still stands as a rather challenging task due to the inherent statistical characteristics of channel fading. On the one hand, minimax/outage-optimal policies are often overconservative and analytically intractable, despite advertising maximally reliable system performance. On the other hand, ergodic-optimal resource allocation policies are often susceptible to the statistical dispersion of heavy-tailed fading channels, leading to relatively frequent drastic performance drops. We investigate a new risk-aware formulation of the classical stochastic resource allocation problem for point-to-point power-constrained communication networks over fading channels with no cross-interference, by leveraging the Conditional Value-at-Risk (CV@R) as a coherent measure of risk. We rigorously derive closed-form expressions for the CV@R-optimal risk-aware resource allocation policy, as well as the optimal associated quantiles of the corresponding user rate functions by capitalizing on the underlying fading distribution, parameterized by dual variables. We then develop a purely dual tail waterfilling scheme, achieving significantly more rapid and assured convergence of dual variables, as compared with the primal-dual tail waterfilling algorithm, recently proposed in the literature. The effectiveness of the proposed scheme is also readily confirmed via detailed numerical simulations.

We consider the variation of two fundamental types of zeta functions that arise in the study of both physical and analytical problems in geometric settings involving conical singularities. These are the Barnes zeta functions and the Bessel zeta functions. Although the series used to define them do not converge at zero, using methods of complex analysis we are able to calculate the derivatives of these zeta functions at zero. These zeta functions depend critically on a certain parameter, and we calculate the variation of these derivatives with respect to the parameter. For integer values of the parameter, we obtain a new expression for the variation of the Barnes zeta function with respect to the parameter in terms of special functions. For the Bessel zeta functions, we obtain two different expressions for the variation via two independent methods. Of course, the expressions should be equal, and we verify this by demonstrating several identities for both special and elementary functions. We encountered these zeta functions while working with determinants of Laplace operators on cones and angular sectors.

The Alperin weight conjecture has been reduced to simple groups by G. Navarro and P. H. Tiep. In this paper, we investigate the Galois-Alperin weight conjecture and give a reduction to simple groups.

We prove that on any closed Riemannian three-manifold $(M,g)$ the time-dependent Euler equations are non-mixing on the space of smooth volume-preserving vector fields endowed with the $C^1$-topology, for any fixed helicity and large enough energy, solving a problem posed by Khesin, Misiolek, and Shnirelman. To prove this, we introduce a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold with an arbitrary fixed metric. This greatly broadens the scope of contact topological methods in hydrodynamics, which so far have had applications only for stationary solutions and without fixing the ambient metric. We further use this framework to prove that spectral invariants obtained from Floer theory, concretely embedded contact homology, define new non-trivial continuous first integrals of the Euler equations in certain regions of the phase space endowed with the $C^{1,s}$-topology, producing countably many disjoint invariant open sets.

In this paper, we deal with Diophantine equations $N = {F_k}^3 + {F_\ell }^3 = {F_m}^3 + {F_n}^3$ and $M = {L_k}^3 + {L_\ell }^3 = {L_m}^3 + {L_n}^3$. In other words, we discover the Fibonacci and Lucas numbers that are also Hardy-Ramanujan numbers.

We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schr\"odinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell--Bloch system, the Maxwell field, current and the population inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a development of suitable methods of differential topology: the transversality and orientation arguments. We also prove the existence of the global compact attractor.

We show that a Hodge class of a complex smooth projective hypersurface is an analytic logarithmic De Rham class. On the other hand we show that for a complex smooth projective variety an analytic logarithmic De Rham class of of type $(d,d)$ is the class of codimension $d$ algebraic cycle. We deduce the Hodge conjecture for smooth projective hypersurfaces.

We study the freeness problem for subgroups of $\operatorname{SL}_2(\mathbb{C})$ generated by two parabolic matrices. For $q = r/p \in \mathbb{Q} \cap (0,4)$, where $p$ is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by the two matrices $A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \text{ and } Q_q = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}.$ We introduce the conjecture that $\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$, the congruence subgroup of $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$ consisting of all matrices with upper right entry congruent to $0$ mod $r$ and diagonal entries congruent to $1$ mod $r$. We prove this conjecture when $r \leq 4$ and for some cases when $r = 5$. Furthermore, conditional on a strong form of Artin's conjecture on primitive roots, we also prove the conjecture when $r \in \{ p-1, p+1, (p+1)/2 \}$. In all these cases, this gives information about the algebraic structure of $\Delta_{r/p}$: it is isomorphic to the fundamental group of a finite graph of virtually free groups, and has finite index $J_2(r)$ in $\operatorname{SL}_2(\mathbb{Z}[\frac{1}{p}])$, where $J_2(r)$ denotes the Jordan totient function.

In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and $F=\mathbb{F}_q$, a self-dual MRD code exists if and only if $q\equiv n\equiv 3 \ [4].$

For every partially ordered sets I, having a finite cofinal subset, and every field K we build a unital, locally matricial and hence unit-regular K-algebra B(I) such that the lattice of all its ideals is order isomorphic to the lattice of all lower subsets of I. We show that the Grothendieck group of B(I), with its natural partial order, is order isomorphic to the restricted Hahn power of Z by I; this gives a contribution to solve the Realization Problem for Dimension Groups with order-unit. Finally we show that the algebra B(I) has the following features: (a) B(I) is prime if and only if I is lower directed; (b) B(I) is primitive if and only if I has a coinitial chain; (c) B(I) is semiartinian if and only if I is artinian, in which the case I is order isomorphic to the primitive spectrum of B(I).

We present a purely analytical inequality which is equivalent to the Riemann hypothesis (RH). The proof of the equivalence is based on a representation of the modulus of the Riemann xi-function. As the first step to analyze the inequality, we consider polynomial approximations. We also show that the RH is equivalent to the statement that some wave functions constructed using the Brownian motion never evolve into perfectly distinguishable states.

We consider the problem of approximating a function from $L^2$ by an element of a given $m$-dimensional space $V_m$, associated with some feature map $\varphi$, using evaluations of the function at random points $x_1,\dots,x_n$. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features $\varphi(x_i)$. We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples $n = O(m\log(m))$, that means that the expected $L^2$ error is bounded by a constant times the best approximation error in $L^2$. Also, further assuming that the function is in some normed vector space $H$ continuously embedded in $L^2$, we further prove that the approximation is almost surely bounded by the best approximation error measured in the $H$-norm. This includes the cases of functions from $L^\infty$ or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

This paper gives an algorithm to determine whether a number in a cyclic quartic field is a sum of two squares, mainly based on local-global principle of isotropy of quadratic forms.

Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k \geq 3$, each infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively irreducible. In this paper, we attempt to make their result effective by building a connection between this problem and the bipartite generalization of the well-studied Diophantine tuples. More precisely, given an integer $k \geq 3$ and a nonzero integer $n$, we call a pair of subsets of positive integers $(A,B)$ \emph{a bipartite Diophantine tuple with property $BD_k(n)$} if $|A|,|B| \geq 2$ and $AB+n \subset \{x^k: x \in \mathbb{N}\}$. We show that $\min \{|A|, |B|\} \ll \log |n|$, extending a celebrated work of Bugeaud and Dujella (where they considered the case $n=1$). We also provide an upper bound on $|A||B|$ in terms of $n$ and $k$ under the assumption $\min \{|A|,|B|\}\geq 4$ and $k \geq 6$. Specializing our techniques to Diophantine tuples, we significantly improve several results by B\'{e}rczes-Dujella-Hajdu-Luca, Bhattacharjee-Dixit-Saikia, and Dixit-Kim-Murty.

Based on the construction of polytope functions and several results about them in [LP], we take a deep look on their mutation behaviors to find a link between a face of a polytope and a sub-cluster algebra of the corresponding cluster algebra. This find provides a way to induce a mutation sequence in a sub-cluster algebra from that in the cluster algebra in totally sign-skew-symmetric case analogous to that achieved via cluster scattering diagram in skew-symmetrizable case by [GHKK] and [M].

With this, we are able to generalize compatibility degree in [CL] and then obtain an equivalent condition of compatibility which does not rely on clusters and thus can be generalized for all polytope functions. Therefore, we could regard compatibility as an intrinsic property of variables, which explains the unistructurality of cluster algebras. According to such cluster structure of polytope functions, we construct a fan $\mathcal{C}$ containing all cones in the $g$-fan.

On the other hand, we also find a realization of $G$-matrices and $C$-matrices in polytopes by the mutation behaviors of polytopes, which helps to generalize the dualities between $G$-matrices and $C$-matrices introduced in [NZ] and leads to another polytope explanation of cluster structures. This allows us to construct another fan $\mathcal{N}$ which also contains all cones in the $g$-fan.

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schr\"odinger representations. Each pattern of Sobolev norm behavior corresponds to a specific $n-$dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2.

When applied to periodically or quasi-periodically forced $n-$dimensional quantum harmonic oscillators, we identify novel growth rates for the $\mathcal{H}^s-$norm as $t$ tends to infinity, such as $t^{(n-1)s}e^{\lambda st}$ (with $\lambda>0$) and $t^{(2n-1)s}+ \iota t^{2ns}$ (with $\iota\geq 0$). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3.

As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the fastest" growth, as articulated in Theorem 1.4.

It is believed that for metric-like models in the KPZ class the following property holds: with probability one, starting from any point, there are at most two semi-infinite geodesics with the same direction that do not coalesce. Until now, such a result was only proved for one model - exponential LPP (Coupier 11') using its inherent connection to the totally asymmetric exclusion process. We prove that the above property holds for the directed landscape, the universal scaling limit of models in the KPZ class. Our proof reduces the problem to one on line ensembles and therefore paves the way to show similar results for other metric-like models in the KPZ class. Finally, combining our result with the ones in (Busani, Seppalainen,Sorensen 22', Bhatia 23') we obtain the full qualitative geometric description of infinite geodesics in the directed landscape.

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots $\left\{ \rho_{n}\right\} _{n=1}^{\infty}$, as $n\to+\infty$ the moduli $\left|\rho_{n}\right|$ grow at least exponentially. Applications to entire $q$-functions defined by series expansions are provided. These functions include the $q$-analogue of the plane wave function $\mathcal{E}_{q}(z,t)$.

A Heron triangle is one that has all integer side lengths and integer area, which takes its name from Heron of Alexandria's area formula. From a more relaxed point of view, if rescaling is allowed, then one can define a Heron triangle to be one whose side lengths and area are all rational numbers. A perfect triangle is a Heron triangle with all three medians being rational. According to a longstanding conjecture, no such triangle exists, so perfect triangles are as rare as unicorns.

However, if perfect is the enemy of good, then perhaps it is best to insist on only two of the medians being rational. Buchholz and Rathbun found an infinite family of Heron triangles with two rational medians, which they were able to associate with the set of rational points on an elliptic curve $E(\mathbb{Q})$. Here we describe a recently discovered explicit formula for the sides, area and medians of these (almost perfect) triangles, expressed in terms of a pair of integer sequences: these are Somos sequences, which first became popular thanks to David Gale's column in Mathematical Intelligencer.

In this paper, for any fixed integer $q>2$, we construct $q$-ary codes correcting a burst of at most $t$ deletions with redundancy $\log n+8\log\log n+o(\log\log n)+\gamma_{q,t}$ bits and near-linear encoding/decoding complexity, where $n$ is the message length and $\gamma_{q,t}$ is a constant that only depends on $q$ and $t$. In previous works there are constructions of such codes with redundancy $\log n+O(\log q\log\log n)$ bits or $\log n+O(t^2\log\log n)+O(t\log q)$. The redundancy of our new construction is independent of $q$ and $t$ in the second term.

We propose a connection between the newly formulated Virasoro minimal string and the well-established $(2,2m-1)$ minimal string by deriving the string equation of the Virasoro minimal string using the expansion of its density of states in powers of $E^{m+1/2}$. This string equation is expressed as a power series involving double-scaled multicritical matrix models, which are dual to $(2,2m-1)$ minimal strings. This reformulation of Virasoro minimal strings enables us to employ matrix theory tools to compute its $n$-boundary correlators. We analyze the scaling behavior of $n$-boundary correlators and quantum volumes $V^{(b)}_{0,n}(\ell_1,\dots,\ell_n)$ in the JT gravity limit.

We introduce the notion of a categorical valuative invariant of polyhedra or matroids, in which alternating sums of numerical invariants are replaced by split exact sequences in an additive category. We provide categorical lifts of a number of valuative invariants of matroids, including the Poincare polynomial, the Chow and augmented Chow polynomials, and certain two-variable extensions of the Kazhdan--Lusztig polynomial and Z-polynomial. These lifts allow us to perform calculations equivariantly with respect to automorphism groups of matroids.

Based on supermodularity ordering properties, we show that convex risk measures of credit losses are nondecreasing w.r.t. credit-credit and, in a wrong-way risk setup, credit-market, covariances of elliptically distributed latent factors. These results support the use of such setups for computing credit provisions and economic capital or for conducting stress test exercises and risk management analysis.

We give complete presentations for the dagger-compact props of affine Lagrangian and coisotropic relations over an arbitrary field. This provides a unified family of graphical languages for both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser quantum circuits. To this end, we present affine Lagrangian relations by a particular class of undirected coloured graphs. In order to reason about composite systems, we introduce a powerful scalable notation where the vertices of these graphs are themselves coloured by graphs. In the setting of stabiliser quantum mechanics, this scalable notation gives an extremely concise description of graph states, which can be composed via phased spider fusion.'' Likewise, in the classical mechanical setting of electrical circuits, we show that impedance matrices for reciprocal networks are presented in essentially the same way.

Time-uniform log-Sobolev inequalities (LSI) satisfied by solutions of semi-linear mean-field equations have recently appeared to be a key tool to obtain time-uniform propagation of chaos estimates. This work addresses the more general settings of time-inhomogeneous Fokker-Planck equations. Time-uniform LSI are obtained in two cases, either with the bounded-Lipschitz perturbation argument with respect to a reference measure, or with a coupling approach at high temperature. These arguments are then applied to mean-field equations, where, on the one hand, sharp marginal propagation of chaos estimates are obtained in smooth cases and, on the other hand, time-uniform global propagation of chaos is shown in the case of vortex interactions with quadratic confinement potential on the whole space. In this second case, an important point is to establish global gradient and Hessian estimates, which is of independent interest. We prove these bounds in the more general situation of non-attractive logarithmic and Riesz singular interactions.

In this chapter, we investigate the mathematical foundation of the modeling and design of reconfigurable intelligent surfaces (RIS) in both the far- and near-field regimes. More specifically, we first present RIS-assisted wireless channel models for the far- and near-field regimes, discussing relevant phenomena, such as line-of-sight (LOS) and non-LOS links, rich and poor scattering, channel correlation, and array manifold. Subsequently, we introduce two general approaches for the RIS reflective beam design, namely optimization-based and analytical, which offer different degrees of design flexibility and computational complexity. Furthermore, we provide a comprehensive set of simulation results for the performance evaluation of the studied RIS beam designs and the investigation of the impact of the system parameters.

This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.

In this research paper, we examine an optimal control problem involving a dynamical system governed by a nonlinear Caputo fractional time-delay state equation. The primary objective of this study is to obtain the necessary conditions for optimality, both the first and second order, for the Caputo fractional time-delay optimal control problem. We derive the first-order necessary condition for optimality for the given fractional time-delay optimal control problem. Moreover, we focus on a case where the Pontryagin maximum principle degenerates, meaning that it is satisfied in a tivial manner. Consequently, we proceed to derive the second order optimality conditions specific to the problem under investigation. At the end illustrative examples are provided.

This work considers an asynchronous $\textsf{K}_\text{a}$-active-user unsourced multiple access channel (AUMAC) with the worst-case asynchronicity. The transmitted messages must be decoded within $n$ channel uses, while some codewords are not completely received due to asynchronicities. We consider a constraint of the largest allowed delay of the transmission. The AUMAC lacks the permutation-invariant property of the synchronous UMAC since different permutations of the same codewords with a fixed asynchronicity are distinguishable. Hence, the analyses require calculating all $2^{\textsf{K}_\text{a}}-1$ combinations of erroneously decoded messages. Moreover, transmitters cannot adapt the corresponding codebooks according to asynchronicity due to a lack of information on asynchronicities. To overcome this challenge, a uniform bound of the per-user probability of error (PUPE) is derived by investigating the worst-case of the asynchronous patterns with the delay constraint. Numerical results show the trade-off between the energy-per-bit and the number of active users for different delay constraints. In addition, although the asynchronous transmission reduces interference, the required energy-per-bit increases as the receiver decodes with incompletely received codewords, compared to the synchronous case.

The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. In recent work of the authors, Fox's conjecture (1962) -- stating that the absolute values of the coefficients of the Alexander polynomial for any alternating link are unimodal -- was settled for special alternating links. The present paper is a study of the special combinatorial and discrete geometric properties that Alexander polynomials of special alternating links possess along with a generalization to all Eulerian graphs, introduced by Murasugi and Stoimenow (2003). We prove that the Murasugi and Stoimenow generalized Alexander polynomials can be expressed in terms of volumes of root polytopes of unimodular matrices, building on the beautiful works of Li and Postnikov (2013) and T\'othm\'er\'esz (2022). We conjecture a generalization of Fox's conjecture to the Eulerian graph setting. We also bijectively relate two longstanding combinatorial models for the Alexander polynomials of special alternating links: Crowell's state model (1959) and Kauffman's state model (1982, 2006).

We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Sz\H{u}cs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.

In this paper, we consider multi-robot localization problems with focus on cooperative localization and observability analysis of relative pose estimation. For cooperative localization, there is extra information available to each robot via communication network and message passing. If odometry data of a target robot can be transmitted to the ego-robot then the observability of their relative pose estimation can be achieved by range-only or bearing-only measurements provided both of their linear velocities are non-zero. If odometry data of a target robot is not directly transmitted but estimated by the ego-robot then there must be both range and bearing measurements to guarantee the observability of relative pose estimation. For ROS/Gazebo simulations, we consider four different sensing and communication structures in which extended Kalman filtering (EKF) and pose graph optimization (PGO) estimation with different robust loss functions (filtering and smoothing with different batch sizes of sliding window) are compared in terms of estimation accuracy. For hardware experiments, two Turtlebot3 equipped with UWB modules are used for real-world inter-robot relative pose estimation, in which both EKF and PGO are applied and compared.

Solving feedback Stackelberg games with nonlinear dynamics and coupled constraints, a common scenario in practice, presents significant challenges. This work introduces an efficient method for computing local feedback Stackelberg policies in multi-player general-sum dynamic games, with continuous state and action spaces. Different from existing (approximate) dynamic programming solutions that are primarily designed for unconstrained problems, our approach involves reformulating a feedback Stackelberg dynamic game into a sequence of nested optimization problems, enabling the derivation of Karush-Kuhn-Tucker (KKT) conditions and the establishment of a second-order sufficient condition for local feedback Stackelberg policies. We propose a Newton-style primal-dual interior point method for solving constrained linear quadratic (LQ) feedback Stackelberg games, offering provable convergence guarantees. Our method is further extended to compute local feedback Stackelberg policies for more general nonlinear games by iteratively approximating them using LQ games, ensuring that their KKT conditions are locally aligned with those of the original nonlinear games. We prove the exponential convergence of our algorithm in constrained nonlinear games. In a feedback Stackelberg game with nonlinear dynamics and (nonconvex) coupled costs and constraints, our experimental results reveal the algorithm's ability to handle infeasible initial conditions and achieve exponential convergence towards an approximate local feedback Stackelberg equilibrium.

In this paper, we study the sign-changing radial solutions of the following coupled Schr\"odinger system $$

\left\{

\begin{array}{lr}

-{\Delta}u_j+\lambda_j u_j=\mu_j u_j^3+\sum_{i\neq j}\beta_{ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber

u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber

\end{array}

\right.$$ Here, $\lambda_j,\,\mu_j>0$ and $\beta_{ij}=\beta_{ji}$ are constants for $i,j=1,\cdots,N$ and $i\neq j$. $B_1$ denotes the unit ball in the Euclidean space $\mathbb{R}^3$ centred at the origin. For any $P_1,\cdots,P_N\in\mathbb{N}$, we prove the uniqueness of the radial solution $(u_1,\cdots,u_j)$ with $u_j$ changes its sign exactly $P_j$ times for any $j=1,\cdots,N$ in the following case: $\lambda_j\geq1$ and $|\beta_{ij}|$ are small for $i,j=1,\cdots,N$ and $i\neq j$. New Liouville-type theorems and boundedness results are established for this purpose.

In the field of causal modeling, potential outcomes (PO) and structural causal models (SCMs) stand as the predominant frameworks. However, these frameworks face notable challenges in practically modeling counterfactuals, formalized as parameters of the joint distribution of potential outcomes. Counterfactual reasoning holds paramount importance in contemporary decision-making processes, especially in scenarios that demand personalized incentives based on the joint values of $(Y(0), Y(1))$. This paper begins with an investigation of the PO and SCM frameworks for modeling counterfactuals. Through the analysis, we identify an inherent model capacity limitation, termed as the degenerative counterfactual problem'', emerging from the consistency rule that is the cornerstone of both frameworks. To address this limitation, we introduce a novel \textit{distribution-consistency} assumption, and in alignment with it, we propose the Distribution-consistency Structural Causal Models (DiscoSCMs) offering enhanced capabilities to model counterfactuals. To concretely reveal the enhanced model capacity, we introduce a new identifiable causal parameter, \textit{the probability of consistency}, which holds practical significance within DiscoSCM alone, showcased with a personalized incentive example. Furthermore, we provide a comprehensive set of theoretical results about the Ladder of Causation'' within the DiscoSCM framework. We hope it opens new avenues for future research of counterfactual modeling, ultimately enhancing our understanding of causality and its real-world applications.

Recently, DNA storage has surfaced as a promising alternative for data storage, presenting notable benefits in terms of storage capacity, cost-effectiveness in maintenance, and the capability for parallel replication. Mathematically, the DNA storage process can be conceptualized as an insertion, deletion, and substitution (IDS) channel. Due to the mathematical complexity associated with the Levenshtein distance, creating a code that corrects for IDS remains a challenging task. In this paper, we propose a bottom-up generation approach to grow the required codebook based on the computation of Edit Computational Graph (ECG) which differs from the algebraic constructions by incorporating the Derivative-Free Optimization (DFO) method. Specifically, this approach is regardless of the type of errors. Compared the results with the work for 1-substitution-1-deletion and 2-deletion, the redundancy is reduced by about 30-bit and 60-bit, respectively. As far as we know, our method is the first IDS-correcting code designed using classical Natural Language Process (NLP) techniques, marking a turning point in the field of error correction code research. Based on the codebook generated by our method, there may be significant breakthroughs in the complexity of encoding and decoding algorithms.

Guessing random additive noise decoding (GRAND) is a recently proposed decoding paradigm particularly suitable for codes with short length and high rate. Among its variants, ordered reliability bits GRAND (ORBGRAND) exploits soft information in a simple and effective fashion to schedule its queries, thereby allowing efficient hardware implementation. Compared with maximum likelihood (ML) decoding, however, ORBGRAND still exhibits noticeable performance gap in terms of block error rate (BLER). In order to improve the performance of ORBGRAND while still retaining its amenability to hardware implementation, a new variant of ORBGRAND termed RS-ORBGRAND is proposed, whose basic idea is to reshuffle the queries of ORBGRAND so that the expected number of queries is minimized. Numerical simulations show that RS-ORBGRAND leads to noticeable gains compared with ORBGRAND and its existing variants, and is only 0.1dB away from ML decoding, for BLER as low as $10^{-6}$.

We consider channels with synchronization errors modeled as insertions and deletions. A classical result for such channels is the information stability of such channels, hence the existence of the Shannon capacity, when the synchronization errors are memoryless. In this paper, we extend this result to the case where the insertions and deletions have memory. Specifically, we assume that the synchronization errors are governed by a stationary and ergodic finite state Markov chain, and prove that mutual information capacity of such channels exist, and it is equal to its coding capacity, showing that there exists a coding scheme which achieves this limit.

We develop recent ideas of Elsholtz, Proske, and Sauermann to construct denser subsets of $\{1,\dots,N\}$ that lack arithmetic progressions of length $3$. This gives the first quasipolynomial improvement since the original construction of Behrend.

In this review, we have reached from the most basic definitions in the theory of groups, group structures, etc. to representation theory and irreducible representations of the Poincar'e group. Also, we tried to get a more comprehensible understanding of group theory by presenting examples from the nature around us to examples in mathematics and physics and using them to examine more important groups in physics such as the Lorentz group and Poincar'e group and representations It is achieved in the physical fields that are used in the quantum field theory.

This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavorably with the number of evaluation points, which can result in computational inefficiency. To address this issue, we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple, we focus only on one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to more complicated two- and three-dimensional problems with appropriate modifications.

An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. Bollob\'as, Leader and Malvenuto, and also Bukh, conjectured that the Tur\'an density of the $r$-daisy tends to zero as $r \to \infty$. In this paper we disprove this conjecture.

Adapting our construction, we are also able to disprove a folklore conjecture about Tur\'an densities of hypercubes. For fixed $d$ and large $n$, we show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for all $d \geq 8$. In fact, we show that this asymptotic density is at most $c^d$, for some constant $c<1$. As a consequence, we obtain similar bounds for the edge-Tur\'an densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities.

This paper delves into the degradability of quantum channels, with a specific focus on high-dimensional extensions of qubit depolarizing channels in low-noise regimes. We build upon the foundation of $\eta$-approximate degradable channels, as established by Sutter et al. and Leditzky et al., to introduce and examine the Modified Landau-Streater (MLS) channels. These channels expand upon the qubit depolarizing and the recently proposed modified Werner-Holevo channels by Roofeh and Karimipour, extending them to higher-dimensional Hilbert spaces (with dimension $d=2j+1$, where $j$ are positive half-integers). Our investigation centers on their conformity to the $O(\varepsilon^2)$ degradability pattern, aligning with and extending Leditzky et al.'s findings in the $d=2$ case. By replacing the SU($2$) generators with SU($d$) in our treatment, we may explore the potential inclusion of generalized Gell-Mann matrices in future research. Our results enhance the understanding of super-additivity in quantum channels within the low-noise regime and lay the groundwork for future explorations into conditions and structures that could lead to $O(\varepsilon^2)$ degradability across a broader spectrum of quantum channels.

Decomposing a matrix into a weighted sum of Pauli strings is a common chore of the quantum computer scientist, whom is not easily discouraged by exponential scaling. But beware, a naive decomposition can be cubically more expensive than necessary! In this manuscript, we derive a fixed-memory, branchless algorithm to compute the inner product between a 2^N-by-2^N complex matrix and an N-term Pauli tensor in O(2^N) time, by leveraging the Gray code. Our scheme permits the embarrassingly parallel decomposition of a matrix into a weighted sum of Pauli strings in O(8^N) time. We implement our algorithm in Python, hosted open-source on Github, and benchmark against a recent state-of-the-art method called the "PauliComposer" which has an exponentially growing memory overhead, achieving speedups in the range of 1.5x to 5x for N < 8. Note that our scheme does not leverage sparsity, diagonality, Hermitivity or other properties of the input matrix which might otherwise enable optimised treatment in other methods. As such, our algorithm is well-suited to decomposition of dense, arbitrary, complex matrices which are expected dense in the Pauli basis, or for which the decomposed Pauli tensors are a priori unknown.

Bisimulation is a concept that captures behavioural equivalence. It has been studied extensively on nonprobabilistic systems and on discrete-time Markov processes and on so-called continuous-time Markov chains. In the latter time is continuous but the evolution still proceeds in jumps. We propose two definitions of bisimulation on continuous-time stochastic processes where the evolution is a \emph{flow} through time. We show that they are equivalent and we show that when restricted to discrete-time, our concept of bisimulation encompasses the standard discrete-time concept. The concept we introduce is not a straightforward generalization of discrete-time concepts.

Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting where the notion of a step is fundamental. In our setting we are considering "flow"-processes emphasizing that they evolve in continuous time. In such continuous-time settings, the concepts are not straightforward adaptations of their discrete-time analogues and we restrict our study to diffusions that do not lose mass over time and with additional regularity constraints.

In previous work we proposed different definitions of behavioural equivalences for continuous-time stochastic processes where the evolution is a flow through time. That work only addressed equivalences. In this work, we aim at quantifying how differently processes behave. We present two pseudometrics for diffusion-like processes. These pseudometrics are fixpoints of two different functionals on the space of 1-bounded pseudometrics on the state space. We also characterize these pseudometrics in terms of real-valued modal logics; this is a quantitative analogue of the notion of logical characterization of bisimulation. These real-valued modal logics indicate that the two pseudometrics are different and thus yield different notions of behavioural equivalence.

We address the problem of stability of one-dimensional non-periodic ground-state configurations with respect to finite-range perturbations of interactions in classical lattice-gas models. We show that a relevant property of non-periodic ground-state configurations in this context is their homogeneity. The so-called strict boundary condition says that the number of finite patterns of a configuration have bounded fluctuations on any finite subsets of the lattice. We show that if the strict boundary condition is not satisfied, then in order for non-periodic ground-state configurations to be stable, interactions between particles should not decay faster than $1/r^{\alpha}$ with $\alpha>2$. In the Thue-Morse ground state, number of finite patterns may fluctuate as much as the logarithm of the lenght of a lattice subset. We show that the Thue-Morse ground state is unstable for any $\alpha >1$ with respect to arbitrarily small two-body interactions favoring the presence of molecules consisting of two spins up or down. We also investigate Sturmian systems defined by irrational rotations on the circle. They satisfy the strict boundary condition but nevertheless they are unstable for $\alpha>3$.