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



Enlarging on Parts I and II we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of full dynamic equations, including equations where the medium is moving or otherwise changing in time. The motivation is that results and methods in the theory of composites then extend to these equations.

Following some past advances, we reformulate a large class of linear continuum physics equations in the format of the extended abstract theory of composites so that we can apply this theory to better understand and efficiently solve those physical equations. Here in part II we elucidate the form for many time-harmonic equations that do not involve higher order gradients.

We provide an explicit construction of Ricci-flat K3 metrics. It employs the technology of D-geometry, which in the case of interest is equivalent to a hyper-K\"ahler quotient. We relate it to the construction of arXiv:1810.10540, and in particular show that it contains the solution to the BPS state counting problem (that of computing the BPS index of a heterotic little string theory compactified on $T^2$) discussed therein, which is the data needed for this second construction of K3 metrics.

In the presence of Lindblad decoherence, i.e. dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities for "mass" and "flavor" eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators $L^{}_j$ are pseudo-Hermitian, namely, $\eta L^{}_j \eta^{-1} = L^\dagger_j$ with $\eta$ being a linear and positive-definite metric, and respect the PT symmetry as well. We demonstrate that the generalized density matrix $\rho^{}_{\rm G}(t) \equiv \rho(t) \eta$, instead of the normalized density matrix $\rho^{}_{\rm N}(t) \equiv \rho(t)/{\rm tr}\left[\rho(t)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.

We prove that for any $k\in \mathbb{R},$ $v>0,$ and $D>0$ there are only finitely many diffeomorphism types of closed Riemannian $4$-manifolds with sectional curvature $\geq k,$ volume $\geq v,$ and diameter $\leq D.$

Let $Y$ be a subset of a metric space $X.$ We say that $Y$ is $\eta$-Gromov provided $Y$ is $\eta$-separated and not properly contained in any other $\eta$-separated subset of $X.$ In this paper, we review a result of Chew which says that any $\eta$-Gromov subset of $\mathbb{R}^{2}$ admits a triangulation $\mathcal{T}$ whose smallest angle is at least $\pi /6$ and whose edges have length between $\eta$ and $2\eta .$ We then show that given any $k = 1,2,3\ldots$, there is a subdivision $\mathcal{T} _{k}$ of $\mathcal{T}$ whose edges have length in $\left[ \frac{\eta}{10 k},\frac{2\eta}{10 k} \right]$ and whose minimum angle is also $\pi /6$.

These results are used in the proof of the following theorem in [10]: For any $k\in R,v>0,$ and $D>0,$ the class of closed Riemannian $4$-manifolds with sectional curvature $\geq k,$ volume $\geq v,$ and diameter $\leq D$ contains at most finitely many diffeomorphism types. Additionally, these results imply that for any $\varepsilon >0$, if $\eta >0$ is sufficiently small, any $\eta$-Gromov subset of a compact Riemannian $2$-manifold admits a geodesic triangulation $\mathcal{T}$ for which all side lengths are in $\left[ \eta \left( 1-\varepsilon \right) ,2\eta \left( 1+\varepsilon \right) \right]$ and all angles are $\geq \frac{\pi }{6}-\varepsilon .$

Geomagnetic disturbances (GMDs), a result of space weather, pose a severe risk to electric grids. When GMDs occur, they can cause geomagnetically-induced currents (GICs), which saturate transformers, induce hot-spot heating, and increase reactive power losses in the transmission grid. Furthermore, uncertainty in the magnitude and orientation of the geo-electric field, and insufficient historical data make the problem of mitigating the effects of uncertain GMDs challenging. In this paper, we propose a novel distributionally robust optimization (DRO) approach that models uncertain GMDs and mitigates the effects of GICs on electric grids. This is achieved via a set of mitigation actions (e.g., line switching, locating blocking devices, generator re-dispatch and load shedding), prior to the GMD event, such that the worst-case expectation of the system cost is minimized. To this end, we develop a column-and-constraint generation algorithm that solves a sequence of mixed-integer second-order conic programs to handle the underlying convex support set of the uncertain GMDs. Also, we present a monolithic exact reformulation of our DRO model when the underlying support set can be approximated by a polytope with three extreme points. Numerical experiments on 'epri-21' system show the efficacy of the proposed algorithms and the exact reformulation of our DRO model.

Let $P_J$ be the standard parabolic subgroup of $SL_n$ obtained by deleting a subset $J$ of negative simple roots, and let $P_J = L_JU_J$ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue $\theta: {\mathcal O}_q(P_J) \to{\mathcal O}_q(L_J) \otimes {\mathcal O}_q(P_J)$ of an induced coaction and the corresponding subalgebra ${\mathcal O}_q(P_J)^{\operatorname{co} \theta} \subseteq {\mathcal O}_q(P_J)$ of coinvariants. It was shown that the smash product algebra ${\mathcal O}_q(L_J)\# {\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ is isomorphic to ${\mathcal O}_q(P_J)$. In view of this, ${\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ -- while it is not a Hopf algebra -- can be viewed as a quantum analogue of the coordinate ring ${\mathcal O}(U_J)$.

In this paper we prove that when $q\in \mathbb{K}$ is nonzero and not a root of unity, ${\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ is isomorphic to a quantum Schubert cell algebra ${\mathcal U}_q^+[w]$ associated to a parabolic element $w$ in the Weyl group of $\mathfrak{sl}(n)$. An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.

In the paper, we experimentally study the inverse problem with the resonant scattering determinant. We analyze the structure of characteristics of perturbed linear waves. Assuming there is the common part of potential perturbation propagating along the same strips, we estimate the common part of the perturbed wave, and its Fourier transform.

We deduce the partial inverse uniqueness from the Nevanlinna type of representation theorem.

In the paper, we study the inverse problem with the resonant data of fast decaying potential $V$. We review Froese' construction of the Born's approximation and Neumann series to analyze the growth of scattering determinant. Assuming all the the resonances are given, we deduce the certain inverse uniqueness on $V$ from the Nevanlinna type of representation theorem.

We study the biased random walk where at each step of a random walk a "controller" can, with a certain small probability, fix the next step. This model was introduced by Azar et al. [STOC1992]; we extend their work to the time dependent setting and consider cover times of this walk. We obtain new bounds on the cover and hitting times and make progress towards resolving a conjecture of Azar et al. on maximising values of the stationary distribution. We also consider the problem of computing an optimal strategy for the controller to minimise the cover time and show that for directed graphs determining the cover time is PSPACE-complete.

We propose a loss function for generative adversarial networks (GANs) using R\'{e}nyi information measures with parameter $\alpha$. More specifically, we formulate GAN's generator loss function in terms of R\'{e}nyi cross-entropy functionals. We demonstrate that for any $\alpha$, this generalized loss function preserves the equilibrium point satisfied by the original GAN loss based on the Jensen-Renyi divergence, a natural extension of the Jensen-Shannon divergence. We also prove that the R\'{e}nyi-centric loss function reduces to the original GAN loss function as $\alpha \to 1$. We show empirically that the proposed loss function, when implemented on both DCGAN (with $L_1$ normalization) and StyleGAN architectures, confers performance benefits by virtue of the extra degree of freedom provided by the parameter $\alpha$. More specifically, we show improvements with regard to: (a) the quality of the generated images as measured via the Fr\'echet Inception Distance (FID) score (e.g., best FID=8.33 for RenyiStyleGAN vs 9.7 for StyleGAN when evaluated over 64$\times$64 CelebA images) and (b) training stability. While it was applied to GANs in this study, the proposed approach is generic and can be used in other applications of information theory to deep learning, e.g., AI bias or privacy.

A numerical analysis of the effect of artificial viscosity is undertaken in order to understand the effect of numerical diffusion on numerical boundary feedback control. The analysis is undertaken on the linear hyperbolic systems discretised using the upwind scheme. The upwind scheme solves the advection-diffusion equation with up to second-order accuracy. The analysis shows that the upwind scheme with CFL equal to one gives the expected theoretical decay up to first-order. On the other hand the upwind scheme with CFL less than one gives decay depending on the second derivative of the data and the CFL number. Further the decay rates deteriorate if the second derivatives of the solution are small. Thus the decay rates computed by the numerical schemes tend to be higher in comparison to the theoretical prediction. Computations on test cases which include isothermal Euler and the St Venant Equations confirm the analytical results.

We study webs in quantum type $C$, focusing on the rank three case. We define a linear pivotal category $\mathbf{Web}(\mathfrak{sp}_6)$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\mathbf{FundRep}(U_q(\mathfrak{sp}_6))$ of quantum $\mathfrak{sp}_6$ representations generated by the fundamental representations, for generic values of the parameter $q$. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\mathbf{Web}(\mathfrak{sp}_6) \rightarrow \mathbf{FundRep}(U_q(\mathfrak{sp}_6))$, that all $\mathrm{Hom}$-spaces in $\mathbf{Web}(\mathfrak{sp}_6)$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\mathbf{Web}(\mathfrak{sp}_6)$ is $1$-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum $\mathfrak{sp}_6$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.

In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in $L^2-$norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.

Black holes in a class of string compactifications, known as STU models, carry four electric and four magnetic charges. Furthermore a duality group, given by the product of three congruence subgroups of $SL(2,\mathbb{Z})$, acts on these integer valued charges. By placing these eight charges at the eight corners of a Bhargava cube, we provide a classification of the duality orbits in these theories.

In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a joint projection reduced-order model in which the part of the system depending only on the input and the one depending only on the initial value are reduced at once and we prove an a priori error bound. With this result at hand, we derive a separate projection procedure in which the two parts are reduced separately. This gives the freedom to choose different reduction orders for the different subsystems. Moreover, we discuss how the reduced-order models can be constructed in practice. Since the error bounds are parameter-dependent we show how they can be optimized efficiently. We conclude this paper by comparing our results with the ones from the literature by a series of numerical experiments.

Internet of Things (IoT) services will use machine learning tools to efficiently analyze various types of data collected by IoT devices for inference, autonomy, and control purposes. However, due to resource constraints and privacy challenges, edge IoT devices may not be able to transmit their collected data to a central controller for training machine learning models. To overcome this challenge, federated learning (FL) has been proposed as a means for enabling edge devices to train a shared machine learning model without data exchanges thus reducing communication overhead and preserving data privacy. However, Google's seminal FL algorithm requires all devices to be directly connected with a central controller, which significantly limits its application scenarios. In this context, this paper introduces a novel FL framework, called collaborative FL (CFL), which enables edge devices to implement FL with less reliance on a central controller. The fundamentals of this framework are developed and then, a number of communication techniques are proposed so as to improve the performance of CFL. To this end, an overview of centralized learning, Google's seminal FL, and CFL is first presented. For each type of learning, the basic architecture as well as its advantages, drawbacks, and usage conditions are introduced. Then, three CFL performance metrics are presented and a suite of communication techniques ranging from network formation, device scheduling, mobility management, and coding is introduced to optimize the performance of CFL. For each technique, future research opportunities are also discussed. In a nutshell, this article will showcase how the proposed CFL framework can be effectively implemented at the edge of large-scale wireless systems such as the Internet of Things.

A hydrogeological model for the spread of pollution in an aquifer is considered. The model consists in a convection-diffusion-reaction equation involving the dispersion tensor which depends nonlinearly of the fluid velocity. We introduce an explicit flux in the model and use a mixed Finite Element Method for the discretization. We provide existence, uniqueness and stability results for the discrete model. A convergence result is obtained for the semi-discretized in time problem and for the fully discretization.

Stein Variational Gradient Descent (SVGD), a popular sampling algorithm, is often described as the kernelized gradient flow for the Kullback-Leibler divergence in the geometry of optimal transport. We introduce a new perspective on SVGD that instead views SVGD as the (kernelized) gradient flow of the chi-squared divergence which, we show, exhibits a strong form of uniform exponential ergodicity under conditions as weak as a Poincar\'e inequality. This perspective leads us to propose an alternative to SVGD, called Laplacian Adjusted Wasserstein Gradient Descent (LAWGD), that can be implemented from the spectral decomposition of the Laplacian operator associated with the target density. We show that LAWGD exhibits strong convergence guarantees and good practical performance.

Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider an ordered pair of $\mathbb F$-linear maps $A: V \to V$ and $A^*:V\to V$ such that (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i+ V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1} = 0$ and $V_{d+1}= 0$; (iii) there exists an ordering $\lbrace V^*_i\rbrace_{i=0}^{\delta}$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_i+ V^*_{i+1}$ for $0 \leq i \leq \delta$, where $V^*_{-1} = 0$ and $V^*_{\delta+1}= 0$; (iv) there does not exist a subspace $U$ of $V$ such that $AU\subseteq U$, $A^*U \subseteq U$, $U\not=0$, $U\not=V$. We call such a pair a tridiagonal pair on $V$. We assume that $A, A^*$ belongs to a family of tridiagonal pairs said to have $q$-Racah type. There is an infinite-dimensional algebra $\boxtimes_q$ called the $q$-tetrahedron algebra; it is generated by four copies of $U_q(\mathfrak{sl}_2)$ that are related in a certain way. Using $A, A^*$ we construct two $\boxtimes_q$-module structures on $V$. In this construction the two main ingredients are the double lowering map $\psi:V\to V$ due to Sarah Bockting-Conrad, and a certain invertible map $W:V\to V$ motivated by the spin model concept due to V. F. R. Jones.

The goal of this article is to study how combinatorial equivalence implies topological conjugacy. For that, we introduce the concept of kneading sequences for nonautonomous discrete dynamical systems and show that these sequences are a complete invariant for topological conjugacy classes.

For certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with $p|f(n)$ for some $n$ only if $p \in R$. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan.

In this work, we consider $M=(\mathbb{B}^3_r,\bar{g})$ as the Euclidean three-ball with radius $r$ equipped with the metric $\bar{g}=e^{2h}\left\langle , \right\rangle$ conformal to the Euclidean metric. We show that if a free boundary CMC surface $\Sigma$ in $M$ satisfies a pinching condition on the length of the traceless second fundamental tensor which involves the support function of $\Sigma$, the positional conformal vector field $\vec{x}$ and its potential function $\sigma,$ then either $\Sigma$ is a disk or $\Sigma$ is an annulus rotationally symmetric. In a particular case, we construct an example of minimal surface with strictly convex boundary in $M$, when $M$ is the Gaussian space, that illustrate our results. These results extend to the CMC case and to many others different conformally Euclidean spaces the main result obtained by Haizhong Li and Changwei Xiong.

We classify finite groups acting by birational transformations of a non-trivial Severi--Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field $K$ of characteristic zero that has no $K$-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

This paper develops a novel Continuous-time Accelerated Proximal Point Algorithm (CAPPA) for $\ell_1$-minimization problems with provable fixed-time convergence guarantees. The problem of $\ell_1$-minimization appears in several contexts, such as sparse recovery (SR) in Compressed Sensing (CS) theory, and sparse linear and logistic regressions in machine learning to name a few. Most existing algorithms for solving $\ell_1$-minimization problems are discrete-time, inefficient and require exhaustive computer-guided iterations. CAPPA alleviates this problem on two fronts: (a) it encompasses a continuous-time algorithm that can be implemented using analog circuits; (b) it betters LCA and finite-time LCA (recently developed continuous-time dynamical systems for solving SR problems) by exhibiting provable fixed-time convergence to optimal solution. Consequently, CAPPA is better suited for fast and efficient handling of SR problems. Simulation studies are presented that corroborate computational advantages of CAPPA.

Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$. In this article we produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper algebraic subgroups of $E^N$. The method gives a totally explicit version of the Manin-Dam'janenko Theorem in the elliptic case and it is a generalisation of previous results only proved when $E$ does not have Complex Multiplication.

This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular, we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327-355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post-collisional momentum $p'$; specifically we calculate the determinant for $p\mapsto u = \theta p'+\left(1-\theta\right)p$ for $\theta \in [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55-68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649-673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.

For area-preserving H\'enon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, the original method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic points in reversible families containing quadratic conservative orientable and nonorientable H\'enon maps as well as the product of two asymmetric H\'enon maps (with the Jacobians $b$ and $b^{-1}$).

We construct differential operators for families of overconvergent Hilbert modular forms by interpolating the Gauss--Manin connection on strict neighborhoods of the ordinary locus. This is related to work done by Harron and Xiao and by Andreatta and Iovita in the case of modular forms and has applications in particular to $p$-adic $L$-functions of CM fields.

While fast multipole methods (FMMs) are in widespread use for the rapid evaluation of potential fields governed by the Laplace, Helmholtz, Maxwell or Stokes equations, their coupling to high-order quadratures for evaluating layer potentials is still an area of active research. In three dimensions, a number of issues need to be addressed, including the specification of the surface as the union of high-order patches, the incorporation of accurate quadrature rules for integrating singular or weakly singular Green's functions on such patches, and their coupling to the oct-tree data structures on which the FMM separates near and far field interactions. Although the latter is straightforward for point distributions, the near field for a patch is determined by its physical dimensions, not the distribution of discretization points on the surface.

Here, we present a general framework for efficiently coupling locally corrected quadratures with FMMs, relying primarily on what are called generalized Gaussian quadratures rules, supplemented by adaptive integration. The approach, however, is quite general and easily applicable to other schemes, such as Quadrature by Expansion (QBX). We also introduce a number of accelerations to reduce the cost of quadrature generation itself, and present several numerical examples of acoustic scattering that demonstrate the accuracy, robustness, and computational efficiency of the scheme. On a single core of an Intel i5 2.3GHz processor, a Fortran implementation of the scheme can generate near field quadrature corrections for between 1000 and 10,000 points per second, depending on the order of accuracy and the desired precision. A Fortran implementation of the algorithm described in this work is available at https://gitlab.com/fastalgorithms/fmm3dbie.

In an electrostatic simulation, an equipotential condition with an undefined/floating potential value has to be enforced on the surface of an isolated conductor. If this conductor is charged, a nonzero charge condition is also required. While implementation of these conditions using a traditional finite element method (FEM) is not straightforward, they can be easily discretized and incorporated within a discontinuous Galerkin (DG) method. However, DG discretization results in a larger number of unknowns as compared to FEM. In this work, a hybridizable DG (HDG) method is proposed to alleviate this problem. Floating potential boundary conditions, possibly with different charge values, are introduced on surfaces of each isolated conductor and are weakly enforced in the global problem of HDG. The unknowns of the global HDG problem are those only associated with the nodes on the mesh skeleton and their number is much smaller than the total number of unknowns required by DG. Numerical examples show that the proposed method is as accurate as DG while it improves the computational efficiency significantly.

The focus of our work is dispersive, second-order effective model describing the low-frequency wave motion in heterogeneous (e.g.~functionally-graded) media endowed with periodic microstructure. For this class of quasi-periodic medium variations, we pursue homogenization of the scalar wave equation in $\mathbb{R}^d$, $d\geqslant 1$ within the framework of multiple scales expansion. When either $d=1$ or $d=2$, this model problem bears direct relevance to the description of (anti-plane) shear waves in elastic solids. By adopting the lengthscale of microscopic medium fluctuations as the perturbation parameter, we synthesize the germane low-frequency behavior via a fourth-order differential equation (with smoothly varying coefficients) governing the mean wave motion in the medium, where the effect of microscopic heterogeneities is upscaled by way of the so-called cell functions. In an effort to demonstrate the relevance of our analysis toward solving boundary value problems (deemed to be the ultimate goal of most homogenization studies), we also develop effective boundary conditions, up to the second order of asymptotic approximation, applicable to one-dimensional (1D) shear wave motion in a macroscopically heterogeneous solid with periodic microstructure. We illustrate the analysis numerically in 1D by considering (i) low-frequency wave dispersion, (ii) mean-field homogenized description of the shear waves propagating in a finite domain, and (iii) full-field homogenized description thereof. In contrast to (i) where the overall wave dispersion appears to be fairly well described by the leading-order model, the results in (ii) and (iii) demonstrate the critical role that higher-order corrections may have in approximating the actual waveforms in quasi-periodic media.

The perfectly matched layer (PML) is one of the most popular domain truncation techniques used in differential-equation based wave-type equation solvers. Practical implementations of PMLs often use smooth-varying attenuation coefficients to reduce the numerical reflection from PMLs and to reduce the thickness required for reaching a certain level of absorption. In discontinuous Galerkin time-domain (DGTD) methods, assigning smooth-varying coefficients calls for storing individual mass matrices for each mesh element, which significantly increases the memory-cost. This work proposes a memory-efficient implementation of smooth-varying PMLs in DGTD. A weight-adjusted approximation is applied to the mass matrices involved in the PML formulation, which yields a low memory-cost algorithm and maintains the high-order accuracy of DGTD. The proposed scheme has the same accuracy as the implementation of storing local mass matrices of each element, and provides a higher mesh flexibility and a smaller reflection than using a constant coefficient in each element.

Topological Data Analysis (TDA) provides novel approaches that allow us to analyze the geometrical shapes and topological structures of a dataset. As one important application, TDA can be used for data visualization and dimension reduction. We follow the framework of circular coordinate representation, which allows us to perform dimension reduction and visualization for high-dimensional datasets on a torus using persistent cohomology. In this paper, we propose a method to adapt the circular coordinate framework to take into account sparsity in high-dimensional applications. We use a generalized penalty function instead of an $L_{2}$ penalty in the traditional circular coordinate algorithm. We provide simulation experiments and real data analysis to support our claim that circular coordinates with generalized penalty will accommodate the sparsity in high-dimensional datasets under different sampling schemes while preserving the topological structures.

In this paper, we employ Cooperative Rate-Splitting (CRS) technique to enhance the Secrecy Sum Rate (SSR) for the Multiple Input Single Output (MISO) Broadcast Channel (BC), consisting of two legitimate users and one eavesdropper, with perfect Channel State Information (CSI) available at all nodes. For CRS based on the three-node relay channel, the transmitter splits and encodes the messages of legitimate users into common and private streams based on Rate-Splitting (RS). With the goal of maximizing SSR, the proposed CRS strategy opportunistically asks the relaying legitimate user to forward its decoded common message. During the transmission, the eavesdropper keeps wiretapping silently. To ensure secure transmission, the common message is used for the dual purpose, serving both as a desired message and Artificial Noise (AN) without consuming extra transmit power comparing to the conventional AN design. Taking into account the total power constraint and the Physical Layer (PHY) security, the precoders and time-slot allocation are jointly optimized by solving the non-convex SSR maximization problem based on Sequential Convex Approximation (SCA) algorithm. Numerical results show that the proposed CRS secure transmission scheme outperforms existing Multi-User Linear Precoding (MU-LP) and Cooperative Non-Orthogonal Multiple Access (C-NOMA) strategies. Therefore, CRS is a promising strategy to enhance the PHY security in multi-antenna BC systems.

We study a stochastic Nonlinear Schroedinger Equation (NLSE), with additive white Gaussian noise, by means of the Nonlinear Fourier Transform (NFT). In particular, we focus on the propagation of discrete eigenvalues along a focusing fiber. Since the stochastic NLSE is not exactly integrable by means of the NFT, then we use a perturbation approach, where we assume that the signal-to-noise ratio is high. The zeroth-order perturbation leads to the deterministic NLSE while the first-order perturbation allows to describe the statistics of the discrete eigenvalues. This is important to understand the properties of the channel for recently devised optical transmission techniques, where the information is encoded in the nonlinear Fourier spectrum.

Anderson acceleration (AA) is a popular method for accelerating fixed-point iterations, but may suffer from instability and stagnation. We propose a globalization method for AA to improve its stability and achieve global and local convergence. Unlike existing AA globalization approaches that often rely on safeguarding operations and might hinder fast local convergence, we adopt a nonmonotone trust-region framework and introduce an adaptive quadratic regularization together with a tailored acceptance mechanism. We prove global convergence and show that our algorithm attains the same local convergence as AA under appropriate assumptions. The effectiveness of our method is demonstrated in several numerical experiments.

We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra $U(1)^c \times U(1)^c$, or equivalently the linear programming bound for sphere packing in $2c$ dimensions. We give a more detailed picture of the behavior for finite $c$ than was previously available, and we extrapolate as $c \to \infty$. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimensions. Furthermore, we study when these bounds can be tight. Besides the known cases $c=1/2$, $4$, and $12$ and the conjectured case $c=1$, our calculations numerically rule out sharp bounds for all other $c<90$, by combining the modular bootstrap with linear programming bounds for spherical codes.

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term "Fourier sums" should be understood as "partial Fourier integrals". A certain weighted version of the result is also provided. This version leads to a new Men$'$shov-type correction theorem.

We study the Ricci flow of the four-parameter family of Sp(n+1)-invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. These (non-isometric) ancient solutions all have a larger isometry group, namely Sp(n+1)Sp(1), Sp(n+1)U(1), or U(2n+2). Two ancient solutions are non-collapsed and converge, under the backwards flow, to Jensen's second Einstein metric. One solution parametrizes the well known Berger metrics. The rest are new and collapse, under a rescaling of the backwards flow, to Ziller's second homogeneous Einstein metric on complex projective space.

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect from a probabilistic perspective.

In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behavior of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results.

Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersection between these two schools of thought in OT by proving that the entropy-regularized optimal transport problem between two Gaussian measures admits a closed form. Contrary to the unregularized case, for which the explicit form is given by the Wasserstein-Bures distance, the closed form we obtain is differentiable everywhere, even for Gaussians with degenerate covariance matrices. We obtain this closed form solution by solving the fixed-point equation behind Sinkhorn's algorithm, the default method for computing entropic regularized OT. Remarkably, this approach extends to the generalized unbalanced case -- where Gaussian measures are scaled by positive constants. This extension leads to a closed form expression for unbalanced Gaussians as well, and highlights the mass transportation / destruction trade-off seen in unbalanced optimal transport. Moreover, in both settings, we show that the optimal transportation plans are (scaled) Gaussians and provide analytical formulas of their parameters. These formulas constitute the first non-trivial closed forms for entropy-regularized optimal transport, thus providing a ground truth for the analysis of entropic OT and Sinkhorn's algorithm.

This paper concerns the ethics and morality of algorithms and computational systems, and has been circulating internally at Facebook for the past couple years. The paper reviews many Nobel laureates' work, as well as the work of other prominent scientists such as Richard Dawkins, Andrei Kolmogorov, Vilfredo Pareto, and John von Neumann. The paper draws conclusions based on such works, as summarized in the title. The paper argues that the standard approach to modern machine learning and artificial intelligence is bound to be biased and unfair, and that longstanding traditions in the professions of law, justice, politics, and medicine should help.

We consider speeding up stochastic gradient descent (SGD) by parallelizing it across multiple workers. We assume the same data set is shared among $n$ workers, who can take SGD steps and coordinate with a central server. Unfortunately, this could require a lot of communication between the workers and the server, which can dramatically reduce the gains from parallelism. The Local SGD method, proposed and analyzed in the earlier literature, suggests machines should make many local steps between such communications. While the initial analysis of Local SGD showed it needs $\Omega ( \sqrt{T} )$ communications for $T$ local gradient steps in order for the error to scale proportionately to $1/(nT)$, this has been successively improved in a string of papers, with the state-of-the-art requiring $\Omega \left( n \left( \mbox{ polynomial in log } (T) \right) \right)$ communications. In this paper, we give a new analysis of Local SGD. A consequence of our analysis is that Local SGD can achieve an error that scales as $1/(nT)$ with only a fixed number of communications independent of $T$: specifically, only $\Omega(n)$ communications are required.

Online mirror descent (OMD) and dual averaging (DA) are two fundamental algorithms for online convex optimization. They are known to have very similar (or even identical) performance guarantees in most scenarios when a \emph{fixed} learning rate is used. However, for \emph{dynamic} learning rates OMD is provably inferior to DA. It is known that, with a dynamic learning rate, OMD can suffer linear regret, even in common settings such as prediction with expert advice. This hints that the relationship between OMD and DA is not fully understood at present.

In this paper, we modify the OMD algorithm by a simple technique that we call stabilization. We give essentially the same abstract regret bound for stabilized OMD and DA by modifying the classical OMD convergence analysis in a careful and modular way, yielding proofs that we believe to be clean and flexible. Simple corollaries of these bounds show that OMD with stabilization and DA enjoy the same performance guarantees in many applications even under dynamic learning rates. We also shed some light on the similarities between OMD and DA and show simple conditions under which stabilized OMD and DA generate the same iterates.

Let $L^2(D)$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a(D)$ be the Bergman space, i.e., the (closed) subspace of analytic functions in $L^2(D)$. $P_+$ stays for the orthogonal projection going from $L^2(D)$ to $L^2_a(D)$. For a function $\varphi\in L^\infty(D)$, the Toeplitz operator $T_\varphi: L^2_a(D)\to L^2_a(D)$ is defined as $$T_\varphi f=P_+\varphi f, \quad f\in L^2_a(D).$$ The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is $$\varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0,$$ where $z=re^{i\theta}$ and $\varphi_1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible.

We focus on the problem of constructing a Hamilton cycle in as few rounds as possible. In particular, we present a novel strategy for the player which achieves a Hamiltonian cycle in $(2+4e^{-2}+0.07+o(1)) \, n < 2.61135 \, n$ rounds, assuming that a specific non-convex optimization problem has a negative solution (a premise we numerically support). Assuming that this technical condition holds, this improves upon the previously best known upper bound of $3 \, n$ rounds. We also show that the previously best lower bound of $(\ln 2 + \ln (1+\ln 2) + o(1)) \, n$ is not tight.

Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we study Gallai-Ramsey numbers for graphs with chromatic number three such as $\widehat{K}_m$ for $m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from the join of $K_1$ and $P_m$, and a class of graphs with five vertices, denoted by $\mathscr{H}$. We first study the general lower bound of such graphs and propose a conjecture for the exact value of $GR_k(\widehat{K}_m)$. Then we give a unified proof to determine the Gallai-Ramsey numbers for many graphs in $\mathscr{H}$ and obtain the exact value of $GR_k(\widehat{K}_4)$ for $k\ge1$. Our outcomes not only indicate that the conjecture on $GR_k(\widehat{K}_m)$ is true for $m\le4$, but also imply several results on $GR_k(H)$ for some $H\in \mathscr{H}$ which are proved individually in different papers.

Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.

Machine learning is poised as a very powerful tool that can drastically improve our ability to carry out scientific research. However, many issues need to be addressed before this becomes a reality. This article focuses on one particular issue of broad interest: How can we integrate machine learning with physics-based modeling to develop new interpretable and truly reliable physical models? After introducing the general guidelines, we discuss the two most important issues for developing machine learning-based physical models: Imposing physical constraints and obtaining optimal datasets. We also provide a simple and intuitive explanation for the fundamental reasons behind the success of modern machine learning, as well as an introduction to the concurrent machine learning framework needed for integrating machine learning with physics-based modeling. Molecular dynamics and moment closure of kinetic equations are used as examples to illustrate the main issues discussed. We end with a general discussion on where this integration will lead us to, and where the new frontier will be after machine learning is successfully integrated into scientific modeling.

Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $\varphi\in(0,\pi)\setminus\mathbb{Q}\pi$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_\theta$ of singular hyperbolic metrics on a torus with a single cone point of angle $\theta=2(\pi-\varphi)$, and answer an analogous question for the holonomy map $\rho_\xi$ of such a hyperbolic structure $\xi$. In an appendix by X.~Gao, concrete examples of $\theta$ and $\xi\in\mathfrak{C}_\theta$ are given where the image of each $\rho_\xi$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds.

In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multi-variate polynomials, and only a small "admissible" subset of these linear extensions, determined implicitly by the evaluation map, are of interest. This seemingly novel problem arises in the study of global dynamics of gene regulatory networks in which case the poset is a Boolean lattice. We provide an algorithm for solving this problem using linear programming for arbitrary partial orders of linear polynomials.

This algorithm exploits this additional algebraic structure inherited from the polynomials to efficiently compute the admissible linear extensions. The biologically relevant problem involves multi-linear polynomials and we provide a construction for embedding it into an instance of the linear problem.

We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.

We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of a SDE with a time changed Brownian motion, dated back to Doeblin (1940). In cases where the diffusion coefficient is bounded and $\beta$-H\"{o}lder continuous with $0 < \beta \leq 1$, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat a SDE with no strong solution. Our scheme is the first to achieve the strong convergence for the case $0 < \beta < 1/2$.

We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra $A^{(2)}_{2}$. We also give conjectural series for some level 2 modules of $A^{(2)}_{13}$.

We analyze the dynamics of the COVID-19 epidemic taking into account the role of the unreported cases. After a first section in which we deal with a framework of very slow test capacity, we turn to the model recently introduced/implemented by Liu, Magal, Seydi and Webb. First, we prove some basic structural results for the corresponding ODE, as for instance the convergence of S(t) to a positive limit. These are similar to those of the classical SIR model, although the maxima of the corresponding curves are not necessarily unique. Finally, we implement the model -- but with a variable transmission rate -- in the Chilean context. A key parameter adjustment (namely, the fraction of unreported cases) is done via an argument using mortality rates. We conclude with several conclusions and lines of future research.

For the problem of nonlinear equations, the homotopy methods (continuation methods) are popular in engineering fields since their convergence regions are large and they are fairly reliable to find a solution. The disadvantage of the classical homotopy methods is that their consumed time is heavy since they need to solve many auxiliary systems of nonlinear equations during the intermediate continuation processes. In order to overcome this shortcoming, we consider the special continuation method based on the Newton flow and follow its trajectory with the new time-stepping scheme based on the trust-region technique. Furthermore, we analyze the global convergence and local superlinear convergence of the new method. Finally, the promising numerical results of the new method for some real-world problems are also reported, with comparison to the traditional trust-region method (the built-in subroutine fsolve.m of the MATLAB environment \cite{MATLAB,More1978}) and the classical homotopy continuation methods (HOMPACK90 \cite{WSMMW1997} and the built-in subroutines psolve.m for polynomial systems, GaussNewton.m for non-polynomial systems of the NAClab environment \cite{ZL2013}).

In this paper, we complete Jantzen's algorithm to compute the highest derivatives of irreducible representations of $p$-adic odd special orthogonal groups or symplectic groups. As an application, we give some examples of the Langlands data of the Aubert duals of irreducible representations, which are in the integral reducibility case.

Motivated by the prevalent data science applications of processing and mining large-scale graph data such as social networks, web graphs, and biological networks, as well as the high I/O and communication costs of storing and transmitting such data, this paper investigates lossless compression of data appearing in the form of a labeled graph. A universal graph compression scheme is proposed, which does not depend on the underlying statistics/distribution of the graph model. For graphs generated by a stochastic block model, which is a widely used random graph model capturing the clustering effects in social networks, the proposed scheme achieves the optimal theoretical limit of lossless compression without the need to know edge probabilities, community labels, or the number of communities.

The key ideas in establishing universality for stochastic block models include: 1) block decomposition of the adjacency matrix of the graph; 2) generalization of the Krichevsky-Trofimov probability assignment, which was initially designed for i.i.d. random processes. In four benchmark graph datasets (protein-to-protein interaction, LiveJournal friendship, Flickr, and YouTube), the compressed files from competing algorithms (including CSR, Ligra+, PNG image compressor, and Lempel-Ziv compressor for two-dimensional data) take 2.4 to 27 times the space needed by the proposed scheme.

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence.

Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces.

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables).

This article studies connections between group actions and their corresponding vector spaces. Given an action of a group $G$ on a nonempty set $X$, we examine the space $L(X)$ of scalar-valued functions on $X$ and its fixed subspace: $$L^G(X) = \{f\in L(X)\colon f(a\cdot x) = f(x) \textrm{ for all }a\in G, x\in X\}.$$ In particular, we show that $L^G(X)$ is an invariant of the action of $G$ on $X$. In the case when the action is finite, we compute the dimension of $L^G(X)$ in terms of fixed points of $X$ and prove several prominent results for $L^G(X)$, including Bessel's inequality and Frobenius reciprocity.

We prove that the spectrum of every Rota-Baxter operator of weight $\lambda$ on a unital algebraic (not necessarily associative) algebra over a field of characteristic zero is a subset of $\{0,-\lambda\}$. For a finite-dimensional unital algebra the same statement is shown to hold without a restriction on the characteristic of the ground field. Based on these results, we define the Rota-Baxter $\lambda$-index $\mathrm{rb}_\lambda(A)$ of an algebra $A$ as the infimum of the degrees of minimal polynomials of all Rota-Baxter operators of weight $\lambda$ on $A$. We calculate the Rota-Baxter $\lambda$-index for the matrix algebra $M_n(F)$, $\mathrm{char}\,F = 0$: it is shown that $\mathrm{rb}_\lambda(M_n(F)) = 2n-1$.

The Riccati equation method is used to establish some new stability criteria for systems of two linear first-order ordinary differential equations. It is shown that two of these criteria in the two dimensional case imply the Routh - Hurwitz's criterion.

We consider a change-point test based on the Hill estimator to test for structural changes in the tail index of Long Memory Stochastic Volatility time series. In order to determine the asymptotic distribution of the corresponding test statistic, we prove a uniform reduction principle for the tail empirical process in a two-parameter Skorohod space. It is shown that such a process displays a dichotomous behavior according to an interplay between the Hurst parameter, i.e., a parameter characterizing the dependence in the data, and the tail index. Our theoretical results are accompanied by simulation studies and the analysis of financial time series with regard to structural changes in the tail index.

The $1/2$-conjecture on the domination game asserts that if $G$ is a traceable graph, then the game domination number $\gamma_g(G)$ of $G$ is at most $\left\lceil \frac{n(G)}{2} \right\rceil$. A traceable graph is a $1/2$-graph if $\gamma_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ holds. It is proved that the so-called hatted cycles are $1/2$-graphs and that unicyclic graphs fulfill the $1/2$-conjecture. Several additional families of graphs that support the conjecture are determined and computer experiments related to the conjecture described.

In this paper, we present a new type of $\alpha-$Bernstein-P\u{a}lt\u{a}nea operators having a better order of approximation than itself. We establish some approximation results concerning the rate of convergence, error estimation and asymptotic formulas for the new modifications. Also, the theoretical results are verified by using MAPLE algorithms.

We give an example of a linear, time-dependent, Schr{\"o}dinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.

In this article we compute Galois groupoid of discret Painlev{\'e} equations. Our main tool is a semi-continuity theorem for the Galois groupoid in a confluence situation of a diffrence equation to a differential equation.

In the previous work \cite{Wan20} we found a necessary and sufficient condition for the graded algebra of modular forms on a symmetric domain of type IV to be free. Using the sufficient condition, in this paper we construct 16 free algebras of such modular forms for reflection groups related to the eight lattices $A_1(2)$, $A_1(3)$, $A_1(4)$, $2A_1(2)$, $A_2(2)$, $A_2(3)$, $A_3(2)$, $D_4(2)$.

We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three phases: as well as a phase with an almost sure limit where each of the three colours is equally represented and a phase with almost sure convergence to an asymmetric limit, which both occur with two types, there is also an intermediate phase where both symmetric and asymmetric limits are possible. In a model with anti-symmetric interactions between the types, we show the existence of a phase where the proportions of the three colours cycle and do not converge to a limit, alongside a phase where the proportions of the three colours can converge to a limit where each of the three is equally represented.

This paper deals with the problem of state estimation for a class of linear time-invariant systems with quadratic output measurements. An immersion-type approach is presented that transforms the system into a state-affine system by adding a finite number of states to the original system. Under suitable persistence of excitation conditions on the input and its higher derivatives, global state estimation is exhibited by means of a Kalman-type observer. A numerical example is provided to illustrate the applicability of the proposed observer design for the problem of position and velocity estimation for a vehicle navigating in the $n-$dimensional Euclidean space using a single position range measurement.

We consider an infinite homogeneous tree $\mathcal V$ endowed with the usual metric $d$ defined on graphs and a weighted measure $\mu$. The metric measure space $(\mathcal V,d,\mu)$ is nondoubling and of exponential growth, hence the classical theory of Hardy and $BMO$ spaces does not apply in this setting. We introduce a space $BMO(\mu)$ on $(\mathcal V,d,\mu)$ and investigate some of its properties. We prove in particular that $BMO(\mu)$ can be identified with the dual of a Hardy space $H^1(\mu)$ introduced in a previous work and we investigate the sharp maximal function related with $BMO(\mu)$.

We analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity $\varepsilon>0$. A resonator volume of thickness $\varepsilon$ is connected with thin channels (opening $\varepsilon^3$) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit $\varepsilon\to 0$ and find that the effective system can describe sound absorption.

Motivated by the computation of certain Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments $\int_0^\infty I_0(t)^i K_0(t)^{k-i}t^{2j-1}\,\mathrm{d}t \qquad (i, j=1, \ldots, \lfloor (k-1)/2\rfloor),$ where $k \geq 1$ is a fixed integer and $I_0$ and $K_0$ denote the modified Bessel functions. In this paper, we interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed in arXiv:2005.11525, this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne's conjecture explicit, thus explaining many evaluations of critical values of $L$-functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments.

The main problem considered in this paper is construction and theoretical study of efficient $n$-point coverings of a $d$-dimensional cube $[-1,1]^d$. Targeted values of $d$ are between 5 and 50; $n$ can be in hundreds or thousands and the designs (collections of points) are nested. This paper is a continuation of our paper \cite{us}, where we have theoretically investigated several simple schemes and numerically studied many more. In this paper, we extend the theoretical constructions of \cite{us} for studying the designs which were found to be superior to the ones theoretically investigated in \cite{us}. We also extend our constructions for new construction schemes which provide even better coverings (in the class of nested designs) than the ones numerically found in \cite{us}. In view of a close connection of the problem of quantization to the problem of covering, we extend our theoretical approximations and practical recommendations to the problem of construction of efficient quantization designs in a cube $[-1,1]^d$. In the last section, we discuss the problems of covering and quantization in a $d$-dimensional simplex; practical significance of this problem has been communicated to the authors by Professor Michael Vrahatis, a co-editor of the present volume.

It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval $[0,T]$. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the $L^2$ space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.

We construct exact t Hooft-Polyakov monopole solutions in a non-Hermitian field theory with local non-Abelian SU(2) gauge symmetry and a modified antilinear CPT symmetry. The solutions are obtained in a fourfold Bogomolny-Prasad-Sommerfield scaling limit giving rise to two different types of monopole masses that saturate the lower energy bound. These two masses only coincide in the Hermitian limit and in the limit in which the symmetry breaking vacuum tends to the trivial symmetry preserving vacuum. In the two theories corresponding to the two known Dyson maps these two masses are exchanged, unlike the Higgs and the gauge masses, which remain the same in both theories. We identify three separate regions in parameter space bounded by different types of exceptional points. In the first region the monopole masses are finite and tend both to zero at the boundary exceptional point, in the second the monopole masses become complex and in the third only one of the monopole masses becomes zero at the boundary exceptional point, whereas the other tends to infinity. We find a self-dual point in parameter space at which the gauge mass becomes exactly identical to the monopole mass.

The purpose of this paper is to construct a new stochastic process "Brownian house-moving," which is a Brownian bridge that stays between its starting point and its terminal point. To construct this process, statements are prepared on the weak convergence of conditioned Brownian motion, a conditioned Brownian bridge, a conditioned Brownian meander, and a conditioned three-dimensional Bessel bridge. Also studied are the sample path properties of Brownian house-moving and the decomposition formula for its distribution.

A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$ such that for every subset $S' \subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably generated (FIG) if, in addition, one can chose such a subset $S$ to be finite.

In this note we construct a FIG group $G$ with an index $2$ subgroup $N \lhd G$ such that $N$ is not IG. This shows that neither property IG nor FIG is stable under passing to subgroups of finite index, answering questions of Wiegold and Kantor, Lubotzky, Shalev. We also produce the first examples of finitely generated IG groups that are not FIG, answering a question of Cox.

We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian $(-\Delta)^{\alpha/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent $\alpha$ of the fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for the interface speed and time-to-collision in terms of $\alpha$ and a scaling parameter $\varepsilon$. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for $\alpha\in(0.2,2]$. A comparison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

We determine the group of endotrivial modules (as an abstract group) for $G$ a (quasi)simple group of sporadic type, extending previous results in the literature. In many sporadic cases we directly construct the subgroup of trivial-source endotrivial modules. We also resolve the question of whether certain simple modules for sporadic groups are endotrivial, posed by Lassueur, Malle and Schulte, in the majority of open cases. The results rely heavily on a recent description of the group of trivial-source endotrivial modules due to Grodal.

In this paper, we investigate a general class of stochastic gradient descent (SGD) algorithms, called conditioned SGD, based on a preconditioning of the gradient direction. Under some mild assumptions, namely the $L$-smoothness of the objective function and some weak growth condition on the noise, we establish the almost sure convergence and the asymptotic normality for a broad class of conditioning matrices. In particular, when the conditioning matrix is an estimate of the inverse Hessian at the optimal point, the algorithm is proved to be asymptotically optimal. The benefits of this approach are validated on simulated and real datasets.

The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on $\mathbb{G}$ and $\widehat{\mathbb{G}}$ act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group $\widehat{\mathrm{SU}_q(2)}$ and the quantum $az+b$ group.

Dynamic real-time optimization (DRTO) is a challenging task due to the fact that optimal operating conditions must be computed in real time. The main bottleneck in the industrial application of DRTO is the presence of uncertainty. Many stochastic systems present the following obstacles: 1) plant-model mismatch, 2) process disturbances, 3) risks in violation of process constraints. To accommodate these difficulties, we present a constrained reinforcement learning (RL) based approach. RL naturally handles the process uncertainty by computing an optimal feedback policy. However, no state constraints can be introduced intuitively. To address this problem, we present a chance-constrained RL methodology. We use chance constraints to guarantee the probabilistic satisfaction of process constraints, which is accomplished by introducing backoffs, such that the optimal policy and backoffs are computed simultaneously. Backoffs are adjusted using the empirical cumulative distribution function to guarantee the satisfaction of a joint chance constraint. The advantage and performance of this strategy are illustrated through a stochastic dynamic bioprocess optimization problem, to produce sustainable high-value bioproducts.

This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. % Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the \textsf{LDU} factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation much greater than this which we speculate is in accordance with the influence of the Euler product.

We study the differential and Riemannian geometry of algebras $A$ endowed with an action of a triangular Hopf algebra $H$ and noncomutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a symmetric ribbon category of $H$-modules $A$-bimodules, whose internal morphisms correspond to tensor fields. Different approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) $A$-module connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary braided symmetric pseudo-Riemannian metrics is proven.

We study graph-based Laplacian semi-supervised learning at low labeling rates. Laplacian learning uses harmonic extension on a graph to propagate labels. At very low label rates, Laplacian learning becomes degenerate and the solution is roughly constant with spikes at each labeled data point. Previous work has shown that this degeneracy occurs when the number of labeled data points is finite while the number of unlabeled data points tends to infinity. In this work we allow the number of labeled data points to grow to infinity with the number of labels. Our results show that for a random geometric graph with length scale $\varepsilon>0$ and labeling rate $\beta>0$, if $\beta \ll\varepsilon^2$ then the solution becomes degenerate and spikes form, and if $\beta\gg \varepsilon^2$ then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of $O(\varepsilon\beta^{-1/2})$ for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study $p$-Laplacian regularization and show the same degeneracy result when $\beta \ll \varepsilon^p$. The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use $\Gamma$-convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results.

In recent years, there have been many contributions to the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto [Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample. J. Math. Pures Appl. (9) 128 (2019), 330-338] has shown an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give an example of the nonlinear monotone system of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the whole family convergence of the solutions does not hold.

We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain $\Omega \subset \mathbb{R}^2$ that vanishes on a compact set $E \subset \Omega$ and satisfies mild assumptions. Our main question is to determine when such a space is quasiconformally equivalent to a planar domain. We give a characterization in terms of the notion of planar sets that are removable for conformal mappings. We also study the question of when a quasiconformal mapping can be factored as a 1-quasiconformal mapping precomposed with a bi-Lipschitz map.

In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex plane, provided that $V$ is sufficiently small with respect to the mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on $V$, and in particular the spectrum is the same of the unperturbed operator, namely $\sigma(\mathscr{D}_0+V)=\sigma(\mathscr{D}_0)=\mathbb{R}$. The main tools we employ are an abstract version of the Birman-Schwinger principle, which include also the study of embedded eigenvalues, and suitable resolvent estimates for the Schr\"odinger operator.

We provide a Rademacher theorem for intrinsically Lipschitz functions $\phi:U\subseteq \mathbb W\to \mathbb L$, where $U$ is a Borel set, $\mathbb W$ and $\mathbb L$ are complementary subgroups of a Carnot group, where we require that $\mathbb W$ is a Carnot subgroup and $\mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $\mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.

For $h \ge 2$ and an infinite set of positive integers $A$, let $R_{A,h}(n)$ denote the number of solutions of the equation $a_{1} + a_{2} + \dots{} + a_{h} = n, a_{1} \in A, \dots{} ,a_{h} \in A, a_{1} < a_{2} < \dots{} < a_{h}.$ In this paper we prove the existence of a set $A$ formed by perfect powers with almost possible maximal density such that $R_{A,h}(n)$ is bounded by using probabilistic methods.

A new model description and type classification carried out on its base of a wide variety of practical hysteresis loops are suggested. An analysis of the loop approximating function was carried out; the parameters and characteristics of the model were defined - coersitivity, remanent polarization, value of hysteresis, spontaneous polarization, induced piezocoefficients, value of saturation, hysteresis losses of energy per cycle. It was shown that with piezomanipulators of certain hysteresis loop types, there is no difference in heat production. The harmonic linearization coefficients were calculated, and the harmonically linearized transfer function of a nonlinear hysteresis element was deduced. The hysteresis loop type was defined that possesses minimum phase shift. The average relative approximation error of the model has been evaluated as 1.5%-6% for real hysteresis loops. A procedure for definition of the model parameters by experimental data is introduced. Examples of using the results in a scan unit of a scanning tunneling microscope for compensation of raster distortion are given.

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approach is essential for our setting since the stochastic evolution equation cannot be transformed into a family of PDEs with random coefficients via the stationary Ornstein-Uhlenbeck process.

Let $q=p^r$ be the power of a prime $p$ and $(\beta_1,\ldots ,\beta_r)$ be an ordered basis of $\mathbb{F}_q$ over $\mathbb{F}_p$. For $$\xi=\sum\limits_{j=1}^r x_j\beta_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p,$$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$R(\xi)=\sum\limits_{i=1}^{r-1} x_ix_{i+1}, \quad \xi\in \mathbb{F}_q.$$ For a non-constant polynomial $f(X)\in \mathbb{F}_q[X]$ and $c\in \mathbb{F}_p$ we study the number of solutions $\xi\in \mathbb{F}_q$ of $R(f(\xi))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\rightarrow \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.

Non-Markovian effects in quantum evolution appear when the system is strongly coupled to the environment and interacts with it for long periods of time. To include memory effects in the master equations, one usually incorporates time-local generators or memory kernels. However, it turns out that non-Markovian evolution with eternally negative decoherence rates arises from a simple mixture of Markovian semigroups. Moreover, one can have as many as $(d-1)^2$ always negative rates out of $d^2-1$ total, and the quantum evolution is still legitimate.

In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For this new combinatorial sequence we derive its exponential generating function, some recurrence relations, and combinatorial identities. We prove and present results using combinatorial arguments, generating functions, the symbolic method and Riordan arrays. For some specific cases we provide a combinatorial interpretation for the inverse matrix of the generalized Lah numbers by means of two families of posets.

We study codimension-even conical defects that contain a deficit solid angle around each point along the defect. We show that they lead to a delta function contribution to the Lovelock scalar and we compute the contribution by two methods. We then show that these codimension-even defects appear as Euclidean brane solutions in higher dimensional topological AdS gravity which is Lovelock-Chern-Simons gravity without torsion. The theory possesses a holographic Weyl anomaly that is purely of type-A and proportional to the Lovelock scalar. Using the formula for the defect contribution, we prove a holographic duality between codimension-even defect partition functions and codimension-even brane on-shell actions in Euclidean signature. More specifically, we find that the logarithmic divergences match, because the Lovelock-Chern-Simons action localizes on the brane exactly. We demonstrate the duality explicitly for a spherical defect on the boundary which extends as a codimension-even hyperbolic brane into the bulk. For vanishing brane tension, the geometry is a foliation of Euclidean AdS space that provides a one-parameter generalization of AdS-Rindler space.

It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer ring where k is the dimension of the manifold, then this is also true. As a specific study, for 4-dimensional closed manifolds, the topologies (genera) of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example.

In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems. We present an affirmative answer via important theory in the singularity theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.

In a multi-index model with $k$ index vectors, the input variables are transformed by taking inner products with the index vectors. A transfer function $f: \mathbb{R}^k \to \mathbb{R}$ is applied to these inner products to generate the output. Thus, multi-index models are a generalization of linear models. In this paper, we consider monotone multi-index models. Namely, the transfer function is assumed to be coordinate-wise monotone. The monotone multi-index model therefore generalizes both linear regression and isotonic regression, which is the estimation of a coordinate-wise monotone function. We consider the case of nonnegative index vectors. We provide an algorithm based on integer programming for the estimation of monotone multi-index models, and provide guarantees on the $L_2$ loss of the estimated function relative to the ground truth.

We show that the Liechti-Strenner's example for the closed nonorientable surface in \cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial of the action induced on the first cohomology nonpositive. We also show that the Liechti-Strenner's example of orientation-reversing homeomorphism for the closed orientable surface in \cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial $p(x)$ of the action induced on the first cohomology nonpositive or all but the first coefficient of $p(x) (x \pm 1)^2$, $p(x) (x^2 \pm 1)$, or $p(x) (x^2 \pm x + 1)$ nonpositive.

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_\mu$ allows us to derive its behavior in the two limits $\mu\to0$ and $\mu\to\mu_*$ where $\mu_*$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $L^2$ mass of $u_\mu$ in terms of $\mu$, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $u_\mu$.

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number $k$, we establish global $\mathcal{O} (1/ \sqrt{k})$ pointwise and $\mathcal{O} (1/ {k})$ ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM.

Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $\Gamma$. In the first part of this paper we identify the automorphism group of the solenoid associated with $\Gamma$ with the Belyaev completion of its commensurator $\mathrm{Comm}(\Gamma)$ and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of $\mathrm{Gal}(\mathbb C/\mathbb Q)$ on algebraic curves. In turn this fact yields a proof of the Galois invariance of the arithmeticity of $\Gamma$ independent of Kazhhdan's. In the second part we focus on the case in which $\Gamma$ is arithmetic. The list of further Galois invariants we find includes: i) the periods of $\mathrm{Comm}(\Gamma)$, ii) the solvability of the equations $X^2+\sin^2 \frac{2\pi}{2k+1}$ in the invariant quaternion algebra of $\Gamma$ and iii) the property of $\Gamma$ being a congruence subgroup.

We show that Szczarba's twisting cochain for a twisted Cartesian product is essentially the same as the one constructed by Shih. More precisely, Szczarba's twisting cochain can be obtained via the basic perturbation lemma if one uses a 'reversed' version of the classical Eilenberg-MacLane homotopy for the Eilenberg-Zilber contraction. Along the way we prove several new identities involving these homotopies.

For a finite point set in $\mathbb{R}^d$, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number $L(X)$ of a given point set $X$ is defined as the number of steps of the peeling process in order to delete all points in $X$. It is known that if $X$ is a set of random points in $\mathbb{R}^d$, then the expectation of $L(X)$ is $\Theta(|X|^{2/(d+1)})$, and recently it was shown that if $X$ is a point set of the square grid on the plane, then $L(X)=\Theta(|X|^{2/3})$.

In this paper, we investigate the layer number of $\alpha$-evenly distributed point sets for $\alpha>1$; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an $\alpha$-evenly distributed point set for some $\alpha>1$. We find an upper bound of $O(|X|^{3/4})$ for the layer number of an $\alpha$-evenly distributed point set $X$ in a unit disk on the plane for some $\alpha>1$, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of $O(|X|^{\frac{d+1}{2d}})$ for the layer number of an $\alpha$-evenly distributed point set $X$ in a unit ball in $\mathbb{R}^d$ for some $\alpha>1$ and $d\geq 3$.

For a subset $S$ of vertices in a graph $G$, a vertex $v \in S$ is an enclave of $S$ if $v$ and all of its neighbors are in $S$, where a neighbor of $v$ is a vertex adjacent to $v$. A set $S$ is enclaveless if it does not contain any enclaves. The enclaveless number $\Psi(G)$ of $G$ is the maximum cardinality of an enclaveless set in $G$. As first observed in 1997 by Slater [J. Res. Nat. Bur. Standards 82 (1977), 197--202], if $G$ is a graph with $n$ vertices, then $\gamma(G) + \Psi(G) = n$ where $\gamma(G)$ is the well-studied domination number of $G$. In this paper, we continue the study of the competition-enclaveless game introduced in 2001 by Phillips and Slater [Graph Theory Notes N. Y. 41 (2001), 37--41] and defined as follows. Two players take turns in constructing a maximal enclaveless set $S$, where one player, Maximizer, tries to maximize $|S|$ and one player, Minimizer, tries to minimize~$|S|$. The competition-enclaveless game number $\Psi_g^+(G)$ of $G$ is the number of vertices played when Maximizer starts the game and both players play optimally. We study among other problems the conjecture that if $G$ is an isolate-free graph of order $n$, then $\Psi_g^+(G) \ge \frac{1}{2}n$. We prove this conjecture for regular graphs and for claw-free graphs.

We show that a finitely generated group G such that $\mathrm{H}_2(G,\mathbb Z)$ is finitely generated possesses a representation group $\tilde{G}$ which is finitely generated. As applications we provide a characterization of monomial projective representations of finitely generated nilpotent groups and a characterization of polycyclic groups whose projective representations are finite dimensional.

This paper reviews and offers remarks upon Viro's definition of the Khovanov homology of the Kauffman bracket of unoriented framed tangles (Sec. 2). The review is based on a file of his talk. This definition contains an exposition of the relation between the $R$-matrix and the Kauffman bracket (Sec. 2.2).

We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and partial regularity of volume-constrained minimizers. We also derive the Euler--Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal $\frac12$-derivative of the capacitary potential.

Let $n \geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n \times n$ symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to $d_{ij}$. In this paper, we find a positive semidefinite matrix $\widetilde{L}$ such that ${\rm rank}(\widetilde{L})=n-1$, all row sums of $\widetilde{L}$ equal to zero and a rank one matrix $ww^T$ such that $D^{-1}=-\frac{1}{2}\widetilde{L} + \frac{4}{n-1}ww^T.$ An interlacing property between the eigenvalues of $D$ and $\widetilde{L}$ is also proved.

We find a sufficient condition for a nerve of a hyperbolic right-angled Coxeter group, under which the boundary of the group is homeomorphic to the Menger curve. The main problem appears to be to satisfy the condition that no open set in the boundary is planar -- to this end we learn how to embed non-planar graphs in the boundary starting from a particularly placed non-planar graph in the nerve. Then we apply this condition to finding triangulations of surfaces and disks $D^n$ that give, as nerves, right-angled Coxeter groups with Menger curve boundaries.

Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.

In this article, the shape optimization of a linear elastic body subject to frictional (Tresca) contact is investigated. Due to the projection operators involved in the formulation of the contact problem, the solution is not shape differentiable in general. Moreover, shape optimization of the contact zone requires the computation of the gap between the bodies in contact, as well as its shape derivative. Working with directional derivatives, sufficient conditions for shape differentiability are derived. %The problem is addressed in the general framework of two bodies with smooth boundaries. Then, some numerical results, obtained with a gradient descent algorithm based on those shape derivatives, are presented.

In this paper, we study the optimal investment problem of an insurer whose surplus process follows the diffusion approximation of the classical Cramer-Lundberg model. Investment in the foreign market is allowed, and therefore, the foreign exchange rate model is considered and incorporated. It is assumed that the instantaneous mean growth rate of foreign exchange rate price follows an Ornstein-Uhlenbeck process. Dynamic programming method is employed to study the problem of maximizing the expected exponential utility of terminal wealth. By soloving the correspoding Hamilton-Jacobi-Bellman equations, the optimal investment strategies and the value functions are obtained. Finally, numerical analysis is presented.

We consider three-dimensional (3D) localization and imaging of space debris from only one two-dimensional (2D) snapshot image. The technique involves an optical imager that exploits off-center image rotation to encode both the lateral and depth coordinates of point sources, with the latter being encoded in the angle of rotation of the PSF. We formulate 3D localization into a large-scale sparse 3D inverse problem in the discretized form. A recently developed penalty called continuous exact l0 (CEL0) is applied in this problem for the Gaussian noise model. Numerical experiments and comparisons illustrate the efficiency of the algorithm.

Commutation formulae with respect to a non-symmetric affine connection are obtained in this paper. The components of commutation formulae in this paper are covariant derivatives of tensors with respect to symmetric and non-symmetric affine connection.

In this paper we introduce the $g-$Navier-Stokes equations with time-fractional derivative of order $\alpha\in(0,1)$ in domains of $\mathbb R^2$. We then study the existence and uniqueness of weak solutions by means of Galerkin approximation. Finally, an optimal control problem is considered and solved.

We revisit the hypothesis testing problem against independence over a noisy channel and prove a strong converse theorem. In particular, under the Neyman-Pearson formulation, we derive a non-asymptotic upper bound on the type-II exponent of any encoding-decoding functions which can ensure that the type-I error probability is upper bounded by a constant. The strong converse theorem for the problem follows as a corollary as our result. Our proof is based on the recently proposed strong converse technique by Tyagi and Watanabe (TIT 2020) which is based on the change of measure technique. Our work is the first application of the strong converse technique by Tyagi and Watanabe to a hypothesis testing problem over a noisy channel and thus further demonstrates the generality of the technique.

Available alternative routes on which traffic can be rerouted in the case of disruptions are vital for transportation networks. Line sections with less traffic under normal operational conditions but with increased importance in the case of disruptions are identified in the railway network of Hungary by using a weighted directed graph. To describe the goodness of the individual alternative routes the so-called redundancy index is used. The results show that the structure of the network is good, but the lines with the highest redundancy (lines No. 80, 2, 4 and 77 according to the numbering of the national railway operator, M\'AV) are mostly single tracked and in many cases the line speed is low. The building of additional tracks and electrifying these lines while still maintaining the existing diesel locomotives for the case of disruptions of the electric support are the keys to make the performance of the rather dense railway network of Hungary sustainable.

We use ideas from our previous work to obtain some theorems that will allow us to obtain the integer solution of a quadratic polynomial in two variables that represents a natural number

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$

Let c=2^aleph0 denote the cardinality of the continuum and let a,b,k be infinite cardinal numbers with a<b\leq 2^a. We show that there exist precisely 2^b T0-spaces of size a and weight b up to homeomorphism. Among these non-homeomorphic spaces we track down

(1) 2^b zero-dimensional, scattered, paracompact, perfectly normal spaces (which are also extremally disconnected in case that b=2^a); (2) 2^b connected and locally connected Hausdorff spaces; (3) 2^b pathwise connected and locally pathwise connected, paracompact, perfectly normal spaces provided that a\geq c; (4) 2^b connected, nowhere locally connected, totally pathwise disconnected, paracompact, perfectly normal spaces provided that a\geq c; (5) 2^b scattered, compact T1-spaces; (6) 2^b connected, locally connected, compact T1-spaces; (7) 2^b pathwise connected and scattered, compact T0-spaces; (8) 2^b scattered, paracompact P_k-spaces whenever a^{<k}=a and b^{<k}=b and 2^b>2^a.

We show that Somos' constant is universal in sense that is similar to the universality of the Khinchin constant.

We consider homological finiteness properties $FP_n$ of certain $\mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph $\Gamma$. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case.

This article presents a simple port-Hamiltonian formulation of the equations for an RLC electric circuit as a differential-algebraic equation system, and a proof that structural analysis always succeeds on it for a well-posed circuit, thus providing a correct regularisation for numerical solution. The DAE is small - its number of variables/equations is at most the number of edges in the circuit graph.

In this paper, we show that the existing the energy and pointwise decay results for the fields satisfied the tensorial Frackerell-Ipser equations (which are obtained from the Maxwell and spin $\pm 1$ Teukolsky equations) on the Schwarzschild spacetime are sufficient to obtain a conformal scattering theory. This work is continuing the previous work on the conformal scattering theory for linearized gravity fields \cite{Pha2020} which are arised from spin $\pm 2$ Teukolsky equations.

The main goal of 1-bit compressive sampling is to decode $n$ dimensional signals with sparsity level $s$ from $m$ binary measurements. This is a challenging task due to the presence of nonlinearity, noises and sign flips. In this paper, the cardinality constraint least square is proposed as a desired decoder. We prove that, up to a constant $c$, with high probability, the proposed decoder achieves a minimax estimation error as long as $m \geq \mathcal{O}( s\log n)$. Computationally, we utilize a generalized Newton algorithm (GNA) to solve the cardinality constraint minimization problem with the cost of solving a least squares problem with small size at each iteration. We prove that, with high probability, the $\ell_{\infty}$ norm of the estimation error between the output of GNA and the underlying target decays to $\mathcal{O}(\sqrt{\frac{\log n }{m}})$ after at most $\mathcal{O}(\log s)$ iterations. Moreover, the underlying support can be recovered with high probability in $\mathcal{O}(\log s)$ steps provided that the target signal is detectable. Extensive numerical simulations and comparisons with state-of-the-art methods are presented to illustrate the robustness of our proposed decoder and the efficiency of the GNA algorithm.

We present an analytic proof of the existence of phase transition in the large $N$ limit of certain random noncommutaitve geometries. These geometries can be expressed as ensembles of Dirac operators. When they reduce to single matrix ensembles, one can apply the Coulomb gas method to find the empirical spectral distribution. We elaborate on the nature of the large $N$ spectral distribution of the Dirac operator itself. Furthermore, we show that these models exhibit both a single and double cut region for certain values of the order parameter and find the exact value where the transition occurs.

In this paper we investigate the question of when a Noetherian $F$-pure ring is weakly Arf. Under mild conditions we provide an affirmative answer for the question. In addition, we prove that every Stanley-Reisner algebra over a field satisfying the Serre's $(S_2)$ condition is a weakly Arf ring.

In J. Schwenk.(2018) ['What is the Correct Way to Seed a Knockout Tournament?' Retrieved from The American Mathematical Monthly], Schwenk identified a surprising weakness in the standard method of seeding a single elimination (or knockout) tournament. In particular, he showed that for a certain probability model for the outcomes of games it can be the case that the top seeded team would be less likely to win the tournament than the second seeded team. This raises the possibility that in certain situations it might be advantageous for a team to intentionally lose a game in an attempt to get a more optimal (though possibly lower) seed in the tournament. We examine this question in the context of a four team league which consists of a round robin "regular season" followed by a single elimination tournament with seedings determined by the results from the regular season [4]. Using the same probability model as Schwenk we show that there are situations where it is indeed optimal for a team to intentionally lose. Moreover, we show how a team can make the decision as to whether or not it should intentionally lose. We did two detailed analysis. One is for the situation where other teams always try to win every game. The other is for the situation where other teams are smart enough, namely they can also lose some games intentionally if necessary. The analysis involves computations in both probability and (multi-player) game theory.

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.

We introduce sequential warped product submanifolds of Kaehler manifolds, provide examples and establish Chen's inequality for such submanifolds. The equality case is also studied. Moreover, by inspiring Lawson and Simons's integral currrent's theorem on a submanifold, we find a similar pinching inequality for a sequential warped product submanifold and obtain geometric results when the equality case is satisfied.

Our goal is to find an asymptotic behavior as $n\to\infty$ of the orthogonal polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients $a_{n}$ (off-diagonal terms) and $b_{n}$ (diagonal terms). We consider the case $a_{n}\to\infty$, $b_{n}\to\infty$ in such a way that $\sum a_{n}^{-1}<\infty$ $($that is, the Carleman condition is violated$)$ and $\gamma_{n}:=2^{-1}b_{n} (a_{n}a_{n-1})^{-1/2} \to \gamma$ as $n\to\infty$. In the case $|\gamma | \neq 1$ asymptotic formulas for $P_{n}(z)$ are known; they depend crucially on the sign of $| \gamma |-1$. We study the critical case $| \gamma |=1$. The formulas obtained are qualitatively different in the cases $|\gamma_{n}| \to 1-0$ and $|\gamma_{n}| \to 1+0$. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of $P_{n}(z)$ based on a close analogy of the Jacobi difference equations and differential equations of Schr\"odinger type.

We consider stochastic dynamic programming problems with high-dimensional, discrete state-spaces and finite, discrete-time horizons that prohibit direct computation of the value function from a given Bellman equation for all states and time steps due to the "curse of dimensionality". For the case where the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function in the realm of dual dynamic programming. We show that the proposed algorithm terminates after a finite number of iterations. Furthermore, we derive probabilistic guarantees on the value accumulated under the associated policy for a single realisation of the dynamic program and for the expectation of this value. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.

In this paper, we prove several supercongruences via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}{k}}{24^k}\equiv\begin{cases}\binom{(2p-2)/3}{(p-1)/3}\pmod{p^2}\ & if\ p\equiv1\pmod{3},\\ \frac{p}{\binom{(2p+2)/3}{(p+1)/3}}\pmod{p^2}\ & if\ p\equiv2\pmod{3}.\end{cases}$$ As a corollary, we prove that \begin{align*} \sum_{k=0}^{p-1}\frac{\binom{2k}{k}^2\binom{3k}{k}}{(-192)^k}\equiv\begin{cases}x^2-2p\pmod{p^2}\ & if\ p\equiv1\pmod{3}\ \& \ 4p=x^2+27y^2\ (x,y\in\mathbb{Z}),\\ 0\pmod{p^2}\ & if\ p\equiv2\pmod{3}.\end{cases} \end{align*} These confirm two conjectures of Z.-W. Sun. We also obtain some other results of such types.

We explore the Atiyah-Hirzebruch spectral sequence for the $tmf^\bullet[\frac12]$-cohomology of the classifying space $BM_{24}$ of the largest Mathieu group $M_{24}$, twisted by a class $\omega \in H^4(BM_{24};Z[\frac12]) \cong Z_3$. Our exploration includes detailed computations of the $Z_3$-cohomology of $M_{24}$ and of the first few differentials in the AHSS. We are specifically interested in the value of $tmf^\bullet_\omega(BM_{24})[\frac12]$ in cohomological degree $-27$. Our main computational result is that $tmf^{-27}_\omega(BM_{24})[\frac12] = 0$ when $\omega \neq 0$. For comparison, the restriction map $tmf^{-3}_\omega(BM_{24})[\frac12]\to tmf^{-3}(pt)[\frac12] \cong Z_3$ is nonzero for one of the two nonzero values of $\omega$.

Our motivation comes from Mathieu Moonshine. Assuming a well-studied conjectural relationship between $TMF$ and supersymmetric quantum field theory, there is a canonically-defined $Co_1$-twisted-equivariant lifting $[\bar{V}^{f\natural}]$ of the class $\{24\Delta\} \in TMF^{-24}(pt)$, where $Co_1$ denotes Conway's largest sporadic group. We conjecture that the product $[\bar{V}^{f\natural}] \nu$, where $\nu \in TMF^{-3}(pt)$ is the image of the generator of $tmf^{-3}(pt) \cong Z_{24}$, does not vanish $Co_1$-equivariantly, but that its restriction to $M_{24}$-twisted-equivariant $TMF$ does vanish. This conjecture answers some of the questions in Mathieu Moonshine: it implies the existence of a minimally supersymmetric quantum field theory with $M_{24}$ symmetry, whose twisted-and-twined partition functions have the same mock modularity as in Mathieu Moonshine. Our AHSS calculation establishes this conjecture "perturbatively" at odd primes.

An appendix included mostly for entertainment purposes discusses "$\ell$-complexes" and their relation to $\mathrm{SU}(2)$ Verlinde rings. The case $\ell=3$ is used in our AHSS calculations.

Let $S$ and $T$ be two aperiodic commuting automorphisms of a standard Borel space $(X,\mathcal{B})$, and $Cob(S)$, $Cob(T)$ be the sets of their real valued (Borel) coboundaries. We show that $Cob(S) = Cob(T)$ if and only if $S = T^{\pm 1}$. We also prove a weaker form of Rokhlin Lemma for Borel $\mathbb{Z}^d$-actions.

In this paper we estimate the worst rate of exponential decay of degenerate gradient flows $\dot x = -S x$, issued from adaptive control theory. Under persistent excitation assumptions on the positive semi-definite matrix $S$, we provide upper bounds for this rate of decay consistent with previously known lower bounds and analogous stability results for more general classes of persistently excited signals. The strategy of proof consists in relating the worst decay rate to optimal control questions and studying in details their solutions.

As a byproduct of our analysis, we also obtain estimates for the worst $L_2$-gain of the time-varying linear control systems $\dot x=-cc^\top x+u$, where the signal $c$ is persistently excited, thus solving an open problem posed by A. Rantzer in 1999.

We characterize all semigroups $(T(t))_{t\geq0}$ on $L^2(\Omega)$ sandwiched between Dirichlet and Neumann ones, i.e.: \begin{equation*}\label{eq:san}

e^{t\Delta_D}\leq T(t)\leq e^{t\Delta_N}\quad,\text{for all }t\geq0 \end{equation*} in the positive operators sense. The proof uses the well-known Beurling-Deny and Lejan formula to drop the locality assumption made usually on the form associated with $(T(t))_{t\geq 0}$.

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer m > 3, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension m. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

Reservoir Computing (RC) is a well-known strategy for designing Recurrent Neural Networks featured by striking efficiency of training. The crucial aspect of RC is to properly instantiate the hidden recurrent layer that serves as dynamical memory to the system. In this respect, the common recipe is to create a pool of randomly and sparsely connected recurrent neurons. While the aspect of sparsity in the design of RC systems has been debated in the literature, it is nowadays understood mainly as a way to enhance the efficiency of computation, exploiting sparse matrix operations. In this paper, we empirically investigate the role of sparsity in RC network design under the perspective of the richness of the developed temporal representations. We analyze both sparsity in the recurrent connections, and in the connections from the input to the reservoir. Our results point out that sparsity, in particular in input-reservoir connections, has a major role in developing internal temporal representations that have a longer short-term memory of past inputs and a higher dimension.

In this paper we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. The IBVP is first discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order $2p+2$ of accuracy at the stage $p=0,1,2,\cdots$ of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage $p$ of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVP. This fully discretized scheme is unconditionlly stable with order $2p+2$ of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular while an increment of one order is proven for shape-regular and quasi-uniform family of meshes. A numerical test with a bistable reaction-diffusion equation having a strong stiffness ratio is performed and shows that the orders 2,4,6,8 and 10 of accuracy in time are achieved with a very strong stability.

Zeckendorf proved that every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. A natural generalization of this theorem is to look at the sequence defined as follows: for $n\ge 2$, let $F_{n,1} = F_{n,2} = \cdots = F_{n,n} = 1$ and $F_{n, m+1} = F_{n, m} + F_{n, m+1-n}$ for all $m\ge n$. It is known that every positive integer has a unique representation as a sum of $F_{n,m}$'s where the indexes of summands are at least $n$ apart. We call this the $n$-decomposition. Griffiths showed an interesting relationship between the Zeckendorf decomposition and the golden string. In this paper, we continue the work to show a relationship between the $n$-decomposition and the generalized golden string.

Recently, a new kind of multiple zeta value level two $T({\bf k})$ (which is called multiple $T$-values) was introduced and studied by Kaneko and Tsumura. In this paper, we define a kind of alternating version of multiple $T$-values, and study several duality formulas of weighted sum formulas about alternating multiple $T$-values by using the methods of iterated integral representations and series representations. Some special values of alternating multiple $T$-values can also be obtained.

We obtain asymptotic formulas for sums of coefficients over arithmetic progressions of polynomials related to the Borwein conjectures. Let $a_i$ denote the coefficient of $q^i$ in the polynomial $\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s$, where $p$ is an odd prime, and $n, s$ are positive integers. In this note, we prove that

$$\Big|\sum_{i=b\ \text{mod}\ 2pn}a_i-\frac{(p-1)p^{sn-1}}{2n}\Big|\leq p^{sn/2},$$ if $b$ is divisible by $p$, and

$$\Big|\sum_{i=b\ \text{mod}\ 2pn}a_i+\frac{p^{sn-1}}{2n}\Big|\leq p^{sn/2},$$ if $b$ is not divisible by $p$. This improves a recent result of Goswami and Pantangi.

Minimal codewords have applications in decoding linear codes and in cryptography. We study the number of minimal codewords in binary linear codes that arise by appending a unit matrix to the adjacency matrix of a graph.

We study injective objects and injective hulls in the category of $(L,V)$-categories.

This research gauges the ability of deep reinforcement learning (DRL) techniques to assist the optimization and control of fluid mechanical systems. It combines a novel, "degenerate" version of the proximal policy optimization (PPO) algorithm, that trains a neural network in optimizing the system only once per learning episode, and an in-house stabilized finite elements environment implementing the variational multiscale (VMS) method, that computes the numerical reward fed to the neural network. Three prototypical examples of separated flows in two dimensions are used as testbed for developing the methodology, each of which adds a layer of complexity due either to the unsteadiness of the flow solutions, or the sharpness of the objective function, or the dimension of the control parameter space. Relevance is carefully assessed by comparing systematically to reference data obtained by canonical direct and adjoint methods. Beyond adding value to the shallow literature on this subject, these findings establish the potential of single-step PPO for reliable black-box optimization of computational fluid dynamics (CFD) systems, which paves the way for future progress in optimal flow control using this new class of methods.

We show that the variance of a probability measure $\mu$ on a compact subset $X$ of a complete metric space $M$ is bounded by the square of the circumradius $R$ of the canonical embedding of $X$ into the space $P(M)$ of probability measures on $M$, equipped with the Wasserstein metric. When barycenters of measures on $X$ are unique (such as on CAT($0$) spaces), our approach shows that $R$ in fact coincides with the circumradius of $X$ and so this result extends a recent result of Lim-McCann from Euclidean space. Our approach involves bi-linear minimax theory on $P(X) \times P(M)$ and extends easily to the case when the variance is replaced by very general moments. As an application, we provide a simple proof of Jung's theorem on CAT($k$) spaces, a result originally due to Dekster and Lang-Schroeder.

We present an integer programming model to compute the strong rainbow connection number $src(G)$ of any simple graph $G$. We introduce several enhancements to the proposed model, including a fast heuristic, a novel class of valid inequalities, and a variable elimination scheme. Moreover, we present a novel lower bound for $src(G)$ which may be of independent research interest. We evaluate our model with a traditional branch and cut approach as well as an alternative scheme based on iterative lower bound improvement, which we show to be highly effective in practice. To our knowledge, these are the first computational methods for the strong rainbow connection problem. We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs with up to $167$ vertices.

We extend the notion of the transfer matrix of potential scattering to a large class of long-range potentials $v(x)$ and derive its basic properties. We outline a dynamical formulation of the time-independent scattering theory for this class of potentials where we identify their transfer matrix with the $S$-matrix of a certain effective non-unitary two-level quantum system. For sufficiently large values of $|x|$, we express $v(x)$ as the sum of a short-range potential and an exactly solvable long-range potential. Using this result and the composition property of the transfer matrix, we outline an approximation scheme for solving the scattering problem for $v(x)$. To demonstrate the effectiveness of this scheme, we construct an exactly solvable long-range potential and compare the exact values of its reflection and transmission coefficients with those we obtain using our approximation scheme.

This article studies, both theoretically and numerically, a nonlinear drift-diffusion equation describing a gas of fermions in the zero-temperature limit. The equation is considered on a bounded domain whose boundary is divided into an "insulating" part, where homogeneous Neumann conditions are imposed, and a "contact" part, where nonhomogeneous Dirichlet data are assigned. The existence of stationary solutions for a suitable class of Dirichlet data is proven by assuming a simple domain configuration. The long-time behavior of the time-dependent solution, for more complex domain configurations, is investigated by means of numerical experiments.

We consider equations of the form $-L_\mu u +f(u)=0$ in a smooth domain $\Omega$, where $L_\mu=\Delta + \mu\delta^{-2}$ and $\delta(x)$ denotes the distance of the point $x$ to the boundary of the domain. The nonlinear term $f$ is positive, increasing and convex on $(0,\infty)$, satisfies the Keller-Osserman condition and some additional technical assumptions. The conditions are satisfied, in particular, by power and exponential nonlinearities. We discuss the question of existence and uniqueness of large solutions when $\mu>0$.

We present two short proofs for Diestel's criterion that a connected graph has a normal spanning tree provided it contains no subdivision of a countable clique in which every edge has been replaced by uncountably many parallel edges.

In this paper, we show that, on average, the derivatives of $L$-functions of cuspidal Hilbert modular forms with sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$, $\Re(s)\in(\frac{k-1}{2},\frac{k}{2}-\epsilon)\cup(\frac{k}{2}+\epsilon,\frac{k+1}{2})$. This is analogous to the case of classical modular forms.

We show that the cop number of any graph on 18 or fewer vertices is at most 3. This answers a specific case of a question posed by Baird et al. on the minimum order of 4-cop-win graphs, first appearing in 2011. We also find all 3-cop-win graphs on 11 vertices, narrow down the possible 4-cop-win graphs on 19 vertices and get some progress on finding the minimum order of 3-cop-win planar graphs.

We extend the block filtration, defined by Brown based on the work of Charlton, to all motivic multiple zeta values, and study relations compatible with this filtration. We construct a Lie algebra describing relations among motivic multiple zeta values modulo terms of lower block degree, proving Charlton's cyclic insertion conjecture in this structure, and showing the existence of a block shuffle' relation, and a previously unknown dihedral symmetry and differential relation.

The Minor problem, namely the study of the spectrum of a principal submatrix of a Hermitian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on the use of orbital integrals and on the connection with branching coefficients in the decomposition of an irreducible representation of U(n), resp. SU(n), into irreps of U(n-1), resp. SU(n-1).

For $n \ge 2$, we prove that a finite volume complex hyperbolic $n$-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic $3$-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber--Pink conjecture.

Let $s$ and $t$ be positive integers. We use $P_t$ to denote the path with $t$ vertices and $K_{1,s}$ to denote the complete bipartite graph with parts of size $1$ and $s$ respectively. The one-subdivision of $K_{1,s}$ is obtained by replacing every edge $\{u,v\}$ of $K_{1,s}$ by two edges $\{u,w\}$ and $\{v,w\}$ with a new vertex $w$. In this paper, we give a polynomial-time algorithm for the list-three-coloring problem restricted to the class of $P_t$-free graph with no induced 1-subdivision of $K_{1,s}$.

We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.

Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with few exceptions, these hypersurfaces do not admit a mod $p^2$ lift of the Frobenius endomorphism.

Given the recent geometrical classification of 6d $(1,0)$ SCFTs, a major question is how to compute for this large class their elliptic genera. The latter encode the refined BPS spectrum of the SCFTs, which determines geometric invariants of the associated elliptic non-compact Calabi-Yau threefolds. In this paper we establish for all 6d $(1,0)$ SCFTs in the atomic classification blowup equations that fix these elliptic genera to large extent. The latter fall into two types: the unity- and the vanishing blowup equations. For almost all rank one theories, we find unity blowup equations which determine the elliptic genera completely. We develop several techniques to compute elliptic genera and BPS invariants from the blowup equations, including a recursion formula with respect to the number of strings, a Weyl orbit expansion, a refined BPS expansion and an $\epsilon_1,\epsilon_2$ expansion. For higher-rank theories, we propose a gluing rule to obtain all their blowup equations based on those of rank one theories. For example, we explicitly give the elliptic blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of $-2$ curves and conformal matter theories. We also give the toric construction for many elliptic non-compact Calabi-Yau threefolds which engineer 6d $(1,0)$ SCFTs with various matter representations.

We prove power-saving bounds for general Kloosterman sums on $\operatorname{Sp}(4)$ associated to all Weyl elements via a stratification argument coupled with $p$-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of $\operatorname{Sp}(4)$ Poincare series.

Let $M^{n+1}$ be a closed manifold of dimension $3\leq n+1\leq 7$. We show that for a $C^\infty$-generic metric $g$ on $M$, to any connected, closed, embedded, $2$-sided, stable, minimal hypersurface $S\subset (M,g)$ corresponds a sequence of closed, embedded, minimal hypersurfaces $\{\Sigma_k\}$ scarring along $S$, in the sense that the area and Morse index of $\Sigma_k$ both diverge to infinity and, when properly renormalized, $\Sigma_k$ converges to $S$ as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian $3$-manifods.

Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the formulation and numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis (RB) method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters) as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach which exploits deep learning (DL) algorithms to obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. Our DL approach combines deep feedforward neural networks (NNs) and convolutional autoencoders (AEs). We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate three challenging test cases in cardiac electrophysiology and prove that DL-ROM outperforms classical projection-based ROMs.

Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP), based on a single trajectory of Markovian samples induced by a behavior policy. Focusing on a $\gamma$-discounted MDP with state space $\mathcal{S}$ and action space $\mathcal{A}$, we demonstrate that the $\ell_{\infty}$-based sample complexity of classical asynchronous Q-learning -- namely, the number of samples needed to yield an entrywise $\varepsilon$-accurate estimate of the Q-function -- is at most on the order of \begin{equation*} \frac{1}{\mu_{\mathsf{min}}(1-\gamma)^5\varepsilon^2}+ \frac{t_{\mathsf{mix}}}{\mu_{\mathsf{min}}(1-\gamma)} \end{equation*} up to some logarithmic factor, provided that a proper constant learning rate is adopted. Here, $t_{\mathsf{mix}}$ and $\mu_{\mathsf{min}}$ denote respectively the mixing time and the minimum state-action occupancy probability of the sample trajectory. The first term of this bound matches the complexity in the case with independent samples drawn from the stationary distribution of the trajectory. The second term reflects the expense taken for the empirical distribution of the Markovian trajectory to reach a steady state, which is incurred at the very beginning and becomes amortized as the algorithm runs. Encouragingly, the above bound improves upon the state-of-the-art result by a factor of at least $|\mathcal{S}||\mathcal{A}|$. Further, the scaling on the discount complexity can be improved by means of variance reduction.

In large-scale distributed storage systems, erasure codes are used to achieve fault tolerance in the face of node failures. Tuning code parameters to observed failure rates has been shown to significantly reduce storage cost. Such tuning of redundancy requires "code conversion", i.e., a change in code dimension and length on already encoded data. Convertible codes are a new class of codes designed to perform such conversions efficiently. The access cost of conversion is the number of nodes accessed during conversion.

Existing literature has characterized the access cost of conversion of linear MDS convertible codes only for a specific and small subset of parameters. In this paper, we present lower bounds on the access cost of conversion of linear MDS codes for all valid parameters. Furthermore, we show that these lower bounds are tight by presenting an explicit construction for access-optimal linear MDS convertible codes for all valid parameters. En route, we show that, one of the degrees-of-freedom in the design of convertible codes that was inconsequential in the previously studied parameter regimes, turns out to be crucial when going beyond these regimes and adds to the challenge in the analysis and code construction.

Providing high quality-of-service for live communication is a pervasive challenge which is plagued by packet losses during transmission. Streaming codes are a class of erasure codes specifically designed for such low-latency streaming communication settings. We consider the recently proposed setting of streaming codes under variable-size messages which reflects the requirements of applications such as live video streaming. In practice, streaming codes often need to operate in an "online" setting where the sizes of the future messages are unknown. Yet, previously studied upper bounds on the rate apply to "offline" coding schemes with access to all (including future) message sizes.

In this paper, we evaluate whether the optimal offline rate is a feasible goal for online streaming codes when communicating over a burst-only packet loss channel. We identify two broad parameter regimes where, perhaps surprisingly, online streaming codes can, in fact, match the optimal offline rate. For both of these settings, we present rate-optimal online code constructions. For all remaining parameter settings, we establish that it is impossible for online coding schemes to attain the optimal offline rate.

The goal of this paper is to solve a class of stochastic optimal control problems numerically, in which the state process is governed by an It\^o type stochastic differential equation with control process entering both in the drift and the diffusion, and is observed partially. The optimal control of feedback form is determined based on the available observational data. We call this type of control problems the data driven feedback control. The computational framework that we introduce to solve such type of problems aims to find the best estimate for the optimal control as a conditional expectation given the observational information. To make our method feasible in providing timely feedback to the controlled system from data, we develop an efficient stochastic optimization algorithm to implement our computational framework.

Information-theoretic formulations of the private information retrieval (PIR) problem have been investigated under a variety of scenarios. Symmetric private information retrieval (SPIR) is a variant where a user is able to privately retrieve one out of $K$ messages from $N$ non-colluding replicated databases without learning anything about the remaining $K-1$ messages. However, the goal of perfect privacy can be too taxing for certain applications. In this paper, we investigate if the information-theoretic capacity of SPIR (equivalently, the inverse of the minimum download cost) can be increased by relaxing both user and DB privacy definitions. Such relaxation is relevant in applications where privacy can be traded for communication efficiency. We introduce and investigate the Asymmetric Leaky PIR (AL-PIR) model with different privacy leakage budgets in each direction. For user privacy leakage, we bound the probability ratios between all possible realizations of DB queries by a function of a non-negative constant $\epsilon$. For DB privacy, we bound the mutual information between the undesired messages, the queries, and the answers, by a function of a non-negative constant $\delta$. We propose a general AL-PIR scheme that achieves an upper bound on the optimal download cost for arbitrary $\epsilon$ and $\delta$. We show that the optimal download cost of AL-PIR is upper-bounded as $D^{*}(\epsilon,\delta)\leq 1+\frac{1}{N-1}-\frac{\delta e^{\epsilon}}{N^{K-1}-1}$. Second, we obtain an information-theoretic lower bound on the download cost as $D^{*}(\epsilon,\delta)\geq 1+\frac{1}{Ne^{\epsilon}-1}-\frac{\delta}{(Ne^{\epsilon})^{K-1}-1}$. The gap analysis between the two bounds shows that our AL-PIR scheme is optimal when $\epsilon =0$, i.e., under perfect user privacy and it is optimal within a maximum multiplicative gap of $\frac{N-e^{-\epsilon}}{N-1}$ for any $(\epsilon,\delta)$.

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.

Consider any representation $\phi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\hat{S}(g^*)$ of its dual. Consider the tensor product of $\hat{S}(g^*)$ and the exterior algebra $\Lambda(g)$. We show that the representation $\phi$ extends canonically to the representation $\tilde\phi$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the Leibniz rule is deformed.

We show that any open orientable surface S can be properly embedded in H^2xR as an area minimizing surface.

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group.

We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group.

Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves that all maximal subgroups with socle $\mathrm{PSL}_2(p^a)$ inside these finite almost simple groups are known, with three possible exceptions ($p^a=7,8,25$ for $E_7$).

In the three remaining cases we provide considerable information about a potential maximal subgroup.

In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal $\ell$-blocks of finite groups are known for $\ell>71$.

We consider the content delivery problem in a fading multi-input single-output channel with cache-aided users. We are interested in the scalability of the equivalent content delivery rate when the number of users, $K$, is large. Analytical results show that, using coded caching and wireless multicasting, without channel state information at the transmitter (CSIT), linear scaling of the content delivery rate with respect to $K$ can be achieved in some different ways. First, if the multicast transmission spans over $L$ independent sub-channels, e.g., in quasi-static fading if $L = 1$, and in block fading or multi-carrier systems if $L>1$, linear scaling can be obtained when the product of the number of transmit antennas and the number of sub-channels scales logarithmically with $K$. Second, even with a fixed number of antennas, we can achieve the linear scaling with a threshold-based user selection requiring only one-bit feedbacks from the users. When CSIT is available, we propose a mixed strategy that combines spatial multiplexing and multicasting. Numerical results show that, by optimizing the power split between spatial multiplexing and multicasting, we can achieve a significant gain of the content delivery rate with moderate cache size.

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or linked' with each other, and they are closely associated with the linking number between links in $\mathbb{R}^3$. Because we distinguish the time-axis from spatial subspace in $\mathbb{R}^4$, we see that these equivalence relations, will also imply causality.

To a large extent, the stiffness of the bidomain and monodomain models depends on the choice of the ionic model, which varies in terms of complexity and realism. In this paper, we compare and analyze a variety of time-stepping methods: explicit or semi-implicit, operator splitting, exponential, and deferred correction methods. We compare these methods for solving the bidomain model coupled with three ionic models of varying complexity and stiffness: the phenomenological Mitchell-Schaeffer model, the more realistic Beeler-Reuter model, and the stiff and very complex ten Tuscher-Noble-Noble-Panfilov (TNNP) model. For each method, we derive absolute stability criteria of the spatially discretized monodomain model and verify that the theoretical critical time-steps obtained closely match the ones in numerical experiments. We also verify that the numerical methods achieve an optimal order of convergence on the model variables and derived quantities (such as speed of the wave, depolarization time), and this in spite of the local non-differentiability of some of the ionic models. The efficiency of the different methods is also considered by comparing computational times for similar accuracy. Conclusions are drawn on the methods to be used to solve the monodomain model based on the model stiffness and complexity, measured respectively by the eigenvalues of the model's Jacobian and the number of variables, and based on strict stability and accuracy criteria.

Let $V(k)$ be a projective variety over a number field $k\subset\mathbf{C}$ and let $\mathscr{A}_V$ be the Serre $C^*$-algebra of $V(k)$. We construct a functor $F: V(k)\mapsto \mathscr{A}_V$, such that the $\mathbf{C}$-isomorphic ($k$-isomorphic, resp.) varieties $V(k)$ map to the Morita equivalent (isomorphic, resp.) $C^*$-algebras $\mathscr{A}_V$. In other words, the isomorphisms of the algebra $\mathscr{A}_V=F(V(k))$ preserve the $k$-rational points of $V(k)$, while the Morita equivalences of $\mathscr{A}_V$ correspond to the twists of the variety $V(k)$. We apply the result to the arithmetic geometry of the rational elliptic curves.

Internet boards are platforms for online discussions about a variety of topics. On these boards, individuals may start a new thread on a specific matter, or leave comments in an existing discussion. The resulting collective process leads to the formation of discussion trees', where nodes represent a post and comments, and an edge represents a reply-to' relation. The structure of discussion trees has been analysed in previous works, but only from a static perspective. In this paper, we focus on their structural and dynamical properties by modelling their formation as a self-exciting Hawkes process. We first study a Reddit dataset to show that the structure of the trees resemble those produced by a Galton-Watson process with a special root offspring distribution. The dynamical aspect of the model is then used to predict future commenting activity and the final size of a discussion tree. We compare the efficiency of our approach with previous works and show its superiority for the prediction of the dynamics of discussions.

There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general formula for the cohomology class of a $K$-orbit closure in an arbitrary flag variety, where $K$ is a symmetric subgroup of a complex algebraic group. In type A, the terms in this formula are indexed by atoms for permutations. We study the combinatorics of atoms for involutions in the group of signed permutations. In particular, we give a compact description of the atom set for any signed involution and endow it with the structure of a graded poset. Our main result, as an application, is to identify explicitly the terms in Brion's cohomology formula in types B and C. These descriptions apply to all $K$-orbits in these types and are the first of their kind outside of type A.

We compute the effect of concordance surgery, a generalization of knot surgery defined using a self-concordance of a knot, on the Ozsv\'ath-Szab\'o 4-manifold invariant. The formula involves the graded Lefschetz number of the concordance map on knot Floer homology. The proof uses the sutured Floer TQFT, and a version of sutured Floer homology perturbed by a 2-form.

The aim of this paper is to firmly establish subjective fiducial inference as a rival to the more conventional schools of statistical inference, and to show that Fisher's intuition concerning the importance of the fiducial argument was correct. In this regard, methodology outlined in an earlier paper is modified, enhanced and extended to deal with general inferential problems in which various parameters are unknown. As part of this, the joint fiducial distribution of all the parameters of a given model is determined on the basis of the full conditional fiducial distributions of these parameters by using an analytical approach or a Gibbs sampling method, the latter of which does not require these conditional distributions to be compatible. Although the resulting theory is classified as being "subjective", the reason for this is essentially attributed to the argument that all probability statements made about fixed but unknown parameters must be inherently subjective. In particular, it is systematically argued that, in general, there is no need to place a great emphasis on the difference between the fiducial probabilities that can be derived using this theory and objective probabilities. Some important examples of the application of this theory are presented.

We contrast the dynamics of the Horava theory with anisotropic Weyl symmetry with the case when this symmetry is explicitly broken, which is called the kinetic-conformal Horava theory. To do so we perform the canonical formulation of the anisotropic conformal theory at the classical level with a general conformal potential. Both theories have the generator of the anisotropic Weyl transformations as a constraint but it changes from first to second-class when the Weyl symmetry is broken. The FDiff-covariant vector a_i = \partial_i \ln N plays the role of gauge connection for the anisotropic Weyl transformations. A Lagrange multiplier plays also the role of gauge connection, being the time component. The anisotropic conformal theory propagates the same number of degrees of freedom of the kinetic-conformal theory, which in turn are the same of General Relativity. This is due to exchange of a second-class constraint in the kinetic-conformal case by a gauge symmetry in the anisotropic conformal case. An exception occurs if the conformal potential does not depend on the lapse function N, as is the case of the so called Cotton^2 potential, in which case one of the physical modes becomes odd. We develop in detail two explicit anisotropic conformal models. One of them depends on N whereas the other one is the Cotton^2 model. We also study conformally flat solutions in the anisotropic conformal and the kinetic-conformal theories, defining as conformally flat the spatial metric, but leaving for N a form different to the one dictated by the anisotropic Weyl transformations. We find that in both theories these configurations have vanishing canonical momentum and they are critical points of the potential. In the kinetic-conformal theory we find explicitly an exact, nontrivial, conformally flat solution.

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are $\mathscr{C}^{s+1}$ for $s\in (1,\infty]$, where $\mathscr{C}^s$ denotes the Zygmund space of order $s$. We give necessary and sufficient, coordinate-free conditions for the existence of such a coordinate system. Moreover, we present a quantitative study of these coordinate charts. This is the second part in a three-part series of papers. The first part, joint with Stovall, addressed the same question, though the results were not sharp, and showed how such coordinate charts can be viewed as scaling maps in sub-Riemannian geometry. When viewed in this light, these results can be seen as strengthening and generalizing previous works on the quantitative theory of sub-Riemannian geometry, initiated by Nagel, Stein, and Wainger, and furthered by Tao and Wright, the author, and others. In the third part, we prove similar results concerning real analyticity.

This paper pays a visit to a famous contractible open 3-manifold $W^3$ proposed by R. H. Bing in 1950's. By the finiteness theorem \cite{Hak68}, Haken proved that $W^3$ can embed in no compact 3-manifold. However, until now, the question about whether $W^3$ can embed in a more general compact space such as a compact, locally connected and locally 1-connected metric 3-space was not known. Using the techniques developed in Sternfeld's 1977 PhD thesis \cite{Ste77}, we answer the above question in negative. Furthermore, it is shown that $W^3$ can be utilized to produce counterexamples for every contractible open $n$-manifold ($n\geq 4$) embeds in a compact, locally connected and locally 1-connected metric $n$-space.

Let $(S,L)$ be a Lie-Rinehart pair such that $L$ is $S$-projective and let $U$ be its universal enveloping algebra. The purpose of this paper is to present a spectral sequence which converges to the Hochschild cohomology of $U$ and whose second page involves the Lie-Rinehart cohomology of the pair and the Hochschild cohomology of $S$ with values on $U$.

We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class.

We obtain several results for (iterated) planar contact manifolds in higher dimensions: (1) Iterated planar contact manifolds are not weakly symplectically semi-fillable. This generalizes a 3-dimensional result of Etnyre to a higher-dimensional setting. (2) They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds. This generalizes a result by Albers-Bramham-Wendl. (3) They satisfy the Weinstein conjecture, i.e. every contact form admits a closed Reeb orbit. This is proved by an alternative approach as that of [Acu], and is a higher-dimensional generalization of a result of Abbas-Cieliebak-Hofer. The results follow as applications from a suitable symplectic handle attachment, which bears some independent interest.

Periodic event-triggered control (PETC) evaluates the triggering rule periodically and is well-suited for implementation on digital platforms. This paper investigates PETC design for nonlinear systems affected by external disturbances under the impulsive system formulation. Sufficient conditions are provided to ensure the input-to-state stability of the resulting closed-loop system for the state feedback and the observer-based output feedback configurations separately. For each configuration, the sampling period and the triggering functions are provided explicitly. Sufficient conditions in the form of linear matrix inequalities are provided for the PETC design of incrementally quadratic nonlinear systems. Two examples are given to illustrate the effectiveness of the proposed method.

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $\alpha >1/p>0$. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.

We define a scale of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, $p\in[1,\infty]$, that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for $p=1$. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of $\mathbb{R}^{n}$, and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about $L^{p}$-boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.

We consider a family of infinite dimensional product measures with tails between Gaussian and exponential, which we call $p$-exponential measures. We study their measure-theoretic properties and in particular their concentration. Our findings are used to develop a general contraction theory of posterior distributions on nonparametric models with $p$-exponential priors in separable Banach parameter spaces. Our approach builds on the general contraction theory for Gaussian process priors in van der Vaart and van Zanten 2008, namely we use prior concentration to verify prior mass and entropy conditions sufficient for posterior contraction. However, the specific concentration properties of $p$-exponential priors lead to a more complex entropy bound which can influence negatively the obtained rate of contraction, depending on the topology of the parameter space. Subject to the more complex entropy bound, we show that the rate of contraction depends on the position of the true parameter relative to a certain Banach space associated to $p$-exponential measures and on the small ball probabilities of these measures. For example, we apply our theory in the white noise model under Besov regularity of the truth and obtain minimax rates of contraction using (rescaled) $\alpha$-regular $p$-exponential priors. In particular, our results suggest that when interested in spatially inhomogeneous unknown functions, in terms of posterior contraction, it is preferable to use Laplace rather than Gaussian priors.

We define a (mean curvature flow) entropy for Radon measures in $\mathbb R^n$ or in a compact manifold. Moreover, we prove a monotonicity of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen-Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one dimensional standing wave), the limit measure of the parabolic Allen-Cahn equation has unit density for all future time.

Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley--Terry paired comparison model with small undirected graphs and prove that every saturated D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameters space we can prescribe a D-optimal design.

Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact, that we first dealt with in 2009 and 2010, that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one canonically associates with every compact complex manifold $X$, in every degree $k$, a holomorphic vector bundle over $\mathbb{C}$ of rank equal to the $k$-th Betti number of $X$. This vector bundle shows that the degenerating page of the Fr\"olicher spectral sequence of $X$ is the holomorphic limit, as $h\in\mathbb{C}^\star$ tends to $0$, of the $d_h$-cohomology of $X$, where $d_h=h\partial + \bar\partial$. A relative version of this vector bundle is canonically associated with every holomorphic family of compact complex manifolds. The second new ingredient is a relaxation of the notion of strongly Gauduchon (sG) metric that we introduced in 2009. For a given positive integer $r$, a Gauduchon metric $\gamma$ on an $n$-dimensional compact complex manifold $X$ is said to be $E_r$-sG if $\partial\gamma^{n-1}$ represents the zero cohomology class on the $r$-th page of the Fr\"olicher spectral sequence of $X$. Strongly Gauduchon metrics coincide with $E_1$-sG metrics.

This is a survey of the $cd$-index of Eulerian partially ordered sets. The $cd$-index is an encoding of the numbers of chains, specified by ranks, in the poset. It is the most efficient such encoding, incorporating all the affine relations on the flag numbers of Eulerian posets. Eulerian posets include the face posets of regular CW spheres (in particular, of convex polytopes), intervals in the Bruhat order on Coxeter groups, and the lattices of regions of oriented matroids. The paper discusses inequalities on the $cd$-index, connections with other combinatorial parameters, computation, and algebraic approaches.

We compute higher moments of the Siegel--Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W.~Schmidt on the variance of the Siegel transform over $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.

To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type $A$ and as quantum deformations of affine wreath algebras. We study the structure theory of these new algebras and their natural cyclotomic quotients.

This articles first investigates boundary integral operators for the three-dimensional isotropic linear elasticity of a biphasic model with piecewise constant Lam\'e coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material. In the simple case of a spherical inclusion, the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra. Further, in the case of many spherical inclusions with isotropic materials, each with its own set of Lam\'e parameters, we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases.

We consider weakly asymmetric exclusion processes whose initial density profile is a small perturbation of a constant. We show that in the diffusive time-scale, in all dimensions, the density defect evolves as the solution of a viscous Burgers equation.

We prove a theorem concerning the approximation of bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

In this paper we adopt the familiar sparse, high-dimensional linear regression model and focus on the important but often overlooked task of prediction. In particular, we consider a new empirical Bayes framework that incorporates data in the prior in two ways: one is to center the prior for the non-zero regression coefficients and the other is to provide some additional regularization. We show that, in certain settings, the asymptotic concentration of the proposed empirical Bayes posterior predictive distribution is very fast, and we establish a Bernstein--von Mises theorem which ensures that the derived empirical Bayes prediction intervals achieve the targeted frequentist coverage probability. The empirical prior has a convenient conjugate form, so posterior computations are relatively simple and fast. Finally, our numerical results demonstrate the proposed method's strong finite-sample performance in terms of prediction accuracy, uncertainty quantification, and computation time compared to existing Bayesian methods.

The standard solution concept for stochastic games is Markov perfect equilibrium (MPE); however, its computation becomes intractable as the number of players increases. Instead, we consider mean field equilibrium (MFE) that has been popularized in the recent literature. MFE takes advantage of averaging effects in models with a large number of players. We make three main contributions. First, our main result provides conditions that ensure the uniqueness of an MFE. We believe this uniqueness result is the first of its nature in the class of models we study. Second, we generalize previous MFE existence results. Third, we provide general comparative statics results. We apply our results to dynamic oligopoly models and to heterogeneous agent macroeconomic models commonly used in previous work in economics and operations.

Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive $p$-adic groups.

It has long been recognized as a difficult problem to determine whether the observed statistical correlation between two classical variables arise from causality or from common causes. Recent research has shown that in quantum theoretical framework, the mechanisms of entanglement and quantum coherence provide an advantage in tackling this problem. In some particular cases, quantum common causes and quantum causality can be effectively distinguished using observations only. However, these solutions do not apply to all cases. There still exist enormous cases in which quantum common causes and quantum causality can not be distinguished. In this paper, along the line of considering unitary transformation as causality in the quantum world, we formally show quantum common causes and quantum causality are universally separable. Based on the analysis, we further provide a general method to discriminate the two.

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient.

In this paper we develop a framework to study the dependence structure of scrambled point sets, with a focus on scrambled digital $(t,m,s)$-nets. In particular, we show that if for all $(k_1,\dots,k_s)\in\mathbb{N}^s$ the total number of distinct pairs from a given point set that come from the same elementary $(k_1,\dots,k_s)$-interval is less than what is expected with random sampling, then after scrambling the point set will be negative lower/upper orthant dependent (NLOD/NUOD). Using this result, we show that a scrambled digital $(t,m,s)$-net is NLOD/NUOD if and only if $t=0$ and that the first $n$ points of a scrambled $(0,s)$-sequence are always NLOD/NUOD. Finally, we explore the possibility of using our framework to differentiate the quality of point sets that may have the same value for the parameter $t$ but display obvious differences.

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local version of Aldous' skeleton decomposition for size-constrained Galton--Watson trees.

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ coming out of metric spaces $(M,d_0)$ in the sense that $(M,d(g(t)), x_0) \to (M,d_0, x_0)$ as $t\searrow 0$ in the pointed Gromov-Hausdorff sense. In the case that $B_{g(t)}(x_0,1) \Subset M$ for all $t\in (0,T)$ and $d_0$ is generated by a smooth Riemannian metric in distance coordinates, we show using Ricci-harmonic map heat flow, that there is a corresponding smooth solution $\tilde g(t)_{t\in (0,T)}$ to the $\delta$-Ricci-DeTurck flow on an Euclidean ball ${\mathbb B}_{r}(p_0) \subset {\mathbb R}^n$, which can be extended to a smooth solution defined for $t \in [0,T)$. We further show, that this implies that the original solution $g$ can be extended to a smooth solution on $B_{d_0}(x_0,r/2)$ for $t\in [0,T)$, in view of the method of Hamilton.

We combine computable structure theory and algorithmic learning theory to study learning of families of algebraic structures. Our main result is a model-theoretic characterization of the class $\mathbf{InfEx}_{\cong}$, consisting of the structures whose isomorphism types can be learned in the limit. We show that a family of structures $\mathfrak{K}$ is $\mathbf{InfEx}_{\cong}$-learnable if and only if the structures from $\mathfrak{K}$ can be distinguished in terms of their $\Sigma^{\mathrm{inf}}_2$-theories. We apply this characterization to familiar cases and we show the following: there is an infinite learnable family of distributive lattices; no pair of Boolean algebras is learnable; no infinite family of linear orders is learnable.

The introduction of strings into the study of the Riemann Hypothesis provides a visualization of the genesis of zeros for the Zeta function. The method is heuristic and when originally introduced suggested strong visual evidence for the truth of the Riemann Hypothesis. Some sort of organizing principle for strings with similar t values is observed and points towards a region between (1, 0) and (0, 0) on the abscissa, and within order unity along the ordinate. Progress in understanding these observations has been made by expanding the domain of sigma from the critical strip, [0, 1], to the half-line [0, infinity]. The nature of the organizing principle is explained. A generic structure for strings over the expanded domain is proffered. New perspective is gained regarding the truth of the Riemann Hypothesis.

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient:

$\min \left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} = 0 \quad\textrm{ in }\quad \Omega.$ We obtain existence of solutions and prove sharp regularity estimates along the free boundary points, namely $\partial\{u>\phi\} \cap \Omega$. In particular, for the homogeneous case ($f\equiv0$) we get that solutions are $C^{1,1}$ at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also present further features of the solutions and partial results regarding the free boundary.

These are the first results for obstacle problems driven by degenerate type operators in non-divergence form and they are a novelty even for the simpler scenario given by an operator of the form $\mathcal{G}[u] = |Du|^\gamma\Delta u$.

We introduce a presentation of the Chow ring of a matroid by a new set of generators, called "simplicial generators." These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincar\'e duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge-Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of [AHK18]. Our work avoids the use of "flips," the key technical tool employed in [AHK18].

In this paper, we find that the linearized collision operator $L$ of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on $H^k_n$, with $k,n\in\mathbb{R}$. In the theory of Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. The proof is based on pseudo-differential calculus and in general, for a specific class of Weyl quantization, the $L^2$ dissipation implies $H^k_n$ dissipation. This kind of estimate is also known as the G{\aa}rding's inequality.

In this paper, we propose a practical structured constellation for non-coherent communication with a single transmit antenna over Rayleigh flat and block fading channel without instantaneous channel state information. The constellation symbols belong to the Grassmannian of lines and are defined up to a complex scaling. The constellation is generated by partitioning the Grassmannian of lines into a collection of bent hypercubes and defining a mapping onto each of these bent hypercubes such that the resulting symbols are approximately uniformly distributed on the Grassmannian. With a reasonable choice of parameters, this so-called cube-split constellation has higher packing efficiency, represented by the minimum distance, than the existing structured constellations. Furthermore, exploiting the constellation structure, we propose low-complexity greedy symbol decoder and log-likelihood ratio computation, as well as an efficient way to associate it to a multilevel code with multistage decoding. Numerical results show that the performance of the cube-split constellation is close to that of a numerically optimized constellation and better than other structured constellations. It also outperforms a coherent pilot-based scheme in terms of error probability and achievable data rate in the regime of short coherence time and large constellation size.

We consider the non-coherent single-input multiple-output (SIMO) multiple access channel with general signaling under spatially correlated Rayleigh block fading. We propose a novel soft-output multi-user detector that computes an approximate marginal posterior of each transmitted signal using only the knowledge about the channel distribution. Our detector is based on expectation propagation (EP) approximate inference and has polynomial complexity in the number of users, the number of receive antennas, and channel coherence time. We also propose two simplifications of this detector with reduced complexity. With Grassmannian signaling, the proposed detectors outperform a state-of-the-art non-coherent detector with projection-based interference mitigation. With pilot-assisted signaling, the EP detector outperforms, in terms of symbol error rate, some conventional coherent pilot-based detectors, including a sphere decoder and a joint channel estimation-data detection scheme. Our EP-based detectors produce accurate approximates of the true posterior leading to high achievable sum-rates. The gains of these detectors are further observed in terms of the bit error rate when using their soft outputs for a turbo channel decoder.

We study a simple model of DNA evolution in a growing population of cells. Each cell contains a nucleotide sequence which randomly mutates at cell division. Cells divide according to a branching process. Following typical parameter values in bacteria and cancer cell populations, we take the mutation rate to zero and the final number of cells to infinity. We prove that almost every site (entry of the nucleotide sequence) is mutated in only a finite number of cells, and these numbers are independent across sites. However independence breaks down for the rare sites which are mutated in a positive fraction of the population. The model is free from the popular but disputed infinite sites assumption. Violations of the infinite sites assumption are widespread while their impact on mutation frequencies is negligible at the scale of population fractions. Some results are generalised to allow for cell death, selection, and site-specific mutation rates. For illustration we estimate mutation rates in a lung adenocarcinoma.

We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix . We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square prefix, thus proving a 2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.

Recently, Lai and Rohatgi proved a shuffling theorem for lozenge tilings of doubly-dented hexagon, which generalized earlier work of Ciucu. Later, Lai proved an analogous theorem for centrally symmetric tilings, which generalized some other previous work of Ciucu. In this paper, we give a unified proof of these two shuffling theorems. Our arguments do not use the graphical condensation method. Fulmek independently found a proof of Lai and Rohatgi's shuffling theorem that does not use graphical condensation. Our proof also gives a combinatorial explanation for Ciucu's recent conjecture relating the total number and the number of centrally symmetric lozenge tilings.

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $\sigma$-Gr\"obner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.

Let us denote $\lambda$ the Lebesgue measure on $[0,1]$, put$$C(\lambda)=\{f\in C([0,1]);\ \forall~A\subset [0,1], A~\text{Borel}:\ \lambda(A)=\lambda(f^{-1}(A))\}.$$ We endow the set $C(\lambda)$ by the uniform metric $\rho$ and investigate dynamical properties of typical maps in the complete metric space $(C(\lambda),\rho)$.

We investigate the performance of linear consensus algorithms subject to a scaling of the underlying network size. Specifically, we model networked systems with $n^{\text{th}}$ order integrator dynamics over families of undirected, weighted graphs with bounded nodal degrees. In such networks, the algebraic connectivity affects convergence rates, sensitivity, and, for high-order consensus ($n \ge 3$), stability properties. This connectivity scales unfavorably in network size, except in expander families, where consensus performs well regardless of network size. We show, however, that consensus over expander families is fragile to a grounding of the network (resulting in leader-follower consensus). We show that grounding may deteriorate system performance by orders of magnitude in large networks, or cause instability in high-order consensus. Our results, which we illustrate through simulations, also point to a fundamental limitation to the scalability of consensus networks with leaders, which does not apply to leaderless networks.

In this paper we show that a large class of parallel server networks, with $\sqrt{n}$ safety staffing, and no abandonment, in the Halfin-Whitt regime are exponentially ergodic and their invariant probability distributions are tight, uniformly over all stationary Markov controls. This class consists of all networks of tree topology with a single non-leaf server pool, such as the 'N' and 'M' models, as well as networks of any tree topology with class-dependent service rates.

We first define a parameter which characterizes the spare capacity (safety staffing) of the network. If the spare capacity parameter is negative, we show that the controlled diffusion is transient under any stationary Markov control, and that it cannot be positive recurrent when this parameter is zero. Then we show that the limiting diffusion is uniformly exponentially ergodic over all stationary Markov controls if this parameter is positive.

As well known, joint work conservation, that is, keeping all servers busy unless all queues are empty, cannot be always enforced in multiclass multi-pool networks, and as a result the diffusion limit and the prelimit do not "match" on the entire state space. For this reason, we introduce the concept of "system-wide work conserving policies," which are defined as policies that minimize the number of idle servers at all times. We show that, provided the spare capacity parameter is positive, the diffusion-scaled processes are geometrically ergodic and the invariant distributions are tight, uniformly over all system-wide work conserving policies. In addition, when the spare capacity is negative we show that the diffusion-scaled processes are transient under any stationary Markov control, and when it is zero, they cannot be positive recurrent. We use a unified approach in which the same Lyapunov function is used in the study of the prelimit and diffusion limit.

Motivated by the $SO(3)$ Atiyah-Floer conjecture, we consider anti-self-dual instantons on two types of noncompact four-manifolds: the product of the real line with a three-manifold having a cylindrical end and the product of the complex plane with a compact surface. We prove a Gromov-Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a type of singular objects which may have both instanton and holomorphic curve components. This result is the first step towards constructing a natural bounding cochain proposed by Fukaya for the $SO(3)$ Atiyah-Floer conjecture.

Let $n$ be a positive integer and $I$ a $k$-subset of integers in $[0,n-1]$. Given a $k$-tuple $A=(\alpha_0, \cdots, \alpha_{k-1})\in \mathbb{F}^k_{q^n}$, let $M_{A,I}$ denote the matrix $(\alpha_i^{q^j})$ with $0\leq i\leq k-1$ and $j\in I$. When $I=\{0,1,\cdots, k-1\}$, $M_{A,I}$ is called a Moore matrix which was introduced by E. H. Moore in 1896. It is well known that the determinant of a Moore matrix equals $0$ if and only if $\alpha_0,\cdots, \alpha_{k-1}$ are $\mathbb{F}_q$-linearly dependent. We call $I$ that satisfies this property a Moore exponent set. In fact, Moore exponent sets are equivalent to maximum rank-distance (MRD) code with maximum left and right idealisers over finite fields. It is already known that $I=\{0,\cdots, k-1\}$ is not the unique Moore exponent set, for instance, (generalized) Delsarte-Gabidulin codes and the MRD codes recently discovered by Csajb\'ok, Marino, Polverino and the second author both give rise to new Moore exponent sets. By using algebraic geometry approach, we obtain an asymptotic classification result: for $q>5$, if $I$ is not an arithmetic progression, then there exist an integer $N$ depending on $I$ such that $I$ is not a Moore exponent set provided that $n>N$.

We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-H\"ubsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism $s$ of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-H\"ubsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.

In much of the literature on function approximation by deep networks, the function is assumed to be defined on some known domain, such as a cube or a sphere. In practice, the data might not be dense on these domains, and therefore, the approximation theory results are observed to be too conservative. In manifold learning, one assumes instead that the data is sampled from an unknown manifold; i.e., the manifold is defined by the data itself. Function approximation on this unknown manifold is then a two stage procedure: first, one approximates the Laplace-Beltrami operator (and its eigen-decomposition) on this manifold using a graph Laplacian, and next, approximates the target function using the eigen-functions. Alternatively, one estimates first some atlas on the manifold and then uses local approximation techniques based on the local coordinate charts.

In this paper, we propose a more direct approach to function approximation on \emph{unknown}, data defined manifolds without computing the eigen-decomposition of some operator or an atlas for the manifold, and estimate the degree of approximation. Our constructions are universal; i.e., do not require the knowledge of any prior on the target function other than continuity on the manifold. For smooth functions, the estimates do not suffer from the so-called saturation phenomenon. We demonstrate via a property called good propagation of errors how the results can be lifted for function approximation using deep networks where each channel evaluates a Gaussian network on a possibly unknown manifold.

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient at previous step. This methodology extends to the fully nonlinear case the approach recently proposed in \cite{HPW19} for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient, Monge-Amp{\e}re equation and Hamilton-Jacobi-Bellman equation in portfolio optimization.

The harmonic sections of the Kaluza-Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combines the Kaluza-Klein model with the Yang-Mills action and a Dirac action for twisted spinors. In dimension two we show that weak solutions of the Euler-Lagrange system are smooth. For a sequence of approximate solutions on surfaces with uniformly bounded energies we obtain compactness modulo bubbles, namely, energy identities and the no-neck property hold.

Estimations and applications of factor models often rely on the crucial condition that the number of latent factors is consistently estimated, which in turn also requires that factors be relatively strong, data are stationary and weak serial dependence, and the sample size be fairly large, although in practical applications, one or several of these conditions may fail. In these cases it is difficult to analyze the eigenvectors of the data matrix. To address this issue, we propose simple estimators of the latent factors using cross-sectional projections of the panel data, by weighted averages with pre-determined weights. These weights are chosen to diversify away the idiosyncratic components, resulting in "diversified factors". Because the projections are conducted cross-sectionally, they are robust to serial conditions, easy to analyze and work even for finite length of time series. We formally prove that this procedure is robust to over-estimating the number of factors, and illustrate it in several applications, including post-selection inference, big data forecasts, large covariance estimation and factor specification tests. We also recommend several choices for the diversified weights.

We study correlations of multiplicative functions taken along deterministic sequences and sequences that satisfy certain linear independence assumptions. The results obtained extend recent results of Tao and Ter\"av\"ainen and results of the author. Our approach is to use tools from ergodic theory in order to effectively exploit feedback from analytic number theory. The results on deterministic sequences crucially use structural properties of measure preserving systems associated with bounded multiplicative functions that were recently obtained by the author and Host. The results on independent sequences depend on multiple ergodic theorems obtained using the theory of characteristic factors and qualitative equidistribution results on nilmanifolds.

In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [26]. We propose a JKO type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.

Let $(A,\sigma)$ be a central simple algebra with an orthogonal involution. It is well-known that $O(A,\sigma)$ contains elements of reduced norm $-1$ if and only if the Brauer class of $A$ is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary.

Let $f\in L^p(\mathbb{R}^d)$, $d\ge 3$, and let $A_t f(x)$ the average of $f$ over the sphere with radius $t$ centered at $x$. For a subset $E$ of $[1,2]$ we prove close to sharp $L^p\to L^q$ estimates for the maximal function $\sup_{t\in E} |A_t f|$. A new feature is the dependence of the results on both the upper Minkowski dimension of $E$ and the Assouad dimension of $E$. The result can be applied to prove sparse domination bounds for a related global spherical maximal function.

We characterize measure spaces such that the canonical map $L_\infty \to L_1^*$ is surjective. In case of $d$ dimensional Hausdorff measure of a complete separable metric space $X$ we give two equivalent conditions. One is in terms of the order completeness of a quotient Boolean algebra associated with measurable sets and with locally null sets. Another one is in terms of the possibility to decompose space in a certain way into sets of nonzero finite measure. We give examples of $X$ and $d$ so that whether these conditions are met is undecidable in ZFC, including one with $d$ equals the Hausdorff dimension of $X$.

This paper introduces a new nonlinear observer for state estimation of linear time invariant systems. The proposed observer contains a (nonlinear) cubic term in its error dynamics.

"For the final version of this article, please refer to the published version in Int. Jr. of Adaptive Control and Signal Processing. The differences between the published and arxiv version are substantial and significant"

This paper presents for the first time, to our knowledge, a framework for verifying neural network behavior in power system applications. Up to this moment, neural networks have been applied in power systems as a black-box; this has presented a major barrier for their adoption in practice. Developing a rigorous framework based on mixed integer linear programming, our methods can determine the range of inputs that neural networks classify as safe or unsafe, and are able to systematically identify adversarial examples. Such methods have the potential to build the missing trust of power system operators on neural networks, and unlock a series of new applications in power systems. This paper presents the framework, methods to assess and improve neural network robustness in power systems, and addresses concerns related to scalability and accuracy. We demonstrate our methods on the IEEE 9-bus, 14-bus, and 162-bus systems, treating both N-1 security and small-signal stability.

We investigate and prove Lieb-Oxford bounds in one dimension by studying convex potentials that approximate the ill-defined Coulomb potential. A Lieb-Oxford inequality establishes a bound of the indirect interaction energy for electrons in terms of the one-body particle density $\rho_\psi$ of a wave function $\psi$. Our results include modified soft Coulomb potential and regularized Coulomb potential. For these potentials, we establish Lieb-Oxford-type bounds utilizing logarithmic expressions of the particle density. Furthermore, a previous conjectured form $I_\mathrm{xc}(\psi)\geq - C_1 \int_{\mathbb R} \rho_\psi(x)^{2} \mathrm{d}x$ is discussed for different convex potentials.

Let $v$ be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in abelian group $G$ of order $v$, except possibly when $G=C_p^3$ and $p$ is a prime greater than $3 \times 10^{12}$.

We study the vertex classification problem on a graph whose vertices are in $k\ (k\geq 2)$ different communities, edges are only allowed between distinct communities, and the number of vertices in different communities are not necessarily equal. The observation is a weighted adjacency matrix, perturbed by a scalar multiple of the Gaussian Orthogonal Ensemble (GOE), or Gaussian Unitary Ensemble (GUE) matrix. For the exact recovery of the maximum likelihood estimation (MLE) with various weighted adjacency matrices, we prove sharp thresholds of the intensity $\sigma$ of the Gaussian perturbation. These weighted adjacency matrices may be considered as natural models for the electric network. Surprisingly, these thresholds of $\sigma$ do not depend on whether the sample space for MLE is restricted to such classifications that the number of vertices in each group is equal to the true value. In contrast to the $\ZZ_2$-synchronization, a new complex version of the semi-definite programming (SDP) is designed to efficiently implement the community detection problem when the number of communities $k$ is greater than 2, and a common region (independent of $k$) for $\sigma$ such that SDP exactly recovers the true classification is obtained.

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$ and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For $d \geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time $n$ will be localized in a ball of volume asymptotically $d\log_{1/p}n$. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest.

Bayesian optimization provides sample-efficient global optimization for a broad range of applications, including automatic machine learning, engineering, physics, and experimental design. We introduce BoTorch, a modern programming framework for Bayesian optimization that combines Monte-Carlo (MC) acquisition functions, a novel sample average approximation optimization approach, auto-differentiation, and variance reduction techniques. BoTorch's modular design facilitates flexible specification and optimization of probabilistic models written in PyTorch, simplifying implementation of new acquisition functions. Our approach is backed by novel theoretical convergence results and made practical by a distinctive algorithmic foundation that leverages fast predictive distributions, hardware acceleration, and deterministic optimization. In experiments, we demonstrate the improved sample efficiency of BoTorch relative to other popular libraries.

Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of hypergeometric E-functions with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively "small" ring H generated by algebraic numbers, $1/\pi$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove (in two steps) that the coefficients of the asymptotic expansion of an hypergeometric E-function with rational parameters are in H. Finally, we prove a similar result for G-functions.

We define a discrete version of the bilinear spherical maximal function, and show bilinear $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ bounds for $d \geq 3$, $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$, $r>\frac{d}{d-2}$ and $p,q\geq 1$. Due to interpolation, the key estimate is an $l^{p}(\mathbb{Z}^d)\times l^{\infty}(\mathbb{Z}^d) \to l^{p}(\mathbb{Z}^d)$ bound, which holds when $d \geq 3$, $p>\frac{d}{d-2}$. A key feature of our argument is the use of the circle method which allows us to decouple the dimension from the number of functions compared to the work of Cook.

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions $d \geq 5$. That is, we show that this operator is bounded on $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$ for $\frac{1}{p} + \frac{1}{q} \geq \frac{1}{r}$ and $r>\frac{d}{2d-2}$ and we show this range is sharp. Our approach mirrors that used by Jeong and Lee in the continuous setting. For dimensions $d=3,4$, our previous work, which used different techniques, still gives the best known bounds. We also prove analogous results for higher degree $k$, $\ell$-linear operators.

We estimate the displacement energy of Lagrangian 3-spheres in a symplectic 6-manifold $X$, by estimating the displacement energy of a one-parameter family $L_{\lambda}$ of Lagrangian tori near the sphere. The proof establishes a new version of Lagrangian Floer theory with cylinder corrections, which is motivated by the change of open Gromov-Witten invariants under the conifold transition. We also make observations and computations on the classical Floer theory by using symplectic sum formula and Welschinger invariants.

When the genus $g$ is even, we extend the computation of the cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure.

In the appendix, we use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic.

A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$ and corresponding data on an associated $q$-monodromy surface. We also construct the moduli space of $q\textrm{P}_{\textrm{IV}}$ explicitly.

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions.

We study three different topologies on the moduli space $\mathscr{H}^{\rm loc}_m$ of equivariant isometry classes of $m$-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed $\mathcal{C}^{k,\alpha}$-topology, for some $k>1$, which do not admit any convergent subsequence in the pointed $\mathcal{C}^{k+1}$-topology.

The properties of gradient techniques for the phase retrieval problem have received a considerable attention in recent years. In almost all applications, however, the phase retrieval problem is solved using a family of algorithms that can be interpreted as variants of Douglas-Rachford splitting. In this work, we establish a connection between Douglas-Rachford and gradient algorithms. Specifically, we show that in some cases a generalization of Douglas-Rachford, called relaxed-reflect-reflect (RRR), can be viewed as gradient descent on a certain objective function. The solutions coincide with the critical points of that objective, which---in contrast to standard gradient techniques---are not its minimizers. Using the objective function, we give simple proofs of some basic properties of the RRR algorithm. Specifically, we describe its set of solutions, show a local convexity around any solution, and derive stability guarantees. Nevertheless, in its present state, the analysis does not elucidate the remarkable empirical performance of RRR and its global properties.

This paper analyzes a nonlinear opinion dynamics model which generalizes the DeGroot model by introducing a bias parameter for each individual. The original DeGroot model is recovered when the bias parameter is equal to zero. The magnitude of this parameter reflects an individual's degree of bias when assimilating new opinions, and depending on the magnitude, an individual is said to have weak, intermediate, and strong bias. The opinions of the individuals lie between 0 and 1. It is shown that for strongly connected networks, the equilibria with all elements equal identically to the extreme value 0 or 1 is locally exponentially stable, while the equilibrium with all elements equal to the neutral consensus value of 1/2 is unstable. Regions of attraction for the extreme consensus equilibria are given. For the equilibrium consisting of both extreme values 0 and 1, which corresponds to opinion polarization according to the model, it is shown that the equilibrium is unstable for all strongly connected networks if individuals all have weak bias, becomes locally exponentially stable for complete and two-island networks if individuals all have strong bias, and its stability heavily depends on the network topology when individuals have intermediate bias. Analysis on star graphs and simulations show that additional equilibria may exist where individuals form clusters.

In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq \mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus \{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of $\mathbb{R}^d$, $\mathbb{Z}^d$, and in $\mathbb{Q}$ under some restrictions on the colouring.

Among other results, we characterise those $(S,s_0)$ with $S\subseteq \mathbb{Z}$ for which every finite colouring of $\mathbb{R}$ without an infinite monochromatic arithmetic progression contains an almost monochromatic similar copy of $(S,s_0)$. We also show that if $S\subseteq \mathbb{Z}^d$ and $s_0$ is outside of the convex hull of $S\setminus \{s_0\}$, then every finite colouring of $\mathbb{R}^d$ without a similar monochromatic copy of $\mathbb{Z}^d$ contains an almost monochromatic similar copy of $(S,s_0)$.

Further, we propose an approach of finding almost-monochromatic sets that might lead to a non-computer assisted proof of $\chi(\R^2)\geq 5$.

We consider the following 2-D Schr\"{o}dinger-Newton equation \begin{eqnarray*} \begin{cases} -\Delta u+u=w|u|^{p-1}u \\ -\Delta w=2 \pi |u|^p \end{cases}\text{in} \; \mathbb{R}^2 \end{eqnarray*} for $p \geq 2$. Using variational method with the Cerami compactness property, we prove the existence of minimal action odd solutions. Also by carefully applying the method of moving plane to a similar but more complex equation on the upper half space, we prove these solutions are in fact axially symmetric. Our results partially can be seen as the counterpart of results in paper \cite{GS} for the 2-D case, or the extension of the results \cite{CW} to the odd solution case.

We extend the inflationary fixed-point logic, IFP, with a new kind of second-order quantifiers which have (poly-)logarithmic bounds. We prove that on ordered structures the new logic $\exists^{\log^{\omega}}\text{IFP}$ captures the limited nondeterminism class $\beta\text{P}$. In order to study its expressive power, we also design a new version of Ehrenfeucht-Fra\"iss\'e game for this logic and show that our capturing result will not hold on the general case, i.e. on all the finite structures.

Stochastic dual dynamic programming is a cutting plane type algorithm for multi-stage stochastic optimization originated about 30 years ago. In spite of its popularity in practice, there does not exist any analysis on the convergence rates of this method. In this paper, we first establish the number of iterations, i.e., iteration complexity, required by a basic dynamic cutting plane method for solving relatively simple multi-stage optimization problems, by introducing novel mathematical tools including the saturation of search points. We then refine these basic tools and establish the iteration complexity for both deterministic and stochastic dual dynamic programming methods for solving more general multi-stage stochastic optimization problems under the standard stage-wise independence assumption. Our results indicate that the complexity of these methods mildly increases with the number of stages $T$, in fact linearly dependent on $T$ for discounted problems. Therefore, they are efficient for strategic decision making which involves a large number of stages, but with a relatively small number of decision variables in each stage. Without explicitly discretizing the state and action spaces, these methods might also be pertinent to the related reinforcement learning and stochastic control areas.

For $1<p<\infty$ we prove an $L^p$-version of the generalized Korn inequality for incompatible tensor fields $P$ in $W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$. More precisely, let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that \begin{equation*} \| P\|_{L^p(\Omega,\mathbb{R}^{3\times3})}\leq c\,\left( \|\operatorname{sym} P\|_{L^p(\Omega,\mathbb{R}^{3\times3})} + \| \operatorname{Curl}P \|_{L^p(\Omega, \mathbb{R}^{3\times3})}\right)\end{equation*} holds for all tensor fields $P\in W^{1,\,p}_0(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$, i.e., for all $P\in W^{1,\,p}(\operatorname{Curl}; \Omega,\mathbb{R}^{3\times3})$ with vanishing tangential trace $P\times \nu=0$ on $\partial\Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial\Omega$. For compatible $P=D u$ this recovers an $L^p$-version of the classical Korn's first inequality $$\|D u \|_{L^p(\Omega,\mathbb{R}^{3\times 3})} \le c\, \|\operatorname{sym}D u\|_{L^p(\Omega,\mathbb{R}^{3\times3})} \quad \text{with }D u \times \nu = 0 \quad \text{on \partial \Omega},$$ and for skew-symmetric $P=A\in\mathfrak{so}(3)$ an $L^p$-version of the Poincar\'{e} inequality $$\|A\|_{L^p(\Omega,\mathfrak{so}(3))}\le c\, \|\operatorname{Curl} A\|_{L^p(\Omega,\mathbb{R}^{3\times3})} \quad \text{with } A \times \nu = 0 \ \Leftrightarrow \ A=0 \quad \text{on \partial \Omega}.$$

Lucas polynomials are polynomials in $s_1$ and $s_2$ defined recursively by $\{0\}=0$, $\{1\}=1$, and $\{m\}=s_1\{m-1\}+s_2\{m-2\}$ for $m \geq 2$. We generalize Lucas polynomials from 2-variable polynomials to multivariable polynomials. This is done by first defining $r$-Lucas polynomials $\{m\}_r$ in the variables $s_1$, $s_r$, and $s_{2r}$. We show that the binomial analogues of the $r$-Lucas polynomials are polynomial and give a combinatorial interpretation for them. We then extend the generalization of Lucas polynomials to an arbitrarily large set of variables. Recursively defined generating functions are given for these multivariable Lucas polynomials. We conclude by giving additional applications and insights.

We examine the first order structure of pregeometries of structures built via Hrushovski constructions. In particular, we show that the class of flat pregeometries is an amalgamation class such that the pregeometry of the unbounded arity Hrushovski construction is precisely its generic. We show that the generic is saturated, provide an axiomatization for its theory, show that the theory is $\omega$-stable, and has quantifier-elimination down to boolean combinations of $\exists\forall$-formulas. We show that the pregeometries of the bounded-arity Hrushovski constructions satisfy the same theory, and that they in fact form an elementary chain.

Polling systems have been widely studied, however most of these studies focus on polling systems with renewal processes for arrivals and random variables for service times. There is a need driven by practical applications to study polling systems with arbitrary arrivals (not restricted to time-varying or in batches) and revealed service time upon a job's arrival. To address that need, our work considers a polling system with generic setting and for the first time provides the worst-case analysis for online scheduling policies in this system. We provide conditions for the existence of constant competitive ratios, and competitive lower bounds for general scheduling policies in polling systems. Our work also bridges the queueing and scheduling communities by proving the competitive ratios for several well-studied policies in the queueing literature, such as cyclic policies with exhaustive, gated or l-limited service disciplines for polling systems.

We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation.

For a real K3-surface $X$, one can introduce areas of connected components of the real point set $\mathbb{R} X$ of $X$ using a holomorphic symplectic form of $X$. These areas are defined up to simultaneous multiplication by a positive real number, so the areas of different components can be compared. In particular, it turns out that the area of a non-spherical component of $\mathbb{R} X$ is always greater than the area of any spherical component.

In this paper we explore further comparative restrictions on the area for real K3-surfaces admitting a suitable polarization of degree $2g - 2$ (where $g$ is a positive integer) and such that $\mathbb{R} X$ has one non-spherical component and at least $g$ spherical components. For this purpose we introduce and study the notion of simple Harnack curves in real K3-surfaces, generalizing planar simple Harnack curves.

Let $\hat{A}_n$ be the completion by the degree of a differential operator of the $n$-th Weyl algebra with generators $x_1,\ldots,x_n,\partial^1,\ldots,\partial^n$. Consider $n$ elements $X_1,\ldots,X_n$ in $\hat{A}_n$ of the form

$$X_i = x_i + \sum_{K = 1}^\infty \sum_{l = 1}^n\sum_{j = 1}^n x_l p_{ij}^{K-1,l}(\partial)\partial^j,$$ where $p^{K-1,l}_{ij}(\partial)$ is a degree $(K-1)$ homogeneous polynomial in $\partial^1,\ldots,\partial^n$, antisymmetric in subscripts $i,j$. Then for any natural $k$ and any function $i : \{1,\ldots,k\}\to\{1,\ldots,n\}$ we prove

$$\sum_{\sigma \in \Sigma(k)} X_{i_{\sigma(1)}}\cdots X_{i_{\sigma(k)}}\triangleright 1 = k! \,x_{i_1}\cdots x_{i_k},$$ where $\Sigma(k)$ is the symmetric group on $k$ letters and $\triangleright$ denotes the Fock action of the $\hat{A}_n$ on the space of (commutative) polynomials.

This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a "broken exponential martingale" expansion that generalizes and unifies the exponentiation result of Al{\o}s, Gatheral, and Radoi\v{c}i\'c and the cumulant recursion formula of Lacoin, Rhodes, and Vargas. Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula, as required by many of our examples, including iterated stochastic integrals, L\'evy area, Bessel processes, KPZ with smooth noise, Wiener-It\^o chaos and "rough" stochastic (forward) variance models.

Convolutional neural networks (CNNs) often perform well, but their stability is poorly understood. To address this problem, we consider the simple prototypical problem of signal denoising, where classical approaches such as nonlinear diffusion, wavelet-based methods and regularisation offer provable stability guarantees. To transfer such guarantees to CNNs, we interpret numerical approximations of these classical methods as a specific residual network (ResNet) architecture. This leads to a dictionary which allows to translate diffusivities, shrinkage functions, and regularisers into activation functions, and enables a direct communication between the four research communities. On the CNN side, it does not only inspire new families of nonmonotone activation functions, but also introduces intrinsically stable architectures for an arbitrary number of layers.

Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic $\chi$ of metric graphs to the spectrum of their standard Laplacian. These ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured.

In the present work we analyse sufficient hypotheses which guarantee the successful recovery of $\chi$. We also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required. Finally, we compare our findings with numerical examples---surprisingly, just a few dozens of eigenvalues can be enough.

One can use the number theoretic idea of the notion of natural density \cite{B1} to define topological s-torsion elements (which form the statistically characterized subgroups, recently developed in \cite{DPK}) extending Armacost's idea of topological torsion elements. We follow in the line of Armacost who had posed the famous classical problem for "description of topological torsion elements" of the circle group. In this note we consider the natural density version of Armacost's problem and present a complete description of topological s-torsion elements in terms of the support, for all arithmetic sequences which also provides the solution of Problem 6.10 posed in \cite{DPK} .

This paper considers a compressed-coding scheme that combines compressed sensing with forward error control coding. Approximate message passing (AMP) is used to decode the message. Based on the state evolution analysis of AMP, we derive the performance limit of compressed-coding. We show that compressed-coding can approach Gaussian capacity at a very low compression ratio. Further, the results are extended to systems involving non-linear effects such as clipping. We show that the capacity approaching property can still be maintained when generalized AMP is used to decode the message.

To approach the capacity, a low-rate underlying code should be designed according to the curve matching principle, which is complicated in practice. Instead, analog spatial-coupling is used to avoid sophisticated low-rate code design. In the end, we study the coupled scheme in a multiuser environment, where spatial-coupling can be realized in a distributive way. The overall block length can be shared by many users, which reduces block length per-user.

We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.

We present Asynchronous Stochastic Parallel Pose Graph Optimization (ASAPP), the first asynchronous algorithm for distributed pose graph optimization (PGO) in multi-robot simultaneous localization and mapping. By enabling robots to optimize their local trajectory estimates without synchronization, ASAPP offers resiliency against communication delays and alleviates the need to wait for stragglers in the network. Furthermore, ASAPP can be applied on the rank-restricted relaxations of PGO, a crucial class of non-convex Riemannian optimization problems that underlies recent breakthroughs on globally optimal PGO. Under bounded delay, we establish the global first-order convergence of ASAPP using a sufficiently small stepsize. The derived stepsize depends on the worst-case delay and inherent problem sparsity, and furthermore matches known result for synchronous algorithms when there is no delay. Numerical evaluations on simulated and real-world datasets demonstrate favorable performance compared to state-of-the-art synchronous approach, and show ASAPP's resilience against a wide range of delays in practice.

Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.

In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. The dynamics is governed by an assumption of a Lagrangian formulation, with the velocity time derivatives as the momenta conjugate to the velocity configurations. From these definitions and assumptions, we show mathematically that a maximum entropy production principle selects the physical measure as a solution of the fluid Navier-Stokes and Euler equations from among alternate measures on the configuration space. The physical solution is shown to be a solution of the Navier-Stokes or Euler equations in the Lagrangian frame, but its transformation to an Eulerian frame is not established, and depends on the regularity of the solution, a property which is known to be difficult to establish.

Which quantum states can be reached by controlling open Markovian $n$-level quantum systems? Here, we address reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath of temperature $T$. The core problem reduces to a toy model of studying points in the standard simplex allowing for two types of controls: (i) permutations within the simplex, (ii) contractions by a dissipative semigroup. By illustration, we put the problem into context and show how toy-model solutions pertain to the reachable set of the original controlled Markovian quantum system. Beyond the case $T=0$ (amplitude damping) we present new results for $0 <T < \infty$ using methods of $d$-majorisation.

In this note, we give a simple, counting based proof of Fisher's Inequality that does not use any tools from linear algebra.

In this note, we mainly show the analogue of one of Alladi's formulas over $\mathbb{Q}$ with respect to the Dirichlet convolutions involving the M\"{o}bius function $\mu(n)$, which is related to the natural densities of sets of primes by recent work of Dawsey, Sweeting and Woo, and Kural et al. This would give us several new analogues. In particular, we get that if $(k, \ell)=1$, then $$-\sum_{\begin{smallmatrix}n\geq 2\\ p(n)\equiv \ell (\operatorname{mod} k) \end{smallmatrix}} \frac{\mu(n)}{\varphi(n)} = \frac1{\varphi(k)},$$ where $p(n)$ is the smallest prime divisor of $n$, and $\varphi(n)$ is Euler's totient function. This refines one of Hardy's formulas in 1921. At the end, we give some examples for the $\varphi(n)$ replaced by functions "near $n$", which include the sum-of-divisors function.

Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau, and realize its cluster category as a triangulated hull of an orbit category of a derived category, and as the singularity category of a finite dimensional Iwanaga-Gorenstein algebra. Along the way, we give two results which stand on their own. First, we show that the derived category of coherent sheaves over a Calabi-Yau algebra has a natural cluster tilting subcategory whose dimension is determined by the Calabi-Yau dimension and the $a$-invariant of the algebra. Secondly, we prove that two DG orbit categories obtained from a DG endofunctor and its homotopy inverse are quasi-equivalent. As an application, we show that the higher cluster category of a higher representation infinite algebra is triangle equivalent to the singularity category of an Iwanaga-Gorenstein algebra which is explicitly described. Also, we demonstrate that our results generalize the context of Keller--Murfet--Van den Bergh on the derived orbit category involving a square root of the AR translation.

Taking inspiration from a recent paper by Bergounioux, Leaci, Nardi and Tomarelli we study the Riemann-Liouville fractional Sobolev space $W^{s, p}_{RL, a+}(I)$, for $I = (a, b)$ for some $a, b \in \mathbb{R}, a < b$, $s \in (0, 1)$ and $p \in [1, \infty]$; that is, the space of functions $u \in L^{p}(I)$ such that the left Riemann-Liouville $(1 - s)$-fractional integral $I_{a+}^{1 - s}[u]$ belongs to $W^{1, p}(I)$. We prove that the space of functions of bounded variation and the fractional Sobolev space, $BV(I)$ and $W^{s, 1}(I)$, continuously embed into $W^{s, 1}_{RL, a+}(I)$. In addition, we define the space of functions with left Riemann-Liouville $s$-fractional bounded variation, $BV^{s}_{RL,a+}(I)$, as the set of functions $u \in L^{1}(I)$ such that $I^{1 - s}_{a+}[u] \in BV(I)$, and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.

Surjective Constraint Satisfaction Problem (SCSP) is the problem of deciding whether there exists a surjective assignment to a set of variables subject to some specified constraints. In this paper we show that one of the most popular variants of the SCSP, called No-Rainbow Problem, is NP-Hard. Additionally, we disprove the conjecture saying that SCSP over a constraint language $\Gamma$ is equivalent to CSP over the same language with constants. Our counter example also shows that the complexity of SCSP cannot be described in terms of polymorphisms of the constraint language.

In the study of dynamical systems on networks/graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that instead of microscopic details of the individual nodes/vertices rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As for instance in the theory of coupled map lattices, we study Laplace type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of new dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.

We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.

Dimer models have been the focus of intense research efforts over the last years. This paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal (simply or multiply connected) domains much studied in numerical simulations and elsewhere. Our classification of the geometries of frozen boundaries can be seen as geometric universality result for all dimer models. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realised already by the lozenge model. To achieve this we present a new boundary regularity study for a class of Monge-Amp\`ere equations in non-strictly convex domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of gas domains for minimizers. Our results are related to the seminal paper "Limit shapes and the complex Burgers" equation where R. Kenyon and A. Okounkov studied the asymptotic height function in the special class of lozenge tilings and domains. Part of the motivation for development of the new methods in this paper is that it seems difficult to extend those methods to cover more general dimer models, in particular domino tilings, as we do in the present paper. Indeed, our methods prove new and sharper results already for the lozenge model.

In the first part of this paper, we prove a theorem which is the $q$-analogue of a generalized modular Ray-Chaudhuri-Wilson Theorem shown in [Alon, Babai, Suzuki, J. Combin. Theory Series A, 1991]. It is also a generalization of the main theorem in [Frankl and Graham, European J. Combin. 1985] under certain circumstances.

In the second part of this paper, we prove $q$-analogues of results on a recent notion called \emph{fractional $L$-intersecting family} for families of subspaces of a given vector space. We use the above theorem to obtain a general upper bound to the cardinality of such families. We give an improvement to this general upper bound in certain special cases.

Given two planar, conformal, smooth open sets $\Omega$ and $\omega$, we prove the existence of a sequence of smooth sets $\Omega_n$ which geometrically converges to $\Omega$ and such that the (perimeter normalized) Steklov eigenvalues of $\Omega_n$ converge to the ones of $\omega$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.

The proper initialization of weights is crucial for the effective training and fast convergence of deep neural networks (DNNs). Prior work in this area has mostly focused on balancing the variance among weights per layer to maintain stability of (i) the input data propagated forwards through the network and (ii) the loss gradients propagated backwards, respectively. This prevalent heuristic is however agnostic of dependencies among gradients across the various layers and captures only firstorder effects. In this paper, we propose and discuss an initialization principle that is based on a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. The proposed approach is more systematic and recovers previous results for DNN activations such as smooth functions, dropouts, and ReLU. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool which helps to more rigorously initialize weights

This article concerns second-order time discretization of subdiffusion equations with time-dependent diffusion coefficients. High-order differentiability and regularity estimates are established for subdiffusion equations with time-dependent coefficients. Using these regularity results and a perturbation argument of freezing the diffusion coefficient, we prove that the convolution quadrature generated by the second-order backward differentiation formula, with proper correction at the first time step, can achieve second-order convergence for both nonsmooth initial data and incompatible source term. Numerical experiments are consistent with the theoretical results.

In this paper, we investigate the ruin probabilities of non-homogeneous risk models. By employing martingale method, the Lundberg-type inequalities of ruin probabilities of non-homogeneous renewal risk models are obtained under weak assumptions. In addition, for the periodic and quasi-periodic risk models the adjustment coefficients of the Lundberg-type inequalities are obtained. Finally, examples are presented to show that estimations obtained in this paper are more accurate and the ruin probability in non-homogeneous risk models may be fast decreasing which is impossible for the case of homogeneity.

We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-K\"ahler metric with K\"ahler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system.

Model-Free Control (MFC), which is easy to implement both from software and hardware viewpoints, permits the introduction of a high level control synthesis for the Industrial Internet of Things (IIoT) and the Industry 4.0. The choice of the User Diagram Protocol (UDP) as the Internet Protocol permits to neglect the latency. In spite of most severe packet losses, convincing computer simulations and laboratory experiments show that MFC exhibits a good Quality of Service (QoS) and behaves better than a classic PI regulator.

The restricted planar four body problem describes the motion of a massless body under the Newtonian gravitational force of other three bodies (the primaries), of which the motion gives us general solutions of the three body problem.

A trajectory is called {\it oscillatory} if it goes arbitrarily faraway but returns infinitely many times to the same bounded region. We prove the existence of such type of trajectories provided the primaries evolve in suitable periodic orbits.

We study the $q$-analogue of the Haldane-Shastry model, a partially isotropic (XXZ-like) long-range spin chain that enjoys quantum-affine (really: quantum-loop) symmetries at finite size.

We derive the pairwise form of the Hamiltonian, found by one of us building on work of Uglov, via 'freezing' from the affine Hecke algebra. We obtain explicit expressions for the spin-analogue of Macdonald operators. Through freezing these yield the higher Hamiltonians of the spin chain, including a Hamiltonian of the opposite chirality. The sum of the two chiral Hamiltonians has a real spectrum also for $q$ a root of unity.

We clarify the relation between patterns labelling the eigenspaces, known as 'motifs', and the corresponding degeneracies in the crystal limit $q\to\infty$. For each motif we obtain an explicit expression for the exact eigenvector, valid for generic $q$, that has ('pseudo' or 'l-') highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector of $A$ and $D$ and annihilated by $C$. It has a simple component featuring the 'symmetric square' of the $q$-Vandermonde times a Macdonald polynomial - or more precisely its quantum spherical zonal special case. Its other components are obtained through the action of the Hecke algebra, followed by 'evaluation' of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the spectrum is complete.

The model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the $Y$-operators of the affine Hecke algebra. The key step in our diagonalisation is that on a subspace of suitable polynomials the first $M$ 'classical' (i.e. no difference part) $Y$-operators in $N$ variables reduce, upon evaluation as above, to $Y$-operators in $M$ variables with parameters at the quantum zonal spherical point.

In this paper, we extend techniques developed in the context of the Travelling Salesperson Problem for cycle problems. Particularly, we study the shrinking of support graphs and the exact algorithms for subcycle elimination separation problems. The efficient application of the considered techniques has proved to be essential in the Travelling Salesperson Problem when solving large size problems by Branch-and-Cut, and this has been the motivation behind this work. Regarding the shrinking of support graphs, we prove the validity of the Padberg-Rinaldi general shrinking rules and the Crowder-Padberg subcycle-safe shrinking rules. Concerning the subcycle separation problems, we extend two exact separation algorithms, the Dynamic Hong and the Extended Padberg-Gr\"otschel algorithms, which are shown to be superior to the ones used so far in the literature of cycle problems.

The proposed techniques are empirically tested in 24 subcycle elimination problem instances generated by solving the Orienteering Problem (involving up to 15112 vertices) with Branch-and-Cut. The experiments suggest the relevance of the proposed techniques for cycle problems. The obtained average speedup for the subcycle separation problems in the Orienteering Problem when the proposed techniques are used together is around 50 times in medium-sized instances and around 250 times in large-sized instances.

The understanding of probability can be difficult for a few young scientists. Consequently, this new mathematical symbol, related to binomial coefficients and simplicial polytopic numbers, could be helpful to science education. Moreover, one can obtain kinds of remarkable identities and generalize them to a sort of "Newton's binomial theorem". Finally, this symbol could be perhaps useful to other scientific subjects as well, such as computer science.

Symmetrizers for hyperbolic equations are obtained by diagonalizing the Bezoutian matrix of hyperbolic symbols. Such diagonal symmetrizers are applied to the Cauchy problem for hyperbolic operators with triple characteristics. In particular, the V.Ivrii's conjecture concerned with triple effectively hyperbolic characteristics is proved for differential operators with coefficients depending on the time variable.

In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with finite topology and one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. A partial result for asymptotically cylindrical such shrinkers is also given. The mean curvature flow itself is used in the argument.

It is commonly known that $\zeta(2k) = q_{k}\frac{\zeta(2k + 2)}{\pi^2}$ with known rational numbers $q_{k}$. In this work we construct recurrence relations of the form $\sum_{k = 1}^{\infty}r_{k}\frac{\zeta(2k + 1)}{\pi^{2k}} = 0$ and show that series representations for the coefficients $r_{k} \in \mathbb{R}$ can be computed explicitly.

The Prime state of $n$ qubits, $|\mathbb{P}_n\rangle$, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than $2^n$. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of $|\mathbb{P}_n\rangle$ up to $n=30$ qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same, however we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller-Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the $(H^1)'$-norm of the solutions, which can only grow with a square-root law in time. This additional $(H^1)'$-bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.

We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for any orthonormal system of non-overlapping Franklin polynomials. It will also be remarked that $\log n$ is the optimal sequence in this context.

We study the sum $\Sigma_v(U)=\sum_{\substack{d,e\leq U\\(de,v)=1}}\frac{\mu(d)\mu(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, so that a continuous, monotonic and explicit version of Selberg's sieve can be stated.

Thanks to Barban-Vehov (1968), Motohashi (1974) and Graham (1978), it has been long known, but never explicitly, that $\Sigma_1(U)$ asymptotes $\log(U)$. In this article, we discover not only that $\Sigma_q(U)\sim\frac{q}{\varphi(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also, and more importantly, we are able to estimate explicitly $\Sigma_v(U)$ for $v\in\{1,2\}$ by giving rigorously their asymptotic expansion. We thus have $\Sigma_v(U)= \frac{v}{\varphi(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, for some explicit constant $K_v > 0$, where $\mathfrak{s}_1=0.60731\ldots$, and $\mathfrak{s}_2=1.4728\ldots$. Furthermore, we provide a closed-form expression for the constant term $\mathfrak{s}_q$ appearing in the asymptotic expression of $\Sigma_q(U)$, for all $q\in\mathbb{Z}_{>0}$.

As an application, we show how our result gives an explicit version of the Brun-Titchmarsh theorem within a range.

Closed (and simply-connected) manifolds whose dimensions are larger than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed manifolds in such ways. However, the latter work is, in geometric and constructive ways, not so difficult in a sense. The fact that the dimensions are not high enables us to handle the manifolds via diagrams for example. It is difficult to study higher dimensional manifolds in these ways, although it is natural and important. In the present paper, we present such studies via fold maps, which are higher dimensional versions of Morse functions. The author previously constructed fold maps on 7-dimensional closed and simply-connected manifolds satisfying additional conditions on cohomology rings, including exotic homotopy spheres. This paper presents fold maps on such manifolds of new classes.

In this work, we investigate a class of elliptic inverse problems and aim to simultaneously recover multiple inhomogeneous inclusions arising from two different physical parameters, using very limited boundary Cauchy data collected only at one or two measurement events. We propose a new fast, stable and highly parallelable direct sampling method (DSM) for the simultaneous reconstruction process. Two groups of probing and index functions are constructed, and their desired properties are analyzed. In order to identify and decouple the multiple inhomogeneous inclusions of different physical nature, we introduce a new concept of mutually almost orthogonality property that generalizes the important concept of almost orthogonality property in classical DSMs for inhomogeneous inclusions of same physical nature. With the help of this new concept, we develop a reliable strategy to distinguish two different types of inhomogeneous inclusions with noisy data collected at one or two measurement events. We further improve the decoupling effect by choosing an appropriate boundary influx. Numerical experiments are presented to illustrate the robustness and efficiency of the proposed method.

We construct an ${\cal N}{=}\,2$ supersymmetric extension of $n$-particle Ruijsenaars-Schneider models. The guiding feature is a deformation of the phase space. The supercharges have a "free" form linear in the fermions but produce an interacting four-fermion Hamiltonian. A field-dependent unitary transformation maps to standard fermions obeying conventional Poisson brackets. In this frame, the supercharges and Hamiltonian have long "fermionic tails". We also comment on previous attempts in this direction.

We provide a proof that the classes of finitely generated Kleinian groups and of three-manifold groups are quasi-isometrically rigid.

The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving.

In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in [MiO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\mathbb R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three-body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set.

Given a functional central limit (fCLT) and a parameter transformation, we use the functional delta method to construct random processes, called functional delta residuals, which asymptotically have the same covariance structure as the transformed limit process. Moreover, we prove a multiplier bootstrap fCLT theorem for these transformed residuals and show how this can be used to construct simultaneous confidence bands for transformed functional parameters. As motivation for this methodology, we provide the formal application of these residuals to a functional version of the effect size parameter Cohen's $d$, a problem appearing in current brain imaging applications. The performance and necessity of such residuals is illustrated in a simulation experiment for the covering rate of simultaneous confidence bands for the functional Cohen's $d$ parameter.

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form $-u^{\prime\prime}+\left(\frac{\ell(\ell+1)}{x^{2}}+q(x)\right)u=\omega^{2}u$ are obtained. The integral kernel is represented as a Fourier-Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to $\omega$. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones from arXiv:1609.06679 and arXiv:1712.01363 for large values of $\omega$ and $\ell$ and for non-integer values of $\ell$.

The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley-Wiener theorem and some results from constructive approximation theory.

We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma_\infty$, where $\Gamma$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide vanishing bounds in each case. Due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, as an application, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

We give a construction which produces irreducible complex rigid local systems on $\Bbb{P}_{\Bbb{C}}^1-\{p_1,\dots,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups $\operatorname{SU}(n)$ (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give (all) possible unitary irreducible rigid local systems.

As a consequence we obtain that the ranks of unitary irreducible rigid local systems (including those with finite global monodromy) on $\Bbb{P}^1-S$ are bounded above if we fix the cardinality of the set $S=\{p_1,\dots,p_s\}$ and require that the local monodromies have orders which divide $n$, for a fixed $n$. We also show that all unitary irreducible rigid local systems on $\Bbb{P}^1_{\Bbb{C}} -S$ with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalising previous works of the author and J. Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for $\operatorname{SU}(n)$.

We consider gradient models on the lattice $\mathbb{Z}^d$. These models serve as effective models for interfaces and are also known as continuous Ising models. The height of the interface is modelled by a random field with an energy which is a non-convex perturbation of the quadratic interaction. We are interested in the Gibbs measure with tilted boundary condition $u$ at inverse temperature $\beta$ of this model. In [AKM16], [Hil16] and [ABKM19] the authors show that for small tilt $u$ and large inverse temperature $\beta$ the surface tension is strictly convex, where the limit is taken on a subsequence. Moreover, it is shown that the scaling limit (again on a subsequence) is the Gaussian free field on the continuum torus. The method of the proof is a rigorous implementation of the renormalisation group method following a general strategy developed by Brydges and coworkers.

In this paper the renormalisation group analysis is extended from the finite-volume flow to an infinite-volume version to eliminate the necessity of the subsequence in the results in [AKM16], [Hil16] and [ABKM19].

The given study uses the methods to identify compactifications of semigroups $S\subset L(X),$ which reside in the space $L(X).$ This method generalizes in some sense the deLeeuw-Glicksberg-Theory to a greater class of functions. The approach provides an abstract approach to several notions of almost periodicity, which mainly involving right semitopological semigroups \cite{RuppertLNM}, and the adjoint theory. Moreover, the given setting is refined to the case of bounded $C_0-$semigroups.

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-K\"ahler case. We derive a Bochner formula for closed $(1, 1)$-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfy our positivity condition, then any class $\alpha \in H^{1, 1}_{BC}(X)$ can be represented by a closed $(1, 1)$-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

Kernel-based methods for support vector machines (SVM) have seen a great advantage in various applications. However, they may incur prohibitive computational costs when the involved sample size is on a large scale. Therefore, reducing the number of support vectors (or say sample reduction) appears to be necessary, which gives rise to the topic of the sparse SVM. Motivated by this, we aim at solving a sparsity constrained kernel SVM optimization, which is capable of controlling the number of the support vectors. Based on the established optimality conditions associated with the stationary equations, a subspace Newton method is cast to tackle the sparsity constrained problem and enjoys one-step convergence property if the starting point is close to a local region of a stationary point, leading to a super-fast computational speed. Numerical comparisons with some other excellent solvers demonstrate that the proposed method performs exceptionally well, especially for datasets with large numbers of samples, in terms of a much fewer number of support vectors and shorter computational time.

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and nuclear physics are discussed. The special case of orthogonal-orthogonal duality is treated in detail. It is shown that, like it was done by Helmers more than half a century ago in the analogous case of symplectic-symplectic duality, one can base a proof of the orthogonal-orthogonal duality theorem and a precise characterization of the relation between the equivalence classes of the dually related irreducible representations on a calculation of characters by combining it with an analysis of the representation of a reflection. Young diagrams for the description of equivalence classes of irreducible representations of orthogonal Lie algebras are introduced. The properties of a reflection of the number non-conserving part in the dual relationship between orthogonal Lie algebras corroborate a picture of an almost perfect symmetry between the partners.

The work establishes the exact performance limits of stochastic coded caching when users share a bounded number of cache states, and when the association between users and caches, is random. Under the premise that more balanced user-to-cache associations perform better than unbalanced ones, our work provides a statistical analysis of the average performance of such networks, identifying in closed form, the exact optimal average delivery time. To insightfully capture this delay, we derive easy to compute closed-form analytical bounds that prove tight in the limit of a large number $\Lambda$ of cache states. In the scenario where delivery involves $K$ users, we conclude that the multiplicative performance deterioration due to randomness -- as compared to the well-known deterministic uniform case -- can be unbounded and can scale as $\Theta\left( \frac{\log \Lambda}{\log \log \Lambda} \right)$ at $K=\Theta\left(\Lambda\right)$, and that this scaling vanishes when $K=\Omega\left(\Lambda\log \Lambda\right)$. To alleviate this adverse effect of cache-load imbalance, we consider various load balancing methods, and show that employing proximity-bounded load balancing with an ability to choose from $h$ neighboring caches, the aforementioned scaling reduces to $\Theta \left(\frac{\log(\Lambda / h)}{ \log \log(\Lambda / h)} \right)$, while when the proximity constraint is removed, the scaling is of a much slower order $\Theta \left( \log \log \Lambda \right)$. The above analysis is extensively validated numerically.

In this article a fractional cross-diffusion system is derived as the rigorous many-particle limit of a multi-species system of moderately interacting particles that is driven by L\'{e}vy noise. The form of the mutual interaction is motivated by the porous medium equation with fractional potential pressure. Our approach is based on the techniques developed by K. Oelschl\"{a}ger, in which the convergence of a regularization of the empirical measure to the solution of a correspondingly regularized macroscopic system is shown. A well-posedness result and the non-negativity of solutions is proved for the regularized macroscopic system, which then yields the same results for the non-regularized fractional cross-diffusion system in the limit.

For a regular scheme and its reduced closed subscheme, the latter being of finite type over a perfect field of positive characteristic, we define its cotangent bundle restricted to the closed subscheme as a family of vector bundles on smooth schemes over the field endowed with morphisms to the closed subscheme factoring through the Frobenius. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the cotangent bundle. We compute the singular supports of certain Kummer sheaves of rank 1.

It is proved without using the classification of finite simple groups that every finite non-abelian group of order $n$ has a non-central element whose centralizer has order exceeding $n^{1/3}$.

Chemical reaction networks describe interactions between biochemical species. Once an underlying reaction network is given for a biochemical system, the system dynamics can be modelled with various mathematical frameworks such as continuous time Markov processes. In this manuscript, the identifiability of the underlying network structure with a given stochastic system dynamics is studied. It is shown that some data types related to the associated stochastic dynamics can uniquely identify the underlying network structure as well as the system parameters. The accuracy of the presented network inference is investigated when given dynamical data is obtained via stochastic simulations.

We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical $\Theta$-point. In the $\Theta$-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, $\gamma_b^{\Theta}$, to that of terminally-attached arches, $\gamma_{11}^{\Theta},$ and to the correlation length exponent $\nu^{\Theta}.$ We find $\gamma_b^{\Theta} = \gamma_{11}^{\Theta}+\nu^{\Theta}.$ In the case of the special transition, we find $\gamma_b^{\Theta}({\rm sp}) = \frac{1}{2}[\gamma_{11}^{\Theta}({\rm sp})+\gamma_{11}^{\Theta}]+\nu^{\Theta}.$ For general networks, the explicit expression of configurational exponents then naturally involve bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm-Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the case of ordinary, mixed and special surface transitions, and of the $\Theta$-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.

We prove the existence of Bialynicki-Birula decomposition for Kempf monoids, which is a large class that contains for example monoids with reductive unit group in all characteristics. This extends the existence statements from previous works of Alper-Hall-Rydh and Jelisiejew-Sienkiewicz.

Reconfigurable intelligent surface (RIS)-assisted communication appears as one of the potential enablers for sixth generation (6G) wireless networks by providing a new degree of freedom in the system design to telecom operators. Particularly, RIS-empowered millimeter wave (mmWave) communication systems can be a remedy to provide broadband and ubiquitous connectivity. This paper aims to fill an important gap in the open literature by providing a physical, accurate, open-source, and widely applicable RIS channel model for mmWave frequencies. Our model is not only applicable in various indoor and outdoor environments but also includes the physical characteristics of wireless propagation in the presence of RISs by considering 5G radio channel conditions. Various deployment scenarios are presented for RISs and useful insights are provided for system designers from the perspective of potential RIS use-cases and their efficient positioning. The scenarios in which the use of an RIS makes a big difference or might not have a big impact on the communication system performance, are revealed. The open-source and comprehensive SimRIS Channel Simulator is also introduced in this paper.