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



This study presents the analytical and finite element formulation of a geometrically nonlinear and fractional-order nonlocal model of an Euler-Bernoulli beam. The finite nonlocal strains in the Euler-Bernoulli beam are obtained from a frame-invariant and dimensionally consistent fractional-order (nonlocal) continuum formulation. The finite fractional strain theory provides a positive definite formulation that results in a unique solution which is consistent across loading and boundary conditions. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal Euler-Bernoulli beam are obtained using variational principles. Further, a nonlinear finite element model for the fractional-order system is developed in order to achieve the numerical solution of the integro-differential nonlinear governing equations. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the geometrically nonlinear response of a nonlocal beam subject to various loading and boundary conditions. Although presented in the context of a 1D beam, this nonlinear f-FEM formulation can be extended to higher dimensional fractional-order boundary value problems.

We describe co-adjoint orbits and Casimir functions for two-step free-nilpotent Lie algebras. The symplectic foliation consists of affine subspaces of the Lie coalgebra of different dimensions.

Further, we consider left-invariant time-optimal problems on two-step Carnot groups, for which the set of admissible velocities is a strictly convex compactum in the first layer of the Lie algebra containing the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of Pontryagin maximum principle. Further, we describe constancy and periodicity of solutions to this subsystem and controls, and characterize its flow, for two-dimensional co-adjoint orbits.

The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $\F$ is at most $k$ over the algebraic closure of $\F$, where $k$ is a given positive integer. We estimate the arithmetic complexity of our algorithm.

We investigate an inertial forward-backward algorithm in connection with the minimization of the sum of a non-smooth and possible non-convex and a non-convex differentiable function. The algorithm is formulated in the spirit of the famous FISTA method, however the setting is non-convex and we allow different inertial terms. We also treat the case when the non-smooth function is convex and we show that in this case a better step-size can be allowed. We prove some abstract convergence results which applied to our numerical scheme allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka-{\L}ojasiewicz property.

Let $U$ be an open subset of $\mathbb{C}$ with boundary point $x_0$ and let $A_{\alpha}(U)$ be the space of functions analytic on $U$ that belong to lip$\alpha(U)$, the "little Lipschitz class". We consider the condition

$S= \displaystyle \sum_{n=1}^{\infty}2^{(t+\lambda+1)n}M_*^{1+\alpha}(A_n \setminus U)< \infty,$ where $t$ is a non-negative integer, $0<\lambda<1$, $M_*^{1+\alpha}$ is the lower $1+\alpha$ dimensional Hausdorff content, and $A_n = \{z: 2^{-n-1}<|z-x_0|<2^{-n}\}$. This is similar to a necessary and sufficient condition for bounded point derivations on $A_{\alpha}(U)$ at $x_0$. We show that $S= \infty$ implies that $x_0$ is a $(t+\lambda)$-spike for $A_{\alpha}(U)$ and that if $S<\infty$ and $U$ satisfies a cone condition, then the $t$-th derivatives of functions in $A_{\alpha}(U)$ satisfy a H\"older condition at $x_0$ for a non-tangential approach.

Both the ellipse and the hyperbola are geometric places that can be defined by establishing a relationship between points $P$ of the plane and two fixed points $A$ and $B$ (which are its foci $F'=A$ and $F=B$). Given two points $A$ and $B$ of the plan (which we no longer call the foci $F$ and $F'$), we are going to present three geometric places associated with $A$ and $B$ other than the ellipse and the hyperbola.

In this paper we present complete description of the elements of Banach space with one-point spectrum. Some applications of these results are also given.

We complete the classification of globally generated vector bundles with small $c_1$ on projective spaces by treating the case $c_1 = 5$ on $\mathbb{P}^n$, $n \geq 4$ (the case $c_1 \leq 3$ has been considered by Sierra and Ugaglia, while the cases $c_1 = 4$ on any projective space and $c_1 = 5$ on $\mathbb{P}^2$ and $\mathbb{P}^3$ have been studied in two of our previous papers). It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on $\mathbb{P}^5$, and its restriction to $\mathbb{P}^4$. We recall, in an appendix, the main results allowing the classification of globally generated vector bundles with $c_1 = 5$ on $\mathbb{P}^3$. Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to $\mathbb{P}^4$ as globally generated vector bundles.

A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density over all the packings by discs of radii $1$ and $r$ is reached for a compact packing (we give it as well as its density).

The notion of Gelfand pair (G, K) can be generalized if we consider homogeneous vector bundles over G/K instead of the homogeneous space G/K and matrix-valued functions instead of scalar-valued functions. This gives the definition of commutative homogeneous vector bundles. Being a Gelfand pair is a necessary condition of being a commutative homogeneous vector bundle. For the case in which G/K is a nilmanifold having square-integrable representations, in a previous article we determined a big family of commutative homogeneous vector bundles. In this paper, we complete that classification.

We give a description of the automorphism group of a Rauzy diagram as a subgroup of the symmetric group. This is based on an example that appear in some personnal notes of Yoccoz that are to be published in the project ''Yoccoz archives''.

In this paper, optimal actuator shape for nonlinear parabolic systems is discussed. The system under study is an abstract differential equation with a locally Lipschitz nonlinear part. A quadratic cost on the state and input of the system is considered. The existence of an optimal actuator shape has been established in the literature. This paper focuses on driving the optimality conditions for actuator shapes belonging to a Banach space. The application of the theory to the optimal actuator shape design for railway track model is considered.

ABSTRACT. In this article we present a point of view that highlights the importance of finding the upper bounds for prime gaps, in order to solve the twin primes conjecture and the Goldbach conjecture. For this purpose, we present a procedure for the determination of the upper bounds for prime gaps different from the most famous and known approaches. The proposed method analyzes the distribution of prime numbers using the set of relative numbers. Using negative numbers too, it becomes intuitive to understand that that the arrangement of 2P+1 consecutive numbers that goes -P to P, is the only arrangement that minimizes the distance between two powers having the same absolute value of the base D, with |D|<=P. This arrangement is considered important because by increasing the number of powers of the prime numbers within a range of consecutive numbers, it is presumed to decrease the overlap between the prime numbers considered. Consequently, by reducing these overlaps, we suppose to obtain an arrangement, in which the prime numbers less than and equal to P and their multiples occupy the greatest possible number of positions within a range of 2P+1 consecutive numbers. If this result could be demonstrated, would imply not only the resolution of the Legendre conjecture, but also a step forward in the resolution of the twin primes conjecture and the Goldbach conjecture.

Transitive local Lie algebras of vector fields can be easily constructed from dilations of $\mathbb{R}^n$ associating with coordinates positive weights (give me a sequence of $n$ positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on $\mathbb{R}^n$). It is interesting that all transitive nilpotent local Lie algebra of vector fields can be obtained as subalgebras of nilpotent algebras of this kind. Starting with a graded nilpotent Lie algebra one constructs graded parts of its Tanaka prolongations inductively as derivations of degree 0, 1, etc. Of course, vector fields of weight $k$ with respect to the dilation define automatically derivations of weight $k$, so the Tanaka prolongation is in this case never finite. Are they all such derivations given by vector fields or there are additional strange'ones? We answer this question. Except for special cases, derivations of degree 0 are given by vector fields of degree 0 and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation. However, in some particular cases of dilations we can find strange' derivations which we describe in detail

The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in $\mathbb H^1$ as limit of Gauss-Bonnet theorems in $(\mathbb H^1,<,>_L)$ when $L$ goes to infinity. This construction was extended by Wang-Wei in arXiv:1912.00302 to the affine group and the group of rigid motions of the Minkowski plane. We generalize constructions of both papers to surfaces in sub-Riemannian three dimensional manifolds following the approach of arXiv:1909.13341, and prove analogous Gauss-Bonnet theorem.

The classical Hahn-Banach theorem is based on a successive point-by-point procedure of extending bounded linear functionals. In the setting of a general metric domain, the conditions are less restrictive and the extension is only required to be Lipschitz with the same Lipschitz constant. In this case, the successive procedure can be replaced by a much simpler one which was done by McShane and Whitney in the 1930s. Using virtually the same construction, Czipszer and Geh\'er showed a similar extension property for pointwise Lipschitz functions. In the present paper, we relate this construction to another classical result obtained previously by Hausdorff and dealing with pointwise Lipschitz approximations of semi-continuous functions. Moreover, we furnish complementary extension-approximation results for locally Lipschitz functions which fit naturally in this framework.

We study the positive Hermitian curvature flow of left-invariant metrics on complex 2-step nilpotent Lie groups. In this setting we completely characterize the long-time behaviour of the flow, showing that normalized solutions to the flow subconverge to a non-flat algebraic soliton, in Cheeger- Gromov topology. We also exhibit a uniqueness result for algebraic solitons on such Lie groups.

We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs.

Using this method we can show two results involving high girth spectral expander graphs. First, we show that given $d \geq 3$ and $n$, there exists an explicit distribution of $d$-regular $\Theta(n)$-vertex graphs where with high probability its samples have girth $\Omega(\log_{d - 1} n)$ and are $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d - 1} + \epsilon$ (excluding the single trivial eigenvalue of $d$). Then, for every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic poly$(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$-vertices that is $\epsilon$-near-Ramanujan and has girth $\Omega(\sqrt{\log n})$, based on the work of arXiv:1909.06988 .

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$ is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.

We give explicit examples of pairs of one-ended, open 4-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding the nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of the summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

We compute non-extremal three-point functions of scalar operators in $\mathcal{N}=4$ super Yang-Mills at tree-level in $g_{YM}$ and at finite $N_c$, using the operator basis of the restricted Schur characters. We make use of the diagrammatic methods called quiver calculus to simplify the three-point functions. The results involve an invariant product of the generalized Racah-Wigner tensors ($6j$ symbols). Assuming that the invariant product is written by the Littlewood-Richardson coefficients, we show that the non-extremal three-point functions satisfy the large $N_c$ background independence; correspondence between the string excitations on $AdS_5 \times S^5$ and those in the LLM geometry.

Rate splitting (RS) is a potentially powerful and flexible technique for multi-antenna downlink transmission. In this paper, we address several technical challenges towards its practical implementation for beyond 5G systems. To this end, we focus on a single-cell system with a multi-antenna base station (BS) and K single-antenna receivers. We consider RS in its most general form, and joint decoding to fully exploit the potential of RS. First, we investigate the achievable rates under joint decoding and formulate the precoder design problems to maximize a general utility function, or to minimize the transmit power under pre-defined rate targets. Building upon the concave-convex procedure (CCCP), we propose precoder design algorithms for an arbitrary number of users. Our proposed algorithms approximate the intractable non-convex problems with a number of successively refined convex problems, and provably converge to stationary points of the original problems. Then, to reduce the decoding complexity, we consider the optimization of the precoder and the decoding order under successive decoding. Further, we propose a stream selection algorithm to reduce the number of precoded signals. With a reduced number of streams and successive decoding at the receivers, our proposed algorithm can even be implemented when the number of users is relatively large, whereas the complexity was previously considered as prohibitively high in the same setting. Finally, we propose a simple adaptation of our algorithms to account for the imperfection of the channel state information at the transmitter. Numerical results demonstrate that the general RS scheme provides a substantial performance gain as compared to state-of-the-art linear precoding schemes, especially with a moderately large number of users.

The equations governing the gravitational and electromagnetic perturbations of Kerr-Newman spacetime are here derived. They generalize the Teukolsky equation in Kerr and the Teukolsky-like system of equations in Reissner-Nordstr\"om spacetime. Through their Chandrasekhar transformation, we obtain a system of physical-space coupled wave equations. In particular, the physical-space analysis of this system will solve the issue of the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields in the perturbed spacetime", as put by Chandrasekhar. The derivation of the equations here obtained makes use of the formalism introduced in a recent work by Giorgi-Klainerman-Szeftel for Kerr, and represents the first step towards an analytical proof of the stability of the Kerr-Newman black hole.

Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel $\mathcal{K}$, whereas the second achieves the same using a time-local generator $\mathcal{G}$. Here we show that the two are connected by a simple yet general fixed-point relation: $\mathcal{G} = \hat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial relations between the two completely different ways of computing the time-evolution and combine their strengths. We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions. We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally "samples" it at nonzero characteristic frequencies. This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies. Second, we prove that the fixed-point equation sums up the time-domain gradient / Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects. Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel. For the transient generator this produces non-semigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.

We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a stochastic partial differential equation (SPDE) coupled to an ordinary differential equation (ODE). This approach has been employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs that are not self-adjoint. This provides a major generalization as these systems are prevalent in many applications.

We construct an explicit bijection between bipartite pointed maps on an arbitrary surface $\mathcal{S}$, and specific unicellular blossoming maps on the same surface. Our bijection gives access to the degrees of all the faces, and distances from the pointed vertex in the initial map. The main construction generalizes recent work of the second author which covered the case of orientable surface. Our bijection gives rise to a first combinatorial proof of a parametric rationality result concerning the bivariate generating series of maps on a given surface with respect to their numbers of faces and vertices. In particular, it provides a combinatorial explanation of the structural difference between the aforementioned bivariate parametric generating series in the case of orientable and non-orientable maps.

In this article we present a class of codes with few weights arising from special type of linear sets. We explicitly show the weights of such codes, their weight enumerator and possible choices for their generator matrices. In particular, our construction yields also to linear codes with three weights and, in some cases, to almost MDS codes. The interest for these codes relies on their applications to authentication codes and secret schemes, and their connections with further objects such as association schemes and graphs.

We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erd\H{o}s-Hajnal Conjecture, states that for every tournament $S$ there exists $\epsilon(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{\epsilon(S)}$ vertices. Let $C_5$ be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon-Pach-Solymosi conjecture is known to be true for the case when $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament $T$ with no subtorunament isomorphic to $C_5$ there exist disjoint vertex subsets $A$ and $B$, each containing a linear proportion of the vertices of $T$, and such that every vertex of $A$ is adjacent to every vertex of $B$.

We prove that if $q_1, \ldots, q_m: {\Bbb R}^n \longrightarrow {\Bbb R}$ are quadratic forms in variables $x_1, \ldots, x_n$ such that each $q_k$ depends on at most $r$ variables and each $q_k$ has common variables with at most $r$ other forms, then the average value of the product $\left(1+ q_1\right) \cdots \left(1+q_m\right)$ with respect to the standard Gaussian measure in ${\Bbb R}^n$ can be approximated within relative error $\epsilon >0$ in quasi-polynomial $n^{O(1)} m^{O(\ln m -\ln \epsilon)}$ time, provided $|q_k(x)| \leq \gamma \|x\|^2 /r$ for some absolute constant $\gamma > 0$ and $k=1, \ldots, m$. When $q_k$ are interpreted as pairwise squared distances for configurations of points in Euclidean space, the average can be interpreted as the partition function of systems of particles with mollified logarithmic potentials. We sketch a possible application to testing the feasibility of systems of real quadratic equations.

We elaborate Legendre's original proof of his discovery of the first singular modulus in the history of mathematics, as well as its appearance in Ramanujan's formula for the arc length of an ellipse with said singular modulus as eccentricity, and its appearance in the three-body choreography of Bernoulli's lemniscate.

We consider combinatorial semi-bandits over a set of arms ${\cal X} \subset \{0,1\}^d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = {\cal O}\Big( {d (\ln m)^2 (\ln T) \over \Delta_{\min} }\Big)$, but it has computational complexity ${\cal O}(|{\cal X}|)$ which is typically exponential in $d$, and cannot be used in large dimensions. We propose the first algorithm which is both computationally and statistically efficient for this problem with regret $R(T) = {\cal O} \Big({d (\ln m)^2 (\ln T)\over \Delta_{\min} }\Big)$ and computational complexity ${\cal O}(T {\bf poly}(d))$. Our approach involves carefully designing an approximate version of ESCB with the same regret guarantees, showing that this approximate algorithm can be implemented in time ${\cal O}(T {\bf poly}(d))$ by repeatedly maximizing a linear function over ${\cal X}$ subject to a linear budget constraint, and showing how to solve this maximization problems efficiently.

We introduce a novel approach to optimize the architecture of deep neural networks by identifying critical neurons and removing non-critical ones. The proposed approach utilizes a mixed integer programming (MIP) formulation of neural models which includes a continuous importance score computed for each neuron in the network. The optimization in MIP solver minimizes the number of critical neurons (i.e., with high importance score) that need to be kept for maintaining the overall accuracy of the model. Further, the proposed formulation generalizes the recently considered lottery ticket optimization by identifying multiple "lucky" sub-networks resulting in optimized architecture that not only perform well on a single dataset, but also generalize across multiple ones upon retraining of network weights. Finally, the proposed framework provides significant improvement in scalability of automatic sparsification of deep network architectures compared to previous attempts. We validate the performance and generalizability of our approach on MNIST, Fashion-MNIST, and CIFAR-10 datasets, using three different neural networks: LeNet 5 and two ReLU fully connected models.

A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g. Plug-Pichler - stochastic programming, Hellwig - game theory, Aldous - stability of optimal stopping, Hoover-Keisler - model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality. A natural compatible metric for the weak adapted topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug-Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.

This paper provides necessary and sufficient conditions for the existence of a pair of complex conjugate roots, each of multiplicity two, in the spectrum of a linear time-invariant single-delay equation of retarded type. This pair of roots is also shown to be always strictly dominant, determining thus the asymptotic behavior of the system. The proof of this result is based on the corresponding result for real roots of multiplicity four, continuous dependence of roots with respect to parameters, and a detailed study of crossing imaginary roots.

In this article we are investigating the computers development process in the past decades in order to identify the factors that influence it the most. We describe such factors and use them to predict the direction of further development. To solve these problems, we use the concept of the Computer Capacity, which allows us to estimate the performance of computers theoretically, relying only on the description of its architecture.

We define the Ihara zeta function $\zeta(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We show that the $L$-function evaluated at the regular representation is equal to $\zeta(u,X)$ and that $\zeta(u,X//G)$ divides $\zeta(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of quotients of $K_4$.

We construct a Riemannian metric of positive sectional curvature on the $3$-dimensional projective space with a two-sided closed embedded minimal surface of genus $3$, index $1$ and nullity $0$.

We study convex empirical risk minimization for high-dimensional inference in binary models. Our first result sharply predicts the statistical performance of such estimators in the linear asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit in order to prove a bound on the best achievable performance among them. Notably, we show that the proposed bound is tight for popular binary models (such as Signed, Logistic or Probit), by constructing appropriate loss functions that achieve it. More interestingly, for binary linear classification under the Logistic and Probit models, we prove that the performance of least-squares is no worse than 0.997 and 0.98 times the optimal one. Numerical simulations corroborate our theoretical findings and suggest they are accurate even for relatively small problem dimensions.

We study a class of chainable continua which contains, among others, all inverse limit spaces generated by a single interval bonding map which is piecewise monotone and locally eventually onto. Such spaces are realized as attractors of non-hyperbolic surface homeomorphisms. Using dynamical properties of the bonding map, we give conditions for existence of endpoints, characterize the set of local inhomogeneities, and determine when it consists only of endpoints. As a side product we also obtain a characterization of arcs as inverse limits for piecewise monotone bonding maps, which is interesting in its own right.

In this article, we study the complex symmetry of compositions operators $C_{\phi}f=f\circ \phi$ induced on weighted Bergman spaces $A^2_{\beta}(\mathbb{D}),\ \beta\geq -1,$ by analytic self-maps of the unit disk. One of ours main results shows that $\phi$ has a fixed point in $\mathbb{D}$ whenever $C_{\phi}$ is complex symmetric. Our works establishes a strong relation between complex symmetry and cyclicity. By assuming $\beta\in \mathbb{N}$ and $\phi$ is an elliptic automorphism of $\mathbb{D}$ which not a rotation, we show that $C_{\phi}$ is not complex symmetric whenever $\phi$ has order greater than $2(3+\beta).$

We develop two new stochastic Gauss-Newton algorithms for solving a class of stochastic nonconvex compositional optimization problems frequently arising in practice. We consider both the expectation and finite-sum settings under standard assumptions. We use both classical stochastic and SARAH estimators for approximating function values and Jacobians. In the expectation case, we establish $\mathcal{O}(\varepsilon^{-2})$ iteration complexity to achieve a stationary point in expectation and estimate the total number of stochastic oracle calls for both function values and its Jacobian, where $\varepsilon$ is a desired accuracy. In the finite sum case, we also estimate the same iteration complexity and the total oracle calls with high probability. To our best knowledge, this is the first time such global stochastic oracle complexity is established for stochastic Gauss-Newton methods. We illustrate our theoretical results via numerical examples on both synthetic and real datasets.

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. Then the absolutely continuous part of the Laplace operator $\Delta$ with Dirichlet boundary conditions and the free Laplace operator $\Delta_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(\Delta)-g(\Delta_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators $\Delta_1$ and $\Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(\Delta) - g(\Delta_1) - g(\Delta_2) + g(\Delta_0)$ is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case $g(x)=x^\frac{1}{2}$ the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation.

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space. We describe a homeomomorphism between the "Hitchin component" of wild $\mathrm{Sp}(4,\mathbb{R})$-Higgs bundles over $\mathbb{CP}^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\mathbb{H}^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\mathbb{R}^{4}$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\mathbb{H}^{2,2}$ associated to $\mathrm{Sp}(4,\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials.

Nonlinearities in piezoelectric systems can arise from internal factors such as nonlinear constitutive laws or external factors like realizations of boundary conditions. It can be difficult or even impossible to derive detailed models from the first principles of all the sources of nonlinearity in a system. As a specific example, in traditional modeling techniques that use electric enthalpy density with higher-order terms, it can be problematic to choose which polynomial nonlinearities are essential. This paper introduces adaptive estimator techniques to estimate the nonlinearities that can arise in certain piezoelectric systems. Here an underlying assumption is that the nonlinearities can be modeled as functions in a reproducing kernel Hilbert space (RKHS). Unlike traditional modeling approaches, the approach discussed in this paper allows the development of models without knowledge of the precise form or structure of the nonlinearity. This approach can be viewed as a data-driven method to approximate the unknown nonlinear system. This paper introduces the theory behind the adaptive estimator and studies the effectiveness of this approach numerically for a class of nonlinear piezoelectric composite beams.

While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of the universe of sets is analogous to the canonical inner models. This note provides two new instances of this phenomenon.

We consider Hamilton cycles in the random digraph $D_{n,m}$ where the orientation of edges follows a pattern other than the trivial orientation in which the edges are oriented in the same direction as we traverse the cycle. We show that if the orientation forms a periodic pattern, other than the trivial pattern, then approximately half the usual $n\log n$ edges are needed to guarantee the existence of such Hamilton cycles a.a.s.

We consider four examples of short step lattice paths confined to the quarter plane. These are the Kreweras, Reverse Kreweras, Gessel, and Mishna-Rechnitzer lattice paths.The Reverse Kreweras are straightforward to solve and thus interesting as a contrast to the Kreweras paths and Gessel paths as the latter two have historically been significantly more difficult to solve. The Mishna-Rechnitzer paths are interesting as they are associated with an infinite order group. We will give some geometrical insight into all these properties by considering the Weyl chambers associated with their step sets.For Reverse Kreweras paths the Weyl chamber walls coincide with the quarter plane boundary and hence the problem is readily solvable by Bethe Ansatz or by using the Gessel-Zeilberger Theorem. For Kreweras paths the quarter plane corresponds to the union of two adjacent Weyl Chambers and hence neither the Bethe Ansatz nor the Gessel-Zeilberger Theorem are directly applicable making the problem considerably more difficult to solve. Similarly, the quarter plane for Gessel paths is the union of three Weyl chambers. For Mishna-Rechnitzer paths the step set has non-zero barycenter leading to an affine dihedral reflection group. The affine structure corresponds to the drift in the random walk. The quarter plane is the union of an infinite number of Weyl alcoves.

Randomized Kaczmarz (RK), Motzkin Method (MM) and Sampling Kaczmarz Motzkin (SKM) algorithms are commonly used iterative techniques for solving linear system of inequalities (i.e., $Ax \leq b$). As linear systems of equations represents a modeling paradigm for solving many optimization problems, these randomized and iterative techniques are gaining popularity among researchers in different domains. In this work, we propose a Generalized Sampling Kaczamrz Motzkin (GSKM) method that unifies the iterative methods into a single framework. In addition to the general framework, we propose a Nesterov type acceleration scheme in the SKM method called as Probably Accelerated Sampling Kaczamrz Motzkin (PASKM). We prove the convergence theorems for both GSKM and PASKM algorithms in the L2 norm perspective with respect to the proposed sampling distribution. Furthermore, from the convergence theorem of GSKM algorithm, we find the convergence results of several well known algorithms like Kaczmarz method, Motzkin method and SKM algorithm. We perform thorough numerical experiments using both randomly generated and real life (classification with support vector machine and Netlib LP) test instances to demonstrate the efficiency of the proposed methods. We compare the proposed algorithms with SKM, Interior Point Method (IPM) and Active Set Method (ASM) in terms of computation time and solution quality. In majority of the problem instances, the proposed generalized and accelerated algorithms significantly outperform the state-of-the-art methods.

We investigate the problem of the successive refinement for Wyner-Ziv coding with degraded side information and obtain a complete characterization of the rate region for the quadratic vector Gaussian case. The achievability part is based on the evaluation of the Tian-Diggavi inner bound that involves Gaussian auxiliary random vectors. For the converse part, a matching outer bound is obtained with the aid of a new extremal inequality. Herein, the proof of this extremal inequality depends on the integration of the monotone path argument and the doubling trick as well as information-estimation relations.

Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some case, $t(K^{\star})\leq t(K)$. So it is interesting to know when $t(K^{\star})= t(K)+1$.

After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot $K^{\star}$ and its companion $K$, $t(K^{\star})\geq t(K)$; (2) if either $K$ admits a high distance Heegaard splitting or $p/q$ is far away from a fixed subset in the Farey graph, then $t(K^{\star})= t(K)+1$. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

A general shape identification inverse problem is studied in a Bayesian framework. This problem requires the determination of the unknown shape of a domain in the Euclidean space from finite-dimensional observation data with some Gaussian random noise. Then, the stability of posterior is studied for observation data. For each point of the space, the conditional probability that the point is included in the unknown domain given the observation data is considered. The stability is also studied for this probability distribution. As a model problem for our inverse problem, a heat inverse problem is considered. This problem requires the determination of the unknown shape of cavities in a heat conductor from temperature data of some portion of the surface of the heat conductor. To apply the above stability results to this model problem, one needs the measurability and some boundedness of the forward operator. These properties are shown.

This letter studies the problem of maintaining information freshness under passive eavesdropping attacks. The classical three-node wiretap channel model is considered, in which a source aims to send its latest status wirelessly to its intended destination, while protecting the message from being overheard by an eavesdropper. Considering that conventional channel capacity-based secrecy metrics are no longer adequate to measure the information timeliness in status update systems, we define two new age of information-based metrics to characterize the secrecy performance of the considered system. We further propose, analyze, and optimize a randomized stationary transmission policy implemented at the source for further enhancing the secrecy performance. Simulation results are provided to validate our analysis and optimization.

The rapid development of the fifth generation mobile communication systems accelerates the implementation of vehicle-to-everything communications. Compared with the other types of vehicular communications, vehicle-to-vehicle (V2V) communications mainly focus on the exchange of driving safety information with neighboring vehicles, which requires ultra-reliable and low-latency communications (URLLCs). However, the frame size is significantly shortened in V2V URLLCs because of the rigorous latency requirements, and thus the overhead is no longer negligible compared with the payload information from the perspective of size. In this paper, we investigate the frame design and resource allocation for an urban V2V URLLC system in which the uplink cellular resources are reused at the underlay mode. Specifically, we first analyze the lower bounds of performance for V2V pairs and cellular users based on the regular pilot scheme and superimposed pilot scheme. Then, we propose a frame design algorithm and a semi-persistent scheduling algorithm to achieve the optimal frame design and resource allocation with the reasonable complexity. Finally, our simulation results show that the proposed frame design and resource allocation scheme can greatly satisfy the URLLC requirements of V2V pairs and guarantee the communication quality of cellular users.

We introduce the notions of $(G,q)$-opers and Miura $(G,q)$-opers, where $G$ is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of $(G,q)$-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a $q$DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ($q$-differential equations). If $\mathfrak{g}$ is simply-laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra $U_q \widehat{\mathfrak{g}}$. However, if $\mathfrak{g}$ is non-simply laced, then these equations correspond to the XXZ-type model associated not to $U_q \widehat{\mathfrak{g}}$ but to a twisted affine algebra naturally obtained from $\mathfrak{g}$. A key element in this $q$DE/IM correspondence is the $QQ$-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category ${\mathcal O}$ of the relevant quantum affine algebra.

Area under ROC curve (AUC) is a widely used performance measure for classification models. We propose a new distributionally robust AUC maximization model (DR-AUC) that relies on the Kantorovich metric and approximates the AUC with the hinge loss function. We use duality theory to reformulate the DR-AUC model as a tractable convex quadratic optimization problem. The numerical experiments show that the proposed DR-AUC model -- benchmarked with the standard deterministic AUC and the support vector machine models - improves the out-of-sample performance over the majority of the considered datasets. The results are particularly encouraging since our numerical experiments are conducted with training sets of small size which have been known to be conducive to low out-of-sample performance.

Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of $H$ such that $f(xy) \cap V(F) = \{x,y\}$ holds for each edge $xy$ of $F$. In other words, $H$ contains $F$ as a trace.

Let $ex_{r}(n,B_{ind} F)$ denote the maximum number of edges in an $r$-uniform hypergraph with no induced Berge $F$. Let $ex(n,K_r, F)$ denote the maximum number of $K_r$'s in an $F$-free graph on $n$ vertices. We show that these two Tur\'an type functions are strongly related.

We consider the radial focusing energy critical nonlinear wave equation in three spatial dimensions. Our main result proves the stability of the ODE-blowup under random perturbations below the energy space. To the best of our knowledge, this is the first study of blowup in dispersive equations with random initial data. The argument relies on probabilistic Strichartz estimates in similarity coordinates for the linearized evolution around the ODE-blowup.

To efficiently support the real-time control applications, networked control systems operating with ultra-reliable and low-latency communications (URLLCs) become fundamental technology for future Internet of things (IoT). However, the design of control, sensing and communications is generally isolated at present. In this paper, we propose the joint optimization of control cost and energy consumption for a centralized wireless networked control system. Specifically, with the sensing-then-control'' protocol, we first develop an optimization framework which jointly takes control, sensing and communications into account. In this framework, we derive the spectral efficiency, linear quadratic regulator cost and energy consumption. Then, a novel performance metric called the \textit{energy-to-control efficiency} is proposed for the IoT control system. In addition, we optimize the energy-to-control efficiency while guaranteeing the requirements of URLLCs, thereupon a general and complex max-min joint optimization problem is formulated for the IoT control system. To optimally solve the formulated problem by reasonable complexity, we propose two radio resource allocation algorithms. Finally, simulation results show that our proposed algorithms can significantly improve the energy-to-control efficiency for the IoT control system with URLLCs.

Orthogonal blinding based schemes for wireless physical layer security aim to achieve secure communication by injecting noise into channels orthogonal to the main channel and corrupting the eavesdropper's signal reception. These methods, albeit practical, have been proven vulnerable against multi-antenna eavesdroppers who can filter the message from the noise. The vulnerability is rooted in the fact that the main channel state remains static in spite of the noise injection, which allows an eavesdropper to estimate it promptly via known symbols and filter out the noise. Our proposed scheme leverages a reconfigurable antenna for Alice to rapidly change the channel state during transmission and a compressive sensing based algorithm for her to predict and cancel the changing effects for Bob. As a result, the communication between Alice and Bob remains clear, whereas randomized channel state prevents Eve from launching the known-plaintext attack. We formally analyze the security of the scheme against both single and multi-antenna eavesdroppers and identify its unique anti-eavesdropping properties due to the artificially created fast-changing channel. We conduct extensive simulations and real-world experiments to evaluate its performance. Empirical results show that our scheme can suppress Eve's attack success rate to the level of random guessing, even if she knows all the symbols transmitted through other antenna modes.

Small cancellation groups form an interesting class with many desirable properties.

It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no information about small'' group presentations.

In this note, we give closed form formulas for both lower and upper bounds on the density of small cancellation presentations, and compare our results with experimental data.

Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$ based on $\{0,1,\cdots,nd\}$ is called \emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element of $\{1,2,\cdots,nd\}$ occurs exactly one entry of $A$, and its row-sums, column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive integers. An SAMS$(n,d)$ is called \emph{regular} if there are exactly $d$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $n\equiv1,5\pmod 6$, and it is proved that for any $n\equiv1,5\pmod 6$, there exists a regular SAMS$(n,d)$ if and only if $2\leq d\leq n-1$.

In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

In this paper, we propose a structured linear parameterization of a feedback policy to solve the model-free stochastic optimal control problem. This parametrization is corroborated by a decoupling principle that is shown to be near-optimal under a small noise assumption, both in theory and by empirical analyses. Further, we incorporate a model-free version of the Iterative Linear Quadratic Regulator (ILQR) in a sample-efficient manner into our framework. Simulations on systems over a range of complexities reveal that the resulting algorithm is able to harness the superior second-order convergence properties of ILQR. As a result, it is fast and is scalable to a wide variety of higher dimensional systems. Comparisons are made with a state-of-the-art reinforcement learning algorithm, the Deep Deterministic Policy Gradient (DDPG) technique, in order to demonstrate the significant merits of our approach in terms of training-efficiency.

The author is mainly interest in the Gr\"{o}bner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the presentation of group determines the Gr\"{o}bner-Shirshov bases of group. In this paper, the author rearranges the generators, marks them on Coxeter-Dynkin diagrams, and gets a simple presentation of the Gr\"{o}bner-Shirshov bases for Coxeter groups of types $G_2, F_4, E_6$ and $E_7$. This article also gives the Gr\"{o}bner-Shirshov basis of Coxeter group of type $E_8$.

The rateless and information additive properties of fountain codes make them attractive for use in broadcast/multicast applications, especially in radio environments where channel characteristics vary with time and bandwidth is expensive. Conventional schemes using a combination of ARQ (Automatic Repeat reQuest) and FEC (Forward Error Correction) suffer from serious drawbacks such as feedback implosion at the transmitter, the need to know the channel characteristics apriori so that the FEC scheme is designed to be effective and the fact that a reverse channel is needed to request retransmissions if the FEC fails. This paper considers the assessment of fountain codes over radio channels. The performance of fountain codes, in terms of the associated overheads, over radio channels of the type experienced in GPRS (General Packet Radio Service) is presented. The work is then extended to assessing the performance of Fountain codes in combination with the GPRS channel coding schemes in a radio environment.

This paper considers the distributed optimization problem over a network where the global objective is to optimize a sum of local functions using only local computation and communication. Since the existing algorithms either adopt a linear consensus mechanism, which converges at best linearly, or assume that each node starts sufficiently close to an optimal solution, they cannot achieve globally superlinear convergence. To break through the linear consensus rate, we propose a finite-time set-consensus method, and then incorporate it into Polyak's adaptive Newton method, leading to our distributed adaptive Newton algorithm (DAN). To avoid transmitting local Hessians, we adopt a low-rank approximation idea to compress the Hessian and design a communication-efficient DAN-LA. Then, the size of transmitted messages in DAN-LA is reduced to $O(p)$ per iteration, where $p$ is the dimension of decision vectors and is the same as the first-order methods. We show that DAN and DAN-LA can globally achieve quadratic and superlinear convergence rates, respectively. Numerical experiments on logistic regression problems are finally conducted to show the advantages over existing methods.

Network function virtualization is a promising technology to simultaneously support multiple services with diverse characteristics and requirements in the fifth generation and beyond networks. In practice, each service consists of a predetermined sequence of functions, called a service function chain (SFC), running on a cloud environment. To make different service slices work properly in harmony, it is crucial to select the cloud nodes to deploy the functions in the SFC and flexibly route the flow of the services such that these functions are processed in sequence, the end-to-end (E2E) latency constraints of all services are guaranteed, and all resource constraints are respected. In this paper, we propose a new (mixed binary linear program) formulation of the above network slicing problem that optimizes the system energy efficiency while jointly considers the resource budget, functional instantiation, flow routing, and E2E latency requirement. Numerical results show the advantage of the proposed formulation compared to the existing ones.

In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability of isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973), to manifolds with local bounded Ricci covering geometry. Our two generalized results have been used in a recent work of Xiaochun Rong in generalizing Gromov's almost flat manifolds theorem to maximally collapsed manifolds with local bounded Ricci covering geometry.

Self-supervision is key to extending use of deep learning for label scarce domains. For most of self-supervised approaches data transformations play an important role. However, up until now the impact of transformations have not been studied. Furthermore, different transformations may have different impact on the system. We provide novel insights into the use of data transformation in self-supervised tasks, specially pertaining to clustering. We show theoretically and empirically that certain set of transformations are helpful in convergence of self-supervised clustering. We also show the cases when the transformations are not helpful or in some cases even harmful. We show faster convergence rate with valid transformations for convex as well as certain family of non-convex objectives along with the proof of convergence to the original set of optima. We have synthetic as well as real world data experiments. Empirically our results conform with the theoretical insights provided.

This paper mainly illustrates the Bit error rate performance of M-ary QAM and M-ary PSK for different values of SNR over Rician Fading channel. A signal experiences multipath propagation in the wireless communication system which causes expeditious signal amplitude fluctuations in time, is defined as fading. Rician Fading is a small signal fading. Rician fading is a hypothetical model for radio propagation inconsistency produced by fractional cancellation of a radio signal by itself and as a result the signal reaches in the receiver by several different paths. In this case, at least one of the destination paths is being lengthened or shortened. From this paper , it can be observed that the value of Bit error rate decreases when signal to noise ratio increases in decibel for Mary QAM and M-ary PSK such as 256 QAM, 64 PSK etc. Constellation diagrams of M-QAM and M-PSK have also been showed in this paper using MATLAB Simulation. The falling of Bit error rate with the increase of diversity order for a fixed value of SNR has also been included in this paper. Diversity is a influential receiver system which offers improvement over received signal strength.

A $\gamma$-deformed version of $su(2)$ algebra with non-hermitian generators has been obtained from a bi-orthogonal system of vectors in $\bf{C^2}$. The related Jordan-Schwinger(J-S) map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type. Finally the notion of partial $\mathcal{PT}$ symmetry has been introduced as characteristic feature of some operators as well as their eigenfunctions. The possibility of partial $\mathcal{PT}$-symmetry breaking is also discussed. The deformation parameter $\gamma$ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration.

Federated learning is a new distributed machine learning framework, where a bunch of heterogeneous clients collaboratively train a model without sharing training data. In this work, we consider a practical and ubiquitous issue in federated learning: intermittent client availability, where the set of eligible clients may change during the training process. Such an intermittent client availability model would significantly deteriorate the performance of the classical Federated Averaging algorithm (FedAvg for short). We propose a simple distributed non-convex optimization algorithm, called Federated Latest Averaging (FedLaAvg for short), which leverages the latest gradients of all clients, even when the clients are not available, to jointly update the global model in each iteration. Our theoretical analysis shows that FedLaAvg attains the convergence rate of $O(1/(N^{1/4} T^{1/2}))$, achieving a sublinear speedup with respect to the total number of clients. We implement and evaluate FedLaAvg with the CIFAR-10 dataset. The evaluation results demonstrate that FedLaAvg indeed reaches a sublinear speedup and achieves 4.23% higher test accuracy than FedAvg.

Consider a distributed graph where each vertex holds one of two distinct opinions. In this paper, we are interested in synchronous voting processes where each vertex updates its opinion according to a predefined common local updating rule. For example, each vertex adopts the majority opinion among 1) itself and two randomly picked neighbors in best-of-two or 2) three randomly picked neighbors in best-of-three. Previous works intensively studied specific rules including best-of-two and best-of-three individually.

In this paper, we generalize and extend previous works of best-of-two and best-of-three on expander graphs by proposing a new model, quasi-majority functional voting. This new model contains best-of-two and best-of-three as special cases. We show that, on expander graphs with sufficiently large initial bias, any quasi-majority functional voting reaches consensus within $O(\log n)$ steps with high probability. Moreover, we show that, for any initial opinion configuration, any quasi-majority functional voting on expander graphs with higher expansion (e.g., Erd\H{o}s-R\'enyi graph $G(n,p)$ with $p=\Omega(1/\sqrt{n})$) reaches consensus within $O(\log n)$ with high probability. Furthermore, we show that the consensus time is $O(\log n/\log k)$ of best-of-$(2k+1)$ for $k=o(n/\log n)$.

In this paper we study a linear pursuit differential game described by an infinite system of first-order differential equations in Hilbert space. The control functions of players are subject to geometric constraints. The pursuer attempts to bring the system from a given initial state to the origin for a finite time and the evader's purpose is opposite. We obtain a guaranteed pursuit time and construct a strategy for pursuer.

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous.

Upon a consistent topological statistical theory the application of structural statistics requires a quantification of the proximity structure of model spaces. An important tool to study these structures are (Pseudo-)Riemannian metrices, which in the category of statistical models are induced by statistical divergences. The present article is intended to extend the notation of topological statistical models by a differential structure to statistical manifolds and to introduce the differential geometric foundations to study specific families of probability distributions. In this purpose the article successively incorporates the structures of differential-, Riemannian- and symplectic geometry within an underlying topological statistical model. The last section addresses a specific structural category, termed a dually flat statistical manifold, which can be used to study the properties of exponential families, which are of particular importance in machine learning and deep learning.

We study the Hartree-Fock equation universally used in many-electron problems. We prove that for any negative constant there are only finite number of critical values of the Hartree-Fock's functional associated with eigenvalues less than the constant. We also prove that a negative accumulation point of negative eigenvalues is an eigenvalue.

We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $\rightarrow$ +$\infty$. MSC2010: 35F55, 35L65. Notations. We denote $\times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its norm is denoted $\times$ M. The Dirac mass at X $\in$ R n is $\delta$ X or $\delta$ x=X. If $\nu$ $\in$ M (R m) and $\mu$ $\in$ M (R q), then $\nu$ $\otimes$ $\mu$ is the measure over R m+q uniquely defined by $\nu$ $\otimes$ $\mu$, $\psi$ = $\nu$, f $\mu$, g whenever $\psi$(x, y) $\not\equiv$ f (x)g(y). The closed halves of the real line are denoted R + and R --. * U.M.P.A., UMR CNRS-ENSL \# 5669. 46 all{\'e}e d'Italie,

We establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Tur\'{a}n type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalise a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.

In this paper, we aim to give a theoretical approximation for the penalty level of $\ell_{1}$-regularization problems. This can save much time in practice compared with the traditional methods, such as cross-validation. To achieve this goal, we develop two Gaussian approximation methods, which are based on a moderate deviation theorem and Stein's method respectively. Both of them give efficient approximations and have good performances in simulations. We apply the two Gaussian approximation methods into three types of ultra-high dimensional $\ell_{1}$ penalized regressions: lasso, square-root lasso, and weighted $\ell_{1}$ penalized Poisson regression. The numerical results indicate that our two ways to estimate the penalty levels achieve high computational efficiency. Besides, our prediction errors outperform that based on the 10-fold cross-validation.

We extend Korevaar-Schoen's theory of metric valued Sobolev maps to cover the case of the source space being an RCD space. In this situation it appears that no version of the subpartition lemma' holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on:

- the fact that such spaces are strongly rectifiable' a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim's metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density,

- the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential.

When the target space is CAT(0) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.

We prove the consistency of the $\ell_1$ penalized negative binomial regression (NBR). A real data application about German health care demand shows that the $\ell_1$ penalized NBR produces a more concise but more accurate model, comparing to the classical NBR.

Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chromatic number of hierarchical products with two factors determined. An application of dominated colorings in genetic networks is also proposed.

In this short review we first recall combinatorial or ($0-$dimensional) quantum field theory (QFT). We then give the main idea of a standard QFT method, called the intermediate field method, and we review how to apply this method to a combinatorial QFT reformulation of the celebrated Jacobian conjecture on the invertibility of polynomial systems. This approach establishes a related theorem concerning partial elimination of variables that implies a reduction of the generic case to the quadratic one. Note that this does not imply solving the Jacobian conjecture, because one needs to introduce a supplementary parameter for the dimension of a certain linear subspace where the system holds.

We consider practical data characteristics underlying federated learning, where unbalanced and non-i.i.d. data from clients have a block-cyclic structure: each cycle contains several blocks, and each client's training data follow block-specific and non-i.i.d. distributions. Such a data structure would introduce client and block biases during the collaborative training: the single global model would be biased towards the client or block specific data. To overcome the biases, we propose two new distributed optimization algorithms called multi-model parallel SGD (MM-PSGD) and multi-chain parallel SGD (MC-PSGD) with a convergence rate of $O(1/\sqrt{NT})$, achieving a linear speedup with respect to the total number of clients. In particular, MM-PSGD adopts the block-mixed training strategy, while MC-PSGD further adds the block-separate training strategy. Both algorithms create a specific predictor for each block by averaging and comparing the historical global models generated in this block from different cycles. We extensively evaluate our algorithms over the CIFAR-10 dataset. Evaluation results demonstrate that our algorithms significantly outperform the conventional federated averaging algorithm in terms of test accuracy, and also preserve robustness for the variance of critical parameters.

In this note we show that, when the delay goes to zero, the solution of multidimensional delay differential equations driven by a H\"older continuous function of order $\beta \in (\frac13,\frac12)$ converges with the supremum norm to the solution for the equation without delay. As an application, we discuss the applications to stochastic differential equations.

Full indefinite Stieltjes moment problem is studied via the step-by-step Schur algorithm. Naturally associated with indefinite Stieltjes moment problem are generalized Stieltjes continued fraction and a system of difference equations, which, in turn, lead to factorization of resolvent matrices of indefinite Stieltjes moment problem. A criterion for such a problem to be indeterminate in terms of continued fraction is found and a complete description of its solutions is given in the indeterminate case. Explicit formulae for diagonal and sub-diagonal Pad\'{e} approximants for formal power series corresponding to indefinite Stieltjes moment problem and convergence results for Pad\'{e} approximants are presented.

The tensor train approximation of electronic wave functions lies at the core of the QC-DMRG (Quantum Chemistry Density Matrix Renormalization Group) method, a recent state-of-the-art method for numerically solving the $N$-electron Schr\"odinger equation. It is well known that the accuracy of TT approximations is governed by the tail of the associated singular values, which in turn strongly depends on the ordering of the one-body basis.

Here we find that the singular values $s_1\ge s_2\ge ... \ge s_d$ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, $s_1s_d=s_2s_{d-1}=s_3s_{d-2}=...$, thus reducing the tail behaviour to a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wavefunctions, we find that the tail is upper bounded by a suitable superposition of the invariants. Optimizing the invariants or their superposition thus provides a new ordering scheme for QC-DMRG. Numerical tests on simple examples, i.e. linear combinations of a few Slater determinants, show that the new scheme reduces the tail of the singular values by several orders of magnitudes over existing methods, including the widely used Fiedler order.

A linear isometry $R$ of $\mathbb{R}^d$ is called a similarity isometry of a lattice $\Gamma \subseteq \mathbb{R}^d$ if there exists a positive real number $\beta$ such that $\beta R\Gamma$ is a sublattice of (finite index in) $\Gamma$. The set $\beta R\Gamma$ is referred to as a similar sublattice of $\Gamma$. A (crystallographic) point packing generated by a lattice $\Gamma$ is a union of $\Gamma$ with finitely many shifted copies of $\Gamma$. In this study, the notion of similarity isometries is extended to point packings. We provide a characterization for the similarity isometries of point packings and identify the corresponding similar subpackings. Planar examples will be discussed, namely, the $1 \times 2$ rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, we also consider similarity isometries of point packings about points different from the origin. In particular, similarity isometries of a certain shifted hexagonal packing will be computed and compared with that of the hexagonal packing.

Given two distinct subsets $A,B$ in the state space of some dynamical system, Transition Path Theory (TPT) was successfully used to describe the statistical behavior of transitions from $A$ to $B$ in the ergodic limit of the stationary system. We derive generalizations of TPT that remove the requirements of stationarity and of the ergodic limit, and provide this powerful tool for the analysis of other dynamical scenarios: periodically forced dynamics and time-dependent finite-time systems. This is partially motivated by studying applications such as climate, ocean, and social dynamics. On simple model examples we show how the new tools are able to deliver quantitative understanding about the statistical behavior of such systems. We also point out explicit cases where the more general dynamical regimes show different behaviors to their stationary counterparts, linking these tools directly to bifurcations in non-deterministic systems.

he classical theorem of Erd\H os \& Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under the assumption that the conditions in the criterion are fulfilled. We also investigate the case of a conditional distribution.

This paper compares different implementations of monetary policy in a new-Keynesian setting. We can show that a shift from Ramsey optimal policy under short-term commitment (based on a negative feedback mechanism) to a Taylor rule (based on a positive feedback mechanism) corresponds to a Hopf bifurcation with opposite policy advice and a change of the dynamic properties. This bifurcation occurs because of the ad hoc assumption that interest rate is a forward-looking variable when policy targets (inflation and output gap) are forward-looking variables in the new-Keynesian theory.

In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann-Lebesgue lemma and the Cauchy problem for the time-fractional Schr\"{o}dinger equation are considered.

For any smooth connected linear algebraic group G over an algebraically closed field k, we describe the Picard group of the universal moduli stack of principal G-bundles over pointed smooth k-projective curves

Let $k$ be a number field, $\mathbf{G}$ an algebraic group defined over $k$, and $\mathbf{G}(k)$ the group of $k$-rational points in $\mathbf{G}.$ We determine the set of functions on $\mathbf{G}(k)$ which are of positive type and conjugation invariant, under the assumption that $\mathbf{G}(k)$ is generated by its unipotent elements. An essential step in the proof is the classification of the $\mathbf{G}(k)$-invariant ergodic probability measures on an adelic solenoid naturally associated to $\mathbf{G}(k);$ this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of $S$-adic Lie groups.

We consider a diffuse interface approach for solving an elliptic PDE on a given closed hypersurface. The method is based on a (bulk) finite element scheme employing numerical quadrature for the phase field function and hence is very easy to implement compared to other approaches. We estimate the error in natural norms in terms of the spatial grid size, the interface width and the order of the underlying quadrature rule. Numerical test calculations are presented which confirm the form of the error bounds.

We introduce bi-slant $\xi^{\perp}$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of slant and semi-slant $\xi^{\perp}$-Riemannian submersion. We give an example and investigate the geometry foliations. After we obtain necessary and sufficient conditions related to totally geodesicness of submersion. Finally we give decomposition theorems for total manifold of such submersions.

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free energy functional due to the volume exclusion.

Let $K$ be a imaginary quadratic field where the prime $p$ splits. Our goal in this article is to prove results towards the Iwasawa main conjectures for $p$-nearly-ordinary families associated to $\mathrm{GL}_2\times\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$ with a minimal set of assumptions. The main technical input is an improvement on the locally restricted Euler system machinery that allows the treatment of residually reducible cases, which we apply with the Beilinson--Flach Euler system.

In earlier work, we analyzed the impossibility of codimension-one collapse for surfaces of negative Euler characteristic under the condition of a lower bound for the Gaussian curvature. Here we show that, under similar conditions, the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can collapse to a segment; we show that in such a situation, the loops in a short basis for homology must stay a uniform distance apart.

We prove convergence of a finite difference approximation of the compressible Navier--Stokes system towards the strong solution in $R^d,$ $d=2,3,$ for the adiabatic coefficient $\gamma>1$. Employing the relative energy functional, we find a convergence rate which is \emph{uniform} in terms of the discretization parameters for $\gamma \geq d/2$. All results are \emph{unconditional} in the sense that we have no assumptions on the regularity nor boundedness of the numerical solution. We also provide numerical experiments to validate the theoretical convergence rate. To the best of our knowledge this work contains the first unconditional result on the convergence of a finite difference scheme for the unsteady compressible Navier--Stokes system in multiple dimensions.

We are interested in the question of the existence of flat manifolds for which all $\mathbb R$-irreducible components of the holonomy representation are either absolutely irreducible, of complex or of quaternionic type. In the first two cases such examples are well known. But the existence of the third type of flat manifolds was unknown to the authors. In this article we construct such an example. Moreover, we present a list of finite groups for which a construction of manifolds of quaternionic type is impossible.

We give a lower bound for the numerical index of two-dimensional real spaces with absolute and symmetric norm. This allows us to compute the numerical index of the two-dimensional real $L_p$-space for $3/2\leq p\leq 3$.

In this article we use the method of the Bellman function to characterize the measures for which the weighted dual Hardy's inequality holds on dyadic trees. We also give an explicit interpretation of the corresponding Bellman function in terms of the theory of stochastic optimal control.

We obtain sufficient conditions for solutions of the $m$th-order differential inequality $$\sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, u) \ge f (x) g (|u|) \quad \mbox{in } B_1 \setminus \{ 0 \}$$ to have a removable singularity at zero, where $a_\alpha$, $f$, and $g$ are some functions, and $B_1 = \{ x : |x| < 1 \}$ is a unit ball in ${\mathbb R}^n$.

Constructed examples demonstrate the exactness of these conditions.

We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in *M.

We provide a new description of the scaling limit of dimer fluctuations in homogeneous Aztec diamonds via the intrinsic conformal structure of a Lorentz-minimal surface in $\mathbb{R}^{2+1}$. This surface naturally appears as the limit of the graphs of origami maps associated to symmetric t-embeddings (Coulomb gauges) of Aztec diamonds, fitting the framework of [CLR1,CLR2].

Insect-borne diseases are diseases carried by insects affecting humans, animals or plants. They have the potential to generate massive outbreaks such as the Zika epidemic in 2015-2016 mostly distributed in the Americas, the Pacific and Southeast Asia, and the multi-foci outbreak caused by the bacterium {\it Xylella fastidiosa} in Europe in the 2010s. In this article, we propose and analyze the behavior of a spatially-explicit compartmental model adapted to pathosystems with fixed hosts and mobile vectors disseminating the disease. The behavior of this model based on a system of partial differential equations is complementarily characterized via a theoretical study of its equilibrium states and a numerical study of its transitive phase using global sensitivity analysis. The results are discussed in terms of implications concerning the surveillance and control of the disease over a medium-to-long temporal horizon.

The spectral deferred correction (SDC) method is class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well-known, for suitable problems it gains one order per iteration up to the order of the quadrature method of the collocation problem provided. This appealing feature enables an easy creation of flexible, high-order accurate methods for ODEs. A variation of SDC are multi-level spectral deferred corrections (MLSDC). Here, iterations are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. While there are several numerical examples which show its capabilities and efficiency, a theoretical convergence proof is still missing. This paper addresses this issue. A proof of the convergence of MLSDC, including the determination of the convergence rate in the time-step size, will be given and the results of the theoretical analysis will be numerically demonstrated. It turns out that there are restrictions for the advantages of this method over SDC regarding the convergence rate.

For a Dedekind domain $D$, let $\mathcal{P}(D)$ be the set of ideals of $D$ that are radical of a principal ideal. We show that, if $D,D'$ are Dedekind domains and there is an order isomorphism between $\mathcal{P}(D)$ and $\mathcal{P}(D')$, then the rank of the class groups of $D$ and $D'$ is the same.

A Borel probability measure $\mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(\mu)$. In this paper, we characterize all spectral measures in the field $\mathbb{Q}_p$ of $p$-adic numbers.

We discuss boundary control of a wave equation with a non-linear anti-damping boundary condition. We design structured finite-dimensional $H_\infty$-output feedback controllers which stabilize the infinite dimensional system exponentially in closed loop. The method is applied to control torsional vibrations in drilling systems with the goal to avoid slip-stick.

In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure $\mu$, we construct a multiresolution family $\mathscr{C}^\mu$ of windows, and then we use a weighted Jones' function $\hat{J}_2(\mu, x)$ to record how well lines approximate the distribution of mass in each window. We show that when $\mu$ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the converse also holds. Additionally, we present an algorithm for the construction of a rectifiable curve using appropriately chosen $\delta$-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many $\delta$-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.

A marked free monoid morphism is a morphism for which the image of each generator starts with a different letter, and immersions are the analogous maps in free groups. We show that the (simultaneous) PCP is decidable for immersions of free groups, and provide an algorithm to compute bases for the sets, called equalisers, on which the immersions take the same values. We also answer a question of Stallings about the rank of the equaliser.

Analogous results are proven for marked morphisms of free monoids.

The proliferation of connected devices and emergence of internet-of-everything represent a major challenge for broadband wireless networks. This requires a paradigm shift towards the development of innovative technologies for next generation wireless systems. One of the key challenges is the scarcity of spectrum, owing to the unprecedented broadband penetration rate in recent years. A promising solution is the proposal of visible light communications (VLC), which explores the unregulated visible light spectrum to enable high-speed communications, in addition to efficient lighting. This solution offers a wider bandwidth that can accommodate ubiquitous broadband connectivity to indoor users and offload data traffic from cellular networks. Although VLC is secure and able to overcome the shortcomings of RF systems, it suffers from several limitations, e.g., limited modulation bandwidth. In this respect, solutions have been proposed recently to overcome this limitation. In particular, most common orthogonal and non-orthogonal multiple access techniques initially proposed for RF systems, e.g., space-division multiple access (SDMA) and NOMA, have been considered in the context of VLC. In spite of their promising gains, the performance of these techniques is somewhat limited. Consequently, in this article a new and generalized multiple access technique, called rate-splitting multiple access (RSMA), is introduced and investigated for the first time in VLC networks. We first provide an overview of the key multiple access technologies used in VLC systems. Then, we propose the first comprehensive approach to the integration of RSMA with VLC systems. In our proposed framework, SINR expressions are derived and used to evaluate the weighted sum rate (WSR) of a two-user scenario. Our results illustrate the flexibility of RSMA in generalizing NOMA and SDMA, and its WSR superiority in the VLC context.

This is a brief historical note about famous Legendre's criterium for convergent of continued fraction expansion. The paper is written in Russian.

The power of Calder\'on transfer principle is well known when proving strong type and weak type inequalities for certain type of operators in ergodic theory. In this article we show that Calder\'on's argument can be extended to have a transfer principle to be able to prove weighted inequalities for those operators satisfying the condition of Calder\'on transfer principle. We also include some applications of our result.

Massive multiple-input multiple-output (MIMO) systems have attracted much attention lately due to the many advantages they provide over single-antenna systems. Owing to the many antennas, low-cost implementation and low power consumption per antenna are desired. To that end, massive MIMO structures with low-resolution analog-to-digital converters (ADC) have been investigated in many studies. However, the effect of a strong interferer in the adjacent band on quantized massive MIMO systems have not been examined yet. In this study, we analyze the performance of uplink massive MIMO with low-resolution ADCs under frequency selective fading with orthogonal frequency division multiplexing in the perfect and imperfect receiver channel state information cases. We derive analytical expressions for the bit error rate and ergodic capacity. We show that the interfering band can be suppressed by increasing the number of antennas or the oversampling rate when a zero-forcing receiver is employed.

The start up costs in many kinds of generators lead to complex cost structures, which in turn yield severe market loopholes in the locational marginal price (LMP) scheme. Convex hull pricing (a.k.a. extended LMP) is proposed to improve the market efficiency by providing the minimal uplift payment to the generators. In this letter, we consider a stylized model where all generators share the same generation capacity. We analyze the generators' possible strategic behaviors in such a setting, and then propose an index for market power quantification in the convex hull pricing schemes.

Inspired by the recent advances in deep learning (DL), this work presents a deep neural network aided decoding algorithm for binary linear codes. Based on the concept of deep unfolding, we design a decoding network by unfolding the alternating direction method of multipliers (ADMM)-penalized decoder. In addition, we propose two improved versions of the proposed network. The first one transforms the penalty parameter into a set of iteration-dependent ones, and the second one adopts a specially designed penalty function, which is based on a piecewise linear function with adjustable slopes. Numerical results show that the resulting DL-aided decoders outperform the original ADMM-penalized decoder for various low density parity check (LDPC) codes with similar computational complexity.

In this paper, a new take on the concept of an active subspace for reducing the dimension of the design parameter space in a multidisciplinary analysis and optimization (MDAO) problem is proposed. The new approach is intertwined with the concepts of adaptive parameter sampling, projection-based model order reduction, and a database of linear, projection-based reduced-order models equipped with interpolation on matrix manifolds, in order to construct an efficient computational framework for MDAO. The framework is fully developed for MDAO problems with linearized fluid-structure interaction constraints. It is applied to the aeroelastic tailoring, under flutter constraints, of two different flight systems: a flexible configuration of NASA's Common Research Model; and NASA's Aeroelastic Research Wing #2 (ARW-2). The obtained results illustrate the feasibility of the computational framework for realistic MDAO problems and highlight the benefits of the new approach for constructing an active subspace in both terms of solution optimality and wall-clock time reduction

We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.

Although freelancing work has grown substantially in recent years, in part facilitated by a number of online labor marketplaces, (e.g., Guru, Freelancer, Amazon Mechanical Turk), traditional forms of "in-sourcing" work continue being the dominant form of employment. This means that, at least for the time being, freelancing and salaried employment will continue to co-exist. In this paper, we provide algorithms for outsourcing and hiring workers in a general setting, where workers form a team and contribute different skills to perform a task. We call this model team formation with outsourcing. In our model, tasks arrive in an online fashion: neither the number nor the composition of the tasks is known a-priori. At any point in time, there is a team of hired workers who receive a fixed salary independently of the work they perform. This team is dynamic: new members can be hired and existing members can be fired, at some cost. Additionally, some parts of the arriving tasks can be outsourced and thus completed by non-team members, at a premium. Our contribution is an efficient online cost-minimizing algorithm for hiring and firing team members and outsourcing tasks. We present theoretical bounds obtained using a primal-dual scheme proving that our algorithms have a logarithmic competitive approximation ratio. We complement these results with experiments using semi-synthetic datasets based on actual task requirements and worker skills from three large online labor marketplaces.

In the inverse Gaussian sequence space model with additional noisy observations of the operator, we derive nonasymptotic minimax radii of testing for ellipsoid-type alternatives simultaneously for both the signal detection problem (testing against zero) and the goodness-of-fit testing problem (testing against a prescribed sequence) without any regularity assumption on the null hypothesis. The radii are the maximum of two terms, each of which only depends on one of the noise levels. Interestingly, the term involving the noise level of the operator explicitly depends on the null hypothesis and vanishes in the signal detection case. The minimax radii are established by first showing a lower bound for arbitrary null hypotheses and noise levels. For the upper bound we consider two testing procedures, a direct test based on estimating the energy in the image space and an indirect test. Under mild assumptions, we prove that the testing radius of the indirect test achieves the lower bound, which shows the minimax optimality of the radius and the test. We highlight the assumptions under which the direct test also performs optimally. Furthermore, we apply a classical Bonferroni method for making both the indirect and the direct test adaptive with respect to the regularity of the alternative. The radii of the adaptive tests are deteriorated by an additional log-factor, which we show to be unavoidable. The results are illustrated considering Sobolev spaces and mildly or severely ill-posed inverse problems.

Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general structured principal subspace estimation problem which includes as special cases non-negative PCA/SVD, sparse PCA/SVD, subspace constrained PCA/SVD, and spectral clustering. General minimax lower and upper bounds are established to characterize the interplay between the information-geometric complexity of the structural set for the principal subspaces, the signal-to-noise ratio (SNR), and the dimensionality. The results yield interesting phase transition phenomena concerning the rates of convergence as a function of the SNRs and the fundamental limit for consistent estimation. Applying the general results to the specific settings yields the minimax rates of convergence for those problems, including the previous unknown optimal rates for non-negative PCA/SVD, sparse SVD and subspace constrained PCA/SVD.

In this paper, we propose to use the discretized version of the so-called Enhanced Gaussian Noise (EGN) model to estimate the non-linearity effects of fiber on the performance of optical coherent and uncompensated transmission (CUT) systems. By computing the power of non-linear interference noise and considering optical amplifier noise, we obtain the signal-to-noise (SNR) ratio and achievable rate of CUT. To allocate the power of each CUT channel, we consider two optimization problems with the objectives of maximizing minimum SNR margin and achievable rate. We show that by using the discretized EGN model, the complexity of the introduced optimization problems is reduced compared with the existing optimization problems developed based on the so-called discretized Gaussian Noise (GN) model. In addition, the optimization based on the discretized EGN model leads to a better SNR and achievable rate. We validate our analytical results with simulations and experimental results. We simulate a five-channel coherent system on OptiSystem software, where a close agreement is observed between optimizations and simulations. Furthermore, we measured SNR of commercial 100Gbps coherent transmitter over 300 km single-mode fiber (SMF) and non-zero dispersion-shifted fiber (NZDSF), by considering single-channel and three-channel coherent systems. We observe there are performance gaps between experimental and analytical results, which is mainly due to other sources of noise such as transmitter imperfection noise, thermal noise, and shot noise, in experiments. By including these sources of noise in the analytical model, the gaps between analytical and experimental results are reduced.

Iterative linear quadradic regulator(iLQR) has become a benchmark method to deal with nonlinear stochastic optimal control problem. However, it does not apply to delay system. In this paper, we extend the iLQR theory and prove new theorem in case of input signal with fixed delay. Which could be beneficial for machine learning or optimal control application to real time robot or human assistive device.

We consider the problem of downlink power control in wireless networks, consisting of multiple transmitter-receiver pairs communicating with each other over a single shared wireless medium. To mitigate the interference among concurrent transmissions, we leverage the network topology to create a graph neural network architecture, and we then use an unsupervised primal-dual counterfactual optimization approach to learn optimal power allocation decisions. We show how the counterfactual optimization technique allows us to guarantee a minimum rate constraint, which adapts to the network size, hence achieving the right balance between average and $5^{th}$ percentile user rates throughout a range of network configurations.

The goal of this paper is to develop a novel numerical method for efficient multiplicative noise removal. The nonlocal self-similarity of natural images implies that the matrices formed by their nonlocal similar patches are low-rank. By exploiting this low-rank prior with application to multiplicative noise removal, we propose a nonlocal low-rank model for this task and develop a proximal alternating reweighted minimization (PARM) algorithm to solve the optimization problem resulting from the model. Specifically, we utilize a generalized nonconvex surrogate of the rank function to regularize the patch matrices and develop a new nonlocal low-rank model, which is a nonconvex nonsmooth optimization problem having a patchwise data fidelity and a generalized nonlocal low-rank regularization term. To solve this optimization problem, we propose the PARM algorithm, which has a proximal alternating scheme with a reweighted approximation of its subproblem. A theoretical analysis of the proposed PARM algorithm is conducted to guarantee its global convergence to a critical point. Numerical experiments demonstrate that the proposed method for multiplicative noise removal significantly outperforms existing methods such as the benchmark SAR-BM3D method in terms of the visual quality of the denoised images, and the PSNR (the peak-signal-to-noise ratio) and SSIM (the structural similarity index measure) values.

Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs using dimension reduction. A succinct way to state our result is as a step toward the following conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G \cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any element $g\in [g]$. We prove a version of the conjecture for the case of closed manifolds and the case of torus links labeled by pure fluxes.

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal.

In this article we show that $(\mathbb{Z}_2\oplus \mathbb{Z}_2)$-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

For the purpose of addressing the multi-objective optimal reactive power dispatch (MORPD) problem, a two-step approach is proposed in this paper. First of all, to ensure the economy and security of the power system, the MORPD model aiming to minimize active power loss and voltage deviation is formulated. And then the two-step approach integrating decision-making into optimization is proposed to solve the model. Specifically speaking, the first step aims to seek the Pareto optimal solutions (POSs) with good distribution by using a multi-objective optimization (MOO) algorithm named classification and Pareto domination based multi-objective evolutionary algorithm (CPSMOEA). Furthermore, the reference Pareto-optimal front is generated to validate the Pareto front obtained using CPSMOEA; in the second step, integrated decision-making by combining fuzzy c-means algorithm (FCM) with grey relation projection method (GRP) aims to extract the best compromise solutions which reflect the preferences of decision-makers from the POSs. Based on the test results on the IEEE 30-bus and IEEE 118-bus test systems, it is demonstrated that the proposed approach not only manages to address the MORPD issue but also outperforms other commonly-used MOO algorithms including multi-objective particle swarm optimization (MOPSO), preference-inspired coevolutionary algorithm (PICEAg) and the third evolution step of generalized differential evolution (GDE3).

We demonstrate how the key notions of Tononi et al.'s Integrated Information Theory (IIT) can be studied within the simple graphical language of process theories, i.e. symmetric monoidal categories. This allows IIT to be generalised to a broad range of physical theories, including as a special case the Quantum IIT of Zanardi, Tomka and Venuti.

Integrated Information Theory is one of the leading models of consciousness. It aims to describe both the quality and quantity of the conscious experience of a physical system, such as the brain, in a particular state. In this contribution, we propound the mathematical structure of the theory, separating the essentials from auxiliary formal tools. We provide a definition of a generalized IIT which has IIT 3.0 of Tononi et. al., as well as the Quantum IIT introduced by Zanardi et. al. as special cases. This provides an axiomatic definition of the theory which may serve as the starting point for future formal investigations and as an introduction suitable for researchers with a formal background.

We consider a system of linear equations, whose coefficients depend linearly on interval parameters. Its solution set is defined as the set of all solutions of all admissible realizations of the parameters. We study unbounded directions of the solution set and its relation with its kernel. The kernel of a matrix characterizes unbounded direction in the real case and in the case of ordinary interval systems. In the general parametric case, however, this is not completely true. There is still a close relation preserved, which we discuss in the paper. Nevertheless, we identify several special sub-classes, for which the characterization remains valid. Next, we extend the results to the so called AE parametric systems, which are defined by forall-exists quantification.

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. Boundary-domain integral equations are formulated in terms of parametrix-based potential type integral operators defined on the boundary and the domain. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

In the present paper, we obtain the basic Chen inequalities for submanifolds of quaternion Kaehler-like statistical manifolds. Also, we discuss the same inequality for Lagrangian submanifolds.

We prove an interior Schauder estimate for the Laplacian on metric products of two dimensional cones with a Euclidean factor, generalizing the work of Donaldson and reproving the Schauder estimate of Guo-Song. We characterize the space of homogeneous subquadratic harmonic functions on products of cones, and identify scales at which geodesic balls can be well approximated by balls centered at the apex of an appropriate model cone. We then locally approximate solutions by subquadratic harmonic functions at these scales to measure the H\"older continuity of second derivatives.

In 1913 A.D. Bilimovich observed that rheonomic linear and homogeneous in generalized velocities constraints are ideal. As a typical example, he considered rheonomic nonholonomic deformation of the Euler equations which scleronomic version is equivalent to the nonholonomic Suslov system. For the Bilimovich system equations of motion are reduced to quadrature, which is discussed in rheonomic and scleronomic cases.

In this paper we characterize the performance of a class of maximum-a-posteriori (MAP) detectors for network systems driven by unknown stochastic inputs, as a function of the location of the sensors and the topology of the network. We consider two scenarios: one in which the changes occurs in the mean of the input, and the other where the changes are allowed to happen in the covariance (or power) of the input. In both the scenarios, to detect the changes, we associate suitable MAP detectors for a given set of sensors, and study its detection performance as function of the network topology, and the graphical distance between the input nodes and the sensors location. When the input and measurement noise follow a Gaussian distribution, we show that, as the number of measurements goes to infinity, the detectors' performance can be studied using the input to output gain of the transfer function of the network system. Using this characterization, we derive conditions under which the detection performance obtained when the sensors are located on a network cut is not worse (resp. not better) than the performance obtained by measuring all nodes of the subnetwork induced by the cut and not containing the input nodes. Our results provide structural insights into the sensor placement from a detection-theoretic viewpoint. Finally, we illustrate our findings via numerical examples.

Kolmogorov complexity is the length of the ultimately compressed version of a file (that is, anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. We discuss the incomputabilty of Kolmogorov complexity, which formal loopholes this leaves us, recent approaches to compute or approximate Kolmogorov complexity, which approaches fail and which approaches are viable.

We consider extension of a closure system on a finite set S as a closure system on the same set S containing the given one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set of its meet-irreducible elements. When a closure system is described by an implicational base, we provide a characterization of the implicational base for the largest extension. We also show that the largest extension can be handled by a small modification of the implicational base of the input closure system. This answers a question asked in [12]. Second, we are interested in computing the largest extension when the closure system is given by the set of all its meet-irreducible elements. We give an incremental polynomial time algorithm to compute the largest extension of a closure system, and left open if the number of meet-irreducible elements grows exponentially.

Existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on R^n.

We give a topological proof that a free inverse monoid on one or more generators is neither of type left-$FP_2$ nor right-$FP_2$. This strengthens a classical result of Schein that such monoids are not finitely presented as monoids.

We consider cache-aided wireless communication scenarios where each user requests both a file from an a-priori generated cacheable library (referred to as 'content'), and an uncacheable 'non-content' message generated at the start of the wireless transmission session. This scenario is easily found in real-world wireless networks, where the two types of traffic coexist and share limited radio resources. We focus on single-transmitter, single-antenna wireless networks with cache-aided receivers, where the wireless channel is modelled by a degraded Gaussian broadcast channel (GBC). For this setting, we study the delay-rate trade-off, which characterizes the content delivery time and non-content communication rates that can be achieved simultaneously. We propose a scheme based on the separation principle, which isolates the coded caching and multicasting problem from the physical layer transmission problem. We show that this separation-based scheme is sufficient for achieving an information-theoretically order optimal performance, up to a multiplicative factor of 2.01 for the content delivery time, when working in the generalized degrees of freedom (GDoF) limit. We further show that the achievable performance is near-optimal after relaxing the GDoF limit, up to an additional additive factor of 2 bits per dimension for the non-content rates. A key insight emerging from our scheme is that in some scenarios considerable amounts of non-content traffic can be communicated while maintaining the minimum content delivery time, achieved in the absence of non-content messages; compliments of 'topological holes' arising from asymmetries in wireless channel gains.

We introduce the geodesic complexity of a metric space, inspired by the topological complexity of a topological space. Both of them are numerical invariants, but, while the TC only depends on the homotopy type, the GC is an invariant under isometries. We show that in some cases they coincide but we also develop tools to distinguish the two in a range of examples. To this end, we study what we denote the total cut locus, which does not appear to have been explicitly considered in the literature. To the knowledge of the author, the GC is a new invariant of a metric space. Furthermore, just like the TC, the GC has potential applications to the field of robotics.

A cycle $C$ of a graph $G$ is \emph{isolating} if every component of $G-V(C)$ is a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle $C$ of length $8 \leq |E(C)| < \frac{2}{3}(|V(G)|+3)$ implies an isolating cycle $C'$ of larger length that contains $V(C)$. By hopping'' iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving and computing long cycles (we will give an algorithm with running time $O(n^2)$). This provides a method to prove lower bounds on Tutte cycles, as $C'$ will be a Tutte cycle of $G$ if $C$ is. We also prove that $E(C') \leq E(C)+3$ if $G$ does not contain faces of size five, which gives a new tool for proving results about cycle spectra and evidence that these face sizes obstruct long cycles. As a sample application, we test our motor on a conjecture on essentially 4-connected graphs. A planar graph is \emph{essentially $4$-connected} if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Essentially $4$-connected graphs have been thoroughly investigated throughout literature as the subject of Hamiltonicity studies. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2}{5}(n+2)$, and this result has recently been improved multiple times, culminating in the lower bound $\frac{5}{8}(n+2)$. However, the best known upper bound is given by an infinite family of such graphs in which every graph $G$ on $n$ vertices has no cycle longer than $\frac{2}{3}(n+4)$; this upper bound is still unmatched. Using isolating cycles, we improve the lower bound to match the upper (up to a summand $+1$). This settles the long-standing open problem of determining the circumference of essentially 4-connected planar graphs.

In this study, we propose a differentiable layer for OFDM-based autoencoders (OFDM-AEs) to avoid high instantaneous power without regularizing the cost function used during the training. The proposed approach relies on the manipulation of the parameters of a set of functions that yield complementary sequences (CSs) through a deep neural network (DNN). We guarantee the peak-to-average-power ratio (PAPR) of each OFDM-AE symbol to be less than or equal to 3 dB. We also show how to normalize the mean power by using the functions in addition to PAPR. The introduced layer admits auxiliary parameters that allow one to control the amplitude and phase deviations in the frequency domain. Numerical results show that DNNs at the transmitter and receiver can achieve reliable communications under this protection layer at the expense of complexity.

We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant,

$$\lambda_{s,p}(\Omega):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(\Omega),\,\, \|u\|_{L^p(\Omega)}=1 \right\},$$

where $s\in (0,1)$, $\Omega$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $\Omega\mapsto \lambda_{s,p}(\Omega)$ under domain perturbations. In the case where $\lambda_{s,p}(\Omega)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal version of the classical variational Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on $\Omega$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $\lambda_{s,p}(\Omega)$ under volume-preserving deformations, and we find that they are balls if $p\in \{1\}\cup [2,\infty)$. Finally, we consider the maximization problem for $\lambda_{s,p}(\Omega)$ among annular-shaped domains of fixed volume of the type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball whose position is varied within $B$. We prove that, for $p\in \{1,2\}$, the value $\lambda_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.

We introduce and study a new optimal transport problem on a bounded domain $\Omega\subset \mathbb{R}^d$, defined via a dynamical Benamou-Brenier formulation. The model handles differently the motion in theinterior and on the boundary, and penalizes the transfer of mass between the two. The resulting distance interpolates between classical optimal transport on $\bar\Omega$ on the one hand, and on the other hand between two independent optimal transport problems set on $\Omega$ and $\partial\Omega$.

We develop a generic data-driven method for estimator selection in off-policy policy evaluation settings. We establish a strong performance guarantee for the method, showing that it is competitive with the oracle estimator, up to a constant factor. Via in-depth case studies in contextual bandits and reinforcement learning, we demonstrate the generality and applicability of the method. We also perform comprehensive experiments, demonstrating the empirical efficacy of our approach and comparing with related approaches. In both case studies, our method compares favorably with existing methods.

Let $k$ be an algebraically closed field of positive characteristic. For any integer $m \geq 2$, we show that the Hodge numbers of a smooth projective $k$-variety can take on any combination of values modulo $m$, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

We consider level-sets of the Gaussian free field on $\mathbb Z^d$, for $d\geq 3$, above a given real-valued height parameter $h$. As $h$ varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, namely $h_{**}(d)$, $h_{*}(d)$ and $\bar h(d)$, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide, i.e. $h_{**}(d)=h_{*}(d)= \bar h(d)$ for any $d \geq 3$. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control so-called large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.

We highlight what seems to be a remaining subtlety in the argument for the cancellation of the total anomaly associated with the M5-brane in M-theory. Then we prove that this subtlety is resolved under the hypothesis that the C-field flux is charge-quantized in the generalized cohomology theory called J-twisted Cohomotopy.

In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to [15], piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $\mathbf{No}$, which includes $\mathbf{No}$ itself, extend to canonical exponential functions on their surcomplex counterparts. This uses the precursory result that trigonometric-exponential initial subfields of $\mathbf{No}$ and trigonometric ordered initial subfields of $\mathbf{No}$, more generally, admit canonical sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $\mathbf{No}$, to the image of the canonical map of the ordered exponential field $\mathbb{T}$ of transseries into $\mathbf{No}$, which is shown to be initial, and to the ordered exponential fields $\mathbb{R}((\omega))^{EL}$ and $\mathbb{R}\langle\langle\omega\rangle \rangle$, which are likewise shown to be initial.

We give two proofs of a level-rank duality for braided fusion categories obtained from quantum groups of type $C$ at roots of unity. The first proof uses conformal embeddings, while the second uses a classification of braided fusion categories associated with quantum groups of type $C$ at roots of unity. In addition we give a similar result for non-unitary braided fusion categories quantum groups of types $B$ and $C$ at odd roots of unity.

We consider N-fold integer programming problems. After a decade of continuous progress, the currently fastest algorithm for N-fold integer programming by Jansen et al. (2019) has a running time of $(rs\Delta)^{O(r^2s + s^2)} {\phi}^2 \cdot nt \log^{O(1)}(nt)$. Here ${\phi}$ is the largest binary encoding length of a number in the input. This algorithm, like its predecessors are based on the augmentation framework, a tailored integer programming variant of local search. In this paper we propose a different approach that is not based on augmentation. Our algorithm relies on a stronger LP-relaxation of the N-fold integer program instead. This relaxation can be solved in polynomial time with parameter dependence $(s{\Delta})^{O(s^2)}$ by resorting to standard techniques from convex optimization. We show that, for any given optimal vertex solution $x^*$ of this relaxation, there exists an optimal integer solution $z^*$ that is within short $\ell_1$-distance, namely $\|x^* - z^*\|_{1} \leq (rs\Delta)^{O(rs)}$. With dynamic programming one can then find an optimal integer solution of the N-fold IP in time $(rs\Delta)^{O(r^2s + s^2)} \,nt$. This, together with an off-the-shelf-method from convex optimization, results in the currently fastest algorithm for N-fold integer programming.

A secure multi-party batch matrix multiplication problem (SMBMM) is considered, where the goal is to allow a master node to efficiently compute the pairwise products of two batches of massive matrices that originate at external source nodes, by distributing the computation across $S$ honest but curious servers. Any group of up to $X$ colluding servers should gain no information about the input matrices, and the master should gain no additional information about the input matrices beyond the product. A solution called Generalized Cross Subspace Alignment codes with Noise Alignment (GCSA-NA in short) is proposed in this work, based on cross-subspace alignment codes. These codes originated in secure private information retrieval, and have recently been applied to distributed batch computation problems where they generalize and improve upon the state of art schemes such as Entangled Polynomial Codes and Lagrange Coded Computing. The prior state of art solution to SMBMM is a coding scheme called polynomial sharing (PS) that was proposed by Nodehi and Maddah-Ali. GCSA-NA outperforms PS codes in several key aspects -- more efficient and secure inter-server communication (which can entirely take place beforehand, i.e., even before the input matrices are determined), flexible inter-server network topology, efficient batch processing, and tolerance to stragglers. The idea of noise-alignment can also be applied to construct schemes for $N$ sources based on $N$-CSA codes, and to construct schemes for symmetric secure private information retrieval to achieve the asymptotic capacity.

To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic curve. We study gonality sequences for graphs of low genus, proving that for genus up to $5$, the gonality sequence is determined by the genus and the first gonality. We then prove that any reasonable pair of first two gonalities is achieved by some graph. We also develop a modified version of Dhar's burning algorithm more suited for studying higher gonalities.

Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measurevalued solutions to the two-dimensional isentropic compressible Euler equations, although they are energy admissible, can be discarded as unphysical, as they do not arise as vanishing viscosity limits. In fact, these measure-valued solutions also do not arise from a sequence of weak solutions of the Euler equations, in contrast to the incompressible case. Such a phenomenon has already been observed by Chiodaroli, Feireisl, Kreml, and Wiedemann using an $\mathcal{A}$-free rigidity argument, but only for non-deterministic initial datum. We develop their rigidity result to the case of nonconstant states and combine this with a compression wave solution evolving into infinitely many weak solutions, constructed by Chiodaroli, De Lellis, and Kreml. Hereby, we show that there exist infinitely many generalized measure-valued solutions to the two-dimensional isentropic Euler system with quadratic pressure law, which behave deterministically up to a certain time and which cannot be generated by weak solutions with bounded energy or by vanishing viscosity sequences.

In this paper, we study various asymptotic properties (bias, variance, mean squared error, mean integrated squared error, asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the $d$-dimensional simplex. Our results generalize the ones in Leblanc (2012) and Babu et al. (2002), which treated the case $d = 1$, and significantly extend those found in Tenbusch (1994) for the density estimators when $d = 2$. The density estimator (or smoothed histogram) is closely related to the Dirichlet kernel estimator from Ouimet (2020), and can also be used to analyze compositional data.

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures, whose correlation functions are all given by determinants specified by an integral kernel called the correlation kernel. First we show our new scheme of DPPs in which a notion of partial isometies between a pair of Hilbert spaces plays an important role. Many examples of DPPs in one-, two-, and higher-dimensional spaces are demonstrated, where several types of weak convergence from finite DPPs to infinite DPPs are given. Dynamical extensions of DPP are realized in one-dimensional systems of diffusive particles conditioned never to collide with each other. They are regarded as one-dimensional stochastic log-gases, or the two-dimensional Coulomb gases confined in one-dimensional spaces. In the second section, we consider such interacting particle systems in one dimension. We introduce a notion of determinantal martingale and prove that, if the system has determinantal martingale representation (DMR), then it is a determinantal stochastic process (DSP) in the sense that all spatio-temporal correlation function are expressed by a determinant. In the last section, we construct processes of Gaussian free fields (GFFs) on simply connected proper subdomains of ${\mathbb{C}}$ coupled with interacting particle systems defined on boundaries of the domains. There we use multiple Schramm--Loewner evolutions (SLEs) driven by the interacting particle systems. We prove that, if the driving processes are time-changes of the log-gases studied in the second section, then the obtained GFF with multiple SLEs are stationary. The stationarity defines an equivalence relation of GFFs, which will be regarded as a generalization of the imaginary surface studied by Miller and Sheffield.

We study point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a $(2+\delta)$th moment. We make heavily use of precise large deviation results for sums of iid random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.

The global Gan-Gross-Prasad conjecture predicts that the non-vanishing of certain periods is equivalent to the non-vanishing of the central value of certain Rankin-Selberg $L$-functions. There are two types of Gan-Gross-Prasad conjecture: Bessel case, Fourier-Jacobi case. We prove one direction of the full Fourier-Jacobi GGP conjecture on skew-hermitian unitary groups.

Iommi & Kiwi showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture of Iommi & Kiwi by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question of Iommi & Kiwi by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

The cubic Szego equation has been studied as an integrable model for deterministic turbulence, starting with the foundational work of Gerard and Grellier. We introduce a truncated version of this equation, wherein a majority of the Fourier mode couplings are eliminated while the signature features of the model are preserved, namely, a Lax-pair structure and a nested hierarchy of finite-dimensional dynamically invariant manifolds. Despite the impoverished structure of the interactions, the turbulent behaviors of our new equation are stronger in an appropriate sense than for the original cubic Szego equation. We construct explicit analytic solutions displaying exponential growth of Sobolev norms. We furthermore introduce a family of models that interpolate between our truncated system and the original cubic Szego equation, along with a few other related deformations. All of these models possess Lax pairs, invariant manifolds, and display a variety of turbulent cascades. We additionally mention numerical evidence that shows an even stronger type of turbulence in the form of a finite-time blow-up in some different, closely related dynamical systems.

We prove the absence of positive real resonances for Schr\"odinger operators with finitely many point interactions in $\mathbb{R}^3$ and we discuss such a property from the perspective of dispersive and scattering features of the associated Schr\"odinger propagator.

The Fields Medal, often referred as the Nobel Prize of mathematics, is awarded to no more than four mathematician under the age of 40, every four years. In recent years, its conferral has come under scrutiny of math historians, for rewarding the existing elite rather than its original goal of elevating mathematicians from under-represented communities. Prior studies of elitism focus on citational practices and sub-fields; the structural forces that prevent equitable access remain unclear. Here we show the flow of elite mathematicians between countries and lingo-ethnic identity, using network analysis and natural language processing on 240,000 mathematicians and their advisor-advisee relationships. We found that the Fields Medal helped integrate Japan after WWII, through analysis of the elite circle formed around Fields Medalists. Arabic, African, and East Asian identities remain under-represented at the elite level. Through analysis of inflow and outflow, we rebuts the myth that minority communities create their own barriers to entry. Our results demonstrate concerted efforts by international academic committees, such as prize-giving, are a powerful force to give equal access. We anticipate our methodology of academic genealogical analysis can serve as a useful diagnostic for equality within academic fields.

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Tur\'an's inequality.

We show that the monodromy theorem holds on arbitrary connected free sets for noncommutative free analytic functions. Applications are numerous-- pluriharmonic free functions have globally defined pluriharmonic conjugates, locally invertible functions are globally invertible, and there is no nontrivial cohomology theory arising from analytic continuation on connected free sets. We describe why the Baker-Campbell-Hausdorff formula has finite radius of convergence in terms of monodromy, and solve a related problem of Martin-Shamovich. We generalize the Dym-Helton-Klep-McCullough-Volcic theorem-- a uniformly real analytic free noncommutative function is plurisubharmonic if and only if it can be written as a composition of a convex function with an analytic function. The decomposition is essentially unique. The result is first established locally, and then Free Universal Monodromy implies the global result. Moreover, we see that plurisubharmonicity is a geometric property-- a real analytic free function plurisubharmonic on a neighborhood is plurisubharmonic on the whole domain. We give an analytic Greene-Liouville theorem, an entire free plurisubharmonic function is a sum of hereditary and antihereditary squares.

The design of symbol detectors in digital communication systems has traditionally relied on statistical channel models that describe the relation between the transmitted symbols and the observed signal at the receiver. Here we review a data-driven framework to symbol detection design which combines machine learning (ML) and model-based algorithms. In this hybrid approach, well-known channel-model-based algorithms such as the Viterbi method, BCJR detection, and multiple-input multiple-output (MIMO) soft interference cancellation (SIC) are augmented with ML-based algorithms to remove their channel-model-dependence, allowing the receiver to learn to implement these algorithms solely from data. The resulting data-driven receivers are most suitable for systems where the underlying channel models are poorly understood, highly complex, or do not well-capture the underlying physics. Our approach is unique in that it only replaces the channel-model-based computations with dedicated neural networks that can be trained from a small amount of data, while keeping the general algorithm intact. Our results demonstrate that these techniques can yield near-optimal performance of model-based algorithms without knowing the exact channel input-output statistical relationship and in the presence of channel state information uncertainty.

On 12 February 2020 the Royal Statistical Society hosted a meeting to discuss the forthcoming paper Graphical models for extremes'' by Sebastian Engelke and Adrien Hitz [arXiv:1812.01734]. This short note is a supplement to my discussion contribution. It contains the proofs. It is shown that the traditional notion of extremal independence agrees with the newly introduced notion of extremal independence, which subsequently allows for a meaningful interpretation of disconnected graphs in the context of the discussion paper. The notation and references used in this note are adopted from the discussion paper.

The famous singlet correlations of a composite quantum system consisting of two spatially separated components exhibit notable features of two kinds. The first kind consists of striking certainty relations: perfect correlation and perfect anti-correlation in certain settings. The second kind consists of a number of symmetries, in particular, invariance under rotation, as well as invariance under exchange of components, parity, or chirality. In this note, I investigate the class of correlation functions that can be generated by classical composite physical systems when we restrict attention to systems which reproduce the certainty relations exactly, and for which the rotational invariance of the correlation function is the manifestation of rotational invariance of the underlying classical physics. I call such correlation functions classical EPR-B correlations. It turns out that the other three (binary) symmetries can then be obtained "for free": they are exhibited by the correlation function, and can be imposed on the underlying physics by adding an underlying randomisation level. We end up with a simple probabilistic description of all possible classical EPR-B correlations in terms of a "spinning coloured disk" model, and a research programme: describe these functions in a concise analytic way. We survey open problems, and we show that the widespread idea that "quantum correlations are more extreme than classical physics allows" is at best highly inaccurate, through giving a concrete example of a classical correlation which satisfies all the symmetries and all the certainty relations and which exceeds the quantum correlations over a whole range of settings

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained.

We demonstrate that the Betti numbers associated to an N-graded minimal free resolution of the Stanley-Reisner ring of the (d-1)-skeleton of a simplicial complex of dimension d can be expressed as a Z-linear combination of the corresponding Betti numbers of the complex itself. An immediate implication of our main result is that the projective dimension of the Stanley-Reisner ring of the (d-1)-skeleton is at most one greater than the projective dimension of the Stanley-Reisner ring of the original complex, and it thus provides a new and direct proof of this. Our result extends immediately to matroids and their truncations. A similar result for matroid elongations can not be hoped for, but we do obtain a weaker result for these.

This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at vanishing shear viscosity, both the boundary-layer thickness and the boundary-layer solution are discussed. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are based on the global (uniform) estimates which are achieved by making a full use of the "effective viscous flux", the material derivatives and the structure of the one-dimensional equations.

Given any suitably small, localized, and smooth initial data, in this paper, we prove global regularity for the $3D$ finite depth gravity water wave system. As a byproduct, we rule out the small, localized traveling waves in $3D$, which do exist for the same system in $2D$.

This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. To capture the behavior of the solutions at small shear viscosity, we also discuss the boundary-layer thickness and the boundary-layer solution. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are based on the global (uniform) estimates which are achieved by making a full use of the "effective viscous flux", the material derivatives and the structure of the one-dimensional equations. Moreover, the lower positive bound of the density is obtained by using some new ideas, which are rather different from those in the existing literature.

In this paper we propose a new fast Fourier transform to recover a real nonnegative signal ${\bf x}$ from its discrete Fourier transform. If the signal ${\mathbf x}$ appears to have a short support, i.e., vanishes outside a support interval of length $m < N$, then the algorithm has an arithmetical complexity of only ${\cal O}(m \log m \log (N/m))$ and requires ${\cal O}(m \log (N/m))$ Fourier samples for this computation. In contrast to other approaches there is no a priori knowledge needed about sparsity or support bounds for the vector ${\bf x}$. The algorithm automatically recognizes and exploits a possible short support of the vector and falls back to a usual radix-2 FFT algorithm if ${\bf x}$ has (almost) full support. The numerical stability of the proposed algorithm ist shown by numerical examples.

We present a novel inference approach that we call Sample Out-of-Sample (or SOS) inference. The approach can be used widely, ranging from semi-supervised learning to stress testing, and it is fundamental in the application of data-driven Distributionally Robust Optimization (DRO). Our method enables measuring the impact of plausible out-of-sample scenarios in a given performance measure of interest, such as a financial loss. The methodology is inspired by Empirical Likelihood (EL), but we optimize the empirical Wasserstein distance (instead of the empirical likelihood) induced by observations. From a methodological standpoint, our analysis of the asymptotic behavior of the induced Wasserstein-distance profile function shows dramatic qualitative differences relative to EL. For instance, in contrast to EL, which typically yields chi-squared weak convergence limits, our asymptotic distributions are often not chi-squared. Also, the rates of convergence that we obtain have some dependence on the dimension in a non-trivial way but remain controlled as the dimension increases.

It has been proven in previous papers that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable.

This paper studies models in which hypothesis tests have trivial power, that is, power smaller than size. This testing impossibility, or impossibility type A, arises when any alternative is not distinguishable from the null. We also study settings in which it is impossible to have almost surely bounded confidence sets for a parameter of interest. This second type of impossibility (type B) occurs under a condition weaker than the condition for type A impossibility: the parameter of interest must be nearly unidentified. Our theoretical framework connects many existing publications on impossible inference that rely on different notions of topologies to show models are not distinguishable or nearly unidentified. We also derive both types of impossibility using the weak topology induced by convergence in distribution. Impossibility in the weak topology is often easier to prove, it is applicable for many widely-used tests, and it is useful for robust hypothesis testing. We conclude by demonstrating impossible inference in multiple economic applications of models with discontinuity and time-series models.

Winning probabilities of The Hat Game (Ebert's Hat Problem) with three players and three colors are only known in the symmetric case: all probabilities of the colors are equal. This paper solves the asymmetric case: probabilities may be different. We find winning probabilies and optimal strategies in all cases.

In the problem of compressive phase retrieval, one wants to recover an approximately $k$-sparse signal $x \in \mathbb{C}^n$, given the magnitudes of the entries of $\Phi x$, where $\Phi \in \mathbb{C}^{m \times n}$. This problem has received a fair amount of attention, with sublinear time algorithms appearing in \cite{cai2014super,pedarsani2014phasecode,yin2015fast}. In this paper we further investigate the direction of sublinear decoding for real signals by giving a recovery scheme under the $\ell_2 / \ell_2$ guarantee, with almost optimal, $\Oh(k \log n )$, number of measurements. Our result outperforms all previous sublinear-time algorithms in the case of real signals. Moreover, we give a very simple deterministic scheme that recovers all $k$-sparse vectors in $\Oh(k^3)$ time, using $4k-1$ measurements.

We describe semiparametric estimation and inference for causal effects using observational data from a single social network. Our asymptotic result is the first to allow for dependence of each observation on a growing number of other units as sample size increases. While previous methods have generally implicitly focused on one of two possible sources of dependence among social network observations, we allow for both dependence due to transmission of information across network ties, and for dependence due to latent similarities among nodes sharing ties. We describe estimation and inference for new causal effects that are specifically of interest in social network settings, such as interventions on network ties and network structure. Using our methods to reanalyze the Framingham Heart Study data used in one of the most influential and controversial causal analyses of social network data, we find that after accounting for network structure there is no evidence for the causal effects claimed in the original paper.

In this note we determine all possible dominations between different products of manifolds, when none of the factors of the codomain is dominated by products. As a consequence, we determine the finiteness of every product-associated functorial semi-norm on the fundamental classes of the aforementioned products. These results give partial answers to questions of M. Gromov.

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of commutativity' dc_M(G) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative algebraic consequences of the degree of commutativity being above certain thresholds. For example, if m_n is the distribution of the nth step of a symmetric random walk on G, or if G is amenable and (m_n) is a sequence of almost-invariant measures on G, we show that if dc_M(G) is at least a>0 then G contains a normal subgroup G' of index f(a) and a normal subgroup H of cardinality at most g(a) such that G'/H is abelian. This generalises a result for finite groups due to P. M. Neumann, and generalises and quantifies a result for certain residually finite groups of subexponential growth due to Antolin, Martino and Ventura. We also describe some general conditions on M under which such theorems hold. We close with an application to 'conjugacy ratios' as introduced by Cox.

For a finite subgroup $G$ of $\operatorname{GL}(2, \mathbb C)$, we consider the moduli space ${\mathcal M}_\theta$ of $G$-constellations. It depends on the stability parameter $\theta$ and if $\theta$ is generic it is a resolution of singularities of $\mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C^2/G$ is isomorphic to ${\mathcal M}_\theta$ for some generic $\theta$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.

The article is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of numerical integration of Ito stochastic differential equations. The expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ and expansions of iterated Stratonovich stochastic integrals of multiplicities $1$ to $5$, using multiple Fourier-Legendre and multiple trigonometric Fourier series are obtained. The exact and approximate expressions for the mean-square error of approximation of iterated Ito stochastic integrals are derived.

After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a "symmetry" of a generalized zero curvature equation and derive Backlund and (forward, backward and binary) Darboux transformations from it. We also recall a matrix version of the binary Darboux transformation and, inspired by the so-called Cauchy matrix approach, present an infinite system of equations solved by it. Finally, we sketch recent work on a deformation of the matrix binary Darboux transformation in bidifferential calculus, leading to a treatment of integrable equations with sources.

Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan-Lusztig polynomials in type $\tilde{A}_1$. In this paper we take that observation far beyond its original scope. We conjecture that for $\tilde{A}_n$ there is an equivalence of categories between the characteristic $p$ diagrammatic Hecke category and a "blob category" that we introduce (using certain quotients of KLR algebras called \emph{generalized blob algebras}). Using alcove geometry we prove the "graded degree" part of this equivalence for all $n$ and all prime numbers $p$. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic $p$ give the $p$-Kazhdan Lusztig polynomials in type $\tilde{A}_n$. We prove this for $\tilde{A}_1$, the only case where the $p$-Kazhdan Lusztig polynomials are known.

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field $\Bbbk$ of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter-Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand-Kirillov dimension.

Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on the underlying mixing measure. As our primary application, we apply these results to partition-based clustering, generalizing the notion of a Bayes optimal partition from classical parametric model-based clustering to nonparametric settings. Furthermore, this framework is constructive so that it yields a practical algorithm for learning identified mixtures, which is illustrated through several examples on real data. The key conceptual device in the analysis is the convex, metric geometry of probability measures on metric spaces and its connection to the Wasserstein convergence of mixing measures. The result is a flexible framework for nonparametric clustering with formal consistency guarantees.

In this paper, we introduce a new set of vector fields for the relativistic transport equation, which is applicable for general Vlasov-Wave type coupled systems. By combining the strength of Klainerman vector field method and Fourier method, we prove global regularity and scattering for the $3D$ massive relativistic Vlasov-Nordstr\"om system for small initial data without any compact support assumption, which is widely used in the literature for the study of Vlasov solution.

Given any smooth, suitably small initial data, which decays polynomially at infinity, we prove global regularity for the $3D$ relativistic massive Vlasov-Maxwell system. In particular, the compact support assumption, which is widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector fields method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov-Maxwell system.

This paper studies integral-type event-triggered model predictive control (MPC) of continuous-time nonlinear systems. An integral-type event-triggered mechanism is proposed by incorporating the integral of errors between the actual and predicted state sequences, leading to reduced average sampling frequency. Besides, a new and improved robustness constraint is introduced to handle the additive disturbance, rendering the MPC problem with a potentially enlarged initial feasible region. Furthermore, the feasibility of the designed MPC and the stability of the closed-loop system are rigorously investigated. Several sufficient conditions to guarantee these properties are established, which is related to factors such as the prediction horizon, the disturbance bound, the triggering level, and the contraction rate for the robustness constraint. The effectiveness of the proposed algorithm is illustrated by numerical examples and comparisons.

We study the small data global regularity problem of the $3D$ Vlasov-Poisson system for both the relativistic case and the non-relativistic case. The main goal of this paper is twofold. (i) Based on a Fourier method, which works systematically for both the relativistic case and the non-relativistic case, we give a short proof for the global regularity and the sharp decay estimate for the $3D$ Vlasov-Poisson system. Moreover, we show that the nonlinear solution scatters to a linear solution in both cases. The result of sharp decay estimates for the non-relativistic case is not new, see Hwang-Rendall-Vel\'azquez and Smulevici. (ii) The Fourier method presented in this paper serves as a good comparison for the study of more complicated $3D$ relativistic Vlasov-Nordstr\"om system and $3D$ relativistic Vlasov-Maxwell system.

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Folner sequence if G is amenable. In our first main result we show that if G is finitely generated then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent. This generalises both an earlier result of the second author treating the case k=1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below then G is virtually k-step nilpotent, answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients, generalising a result of Gallagher.

Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander--Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points.

Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers.

This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode $( d+1 )$-dimensional structures for an integer $d \geqslant 1$. They are $( d+2 )$-angulated categories, which belong to the subject of higher homological algebra.

We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general $( d+2 )$-angulated categories) to the integers. Following Palu, we will define a notion of $( d+2 )$-angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

We consider a family of norms (called operator E-norms) on the algebra $B(H)$ of all bounded operators on a separable Hilbert space $H$ induced by a positive densely defined operator $G$ on $H$. Each norm of this family produces the same topology on $B(H)$ depending on $G$. By choosing different generating operator $G$ one can obtain operator E-norms producing different topologies, in particular, the strong operator topology on bounded subsets of $B(H)$. We obtain a generalised version of the Kretschmann-Schlingemann-Werner theorem, which shows continuity of the Stinespring representation of CP linear maps w.r.t. the energy-constrained $cb$-norm (diamond norm) on the set of CP linear maps and the operator E-norm on the set of Stinespring operators.

The operator E-norms induced by a positive operator $G$ are well defined for linear operators relatively bounded w.r.t. the operator $\sqrt{G}$ and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between the operator E-norms and the standard characteristics of $\sqrt{G}$-bounded operators. The operator E-norms allow to obtain simple upper estimates and continuity bounds for some functions depending on $\sqrt{G}$-bounded operators used in applications.

We show that the classical Brink-Schwarz superparticle is a generalized AKSZ field theory. We work in the Batalin-Vilkovisky formalism: the main technical tool is the vanishing of Batalin--Vilkovisky cohomology below degree -1.

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\subset B$ are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where $T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$.

In this paper we introduce the notion of Lebesgue currents. They are a special type of currents involving the Lebesgue measure.

In this paper, we prove the Girsanov formula for $G$-Brownian motion without the non-degenerate condition. The proof is based on the perturbation method in the nonlinear setting by constructing a product space of the $G$-expectation space and a linear space that contains a standard Brownian motion. The estimates for exponential martingale of $G$-Brownian motion are important for our arguments.

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, i.e. non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach's higher-order RM. As to splittings, there are some examples in RM of theorems $A, B, C$ such that $A\leftrightarrow(B\wedge C)$, i.e. $A$ can be split into two independent (fairly natural) parts $B$ and $C$, and the aforementioned topological notions give rise to a number of splittings involving highly natural $A, B, C$. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of e.g. the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory of second-order RM We obtain similarly 'exceptional' results for the Urysohn identity, the Lindel\"of lemma, and partitions of unity.

We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend these estimates to smoothed analysis setting.

Let C/K be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of C over a field F where it becomes semistable. This allows us to describe the Galois action on the l-adic Tate module of the Jacobian of C/K in terms of the special fibre of this model over F.

We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian. We show the existence, uniqueness of bounded solutions and study their further properties. Several new examples of non-local, non-linear operators are provided.

Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f\colon M\to M$, we construct a map of degree one from a finite cover of $M\rtimes_f S^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M\rtimes_f S^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except when $M$ is virtually $(\#_n T^2\rtimes S^1)\#(\#_mS^2\times S^1)$. Li-Ni's conjecture was recently confirmed by Ni with a group theoretic result, namely, by showing that there exists a $\pi_1$-surjection from a finite cover of any mapping torus of a reducible $3$-manifold to a certain mapping torus of $\#_m S^2\times S^1$ and using the fact that free-by-cyclic groups are large when the free group is generated by more than one element.

Consider a wireless network where each communication link has a minimum bandwidth quality-of-service requirement. Certain pairs of wireless links interfere with each other due to being in the same vicinity, and this interference is modeled by a conflict graph. Given the conflict graph and link bandwidth requirements, the objective is to determine, using only localized information, whether the demands of all the links can be satisfied. At one extreme, each node knows the demands of only its neighbors; at the other extreme, there exists an optimal, centralized scheduler that has global information. The present work interpolates between these two extremes by quantifying the tradeoff between the degree of decentralization and the performance of the distributed algorithm. This open problem is resolved for the primary interference model, and the following general result is obtained: if each node knows the demands of all links in a ball of radius $d$ centered at the node, then there is a distributed algorithm whose performance is away from that of an optimal, centralized algorithm by a factor of at most $(2d+3)/(2d+2)$. The tradeoff between performance and complexity of the distributed algorithm is also analyzed. It is shown that for line networks under the protocol interference model, the row constraints are a factor of at most $3$ away from optimal. Both bounds are best possible.

This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite families of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden symmetries grows linearly with volume. This second family realizes the maximal growth rate for the number of hidden symmetries relative to volume for non-arithmetic hyperbolic 3-manifolds. Our work requires a careful analysis of the geometry of these link complements, including their cusp shapes and totally geodesic surfaces inside of these manifolds.

Assuming regularity of the fixed subalgebra, any action of a finite group $G$ on a holomorphic VOA $V$ determines a gauge anomaly $\alpha \in \mathrm{H}^3(G; \boldsymbol{\mu})$, where $\boldsymbol{\mu} \subset \mathbb{C}^\times$ is the group of roots of unity. We show that under Galois conjugation $V \mapsto {^\gamma V}$, the gauge anomaly transforms as $\alpha \mapsto \gamma^2(\alpha)$. This provides an a priori upper bound of $24$ on the order of anomalies of actions preserving a $\mathbb{Q}$-structure, for example the Monster group $\mathbb{M}$ acting on its Moonshine VOA $V^\natural$. We speculate that each field $\mathbb{K}$ should have a "vertex Brauer group" isomorphic to $\mathrm{H}^3(\mathrm{Gal}(\bar{\mathbb{K}}/\mathbb{K}); \boldsymbol{\mu}^{\otimes 2})$. In order to motivate our constructions and speculations, we warm up with a discussion of the ordinary Brauer group, emphasizing the analogy between VOA gauging and quantum Hamiltonian reduction.

Let $g$ be locally homogeneous (LH) Riemannian metric on a differentiable compact manifold $M$, and $K$ be a compact Lie group endowed with an $\mathrm {ad}$-invariant inner product on its Lie algebra $\mathfrak{k}$. A connection $A$ on a principal $K$-bundle $p:P\to M$ on $M$ is locally homogeneous if for any two points $x_1$, $x_2\in M$ there exists an isometry $\varphi:U_1\to U_2$ between open neighborhoods $U_i\ni x_i$ which sends $x_1$ to $x_2$ and admits a $\varphi$-covering bundle isomorphism preserving the connection $A$. This condition is invariant under the action of the automorphism group (gauge group) of the bundle, so the classification problem for LH connections leads to an interesting moduli problem: for fixed objects $(M,g,K)$ as above describe geometrically the moduli space of all LH connections on principal $K$-bundles on $M$ (up to bundle isomorphisms). Note that if $A$ is LH, then the associated connection metric $g_A$ on $P$ is locally homogenous, so it defines a geometric structure (in the sense of Thurston) on the total space of the bundle. Therefore this moduli problem is related to the classification of LH (geometric) Riemannian manifolds which admit a Riemannian submersion onto the given manifold $M$.

Omitting the details, our moduli problem concerns the classification of geometric fibre bundles over a given geometric base. We develop a general method for describing moduli spaces of LH connections on a given base. Using our method we give explicit descriptions of these moduli spaces when the base manifold is a hyperbolic Riemann surface $(M,g)$ and $K\in\{S^1,PU(2)\}$. The case $K=S^1$ leads to a new construction of the moduli spaces of Yang-Mills $S^1$-connections on hyperbolic Riemann surfaces, and the case $K=PU(2)$ leads to a one-parameter family of compact, 5-dimensional geometric manifolds, which we study in detail.

Manifolds endowed with torsion and nonmetricity are interesting both from the physical and the mathematical points of view. In this paper, we generalize some results presented in the literature. We study Einstein manifolds (i.e., manifolds whose symmetrized Ricci tensor is proportional to the metric) in d dimensions with nonvanishing torsion that has both a trace and a traceless part, and analyze invariance under extended conformal transformations of the corresponding field equations. Then, we compare our results to the case of Einstein manifolds with zero torsion and nonvanishing nonmetricity, where the latter is given in terms of the Weyl vector (Einstein-Weyl spaces). We find that the trace part of the torsion can alternatively be interpreted as the trace part of the nonmetricity. The analysis is subsequently extended to Einstein spaces with both torsion and nonmetricity, where we also discuss the general setting in which the nonmetricity tensor has both a trace and a traceless part. Moreover, we consider and investigate actions involving scalar curvatures obtained from torsionful or nonmetric connections, analyzing their relations with other gravitational theories that appeared previously in the literature. In particular, we show that the Einstein-Cartan action and the scale invariant gravity (also known as conformal gravity) action describe the same dynamics. Then, we consider the Einstein-Hilbert action coupled to a three-form field strength and show that its equations of motion imply that the manifold is Einstein with totally antisymmetric torsion.

We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace $\mathcal{K}([0, 1]^{\omega})$ containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace.

We recall that to every analytic subset of $\mathcal{K}([0, 1]^{\omega})$ there exist an equivalent $G_{\delta}$ subset. Then we show that up to the equivalence open subsets of the hyperspace $\mathcal{K}([0, 1]^{\omega})$ correspond to countably many classes of metrizable compacta. Finally we use the structure of open subsets up to equivalence to prove that to every $F_{\sigma}$ subset of $\mathcal{K}([0, 1]^{\omega})$ there exists an equivalent closed subset.

Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of $\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of congruences for $\overline{\mathcal{EO}}(n)$. We next study parity properties of $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely many integers $N$ in every arithmetic progression for which $\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely many integers $M$ in every arithmetic progression for which $\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Very recently, Uncu has treated a different subset of the partitions enumerated by $\mathcal{EO}(n)$. We prove that Uncu's partition function is divisible by $2^k$ for almost all $k$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.

The main purpose of this paper is to study $L^r$ H\"older type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}(\mathbb{R}^{2})$, the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$T_{m_1,m_2}(f,g,h)(x):=\int_{\mathbb{R}^{6}}m_1(\xi,\eta,\zeta)m_2(\eta,\zeta)\hat f(\xi) \hat g(\eta)\hat h(\zeta)e^{2\pi i(\xi+\eta+\zeta)\cdot x}d\xi d\eta d\zeta,$$ when $m_1,m_2$ are two bi-parameter symbols. We will show that our problem can be reduced to establish the $L^r$ estimate for the special multiplier $m_1(\xi_1, \eta_1, \zeta_1) m_2(\eta_2, \zeta_2)$ (see Theorem 1.7).

We also study these $L^r$ estimates for the corresponding bi-parameter trilinear pseudo-differential operators defined by $$T_{ab}(f,g,h)(x):=\int_{\mathbb{R}^6}a(x,\xi,\eta,\zeta)b(x,\eta,\zeta)\hat f(\xi)\hat g(\eta)\hat h(\zeta)e^{2\pi i x(\xi+\eta+\zeta)}d\xi d\eta d\zeta,$$ where the smooth symbols $a,b$ satisfy certain bi-parameter H\"ormander conditions. We will also show that the $L^r$ estimate holds for $T_{ab}$ as long as the $L^r$ estimate for the flag multiplier operator holds when the multiplier has the special form $m_1(\xi_1, \eta_1, \zeta_1) m_2(\eta_2, \zeta_2)$ (see Theorem 1.10).

The bi-parameter and trilinear flag Fourier multipliers considered in this paper do not satisfy the conditions of the classical bi-parameter trilinear Fourier multipliers considered by Muscalu, Tao, Thiele and the second author [21, 22]. They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by Muscalu [18].

In this paper we study the problem of testing if an $L_2-$function $f$ belonging to a certain $l_2$-Sobolev-ball $B_t(R)$ of radius $R>0$ with smoothness level $t>0$ indeed exhibits a higher smoothness level $s>t$, that is, belongs to $B_s(R)$. We assume that only a perturbed version of $f$ is available, where the noise is governed by a standard Brownian motion scaled by $\frac{1}{\sqrt{n}}$. More precisely, considering a testing problem of the form $$H_0:~f\in B_s(R)~~\mathrm{vs.}~~H_1:~f\in B_t(R),~\inf_{h\in B_s}\Vert f-h\Vert_{L_2}>\rho$$ for some $\rho>0$, we approach the task of identifying the smallest value for $\rho$, denoted $\rho^\ast$, enabling the existence of a test $\varphi$ with small error probability in a minimax sense. By deriving lower and upper bounds on $\rho^\ast$, we expose its precise dependence on $n$: $$\rho^\ast\sim n^{-\frac{t}{2t+1/2}}.$$ As a remarkable aspect of this composite-composite testing problem, it turns out that the rate does not depend on $s$ and is equal to the rate in signal-detection, i.e. the case of a simple null hypothesis.

The generalized Prony method introduced by Peter & Plonka (2013) is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator $A$. However, this procedure requires the evaluation of higher powers of the linear operator $A$ that are often expensive to provide.

In this paper we propose two important extensions of the generalized Prony method that simplify the acquisition of the needed samples essentially and at the same time can improve the numerical stability of the method. The first extension regards the change of operators from $A$ to $\varphi(A)$, where $\varphi$ is an analytic function, while $A$ and $\varphi(A)$ possess the same set of eigenfunctions. The goal is now to choose $\varphi$ such that the powers of $\varphi(A)$ are much simpler to evaluate than the powers of $A$. The second extension concerns the choice of the sampling functionals. We show, how new sets of different sampling functionals $F_{k}$ can be applied with the goal to reduce the needed number of powers of the operator $A$ (resp. $\varphi(A)$) in the sampling scheme and to simplify the acquisition process for the recovery method.

Let $\{D_M\}_{M\geq 0}$ be the $n$-vertex random directed graph process, where $D_0$ is the empty directed graph on $n$ vertices, and subsequent directed graphs in the sequence are obtained by the addition of a new directed edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any directed graph $D_M$ with minimum in- and out-degree at least 1 is not only Hamiltonian (as shown by Frieze), but remains Hamiltonian when edges are removed, as long as at most $(1/2-\varepsilon)$ of both the in- and out-edges incident to each vertex are removed. We say such a directed graph is $(1/2-\varepsilon)$-resiliently Hamiltonian. Furthermore, for each $\varepsilon>0$, we show that, almost surely, each directed graph $D_M$ in the sequence is not $(1/2+\varepsilon)$-resiliently Hamiltonian.

This improves a result of Ferber, Nenadov, Noever, Peter and \v{S}kori\'{c}, who showed, for each $\varepsilon>0$, that the binomial random directed graph $D(n,p)$ is almost surely $(1/2-\varepsilon)$-resiliently Hamiltonian if $p=\omega(\log^8n/n)$.

This paper presents a computational solution to determine if a chemical reaction network endowed with power-law kinetics (PLK system) has the capacity for multistationarity, i.e., whether there exist positive rate constants such that the corresponding differential equations admit multiple positive steady states within a stoichiometric class. The approach, which is called the "Multistationarity Algorithm for PLK systems" (MSA), combines (i) the extension of the "higher deficiency algorithm" of Ji and Feinberg for mass action to PLK systems with reactant-determined interactions, and (ii) a method that transforms any PLK system to a dynamically equivalent one with reactant-determined interactions. Using this algorithm, we obtain two new results: the monostationarity of a popular model of anaerobic yeast fermentation pathway, and the multistationarity of a global carbon cycle model with climate engineering, both in the generalized mass action format of biochemical systems theory. We also provide examples of the broader scope of our approach for deficiency one PLK systems in comparison to the extension of Feinberg's "deficiency one algorithm" to such systems.

The persistent walk is a classical model in kinetic theory, which has also been studied as a toy model for MCMC questions. Its continuous limit, the telegraph process, has recently been extended to various velocity jump processes (Bouncy Particle Sampler, Zig-Zag process, etc.) in order to sample general target distributions on $\mathbb R^d$. This paper studies, from a sampling point of view, general kinetic walks that are natural discrete-time (and possibly discrete-space) counterparts of these continuous-space processes. The main contributions of the paper are the definition and study of a discrete-space Zig-Zag sampler and the definition and time-discretisation of hybrid jump/diffusion kinetic samplers for multi-scale potentials on $\mathbb R^d$.

To efficiently support safety-related vehicular applications, the ultra-reliable and low-latency communication (URLLC) concept has become an indispensable component of vehicular networks (VNETs). Due to the high mobility of VNETs, exchanging near-instantaneous channel state information (CSI) and making reliable resource allocation decisions based on such short-term CSI evaluations are not practical. In this paper, we consider the downlink of a vehicle-to-infrastructure (V2I) system conceived for URLLC based on idealized perfect and realistic imperfect CSI. By exploiting the benefits of the massive MIMO concept, a two-stage radio resource allocation problem is formulated based on a novel twin-timescale perspective for avoiding the frequent exchange of near-instantaneous CSI. Specifically, based on the prevalent road-traffic density, Stage 1 is constructed for minimizing the worst-case transmission latency on a long-term timescale. In Stage 2, the base station allocates the total power at a short-term timescale according to the large-scale fading CSI encountered for minimizing the maximum transmission latency across all vehicular users. Then, a primary algorithm and a secondary algorithm are conceived for our V2I URLLC system to find the optimal solution of the twin-timescale resource allocation problem, with special emphasis on the complexity imposed. Finally, our simulation results show that the proposed resource allocation scheme significantly reduces the maximum transmission latency, and it is not sensitive to the fluctuation of road-traffic density.

One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases.

In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into $k$ common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when $k=2$.

Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when $k=2$, one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.

We formulate a composite problem involving the decision making of the optimal entry time and dynamic consumption afterwards: in stage-1, the investor has access to full market information subjecting to some information costs and needs to choose an optimal stopping time to initiate stage-2; in stage-2 starting from the chosen stopping time, the investor terminates the costly full information acquisition and starts dynamic investment and consumption under partial observations of free public stock prices. The habit formation preference is employed, in which the past consumption affects the investor's current decisions. The value function of the composite problem is proved to be the unique viscosity solution of some variational inequalities.

We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. The phrase well-balancing is used in a wider sense, since the method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained. The potentially high order of accuracy of the method is maintained under the modification. We show numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics. Different grid geometries and reconstruction methods are used. We demonstrate high order convergence numerically.

On the Sierpinski gasket $\mathcal{SG}$, we consider Sobolev spaces $L^2_\sigma(\mathcal{SG})$ associated with the standard Laplacian $\Delta$ with order $\sigma\geq 0$. When $\sigma\in\mathbb{Z}^+$, $L^2_\sigma(\mathcal{SG})$ consists of functions equipped with $L^2$ norms of the function itself and its Laplacians up to $\sigma$ order; when $\sigma\notin \mathbb{Z}^+$, we fill up the gaps between integer orders by using complex interpolation. Let $L^2_{\sigma, D}(\mathcal{SG})=(I-\Delta_D)^{-\sigma}L^2(\mathcal{SG})$ where $\Delta_D$ is the Dirichlet Laplacian associated with $\Delta$. Let $\{p_n\}_{n\geq 0}$ be a collection of countably many points located along one of the symmetrical axes of $\mathcal{SG}$. We make a full characterization of the trace spaces of $L^2_\sigma(\mathcal{SG})$ and $L^2_{\sigma,D}(\mathcal{SG})$ to $\{p_n\}_{n\geq 0}$. Using this, we get a full description of the relationship between $L^2_\sigma(\mathcal{SG})$ and $L^2_{\sigma,D}(\mathcal{SG})$ for $\sigma\geq 0$. The result indicates that when $\sigma-\frac{\log 3}{2\log5}\in \mathbb{Z}^+$, $L^2_{\sigma, D}(\mathcal{SG})$ is not closed in $L^2_{\sigma}(\mathcal{SG})$ and has an infinite codimension. Otherwise, $L^2_{\sigma, D}(\mathcal{SG})$ is closed in $L^2_{\sigma}(\mathcal{SG})$ with a finite codimension. Similar result holds for the Neumann case. Another consequence of the trace result is that the Sobolev spaces $L^2_\sigma(\mathcal{SG})$ are stable under complex interpolation for $\sigma\geq 0$ although they are defined by piecewise interpolation between integer orders.

Following the strategy proposed by Makarov and Smirnov in arXiv:0909.5377, we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up of arXiv:0909.5377 appeared since then, we believe that such a treatment might be of interest for the community. We do not require any regularity of the limiting planar domain $\Omega$ near its degenerate prime ends $a$ and $b$ except that $(\Omega^\delta,a^\delta,b^\delta)$ are assumed to be close discrete approximations' to $(\Omega,a,b)$ near $a$ and $b$ in the sense of a recent work arXiv:1810.05608.

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic.

We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily-Borel and toroidal compactifications of the ball quotient model, due to Allcock-Carlson-Toledo. Our starting point is Kirwan's method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli of cubic surfaces is discussed in an appendix.

We investigate the phase-field approximation of the Willmore flow. This is a fourth-order diffusion equation with a parameter $\epsilon>0$ that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as $\epsilon$ trends to zero the level-set of solution will converge to motion by Willmore flow before the singularity of the later occurs. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step and also the major contribution of this work is to show a spectrum condition of the linearized operator at the optimal profile. This is a fourth-order operator written as the sum of the squared Allen-Cahn operator and a singular linear perturbation.

We study the quantization of coupled K\"ahler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the "balanced metrics". We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence.

If $\Sigma$ and $\Sigma'$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer-valued notions of distance between the embeddings: the singularity distance $d_{\text{sing}}(\Sigma,\Sigma')$ and the stabilisation distance $d_{\text{st}}(\Sigma,\Sigma')$. Using techniques similar to those used by Gabai in his proof of the 4-dimensional light-bulb theorem, we prove that $d_{\text{st}}(\Sigma,\Sigma')\leq d_{\text{sing}}(\Sigma,\Sigma')+1$.

We discuss the problem of classifying Dembowski-Ostrom polynomials from the composition of reversed Dickson polynomials of arbitrary kind and monomials over finite fields of odd characteristic. Moreover, by using a variant of Weil bound for the number of points of affine algebraic curves over finite fields or otherwise, we discuss the planarity of all such Dembowski-Ostrom polynomials.

This is the second installment of a two part series of papers studying free globularly generated double categories. We introduce the canonical double projection construction. The canonical double projection translates information from free globularly generated double categories to double categories defined through the same set of globular and vertical data. We use the canonical double projection to define compatible formal linear functorial extensions of the Haagerup standard form and the Connes fusion operation to possibly-infinite index morphisms between factors. We use the canonical double projection to prove that the free globularly generated double category construction is left adjoint to decorated horizontalization. We thus interpret free globularly generated double categories as formal decorated analogs of double categories of quintets and as generators for internalizations.

The isometries of an exact plane gravitational wave are symmetries for both massive and massless particles. Their conformal extensions are in fact chrono-projective transformations introduced earlier by Duval et al are symmetries for massless particles. Homotheties are universal chrono-projective symmetries for any profile. Chrono-projective transformations also generate new conserved quantities for the underlying non-relativistic systems in the Bargmann framework. Homotheties play a similar role for the lightlike "vertical" coordinate as isometries play for the transverse coordinates.

Predicting scalar outcomes using functional predictors is a classic problem in functional data analysis. In many applications, however, only specific locations or time-points of the functional predictors have an impact on the outcome. Such points of impact'' are typically unknown and have to be estimated in addition to estimating the usual model components. We show that our points of impact estimator enjoys a super-consistent convergence rate and does not require knowledge or pre-estimates of the unknown model components. This remarkable result facilitates the subsequent estimation of the remaining model components as shown in the theoretical part, where we consider the case of nonparametric models and the practically relevant case of generalized linear models. The finite sample properties of our estimators are assessed by means of a simulation study. Our methodology is motivated by data from a psychological experiment in which the participants were asked to continuously rate their emotional state while watching an affective video eliciting a varying intensity of emotional reactions.

We study gradient compression methods to alleviate the communication bottleneck in data-parallel distributed optimization. Despite the significant attention received, current compression schemes either do not scale well or fail to achieve the target test accuracy. We propose a new low-rank gradient compressor based on power iteration that can i) compress gradients rapidly, ii) efficiently aggregate the compressed gradients using all-reduce, and iii) achieve test performance on par with SGD. The proposed algorithm is the only method evaluated that achieves consistent wall-clock speedups when benchmarked against regular SGD with an optimized communication backend. We demonstrate reduced training times for convolutional networks as well as LSTMs on common datasets. Our code is available at https://github.com/epfml/powersgd.

A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of Tangles with a given length or area grows exponentially, and we show the existence of their growth constants by comparing Tangles to two families of polyominoes.

Motivated by works on extension sets in standard domains we introduce a notion of the Carath\'eodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a class of two dimensional submanifolds in the tridisc which not only turns out to be Carath\'eodory but also provides examples of two dimensional domains for which the celebrated Lempert Theorem holds. Additionally, a recently introduced notion of universal sets for the Carath\'eodory extremal problem is studied and new results on domains admitting (no) finite universal sets are given.

We propose a new estimator, the quadratic form estimator, of the Kronecker product model for covariance matrices. We show that this estimator has good properties in the large dimensional case (i.e., the cross-sectional dimension $n$ is large relative to the sample size $T$). In particular, the quadratic form estimator is consistent in a relative Frobenius norm sense provided $\log^{3}n/(n^{2-\beta_1}T)\rightarrow0$ and $\log^2n/T\to 0$, where $\beta_1$ is some measure of cross-sectional dependence. We obtain the limiting distributions of Lagrange multiplier (LM) and Wald tests under both the null and local alternatives concerning the mean vector $\mu$. Testing linear restrictions of $\mu$ is also investigated. Finally, our methodology performs well in the finite-sample situations both when the Kronecker product model is true, and when it is not true.

We provide a discussion of several recent results which, in certain scenarios, are able to overcome a barrier in distributed stochastic optimization for machine learning. Our focus is the so-called asymptotic network independence property, which is achieved whenever a distributed method executed over a network of n nodes asymptotically converges to the optimal solution at a comparable rate to a centralized method with the same computational power as the entire network. We explain this property through an example involving the training of ML models and sketch a short mathematical analysis for comparing the performance of distributed stochastic gradient descent (DSGD) with centralized stochastic gradient decent (SGD).

We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold $M$ with a time-like boundary $\partial M$ that was proposed by Donnelly and Freidel [JHEP 1609, 102 (2016)]. This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature. Our construction also applies to linear Chern-Simons theory, in which case we obtain the extended phase space introduced by Geiller [Nucl. Phys. B 924, 312 (2017)].

The multigroup neutron transport criticality calculations using modern supercomputers have been widely employed in a nuclear reactor analysis for studying whether or not a system is self-sustaining. However, the design and development of efficient parallel algorithms for the transport criticality calculations is challenging especially when the number of processor cores is large and an unstructured mesh is adopted. In particular, both the compute time and memory usage have to be carefully taken into consideration due to the dimensionality of the neutron transport equations. In this paper, we study a monolithic multilevel Schwarz preconditioner for the transport criticality calculations based on a nonlinear diffusion acceleration (NDA) method. We propose a monolithic multilevel Schwarz method that is capable of efficiently handling the systems of linear equations for both the transport system and the diffusion system. However, in the multilevel method, algebraically constructing coarse spaces is expensive and often unscalable. We study a subspace-based coarsening algorithm to address such a challenge by exploring the matrix structures of the transport equations and the nonlinear diffusion equations. We numerically demonstrate that the monolithic multilevel preconditioner with the subspace-based coarsening algorithm is twice as fast as that equipped with an unmodified coarsening approach on thousands of processor cores for an unstructured mesh neutron transport problem with billions of unknowns.

We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to the square root of the spectral radius of the variance matrix of $X$ when $n$ tends to infinity. We also establish the optimal rate of convergence, that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law [arXiv:1612.07776] at the spectral edge.

In the first part of this paper we consider expanding vacuum cosmological spacetimes with a free $T^N$-action. Among them, we give evidence that Gowdy spacetimes have AVTD (asymptotically velocity term dominated) behavior for their initial geometry, in any dimension. We then give sufficient conditions to reach a similar conclusion about a $T^2$-invariant four dimensional nonGowdy spacetime. In the second part of the paper we consider vacuum cosmological spacetimes with crushing singularities. We introduce a monotonic quantity to characterize Kasner spacetimes. Assuming scale-invariant curvature bounds and local volume bounds, we give results about causal pasts.

We study the algorithmic problem of estimating the mean of heavy-tailed random vector in $\mathbb{R}^d$, given $n$ i.i.d. samples. The goal is to design an efficient estimator that attains the optimal sub-gaussian error bound, only assuming that the random vector has bounded mean and covariance. Polynomial-time solutions to this problem are known but have high runtime due to their use of semi-definite programming (SDP). Conceptually, it remains open whether convex relaxation is truly necessary for this problem.

In this work, we show that it is possible to go beyond SDP and achieve better computational efficiency. In particular, we provide a spectral algorithm that achieves the optimal statistical performance and runs in time $\widetilde O\left(n^2 d \right)$, improving upon the previous fastest runtime $\widetilde O\left(n^{3.5}+ n^2d\right)$ by Cherapanamjeri el al. (COLT '19). Our algorithm is spectral in that it only requires (approximate) eigenvector computations, which can be implemented very efficiently by, for example, power iteration or the Lanczos method.

At the core of our algorithm is a novel connection between the furthest hyperplane problem introduced by Karnin et al. (COLT '12) and a structural lemma on heavy-tailed distributions by Lugosi and Mendelson (Ann. Stat. '19). This allows us to iteratively reduce the estimation error at a geometric rate using only the information derived from the top singular vector of the data matrix, leading to a significantly faster running time.

The fundamental decomposition of a chemical reaction network (also called its "$\mathscr{F}$-decomposition") is the set of subnetworks generated by the partition of its set of reactions into the "fundamental classes" introduced by Ji and Feinberg in 2011 as the basis of their "higher deficiency algorithm" for mass action systems. The first part of this paper studies the properties of the $\mathscr{F}$-decomposition, in particular, its independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and its incidence-independence (i.e., the image of the network's incidence map is the direct sum of the incidence maps' images of the subnetworks). We derive necessary and sufficient conditions for these properties and identify network classes where the $\mathscr{F}$-decomposition coincides with other known decompositions. The second part of the paper applies the above-mentioned results to improve the Multistationarity Algorithm for power-law kinetic systems (MSA), a general computational approach that we introduced in previous work. We show that for systems with non-reactant determined interactions but with an independent $\mathscr{F}$-decomposition, the transformation to a dynamically equivalent system with reactant-determined interactions -- required in the original MSA -- is not necessary. We illustrate this improvement with the subnetwork of Schmitz's carbon cycle model recently analyzed by Fortun et al.

Let $(X,\Delta)$ be a projective log canonical pair of dimension $n$ such that $X$ is uniruled. If $X$ is not rationally connected, then $(X,\Delta)$ has a good model, assuming the Minimal Model Program in dimension $n-1$. If $X$ is rationally connected, then the existence of a good model for $(X,\Delta)$ follows from a nonexistence conjecture for a very specific class of rationally connected pairs of Calabi-Yau type.

As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends the Drinfeld-Jimbo coproduct on the horizontal quantum affine subalgebra.

We classify $N{=}1$ SVOAs with no free fermions and with bosonic subalgebra a simply connected WZW algebra which is not of type $\mathrm{E}$. The latter restriction makes the classification tractable; the former restriction implies that the $N{=}1$ automorphism groups of the resulting SVOAs are finite. We discover two infinite families and nine exceptional examples. The exceptions are all related to the Leech lattice: their automorphism groups are the larger groups in the Suzuki chain ($\mathrm{Co}_1$, $\mathrm{Suz}{:}2$, $\mathrm{G}_2(4){:}2$, $\mathrm{J}_2{:}2$, $\mathrm{U}_3(3){:}2$) and certain large centralizers therein ($2^{10}{:}\mathrm{M}_{12}{:}2$, $\mathrm{M}_{12}{:}2$, $\mathrm{U}_4(3){:}D_8$, $\mathrm{M}_{21}{:}2^2$). Along the way, we elucidate fermionic versions of a number of VOA operations, including simple current extensions, orbifolds, and 't Hooft anomalies.

The aim of this work is to link the conformal geometry of a Euclidean domain $U$ to the spectral properties of its energy integral $(D,H^{1,2}(U))$, through the algebra of multipliers $M(H^{1,2}(U))$ of the Sobolev space $H^{1,2}(U)$. The M\"obius group $G(R^n)$ for $n\ge 3$, is shown to act isometrically on the algebra of multipliers $M(H^{1,2}_e(R^n))$ of the extended Sobolev space $H^{1,2}_e(R^n)$ and the Dirichlet integral $\mathcal{D}$ is proved to be a closable on the space $L^2(R^n,\Gamma[a])$ of the energy measure $\Gamma[a]=|\nabla a|^2\, dx$ of any $a\in M(H^{1,2}_e(R^n))$. We then show that, for any $\gamma\in G(R^n)$, the form closure $(D,F^a)$ of $(D,H^{1,2}(R^n))$ is a Dirichlet form on $L^2(R^n,\Gamma[a])$, naturally unitarely equivalent to the one $(D,F^{a\circ\gamma})$ on $L^2(R^n,\Gamma[a\circ\gamma])$. These Dirichlet forms thus share the same spectrum. In a converse direction, we prove that a homeomorphism $\gamma:U\to\gamma(U)$ between Euclidean domains, giving rise to an algebraic isomorphism $a\mapsto a\circ\gamma$ between the algebras of multipliers $M(H^{1,2}(\gamma(A)))$ and $M(H^{1,2}(A))$ of any relatively compact subdomain $A\subseteq U$ and leaving invariant the corresponding fundamental tones $\mu_1(\gamma(A),a)=\mu_1(A,a\circ\gamma)\, ,$ is necessarily the restriction of a M\"obius transformation to $U$. Companion results hold true for quasi-conformal and bounded distortion maps. Connections between fundamental tones and ergodic properties of multipliers are studied too. In particular, it is shown that the existence of a spectral gap for $(D,H^{1,2}(U))$ on $L^2(U,dx)$ implies the non vanishing of the fundamental tone of $(D,F^a)$ on $L^2(U,\Gamma[a])$, for any multiplier having full support.

We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, ${\bf K}$, and then to associate to ${\bf K}$ a topological space--the Balmer spectrum $\operatorname{Spc} {\bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\operatorname{Spc} {\bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$\Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${\bf K}$, which in turn can be applied to classify the thick two-sided ideals and $\operatorname{Spc} {\bf K}$.

As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon.

In this paper, we study direct images of pluricanonical bundles in positive characteristic, assuming that generic fibers have finitely generated canonical rings and sufficiently large Frobenius stable canonical rings. We treat a Fujita-type conjecture due to Popa and Schnell, and prove it under some additional hypotheses. As its application, we show the subadditivity of Kodaira dimensions in some new cases. We also prove an analog of Fujino's result concerning his Fujita-type conjecture.

In this paper, we deal with two logarithmic fourth order differential equations: the extended one-dimensional DLSS equation and its multi-dimensional analog. We show the global existence of solution in critical spaces, its convergence to equilibrium and the gain of spatial analyticity for these two equations in a unified way.

We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their word problem is $\mathsf{NC}^1$-hard. For some of these groups (including Grigorchuk's group and Thompson's groups) we prove that the compressed word problem (which is equivalent to the circuit evaluation problem) is $\mathsf{PSPACE}$-complete.

Kernelization is the fundamental notion for polynomial-time data reduction with performance guarantees. Kernelization for weighted problems particularly requires to also shrink weights. Marx and V\'egh [ACM Trans. Algorithms 2015] and Etscheid et al. [J. Comput. Syst. Sci. 2017] used a technique of Frank and Tardos [Combinatorica 1987] to obtain polynomial-size kernels for weighted problems, mostly with additive goal functions. We lift the technique to linearizable functions, a function type that we introduce and that also contains non-additive functions. Using the lifted technique, we obtain kernelization results for natural problems in graph partitioning, network design, facility location, scheduling, vehicle routing, and computational social choice, thereby improving and generalizing results from the literature.

We show that the Kottman constant $K(\cdot)$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $X$.

We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [CaGoPa17], namely that the isomorphic Kottman constant of a twisted-sum space is the maximum of the constants of the respective summands. Consequently, the Kalton--Peck space may be renormed to have Kottman's constant arbitrarily close to $\sqrt{2}$. For other classical parameters, such as the Whitley and the James constants, we prove the continuity with respect to the Kadets metric.

We study weak approximation on rationally connected varieties under an assumption of strong approximation for a "simple" variety or under Schinzel's hypothesis. We also get some unconditional results.

We show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms by giving an explicit counterexample involving large symmetric groups. We also show that, in contrast, the spheres jointly reflect equivalences in the homotopy 2-category of spaces. The non-existence of such a set in the homotopy category was originally claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough transfinite sequential diagrams admit weak colimits which are privileged with respect to the given set. Using the theory of graphs of groups, we show that this statement is false, by proving that for every ordinal with uncountable cofinality, there is a diagram indexed by that ordinal which admits no weak colimit that is privileged with respect to the spheres.

The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as direct product of the corresponding regular groups of the inducing subextensions. We describe here the attached Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. We give a general matrix description of the Hopf action which is useful to compute bases of associated orders. In case of an induced Hopf Galois structures it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.

A rapid change of channels in high-speed mobile communications will lead to difficulties in channel estimation and tracking but can also provide Doppler diversity. In this paper, the performance of a multiple-input multiple-output system with pilot-assisted repetition coding and spatial multiplexing is studied. With minimum mean square error (MMSE) channel estimation, an equivalent channel model and the corresponding system model are presented. Based on random matrix theory, asymptotic expressions of the normalized achievable sum rate of the linear receivers, such as the maximal ratio combining (MRC) receiver, MMSE receiver and MRC-like receiver, are derived. In addition, according to the symbol error rate of the MRC-like receiver, the maximum normalized Doppler diversity order and the minimum coding gain loss can be achieved when the repetition number and signal-to-noise ratio tend to infinity, and the corresponding conditions are derived. Based on the theoretical results, the impacts of different system configurations and channel parameters on the system performance are demonstrated.

We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramm's transboundary extremal lengths.

A robust observer for performing power system dynamic state estimation (DSE) of a synchronous generator is proposed. The observer is developed using the concept of $\mathcal{L}_{\infty}$ stability for uncertain, nonlinear dynamic generator models. We use this concept to (i) design a simple, scalable, and robust dynamic state estimator and (ii) obtain a performance guarantee on the state estimation error norm relative to the magnitude of uncertainty from unknown generator inputs, and process and measurement noises. Theoretical methods to obtain upper and lower bounds on the estimation error are also provided. Numerical tests validate the performance of the $\mathcal{L}_{\infty}$-based estimator in performing DSE under various scenarios. The case studies reveal that the derived theoretical bounds are valid for a variety of case studies and operating conditions, while yielding better performance than existing power system DSE methods.

Multiple access technology plays an important role in wireless communication in the last decades: it increases the capacity of the channel and allows different users to access the system simultaneously. However, the conventional multiple access technology, as originally designed for current human-centric wireless networks, is not scalable for future machine-centric wireless networks.

Massive access (also known as Massive-device Multiple Access, Unsourced Massive Random Access, Massive Connectivity, Massive Machine-type Communication, and Many-Access Channels) exhibits a clean break with current networks by potentially supporting millions of devices in each cellular network. The tremendous growth in the number of connected devices requires a fundamental rethinking of the conventional multiple access technologies in favor of new schemes suited to massive random access. Among the the many new challenges arising in this setting, the most relevant are: the fundamental limits of communication from a massive number of bursty devices transmitting simultaneously with short packets, the design of low complexity and energy-efficient massive access coding and communication schemes, efficient methods for the detection of a relatively small number of active users among a large number of potential user devices with sporadic transmission pattern, and the integration of massive access with massive MIMO and other important wireless communication technologies. This paper presents an overview of the concept of massive access wireless communication and of the contemporary research on this important topic.

Generative Adversarial Networks (GANs) are powerful class of generative models in the deep learning community. Current practice on large-scale GAN training~\citep{brock2018large} utilizes large models and distributed large-batch training strategies, and is implemented on deep learning frameworks (e.g., TensorFlow, PyTorch, etc.) designed in a centralized manner. In the centralized network topology, every worker needs to communicate with the central node. However, when the network bandwidth is low or network latency is high, the performance would be significantly degraded. Despite recent progress on decentralized algorithms for training deep neural networks, it remains unclear whether it is possible to train GANs in a decentralized manner. The main difficulty lies at handling the nonconvex-nonconcave min-max optimization and the decentralized communication simultaneously. In this paper, we address this difficulty by designing the \textbf{first gradient-based decentralized parallel algorithm} which allows workers to have multiple rounds of communications in one iteration and to update the discriminator and generator simultaneously, and this design makes it amenable for the convergence analysis of the proposed decentralized algorithm. Theoretically, our proposed decentralized algorithm is able to solve a class of non-convex non-concave min-max problems with provable non-asymptotic convergence to first-order stationary point. Experimental results on GANs demonstrate the effectiveness of the proposed algorithm.

This paper presents a MATLAB toolbox for computing the conformal mapping from a given polygonal multiply connected domain onto a circular multiply connected domain and its inverse. The toolbox can be used for multiply connected domains with high connectivity and complex geometry. It can be employed also for simply connected domains.

Unmanned aerial vehicle (UAV) base stations (BSs) are reliable and efficient alternative to full fill the coverage and capacity requirements when the backbone network fails to provide such requirements due to disasters. In this paper, we consider optimal UAV-deployment problem in 3D space for a mmWave network. The objective is to deploy multiple aerial BSs simultaneously to completely serve the ground users. We develop a novel algorithm to find the feasible positions for a set of UAV-BSs from a predefined set of locations, subject to a signal-to-interference-plus-noise ratio (SINR) constraint of every associated user, UAV-BS's limited hovering altitude constraint and restricted operating zone constraint. We cast this 3D positioning problem as an l_0 minimization problem. This is a combinatorial, NP-hard problem. We approximate the l_0 minimization problem as non-combinatorial l_1-norm problem. Therefore, we provide a suboptimal algorithm to find a set of feasible locations for the UAV-BSs to operate. The analysis shows that the proposed algorithm achieves a set of the location to deploy multiple UVA-BSs simultaneously while satisfying the constraints.

We consider a class of nonsmooth optimization problems over the Stiefel manifold, in which the objective function is weakly convex in the ambient Euclidean space. Such problems are ubiquitous in engineering applications but still largely unexplored. We present a family of Riemannian subgradient-type methods---namely Riemannain subgradient, incremental subgradient, and stochastic subgradient methods---to solve these problems and show that they all have an iteration complexity of ${\cal O}(\varepsilon^{-4})$ for driving a natural stationarity measure below $\varepsilon$. In addition, we establish the local linear convergence of the Riemannian subgradient and incremental subgradient methods when the problem at hand further satisfies a sharpness property and the algorithms are properly initialized and use geometrically diminishing stepsizes. To the best of our knowledge, these are the first convergence guarantees for using Riemannian subgradient-type methods to optimize a class of nonconvex nonsmooth functions over the Stiefel manifold. The fundamental ingredient in the proof of the aforementioned convergence results is a new Riemannian subgradient inequality for restrictions of weakly convex functions on the Stiefel manifold, which could be of independent interest. We also show that our convergence results can be extended to handle a class of compact embedded submanifolds of the Euclidean space. Finally, we discuss the sharpness properties of various formulations of the robust subspace recovery and orthogonal dictionary learning problems and demonstrate the convergence performance of the algorithms on both problems via numerical simulations.

The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $\mathbb{B}^N\times \mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $\eta$-Einstein case.

Performance monitoring is an essential function for margin measurements in live systems. Historically, system budgets have been described by the Q-factor converted from the bit error rate (BER) under binary modulation and direct detection. The introduction of hard-decision forward error correction (FEC) did not change this. In recent years technologies have changed significantly to comprise coherent detection, multilevel modulation and soft FEC. In such advanced systems, different metrics such as (nomalized) generalized mutual information (GMI/NGMI) and asymmetric information (ASI) are regarded as being more reliable. On the other hand, Q budgets are still useful because pre-FEC BER monitoring is established in industry for live system monitoring.

The pre-FEC BER is easily estimated from available information of the number of flipped bits in the FEC decoding, which does not require knowledge of the transmitted bits that are unknown in live systems. Therefore, the use of metrics like GMI/NGMI/ASI for performance monitoring has not been possible in live systems. However, in this work we propose a blind soft-performance estimation method. Based on a histogram of log-likelihood-values without the knowledge of the transmitted bits, we show how the ASI can be estimated.

We examined the proposed method experimentally for 16 and 64-ary quadrature amplitude modulation (QAM) and probabilistically shaped 16, 64, and 256-QAM in recirculating loop experiments. We see a relative error of 3.6%, which corresponds to around 0.5 dB signal-to-noise ratio difference for binary modulation, in the regime where the ASI is larger than the assumed FEC threshold. For this proposed method, the digital signal processing circuitry requires only a minimal additional function of storing the L-value histograms before the soft-decision FEC decoder.

Classes of pairs of ternary quadratic forms parametrize quartic rings by a result of Bhargava. We give an algorithm for finding a pair of ternary quadratic forms that parametrize a given order of a quartic field. We examine a new technique, essential pairs, for obtaining parameters of orders of quartic fields from a number field database. Essential pairs for maximal orders of quartic fields are very common and provide a simple means of obtaining an integral basis for the ring of integers.

We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients we employ the Wiman-Valiron theory. We demonstrate the sharpness of our bounds by studying well-behaved covariance kernels, which we call admissible (after Hayman).

We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, for the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete reductions, studied by Rees, to prove sharper bounds for the mixed multiplicity in low dimensions.

We prove that if $G = G_1\times\dots\times G_n$ acts essentially, properly and cocompactly on a CAT(0) cube complex X, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal dimension of a cube complex the group acts on is strictly larger than that of the minimal dimension of a CAT(0) space upon which the group acts.

This work is motivated by the need of collecting fresh data from power-constrained sensors in the industrial Internet of Things (IIoT) network. A recently proposed metric, the Age of Information (AoI) is adopted to measure data freshness from the perspective of the central controller in the IIoT network. We wonder what is the minimum average AoI the network can achieve and how to design scheduling algorithms to approach it. To answer these questions when the channel states of the network are Markov time-varying and scheduling decisions are restricted to bandwidth constraint, we first decouple the multi-sensor scheduling problem into a single-sensor constrained Markov decision process (CMDP) through relaxation of the hard bandwidth constraint. Next we exploit the threshold structure of the optimal policy for the decoupled single sensor CMDP and obtain the optimum solution through linear programming (LP). Finally, an asymptotically optimal truncated policy that can satisfy the hard bandwidth constraint is built upon the optimal solution to each of the decoupled single-sensor. Our investigation shows that to obtain a small AoI performance: (1) The scheduler exploits good channels to schedule sensors supported by limited power; (2) Sensors equipped with enough transmission power are updated in a timely manner such that the bandwidth constraint can be satisfied.

This paper describes a new algorithm for computing Nonnegative Low Rank Matrix (NLRM) approximation for nonnegative matrices. Our approach is completely different from classical nonnegative matrix factorization (NMF) which has been studied for more than twenty five years. For a given nonnegative matrix, the usual NMF approach is to determine two nonnegative low rank matrices such that the distance between their product and the given nonnegative matrix is as small as possible. However, the proposed NLRM approach is to determine a nonnegative low rank matrix such that the distance between such matrix and the given nonnegative matrix is as small as possible. There are two advantages. (i) The minimized distance by the proposed NLRM method can be smaller than that by the NMF method, and it implies that the proposed NLRM method can obtain a better low rank matrix approximation. (ii) Our low rank matrix admits a matrix singular value decomposition automatically which provides a significant index based on singular values that can be used to identify important singular basis vectors, while this information cannot be obtained in the classical NMF. The proposed NLRM approximation algorithm was derived using the alternating projection on the low rank matrix manifold and the non-negativity property. Experimental results are presented to demonstrate the above mentioned advantages of the proposed NLRM method compared the NMF method.

In this article we construct Laurent polynomial Landau-Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in arXiv:math/0511124. The Laurent polynomial takes a similar shape to the one given in arXiv:alg-geom/9603021 for projective complete intersections, i.e. it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in arXiv:math/0607492, associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in arXiv:1404.4844 and arXiv:1304.4958 for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau-Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

We consider the facilitated exclusion process, which is a nonergodic, kinetically constrained exclusion process. We show that in the hydrodynamic limit, its macroscopic behavior is governed by a free boundary problem. The particles evolve on the one-dimensional lattice according to jump rates which are degenerate, since they can vanish on non-trivial configurations and create distinct phases: indeed, configurations can be totally blocked (they cannot evolve under the dynamics), ergodic (they belong to an irreducible component), or transient (after a transitive period of time they will become either blocked or ergodic). We additionally prove that the microscopic separation into blocked/ergodic phases fully coincides with the moving interface problem given by the hydrodynamic equation.

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size $K$ placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The reservoirs have slow dynamics, in the sense that they intervene at speed $N^{-\theta}$ w.r.t. the bulk dynamics. In the first part of this article, we treated the case $\theta>1$ for which the entropy method can be adapted. We treat here the case where the boundary dynamics is too fast for the Entropy Method to apply. We prove, using duality estimates inspired by previous work by Erignoux, Landim and Xu, that the hydrodynamic limit is given by the heat equation with Dirichlet boundary conditions, where the density at the boundaries is fixed by the parameters of the model.

We show that the classical Shinbrot's criteria to guarantee that a Leray-Hopf solution satisfies the energy equality follows trivially from the $L^4( (0\,,T)\times\Omega))$ Lions-Prodi particular case. Moreover we extend Shinbrot's result to space coefficients $r \in (3,\,4)\,.$ In this last case our condition coincides with Shinbrot condition for $r=4$, but for $r<4$ it is more restrictive than the classical one, $2/p + 2/r = 1\,.$ It looks significant that in correspondence to the extreme values $r=3$ and $r=\infty$, and just for these two values, the conditions become respectively $u \in L^\infty(L^3)$ and $u \in L^2(L^\infty)$, which imply regularity by appealing to classical Ladyzhenskaya-Prodi-Serrin (L-P-S) type conditions. However, for values $r\in (3,\infty)$ the L-P-S condition does not apply, even for the more demanding case $\,3<r<4\,.$ The proofs are quite trivial, by appealing to interpolation, with $L^\infty(L^2)$ in the first case and with $L^2(L^6)$ in the second case. The central position of this old classical problem in Fluid-Mechanics, together with the simplicity of the proofs (in particular the novelty of the second result) looks at least curious. This may be considered a merit of this very short note.

In this paper, we discuss estimates on transition densities for subordinators, which are global in time. We establish the sharp two-sided estimates on the transition densities for subordinators whose L\'evy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on L\'evy measures, we also obtain a precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more.

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov-Fokker-Planck equation coupled with the compressible isentropic Navier-Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier-Stokes system. Our main strategy relies on the relative entropy argument based on the weak-strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier-Stokes system.

We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the K\"ahler classes in dense subsets of the K\"ahler cone. Using the same technique, we show that the K\"ahler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.

Recently, Marko and Litvinov (ML) conjectured that, for all positive integers $n$ and $p$, the $p$th power of $n$ admits the representation $n^p = \sum_{\ell =0}^{p-1} (-1)^{l} c_{p,\ell} F_{n}^{p-\ell}$, where $F_{n}^{p-\ell}$ is the $n$th hyper-tetrahedron number of dimension $p-\ell$ and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets of the $p$-dimensional simplex $x_{\sigma_1} \geq x_{\sigma_2} \geq \cdots \geq x_{\sigma_p}$ (where $\sigma$ is a permutation of $\{ 1, 2, \ldots, p \}$) formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. In this paper we show that the ML conjecture is true for every natural number $p$. Our proof relies on the fact that the validity of the ML conjecture necessarily implies that $c_{p,\ell} = (p-\ell)! S(p, p-\ell)$, where $S(p,p-\ell)$ are the Stirling numbers of the second kind. Furthermore, we provide a number of equivalent formulas expressing the sum of powers $\sum_{i=1}^{n} i^p$ as a linear combination of figurate numbers.

We show that several integrable (i.e., exactly solvable) scalar cosmologies considered by Fr\'e, Sagnotti and Sorin (Nuclear Physics \textbf{B 877}(3) (2013), 1028--1106) can be generalized to include cases where the spatial curvature is not zero and, besides a scalar field, matter or radiation are present with an equation of state $p^{(m)} = w\, \rho^{(m)}$; depending on the specific form of the self-interaction potential for the field, the constant $w$ can be arbitrary or must be fixed suitably.

Invariable generation is a topic that, until relatively recently, was exclusive to finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various authors have investigated the property for particular infinite groups or families of infinite groups.

A group is invariably generated by a subset $S$ if replacing each element of $S$ with any of its conjugates still results in a generating set for $G$. In this paper we investigate how this property behaves with respect to wreath products. Our main work is to deal with the case where the base of $G\wr_X H$ in not invariably generated. We see both positive and negative results here depending on $H$ and its action on $X$.

Diverse inverse problems in imaging can be cast as variational problems composed of a task-specific data fidelity term and a regularization term. In this paper, we propose a novel learnable general-purpose regularizer exploiting recent architectural design patterns from deep learning. We cast the learning problem as a discrete sampled optimal control problem, for which we derive the adjoint state equations and an optimality condition. By exploiting the variational structure of our approach, we perform a sensitivity analysis with respect to the learned parameters obtained from different training datasets. Moreover, we carry out a nonlinear eigenfunction analysis, which reveals interesting properties of the learned regularizer. We show state-of-the-art performance for classical image restoration and medical image reconstruction problems.

Derksen proved that the spectral norm is multiplicative with respect to vertical tensor products (also known as tensor Kronecker products). We will use this result to show that the nuclear norm and other norms of interest are also multiplicative with respect to vertical tensor products.

Three concepts of local times for deterministic c\{a}dl\{a}g paths are developed and the corresponding pathwise Tanaka--Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times.

Large-scale nonconvex optimization problems are ubiquitous in modern machine learning, and among practitioners interested in solving them, Stochastic Gradient Descent (SGD) reigns supreme. We revisit the analysis of SGD in the nonconvex setting and propose a new variant of the recently introduced expected smoothness assumption which governs the behaviour of the second moment of the stochastic gradient. We show that our assumption is both more general and more reasonable than assumptions made in all prior work. Moreover, our results yield the optimal $\mathcal{O}(\varepsilon^{-4})$ rate for finding a stationary point of nonconvex smooth functions, and recover the optimal $\mathcal{O}(\varepsilon^{-1})$ rate for finding a global solution if the Polyak-{\L}ojasiewicz condition is satisfied. We compare against convergence rates under convexity and prove a theorem on the convergence of SGD under Quadratic Functional Growth and convexity, which might be of independent interest. Moreover, we perform our analysis in a framework which allows for a detailed study of the effects of a wide array of sampling strategies and minibatch sizes for finite-sum optimization problems. We corroborate our theoretical results with experiments on real and synthetic data.

Three-dimensional pseudo-spherical submanifolds in $\mathbb R^5$, whose Bianchi transformations are degenerate of rank 2, are studied. A complete description of such submanifolds is obtained in the case where the Bianchi transformations are holonomically degenerate.

Let $\Omega \subseteq \{1,\dots,m\} \times \{1,\dots,n\}$. We consider fibers of coordinate projections $\pi_\Omega : \mathscr{M}_k(r,m \times n) \rightarrow k^{\# \Omega}$ from the algebraic variety of $m \times n$ matrices of rank at most $r$ over an infinite field $k$. For $\#\Omega = \dim \mathscr{M}_k(r,m \times n)$ we describe a class of $\Omega$'s for which there exist non-empty Zariski open sets $\mathscr{U}_\Omega \subset \mathscr{M}_k(r,m \times n)$ such that $\pi_\Omega^{-1}\big(\pi_\Omega(X)\big) \cap \mathscr{U}_\Omega$ is a finite set $\forall X \in \mathscr{U}_\Omega$. For this we interpret matrix completion from a point of view of hyperplane sections on the Grassmannian $\operatorname{Gr}(r,m)$. Crucial is a description by Sturmfels $\&$ Zelevinsky of classes of local coordinates on $\operatorname{Gr}(r,m)$ induced by vertices of the Newton polytope of the product of maximal minors of an $m \times (m-r)$ matrix of variables.

Inspired by the Galerkin and particular method, a new approximation approach is recalled in the Cartesian case. In this paper, we are interested specially by constructing this method, when the domain of consideration is a two dimensional ball, to extend this work to the several dimension. We reduce the number of iterations to calculate integrals and numerical solution of Poisson and the Heat problems (elliptic nd parabolic PDEs), in a very fast way.

We study the boundary value problem of two stationary BGK-type models - the BGK model for fast chemical reaction and the BGK model for slow chemical reaction - and provide a unified argument to establish the existence and uniqueness of stationary flows of reactive BGK models in a slab. For both models, the main difficulty arises in the uniform control of the reactive parameters from above and below, since, unlike the BGK models for non-reactive gases, the reactive parameters for the reactive BGK models are defined through highly nonlinear relations. To overcome this difficulty, we introduce several nonlinear functionals that capture essential structures of such nonlinear relations such as the monotonicity in specific variables, that enable one to derive necessary estimates for the reactive equilibrium coefficients.

Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q})$ of the free loop space of $X$ preserves the Hodge decomposition of $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q})$ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in our earlier work.

The design of reliable path-following controllers is a key ingredient for successful deployment of self-driving vehicles. This controller-design problem is especially challenging for a general 2-trailer with a car-like tractor due to the vehicle's structurally unstable joint-angle kinematics in backward motion and the car-like tractor's curvature limitations which can cause the vehicle segments to fold and enter a jackknife state. Furthermore, optical sensors with a limited field of view have been proposed to solve the joint-angle estimation problem online, which introduce additional restrictions on which vehicle states that can be reliably estimated. To incorporate these restrictions at the level of control, a model predictive path-following controller is proposed. By taking the vehicle's physical and sensing limitations into account, it is shown in real-world experiments that the performance of the proposed path-following controller in terms of suppressing disturbances and recovering from non-trivial initial states is significantly improved compared to a previously proposed solution where the constraints have been neglected.

The aim of this paper is to study bimodule stably Calabi-Yau properties of derivation quotient algebras. We give the definition of a twisted stably Calabi-Yau algebra and show that every twisted derivation quotient algebra $A$ for which the associated bimodule complex gives the beginning of a bimodule resolution for $A$ is bimodule stably twisted Calabi-Yau. In this setting we give a new interpretation of some results by Yu [Yu12], implying that $A$ is almost Koszul of periodic type. Using the characterization of higher preprojective algebras given by Amiot and Oppermann in [AO14], we prove that finite dimensional bigraded derivation quotient algebras with homogeneous potential and exact associated complex are higher preprojective algebras of their degree-zero subalgebra, which is Koszul and $(d-1)$-representation finite.

Given a finite set $X$ and a function $f:X\to X$, we define the degree of noninvertibility of $f$ to be $\displaystyle\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call nibble sort.'' We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that $\max_{\substack{f:X\to X\\ |X|=n}}\frac{\text{deg}(f^k)}{\text{deg}(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})$ for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.