### Essential holonomy of Cantor actions

A group action has essential holonomy if the set of points with non-trivial holonomy has positive measure. If such an action is topologically free, then having essential holonomy is equivalent to the action not being essentially free, that is, the set of points with non-trivial stabilizer has positive measure. In the paper, we investigate the relation between the property of having essential holonomy and structure of the acting group for minimal equicontinuous actions on Cantor sets. We show that if such a group action is locally quasi-analytic and has essential holonomy, then every commutator subgroup in the group lower central series has elements with positive measure set of points with non-trivial holonomy. We deduce that all minimal equicontinuous Cantor actions by nilpotent groups have no essential holonomy. We also introduce a local version of the Farber criterion, which allows to determine when a locally quasi-analytic action has no essential holonomy.

### Poles of degenerate Eisenstein series and Siegel-Weil identities for exceptional split groups

Let $G$ be a linear split algebraic group. The degenerate Eisenstein series associated to a maximal parabolic subgroup $E_{P}(f^{0},s,g)$ with the spherical section $f^{0}$ is studied in the first part of the thesis. In this part, we study the poles of $E_{P}(f^{0},s,g)$ in the region $\operatorname{Re} s >0$. We determine when the leading term in the Laurent expansion of $E_{P}(f^{0},s,g)$ around $s=s_0$ is square integrable. The second part is devoted to finding identities between the leading terms of various Eisenstein series at different points. We present an algorithm to find this data and implement it on \textit{SAGE}. While both parts can be applied to a general algebraic group, we restrict ourself to the case where $G$ is split exceptional group of type $F_4,E_6,E_7$, and obtain new results.

### A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves

We show that a bound of the Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety $X\subseteq\mathbb{P}^r$ is sharp exactly for complete intersections, provided the variety $X$ is cut out scheme-theoretically by several hypersurfaces in $\mathbb{P}^r$. This generalizes a result of Bertram-Ein-Lazarsfeld.

### [4] 2205.06298

The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract incidence algebra of the division poset.

### An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices

We consider random matrices whose entries are obtained by applying a (nonlinear) kernel function to the pairwise inner products between $n$ independent data vectors drawn uniformly from the unit sphere in $\mathbb{R}^d$. Our study of this model is motivated by problems in machine learning, statistics, and signal processing, where such inner-product kernel random matrices and their spectral properties play important roles. Under mild conditions on the kernel function, we establish the weak-limit of the empirical spectral distribution of these matrices when $d, n \to \infty$ such that $n / d^\ell \to \kappa \in (0, \infty)$, for some fixed $\ell \in \mathbb{N}$ and $\kappa \in \mathbb{R}$. This generalizes an earlier result of Cheng and Singer (2013), who studied the same model in the linear scaling regime (with $\ell = 1$ and $n/d \to \kappa$). The main insight of our work is a general equivalence principle: the spectrum of the random kernel matrix is asymptotically equivalent to that of a simpler matrix model, constructed as the linear combination of a (shifted) Wishart matrix and an independent matrix drawn from the Gaussian orthogonal ensemble. The aspect ratio of the Wishart matrix and the coefficients of the linear combination are determined by $\ell$ and by the expansion of the kernel function in the orthogonal Hermite polynomial basis. Consequently, the limiting spectrum of the random kernel matrix can be characterized as the free additive convolution between a Marchenko-Pastur law and a semicircle law.

### Continuity in right semitopological groups

Groups with a topology that is in one way or another consistent with the algebraic structure are considered. Classical groups with topology -- topological, paratopological, semitopological, quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). Are studied when a group with topology is a topological group, research in this direction began with the work of Deane Montgomery and Robert Ellis. The (invariant) semi-neighborhoods of the diagonal are used as a means of study.

### Frucht's theorem in Borel setting

In this paper, we show that Frucht's theorem holds in Borel setting. More specifically, we prove that any standard Borel group can be realized as the Borel automorphism group of a Borel graph. A slight modification of our construction also yields the following result in topological setting: Any Polish group can be realized as the homeomorphic automorphism group of a $\mathbf{\Delta^0_2}$-graph on a Polish space.

### Reversing monoid actions and domination in graphs

Given a graph $G=(V,E)$, a set of vertices $D\subseteq V$ is called a dominating set if every vertex in $V\backslash D$ is adjacent to a vertex in $D$, and a subset $B\subseteq V$ is called a non-blocking set if $V\backslash B$ is a dominating set. In this paper, we introduce a graph dynamical system and establish a one to one correspondence between the periodic points of this system and subsets of $V$ that are simultaneously dominating and non-blocking sets. Besides, by using the action of this graph dynamical system we obtain actions of free monoid on two letters, for which fixed points of the action and more specialized dominating sets, such as, independent dominating sets and perfect dominating sets, coincide.

### Average capacity of quantum entanglement

As an alternative to entanglement entropies, the capacity of entanglement becomes a promising candidate to probe and estimate the degree of entanglement of quantum bipartite systems. In this work, we study the typical behavior of entanglement capacity over major models of random states. In particular, the exact and asymptotic formulas of average capacity have been derived under the Hilbert-Schmidt and Bures-Hall ensembles. The obtained formulas generalize some partial results of average capacity computed recently in the literature. As a key ingredient in deriving the results, we make use of recent advances in random matrix theory pertaining to the underlying orthogonal polynomials and special functions. Numerical study has been performed to illustrate the usefulness of average capacity as an entanglement indicator.

### Solvability of some Stefan type problems

In this paper, we interest on some class of Stefan type problems. We prove the existence and uniqueness of renormalized solution in anisotropic Sobolev spaces with data belongs to $L^1- data,$ based on the properties of the renormalized trunctions and the generalized monotonicity method in the functional spaces.

### Fractional-Step Runge--Kutta Methods: Representation and Linear Stability Analysis

Fractional-step methods are a popular and powerful divide-and-conquer approach for the numerical solution of differential equations. When the integrators of the fractional steps are Runge--Kutta methods, such methods can be written as generalized additive Runge--Kutta (GARK) methods, and thus the representation and analysis of such methods can be done through the GARK framework. We show how the general Butcher tableau representation and linear stability of such methods are related to the coefficients of the splitting method, the individual sub-integrators, and the order in which they are applied. We use this framework to explain some observations in the literature about fractional-step methods such as the choice of sub-integrators, the order in which they are applied, and the role played by negative splitting coefficients in the stability of the method.

### Continuous Interior Penalty stabilization for divergence-free finite element methods

In this paper we propose, analyze, and test numerically a pressure-robust stabilized finite element for a linearized case of the Navier-Stokes' equation in the high Reynolds number regime, also known as Oseen's problem. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

### A description of the Zeta map on Dyck paths area sequences

We give a description of the well known Zeta map on Dyck paths which sends the dinv,area statistics to the area,bounce statistics. Our description uses Dyck paths area sequences and can be implemented easily.

### Regular theory in complex braid groups

In his seminal paper on complex reflection arrangements, Bessis introduces a Garside structure for the braid group of a well-generated irreducible complex reflection group. Using this Garside structure, he establishes a strong connection between regular elements in the reflection group, and roots of the "full twist" element of the pure braid group. He then suggests that it would be possible to extend the conclusion of this theorem to centralizers of regular elements in well-generated groups. In this paper we give a positive answer to this question and we show moreover that these results hold for an arbitrary reflection group.

### Degree Based Topological Indices of a General Random Chain

Let G=(V(G),E(G)) a graph, many important topological indices can be defined as TI(G)= \sum_{vu\in E(G)} h(d_{v},d_{u}). In this paper, we look at one type of general random chains and an alternative approach to study these kinds of topological indices is proposed. In which their explicit analytical expressions of the expected values and variances are obtained. Moreover, the asymptotic normality of topological indices of a random chain is established through the Martingale Central Limit Theorem.

### Determining the collision kernel in the Boltzmann equation near the equilibrium

We consider an inverse problem for the nonlinear Boltzmann equation near the equilibrium. Our goal is to determine the collision kernel in the Boltzmann equation from the knowledge of the Albedo operator. Our approach relies on a linearization technique as well as the injectivity of the Gauss-Weierstrass transform.

### Virtually Full-duplex Cell-Free Massive MIMO with Access Point Mode Assignment

We consider a cell-free massive multiple-input multiple-output (MIMO) network utilizing a virtually full-duplex (vFD) mode, where access points (APs) with a downlink (DL) mode and those with an uplink (UL) mode simultaneously serve DL and UL users (UEs). In order to maximize the sum spectral efficiency (SE) of the DL and UL transmissions, we formulate a mixed-integer optimization problem to jointly design the AP mode assignment and power control. This problem is subject to minimum per-UE SE requirements, per-AP power control, and per-UL UE power constraints. By employing the successive convex approximation technique, we propose an algorithm to obtain a stationary solution of the formulated problem. Numerical results show that the proposed vFD approach can provide a sum SE that is $2.5$ and $1.5$ times larger than the traditional half-duplex and heuristic baseline schemes, respectively, in terms of $95\%$-likely sum SE.

### Optimal Order of Encoding for Gaussian MIMO Multi-Receiver Wiretap Channel

The Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel is studied in this paper. The base station broadcasts confidential messages to K intended users while keeping the messages secret from an eavesdropper. The capacity of this channel has already been characterized by applying dirty-paper coding and stochastic encoding. However, K factorial encoding orders may need to be enumerated for that, which makes the problem intractable. We prove that there exists one optimal encoding order and reduced the K factorial times to a one-time encoding. The optimal encoding order is proved by forming a secrecy weighted sum rate (WSR) maximization problem. The optimal order is the same as that for the MIMO broadcast channel without secrecy constraint, that is, the weight of users' rate in the WSR maximization problem determines the optimal encoding order. Numerical results verify the optimal encoding order.

### Representations of Quantum Coordinate Algebras at Generic $q$ and Wiring Diagrams

This paper is devoted to the representation theory of quantum coordinate algebra $\mathbb{C}_q[G]$, for a semisimple Lie group $G$ and a generic parameter $q$. By inspecting the actions of normal elements on tensor modules, we generalize a result of Levendorski\u{\i} and Soibelman for highest weight modules. For a double Bruhat cell $G^{w_1,w_2}$, we describe the primitive spectra $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$ in a new fashion, and construct a bundle of $(w_1,w_2)$ type simple modules onto $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$, provided $\mathrm{Supp}(w_1)\cap\mathrm{Supp}(w_2)=\varnothing$ or enough pivot elements. The fibers of the bundle are shown to be products of the spectrums of simple modules of 2-dimensional quantum torus $L_q(2)$. As an application of our theory, we deduce an equivalent condition for the tensor module to be simple, and construct some simple modules for each primitive ideal when $G=SL_3(\mathbb{C})$. This completes Dixmier's program for $\mathbb{C}_q[SL_3(\mathbb{C})]$.

### Ballisticity of Random walks in Random Environments on $\mathbb{Z}$ with Bounded Jumps

We characterize ballistic behavior for general i.i.d. random walks in random environments on $\mathbb{Z}$ with bounded jumps. The two characterizations we provide do not use uniform ellipticity conditions. They are natural in the sense that they both relate to formulas for the limiting speed in the nearest-neighbor case. Note: This paper duplicates results from some versions of the preprint "Random walks in Dirichlet random environments on $\mathbb{Z}$ with bounded jumps." (arxiv: 2104.14950). The present paper is being split off for reasons of length, and the plan is to remove these results from a future version of the previous paper and replace them with a citation of the present preprint.

### Scaling limit of a generalized contact process

We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice $\mathbb{Z}^d$. The states at each lattice site can take values in $0,\ldots,k$. These can be interpreted as neuronal membrane potential, with the state $k$ corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state $k$ corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from $0$ to $k$. The rate at which it climbs is determined by other neurons at state $k$, coupled to it through a Kac-type potential, of range $\gamma^{-1}$. The hydrodynamic equations are obtained in the limit $\gamma\rightarrow 0$. Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered.

### On multilevel Monte Carlo methods for deterministic and uncertain hyperbolic systems

In this paper, we evaluate the performance of the multilevel Monte Carlo method (MLMC) for deterministic and uncertain hyperbolic systems, where randomness is introduced either in the modeling parameters or in the approximation algorithms. MLMC is a well known variance reduction method widely used to accelerate Monte Carlo (MC) sampling. However, we demonstrate in this paper that for hyperbolic systems, whether MLMC can achieve a real boost turns out to be delicate. The computational costs of MLMC and MC depend on the interplay among the accuracy (bias) and the computational cost of the numerical method for a single sample, as well as the variances of the sampled MLMC corrections or MC solutions. We characterize three regimes for the MLMC and MC performances using those parameters, and show that MLMC may not accelerate MC and can even have a higher cost when the variances of MC solutions and MLMC corrections are of the same order. Our studies are carried out by a few prototype hyperbolic systems: a linear scalar equation, the Euler and shallow water equations, and a linear relaxation model, the above statements are proved analytically in some cases, and demonstrated numerically for the cases of the stochastic hyperbolic equations driven by white noise parameters and Glimm's random choice method for deterministic hyperbolic equations.

### A quantitative Khintchine-Groshev theorem for S-arithmetic Diophantine approximation

In his 1960 paper, Schmidt studied a quantitative type of Khintchine-Groshev theorem. Recently, a new proof of the theorem was found, which made it possible to relax the dimensional constraint and more generally, to add on the congruence condition by M. Alam, A. Ghosh, and S. Yu. In this paper, we generalize this new approach to S-arithmetic spaces and obtain a quantitative version of an S-arithmetic Khintchine-Groshev theorem. In fact, we consider a new S-arithmetic analog of Diophantine approximation, which is different from the one formerly established (see the 2007 paper of D. Kleinbock and G. Tomanov). Hence for the sake of completeness, we also deal with the convergence case of the Khintchine-Groshev theorem, based on this new generalization.

### Double crossed biproducts and related structures

Let $H$ be a bialgebra. Let $\sigma: H\otimes H\to A$ be a linear map, where $A$ is a left $H$-comodule coalgebra, and an algebra with a left $H$-weak action $\triangleright$. Let $\tau: H\otimes H\to B$ be a linear map, where $B$ is a right $H$-comodule coalgebra, and an algebra with a right $H$-weak action $\triangleleft$. In this paper, we improve the necessary conditions for the two-sided crossed product algebra $A\#^{\sigma} H~{^{\tau}\#} B$ and the two-sided smash coproduct coalgebra $A\times H\times B$ to form a bialgebra (called double crossed biproduct) such that the condition $b_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b$ in Majid's double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzez\'nski's crossed product and give some applications.

### Heat kernel estimate in a conical singular space

Let $(X,g)$ be a product cone with the metric $g=dr^2+r^2h$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. We study the upper boundedness of heat kernel associated with the operator $L_V=-\Delta_g+V_0 r^{-2}$, where $-\Delta_g$ is the positive Friedrichs extension Laplacian on $X$ and $V=V_0(y) r^{-2}$ and $V_0\in\mathcal{C}^\infty(Y)$ is a real function such that the operator $-\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$.The new ingredient of the proof is the Hadamard parametrix and finite propagation speed of wave operator on $Y$.

### Rigid and Separable Algebras in Compact Semisimple Monoidal 2-Categories

Given a monoidal 2-category that has right and left adjoints, we prove that the 2-categories of modules and of bimodules over a rigid algebra have right and left adjoints. Given a compact semisimple monoidal 2-category, we use this result to prove that the 2-categories of modules and of bimodules over a separable algebra are compact semisimple. Finally, we define the dimension of a connected rigid algebra in a compact semisimple monoidal 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.

### Blind Deconvolution with Non-smooth Regularization via Bregman Proximal DCAs

Blind deconvolution is a technique to recover an original signal without knowing a convolving filter. It is naturally formulated as a minimization of a quartic objective function under some assumption. Because its differentiable part does not have a Lipschitz continuous gradient, existing first-order methods are not theoretically supported. In this letter, we employ the Bregman-based proximal methods, whose convergence is theoretically guaranteed under the $L$-smad property. We first reformulate the objective function as a difference of convex (DC) functions and apply the Bregman proximal DC algorithm (BPDCA). This DC decomposition satisfies the $L$-smad property. The method is extended to the BPDCA with extrapolation (BPDCAe) for faster convergence. When our regularizer has a sufficiently simple structure, each iteration is solved in a closed-form expression, and thus our algorithms solve large-scale problems efficiently. We also provide the stability analysis of the equilibriums and demonstrate the proposed methods through numerical experiments on image deblurring. The results show that BPDCAe successfully recovered the original image and outperformed other existing algorithms.

### Disjoint Total Dominating Sets in Near-Triangulations

We show that every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88 (2018) 174-191].

### Strongly First Order, Domain Independent Dependencies: the Union-Closed Case

Team Semantics generalizes Tarski's Semantics by defining satisfaction with respect to sets of assignments rather than with respect to single assignments. Because of this, it is possible to use Team Semantics to extend First Order Logic via new kinds of connectives or atoms - most importantly, via dependency atoms that express dependencies between different assignments. Some of these extensions are more expressive than First Order Logic proper, while others are reducible to it. In this work, I provide necessary and sufficient conditions for a dependency atom that is domain independent (in the sense that its truth or falsity in a relation does not depend on the existence in the model of elements that do not occur in the relation) and union closed (in the sense that whenever it is satisfied by all members of a family of relations it is also satisfied by their union) to be strongly first order, in the sense that the logic obtained by adding them to First Order Logic is no more expressive than First Order Logic itself.

### Extinction of multiple shocks in the modular Burgers equation

A traveling viscous shock was previously studied in the Burgers equation with the modular advection term. It was shown that small, smooth, and exponentially decaying in space perturbations to the viscous shock decay in time. The present work addresses multiple shocks of the same model. We first prove that no traveling viscous shocks with multiple interfaces exist. We then suggest with the help of a priori energy estimates and numerical simulations that the evolution of viscous shocks with multiple interfaces leads to the finite-time extinction of compact regions between two consequent interfaces. We specify a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulations.

### A class of few-Lee weight $\mathbb{Z}_2[u]$-linear codes using simplicial complexes and minimal codes via Gray map

Recently some mixed alphabet rings are involved in constructing few-Lee weight codes with optimal or minimal Gray images using suitable defining sets or down-sets. Inspired by these works, we choose the mixed alphabet ring $\mathbb{Z}_2\mathbb{Z}_2[u]$ to construct a special class of linear code $C_L$ over $\mathbb{Z}_2[u]$ with $u^2=0$ by employing simplicial complexes generated by a single maximal element. We show that $C_L$ has few-Lee weights by determining the Lee weight distribution of $C_L$. Also we have an infinite family of minimal codes over $\mathbb{Z}_2$ via Gray map, which can be used to secret sharing schemes.

### Data-Driven Upper Bounds on Channel Capacity

We consider the problem of estimating an upper bound on the capacity of a memoryless channel with unknown channel law and continuous output alphabet. A novel data-driven algorithm is proposed that exploits the dual representation of capacity where the maximization over the input distribution is replaced with a minimization over a reference distribution on the channel output. To efficiently compute the required divergence maximization between the conditional channel and the reference distribution, we use a modified mutual information neural estimator that takes the channel input as an additional parameter. We evaluate our approach on different memoryless channels and show that the estimated upper bounds closely converge either to the channel capacity or to best-known lower bounds.

### Monomial reduction of knot polynomials

For all natural numbers $N$ and prime numbers $p$, we find a knot $K$ whose skein polynomial $P_K(a,z)$ evaluated at $z=N$ has trivial reduction modulo $p$. An interesting consequence of our construction is that all polynomials $P_K(a,N)$ (mod~$p$) with bounded $a$-span are realised by knots with bounded braid index. As an application, we classify all polynomials of the form $P_K(a,1)$ (mod $2$) with $a$-span $\leq 10$.

### Existence and weak-strong uniqueness for Maxwell-Stefan-Cahn-Hilliard systems

A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a bounded domain with no-flux boundary conditions. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott--Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yielding $H^2(\Omega)$ bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

### Frame set for Gabor systems with Haar window

We show the full structure of the frame set for the Gabor system $\mathcal{G}(g;\alpha,\beta):=\{e^{-2\pi i m\beta\cdot}g(\cdot-n\alpha):m,n\in\Bbb Z\}$ with the window being the Haar function $g=-\chi_{[-1/2,0)}+\chi_{[0,1/2)}$. The strategy of this paper is to introduce the piecewise linear transformation $\mathcal{M}$ on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish {a} necessary and sufficient condition for the Gabor system $\mathcal{G}(g;\alpha,\beta)$ to be a frame, i.e., the symmetric invariant set of the transformation $\mathcal{M}$ is empty. Compared with the previous studies, the present paper provides a self-contained environment to study Gabor frames by a new perspective, which includes that the techniques developed here are new and all the proofs could be understood thoroughly by the readers without reference to the known results in the previous literature.

### Opportunistic Routing aided Cooperative Communication Network with Energy-Harvesting Nodes

In this paper, a cooperative communication network based on two energy-harvesting (EH) decode-and-forward (DF) relays which harvest energy from the ambience using buffers with harvest-store-use (HSU) architecture is considered. For improving the data delivery in this network, an opportunistic routing (OR) algorithm considering channel status information (CSI), location and energy buffer status of relays is proposed. For the sake of deriving the theoretical expressions for limiting distribution of energy stored in buffers with discrete-time continuous-state space Markov chain (DCSMC) model, the probabilities that the packet to be forwarded exists in one and more transmitting nodes are obtained based on the state transition matrix (STM). Moreover, the closed-form expressions for outage probability and throughput of the network based on the CSI and the limiting distributions of energy stored in buffers are presented. Numerous experiments have been performed to verify the derived theoretical expressions.

### Asymptotics for connected graphs and irreducible tournaments

We compute the whole asymptotic expansion of the probability that a large uniform labeled graph is connected, and of the probability that a large uniform labeled tournament is irreducible. In both cases, we provide a combinatorial interpretation of the involved coefficients.

### Ordering sequence for link diagrams with respect to Ridemeister moves I and III

We prove that there exist infinitely many pairs of RI-III related (see Definition 2.1 in this paper) trivial knot diagrams that are not transformed into each other by a sequence of Reidemeister moves I, followed by a sequence of Reidemeister moves III, followed by a sequence of Reidemeister moves I. To create a simple sequence for RI-III related link diagrams instead of the ordinary ordering sequence, we prove that RI-III related link diagrams are always transformed into each other by applying an I-generalized ordering sequence.

### Regularization Theory of the Analytic Deep Prior Approach

The analytic deep prior (ADP) approach was recently introduced for the theoretical analysis of deep image prior (DIP) methods with special network architectures. In this paper, we prove that ADP is in fact equivalent to classical variational Ivanov methods for solving ill-posed inverse problems. Besides, we propose a new variant which incorporates the strategy of early stopping into the ADP model. For both variants, we show how classical regularization properties (existence, stability, convergence) can be obtained under common assumptions.

### Coded Caching at the Edge of Satellite Networks

Caching multimedia contents at the network edge is a key solution to decongest the amount of traffic in the backhaul link. In this paper, we extend and analyze the coded caching technique [1] in an unexplored scenario, i.e. at the edge of two-tier heterogeneous networks with an arbitrary number of users. We characterize the performance of such scheme by deriving a closed-form expression of the average backhaul load and reveal a significant gain compared to other benchmark caching schemes proposed in the literature.

### On Fekete Points for a Real Simplex

We survey what is known about Fekete points/optimal designs for a simplex in $\R^d.$ Several new results are included. The notion of Fej\'er exponenet for a set of interpolation points is introduced.

### The Prime Number Theorem and Pair Correlation of Zeros of the Riemann Zeta-Function

We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms.

### Characters and projective characters of alternating and symmetric groups determined by values on $l'$-classes

This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer $l$, for $l \neq 3$. We do the same for the double covers of the symmetric and alternating groups. The only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by $l$. When $l$ is prime, this implies that the rows of the $l$-modular decomposition matrix are distinct except for the rows labelled by these pairs. When $l=3$ we exhibit many additional examples of such pairs of characters.

### When all Permutations are Combinatorial Similarities

Let $(X, d)$ be a semimetric space. A permutation $\Phi$ of the set $X$ is a combinatorial self similarity of $(X, d)$ if there is a bijective function $f \colon d(X^2) \to d(X^2)$ such that $$d(x, y) = f(d(\Phi(x), \Phi(y)))$$ for all $x$, $y \in X$. We describe the set of all semimetrics $\rho$ on an arbitrary nonempty set $Y$ for which every permutation of $Y$ is a combinatorial self similarity of $(Y, \rho)$.

### An affine Birkhoff--Kellogg type result in cones with applications to functional differential equations

In this short note we prove, by means of classical fixed point index, an affine version of a Birkhoff--Kellogg type theorem in cones. We apply our result to discuss the solvability of a class of boundary value problems for functional differential equations subject to functional boundary conditions. We illustrate our theoretical results in an example.

### Representations of the Kottwitz gerbes

Let $F$ be a local or global field and let $G$ be a linear algebraic group over $F$. We study Tannakian categories of representations of the Kottwitz gerbes $\text{Rep}(\text{Kt}_{F})$ and the functor $G\mapsto B(F, G)$ defined by Kottwitz in [28]. In particular, we show that if $F$ is a function field of a curve over $\mathbb{F}_q$, then $\text{Rep}(\text{Kt}_F)$ is equivalent to the category of Drinfeld isoshtukas. In the case of number fields, we establish the existence of various fiber functors on $\text{Rep}(\text{Kt}_{\mathbb{Q}})$ and its subcategories and show that Scholze's conjecture [41, Conjecture 9.5] follows from the full Tate conjecture over finite fields [47].

### An Information-theoretic Method for Collaborative Distributed Learning with Limited Communication

In this paper, we study the information transmission problem under the distributed learning framework, where each worker node is merely permitted to transmit a $m$-dimensional statistic to improve learning results of the target node. Specifically, we evaluate the corresponding expected population risk (EPR) under the regime of large sample sizes. We prove that the performance can be enhanced since the transmitted statistics contribute to estimating the underlying distribution under the mean square error measured by the EPR norm matrix. Accordingly, the transmitted statistics correspond to the eigenvectors of this matrix, and the desired transmission allocates these eigenvectors among the statistics such that the EPR is minimal. Moreover, we provide the analytical solution of the desired statistics for single-node and two-node transmission, where a geometrical interpretation is given to explain the eigenvector selection. For the general case, an efficient algorithm that can output the allocation solution is developed based on the node partitions.

### Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities

In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems.

### Simplex Closing Probabilities in Directed Graphs

Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brains sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions - in contrast to Erd\H{o}s-R\'enyi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices. We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimised code for this new method is publicly available.

### Deterministic Identification over Channels without CSI

Identification capacities of randomized and deterministic identification were proved to exceed channel capacity for Gaussian channels \emph{with} channel side information (CSI). In this work, we extend deterministic identification to the block fading channels without CSI by applying identification codes for both channel estimation and user identification. We prove that identification capacity is asymptotically higher than transmission capacity even in the absence of CSI. And we also analyze the finite-length performance theoretically and numerically. The simulation results verify the feasibility of the proposed blind deterministic identification in finite blocklength regime.

### Valuatuions and orderings on the real Weyl algebra


### Characterization of Lipschitz Functions via the Commutators of Maximal Function on Stratified Lie Groups

In this paper, the main aim is to consider the boundedness of the Hardy-Littlewood maximal commutator $M_{b}$ and the nonlinear commutator $[b, M]$ on the Lebesgue spaces and Morrey spaces over some stratified Lie group $\mathbb{G}$ when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces on Lie group are given.

### Reduced modelling and optimal control of epidemiological individual-based models with contact heterogeneity

Modelling epidemics via classical population-based models suffers from shortcomings that so-called individual-based models are able to overcome, as they are able to take heterogeneity features into account, such as super-spreaders, and describe the dynamics involved in small clusters. In return, such models often involve large graphs which are expensive to simulate and difficult to optimize, both in theory and in practice. By combining the reinforcement learning philosophy with reduced models, we propose a numerical approach to determine optimal health policies for a stochastic epidemiological graph-model taking into account super-spreaders. More precisely, we introduce a deterministic reduced population-based model involving a neural network, and use it to derive optimal health policies through an optimal control approach. It is meant to faithfully mimic the local dynamics of the original, more complex, graph-model. Roughly speaking, this is achieved by sequentially training the network until an optimal control strategy for the corresponding reduced model manages to equally well contain the epidemic when simulated on the graph-model. After describing the practical implementation of this approach, we will discuss the range of applicability of the reduced model and to what extent the estimated control strategies could provide useful qualitative information to health authorities.

### Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy

We consider a family of vectorial models for cohesive fracture, which may incorporate $\mathrm{SO}(n)$-invariance. The deformation belongs to the space of generalized functions of bounded variation and the energy contains an (elastic) volume energy, an opening-dependent jump energy concentrated on the fractured surface, and a Cantor part representing diffuse damage. We show that this type of functional can be naturally obtained as $\Gamma$-limit of an appropriate phase-field model. The energy densities entering the limiting functional can be expressed, in a partially implicit way, in terms of those appearing in the phase-field approximation.

### Extension Operators for Trimmed Spline Spaces

We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree $p$ with $k$ continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.

### On sets with sum and difference structure

For nonempty sets $A,B$ of nonnegative integers and an integer $n$, let $r_{A,B}(n)$ be the number of representations of $n$ as $a+b$ and $d_{A,B}(n)$ be the number of representations of $n$ as $a-b$, where $a\in A, b\in B$. In this paper, we determine the sets $A,B$ such that $r_{A,B}(n)=1$ for every nonnegative integer $n$. We also consider the \emph{difference} structure and prove that: there exist sets $A$ and $B$ of nonnegative integers such that $r_{A,B}(n)\ge 1$ for all large $n$, $A(x)B(x)=(1+o(1))x$ and for any given nonnegative integer $c$, we have $d_{A,B}(n)=c$ for infinitely many positive integers $n$. Other related results are also contained.

### From octonions to composition superalgebras via tensor categories

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal Magic Square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

### Random cluster model on regular graphs

For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum_{A\subseteq E(G)}q^{k(A)}w^{|A|},$$ where $k(A)$ denotes the number of connected components of the graph $(V,A)$. In this paper we show that if $(G_n)_n$ is a sequence of $d$-regular graphs such that the girth $g(G_n)\to \infty$, the length of the shortest cycle, then the limit $$\lim_{n\to \infty} \frac{1}{v(G_n)}\ln Z_{G_n}(q,w)=\ln \Phi_{d,q,w}$$ exists if $q\geq 2$ and $w\geq 0$, and is equal to $$\Phi_{d,q,w}:=\max_{t\in [-\pi,\pi]}\Phi_{d,q,w}(t),$$ where $$\Phi_{d,q,w}(t):=\left(\sqrt{1+\frac{w}{q}}\cos(t)+\sqrt{\frac{(q-1)w}{q}}\sin(t)\right)^{d}+(q-1)\left(\sqrt{1+\frac{w}{q}}\cos(t)-\sqrt{\frac{w}{q(q-1)}}\sin(t)\right)^{d}.$$ The same conclusion holds true for a sequence of random $d$-regular graphs with probability $1$. We extend the work of Dembo, Montanari, Sly and Sun for the Potts model (integer $q$) and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed $q$.

### Comparison bounds for perturbed Schrödinger operators with single-well potentials

We prove bounds on the sum of the differences between the eigenvalues of a Schr\"odinger operator and its perturbation. Our results hold for operators in one dimension with single-well potentials. We rely on a variation of the well-known factorisation method. In the P\"oschl-Teller and Coulomb cases we are able to use the explicit factorisations to establish improved bounds.

### A dynamic approach to heterogeneous elastic wires

We consider closed planar curves with fixed length whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness and global existence of solutions for initial data in a weak regularity class and with arbitrary winding number.

### Energy bounds of sign-changing solutions to Yamabe equations on manifolds with boundary

We study the Yamabe equation in the Euclidean half-space. We prove that any sign-changing solution has at least twice the energy of a standard bubble. Moreover, a sharper energy lower bound of the sign-changing solution set is also established via the method of moving planes. This bound increases the energy range for which Palais-Smale sequences of related variational problem has a non-trivial weak limit.

### Discrete density comonads and graph parameters

Game comonads have brought forth a new approach to studying finite model theory categorically. By representing model comparison games semantically as comonads, they allow important logical and combinatorial properties to be exressed in terms of their Eilenberg-Moore coalgebras. As a result, a number of results from finite model theory, such as preservation theorems and homomorphism counting theorems, have been formalised and parameterised by comonads, giving rise to new results simply by varying the comonad. In this paper we study the limits of the comonadic approach in the combinatorial and homomorphism-counting aspect of the theory, regardless of whether any model comparison games are involved. We show that any standard graph parameter has a corresponding comonad, classifying the same class. This comonad is constructed via a simple Kan extension formula, making it the initial solution to this problem and, furthermore, automatically admitting a homomorphism-counting theorem.

### Urysohn and Hammerstein operators on H"older spaces

We present an application-oriented approach to Urysohn and Hammerstein integral operators acting between spaces of H"older continuous functions over compact metric spaces. These nonlinear mappings are formulated by means of an abstract measure theoretical integral involving a finite measure. This flexible setting creates a common framework to tackle both such operators based on the Lebesgue integral like frequently met in applications, as well as e.g.\ their spatial discretization using stable quadrature/cubature rules (Nystr"om methods). Under suitable Carath{\'e}odory conditions on the kernel functions, properties like well-definedness, boundedness, (complete) continuity and continuous differentiability are established. Furthermore, the special case of Hammerstein operators is understood as composition of Fredholm and Nemytskii operators. While our differentiability results for Urysohn operators appear to be new, the section on Nemytskii operators has a survey character. Finally, an appendix provides a rather comprehensive account summarizing the required preliminaries for H\"older continuous functions defined on metric spaces.

### On varieties with Ulrich twisted normal bundles

We characterize smooth irreducible varieties with Ulrich twisted normal bundle.

### $\times a$ and $\times b$ empirical measures, the irregular set and entropy

We consider the $\times a$ and $\times b$ maps: $T_a$ and $T_b$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ for integers $a$ and $b\geq 2$. It is known that, if $a$ and $b$ are multiplicatively independent, then the only $T_a,T_b$-invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, whether there exists a nontrivial $T_a,T_b$-invariant and ergodic measure is not known. In this paper, we study the empirical measures of $x\in\mathbb{T}$ with respect to the $T_a,T_b$-action and show that the set of $x$ such that the empirical measures of $x$ do not converge to any measure has Hausdorff dimension $1$ and the set of $x$ such that the empirical measures can approach a nontrivial $T_a,T_b$-invariant measure has Hausdorff dimension $0$. Furthermore, we obtain some equidistribution result about the $T_a,T_b$-orbit of $x$ in the complement of a set of Hausdorff dimension $0$.

### Smooth $l$-Fano weighted complete intersections

In this paper we prove that for $n$-dimensional smooth $l$-Fano well formed weighted complete intersections, which is not isomorphic to a usual projective space, the upper bound for $l$ is equal to $\lceil \log_2(n+2) \rceil-1 .$ We also prove that the only $l$-Fano of dimension $n$ among such manifolds with inequalities $\lceil \log_3(n+2) \rceil \leqslant l \leqslant \lceil \log_2(n+2) \rceil -1$ is a complete intersection of quadrics in a usual projective space.

### Fine Selmer groups of modular forms

We compare the Iwasawa invariants of fine Selmer groups of $p$-adic Galois representations over admissible $p$-adic Lie extensions of a number field $K$ to the Iwasawa invariants of ideal class groups along these Lie extensions. More precisely, let $K$ be a number field, let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$, and choose a $G_K$-invariant lattice ${T \subseteq V}$. We study the fine Selmer groups of ${A = V/T}$ over suitable $p$-adic Lie extensions $K_\infty/K$, comparing their corank and $\mu$-invariant to the corank and the $\mu$-invariant of the Iwasawa module of ideal class groups in $K_\infty/K$. In the second part of the article, we compare the Iwasawa $\mu$- and $l_0$-invariants of the fine Selmer groups of CM modular forms on the one hand and the Iwasawa invariants of ideal class groups on the other hand over trivialising multiple $\mathbb{Z}_p$-extensions of $K$.

### Existence of real algebraic hypersurfaces with many prescribed components

Given a real algebraic variety $X$ of dimension $n$, a very ample divisor $D$ on $X$ and a smooth closed hypersurface $\Sigma$ of $\mathbf{R}^n$, we construct real algebraic hypersurfaces in the linear system $|mD|$ whose real locus contains many connected components diffeomorphic to $\Sigma$. As a consequence, we show the existence of real algebraic hypersurfaces in the linear system $|mD|$ whose Betti numbers grow by the maximal order, as $m$ goes to infinity. As another application, we recover a result by D. Gayet on the existence of many disjoint lagrangians with prescribed topology in any smooth complex hypersurface of $\mathbf{C}\mathbf{P}^n$. The results in the paper are proved more generally for complete intersections. The proof of our main result uses probabilistic tools.

### The probability of random trees being isomorphic

We show that the probability that two randomly chosen trees are isomorphic decays exponentially for rooted labelled trees as well as Galton--Watson trees with bounded degrees. In the former case a full asymptotic expansion is derived. We also show that, in general, we cannot obtain exponential decay for Galton--Watson trees. Lastly, we prove joint convergence to a multivariate normal distribution for vertices of given degrees in pairs of labelled trees conditioned on being isomorphic.

### The disguised toric locus and affine equivalence of reaction networks

Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a canonical bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially.

### Controlled continuous $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules

The frame theory is dynamic and exciting with various pure and applied mathematics applications. In this paper, we introduce and study the concept of Controlled Continuous $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules, which is a generalization of discrete controlled $\ast$-$g$-Frames in Hilbert $C^{\ast}$-Modules. Also, we give some properties.

### A Polar Subcode Approach to Belief Propagation List Decoding

Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed algorithm outperforms the previously best performing iterative CRC-aided belief propagation list (CA-BPL) decoder both in error-rate performance and decoding latency.

### Scattered linear sets in a finite projective line and translation planes

Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in $\mathrm{PG}(1,q^t)$. In this paper a similar construction of a translation plane $\mathcal A_f$ obtained from any scattered linearized polynomial $f(x)$ in $\mathbb F_{q^t}[x]$ is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by $U_f$ the $\mathbb F_q$-subspace of $\mathbb F_{q^t}^2$ associated with $f(x)$. If $f(x)$ and $f'(x)$ are scattered, then $\mathcal A_f$ and $\mathcal A_{f'}$ are isomorphic if and only if $U_f$ and $U_{f'}$ belong to the same orbit under the action of $\Gamma\mathrm L(2,q^t)$. This gives rise to as many distinct translation planes as there are inequivalent scattered linearized polynomials. In particular, for any scattered linear set $L$ of maximum rank in $\mathrm{PG}(1,q^t)$ there are $c_\Gamma(L)$ pairwise non-isomorphic translation planes, where $c_\Gamma(L)$ denotes the $\Gamma\mathrm L$-class of $L$, as defined by Csajb\'ok, Marino and Polverino. A result by Jha and Johnson allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined by Lunardon and Polverino.

### Robust Fundamental Lemma for Data-driven Control

The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and control. Under the hood, the fundamental lemma works by applying a persistently exciting input to the system. This ensures that the Hankel matrix of resulting input/output data has the "right" rank, meaning that its columns span the entire subspace of trajectories. However, such binary rank conditions are known to be fragile in the sense that a small additive noise could already cause the Hankel matrix to have full rank. Therefore, in this extended abstract we present a robust version of the fundamental lemma. The idea behind the approach is to guarantee certain lower bounds on the singular values of the data Hankel matrix, rather than mere rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of a deeper input Hankel matrix is sufficiently large. This inspires a new quantitative and robust notion of persistency of excitation. The relevance of the result for data-driven control will also be highlighted through comparing the predictive control performance for varying degrees of persistently exciting data.

### A scalable space-time domain decomposition approach for solving large-scale nonlinear regularized inverse ill-posed problems in 4D variational data assimilation

We develop innovative algorithms for solving the strong-constraint formulation of four-dimensional variational data assimilation in large-scale applications. We present a space-time decomposition approach that employs domain decomposition along both the spatial and temporal directions in the overlapping case and involves partitioning of both the solution and the operators. Starting from the global functional defined on the entire domain, we obtain a type of regularized local functionals on the set of subdomains providing the order reduction of both the predictive and the data assimilation models. We analyze the algorithm convergence and its performance in terms of reduction of time complexity and algorithmic scalability. The numerical experiments are carried out on the shallow water equation on the sphere according to the setup available at the Ocean Synthesis/Reanalysis Directory provided by Hamburg University.

### Turning grain maps into diagrams

The present paper studies mathematical models for representing, imaging, and analyzing polycrystalline materials. We introduce various techniques for converting grain maps into diagram or tessellation representations that rely on constrained clustering. In particular, we show how to significantly accelerate the generalized balanced power diagram method from [1] and how to extend it to allow for optimization over all relevant parameters. A comparison of the accuracies of the proposed approaches is given based on a 3D real-world data set of $339\times 339 \times 599$ voxels.

### Univalent typoids

A typoid is a type equipped with an equivalence relation, such that the terms of equivalence between the terms of the type satisfy certain conditions, with respect to a given equivalence relation between them, that generalise the properties of the equality terms. The resulting weak 2-groupoid structure can be extended to every finite level. The introduced notions of typoid and typoid function generalise the notions of setoid and setoid function. A univalent typoid is a typoid satisfying a general version of the univalence axiom. We prove some fundamental facts on univalent typoids, their product and exponential. As a corollary, we get an interpretation of propositional truncation within the theory of typoids. The couple typoid and univalent typoid is a weak groupoid-analogue to the couple precategory and category in homotopy type theory.

### Pullback and forward attractors of contractive difference equations

The construction of attractors of a dissipative difference equation is usually based on compactness assumptions. In this paper, we replace them with contractivity assumptions under which the pullback and forward attractors are identical. As a consequence, attractors degenerate to unique bounded entire solutions. As an application, we investigate attractors of integrodifference equations which are popular models in theoretical ecology.

### Discrete m-functions with Doubly Palindromic Continued Fraction Coefficients

We demonstrate that discrete m-functions with eventually periodic continued fraction coefficients have an algebraic relationship to their second solution if and only if the periodic part of the sequence of continued fraction coefficients is doubly palindromic. In this setting, doubly palindromic means that each sequence is a repeated concatenation of two palindromes and a compatibility condition between the lengths of these palindromes is satisfied.

### Complete monotonicity of time-changed Lévy processes at first passage

We consider the class of (possibly killed) spectrally positive L\'evy process that have been time-changed by the inverse of an integral functional. Within this class we characterize the family of those processes which observe the following property: as functions of point of issue, the Laplace transforms of their first-passage times downwards are completely monotone. A wide (dense, in a sense) subfamily of this family admits closed form expressions for said Laplace transforms.

### Dynamical boundary conditions for time-dependent fractional operators on extension domains

We consider a parabolic semilinear non-autonomous problem $(\tilde P)$ for a fractional time-dependent operator $\mathcal{B}^{s,t}_\Omega$ with Venttsel'-type boundary conditions in an extension domain $\Omega\subset\mathbb{R}^N$ having as boundary a $d$-set. We prove existence and uniqueness of the mild solution of the associated semilinear abstract Cauchy problem via an evolution family $U(t,\tau)$. We then prove that the mild solution of the abstract problem actually solves problem $(\tilde P)$ via a generalized fractional Green formula.

### A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks

In this paper, a new weighted first-order formulation is proposed for solving the anisotropic diffusion equations with deep neural networks. For many numerical schemes, the accurate approximation of anisotropic heat flux is crucial for the overall accuracy. In this work, the heat flux is firstly decomposed into two components along the two eigenvectors of the diffusion tensor, thus the anisotropic heat flux approximation is converted into the approximation of two isotropic components. Moreover, to handle the possible jump of the diffusion tensor across the interface, the weighted first-order formulation is obtained by multiplying this first-order formulation by a weighted function. By the decaying property of the weighted function, the weighted first-order formulation is always well-defined in the pointwise way. Finally, the weighted first-order formulation is solved with deep neural network approximation. Compared to the neural network approximation with the original second-order elliptic formulation, the proposed method can significantly improve the accuracy, especially for the discontinuous anisotropic diffusion problems.

### Long term analysis of splitting methods for charged-particle dynamics

In this paper, we rigorously analyze the energy, momentum and magnetic moment behaviours of two splitting methods for solving charged-particle dynamics. The near-conservations of these invariants are given for the system under constant magnetic field or quadratic electric potential. By the approach named as backward error analysis, we derive the modified equations and modified invariants of the splitting methods and based on which, the near-conservations over long times are proved. Some numerical experiments are presented to demonstrate these long time behaviours.

### Injectivity for algebras and categories with quantum symmetry

We study completely positive maps and injectivity for Yetter-Drinfeld algebras over compact quantum groups, and module categories over rigid C*-tensor categories. This gives a generalization of Hamana's theory of injective envelope to the framework of dynamical systems over quantum groups. As a byproduct we also establish a duality between the Yetter-Drinfeld algebras and certain bimodule categories with central generators.

### Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck and of Faisant. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p,$ extending and simplifying earlier work of Zhi-Hong Sun (2003).

### Pebble trees

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.

### Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees

We define the co-spectral radius of inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving equivalence relations, as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. The almost sure existence of the sampling exponent is already new for i.i.d. percolation clusters on countable groups. For the proof, we develop a general method called 2-3-method that is based on the mass-transport principle. As a byproduct, we show that the growth of a unimodular random rooted tree of bounded degree always exists, assuming its upper growth passes a critical threshold. This complements Timar's work who showed the possible nonexistence of growth below this threshold. We also show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for Property (T) and hyperfinite relations.

### On the use of a local R-hat to improve MCMC convergence diagnostic

Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions.

### Metric lines in Jet Space

The space $J^k$ of $k$-jets of a real function of one real variable $x$ admits the structure of a Carnot group. $J^k$ has a natural \sR submersion onto the Euclidean plane and curves in the Euclidean plane can be horizontal lifted to $J^k$. The horizontal lifts of Euclidean lines are thus metric lines in $J^k$. Are other metric lines in $J^k$ besides the horizontal lifts lines? This work is the first of a sequence of papers, where we attempt to make a complete classification of the metric lines in $J^k$.

### Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems

We consider physics-informed neural networks [Raissi et al., J. Comput. Phys. 278 (2019) 686-707] for forward physical problems. In order to find optimal PINNs configuration, we introduce a hyper-parameter tuning procedure via Gaussian processes-based Bayesian optimization. We apply the procedure to Helmholtz problems for bounded domains and conduct a thorough study, focusing on: (i) performance, (ii) the collocation points density $r$ and (iii) the frequency $\kappa$, confirming the applicability and necessity of the method. Numerical experiments are performed in two and three dimensions, including comparison to finite element methods.

### The Capacity of Causal Adversarial Channels

We characterize the capacity for the discrete-time arbitrarily varying channel with discrete inputs, outputs, and states when (a) the encoder and decoder do not share common randomness, (b) the input and state are subject to cost constraints, (c) the transition matrix of the channel is deterministic given the state, and (d) at each time step the adversary can only observe the current and past channel inputs when choosing the state at that time. The achievable strategy involves stochastic encoding together with list decoding and a disambiguation step. The converse uses a two-phase "babble-and-push" strategy where the adversary chooses the state randomly in the first phase, list decodes the output, and then chooses state inputs to symmetrize the channel in the second phase. These results generalize prior work on specific channels models (additive, erasure) to general discrete alphabets and models.

### Linesearch Newton-CG methods for convex optimization with noise

This paper studies the numerical solution of strictly convex unconstrained optimization problems by linesearch Newton-CG methods. We focus on methods employing inexact evaluations of the objective function and inexact and possibly random gradient and Hessian estimates. The derivative estimates are not required to satisfy suitable accuracy requirements at each iteration but with sufficiently high probability. Concerning the evaluation of the objective function we first assume that the noise in the objective function evaluations is bounded in absolute value. Then, we analyze the case where the error satisfies prescribed dynamic accuracy requirements. We provide for both cases a complexity analysis and derive expected iteration complexity bounds. We finally focus on the specific case of finite-sum minimization which is typical of machine learning applications.

### The height of binomial ideals and toric $K$-algebras with isolated singularity

We give an upper bound for the height of an arbitrary binomial ideal $I$ in terms of the dimension of a vector space spanned by integer vectors corresponding to a set of binomial generators of $I$. When $I$ is an unmixed binomial ideal, this dimension is precisely the height of $I$. Applying this result to the ideal of inner $2$-minors $I_\Pc$ of a finite set of cells $\Pc$, one gets a nice interpretation for $\height I_\Pc$ for an unmixed ideal $I_\Pc$, in terms of the number of cells of $\Pc$. Moreover, we study some families of toric $K$-algebras such as Hibi rings and toric rings defined by inner $2$-minors to determine when they have isolated singularity.

### The existence of $m$-tree-connected $(g,f+f'-m)$-factors using $(g,f)$-factors and $m$-tree-connected $(m,f')$-factors

Let $G$ be a graph and let $g$, $f$, and $f'$ be three positive integer-valued functions on $V(G)$ with $g\le f$. Tokuda, Xu, and Wang (2003) showed that if $G$ contains a $(g,f)$-factor and a spanning $f'$-tree, then $G$ also contains a connected $(g,f+f'-1)$-factor. In this note, we develop their result to a tree-connected version by proving that if $G$ contains a $(g,f)$-factor and an $m$-tree-connected $(m,f')$-factor, then $G$ also contains an $m$-tree-connected $(g,f+f'-m)$-factor, provided that $f\ge m$. In addition, we show that $g$ allows to be nonnegative.

### Equivalent Boundary Conditions for an Elasto-Acoustic Problem set in a Domain with a Thin Layer

We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

### Proofs For Progressively Generalized Fibonacci Identities Using Maximal Independent Sets of Tree Graphs

This paper generalizes a graph theoretic proof technique for a Fibonacci identity proposed by Lee Knisley Sanders, and explores characteristics of these generalized theorems ad infinitum.

### Multi-Marginal Gromov-Wasserstein Transport and Barycenters

Gromov-Wasserstein (GW) distances are generalizations of Gromov-Haussdorff and Wasserstein distances. Due to their invariance under certain distance-preserving transformations they are well suited for many practical applications. In this paper, we introduce a concept of multi-marginal GW transport as well as its regularized and unbalanced versions. Then we generalize a bi-convex relaxation of the GW transport to our multi-marginal setting which is tight if the cost function is conditionally negative definite in a certain sense. The minimization of this relaxed model can be done by an alternating algorithm, where each step can be performed by a Sinkhorn scheme for a multi-marginal transport problem. We show a relation of our multi-marginal GW problem for a tree-structured cost function to an (unbalanced) GW barycenter problem and present different proof-of-concept numerical results.

### Hybridized Discontinuous Galerkin Methods for a Multiple Network Poroelasticity Model with Medical Applications

The quasi-static multiple network poroelastic theory (MPET) model, first introduced in the context of geomechanics, has recently found new applications in medicine. In practice, the parameters in the MPET equations can vary over several orders of magnitude which makes their stable discretization and fast solution a challenging task. Here, a new efficient parameter-robust hybridized discontinuous Galerkin method, which also features fluid mass conservation, is proposed for the MPET model. Its stability analysis which is crucial for the well-posedness of the discrete problem is performed and cost-efficient fast parameter-robust preconditioners are derived. We present a series of numerical computations for a 4-network MPET model of a human brain which support the performance of the new algorithms.

### Dynamical maps and symmetroids

Starting from the canonical symmetroid $\mathcal{S}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid $\mathcal{S}(G)$, the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.

### A model of invariant control system using mean curvature drift from Brownian motion under submersions

Given a submersion $\phi: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=\phi(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $\pi: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincar\'e's upper half plane and the Bures-Wasserstein geometry on $\pdef$, on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson's Brownian motion. By choosing the background metric via natural $GL(n)$ action, we arrive at an invariant control system on the $GL(n)$-homogenous space $GL(n)/O(n)$. We investigate feasibility of developing stochastic algorithms using the mean curvature flow. KEY WORDS: mean curvature flow, gradient flow, Brownian motion, Riemannian submersion, random matrix, eigenvalue processes, geometry of positive definite matrices, stochastic algorithm, control theory on homogeneous space

### Corner asymptotics of the magnetic potential in the eddy-current model

In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator.

### A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,\xi)$ is quadratic and convex in $\xi$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,\xi)$, depending smoothly on $\xi$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$.

### On the size and local equations of fibres of general projections

For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1}$ and the fibres are locally defined by linear and quadratic equations.

### Twisted Gan-Gross-Prasad conjecture for certain tempered L-packets

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

### Codes for Preventing Zeros at Partially Defect Memory Positions

This work deals with error correction for non-volatile memories that are partially defect-at some levels. Such memory cells can only store incomplete information since some of their levels cannot be utilized entirely due to, e.g. wearout. On top of that, this paper corrects random errors $t\geq 1$ that could happen among $u$ partially defective cells while preserving their constraints. First, we show that the probability of violating the partially defective cells' restriction due to random errors is not trivial. Next, we update the models in [1] such that the coefficients of the output encoded vector plus the error vector at the partially defect positions are nonzero. Lastly, we state a simple theorem for masking the partial defects using a code with a minimum distance $d$ such that $d\geq (u+t)+1$. "Masking" means selecting a word whose entries correspond to writable levels in the (partially) defect positions. A comparison shows that, for a certain BCH code, masking $u$ cells by this theorem is as good as using the complicated coding scheme proven in [1, Theorem 1].

### Symplectic geometry and Toeplitz operators on Cartan domains of type IV

Let us consider, for $n \geq 3$, the Cartan domain $\mathrm{D}_n^{\mathrm{IV}}$ of type IV. On the weighted Bergman spaces $\mathcal{A}^2_\lambda(\mathrm{D}_n^{\mathrm{IV}})$ we study the problem of the existence of commutative $C^*$-algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $\mathrm{SO}(n) \times \mathrm{SO}(2)$ of biholomorphisms of $\mathrm{D}_n^{\mathrm{IV}}$ that fix the origin. The $\mathrm{SO}(n) \times \mathrm{SO}(2)$-invariant symbols yield Toeplitz operators that generate commutative $C^*$-algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $\mathrm{SO}(2)$. We compute the moment map $\mu^{\mathrm{SO}(2)}$-action for the $\mathrm{SO}(2)$-action on $\mathrm{D}_n^{\mathrm{IV}}$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $a = f \circ \mu^{\mathrm{SO}(2)}$, denoted by $L^\infty(\mathrm{D}_n^{\mathrm{IV}})^{\mu^{\mathrm{SO}(2)}}$, yield Toeplitz operators that generate commutative $C^*$-algebras. Spectral integral formulas for these Toeplitz operators are also obtained.

### Encodings of trajectories and invariant measures

We consider a discrete dynamical system on a compact manifold M generated by a homeomorphism f. Let C = {M(i)} be a finite covering of M by closed cells. The symbolic image of a dynamical system is a directed graph G with vertices corresponding to cells in which vertices i and j are joined by an arc i to j if the image f(M(i)) intersects M(j). We show that the set of paths of the symbolic image converges to the set of trajectories of the system in the Tychonoff topology as the diameter of the covering tends to zero. For a cycle on G going through different vertices, a simple flow is by definition a uniform distribution on arcs of this cycle. We show that simple flows converge to ergodic measures in the weak topology as the diameter of the covering tends to zero.

### Partitioning through projections: strong SDP bounds for large graph partition problems

The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by polyhedral cuts for two variations of the GPP: the $k$-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting planes. We are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds.

### Schur-Sato theory for quasi-elliptic rings

The notion of quasi-elliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators found in the theory of integrable systems, such as rings of commuting differential, difference, differential-difference, etc. operators. They are contained in a certain non-commutative "universal" ring - a purely algebraic analogue of the ring of pseudodifferential operators on a manifold, and admit (under certain mild restrictions) a convenient algebraic-geometric description. An important algebraic part of this description is the Schur-Sato theory - a generalisation of the well known theory for ordinary differential operators. Some parts of this theory were developed earlier in a series of papers, mostly for dimension two. In this paper we present this theory in arbitrary dimension. We apply this theory to prove two classification theorems of quasi-elliptic rings in terms of certain pairs of subspaces (Schur pairs). They are necessary for the algebraic-geometric description of quasi-elliptic rings mentioned above. The theory is effective and has several other applications, among them is a new proof of the Abhyankar inversion formula.

### Bandwidth Cost of Code Conversions in the Split Regime

Distributed storage systems must store large amounts of data over long periods of time. To avoid data loss due to device failures, an $[n,k]$ erasure code is used to encode $k$ data symbols into a codeword of $n$ symbols that are stored across different devices. However, device failure rates change throughout the life of the data, and tuning $n$ and $k$ according to these changes has been shown to save significant storage space. Code conversion is the process of converting multiple codewords of an initial $[n^I,k^I]$ code into codewords of a final $[n^F,k^F]$ code that decode to the same set of data symbols. In this paper, we study conversion bandwidth, defined as the total amount of data transferred between nodes during conversion. In particular, we consider the case where the initial and final codes are MDS and a single initial codeword is split into several final codewords ($k^I=\lambda^F k^F$ for integer $\lambda^F \geq 2$), called the split regime. We derive lower bounds on the conversion bandwidth in the split regime and propose constructions that significantly reduce conversion bandwidth and are optimal for certain parameters.

### On degenerate blow-up profiles for the subcritical semilinear heat equation

We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.

### A note on the involutive concordance invariants for certain (1,1)-knots

We compute the involutive concordance invariants for the 10- and 11-crossing (1,1)-knots.

### Global Convergence of Hessenberg Shifted QR III: Approximate Ritz Values via Shifted Inverse Iteration

We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in $O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point operations using $O(\log^2(n/\delta))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-Vargas, Srivastava, 2022).

### Global Convergence of Hessenberg Shifted QR II: Numerical Stability

We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap.

### Expectation of the Maximum of a Pair of Random Variables with Zero-Truncated Bivariate Normal Distribution

Calculating the expectation of the maximum of normally distributed variables arises in many applications. We derive a closed-form solution for the expectation of the maximum of a zero-truncated bivariate normal distribution, and conduct a simulation study to numerically confirm the solution by comparing it with a Monte-Carlo method.

### $α$-GAN: Convergence and Estimation Guarantees

We prove a two-way correspondence between the min-max optimization of general CPE loss function GANs and the minimization of associated $f$-divergences. We then focus on $\alpha$-GAN, defined via the $\alpha$-loss, which interpolates several GANs (Hellinger, vanilla, Total Variation) and corresponds to the minimization of the Arimoto divergence. We show that the Arimoto divergences induced by $\alpha$-GAN equivalently converge, for all $\alpha\in \mathbb{R}_{>0}\cup\{\infty\}$. However, under restricted learning models and finite samples, we provide estimation bounds which indicate diverse GAN behavior as a function of $\alpha$. Finally, we present empirical results on a toy dataset that highlight the practical utility of tuning the $\alpha$ hyperparameter.

### Learning Based User Scheduling in Reconfigurable Intelligent Surface Assisted Multiuser Downlink

Reconfigurable intelligent surface (RIS) is capable of intelligently manipulating the phases of the incident electromagnetic wave to improve the wireless propagation environment between the base-station (BS) and the users. This paper addresses the joint user scheduling, RIS configuration, and BS beamforming problem in an RIS-assisted downlink network with limited pilot overhead. We show that graph neural networks (GNN) with permutation invariant and equivariant properties can be used to appropriately schedule users and to design RIS configurations to achieve high overall throughput while accounting for fairness among the users. As compared to the conventional methodology of first estimating the channels then optimizing the user schedule, RIS configuration and the beamformers, this paper shows that an optimized user schedule can be obtained directly from a very short set of pilots using a GNN, then the RIS configuration can be optimized using a second GNN, and finally the BS beamformers can be designed based on the overall effective channel. Numerical results show that the proposed approach can utilize the received pilots more efficiently than the conventional channel estimation based approach, and can generalize to systems with an arbitrary number of users.

### Counting unique molecular identifiers in sequencing using a decomposable multitype branching process with immigration

Detection of extremely rare variant alleles, such as tumour DNA, within a complex mixture of DNA molecules is difficult. Barcoding of DNA template molecules early in the next-generation sequencing library construction provides a way to identify and bioinformatically remove polymerase errors. During the PCR-based barcoding procedure consisting of $t$ consecutive PCR-cycles, DNA molecules become barcoded by random nucleotide sequences. Previously, values 2 and 3 of $t$ have been used, however even larger values of $t$ might be relevant. This paper proposes using a multi-type branching process with immigration as a model describing the random outcome of imperfect PCR-barcoding procedure, with variable $t$ treated as the time parameter. For this model we focus on the expected numbers of clusters of molecules sharing the same unique molecular identifier.

### Skew-sparse matrix multiplication

Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main feature of our method is the acceleration for matrix multiplication if the product is skew-sparse. Based on the new method, we design a deterministic algorithm with complexity $O(T^{\omega-2} p^2)$, where $T\le p-1$ is a parameter determined by the skew-sparsity of input matrices and $\omega$ is the asymptotic exponent of matrix multiplication. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity $O^\thicksim(t^{\omega-2}p^2+p^2\log\frac{1}{\nu})$, where $t\le p-1$ is the skew-sparsity of the product and $\nu$ is the probability parameter.

### Continuous-time mean-variance portfolio selection under non-Markovian regime-switching model with random horizon

In this paper, we consider a continuous-time mean-variance portfolio selection with regime-switching and random horizon. Unlike previous works, the dynamic of assets are described by non-Markovian regime-switching models in the sense that all the market parameters are predictable with respect to the filtration generated jointly by Markov chain and Brownian motion. We formulate this problem as a constrained stochastic linear-quadratic optimal control problem. The Markov chain is assumed to be independent of the Brownian motion. So the market is incomplete. We derive closed-form expressions for both the optimal portfolios and the efficient frontier. All the results are different from those in the problem with fixed time horizon.

### Exponential Integral Solutions for Fixation Time in Wright-Fisher Model With Selection

In this work we derive new analytic expressions for fixation time in Wright-Fisher model with selection. The three standard cases for fixation are considered: fixation to zero, to one or both. Second order differential equations for fixation time are obtained by a simplified approach using only the law of total probability and Taylor expansions. The obtained solutions are given by a combination of exponential integral functions with elementary functions. We then state approximate formulas involving only elementary functions valid for small selection effects. The quality of our results are explored throughout an extensive simulation study. We show that our results approximate the discrete problem very accurately even for small population size (a few hundreds) and large selection coefficients.

### Convergence of Deep Neural Networks with General Activation Functions and Pooling

Deep neural networks, as a powerful system to represent high dimensional complex functions, play a key role in deep learning. Convergence of deep neural networks is a fundamental issue in building the mathematical foundation for deep learning. We investigated the convergence of deep ReLU networks and deep convolutional neural networks in two recent researches (arXiv:2107.12530, 2109.13542). Only the Rectified Linear Unit (ReLU) activation was studied therein, and the important pooling strategy was not considered. In this current work, we study the convergence of deep neural networks as the depth tends to infinity for two other important activation functions: the leaky ReLU and the sigmoid function. Pooling will also be studied. As a result, we prove that the sufficient condition established in arXiv:2107.12530, 2109.13542 is still sufficient for the leaky ReLU networks. For contractive activation functions such as the sigmoid function, we establish a weaker sufficient condition for uniform convergence of deep neural networks.

### Convergence Analysis of Deep Residual Networks

Various powerful deep neural network architectures have made great contribution to the exciting successes of deep learning in the past two decades. Among them, deep Residual Networks (ResNets) are of particular importance because they demonstrated great usefulness in computer vision by winning the first place in many deep learning competitions. Also, ResNets were the first class of neural networks in the development history of deep learning that are really deep. It is of mathematical interest and practical meaning to understand the convergence of deep ResNets. We aim at characterizing the convergence of deep ResNets as the depth tends to infinity in terms of the parameters of the networks. Toward this purpose, we first give a matrix-vector description of general deep neural networks with shortcut connections and formulate an explicit expression for the networks by using the notions of activation domains and activation matrices. The convergence is then reduced to the convergence of two series involving infinite products of non-square matrices. By studying the two series, we establish a sufficient condition for pointwise convergence of ResNets. Our result is able to give justification for the design of ResNets. We also conduct experiments on benchmark machine learning data to verify our results.

### Blind Image Inpainting with Sparse Directional Filter Dictionaries for Lightweight CNNs

Blind inpainting algorithms based on deep learning architectures have shown a remarkable performance in recent years, typically outperforming model-based methods both in terms of image quality and run time. However, neural network strategies typically lack a theoretical explanation, which contrasts with the well-understood theory underlying model-based methods. In this work, we leverage the advantages of both approaches by integrating theoretically founded concepts from transform domain methods and sparse approximations into a CNN-based approach for blind image inpainting. To this end, we present a novel strategy to learn convolutional kernels that applies a specifically designed filter dictionary whose elements are linearly combined with trainable weights. Numerical experiments demonstrate the competitiveness of this approach. Our results show not only an improved inpainting quality compared to conventional CNNs but also significantly faster network convergence within a lightweight network design.

### Rectangular mesh contour generation algorithm for finite differences calculus

In this work, a 2D contour generation algorithm is proposed for irregular regions. The contour of the physical domain is approximated by mesh segments using the known coordinates of the contour. For this purpose, the algorithm uses a repeating structure that analyzes the known irregular contour coordinates to approximate the physical domain contour by mesh segments. To this end, the algorithm calculates the slope of the line defined by the known point of the irregular contours and the neighboring vertices. In this way, the algorithm calculates the points of the line and its distance to the closest known nodes of the mesh, allowing to obtain the points of the approximate contour. This process is repeated until the approximate contour is obtained. Therefore, this approximate contour generation algorithm, from known nodes of a mesh, is suitable for describing meshes involving geometries with irregular contours and for calculating finite differences in numerical simulations. The contour is evaluated through three geometries, the difference between the areas delimited by the given contour and the approximate contour, the number of nodes and the number of internal points. It can be seen that the increase in geometry complexity implies the need for a greater number of nodes in the contour, generating more refined meshes that allow reaching differences in areas below 2%.

### Improved Upper Bound on Independent Domination Number for Hypercubes

We revisit the problem of determining the independent domination number in hypercubes for which the known upper bound is still not tight for general dimensions. We present here a constructive method to build an independent dominating set $S_n$ for the $n$-dimensional hypercube $Q_n$, where $n=2p+1$, $p$ being a positive integer $\ge 1$, provided an independent dominating set $S_p$ for the $p$-dimensional hypercube $Q_p$, is known. The procedure also computes the minimum independent dominating set for all $n=2^k-1$, $k>1$. Finally, we establish that the independent domination number $\alpha_n\leq 3 \times 2^{n-k-2}$ for $7\times 2^{k-2}-1\leq n<2^{k+1}-1$, $k>1$. This is an improved upper bound for this range as compared to earlier work.

### Heavy-Tail Phenomenon in Decentralized SGD

Recent theoretical studies have shown that heavy-tails can emerge in stochastic optimization due to `multiplicative noise', even under surprisingly simple settings, such as linear regression with Gaussian data. While these studies have uncovered several interesting phenomena, they consider conventional stochastic optimization problems, which exclude decentralized settings that naturally arise in modern machine learning applications. In this paper, we study the emergence of heavy-tails in decentralized stochastic gradient descent (DE-SGD), and investigate the effect of decentralization on the tail behavior. We first show that, when the loss function at each computational node is twice continuously differentiable and strongly convex outside a compact region, the law of the DE-SGD iterates converges to a distribution with polynomially decaying (heavy) tails. To have a more explicit control on the tail exponent, we then consider the case where the loss at each node is a quadratic, and show that the tail-index can be estimated as a function of the step-size, batch-size, and the topological properties of the network of the computational nodes. Then, we provide theoretical and empirical results showing that DE-SGD has heavier tails than centralized SGD. We also compare DE-SGD to disconnected SGD where nodes distribute the data but do not communicate. Our theory uncovers an interesting interplay between the tails and the network structure: we identify two regimes of parameters (stepsize and network size), where DE-SGD %addition of network structure can have lighter or heavier tails than disconnected SGD depending on the regime. Finally, to support our theoretical results, we provide numerical experiments conducted on both synthetic data and neural networks.

### Sparsity and $\ell_p$-Restricted Isometry

A matrix $A$ is said to have the $\ell_p$-Restricted Isometry Property ($\ell_p$-RIP) if for all vectors $x$ of up to some sparsity $k$, $\|Ax\|_p$ is roughly proportional to $\|x\|_p$. It is known that with high probability, random dense $m\times n$ matrices (e.g., with i.i.d.\ $\pm 1$ entries) are $\ell_2$-RIP with $k \approx m/\log n$, and sparse random matrices are $\ell_p$-RIP for $p \in [1,2)$ when $k, m = \Theta(n)$. However, when $m = \Theta(n)$, sparse random matrices are known to \emph{not} be $\ell_2$-RIP with high probability. With this backdrop, we show that there are no sparse matrices with $\pm 1$ entries that are $\ell_2$-RIP. On the other hand, for $p \neq 2$, we show that any $\ell_p$-RIP matrix \emph{must} be sparse. Thus, sparsity is incompatible with $\ell_2$-RIP, but necessary for $\ell_p$-RIP for $p \neq 2$.

### Artificial Intelligence-Assisted Optimization and Multiphase Analysis of Polygon PEM Fuel Cells

This article presents new PEM fuel cell models with hexagonal and pentagonal designs. After observing cell performance improvement in these models, we optimized them. Inlet pressure and temperature were used as input parameters, and consumption and output power were the target parameters of the multi-objective optimization algorithm. Then we used artificial intelligence techniques, including deep neural networks and polynomial regression, to model the data. Next, we employed the RSM (Response Surface Method) method to derive the target functions. Furthermore, we applied the NSGA-II multi-objective genetic algorithm to optimize the targets. Compared to the base model (Cubic), the optimized Pentagonal and Hexagonal models averagely increase the output current density by 21.819% and 39.931%, respectively.

### Principal-Agent Hypothesis Testing

Consider the relationship between the FDA (the principal) and a pharmaceutical company (the agent). The pharmaceutical company wishes to sell a product to make a profit, and the FDA wishes to ensure that only efficacious drugs are released to the public. The efficacy of the drug is not known to the FDA, so the pharmaceutical company must run a costly trial to prove efficacy to the FDA. Critically, the statistical protocol used to establish efficacy affects the behavior of a strategic, self-interested pharmaceutical company; a lower standard of statistical evidence incentivizes the pharmaceutical company to run more trials for drugs that are less likely to be effective, since the drug may pass the trial by chance, resulting in large profits. The interaction between the statistical protocol and the incentives of the pharmaceutical company is crucial to understanding this system and designing protocols with high social utility. In this work, we discuss how the principal and agent can enter into a contract with payoffs based on statistical evidence. When there is stronger evidence for the quality of the product, the principal allows the agent to make a larger profit. We show how to design contracts that are robust to an agent's strategic actions, and derive the optimal contract in the presence of strategic behavior.