We find a Floer theoretic approach to obtain the transpose polynomial $W^T$ of an invertible curve singularity $W$. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical $A_\infty$-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of $W^T$. We find Lagrangians in the Milnor fiber of $W$ that are mirror to the indecomposable matrix factorizations of $W^T$ when $W^T$ is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror. There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian $L$ and corresponding disc potential function $W_L$, we first determine the quotient $X$ by the maximal symmetry group for the Milnor fiber. Second, we define a class $\Gamma$ of symplectic cohomology of $X$ based on the monodromy of the singularity $W$. Another disc counting function, $g$, is defined by the closed-open image of $\Gamma$ on $L$. We demonstrate that restricting to the hypersurface $g = 0$ transforms the disc potential function $W_L$ into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class $\Gamma$.

The theory of motivic superpolynomials is developed, including its extension to algebraic links, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis and DAHA recurrences for iterated torus links. The DAHA construction is provided. Applications are considered to affine Springer fibers and compactified Jacobians of type $A_n$ in the most general case (for arbitrary characteristic polynomials), rho-invariants of algebraic knots, motivic formulas for the DAHA-Jones polynomials, including the case of $A_1$. The key theme is the conjectural coincidence of motivic superpolynomials with the DAHA ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. The 2nd connection conjecture links the superpolynomials to the $L$-functions. DAHA theory is adjusted to the definition and main properties of superpolynomials. The motivic tools are systematically exposed, a certain unification of Schubert Calculus with the theory of plane curve singularities.

In this work, we study the composition operators on the little Lipschitz space ${\mathcal L}_0$ of a rooted tree $T$ defined as the subspace of the Lipschitz space ${\mathcal L}$, introduced in Colonna and Easley [4], that consists of the complex-valued functions $f$ on $T$ such that $$ \lim_{|v|\to\infty}|f(v)-f(v^-)|=0, $$ where $v^-$ is the vertex adjacent to the vertex $v$ in the path from the root to $v$ and $|v|$ denotes the number of edges from the root to $v$. Specifically, we give several sufficient conditions on a Lipschitz self-map $\varphi$ on $T$ for which the composition operator $C_\varphi$ maps ${\mathcal L}_0$ into itself, and estimate its operator norm. In addition we study the spectrum of $C_\varphi$ and the hypercyclicity of a constant multiple of $C_\varphi$.

A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$.

I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies.

Let $M, N$ be free modules over a Noetherian commutative ring $R$ and let $F$ be a field whose cardinality does not exceed the continuum. We prove the following: 1) The Injective Continuum Function Hypothesis, ICF, is equivalent to assertion that [ any two F-vector spaces are isomorphic iff their duals are so.] 2) If the dual of $M$ is a projective $R$-module and rank$_R(M)$ is infinite then the ring $R$ is Artinian. 3) If $R$ is Artinian and card$(R)$ does not exceed the continuum then the dual of $M$ is free. 4) If $M, N$ have isomorphic duals then they are themselves isomorphic (over $R$), when rank$_R(M)$ is not an $\omega$-measurable cardinal and $R$ is a non-Artinian ring that is either Hilbert or countable. 5) If $R$ is a non-local domain then $R$ is a half-slender ring. We also prove that if the powersets of two given sets have equal cardinalities then there is a bijection from the one powerset to the other that preserves the symmetric difference of sets.

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

Let $\mathcal{D}$ be the whole space $\mathbb{R}^d$ or a sphere of radius $R>0$ in $\mathbb{R}^d$ with $1\leq d\leq3$. Consider the following cubic-quintic energy functional \begin{gather*} \begin{aligned} E_{N}(\varphi)=\frac{1}{2}\int_{\mathcal{D}}|\nabla \varphi|^2\,dx-\frac{N}{4}\int_{ \mathcal{D}}|\varphi|^{4}\,dx+\frac{N^{2}}{6}\int_{\mathcal{D}}|\varphi|^6\,dx \end{aligned} \end{gather*} minimized in $\left\lbrace \varphi\in H^1_0(\mathcal{D}): \int_{\mathcal{D}}|\varphi|^2\,dx=1\right\rbrace$. Firstly, we prove the limiting behavior of the ground state $\varphi_N$ as $N\to+\infty$, which corresponds to the \emph{Thomas-Fermi limit}. The limit profile is given by the Thomas-Fermi minimizer $u^{TF}=\sqrt{{3\cdot\mathbb{1}_{\Omega}}/{4}}(x)$ with $\Omega=B(0,\sqrt[d]{4/(3\omega_d)})$. Moreover, we obtain a sharp vanishing rate for $\varphi_N$ that $\|\varphi_{N}\|_{L^\infty(\mathcal{D})}\sim N^{-{1}/{2}}$ as $N\to+\infty$. Finally, we prove that $\left\|N^{{1}/{2}}\varphi_{N}\left(N^{{1}/{d}}x\right)-u^{TF}(x)\right\|_{L^\infty\left(\Omega\right)}\lesssim N^{-1/(2d)}$ as $N\to+\infty$.

Forward-backwards stochastic differential equations (FBSDEs) are central in optimal control, game theory, economics, and mathematical finance. Unfortunately, the available FBSDE solvers operate on \textit{individual} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs as these solvers must be re-run several times. \textit{Neural operators} (NOs) offer an alternative approach for \textit{simultaneously solving} large families of FBSDEs by directly approximating the solution operator mapping \textit{inputs:} terminal conditions and dynamics of the backwards process to \textit{outputs:} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. We confirm that ``small'' NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms, to any prescribed error $\varepsilon>0$ using a depth of $\mathcal{O}(\log(1/\varepsilon))$, a width of $\mathcal{O}(1)$, and a sub-linear rank; i.e. $\mathcal{O}(1/\varepsilon^r)$ for some $r<1$. This result is rooted in our second main contribution, which shows that convolutional NOs of similar depth, width, and rank can approximate the solution operator to a broad class of Elliptic PDEs. A key insight here is that the convolutional layers of our NO can efficiently encode the Green's function associated to the Elliptic PDEs linked to our FBSDEs. A byproduct of our analysis is the first theoretical justification for the benefit of lifting channels in NOs: they exponentially decelerate the growth rate of the NO's rank.

An overview of the authors' ideas about the process of completing a $p$-adically normed space in the setting of vertex operator algebras. We focus in particular on the $p$-adic Heisenberg VOA and its connections with $p$-adic modular forms.

A shape optimization program is developed for the ratio of Riesz capacities $\text{Cap}_q(K)/\text{Cap}_p(K)$, where $K$ ranges over compact sets in $\mathbb{R}^n$. In different regions of the $pq$-parameter plane, maximality is conjectured for the ball, the vertices of a regular simplex, or the endpoints of an interval. These cases are separated by a symmetry-breaking transition region where the shape of maximizers remains unclear. On the boundary of $pq$-parameter space one encounters existing theorems and conjectures, including: Watanabe's theorem minimizing Riesz capacity for given volume, the classical isodiametric theorem that maximizes volume for given diameter, Szeg\H{o}'s isodiametric theorem maximizing Newtonian capacity for given diameter, and the still-open isodiametric conjecture for Riesz capacity. The first quadrant of parameter space contains P\'{o}lya and Szeg\H{o}'s conjecture on maximizing Newtonian over logarithmic capacity for planar sets. The maximal shape for each of these scenarios is known or conjectured to be the ball. In the third quadrant, where both $p$ and $q$ are negative, the maximizers are quite different: when one of the parameters is $-\infty$ and the other is suitably negative, maximality is proved for the vertices of a regular simplex or endpoints of an interval. Much more is proved in dimensions $1$ and $2$, where for large regions of the third quadrant, maximizers are shown to consist of the vertices of intervals or equilateral triangles.

We extend recent computer-assisted design and analysis techniques for first-order optimization over structured functions--known as performance estimation--to apply to structured sets. We prove "interpolation theorems" for smooth and strongly convex sets with Slater points and bounded diameter, showing a wide range of extremal questions amount to structured mathematical programs. Prior function interpolation theorems are recovered as a limit of our set interpolation theory. Our theory provides finite-dimensional formulations of performance estimation problems for algorithms utilizing separating hyperplane oracles, linear optimization oracles, and/or projection oracles of smooth/strongly convex sets. As direct applications of this computer-assisted machinery, we identify the minimax optimal separating hyperplane method and several areas for improvement in the theory of Frank-Wolfe, Alternating Projections, and non-Lipschitz Smooth Optimization. While particular applications and methods are not our primary focus, several simple theorems and numerically supported conjectures are provided.

This paper revisits the classical notion of unicity distance from an enlightening perspective grounded in information theory, specifically by framing the encryption process as a noisy transmission channel. Using results from reliable communication theory, we derive a simple information-theoretic proof of the same unicity distance formula as in Shannon's classical result and a channel transmission interpretation of the unicity distance.

Subfactors of the hyperfinite II$_1$ factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index $\frac{5+\sqrt{13}}{2}$, i.e. the index of the Haagerup subfactor. We are led to a conjecture pertaining to Jones indices of irreducible, hyperfinite subfactors.

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of $q$-Hahn type is provided. The approach relies on the meta $q$-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of $q$-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta $q$-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta $q$-Hahn algebra on various bases.

We establish Struwe-type decompositions of Palais-Smale sequences for a class of critical $p$-Laplace equations of the Caffarelli-Kohn-Nirenberg type in a bounded domain $\Omega\subset\mathbb{R}^n$, $n\ge2$, containing the origin. In doing so, we highlight important differences introduced by the weights and require new rescaling laws to account for this new framework.

In this manuscript, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-H\'{e}non-type, featuring an explicit regularity exponent depending only on universal parameters. Our approach is based on geometric tangential methods and uses a refined oscillation mechanism, compactness, and scaling techniques. In some specific scenarios, we establish higher regularity estimates and non-degeneracy properties, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the $C^{p^{\prime}}$ conjecture for the $p$-Laplacian with a bounded source term. As applications of our results, we address some Liouville-type results for our class of equations. Finally, our results are noteworthy, even in the simplest model case governed by the $p$-Laplacian with regular coefficients: $$ \mathrm{div}\left( |\nabla u|^{p-2}\mathfrak{A}(|x|) \nabla u\right) = |x|^{\alpha}u_+^m(x) \quad \text{in} \quad B_1 $$ under suitable assumptions on the data, with possibly singular weight $\mathfrak{h}(|x|) = |x|^{\alpha}$, which includes the Matukuma and Batt-Faltenbacher-Horst's equations as toy models.

We show that the initial-value problem for the Benjamin-Ono equation on $\mathbb{R}$ with $L^2(\mathbb{R})$ rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables $(t,x)$ and the dispersion coefficient $\epsilon$. This allows for various interesting asymptotic limits to be resolved quite efficiently.

We classify the (filtered) Hopf actions of Hopf-Ore extensions of group algebras on path algebras of quivers, extending results in several other works from special cases to this general setting. Having done this for general Hopf-Ore extensions of group algebras, we demonstrate application by specializing our main result to certain Hopf-Ore extensions including some Noetherian prime Hopf algebras of GK-dimension one and two.

We define the symmetric braid index $b_s(K)$ of a ribbon knot $K$ to be the smallest index of a braid whose closure yields a symmetric union diagram of $K$, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most three. As applications, we show that there exist knots whose symmetric braid index is strictly greater than the braid index, and deduce that every chiral slice knot with determinant one has braid index at least four. We also calculate bounds for $b_s(K)$ for prime ribbon knots with at most 11 crossings.

Many applications require minimizing a family of optimization problems indexed by some hyperparameter $\lambda \in \Lambda$ to obtain an entire solution path. Traditional approaches proceed by discretizing $\Lambda$ and solving a series of optimization problems. We propose an alternative approach that parameterizes the solution path with a set of basis functions and solves a \emph{single} stochastic optimization problem to learn the entire solution path. Our method offers substantial complexity improvements over discretization. When using constant-step size SGD, the uniform error of our learned solution path relative to the true path exhibits linear convergence to a constant related to the expressiveness of the basis. When the true solution path lies in the span of the basis, this constant is zero. We also prove stronger results for special cases common in machine learning: When $\lambda \in [-1, 1]$ and the solution path is $\nu$-times differentiable, constant step-size SGD learns a path with $\epsilon$ uniform error after at most $O(\epsilon^{\frac{1}{1-\nu}} \log(1/\epsilon))$ iterations, and when the solution path is analytic, it only requires $O\left(\log^2(1/\epsilon)\log\log(1/\epsilon)\right)$. By comparison, the best-known discretization schemes in these settings require at least $O(\epsilon^{-1/2})$ discretization points (and even more gradient calls). Finally, we propose an adaptive variant of our method that sequentially adds basis functions and demonstrates strong numerical performance through experiments.

We consider the problem of characterizing extreme points of the convex set of positive linear operators on a possibly infinite-dimensional Hilbert space under linear constraints. We show that even perturbations of points in such sets admit what resembles a Douglas factorization. Using this result, we prove that an operator is extreme iff a corresponding set of linear operators is dense in the space of trace-class self-adjoint operators with range contained in the closure of the range of that operator. If the number of constraints is finite, we show that the extreme point must be of low-rank relative to the number of constraints and derive a purely rank-based characterization of the extreme points. In the finite-dimensional setting, our results lead to a remarkably simple characterization of the elliptope, that is, the set of correlation matrices, in terms of the Hadamard product which allows us to characterize the set of matrices which constitute the equality case of the Hadamard rank inequality when the involved matrices are equal and positive semi-definite. We illustrate the importance of our results using examples from statistics and quantum mechanics.

This paper investigates the structure of product systems of Hilbert spaces derived from Banach space-valued L\'evy processes. We establish conditions under which these product systems are completely spatial and show that Gaussian L\'evy processes with non-degenerate covariance always give rise to product systems of type I. Furthermore, we construct a continuum of non-isomorphic product systems of type \(\rm{II}\sb\infty\) from pure jump L\'evy processes.

In this paper, we consider the contextual robust optimization problem under an out-of-distribution setting. The contextual robust optimization problem considers a risk-sensitive objective function for an optimization problem with the presence of a context vector (also known as covariates or side information) capturing related information. While the existing works mainly consider the in-distribution setting, and the resultant robustness achieved is in an out-of-sample sense, our paper studies an out-of-distribution setting where there can be a difference between the test environment and the training environment where the data are collected. We propose methods that handle this out-of-distribution setting, and the key relies on a density ratio estimation for the distribution shift. We show that additional structures such as covariate shift and label shift are not only helpful in defending distribution shift but also necessary in avoiding non-trivial solutions compared to other principled methods such as distributionally robust optimization. We also illustrate how the covariates can be useful in this procedure. Numerical experiments generate more intuitions and demonstrate that the proposed methods can help avoid over-conservative solutions.

As the number of low Earth orbit (LEO) satellites rapidly increases, the consideration of frequency sharing or cooperation between geosynchronous Earth orbit (GEO) and LEO satellites is gaining attention. In this paper, we consider a hybrid GEO-LEO satellite network where GEO and LEO satellites are distributed according to independent Poisson point processes (PPPs) and share the same frequency resources. Based on the properties of PPPs, we first analyze satellite-visible probabilities, distance distributions, and association probabilities. Then, we derive an analytical expression for the network's coverage probability. Through Monte Carlo simulations, we verify the analytical results and demonstrate the impact of system parameters on coverage performance. The analytical results effectively estimate the coverage performance in scenarios where GEO and LEO satellites cooperate or share the same resource.

We study dynamical models on a self-similar space-time lattice as toy models for multiscale motion in hydrodynamic turbulence. Here an ill-posed ideal system is regularized at small scales and the vanishing regularization (inviscid) limit is considered. By relating the inviscid limit to the dynamics of the RG operator acting on the flow maps, we explain the existence and universality (regularization independence) of the limiting solutions as a consequence of the fixed-point RG attractor. Considering the local linearized dynamics, we show that the approach to the inviscid limit is governed by the universal RG eigenmode. We also demonstrate that the RG attractor undergoes a period-doubling bifurcation with parameter variation, thereby changing the nature of the inviscid limit. In the case of chaotic RG dynamics, we introduce the stochastic RG operator acting on Markov kernels. Then the RG attractor becomes stochastic, which explains the existence and universality of spontaneously stochastic solutions in the limit of vanishing noise. We study a linearized structure (RG eigenmode) of the stochastic RG attractor and its period-doubling bifurcation. Viewed as prototypes of Eulerian spontaneous stochasticity, our models explain its mechanism, universality and potential diversity.

The Cohn-Umans (FOCS '03) group-theoretic framework for matrix multiplication produces fast matrix multiplication algorithms from three subsets of a finite group $G$ satisfying a simple combinatorial condition (the Triple Product Property). The complexity of such an algorithm then depends on the representation theory of $G$. In this paper we extend the group-theoretic framework to the setting of infinite groups. In particular, this allows us to obtain constructions in Lie groups, with favorable parameters, that are provably impossible in finite groups of Lie type (Blasiak, Cohn, Grochow, Pratt, and Umans, ITCS '23). Previously the Lie group setting was investigated purely as an analogue of the finite group case; a key contribution in this paper is a fully developed framework for obtaining bona fide matrix multiplication algorithms directly from Lie group constructions.

Given two associative algebras A, C and a linear space V together with some linear maps R_1, R_2, R_3, E satisfying some conditions, we define an associative algebra structure on A\otimes V\otimes C called a two-sided crossed product. Particular cases of this construction are the iterated twisted tensor product of algebras and the two-sided crossed product over a quasi-bialgebra.

High order exponential integrators require computing linear combination of exponential like $\varphi$-functions of large matrices $A$ times a vector $v$. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix is $A$ is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis $V_m$. In parallel, GS orthogonalization requires \textit{global synchronizations} for inner products and vector normalization in the orthogonalization process. Reducing the amount of global synchronizations is of paramount importance for the efficiency of a numerical algorithm in a massively parallel setting. We improve the parallel strong scaling properties of exponential integrators by addressing the underlying bottleneck in the linear algebra using low-synchronization GS methods. The resulting orthogonalization algorithms have an accuracy comparable to modified Gram-Schmidt yet are better suited for distributed architecture, as only one global communication is required per orthogonalization-step. We present geophysics-based numerical experiments and standard examples routinely used to test stiff time integrators, which validate that reducing global communication leads to better parallel scalability and reduced time-to-solution for exponential integrators.

We present a scalable approach to solve a class of elliptic partial differential equation (PDE)-constrained optimization problems with bound constraints. This approach utilizes a robust full-space interior-point (IP)-Gauss-Newton optimization method. To cope with the poorly-conditioned IP-Gauss-Newton saddle-point linear systems that need to be solved, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE-constrained optimization problems. A block Gauss-Seidel preconditioner is proposed for the GMRES-based solution of the IP-Gauss-Newton linear systems. It is shown, for a large-class of PDE- and bound-constrained optimization problems, that the spectrum of the block Gauss-Seidel preconditioned IP-Gauss-Newton matrix is asymptotically independent of discretization and is not impacted by the ill-conditioning that notoriously plagues interior-point methods. We propose a regularization and log-barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)-based solution of the related IP-Gauss-Newton-Schur complement linear systems. The scalability of the approach is demonstrated on an example problem with bound and nonlinear elliptic PDE constraints. The numerical solution of the optimization problem is shown to require a discretization independent number of IP-Gauss-Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill-conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of preconditioner and linear system matrix applies, achieved with algebraic multigrid based solvers, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE- and bound-constrained problems.

In this work, we deal with Delone sets and their rectifiability under different classes of regularity. By pursuing techniques developed by Rivi\`ere and Ye, and Aliste-Prieto, Coronel and Gambaudo, we give sufficient conditions for a Delone set to be equivalent to the standard lattice by bijections having regularity in between bi-Lipschitz and bi-H\"older-homogeneous. From this criterion, we improve a result of McMullen by showing that, for any dimension $d\geq 1$, there exists a threshold of moduli of continuity $\mathcal{M}_d$, including the class of the H\"{o}lder ones, such that for every $\omega\in\mathcal{M}_d$, any two Delone sets in $\mathbb{R}^d$ cannot be distinguished under bi-$\omega$-equivalence. Also, we extend a result due to Aliste, Coronel, and Gambaudo, which establishes that every linearly repetitive Delone set in $\mathbb{R}^d$ is rectifiable by extending it to a broader class of repetitive behaviors. Moreover, we show that for the modulus of continuity $\omega(t)=t(\log(1/t))^{1/d}$, every $\omega$-repetitive Delone set in $\mathbb{R}^d$ is equivalent to the standard lattice by a bi-$\omega$-homogeneous map. Finally, we address a problem of continuous nature related to the previous ones about finding solutions to the prescribed volume form equation in intermediate regularity, thereby extending the results of Rivi\`ere and Ye.

We consider extended Cartan homotopy formula (ECHF) for higher gauge theory. Firstly, we construct an oriented simplex based on 2-connections and present differential and integral forms of the higher ECHF. Then, we study the higher Chern-Simons-Antoniadis-Savvidy (ChSAS) theory and prove that the higher ECHF can reproduce the higher Chern-Weil theorem and give higher triangle equation. We finally conclude from the higher ECHF that a higher transgression form can be written as the difference of two higher ChSAS forms minus an exact form.

The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group $G$ acts by isometries on two metric spaces $X$ and $Y$. In this article, we study how the Hausdorff and Gromov-Hausdorff distances between $X$ and $Y$ and their quotient spaces $X/G$ and $Y/G$ are related. For the Hausdorff distance, we show that if $X$ and $Y$ are $G$-invariant subsets of a common metric space, then we have $d_{\mathrm{H}}(X,Y)=d_{\mathrm{H}}(X/G,Y/G)$. However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio $\frac{d_{\mathrm{GH}}(X/G,Y/G)}{d_{\mathrm{GH}}(X,Y)}$ both arbitrarily large and arbitrarily small, even if $X$ is an arbitrarily dense $G$-invariant subset of $Y$.

This work focuses on the degree bound of maps between balls with maximum geometric rank and minimum target dimension where this geometric rank occurs. Specifically, we show that rational proper maps between $\mathbb{B}_n$ and $\mathbb{B}_N$ with $n \geq 2$, $N = \frac{n(n+1)}{2}$, and geometric rank $n-1$ cannot have a degree of more than $n+1$.

In this paper, we present an asymptotic view for any $C$-pseudo-cone, which allows us to decompose a $C$-pseudo-cone $E$ into the sum of a $C$-asymptotic set $\mathbb{A}$ and $C$-starting point $z\in C$ of $E$. Combining this with the novel work by Schneider, we introduce the asymptotic weighted co-volume functional $T_\Theta(E)$ of the $C$-pseudo-cone $E$, which is also a generalized function with the singular point $o$ (the origin). Using our convolution formula for $T_\Theta(E)$, we establish a decay estimate for $T_\Theta(E)$ at infinity and present some interesting results. As applications of this asymptotic theory, we prove a weighted Brunn-Minkowski type inequality and study the solutions to the weighted Minkowski problem for pseudo-cones. Moreover, we pose an open problem regarding $T_\Theta(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones.

We establish that the Lavrentiev gap between Sobolev and Lipschitz maps does not occur for a scalar variational problem of the form: \[ \textrm{to minimize} \qquad u \mapsto \int_\Omega f(x,u,\nabla u)\dx \,, \] under a Dirichlet boundary condition. Here, \(\Omega\) is a bounded Lipschitz open set in \(\rn\), \(N\geq 1\) and the function $f$ is required to be measurable with respect to the spacial variable, continuous with respect to the second one, and continuous and comparable to convex with respect to the last variable. Moreover, we assume that $f$ satisfies a natural condition balancing the variations with respect to the first variable and the growth with respect to the last one. Remarkably, typical conditions that are usually imposed on $f$ to discard the Lavrentiev gap are dropped here: we do not require $f$ to be bounded or convex with respect to the second variable, nor impose any condition of $\Delta_2$-kind with respect to the last variable.

The forward problem arising in several hybrid imaging modalities can be modeled by the Cauchy problem for the free space wave equation. Solution to this problems describes propagation of a pressure wave, generated by a source supported inside unit sphere $S$. The data $g$ represent the time-dependent values of the pressure on the observation surface $S$. Finding initial pressure $f$ from the known values of $g$ consitutes the inverse problem. The latter is also frequently formulated in terms of the spherical means of $f$ with centers on~$S$. Here we consider a problem of range description of the wave operator mapping $f$ into $g$. Such a problem was considered before, with data $g$ known on time interval at least $[0,2]$ (assuming the unit speed of sound). Range conditions were also found in terms of spherical means, with radii of integration spheres lying in the range $[0,2]$. However, such data are redundant. We present necessary and sufficient conditions for function $g$ to be in the range of the wave operator, for $g$ given on a half-time interval $[0,1]$. This also implies range conditions on spherical means measured for the radii in the range $[0,1]$.

Let $G$ be a simple connected graph of order $n$ with diameter $d$. Let $m_G(-1)$ denote the multiplicity of the eigenvalue $-1$ of the adjacency matrix of $G$, and let $P = P_{d+1}$ be the diameter path of $G$. If $-1$ is not an eigenvalue of $P$, then by the interlacing theorem, we have $m_G(-1)\leq n - d - 1$. In this article, we characterize the extremal graphs where equality holds. Moreover, for the completeness of the results, we also characterize the graphs $G$ that achieve $m_G(-1) = n - d - 1$ when $-1$ is an eigenvalue of $P$. Thus, we provide a complete characterization of the graphs $G$ for which $m_G(-1) = n - d - 1$.

We study the finite horizon Restless Multi-Armed Bandit (RMAB) problem with $N$ homogeneous arms, focusing on the challenges posed by degenerate RMABs, which are prevalent in practical applications. While previous work has shown that Linear Programming (LP)-based policies achieve exponentially fast convergence relative to the LP upper bound in non-degenerate models, applying these LP-based policies to degenerate RMABs results in slower convergence rates of $O(1/\sqrt{N})$. We construct a diffusion system that incorporates both the mean and variance of the stochastic processes, in contrast to the fluid system from the LP, which only accounts for the mean, thereby providing a more accurate representation of RMAB dynamics. Consequently, our novel diffusion-resolving policy achieves an optimality gap of $O(1/N)$ relative to the true optimal value, rather than the LP upper bound, revealing that the fluid approximation and the LP upper bound are too loose in degenerate settings. These insights pave the way for constructing policies that surpass the $O(1/\sqrt{N})$ optimality gap for any RMAB, whether degenerate or not.

The focus of this work is the study of Parikh matrices with emphasis on two concrete problems. In the first part of our presentation we show that a conjecture by Dick at al. in 2021 only stands in the case of ternary alphabets, while providing counterexamples for larger alphabets. In particular, we show that the only type of distinguishability in the case of 3-letter alphabets is the trivial one. The second part of the paper builds on the notion of Parikh matrices for projections of words, discussed initially in this work, and answers, once more in the case of a ternary alphabet, a question posed by Atanasiu et al. in 2022 with regards to the minimal Hamming distance in between words sharing a congruency class.

One of the tasks in color image processing and computer vision is to recover clean data from partial observations corrupted by noise. To this end, robust quaternion matrix completion (QMC) has recently attracted more attention and shown its effectiveness, whose convex relaxation is to minimize the quaternion nuclear norm plus the quaternion $L_1$-norm. However, there is still room to improve due to the convexity of the convex surrogates. This paper proposes a new nonconvex robust QMC model, in which the nonconvex MCP function and the quaternion $L_p$-norm are used to enhance the low-rankness and sparseness of the low-rank term and sparse term, respectively. An alternating direction method of multipliers (ADMM) algorithm is developed to solve the proposed model and its convergence is given. Moreover, a novel nonlocal-self-similarity-based nonconvex robust quaternion completion method is proposed to handle large-scale data. Numerical results on color images and videos indicate the advantages of the proposed method over some existing ones.

In this article, we consider the problem of unconstrained time-varying convex optimization, where the cost function changes with time. We provide an in-depth technical analysis of the problem and argue why freezing the cost at each time step and taking finite steps toward the minimizer is not the best tracking solution for this problem. We propose a set of algorithms that by taking into account the temporal variation of the cost aim to reduce the tracking error of the time-varying minimizer of the problem. The main contribution of our work is that our proposed algorithms only require the first-order derivatives of the cost function with respect to the decision variable. This approach significantly reduces computational cost compared to the existing algorithms, which use the inverse of the Hessian of the cost. Specifically, the proposed algorithms reduce the computational cost from $O(n^3)$ to $O(n)$ per timestep, where $n$ is the size of the decision variable. Avoiding the inverse of the Hessian also makes our algorithms applicable to non-convex optimization problems. We refer to these algorithms as $O(n)$-algorithms. These $O(n)$-algorithms are designed to solve the problem for different scenarios based on the available temporal information about the cost. We illustrate our results through various examples, including the solution of a model predictive control problem framed as a convex optimization problem with a streaming time-varying cost function.

The purpose of this article is to study the residual Monge-Amp\`{e}re mass of a plurisubharmonic function with an isolated unbounded locus. A general decomposition formula for the residual mass is obtained, under the Sasakian structure of the unit sphere. In complex dimension two, we further obtain an upper-bound estimate, provided with the uniform directional Lipschitz continuity. As an application, the zero mass conjecture is confirmed, if the function further has a single frequency on its alternating part.

The star chromatic index of a graph $G$, denoted by $\chi^\prime_s(G)$, is the smallest integer $k$ for which $G$ admits a proper edge coloring with $k$ colors such that every path and cycle of length four is not bicolored. Let $d$ be the greatest common divisor of $n$ and $k$. Zhu~et~al. (\footnotesize{Discussiones Mathematicae: Graph Theory, 41(2): 1265, 2021}) showed that for every integers $k$ and $n> 2k$ with $d\geq 3$, generalized Petersen graph $GP(n,k)$ admits a 5-star edge coloring, with the exception of the case that $d = 3$, $k\neq d$ and $\frac{n}{3}= 1\pmod{3}$. Also, they conjectured that for every $n>2k$, $\chi^\prime_s(GP(n,k))\leq 5$, except $GP(3,1)$. In this paper, we prove that for every $GP(n,k)$ with $n\geq 2k$ and $d\geq 3$ their conjecture is true. In fact, we provide a 5-star edge coloring of $GP(n,k)$, where $n\geq 2k$ and $d\geq 3$. We also obtain some results for 5-star edge coloring of $GP(n,k)$ with $d=2$. Moreover, Dvo{\v{r}}{\'a}k et al. ({\footnotesize Journal of Graph Theory, 72(3):313-326, 2013}) conjectured that the star of chromatic index of subcubic graphs is at most 6. Thus, our results also prove this conjecture for the generalized Petersen graphs, as a class of subcubic graphs.

Let $\mu$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_{\mu}:f \mapsto f* \mu$ maps the space of smooth $\mathfrak{v}$-radial functions onto itself if and only if the spherical Fourier transform $\widetilde{\mu}(\lambda)$, $\lambda \in \mathbb{C}$, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth $\mathfrak{v}$-radial functions.

We consider hard-potential cutoff multi-species Boltzmann operators modeling microscopic binary elastic collisions and bimolecular reversible chemical reactions inside a gaseous mixture. We prove that the spectral gap estimate derived for the linearized elastic collision operator can be exploited to deduce an explicit negative upper bound for the Dirichlet form of the linearized chemical Boltzmann operator. Such estimate may be used to quantify explicitly the rate of convergence of close-to-equilibrium solutions to the reactive Boltzmann equation toward the global chemical equilibrium of the mixture.

In this paper, we prove the Regev formulae for the characters of the Ariki--Koike algebras by applying the Schur--Sergeev reciprocity between the quantum superalgebras and the Ariki--Koike algebras, which is a generalization of the formulas in (D. Zhao, Israel J. Math. 229 (2019): 67--83 ). As a corollary, we provide the Regev formulae for the characters of the complex reflection group of type $G(m,1,n)$, which is a generalization of the formulas in (A. Regev, Israel J. Math. 195 (2013): 31--35).

In this paper, we study the existence of normalized solutions for the following $(2, q)$-Laplacian equation \begin{equation*}\label{Eq-Equation1} \left\{\begin{array}{l} -\Delta u-\Delta_q u+\lambda u=f(u) \quad x \in \mathbb{R}^N , \\ \int_{\mathbb{R}^N}u^2 d x=c^2, \\ \end{array}\right. \end{equation*} where $1<q<N$, $N\geq3$, $\Delta_q=\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right)$ denotes the $q$-Laplacian operator, $\lambda$ is a Lagrange multiplier and $c>0$ is a constant. The nonlinearity $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, with mass-subcritical growth at the origin, mass-supercritical growth at infinity, and is more general than the sum of two powers. Under different assumptions, we prove the existence of a locally least-energy solution and the existence of a second solution with higher energy.

There is a close relationship between linear codes and $t$-designs. Through their research on a class of narrow-sense BCH codes, Ding and Tang made a breakthrough by presenting the first two infinite families of near MDS codes holding $t$-designs with $t=2$ or 3. In this paper, we present an infinite family of MDS codes over $\mathbb{F}_{2^s}$ and two infinite families of almost MDS codes over $\mathbb{F}_{p^s}$ for any prime $p$, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over $\mathbb{F}_{3^s}$, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost AMDS codes and their dual codes hold infinite families of $3$-designs over \(\mathbb{F}_{p^s}\) for any prime $p$. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.

The soft-output successive cancellation list (SOSCL) decoder provides a methodology for estimating the a-posteriori probability log-likelihood ratios by only leveraging the conventional SCL decoder for polar codes. However, the sequential nature of SCL decoding leads to a high decoding latency for the SO-SCL decoder. In this paper, we propose a soft-output fast SCL (SO-FSCL) decoder by incorporating node-based fast decoding into the SO-SCL framework. Simulation results demonstrate that the proposed SO-FSCL decoder significantly reduces the decoding latency without loss of performance compared with the SO-SCL decoder.

We present a method to rapidly approximate convolution quadrature (CQ) approximations, based on a piecewise polynomial interpolation of the Laplace domain operator, which we call the \emph{parsimonious} convolution quadrature method. For implicit Euler and second order backward difference formula based discretizations, we require $O(\sqrt{N}\log N)$ evaluations in the Laplace domain to approximate $N$ time steps of the convolution quadrature method to satisfactory accuracy. The methodology proposed here differentiates from the well-understood fast and oblivious convolution quadrature \cite{SLL06}, since it is applicable to Laplace domain operator families that are only defined and polynomially bounded on a positive half space, which includes acoustic and electromagnetic wave scattering problems. The methods is applicable to linear and nonlinear integral equations. To elucidate the core idea, we give a complete and extensive analysis of the simplest case and derive worst-case estimates for the performance of parsimonious CQ based on the implicit Euler method. For sectorial Laplace transforms, we obtain methods that require $O(\log^2 N)$ Laplace domain evaluations on the complex right-half space. We present different implementation strategies, which only differ slightly from the classical realization of CQ methods. Numerical experiments demonstrate the use of the method with a time-dependent acoustic scattering problem, which was discretized by the boundary element method in space.

Time delays are ubiquitous in industrial processes, and they must be accounted for when designing control algorithms because they have a significant effect on the process dynamics. Therefore, in this work, we propose a simultaneous approach for numerical optimal control of delay differential equations with distributed time delays. Specifically, we linearize the delayed variables around the current time, and we discretize the resulting implicit differential equations using Euler's implicit method. Furthermore, we transcribe the infinite-dimensional optimal control problem into a finite-dimensional nonlinear program, which we solve using Matlab's fmincon. Finally, we demonstrate the efficacy of the approach using a numerical example involving a molten salt nuclear fission reactor.

H. Gl\"ockner and G. A. Willis have recently shown that locally pro-p contraction groups are nilpotent. The proof hinges on a fixed-point result: if the local field $\mathbb{F}_{p}(\!(t)\!)$ acts on its $d$-th power $\mathbb{F}_{p}(\!(t)\!)^{d}$ additively, continuously, and in an appropriately equivariant manner, then the action has a non-zero fixed point. We provide a short proof of this theorem.

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. A $t$-distinct partition of a positive integer $n$ is a partition of $n$ such that any of its parts can occur at most $t-1$ times. Let $b_{t,k}(n)$ (respectively, $d_{t,k}(n)$) denote the number of hooks of length $k$ in all the $t$-regular (respectively, $t$-distinct) partitions of $n$. In this article, we prove some biases for $b_{t,k}(n)$ for fixed values of $k$. We prove that for any $t\geq2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq0$. We establish that $b_{3,2}(n)\geq b_{2,2}(n)$, for all $n>3$ and $b_{3,3}(n)\geq b_{2,3}(n)$, for all $n\geq0$. We also prove a very general bias $d_{t+1,k}(n)\geq d_{t,k}(n)$ which holds for all $n\geq0$ and for any $t$ and $k$. Our methods of proofs are completely combinatorial. Finally, we state some problems for future works.

Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions.

We prove $L^{\infty}_{t}W^{1,p}_{x}$ Sobolev estimates in the Keller-Segel system by proving a functional inequality, inspired by the Brezis-Gallou\"et-Wainger inequality. These estimates are also valid at the discrete level in the Jordan-Kinderlehrer-Otto (JKO) scheme. By coupling this result with the diffusion properties of a functional according to Bakry-Emery theory, we deduce the $L^2_t H^{2}_{x}$ convergence of the scheme, thereby extending the recent result of Santambrogio and Toshpulatov in the context of the Fokker-Planck equation to the Keller-Segel system.

In this paper, we employ the framework of localization algebras to compute the equivariant K-homology class of the Euler characteristic operator, a central object in studying equivariant index theory on manifolds. This approach provides a powerful algebraic language for analyzing differential operators on equivariant structures and allows for the application of Witten deformation techniques in a K-homological context. Utilizing these results, we establish an equivariant version of the Poincar\'e-Hopf theorem, extending classical topological insights to the equivariant case, inspired by the results of L\"uck-Rosenberg. This work thus offers a new perspective on the localization techniques in the equivariant K-homology, highlighting their utility in deriving explicit formulas for index-theoretic invariants.

We provide a set of conditions that is necessary and sufficient for the $L^{2}$-wellposedness of the Cauchy problem for fifth and sixth order variable-coefficient linear dispersive equations. The necessity of these conditions had been presented by Tarama, and we scrutinized their proof to split the conditions into several parts so that an inductive argument is applicable. This inductive argument simplifies the engineering process of the appropriate pseudodifferential operator needed for the proof of $L^{2}$-wellposedness.

For almost two decades, mixed integer programming (MIP) solvers have used graph-based conflict analysis to learn from local infeasibilities during branch-and-bound search. In this paper, we improve MIP conflict analysis by instead using reasoning based on cuts, inspired by the development of conflict-driven solvers for pseudo-Boolean optimization. Phrased in MIP terminology, this type of conflict analysis can be understood as a sequence of linear combinations, integer roundings, and cut generation. We leverage this MIP perspective to design a new conflict analysis algorithm based on mixed integer rounding cuts, which theoretically dominates the state-of-the-art method in pseudo-Boolean optimization using Chv\'atal-Gomory cuts. Furthermore, we extend this cut-based conflict analysis from pure binary programs to mixed binary programs and-in limited form-to general MIP with also integer-valued variables. We perform an empirical evaluation of cut-based conflict analysis as implemented in the open-source MIP solver SCIP, testing it on a large and diverse set of MIP instances from MIPLIB 2017. Our experimental results indicate that the new algorithm improves the default performance of SCIP in terms of running time, number of nodes in the search tree, and the number of instances solved.

This work extends the study of mean field equations arising in two-dimensional (2D) turbulence by introducing generalized weighted Sobolev operators. Employing variational methods, particularly the mountain pass theorem and a refined blow-up analysis, we establish the existence of nontrivial solutions under broader boundary conditions than those considered in previous studies. In contrast to the unweighted approach developed by Ricciardi (2006), the incorporation of variable weights \(\rho(x)\) enables a more accurate representation of complex geometric domains. This generalization addresses cases where the geometry of the manifold or external factors induce local fluctuations, thereby enhancing the applicability of mean field models to real-world turbulence phenomena. The weighted Sobolev framework also strengthens control over the nonlinearity in the problem, reducing the risk of blow-up and ensuring the convergence of solutions in the \(H^1(M)\) space. The analysis reveals new conditions for the existence of solutions, including situations where classical methods may fail due to a lack of compactness. The significant advancement lies in the flexibility of the model; the inclusion of weights allows for a more precise description of physical systems in varied settings, representing a key advantage over the original unweighted problem. These results contribute to a deeper understanding of the role of geometric and physical constraints in turbulent behavior and pave new pathways for future research on nonlinear partial differential equations in complex domains.

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to some norm, much better lower bounds on $d$ have been obtained, with most research focusing on such lower bounds and on the degree of approximation of the section by a $k$-dimensional Euclidean ball. In this note we focus on another parameter, namely the radius of the approximating ball. We focus on the case of the unit ball of the space $\ell_1^N$ (the so-called cross-polytope), which is relevant to various questions of interest in theoretical computer science. We will also survey other instances where similar questions for other normed spaces (most often $\ell_p$-spaces or their non-commutative analogues) were found relevant to problems in various areas of mathematics and its applications, and state some open problems. Finally, in view of the computer science ramifications, we will comment on the algorithmic aspects of finding nearly spherical sections.

This paper presents a novel model-free method to solve linear quadratic Gaussian mean field social control problems in the presence of multiplicative noise. The objective is to achieve a social optimum by solving two algebraic Riccati equations (AREs) and determining a mean field (MF) state, both without requiring prior knowledge of individual system dynamics for all agents. In the proposed approach, we first employ integral reinforcement learning techniques to develop two model-free iterative equations that converge to solutions for the stochastic ARE and the induced indefinite ARE respectively. Then, the MF state is approximated, either through the Monte Carlo method with the obtained gain matrices or through the system identification with the measured data. Notably, a unified state and input samples collected from a single agent are used in both iterations and identification procedure, making the method more computationally efficient and scalable. Finally, a numerical example is given to demonstrate the effectiveness of the proposed algorithm.

Let $X$ be a smooth projective variety defined over a number field $k$ that has Brauer-Manin obstruction to the existence of rational points. We study the orders of the Brauer classes which capture the obstruction. We prove that one may require elements of arbitrarily large order to capture the obstruction. In other words, the exponent of the group which captures the obstruction has no upper bound.

We develop a robust structure theory for multiple ergodic averages of commuting transformations along Hardy sequences of polynomial growth. We then apply it to derive a number of novel results on joint ergodicity, recurrence and convergence. Specifically, we construct a suitable generalization of Host-Kra and box seminorms that quantitatively controls the aforementioned averages subject to necessary nondegeneracy conditions on the Hardy sequences. Combining this with a variant of the seminorm smoothing argument, we obtain Host-Kra seminorm estimates for averages along all pairwise independent Hardy sequences. In conjunction with joint ergodicity criteria and classical equidistribution results, these estimates yield a number of novel joint ergodicity results that reach far beyond the state of the art. In particular, we prove joint ergodicity for (a) pairwise independent Hardy sequences and weakly mixing transformations, (b) strongly independent Hardy sequences and ergodic transformations, (c) irrationally strongly independent Hardy sequences and totally ergodic transformations. We also positively resolve the joint ergodicity classification problem for all pairwise independent Hardy sequences, of which the aforementioned three families are special cases. Lastly, we use these joint ergodicity results to provide new recurrence results for multidimensional patterns along strongly independent Hardy sequences. While building on recent technical advances (e.g. PET coefficient tracking schemes and joint ergodicity criteria), our work introduces a number of technical developments of its own, including the theory of generalized box seminorms, an ergodic version of the quantitative concatenation argument, improved simultaneous Taylor approximations for Hardy sequences, and a generalization of the seminorm smoothing argument.

We consider a semilinear Schr\"odinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli-Kohn-Nirenberg's inequality. Depending on the parameter $a$ and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. We pose some natural conjectures, which are still open -- such as the radiality of the ground states, the non-degeneracy and most importantly uniqueness.

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.

Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable methodologies, for such a significant portion of real systems, have been historically scarce. Around the year 2000, the backstepping approach to PDE control began to offer not only a less abstract alternative to PDE control techniques replicating optimal and spectrum assignment techniques of the 1960s, but also enabled the methodologies of adaptive and nonlinear control, matured in the 1980s and 1990s, to be extended from ODEs to PDEs, allowing feedback synthesis for physical and engineering systems that are uncertain, nonlinear, and infinite-dimensional. The PDE backstepping literature has grown in its nearly a quarter century of development to many hundreds of papers and nearly a dozen books. This survey aims to facilitate the entry, for a new researcher, into this thriving area of overwhelming size and topical diversity. Designs of controllers and observers, for parabolic, hyperbolic, and other classes of PDEs, in one and more dimensions (in box and spherical geometries), with nonlinear, adaptive, sampled-data, and event-triggered extensions, are covered in the survey. The lifeblood of control are technology and physics. The survey places a particular emphasis on applications that have motivated the development of the theory and which have benefited from the theory and designs: applications involving flows, flexible structures, materials, thermal and chemically reacting dynamics, energy (from oil drilling to batteries and magnetic confinement fusions), and vehicles.

We construct the first examples of genuine ergodic discrete measured groupoids that are not isomorphic to any equivalence relation or transformation groupoid. We use a construction due to B.H. Neumann of an uncountable family of pairwise non-isomorphic 2-generated groups for our result.

We show that any complete local (normal) domain admits a module-finite quasi-Gorenstein normal (complete local) domain extension. Notably our result resolves the previously open case of residual characteristic two.

Motivated by engineering and photonics research on resonators in random or uncertain environments, we study rigorous randomizations of boundary conditions for wave equations of the acoustic-type in Lipschitz domains $\mathcal{O}$. First, a parametrization of essentially all m-dissipative boundary condition by contraction operators in the boundary $L^2$-space is constructed with the use of m-boundary tuples (boundary value spaces). We consider randomizations of these contraction operators that lead to acoustic operators random in the resolvent sense. To this end, the use of Neumann-to-Dirichlet maps and Krein-type resolvent formulae is crucial. We give a description of random m-dissipative boundary conditions that produce acoustic operators with almost surely (a.s.) compact resolvents, and so, also with a.s. discrete spectra. For each particular applied model, one can choose a specific boundary condition from the constructed class either by means of optimization, or on the base of empirical observations. A mathematically convenient randomization is constructed in terms of eigenfunctions of the Laplace-Beltrami operator $\Delta^{\partial \mathcal{O}}$ on the boundary $\mathcal{O}$ of the domain. We show that for this randomization the compactness of the resolvent is connected with the Weyl-type asymptotics for the eigenvalues of $\Delta^{\partial \mathcal{O}}$.

The Clifford algebra of the endomorphisms of the exterior algebra of a countably dimensional vector space induces natural bosonic shadows, i.e. families of linear maps between the cohomologies of complex grassmannians. The main result of this paper is to provide a determinantal formula expressing generating functions of such endomorphisms unifying several classical special cases. For example the action over a point recovers the Jacobi-Trudy formula in the theory of symmetric functions or the Giambelli's one in classical Schubert calculus, whereas the action of degree preserving endomorphisms take into account the finite type version of the Date-Jimbo-Kashiwara-Miwa bosonic vertex operator representation of the Lie algebra $gl(\infty)$. The fermionic actions on (finite type) bosonic spaces is described in terms of the classical theory of symmetric functions. The main guiding principle is the fact that the exterior algebra is a (non irreducible) representation of the ring of symmetric functions, which is the way we use to spell the ``finite type'' Boson-Fermion correspondence.

Consider a high-dimensional Wishart matrix $\bd{W}=\bd{X}^T\bd{X}$ where the entries of $\bd{X}$ are i.i.d. random variables with mean zero, variance one, and a finite fourth moment $\eta$. Motivated by problems in signal processing and high-dimensional statistics, we study the maximum of the largest eigenvalues of any two-by-two principal minors of $\bd{W}$. Under certain restrictions on the sample size and the population dimension of $\bd{W}$, we obtain the limiting distribution of the maximum, which follows the Gumbel distribution when $\eta$ is between 0 and 3, and a new distribution when $\eta$ exceeds 3. To derive this result, we first address a simpler problem on a new object named a deformed Gaussian orthogonal ensemble (GOE). The Wishart case is then resolved using results from the deformed GOE and a high-dimensional central limit theorem. Our proof strategy combines the Stein-Poisson approximation method, conditioning, U-statistics, and the H\'ajek projection. This method may also be applicable to other extreme-value problems. Some open questions are posed.

In this work we consider estimating the probability of many (possibly dependent) binary outcomes which is at the core of many applications, e.g., multi-level treatments in causal inference, demands for bundle of products, etc. Without further conditions, the probability distribution of an M dimensional binary vector is characterized by exponentially in M coefficients which can lead to a high-dimensional problem even without the presence of covariates. Understanding the (in)dependence structure allows us to substantially improve the estimation as it allows for an effective factorization of the probability distribution. In order to estimate the probability distribution of a M dimensional binary vector, we leverage a Bahadur representation that connects the sparsity of its coefficients with independence across the components. We propose to use regularized and adversarial regularized estimators to obtain an adaptive estimator with respect to the dependence structure which allows for rates of convergence to depend on this intrinsic (lower) dimension. These estimators are needed to handle several challenges within this setting, including estimating nuisance parameters, estimating covariates, and nonseparable moment conditions. Our main results consider the presence of (low dimensional) covariates for which we propose a locally penalized estimator. We provide pointwise rates of convergence addressing several issues in the theoretical analyses as we strive for making a computationally tractable formulation. We apply our results in the estimation of causal effects with multiple binary treatments and show how our estimators can improve the finite sample performance when compared with non-adaptive estimators that try to estimate all the probabilities directly. We also provide simulations that are consistent with our theoretical findings.

We introduce a new family of monoidal categories which are cyclotomic quotients of the nil-Brauer category. We construct a monoidal functor from the cyclotomic nil-Brauer category to another monoidal category constructed from singular Soergel bimodules of type D. We conjecture that our functor is an equivalence of categories. Although we can prove neither fullness nor faithfulness at this point, we are able to show that the functor induces an isomorphism at the level of Grothendieck rings. We compute these rings and their canonical bases over any field of characteristic different from 2, and give diagrammatic descriptions of the corresponding primitive idempotents.

This paper contains a study on stochastic Volterra integral equations with fuzzy sets-values and involving on a constant retardation. Moreover, the form of the equation is symmetric in the sense that fuzzy stochastic integrals are placed on both sides of the equation. We show that the considered initial value problem formulated in terms of symmetric fuzzy stochastic Volterra integral equation is well-posed. In particular, we show that there exists a unique solution and this solution depends continuously on the parameters of the equation. The results are achieved with the conditions of Lipschitz continuity of drift and diffusion coefficients, and continuity of kernels

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann spaces can be expressed as theta line bundles on Abelian varieties. We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get results which hold in higher dimension. We also give an application to asymptotics of restriction operators which arises from the asymptotic behavior of Bergman kernels for high tensor powers.

Starting with a smooth $n$-dimensional knot $K\subset\bS^{n+2}$, and a beaded $n$-dimensional necklace subordinated to $K$, we construct a wild knot with a Cantor set of wild points (\ie the knot is not locally flat in these points). The construction uses the conformal Schottky group acting on $\bS^{n+2}$, generated by inversions on the spheres which are the boundary of the``beads''. We show that if $K$ is a fibered knot then the wild knot is also fibered. We also study cyclic branched coverings along the wild knots.

Higher-order spectra (or polyspectra), defined as the Fourier Transform of a stationary process' autocumulants, are useful in the analysis of nonlinear and non Gaussian processes. Polyspectral means are weighted averages over Fourier frequencies of the polyspectra, and estimators can be constructed from analogous weighted averages of the higher-order periodogram (a statistic computed from the data sample's discrete Fourier Transform). We derive the asymptotic distribution of a class of polyspectral mean estimators, obtaining an exact expression for the limit distribution that depends on both the given weighting function as well as on higher-order spectra. Secondly, we use bispectral means to define a new test of the linear process hypothesis. Simulations document the finite sample properties of the asymptotic results. Two applications illustrate our results' utility: we test the linear process hypothesis for a Sunspot time series, and for the Gross Domestic Product we conduct a clustering exercise based on bispectral means with different weight functions.

We prove that a general three-dimensional quartic $V$ in the complex projective space ${\mathbb P}^4$, the only singularity of which is a double point of rank 3, is a birationally rigid variety. Its group of birational self-maps is, up to the finite subgroup of biregular automorphisms, a free product of 25 cyclic groups of order 2. It follows that the complement to the set of birationally rigid factorial quartics with terminal singularities is of codimension at least 3 in the natural parameter space.

We analyze a model of the evolution of a (solid) magnetoelastic material. More specifically, the model we consider describes the evolution of a compressible magnetoelastic material with a non-convex energy and coupled to a gradient flow equation for the magnetization in the quasi-static setting. The viscous dissipation considered in this model induces an extended material derivative in the magnetic force balance. We prove existence of weak solutions based on De Giorgi's minimizing movements scheme, which allows us to deal with the non-convex energy as well as the non-convex state space for the deformation. In the application of this method we rely on the fact that the magnetic force balance in the model can be expressed in terms of the same energy and dissipation potentials as the equation of motion, allowing us to model the functional for the discrete minimization problem based on these potentials.

We consider learning the dynamics of a typical nonholonomic system -- the rolling penny. A nonholonomic system is a system subject to nonholonomic constraints. Unlike holonomic constraints, a nonholonomic constraint does not define a submanifold on the configuration space. Therefore, the inverse problem of finding the constraints has to involve the tangent space. This paper discuss how to learn the dynamics, as well as the constraints for such a system given the data set of discrete trajectories on the tangent bundle $TQ$.

Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp\`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^n\mathrm{MA}(\varphi)$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czy\.{z}--Hiep. Under some natural conditions, we show that $\varphi_\theta$ is comparable to $\theta\varphi$ on much of $\Omega$; especially, it is bounded on the interior of $\{\theta = 0\}$. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on $\mathbb{R}^N$. We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.

Let $\mathbb{U}$ be a Banach Lie group and $S\subseteq \mathbb{U}$ an ad-bounded subset thereof, in the sense that there is a uniform bound on the adjoint operators induced by elements of $S$ on the Lie algebra of $\mathbb{U}$. We prove that (1) $S$-valued continuous maps from compact groups to $\mathbb{U}$ sufficiently close to being morphisms are uniformly close to morphisms; and (2) for any Lie subgroup $\mathbb{G}\le \mathbb{U}$ there is an identity neighborhood $U\ni 1\in \mathbb{U}$ so that $\mathbb{G}\cdot U\cap S$-valued morphisms (embeddings) from compact groups into $\mathbb{U}$ are close to morphisms (respectively embeddings) into $\mathbb{G}$. This recovers and generalizes results of Turing's to the effect that (a) Lie groups arbitrarily approximable by finite subgroups have abelian identity component and (b) if a Lie group is approximable in this fashion and has a faithful $d$-dimensional representation then it is also so approximable by finite groups with the same property. Another consequence is a strengthening of a prior result stating that finite subgroups in a Banach Lie group sufficiently close to a given compact subgroup thereof admit a finite upper bound on the smallest indices of their normal abelian subgroups (an approximate version of Jordan's theorem on finite subgroups of linear groups).

This paper shows how a family of function spaces (coined as Assiamoua spaces) plays a fundamental role in the Fourier analysis of vector-valued functions compact groups. These spaces make it possible to transcribe the classic results of Fourier analysis in the framework of analysis of vector-valued functions and vector measures. The construction of Sobolev spaces of vector-valued functions on compact groups rests heavily on the members of the aforementioned family.

A biclique in a graph $G$ is a complete bipartite subgraph (not necessarily induced), and the least positive integer $k$ for which the vertex set of $G$ can be partitioned into at most $k$ bicliques is the biclique vertex partition number $bp(G)$ of $G$. We prove that the inequality $reg(S(G))\geq |G|-bp(G)$ holds for every graph $G$, where $S(G)$ is the $1$-subdivision graph of $G$ and $reg(S(G))$ denotes the (Castelnuovo-Mumford) regularity of the graph $S(G)$. In particular, we show that the equality $reg(S(B))=|B|-bp(B)$ holds provided that $B$ is a chordal bipartite graph. Furthermore, for every chordal bipartite graph $B$, we prove that the independence complex of $S(B)$ is either contractible or homotopy equivalent to a sphere, and provide a polynomial time checkable criteria for when it is contractible, and describe the dimension of the sphere when it is not.

Tensor train (TT) decomposition represents an $N$-order tensor using $O(N)$ matrices (i.e., factors) of small dimensions, achieved through products among these factors. Due to its compact representation, TT decomposition has found wide applications, including various tensor recovery problems in signal processing and quantum information. In this paper, we study the problem of reconstructing a TT format tensor from measurements that are contaminated by outliers with arbitrary values. Given the vulnerability of smooth formulations to corruptions, we use an $\ell_1$ loss function to enhance robustness against outliers. We first establish the $\ell_1/\ell_2$-restricted isometry property (RIP) for Gaussian measurement operators, demonstrating that the information in the TT format tensor can be preserved using a number of measurements that grows linearly with $N$. We also prove the sharpness property for the $\ell_1$ loss function optimized over TT format tensors. Building on the $\ell_1/\ell_2$-RIP and sharpness property, we then propose two complementary methods to recover the TT format tensor from the corrupted measurements: the projected subgradient method (PSubGM), which optimizes over the entire tensor, and the factorized Riemannian subgradient method (FRSubGM), which optimizes directly over the factors. Compared to PSubGM, the factorized approach FRSubGM significantly reduces the memory cost at the expense of a slightly slower convergence rate. Nevertheless, we show that both methods, with diminishing step sizes, converge linearly to the ground-truth tensor given an appropriate initialization, which can be obtained by a truncated spectral method.

We compare the essential properties of projections in the L2 and L1 normed spaces by two methods: Projection operators and by minimization of the distance. In Euclidean geometry the orthogonality (L2- conjugacy) plays central role; while in L1 normed space L1- conjugacy (the sign function) plays central role. Furthermore, this fact appears in the Pythagorean Theorem and its Taxicab analogue. We also compare three singular value decompositions: SVD, Taxicab SVD and L1min-SVD.

We study a two-stage online reusable resource allocation problem over T days involving advance reservations and walk-ins. Each day begins with a reservation stage (Stage I), where reservation requests arrive sequentially. When service starts (Stage II), both reserved and walk-in customers arrive to check in and occupy resources for several days. Reserved customers can cancel without penalty before or during a confirmation call initiated by the decision maker (DM) before day's end. The DM must immediately accept or reject each booking or check-in request, potentially overbooking by accepting more reservations than capacity. An overbooking loss occurs if a reserved customer's check-in is rejected in Stage II; a reward is obtained for each occupied resource unit daily. Our goal is to develop an online policy that controls bookings and check-ins to maximize total revenue over the T-day horizon. We show that due to cancellation uncertainties and complex correlations between occupancy durations, any online policy incurs a regret of \Omega(T) compared to the offline optimal policy when the \textit{busy season} assumption does not hold. To address this, we introduce decoupled adaptive safety stocks, which use only single-day information to hedge against overbooking risks and reduce resource idling. Under the busy season condition, our policy decouples the overall offline optimal into single-day offline optimal policies. Consequently, the regret between our policy and the offline optimal decays exponentially with the time between the confirmation call and day's end, suggesting the DM can delay confirmation calls while maintaining near-optimal performance. We validate our algorithm through sythetic experiments and empirical data from an Algarve resort hotel.

The Stefan problem with surface tension is well known to exhibit discontinuities in the associated moving aggregate (i.e., in the domain occupied by the solid), whose structure has only been understood under translational or radial symmetry so far. In this paper, we derive an auxiliary partial differential equation of second-order hyperbolic type, referred to as the cascade equation, that captures said discontinuities in the absence of any symmetry assumptions. Specializing to the one-phase setting, we introduce a novel (global) notion of weak solution to the cascade equation, which is defined as a limit of mean-field game equilibria. For the spatial dimension two, we show the existence of such a weak solution and prove a natural perimeter estimate on the associated moving aggregate.

A degenerate wave equation with time-varying delay in the boundary control input is considered. The well-posedness of the system is established by applying the semigroup theory. The boundary stabilization of the degenerate wave equation is concerned and the uniform exponential decay of solutions is obtained by combining the energy estimates with suitable Lyapunov functionals and an integral inequality under suitable conditions.

We introduce the concept of a pants decomposition for a rose, where a rose is defined as a wedge sum of finitely many circles, and construct the corresponding pants graph of a rose. A pants decomposition of a rose leads to the formation of a simplicial graph, referred to as the pants graph of a rose, consisting of all possible pants decompositions. Recall that the outer automorphism group of a free group is isomorphic to the group of homotopy classes of homotopy equivalences of a rose. The natural isometric action of this group on the pants graph induces a coarsely Lipschitz orbit map. Additionally, we construct a coarsely Lipschitz map from the pants graph to the free splitting complex. These results imply that the pants graph of a rose is both connected and unbounded.

We show the vanishing of the first direct image of the structure sheaf of a normal excellent scheme $X$ which is mapped properly and birationally over an excellent regular scheme of any dimension. On the other hand, for any dimension greater than two, we show examples of a proper birational morphism from a normal and Cohen-Macaulay scheme to a regular scheme such that the second direct image does not vanish and has an isolated support.

Recent experiments have demonstrated the feasibility of storing digital information in macromolecules such as DNA and protein. However, the DNA storage channel is prone to errors such as deletions, insertions, and substitutions. During the synthesis and reading phases of DNA strings, many noisy copies of the original string are generated. The problem of recovering the original string from these noisy copies is known as sequence reconstruction. A key concept in this problem is the error ball, which is the set of all possible sequences that can result from a limited number of errors applied to the original sequence. Levenshtein showed that the minimum number of noisy copies required for a given channel to recover the original sequence is equal to one plus the maximum size of the intersection of two error balls. Therefore, deriving the size of the error ball for any channel and any sequence is essential for solving the sequence reconstruction problem. In DNA storage systems, multiple types of errors such as deletion, insertion and substitution in a string could occur simultaneously. In this work, we aim to derive the size of the error ball for channels with multiple types of errors and at most three edits. Specifically, we consider the channels with single-deletion double-substitution, single-deletion double-insertion and single-insertion single-substitution errors.

We study a pair consisting of a smooth variety over a field of positive characteristic and a multi-ideal with a real exponent. We prove the finiteness of the set of minimal log discrepancies for a fixed exponent if the dimension is less than or equal to three. We also prove that the set of log canonical thresholds (lct for short) of ideals on a smooth variety in positive characteristic is contained in the set of lct's of ideals on a smooth variety over C, assuming the dimension is less than or equal to three. Under the same dimension assumption, it follows that the accumulation points of log canonical thresholds are rational. Our proofs also show the same statements for the higher dimensional case if all such pairs admit log resolutions by a composite of blow-ups by smooth centers.

An elliptic pair $(X, C)$ is a generalization of a rational elliptic fibration $X \to \mathbb{P}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an irreducible curve contained in the smooth locus of $X,$ with $p_a(C)=1$ and $C^2=0.$ These naturally arise as blowups $X:=\text{Bl}_e(\mathbb{P}_\Delta)$ of projective toric surfaces, whose Newton polygon is elliptic. The order of $\mathcal{O}(C)|_C$ in $\text{Pic}^0(C)$ gives a quantitative way to check if $X$ is an elliptic fibration, which is equivalent to finiteness of the order. We call $\Delta$ a Lang-Trotter polygon when this order is infinite, in which case $\overline{\text{Eff}(\text{Bl}_e(\mathbb{P}_\Delta))}$ is non-polyhedral. The paper \cite{lizzie} shows there are exactly $3$ elliptic triangles up to $\text{SL}_2(\mathbb{Z}),$ none of which is Lang-Trotter. The paper \cite{jenia_blowup} gives an infinite family of Lang-Trotter pentagons and heptagons, and various examples of other polygons when $\rho(\mathbb{P}_\Delta)>2.$ Remark 4.7 in the paper asks if any Lang-Trotter quadrilaterals exist, and we answer this in the negative by studying the curves in the Zariski Decomposition of $K_X+C.$

In the present paper, we first revisit the volatility estimation approach proposed by N. Kunitomo and S. Sato, and second, we show that the volatility estimator proposed by P. Malliavin and M.E. Mancino can be understood in a unified way by the approach. Third, we introduce an alternative estimator that might overcome the inconsistency caused by the microstructure noise of the initial observation.

In the spirit of the ground-breaking result of Bourgain--Brezis--Mironescu, we establish some characterizations of Sobolev functions in metric measure spaces including fractals like the Vicsek set, the Sierpi\'{n}ski gasket and the Sierpi\'{n}ski carpet. As corollaries of our characterizations, we present equivalent norms on the Korevaar--Schoen--Sobolev space, and show that the domain of a $p$-energy form is identified with a Besov-type function space under a suitable $(p,p)$-Poincar\'e inequality, capacity upper bound and the volume doubling property.

The block count of an equivalence $\mu\in$ Equ$(A)$ is the number blnum$(\mu)$ of blocks of (the partition corresponding to) $\mu$. We say that $X=\{\mu_1,\mu_2,\mu_3,\mu_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(\mu_{i+1})$ = blnum$(\mu_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices.

We present a computer assisted proof for a result concerning a three player betting game, introduced by Angel and Holmes. The three players start with initial capital $x, y, z > 0$ respectively. At each step of this game two players are selected at random to bet on the outcome of a fair coin toss, with the size of the bet being the largest possible, namely the total capital held by the poorer of the two players at that time. The main quantity of interest is the probability of player 1 being eliminated (reaching 0 capital) first. Angel and Holmes have shown that this probability is not monotone decreasing as a function of the initial capital $x$ of player 1. They conjecture that if $x < y < z$ then player 1 would be better off (less likely to be eliminated first) by swapping their capital with another player. In this paper we present a computer-assisted proof of this conjecture. To achieve this, we introduce the theoretical framework MeshItUp, and then perform a two-stage reduction to make MeshItUp computationally feasible, through the use of mixed-integer programming.

In this work, we derived the three-dimensional scaled boundary finite element formulation for thermal conduction problems. By introducing Wachspress shape functions, we proposed a novel polyhedral scaled boundary finite element method (PSBFEM) to address thermal conduction problems. The proposed method effectively addresses the challenges associated with complex geometries by integrating the polyhedral mesh and the octree mesh. The presented formulation handles both steady-state and transient thermal conduction analyses. Through a series of numerical examples, the accuracy and convergence of the proposed method were validated. The results demonstrate that mesh refinement leads to superior accuracy for the PSBFEM compared to the FEM. Moreover, Polyhedral elements provide an effective and efficient approach for complex simulations that substantially reduces computational costs.

The derived McKay correspondence conjecture says that there is an equivalence of triangulated categories between the bounded derived categories of commutative and non-commutative crepant resolutions of a Gorenstein singularity. We will prove that this derived equivalence extends between the semi-universal non-commutative deformations of the commutative and the non-commutative crepant resolutions of a toric surface singularity.

Static manifolds with boundary were recently introduced to mathematics. This kind of manifolds appears naturally in the prescribed scalar curvature problem on manifolds with boundary, when the mean curvature of the boundary is also prescribed. They are also interesting from the point of view of General Relativity. For example, the (time-slice of the) photon sphere on the Riemannian Schwarzschild manifold splits it into static manifolds with boundary. In this paper we prove a number of theorems, which relate the topology and geometry of a given static manifold with boundary to some properties of the zero-level set of its potential (such as connectedness and closedness). Also, we characterize the round ball in the Euclidean 3-space with standard potential as the only scalar-flat static manifold with mean-convex boundary, whose zero-level set of the potential has Morse index one. This result follows from a general isoperimetric inequality for 3-dimensional static manifolds with boundary, whose zero-level set of the potential has Morse index one. Finally, we prove some uniqueness theorem for the domains bounded by the photon sphere on the Riemannian Schwarzschild manifold.

The paper explores the effect of powerful class of Sylow $p$-subgroups of a given finite group on control of transfer or fusion. We also find an explicit bound for the $p$-length of a $p$-solvable group in terms of the poweful class of a Sylow $p$-subgroup.

In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method utilizes Hutchinson's stochastic trace estimator together with the Lanczos algorithm to approximate quadratic forms involving spectral projectors. Instead of focusing on determining the gap between two particular consecutive eigenvalues of $A$, we aim to find all gaps that are wider than a specified threshold. By examining the problem from this perspective, and thoroughly analyzing both the Hutchinson and the Lanczos components of the algorithm, we obtain error bounds that allow us to determine the numbers of Hutchinson's sample vectors and Lanczos iterations needed to ensure the detection of all gaps above the target width with high probability. In particular, we conclude that the most efficient strategy is to always use a single random sample vector for Hutchinson's estimator and concentrate all computational effort in the Lanczos algorithm. Our numerical experiments demonstrate the efficiency and reliability of this approach.

This paper studies the number of centers and limit cycles of the family of planar quartic polynomial vector fields that has the invariant algebraic curve $(4x^2-1)(4y^2-1)=0.$ The main interest for this type of vector fields comes from their appearance in some mathematical models in Game Theory composed by two players. In particular, we find examples with five nested limit cycles surrounding the same singularity, as well as examples with four limit cycles formed by two disjoint nests, each one of them with two limit cycles. We also prove a Berlinski\u \i's type result for this family of vector fields.

We define an independence system associated with simple graphs. We prove that the independence system is a matroid for certain families of graphs, including trees, with bases as minimal resolving sets. Consequently, the greedy algorithm on the matroid can be used to find the minimum-cost resolving set of weighted graphs, wherein the independent system is a matroid. We also characterize hyperplanes of the matroid for trees and prove that its dual matroid is loop-free.

This paper demonstrates how to explicitly construct a bidiagonal factorization of the banded recurrence matrix that appears in mixed multiple orthogonality on the step-line in terms of the coeffcients of the mixed multiple orthogonal polynomials. The construction is based on the \(LU\) factorization of the moment matrix and Christoffel transformations applied to the matrix of measures and the associated mixed multiple orthogonal polynomials.

We revisit FedExProx - a recently proposed distributed optimization method designed to enhance convergence properties of parallel proximal algorithms via extrapolation. In the process, we uncover a surprising flaw: its known theoretical guarantees on quadratic optimization tasks are no better than those offered by the vanilla Gradient Descent (GD) method. Motivated by this observation, we develop a novel analysis framework, establishing a tighter linear convergence rate for non-strongly convex quadratic problems. By incorporating both computation and communication costs, we demonstrate that FedExProx can indeed provably outperform GD, in stark contrast to the original analysis. Furthermore, we consider partial participation scenarios and analyze two adaptive extrapolation strategies - based on gradient diversity and Polyak stepsizes - again significantly outperforming previous results. Moving beyond quadratics, we extend the applicability of our analysis to general functions satisfying the Polyak-Lojasiewicz condition, outperforming the previous strongly convex analysis while operating under weaker assumptions. Backed by empirical results, our findings point to a new and stronger potential of FedExProx, paving the way for further exploration of the benefits of extrapolation in federated learning.

We perform a systematic study of the base change conductor for Jacobians. Through the lens of intersection theory and Deligne's Riemann-Roch theorem, we present novel computational approaches for both the tame and wild parts of the base change conductor. Our key results include a general formula of the tame part, as well as a computation of the wild part in terms of Galois quotients of semistable models of the curves. We treat in detail the case of potential good reduction when the quotient only has weak wild quotient singularities, relying on recent advances by Obus and Wewers.

The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. M.Viale proved that Martin's Maximum$^{++}$ together with the assumption that there are class many Woodin cardinals implies $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{\Sigma_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ for a generic $\mathbb{G}$ on any stationary preserving $\mathbb{P}$ which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of $(\mathcal{P},\mathcal{H}(\kappa))_{\Sigma_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(\kappa)$), and the existence of the tightly $\mathcal{P}$-Laver-generically huge cardinal. While under ``$\mathcal{P}=$ all stationary preserving posets'', our results are not very much more than Viale's theorem, for other classes of posets, ``$\mathcal{P}=$ all proper posets'' or ``$\mathcal{P}=$ all ccc posets'', for example, our theorems are not at all covered by his theorem. The assumptions (and hence also the conclusion) of Viale's Theorem are compatible with the Ground Axiom. In contrast, we show that the assumptions of our theorems (for most of the common settings of $\mathcal{P}$ and with a modification of the large cardinal property involved) imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom $(\mathcal{P},\mathcal{H}(\kappa))_\Gamma$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(\kappa))_\Gamma$ for $\Gamma=\Pi_2$, $\Sigma_2$, $\Pi_3$.

We define the preprojective algebra of a finite EI quiver. We prove that it is isomorphic to a centain tensor algebra. For a finite EI quiver of Cartan type, we prove that the corresponding preprojective algebra is isomorphic to the generalized preprojective algebra.

We propose a simple definition of a Born geometry in the framework of K\"unneth geometry. While superficially different, this new definition is equivalent to the known definitions in terms of para-quaternionic or generalized geometries. We discuss integrability of Born structures and their associated connections. In particular we find that for integrable Born geometries the Born connection is obtained by a simple averaging under a conjugation from the K\"unneth connection. We also give examples of integrable Born geometries on nilmanifolds.

We show how Bailey pairs can be used to give a simple proof of an identity of Chern, Li, Stanton, Xue, and Yee. The same method yields a number of related identities as well as false theta companions.

Using an approach developed by Melrose to study the geometry at infinity of the Nakajima metric on the reduced Hilbert scheme of points on $\mathbb{C}^2$, we show that the Nakajima metric on a quiver variety is quasi-asymptotically conical (QAC) whenever its defining parameters satisfy an appropriate genericity assumption. As such, it is of bounded geometry and of maximal volume growth. Being QAC is one of two main ingredients allowing us to use the work of Kottke and the second author to compute its reduced $L^2$-cohomology and prove the Vafa-Witten conjecture. The other is a spectral gap for the Hodge-deRham operator associated with an exact $3$-Sasakian wedge metric that we obtain using the Weitzenb\"ock formula of Semmelmann and Weingart on quaternionic-K\"ahler manifolds. This yields in particular a generalization and a different proof of the vanishing theorem in cohomology originally obtained by Galicki and Salamon for closed $3$-Sasakian manifolds.

We study the theory of equations in one variable over polyhedral semirings. The article revolves around a notion of solution to a polynomial equation over a polyhedral semiring. Our main results are a characterisation of local solutions in terms of the coefficients, a local-global principle, and the basics of multiplicity and discriminants. Our primary sources of motivation are tropical geometry and the theory of exceptional points in non-Hermitian physics.

We investigate coding schemes that map source symbols into multisets of an alphabet set. Such a formulation of source coding is an alternative approach to the traditional framework and is inspired by an object tracking problem over proximity sensor networks. We define a \textit{multiset combinatorial Gray code} as a mulitset code with fixed multiset cardinality that possesses combinatorial Gray code characteristic. For source codes that are organized as a grid, namely an integer lattice, we propose a solution by first constructing a mapping from the grid to the alphabet set, the codes are then defined as the images of rectangular blocks in the grid of fixed dimensions. We refer to the mapping as a \textit{color mapping} and the code as a \textit{color multiset code}. We propose the idea of product multiset code that enables us to construct codes for high dimensional grids based on 1-dimensional (1D) grids. We provide a detailed analysis of color multiset codes on 1D grids, focusing on codes that require the minimal number of colors. To illustrate the application of such a coding scheme, we consider an object tracking problem on 2D grids and show its efficiency, which comes from exploiting transmission parallelism. Some numerical results are presented to conclude the paper.

We examine the distribution and popularity of different parameters (such as the number of descents, runs, valleys, peaks, right-to-left minima, and more) on the sets of increasing and flattened permutations. For each parameter, we provide an exponential generating function for its corresponding distribution and popularity.Additionally, we present one-to-one correspondences between these permutations and some classes of simpler combinatorial objects.

We study standard processes with no negative jumps under the entrance boundary condition. Similarly to one-dimensional diffusions, we show that the process can be made into a Feller process by attaching the boundary point to the state space and extending the transition semigroup. We investigate the spectrum of the generator in detail via the scale function, characterizing it as the zeros of an entire function. As an application, we prove that under the strong Feller property, the convergence to the Yaglom limit of the process killed at the boundary is exponentially fast.

Let $X$ be a compact locally symmetric space associated to $SL_n(\mathbb{R})$ and $Y \subset X$ a maximal flat submanifold, not necessarily closed. Using a Euclidean approximation, we give an upper bound in the spectral aspect for Maass forms integrated against a smooth cutoff function on $Y$ .

It is shown that van Suijlekom's technique of imposing a set of conditions on operator system spectral triples ensures Gromov-Hausdorff convergence of sequences of sets of unital completely positive maps (equipped with the BW-topology which is metrizable). This implies that even when only a part of the spectrum of the Dirac operator is available together with a certain truncation of the $C^*$-algebra, information about the geometry can be extracted.

We show that the anti-canonical bundle of any $\mathbb Q$-factorial surface is numerically effective if and only if it is pseudo-effective. To prove this, we establish a numerical non-vanishing theorem for surfaces polarized with pseudo-effective divisors. The latter answers a question of C. Fontanari.

We study the reflexivity and strong subdifferentiability within the framework of group invariant mappings. We show that a Banach space is G-reflexive if the norm of its dual is G-strong subdifferentiable. To do this, we extend numerous classical concepts in functional analysis such as weak and weak-star topologies, the polar of a set, duality mapping, to the framework of group invariant mappings. We also extend many classical results in functional analysis including Banach-Alaoglu-Bourbaki's theorem, James' theorem, Moreau's maximum formula, and Krein-Smulian's theorem, to this context. To conclude, we provide an application of these new results by providing sufficient conditions to ensure the existence of closed Banach spaces inside the set of norm-attaining functionals of a Banach space.

We consider one-parameter families of smooth circle cocycles over an ergodic transformation in the base, and show that their rotation numbers must be log-H\"older regular with respect to the parameter. As an immediate application, we get a dynamical proof of 1D version of the Craig-Simon theorem that establishes that the integrated density of states of an ergodic Schr\"odinger operator must be log-H\"older.

We prove a quenched version of the large deviation principle for Birkhoff-like sums along a sequence of random quantum measurements driven by an ergodic process. We apply the result to the study of entropy production in the two-time measurement framework.

This is my (reviewed) PhD manuscript. It contains 6 Chapters, which contains mostly already published work, except for Chapter 5 which is new. Chapter 1 introduce basic notions on fractal geometry: the Fourier dimension, the thermodynamical formalism and additive combinatorics. Chapter 2 is a generalized version of arXiv:2211.08088. Chapter 3 is a slightly upgraded version of arXiv:2112.00701. Chapter 4 is a generalized version of arXiv:2301.10623. Chapter 5 and Chapter 4 together contains the first proof of the positivity of the Fourier dimension for basic sets of nonlinear, area-preserving, smooth Axiom A diffeomorphisms on surfaces. This is possible by adapting some ideas that can be found in Tsujii-Zhang's work arXiv:2006.04293. Some future work still needs to be done to prove that the nonlinearity conditions are generic in this setting: this should become a proper research article in the future. Chapter 6 is a slighty upgraded version of arXiv:2307.10755.

From the classical theory of Lie algebras, it is well-known that the bilinear form $B(X,Y)={\rm tr}(XY)$ defines a non-degenerate scalar product on the simple Lie algebra ${\mathfrak{sl}}(n,{\mathbb R})$. Diagonalizing the Gram matrix $Gr$ associated with this scalar product we find a basis of ${\mathfrak{sl}}(n,{\mathbb R})$ of eigenvectors of $Gr$ which produces a family of bracket generating distributions on ${\rm SL}(n,{\mathbb R})$. Consequently, the bilinear form $B$ defines sub-pseudo-Riemannian structures on these distributions. Each of these geometric structures naturally carries a metric quadratic Hamiltonian. In the present paper, we construct in detail these manifolds, study Poisson-commutation relations between different Hamiltonians, and present some explicit solutions of the corresponding Hamiltonian system for $n=2$.

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

We prove the global well-posedness of the dynamical sine-Gordon model up to the third threshold, i.e., for parameters $\beta^2 < 6\pi$. The key novelty in our approach is the introduction of the so-called resonant equation, whose solution is entirely deterministic and completely captures the size of the solution to the dynamical sine-Gordon model. The probabilistic fluctuations in the dynamical sine-Gordon model are then controlled using uniform estimates for modified stochastic objects.

Enumerating the extremal submodular functions defined on subsets of a fixed base set has only been done for base sets up to five elements. This paper reports the results of attempting to generate all such functions on a six-element base set. Using improved tools from polyhedral geometry, we have computed 360 billion of them, and provide the first reasonable estimate of their total number, which is expected to be between 1,000 and 10,000 times this number. The applied Double Description and Adjacency Decomposition methods require an insertion order of the defining inequalities. We introduce two novel orders, which speed up the computations significantly, and provide additional insight into the highly symmetric structure of submodular functions. We also present an improvement to the combinatorial test used as part of the Double Description method, and use statistical analyses to estimate the degeneracy of the polyhedral cone used to describe these functions. The statistical results also highlight the limitations of the applied methods.

We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding algorithms. Our contributions are as follows. 1. Explicit Near-Optimal Linear Time Uniquely Decodable Codes: We construct a family of explicit $\mathbb{F}_2$-linear codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of correcting $e$ errors and $s$ erasures whenever $2e + s < (1 - \varepsilon)n$ in linear-time. 2. Explicit Near-Optimal List Decodable Codes: We construct a family of explicit list decodable codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of list decoding from $1-\varepsilon$ fraction of errors with a list size $L = \exp\exp\exp(\log^{\ast}n)$ in polynomial time. 3. List Decodable Code with Near-Optimal List Size: We construct a family of explicit list decodable codes with an optimal list size of $O(1/\varepsilon)$, albeit with a suboptimal rate of $O(\varepsilon^2)$, capable of list decoding from $1-\varepsilon$ fraction of errors in polynomial time. Furthermore, we introduce a new combinatorial object called multi-set disperser, and use it to give a family of list decodable codes with near-optimal rate $\frac{\varepsilon}{\log^2(1/\varepsilon)}$ and list size $\frac{\log^2(1/\varepsilon)}{\varepsilon}$, that can be constructed in probabilistic polynomial time and decoded in deterministic polynomial time. We also introduce new decoding algorithms that may prove valuable for other graph-based codes.

In this paper we provide a complete characterisation of coisotropic embeddings of precosymplectic manifolds into cosymplectic manifolds. This result extends a theorem of Gotay about coisotropic embeddings of presymplectic manifolds. We also extend to the cosymplectic case some results of A. Weinstein which generalise the Darboux theorem. While symplectic geometry is the natural framework for developing Hamiltonian mechanics, cosymplectic geometry is the corresponding framework for time-dependent Hamiltonian mechanics. The motivation behind proving this theorem is to generalise known results for symplectic geometry to cosymplectic geometry, so that they can be used to study time-dependent systems, for instance for the regularization problem of singular Lagrangian systems.

This paper presents and analyzes two robust, efficient, and optimally accurate fully discrete finite element algorithms for computing the parameterized Navier-Stokes Equations (NSEs) flow ensemble. The timestepping algorithms are linearized, use the backward-Euler method for approximating the temporal derivative, and Ensemble Eddy Viscosity (EEV) regularized. The first algorithm is a coupled ensemble scheme, and the second algorithm is decoupled using projection splitting with grad-div stabilization. We proved the stability and convergence theorems for both algorithms. We have shown that for sufficiently large grad-div stabilization parameters, the outcomes of the projection scheme converge to the outcomes of the coupled scheme. We then combine the Stochastic Collocation Methods (SCMs) with the proposed two Uncertainty Quantification (UQ) algorithms. A series of numerical experiments are given to verify the predicted convergence rates and performance of the schemes on benchmark problems, which shows the superiority of the splitting algorithm.

We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in Carlsson-Chou (2024+). Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_\mu[X;q]$, due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of $R_{\mu}$ with respect to a Young subgroup $S_\gamma$ is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of $R_{\mu}$ is given by the catabolizability formula for $\tilde{H}_\mu[X;q]$.

This paper presents a convolution tensor decomposition based model reduction method for solving the Allen-Cahn equation. The Allen-Cahn equation is usually used to characterize phase separation or the motion of anti-phase boundaries in materials. Its solution is time-consuming when high-resolution meshes and large time scale integration are involved. To resolve these issues, the convolution tensor decomposition method is developed, in conjunction with a stabilized semi-implicit scheme for time integration. The development enables a powerful computational framework for high-resolution solutions of Allen-Cahn problems, and allows the use of relatively large time increments for time integration without violating the discrete energy law. To further improve the efficiency and robustness of the method, an adaptive algorithm is also proposed. Numerical examples have confirmed the efficiency of the method in both 2D and 3D problems. Orders-of-magnitude speedups were obtained with the method for high-resolution problems, compared to the finite element method. The proposed computational framework opens numerous opportunities for simulating complex microstructure formation in materials on large-volume high-resolution meshes at a deeply reduced computational cost.

We prove, for a class of first order differential operators that contains the Stein-Weiss, Dirac and Penrose twistor operators, a family of Kato inequalities that interpolates between the classical and the refined Kato. For the Hodge-de-Rham operator we get a more detailed result. As a corollary, we get various Kato inequalities from the literature.

In this paper, we compute the Morse index of rotationally symmetric minimal Y-singular surfaces under assumption that the Y-singularities form a single circle. This computation is carried out by utilizing information from two simpler problems: the first deals with the fixed boundary problem on singularities, and the second focuses on the Dirichlet-to-Neumann map associated with the stability operator. Notably, our findings reveal that the index of the Y -catenoid is one.

In this note, we organize minimal annuli in a slab based on the winding number of the circles that foliate them and study the area of minimal annuli with given winding number. Specifically, we deduce some results regarding the convexity of the length function of corresponding level curves, and apply them to estimate the area of the annui by comparing to the area of waist of covers of catenoids.

We prove that the space $\mathcal{H}_\infty$ of framed infinite volume hyperbolic $3$-manifolds is connected but not path connected. Two proofs of connectivity of this space, which is equipped with the geometric topology, are given, each utilizing the density theorem for Kleinian groups. In particular, we decompose $\mathcal{H}_\infty$ into a union of leaves, each leaf corresponding to the set of framings on a fixed manifold, and construct a leaf which is dense in $\mathcal{H}_\infty$. Examples of paths in $\mathcal{H}_\infty$ are discussed, and machinery for analyzing general paths is developed, allowing for a description of paths of convex cocompact framed manifolds. The discussion of paths culminates in describing an infinite family of non-tame hyperbolic $3$-manifolds whose corresponding leaves are path components of $\mathcal{H}_\infty$, which establishes that $\mathcal{H}_\infty$ is not path connected.

The category $\bcalNT$ was introduced in \cite{Lobos2} in order to provide a structural setting to the study of the many Gelfand-Tsetlin subalgebras appearing in the context of the diagrammatic Soergel category of Elias and Williamson \cite{EW}. The category $\bcalNT$ have as objects, all that we called \emph{nil graded algebras associated to a triangular matrix} and as morphisms, all the \emph{preserving degree} homomorphisms of graded algebras between them. In this article we develop a series of \emph{isomorphism criteria} on $\bcalNT$ that we used later to find out relevant information on the Gelfand-Tsetlin subalgebras of the diagrammatic Soergel category.

We establish a direct log-epiperimetric inequality for Yang--Mills fields in arbitrary dimension and we leverage on it to prove uniqueness of tangent cones with isolated singularity for energy minimizing Yang--Mills fields and $\omega$-ASD connections (where $\omega$ is not necessarily closed) satisfying some suitable regularity assumptions. En route to this we establish a Luckhaus type lemma for Yang--Mills connections to exclude curvature concentration along blow-up sequences.

[Connelly and Servatius, 1994] shows the difficulty of properly defining n-th order rigidity and flexiblity of a bar-and-joint framework for higher order (n >= 3) through the introduction of a cusp mechanism. The author proposes a "proper" definition of the order of rigidity by the order of elongation of the bars with respect to the arclength along the path in the configuration space. We show that the classic definition using formal n-th derivative of the length constraint is a sufficient condition for the n-th flexiblity in the proposed definition and also a necessary condition only for n = 1, 2.

We study the Dirichlet energy of some smooth maps appearing in a collapsing family of hyper-K\"ahler metrics on the $K3$ surface constructed by Foscolo. We introduce an invariant for homotopy classes of smooth maps from the $K3$ surface with a hyper-K\"ahler metric to a flat Riemannian orbifold of dimension $3$, then show that it gives a lower bound of the energy. Moreover, we show that the ratio of the energy to the invariant converges to $1$ for Foscolo's collapsing families.

We show that iterating Nash blowups resolve the singularities of normal toric surfaces satisfying the following property: the minimal generating set of the corresponding semigroup is contained in one or two segments. We also provide examples with an arbitrary number of segments for which the same result holds.

Solovay reducibility $\redsolovay$ was introduced by Robert M. Solovay in 1975 via translation functions on rationals. In 2022, its totalized version $\redsolovaytotal$ (i.e., Solovay reducibility via a total function on rationals) has been examined by Merkle and Titov (arXiv:2407.14869). In 2020, Kumabe, Miyabe, Mizusawa and Suzuki (arXiv:1903.08625) have discovered that Solovay reducibility can be characterized on left-c.e.\ reals using the notion of a translation function on reals. In 2024, Kumabe, Miyabe, and Suzuki (DOI: 10.3233/COM-230486) have introduced a new reducibility $\redclopen$ on all reals, that uses the notion of a translation function on reals, and its totalized version $\redcllocal$. %They have also shown that $\redcllocal$ implies $\redclopen$, wherein the converse is not true even for left-c.e. reals. In this work, we show that $\redsolovayreal$ implies $\redclopen$ and $\redsolovaytotal$ implies $\redcllocal$ on all reals.

In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.

We assign a new polynomial to any checkerboard-colorable 4-valent virtual graph in terms of its Euler circuit expansion. This provides a new combinatorial formulation of the Kauffman-Jones polynomial for checkerboard-colorable virtual links.

We present some aspects of the theory of finite element exterior calculus as applied to partial differential equations on manifolds, especially manifolds endowed with an approximate metric called a Regge metric. Our treatment is intrinsic, avoiding wherever possible the use of preferred coordinates or a preferred embedding into an ambient space, which presents some challenges but also conceptual and possibly computational advantages. As an application, we analyze and implement a method for computing an approximate Levi-Civita connection form for a disc whose metric is itself approximate.

In this paper, we propose a distributed first-order algorithm with backtracking linesearch for solving multi-agent minimisation problems, where each agent handles a local objective involving nonsmooth and smooth components. Unlike existing methods that require global Lipschitz continuity and predefined stepsizes, our algorithm adjusts stepsizes using distributed linesearch procedures, making it suitable for problems where global constants are unavailable or difficult to compute. The proposed algorithm is designed within an abstract linesearch framework for a primal-dual proximal-gradient method to solve min-max convex-concave problems, enabling the consensus constraint to be decoupled from the optimisation task. Our theoretical analysis allows for gradients of functions to be locally Lipschitz continuous, relaxing the prevalent assumption of globally Lipschitz continuous gradients.

We study a variant of the Erd\H{o}s Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erd\H{o}s-R\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We show that when $n\gg k^{2}s$ and $p$ is not too small, with high probability, the maximum number of edges in a sub-hypergraph of $\mathcal{K}_p(n,k)$ with matching number $s$ is obtained by the trivial sub-hypergraphs, i.e. the sub-hypergraph consisting of all edges containing at least one vertex in a fixed set of $s$ vertices.

This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the knot type via its knot-type knotoids. We prove that the pure knotoids in the knotoid spectra of a knot, which are individually agnostic of the knot type, can distinguish knots of Gordian distance greater than one. We also prove that the neighborhood of some embeddings of the unknot can be distinguished from any embedding of any non-trivial knot that satisfies the cosmetic crossing conjecture. Topological invariants of knots can be extended to their open curve neighborhood to define continuous functions in the neighborhood of knots. We discuss their properties and prove that invariants in the neighborhood of knots may be able to distinguish more knots than their application to the knots themselves. For example, we prove that an invariant of knots that fails to distinguish mutant knots (and mutant knotoids), can distinguish them by their neighborhoods, unless it also fails to distinguish non-mutant pure knotoids in their spectra. Studying the neighborhood of knots opens the possibility of answering questions, such as if an invariant can detect the unknot, via examining possibly easier questions, such as whether it can distinguish height one knotoids from the trivial knotoid.

Let F be a complete discretely valued field with residue field a global field or a local field with no real orderings. Let G be an absolutely simple simply connected group of outer type A_n. If 2 and the index of the underlying algebra of G are coprime to the characteristic of the residue field of F, then we prove that the Rost invariant map from the first Galois cohomology set of G to the degree three Galois cohomology group is injective. Let L be the function field of a curve over a local field K and G an absolutely simple simply connected linear algebraic group over L of classical type. Suppose that the characteristic of the residue field of K is a good prime for G. As a consequence of our result and some known results we conclude that the Rost invariant of G is injective.

We present a unifying approach to studying the replica symmetric solution in general diluted spin glass models on random $p$-uniform hypergraphs with sparsity parameter $\alpha$. Our result shows that there exist two key regimes in which the model exhibits replica symmetry and the free energy can be explicitly represented as the evaluation of an energy functional at the unique fixed point of a recursive distributional equation. One is called the high temperature regime, where the temperature and the sparsity parameter are essentially inversely proportional to each other; the other is the subcritical regime defined as $\alpha p (p-1)\leq 1$. In particular, the fact that the second regime is independent of the temperature parameter further allows us to deduce an analogous representation of the ground state energy in the subcritical regime. Along the way, we revisit several well-known formulas and also derive new ones for the free and ground state energies in the constraint satisfaction problem, Potts model, XY model, and continuous hardcore model.

The scheduling problem in additive manufacturing is receiving increasing attention; however, few have considered the effect of scheduling decisions on machine energy consumption. This research focuses on the nesting and scheduling problem of a single selective laser melting (SLM) machine to reduce total energy consumption. Based on an energy consumption model, a nesting and scheduling problem is formulated, and a mixed integer linear programming model is proposed. This model simultaneously determines part-to-batch assignments, part placement in the batch, and the choice of build orientation to reduce the total energy consumption of the SLM machine. The energy-saving potential of the model is validated through numerical experiments. Additionally, the effect of the number of alternative build orientations on energy consumption is explored.

Given suitable functions $V, \Psi:[0,\infty) \to [0,\infty)$, we obtain necessary and sufficient conditions on $V,\Psi$ for the existence of a metric measure space and a symmetric diffusion process that satisfies sub-Gaussian heat kernel estimates with volume growth profile $V$ and escape time profile $\Psi$. We prove sufficiency by constructing a new family of diffusions. Special cases of this construction also leads to a new family of infinite graphs whose simple random walks satisfy sub-Gaussian heat kernel estimates with prescribed volume growth and escape time profiles. In particular, these random walks on graphs generalizes earlier results of Barlow who considered the case $V(r)=r^\alpha$ and $\Psi(r)=r^\beta$ (Rev Mat Iberoam 2004). The family of diffusions we construct have martingale dimension one but can have arbitrarily high spectral dimension. Therefore our construction shows the impossibility of obtaining non-trivial \emph{lower} bounds on martingale dimension in terms of spectral dimension which is in contrast with upper bounds on martingale dimension using spectral dimension obtained by Hino (Probab Theory Relat Fields 2013).

Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and a semilinear wave equation on the irregular domain.

In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$. This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from $29$ to $7$ in $\mathbb{R}^{4+1}$. As in [SWZ2024a,SWZ2024b], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result in [SWZ2024a] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.

Let $H$ be a monoid (written multiplicatively). We call $H$ Archimedean if, for all $a, b \in H$ such that $b$ is a non-unit, there is an integer $k \ge 1$ with $b^k \in HaH$; strongly Archimedean if, for each $a \in H$, there is an integer $k \ge 1$ such that $HaH$ contains any product of any $k$ non-units of $H$; and duo if $aH = Ha$ for all $a \in H$. We prove that the ideals of two strongly Archimedean, cancellative, duo monoids make up isomorphic semigroups under the induced operation of setwise multiplication if and only if the monoids themselves are isomorphic up to units; and the same holds upon restriction to finitely generated ideals in Archimedean, cancellative, duo monoids. Then we use the previous results to tackle a new case of a problem of Tamura and Shafer from the late 1960s.

In this paper, we will prove a Poisson summation formula on the Kuznetsov quotient that will be responsible for the functional equation of the standard $L$-functions of $\GL_2$.

In this paper, I will bijectively prove Andrews' refinement of the Alladi-Schur theorem. I will also reproduce Andrews' recursive relations for the Alladi-Schur polynomials.

We prove a number of new results concerning the topology and Hodge theory of singular varieties. A common theme is that concrete conditions on the complexity of the singularities are closely related to the symmetries of the Hodge-Du Bois diamond.

We obtain Hanson-Wright inequalities for the quadratic form of a random vector with independent sparse random variables. Specifically, we consider cases where the components of the random vector are sparse $\alpha$-sub-exponential random variables with $\alpha>0$. Our proof relies on a novel combinatorial approach to estimate the moments of the random quadratic form.

Integrated sensing and communication (ISAC) is a very promising technology designed to provide both high rate communication capabilities and sensing capabilities. However, in Massive Multi User Multiple-Input Multiple-Output (Massive MU MIMO-ISAC) systems, the dense user access creates a serious multi-user interference (MUI) problem, leading to degradation of communication performance. To alleviate this problem, we propose a decentralized baseband processing (DBP) precoding method. We first model the MUI of dense user scenarios with minimizing Cramer-Rao bound (CRB) as an objective function.Hybrid precoding is an attractive ISAC technique, and hybrid precoding using Partially Connected Structures (PCS) can effectively reduce hardware cost and power consumption. We mitigate the MUI between dense users based on ThomlinsonHarashima Precoding (THP). We demonstrate the effectiveness of the proposed method through simulation experiments. Compared with the existing methods, it can effectively improve the communication data rates and energy efficiency in dense user access scenario, and reduce the hardware complexity of Massive MU MIMO-ISAC systems. The experimental results demonstrate the usefulness of our method for improving the MUI problem in ISAC systems for dense user access scenarios.

We consider a free boundary problem for the heat equation with a given non-negative external heat source. On the free boundary, we impose the zero Dirichlet condition and the fixed normal derivative so that heat escapes from the boundary. In various settings, we show that there exist no solutions when the initial temperature equals the fixed temperature no matter where the initial location of the free boundary is given provided that the external heat source is bounded from above. We also note that there is a chance to have a solution when the external temperature is unbounded as time tends to zero by giving a self-similar solution.

It is well known that the chain map between the de Rham and Poisson complexes on a Poisson manifold also maps the Koszul bracket of differential forms into the Schouten bracket of multivector fields. In the generalized case of a $P_\infty$-structure, where a Poisson bivector $P$ is replaced by an arbitrary even multivector obeying $[[P,P]]=0$, an analog of the chain map and an $L_\infty$-morphism from the higher Koszul brackets into the Schouten bracket are also known; however, they differ significantly in nature. In the present paper, we address the problem of quantizing this picture. In particular, we show that the $L_\infty$-morphism is quantized into a single linear operator, which is a formal Fourier integral operator. This paper employs Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of $L_\infty$-algebroids.

A few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter $q.$ It is known now that the golden ratio has minimal radius among all these series. We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 2000. We also comment on the irrational case and on the relation with braids, rational knots and links.

Total variation integer optimal control problems admit solutions and necessary optimality conditions via geometric variational analysis. In spite of the existence of said solutions, algorithms which solve the discretized objective suffer from high numerical cost associated with the combinatorial nature of integer programming. Hence, such methods are often limited to small- and medium-sized problems. We propose a globally convergent, coordinate descent-inspired algorithm that allows tractable subproblem solutions restricted to a partition of the domain. Our decomposition method solves relatively small trust-region subproblems that modify the control variable on a subdomain only. Given sufficient subdomain overlap, we prove that first-order optimality is equivalent to first-order optimality per subdomain. We additionally show that sufficient decrease is achieved on a single subdomain by way of a trust region subproblem solver. We utilize this to prove convergence of our algorithm, which operates via greedy patch selection. Finally, our method demonstrates the capacity to accelerate large PDE-constrained integer control problems.

Let $n,k,s$ be three integers such that $k\geq 2$ and $n\geq s\geq 1$. Let $H$ be a $k$-partite $k$-uniform hypergraph with $n$ vertices in each class. Aharoni (2017) showed that if $e(H)>(s-1)n^{k-1}$, then $H$ has a matching of size $s$. In this paper, we gave a stability result for 3-partite 3-uniform hypergraphs: if $G$ is a $3$-partite $3$-uniform hypergraph with $n\geq 162$ vertices in each class, $e(G)\geq (s-1)n^2+3n-s$ and $G$ contains no matching of size $s+1$, then $G$ has a vertex cover of size $s$. Our bound is also tight.

We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can solve large-scale distance geometry problems with adjacencies, they are severely challenged in the absence thereof.

In this paper, we look at the nonlinear bending theory for thin prestrained elastic sheets with an oscillating periodic structure. We show that all limiting (in the sense of $\Gamma-$convergence) energy functionals which formally correspond to an infinite thickness-to-period ratio are equivalent. In contrast, there are several different limit models that formally correspond to a $0$ thickness-to-period ratio.

Increased attention has been given recently to the statistical analysis of variables with values on nonlinear manifolds. A natural but nontrivial problem in that context is the definition of quantile concepts. We are proposing a solution for compact Riemannian manifolds without boundaries; typical examples are polyspheres, hyperspheres, and toro\"{\i}dal manifolds equipped with their Riemannian metrics. Our concept of quantile function comes along with a concept of distribution function and, in the empirical case, ranks and signs. The absence of a canonical ordering is offset by resorting to the data-driven ordering induced by optimal transports. Theoretical properties, such as the uniform convergence of the empirical distribution and conditional (and unconditional) quantile functions and distribution-freeness of ranks and signs, are established. Statistical inference applications, from goodness-of-fit to distribution-free rank-based testing, are without number. Of particular importance is the case of quantile regression with directional or toro\"{\i}dal multiple output, which is given special attention in this paper. Extensive simulations are carried out to illustrate these novel concepts.

An inductive mean is a mean defined as a limit of a convergence sequence of other means. Historically, this notion of inductive means obtained as limits of sequences was pioneered independently by Lagrange and Gauss for defining the arithmetic-geometric mean. In this note, we first explain several generalizations of the scalar geometric mean to symmetric positive-definite matrices, and then present several inductive mean mechanisms for sets of symmetric positive-definite matrices.

The goal of this paper is to establish a decomposition of the network based on the maximum flow problem.

In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and vertices are said to be active independently with probability $p \in [0, 1]$. We consider $W$ to be the reunion of random balls with an active center. In certain circumstances, the model does not exhibit a phase transition, in the sense that $W$ almost surely contains an infinite component for all $p > 0$, or even $W$ covers the entire graph. In this paper, we give a sufficient condition on the radius distribution for the existence of a subcritical phase, namely a regime such that almost surely all the connected components of $W$ are finite. Additionally, we provide a sufficient condition for the exponential decay of the size of a typical component.

We study the mixing properties of the white-forced Navier--Stokes system in an infinite periodic pipe $\mathbb{R}\times\mathbb{T}$. Assuming that the noise is sufficiently non-degenerate, we prove the uniqueness of stationary measure and polynomial mixing in the dual-Lipschitz metric. The proof combines the coupling method with a Foia\c{s}--Prodi type estimate and weighted growth estimates.

Solving constrained nonlinear programs (NLPs) is of great importance in various domains such as power systems, robotics, and wireless communication networks. One widely used approach for addressing NLPs is the interior point method (IPM). The most computationally expensive procedure in IPMs is to solve systems of linear equations via matrix factorization. Recently, machine learning techniques have been adopted to expedite classic optimization algorithms. In this work, we propose using Long Short-Term Memory (LSTM) neural networks to approximate the solution of linear systems and integrate this approximating step into an IPM. The resulting approximate NLP solution is then utilized to warm-start an interior point solver. Experiments on various types of NLPs, including Quadratic Programs and Quadratically Constrained Quadratic Programs, show that our approach can significantly accelerate NLP solving, reducing iterations by up to 60% and solution time by up to 70% compared to the default solver.

We provide special values of $K$-theoretic Schur $P$- and $Q$-functions. Using these special values, we show the odd property of the number of shifted set-valued tableaux. Additionally, we generalize these special values to skew cases. Based on these special values, we demonstrate the odd property of the number of shifted skew set-valued tableaux and that they form pairs.

We prove the transitivity of real Anosov diffeomorphisms, which are Anosov diffeomorphisms where stable and unstable spaces decompose into a continuous sum of invariant one-dimensional sub-spaces with uniform contraction/expansion over the ambient manifold. We analyze how the stable/unstable holonomies of Anosov diffeomorphisms behave on Artigue's self-similar hyperbolic metric and prove that if a stable/unstable curve has a well-defined length in a self-similar hyperbolic metric, then it has a globally defined holonomy. We exhibit a self-similar hyperbolic metric with well-defined length of curves for each real Anosov diffeomorphism.

This paper focuses on the existence and multiplicity of normalized solutions for the coupled Schrodinger system with Sobolev critical coupling term. We present several existence and multiplicity results under some explicit conditions. Furthermore, we present a non-existence result for the defocusing case. This paper, together with the paper [T. Bartsch, H. W. Li and W. M. Zou. Calc. Var. Partial Differential Equations 62 (2023) ], provides a more comprehensive understanding of normalized solutions for Sobolev critical systems. We believe our methods can also address the open problem of the multiplicity of normalized solutions for Schrodinger systems with Sobolev critical growth, with potential for future development and broader applicability.

This paper examines the interdependence between green financial instruments, represented by green bonds and green stocks, and a set of major conventional assets, such as Treasury, investment-grade and high-yield corporate bonds, general stocks, crude oil, and gold. To that end, a novel wavelet-based network approach that allows for assessing the degree of interconnection between green financial products and traditional asset classes across different investment horizons is applied. The~empirical results show that green bonds are tightly linked to Treasury and investment-grade corporate bonds, while green stocks are strongly tied to general stocks, regardless of the specific time period and investment horizon considered. However, despite their common climate-friendly nature, there is no a remarkable association between green bonds and green stocks. This means that these green investments constitute basically two independent asset classes, with a distinct risk-return profile and aimed at a different type of investor. Furthermore, green financial products have a weak connection with high-yield corporate bonds and crude oil. These findings can have important implications for investors and policy makers in terms of investment decision, hedging strategies, and sustainability and energy policies.

Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$ whenever $B(f_1, f_2)= (T_1f_1) (T_2 f_2)$ is the product of two singular integral operators satisfying Dini conditions. Additionally, we also establish, as an application, the boundedness of a certain class of bounded variation bilinear Fourier multipliers solving a question posted in [Bilinear Fourier multipliers of bounded variation; Int. Math. Res. Not. (2023), no.24, 21943--21975 by Baena-Miret, Carro, Luque and Sanchez-Pascuala].

We study two continuous and isotropic analogues of the model of greedy lattice animals introduced by Cox, Gandolfi, Griffin and Kesten in 1993. In our framework, animals collect masses scattered on a Poisson point process on $\mathbb R^d$, and are allowed to have vertices outside the process or not, depending on the model. The author recently proved in a more general setting that for all $u$ in the Euclidean open unit ball, the maximal mass of animals with length $\ell$, containing $0$ and $\ell u$ satisfies a law of large numbers. We prove some additional properties in the Poissonian case, including an extension of the functional law of large numbers to the closed unit ball, and study strict monotonicity of the limit function along a radius. Moreover, we prove that a third, penalized model is a suitable interpolation between the former two.

We obtain boundary nondegeneracy and regularity estimates for harmonic functions in $C^1$ domains, providing an explicit modulus of continuity. Our results extend the classical Hopf-Oleinik lemma and boundary Lipschitz regularity for domains with $C^{1,\mathrm{Dini}}$ boundaries, while also recovering the known $C^{1-\varepsilon}$ regularity for flat Lipschitz domains, unifying both theories with a single proof.

We give a short and elementary proof of the boundedness of triangular Hilbert transform along non-flat curves definable in a polynomially bounded o-minimal structure. We also provide a criterion on the multiplier to determine whether the associated fiber-wisely defined bilinear operator admits a smoothing inequality.

In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical decision problems only plays a role in counting the number of quantifiers, jumps, or mind-changes. In contrast, we reveal that when the realizability interpretation is combined with many-one reducibility, it becomes possible to classify natural arithmetical problems in a very nontrivial way.

This paper develops an impulsive faecal-oral model with free boundary to in order to understand how the exposure to a periodic disinfection and expansion of the infected region together influences the spread of faecal-oral diseases. We first check that this impulsive model has a unique globally nonnegative classical solution. The principal eigenvalues of the corresponding periodic eigenvalue problem at the initial position and infinity are defined as $\lambda^{\vartriangle}_{1}(h_{0})$ and $\lambda^{\vartriangle}_{1}(\infty)$, respectively. They both depend on the impulse intensity $1-G'(0)$ and expansion capacities $\mu_{1}$ and $\mu_{2}$. The possible long time dynamical behaviours of the model are next explored in terms of $\lambda^{\vartriangle}_{1}(h_{0})$ and $\lambda^{\vartriangle}_{1}(\infty)$: if $\lambda^{\vartriangle}_{1}(\infty)\geq 0$, then the diseases are vanishing; if $\lambda^{\vartriangle}_{1}(\infty)<0$ and $\lambda^{\vartriangle}_{1}(h_{0})\leq 0$, then the disease are spreading; if $\lambda^{\vartriangle}_{1}(\infty)<0$ and $\lambda^{\vartriangle}_{1}(h_{0})> 0$, then for any given $\mu_{1}$, there exists a $\mu_{0}$ such that spreading happens as $\mu_{2}\in( \mu_{0},+\infty)$, and vanishing happens as $\mu_{2}\in(0, \mu_{0})$. Finally, numerical examples are presented to corroborate the correctness of the obtained theoretical findings and to further understand the influence of an impulsive intervention and expansion capacity on the spreading of the diseases. Our results show that both the increase of impulse intensity and the decrease of expansion capacity have a positive contribution to the prevention and control of the diseases.

The sum-rank metric generalizes the Hamming and rank metric by partitioning vectors into blocks and defining the total weight as the sum of the rank weights of these blocks, based on their matrix representation. In this work, we explore support-guessing algorithms for decoding sum-rank-metric codes. Support-guessing involves randomly selecting candidate supports and attempting to decode the error under the assumption that it is confined to these supports. While previous works have focused on worst-case scenarios, we analyze the average case and derive an optimal support-guessing distribution in the asymptotic regime. We show that this distribution also performs well for finite code lengths. Our analysis provides exact complexity estimates for unique decoding scenarios and establishes tighter bounds beyond the unique decoding radius. Additionally, we introduce a randomized decoding algorithm for Linearized Reed--Solomon (LRS) codes. This algorithm extends decoding capabilities beyond the unique decoding radius by leveraging an efficient error-and-erasure decoder. Instead of requiring the entire error support to be confined to the guessed support, the algorithm succeeds as long as there is sufficient overlap between the guessed support and the actual error support. As a result, the proposed method improves the success probability and reduces computational complexity compared to generic decoding algorithms. Our contributions offer more accurate complexity estimates than previous works, which are essential for understanding the computational challenges involved in decoding sum-rank-metric codes. This improved complexity analysis, along with optimized support-guessing distributions, provides valuable insights for the design and evaluation of code-based cryptosystems using the sum-rank metric. This is particularly important in the context of quantum-resistant cryptography.

In this paper, we address three Principal--Agent problems in a moral hazard context and show that they are connected. We start by studying the problem of Principal with multiple Agents in cooperation. The term cooperation is manifested here by the fact that the agents optimize their criteria through Pareto equilibria. We show that as the number of agents tends to infinity, the principal's value function converges to the value function of a McKean--Vlasov control problem. Using the solution to this McKean--Vlasov control problem, we derive a constructive method for obtaining approximately optimal contracts for the principal's problem with multiple agents in cooperation. In a second step, we show that the problem of Principal with multiple Agents turns out to also converge, when the number of agents goes to infinity, towards a new Principal--Agent problem which is the Principal--Agent problem with Mckean--Vlasov dynamics. This is a Principal--Agent problem where the agent--controlled production follows a Mckean-Vlasov dynamics and the contract can depend of the distribution of the production. The value function of the principal in this setting is equivalent to that of the same McKean--Vlasov control problem from the multi--agent scenario. Furthermore, we show that an optimal contract can be constructed from the solution to this McKean--Vlasov control problem. We conclude by discussing, in a simple example, the connection of these problems with the multitask Principal--Agent problem which is a situation when a principal delegates multiple tasks that can be correlated to a single agent.

Examples of stochastic processes whose state space representations involve functions of an integral type structure $$I_{t}^{(a,b)}:=\int_{0}^{t}b(Y_{s})e^{-\int_{s}^{t}a(Y_{r})dr}ds, \quad t\ge 0$$ are studied under an ergodic semi-Markovian environment described by an $S$ valued jump type process $Y:=(Y_{s}:s\in\mathbb{R}^{+})$ that is ergodic with a limiting distribution $\pi\in\mathcal{P}(S)$. Under different assumptions on signs of $E_{\pi}a(\cdot):=\sum_{j\in S}\pi_{j}a(j)$ and tail properties of the sojourn times of $Y$ we obtain different long time limit results for $I^{(a,b)}_{}:=(I^{(a,b)}_{t}:t\ge 0).$ In all cases mixture type of laws emerge which are naturally represented through an affine stochastic recurrence equation (SRE) $X\stackrel{d}{=}AX+B,\,\, X\perp\!\!\!\perp (A, B)$. Examples include explicit long-time representations of pitchfork bifurcation, and regime-switching diffusions under semi-Markov modulated environments, etc.

In this work, we characterise the statistics of Markov chains by constructing an associated sequence of periodic differential operators. Studying the density of states of these operators reveals the absolutely continuous invariant measure of the Markov chain. This approach also leads to a direct proof of convergence to the invariant measure, along with explicit convergence rates. We show how our method can be applied to a class of related Markov chains including the logistic map, the tent map and Chebyshev maps of arbitrary order.

We explore the problem of distributed Hypothesis Testing (DHT) against independence, focusing specifically on Binary Symmetric Sources (BSS). Our investigation aims to characterize the optimal quantizer among binary linear codes, with the objective of identifying optimal error probabilities under the Neyman-Pearson (NP) criterion for short code-length regime. We define optimality as the direct minimization of analytical expressions of error probabilities using an alternating optimization (AO) algorithm. Additionally, we provide lower and upper bounds on error probabilities, leading to the derivation of error exponents applicable to large code-length regime. Numerical results are presented to demonstrate that, with the proposed algorithm, binary linear codes with an optimal covering radius perform near-optimally for the independence test in DHT.

We give a brief introduction to $\tau$-tilting theory [AIR]. In particular, we will see how our theory unifies two different branches of tilting theory, namely, silting theory and cluster tilting theory. We also introduce the history and recent developments.

An accurate approximation of solutions to elliptic problems in infinite domains is challenging from a computational point of view. This is due to the need to replace the infinite domain with a sufficiently large and bounded computational domain, and posing artificial boundary conditions on the boundary of the truncated computational geometry, which will then pollute the solution in an interior region of interest. For elliptic problems with periodically varying coefficients (with a possibly unknown period), a modelling strategy based on exponentially regularized elliptic problem was previously developed and analysed. The main idea was to replace the infinite domain periodic problem with a regularized elliptic problem posed over a finite domain, while retaining an accuracy decaying exponentially with respect to the size of the truncated domain. In this article, we extend the analysis to problems, where no structural assumptions on the coefficient are assumed. Moreover, the analysis here uncovers an interesting property of the right hand side in the Fourier domain for the method to converge fast for problems beyond periodicity.

Recently, Andrews and El Bachraoui (2024) proved three very interesting $q$-series identities, from which three simple looking identities involving certain restricted partitions into distinct even parts and $4$-regular partitions follow. In this short note, we give combinatorial proofs of these identities. We also prove the counterpart identities for the restricted partitions into distinct odd parts.

We study a system of stochastic differential equations with singular drift which describes the dynamics of signed particles in two dimensions interacting by the Coulomb potential. In contrast to the well-studied cases of identical particles that either all repel each other or all attract each other, this system contains both `positive' and `negative' particles. Equal signs repel and opposite signs attract each other; apart from the sign, the potential is the same. We derive results on well-posedness of the system, on the type of collisions that can occur, and on the mean-field limit as the number of particles tends to infinity. Our results demonstrate that the signed system shares features of both the fully repulsive and the fully attractive cases. Our proof method is inspired by the work of Fournier and Jourdain (The Annals of Applied Probability, 27, pp. 2807-2861, 2017) on the fully attractive case; we construct an approximate system of equations, establish uniform estimates, and use tightness to pass to limits.

In this paper, we investigate the convergence of asymptotic systems in non-autonomous Cohen--Grossberg neural network models, which include both infinite discrete time-varying and distributed delays. We derive stability results under conditions where the non-delay terms asymptotically dominate the delay terms. Several examples and a numerical simulation are provided to illustrate the significance and novelty of the main result.

The control barrier function (CBF) has become a fundamental tool in safety-critical systems design since its invention. Typically, the quadratic optimization framework is employed to accommodate CBFs, control Lyapunov functions (CLFs), other constraints and nominal control design. However, the constrained optimization framework involves hyper-parameters to tradeoff different objectives and constraints, which, if not well-tuned beforehand, impact system performance and even lead to infeasibility. In this paper, we propose a hierarchical optimization framework that decomposes the multi-objective optimization problem into nested optimization sub-problems in a safety-first approach. The new framework addresses potential infeasibility on the premise of ensuring safety and performance as much as possible and applies easily in multi-certificate cases. With vivid visualization aids, we systematically analyze the advantages of our proposed method over existing QP-based ones in terms of safety, feasibility and convergence rates. Moreover, two numerical examples are provided that verify our analysis and show the superiority of our proposed method.

We describe those Weil divisors of cyclic quotient surface singularities which are (abstract) $r$--tuple curve singularities.

This paper proposes a general analysis of codeword detection in noncoherent communications. Motivated by the existence of error floors in various regimes, fundamental characteristics of signal design are investigated. In particular, the necessary and sufficient conditions for asymptotically singular detection (i.e. error-free in the limit) are derived from classical results in detection theory. By leveraging tools from linear algebra and the theory of Hilbert spaces, we are able to characterize asymptotic singularity in two main scenarios: the large array and high SNR regimes. The results generalize previous works and extend the notion of unique identification, as well as recontextualize the geometry of Grassmannian constellations from an alternative perspective.

We construct global homotopies to compute differential Hochschild cohomologies in differential geometry. This relies on two different techniques: a symbol calculus from differential geometry and a coalgebraic version of the van Est theorem. Not only do we obtain an improved version of the Hochschild-Kostant-Rosenberg theorem but we also compute Hochschild cohomologies for related scenarios, e.g. for principal bundles, and invariant versions.

A semigroup of binary relations (under composition) on a set $X$ is \emph{complemented} if it is closed under the taking of complements within $X\times X$. We resolve a 1991 problem of Boris Schein by showing that the class of finite unary semigroups that are representable as complemented semigroups of binary relations is undecidable, so composition with complementation forms a minimal subsignature of Tarski's relation algebra signature that has undecidability of representability. In addition we prove similar results for semigroups of binary relations endowed with unary operations returning the kernel and cokernel of a relation. We generalise to signatures which may include arbitrary, definable operations and provide a chain of weaker and weaker signatures, each definable in the previous signature, each having undecidability of representability, but whose limit signature includes composition only, which corresponds to the well known, decidable and finitely axiomatised variety of semigroups. All these results are also proved for representability as binary relations over a finite set.

The object of this paper is to study and develop a Poisson distribution in generalized Wright function form.

Numerous Optimization Algorithms have a time-varying update rule thanks to, for instance, a changing step size, momentum parameter or, Hessian approximation. In this paper, we apply unrolled or automatic differentiation to a time-varying iterative process and provide convergence (rate) guarantees for the resulting derivative iterates. We adapt these convergence results and apply them to proximal gradient descent with variable step size and FISTA when solving partly smooth problems. We confirm our findings numerically by solving $\ell_1$ and $\ell_2$-regularized linear and logisitc regression respectively. Our theoretical and numerical results show that the convergence rate of the algorithm is reflected in its derivative iterates.

We consider a class of nonlocal Cahn-Hilliard equations in a bounded domain $\Omega\subset\mathbb{R}^{d}$ $(d\in\{2,3\})$, subject to a nonlocal kinetic rate dependent dynamic boundary condition. This diffuse interface model describes phase separation processes with possible long-range interactions both within the bulk material and on its boundary. The kinetic rate $1/L$, with $L\in[0,+\infty]$, distinguishes different types of bulk-boundary interactions. For the initial boundary value problem endowed with general singular potentials, including the physically relevant logarithmic potential, we first establish the existence and uniqueness of global weak solutions when the bulk and boundary chemical potentials are coupled through a Robin-type boundary condition, i.e., $L\in(0,+\infty)$. The proof of existence is based on a Yosida approximation of singular potentials and a suitable Faedo-Galerkin scheme. Subsequently, we investigate asymptotic limits as the kinetic rate approaches zero or infinity, which yield weak well-posedness for the limiting cases $L=+\infty$ and $L=0$, respectively. Under appropriate assumptions on the interaction kernels, we derive explicit convergence rates for the Yosida approximation as $\varepsilon\to0$ and for the asymptotic limits as $L\to0$ or $L\to+\infty$. Finally, we demonstrate that every global weak solution exhibits a propagation of regularity over time and establish the instantaneous strict separation property.

For any compact connected submanifold $K$ of $\mathbb{R}^n$, let $\Lambda_K$ denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of $\mathbb{R}^n$. In this paper, we give examples of pairs $(K_0,K_1)$ of compact connected submanifolds of $\mathbb{R}^n$ such that $\Lambda_{K_0}$ is not Legendrian isotopic to $\Lambda_{K_1}$, although they cannot be distinguished by classical invariants. Here, $K_1$ is the image of an embedding $\iota_f \colon K_0 \to \mathbb{R}^n$ which is regular homotopic to the inclusion map of $K_0$ and the codimension in $\mathbb{R}^n$ is greater than or equal to $4$. As non-classical invariants, we define the strip Legendrian contact homology and a coproduct on it under certain conditions on Legendrian submanifolds. Then, we give a purely topological description of these invariants for $\Lambda_K$ when the codimension of $K$ is greater than or equal to $4$. The main examples $\Lambda_{K_0}$ and $\Lambda_{K_1}$ are distinguished by the coproduct, which is computed by using an idea of string topology.

We show that the lamplighter groups $(\mathbb{Z}/p\mathbb{Z})^n\wr\mathbb{Z}$, where $p$ is prime and $n\ge 1$ is a positive integer, are profinitely rigid.

Building on Buchholz' assignment for ordinals below Bachmann-Howard ordinal, see Buchholz 2003, we introduce systems of fundamental sequences for two kinds of relativized $\vartheta$-function-based notation systems of strength $\Pi^1_1{\operatorname{-CA}_0}$ and prove Bachmann property for these systems, which is essential for monotonicity properties of fast growing hierarchies defined on the basis of fundamental sequences. The central notion of our construction is the notion of localization, which was introduced in Wilken 2007. The first kind of stepwise defined $\vartheta$-functions over ordinal addition as basic function fits the framework of the ordinal arithmetical toolkit developed in Wilken 2007, whereas the second kind of $\vartheta$-functions is defined simultaneously and will allow for further generalization to larger proof-theoretic ordinals, see Weiermann and Wilken 2011. The systems of fundamental sequences given here enable the investigation of fundamental sequences and independence phenomena also in the context of patterns of resemblance, an approach to ordinal notations that is both semantic and combinatorial and was first introduced by Carlson 2001 and further analyzed in Wilken 2006, 2007, and Carlson and Wilken 2012. Our exposition is put into the context of the abstract approach to fundamental sequences developed by Buchholz, Cichon, and Weiermann 1994. The results of this paper will be applied in the theory of Goodstein sequences, extending results of Fern\`andez-Duque and Weiermann 2024.

The Wright-Fisher diffusion is a fundamentally important model of evolution encompassing genetic drift, mutation, and natural selection. Suppose you want to infer the parameters associated with these processes from an observed sample path. Then to write down the likelihood one first needs to know the mutual arrangement of two path measures under different parametrizations; that is, whether they are absolutely continuous, equivalent, singular, and so on. In this paper we give a complete answer to this question by finding the separating times for the diffusion - the stopping time before which one measure is absolutely continuous with respect to the other and after which the pair is mutually singular. In one dimension this extends a classical result of Dawson on the local equivalence between neutral and non-neutral Wright-Fisher diffusion measures. Along the way we also develop new zero-one type laws for the diffusion on its approach to, and emergence from, the boundary. As an application we derive an explicit expression for the joint maximum likelihood estimator of the mutation and selection parameters and show that its convergence properties are closely related to the separating time.

A sequential solver for differential-algebraic equations with embedded optimization criteria (DAEOs) was developed to take advantage of the theoretical work done by Deussen et al. Solvers of this type separate the optimization problem from the differential equation and solve each individually. The new solver relies on the reduction of a DAEO to a sequence of differential inclusions separated by jump events. These jump events occur when the global solution to the optimization problem jumps to a new value. Without explicit treatment, these events will reduce the order of convergence of the integration step to one. The solver implements a "local optimizer tracking" procedure to detect and correct these jump events. Local optimizer tracking is much less expensive than running a deterministic global optimizer at every time step. This preserves the order of convergence of the integrator component without sacrificing performance to perform deterministic global optimization at every time step. The newly developed solver produces correct solutions to DAEOs and runs much faster than sequential DAEO solvers that rely only on global optimization.

The purpose of this paper is to clarify the relations between various constructions of solutions of the Yang-Baxter equation from Leibniz algebras, racks, 3-Leibniz algebras, 3-racks, linear racks, trilinear racks, and give new constructions of solutions of the Yang-Baxter equation. First we show that a 3-Leibniz algebra naturally gives rise to a 3-rack on the underlying vector space, which generalizes Kinyon's construction of racks from Leibniz algebras. Then we show that a trilinear rack naturally gives rise to a linear rack. Combined with Lebed's construction of solutions of the Yang-Baxter equation from linear racks, our results give an intrinsic explanation of Abramov and Zappala's construction of solutions of the Yang-Baxter equation from trilinear racks. Next we show that a 3-Leibniz algebra gives rise to a trilinear rack, which generalizes Abramov and Zappala's construction from 3-Lie algebras. Finally, we construct solutions of the Yang-Baxter equation using central extensions of 3-Leibniz algebras and Leibniz algebras. In particular, given a 3-Leibniz algebra, there are two different approaches to construct solutions of the Yang-Baxter equation, namely either consider the central extension of the Leibniz algebra on the fundamental objects, or consider the Leibniz algebra on the fundamental objects of the central extension of the 3-Leibniz algebra. We also show that there is a homomorphism between the corresponding solutions.

Discrete Empirical Interpolation Method (DEIM) estimates a function from its pointwise incomplete observations. In particular, this method can be used to estimate the state of a dynamical system from observational data gathered by sensors. However, when the number of observations are limited, DEIM returns large estimation errors. Sparse DEIM (S-DEIM) was recently developed to address this problem by introducing a kernel vector which previous DEIM-based methods had ignored. Unfortunately, estimating the optimal kernel vector in S-DEIM is a difficult task. Here, we introduce a data-driven method to estimate this kernel vector from sparse observational time series using recurrent neural networks. Using numerical examples, we demonstrate that this machine learning approach together with S-DEIM leads to nearly optimal state estimations.

In this note, we connect two seemingly unrelated objects: On the one hand is a two-dimensional drift-diffusion process $X$ with divergence-free and time-independent drift $b$. The drift is given by a stationary Gaussian ensemble, and we focus on the critical case where a small-scale cut-off is necessary for well-posedness and the large-scale cancellations lead to a borderline super-diffusive behavior. On the other hand is the natural diffusion $F$ on the Lie group $\textbf{SL}(2)$ of matrices of determinant one. As a consequence of this connection, the strongly non-Gaussian character of $F$ transmits to how $X$ depends on its starting point.

The Robbins-Siegmund theorem is one of the most important results in stochastic optimization, where it is widely used to prove the convergence of stochastic algorithms. We provide a quantitative version of the theorem, establishing a bound on how far one needs to look in order to locate a region of metastability in the sense of Tao. Our proof involves a metastable analogue of Doob's theorem for $L_1$-supermartingales along with a series of technical lemmas that make precise how quantitative information propagates through sums and products of stochastic processes. In this way, our paper establishes a general methodology for finding metastable bounds for stochastic processes that can be reduced to supermartingales, and therefore for obtaining quantitative convergence information across a broad class of stochastic algorithms whose convergence proof relies on some variation of the Robbins-Siegmund theorem. We conclude by discussing how our general quantitative result might be used in practice.

We consider a degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise and establish the essential m-dissipativity on $L^2(\mu^{\Phi})$ of the corresponding Kolmogorov (backwards) operator. Here, $\Phi$ is the potential and $\mu^{\Phi}$ the invariant measure with density $e^{-\Phi}$ with respect to an infinite dimensional non-degenerate Gaussian measure. The main difficulty, besides the non-sectorality of the Kolmogorov operator, is the coverage of a large class of potentials. We include potentials that have neither a bounded nor a Lipschitz continuous gradient. The essential m-dissipativity is the starting point to establish the hypocoercivity of the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the Kolmogorov operator. By using the refined abstract Hilbert space hypocoercivity method of Grothaus and Stilgenbauer, originally introduced by Dolbeault, Mouhot and Schmeiser, we construct a $\mu^{\Phi}$-invariant Hunt process with weakly continuous paths and infinite lifetime, whose transition semigroup is associated with $(T_t)_{t\geq 0}$. This process provides a stochastically and analytically weak solution to the degenerate infinite dimensional stochastic Hamiltonian system with multiplicative noise. The hypocoercivity of $(T_t)_{t\geq 0}$ and the identification of $(T_t)_{t\geq 0}$ with the transition semigroup of the process leads to the exponential ergodicity. Finally, we apply our results to degenerate second order in time stochastic reaction-diffusion equations with multiplicative noise. A discussion of the class of applicable potentials and coefficients governing these equations completes our analysis.

We define a dissipative measure-valued (DMV) solution to the system of equations governing the motion of a general compressible, viscous, electrically and heat conducting fluid driven by non-conservative boundary conditions. We show the stability of strong solutions to the full compressible magnetohydrodynamic system in a large class of these DMV solutions. In other words, we prove a DMV-strong uniqueness principle: a DMV solution coincides with the strong solution emanating from the same initial data as long as the latter exists.

We prove that any compact, orientable 3-manifold with empty or incompressible toral boundary is profinitely almost rigid among all compact, orientable, boundary-incompressible 3-manifolds, i.e. the profinite completion of its fundamental group determines its homeomorphism type up to finitely many possibilities. Moreover, the profinite completion of the fundamental group of a mixed 3-manifold, together with the peripheral structure, uniquely determines the homeomorphism type of its Seifert part (i.e. the maximal graph manifold components in the JSJ-decomposition). On the other hand, without assigning the peripheral structure, the profinite completion of a mixed 3- manifold group may not even determine the fundamental group of its Seifert part. The proof is based on JSJ-decomposition.

We show that for a given DAG $G$, among all observational distributions of Bayesian networks over $G$ with arbitrary outcome spaces, the faithful distributions are `typical': they constitute a dense, open set with respect to the total variation metric. As a consequence, the set of faithful distributions is non-empty, and the unfaithful distributions are nowhere dense. We extend this result to the space of Bayesian networks, where the properties hold for Bayesian networks instead of distributions of Bayesian networks. As special cases, we show that these results also hold for the faithful parameters of the subclasses of linear Gaussian -- and discrete Bayesian networks, giving a topological analogue of the measure-zero results of Spirtes et al. (1993) and Meek (1995). Finally, we extend our topological results and the measure-zero results of Spirtes et al. and Meek to Bayesian networks with latent variables.

We consider a two-dimensional system in which a charged particle is exposed to a homogeneous magnetic field perpendicular to the plane and a potential that is translationally invariant in one dimension. We derive several conditions on such a perturbation under which the Landau levels change into an absolutely continuous spectrum.

We consider the two-dimensional nonlinear Schr\"odinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide a proof by employing a Kato's method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

A field $K$ is $d$-local if there exist fields $K=k_d,\dots,k_0$ with $k_{i+1}$ completely discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of such $K$ with many coefficients, including finite coefficients of any order. Previously, such duality was only known in few cases : as a perfect pairing of finite groups for finite coefficients prime to $\mathrm{char} k_0$ in general, or for any finite coefficients when $k_1$ is $p$-adic ; or as a perfect pairing of locally compact Hausdorff groups for the $\mathrm{fppf}$ cohomology of finite group schemes when $K$ is local. With no obvious reasonable topology available, we abandon perfectness altogether and instead obtain nondegenerate pairings of abstract abelian groups. This is done with new diagram-chasing results for pairings of torsion groups, allowing an approach by d\'evissage which breaks our results down to the study of pairings $K^M_r(K)/p\times H^{d+1-r}_p(K)\to\mathbb Z/p$ using results of Kato.

An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the existence of optimal domains among a class of convex shapes and propose a numerical scheme to approximate the eigenvalue. The stability of the shape optimization among convex, bounded domains in $\mathbb{R}^3$ is proven for an approximation with polyhedral domains under a non-conformal convexity constraint. We prove that on the ball, symmetry breaking of the optimal insulation can be expected in general. To observe how the lower bound affects the breaking of symmetry in the optimal insulation and the shape optimization, the eigenvalue and optimal domains are approximated for several values of mass $m$ and lower bounds $\ell_{\min}\ge0$. The numerical experiments suggest, that in general symmetry breaking still arises, unless $m$ is close to a critical value $m_0$, and $\ell_{\min}$ large enough such that almost all of the mass $m$ is fixed through the lower bound. For $\ell_{\min}=0$, the numerical results are consistent with previous numerical experiments on shape optimization restricted to rotationally symmetric, convex domains.

Let $a_1,\ldots,a_m$ be nonzero integers, $c \in \mathbb Z$ and $r \ge 2$. The Rado number for the equation \[ \sum_{i=1}^m a_ix_i = c \] in $r$ colours is the least positive integer $N$ such that any $r$-colouring of the integers in the interval $[1,N]$ admits a monochromatic solution to the given equation. We introduce the concept of $t$-distributability of sets of positive integers, and determine exact values whenever possible, and upper and lower bounds otherwise, for the Rado numbers when the set $\{a_1,\ldots,a_{m-1}\}$ is $2$-distributable or $3$-distributable, $a_m=-1$, and $r=2$. This generalizes previous works by several authors.

Let $M$ be a closed orientable $(n-1)$-connected $(2n+2)$-manifold, $n\geq 2$. In this paper we combine the Postnikov tower of spheres and the homotopy decomposition of the reduced suspension space $\Sigma M$ to investigate the cohomotopy sets $\pi^\ast(M)$ for $n=2,3,4$, under the assumption that $M$ has $2$-torsion-free homology. All cohomotopy sets $\pi^i(M)$ of such manifolds $M$ are characterized except $\pi^4(M)$ for $n=3,4$.

The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov-Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov-Poisson algebra. Hence all transposed Poisson algebras of Novikov-Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of $2$-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence.

A connected dominating set (CDS) in a graph is a dominating set of vertices that induces a connected subgraph. Having many disjoint CDSs in a graph can be considered as a measure of its connectivity, and has various graph-theoretic and algorithmic implications. We show that $d$-regular (weakly) pseudoreandom graphs contain $(1+o(1))d/\ln d$ disjoint CDSs, which is asymptotically best possible. In particular, this implies that random $d$-regular graphs typically contain $(1+o(1))d/\ln d$ disjoint CDSs.

In this paper we consider the equality problem of generalized Bajraktarevi\'c means, i.e., we are going to solve the functional equation \begin{equation}\label{E0}\tag{*} f^{(-1)}\bigg(\frac{p_1(x_1)f(x_1)+\dots+p_n(x_n)f(x_n)}{p_1(x_1)+\dots+p_n(x_n)}\bigg)=g^{(-1)}\bigg(\frac{q_1(x_1)g(x_1)+\dots+q_n(x_n)g(x_n)}{q_1(x_1)+\dots+q_n(x_n)}\bigg), \end{equation} which holds for all $x=(x_1,\dots,x_n)\in I^n$, where $n\geq 2$, $I$ is a nonempty open real interval, the unknown functions $f,g:I\to\mathbb{R}$ are strictly monotone, $f^{(-1)}$ and $g^{(-1)}$ denote their generalized left inverses, respectively, and the vector-valued weight functions $p=(p_1,\dots,p_n):I\to\mathbb{R}_{+}^n$ and $q=(q_1,\dots,q_n):I\to\mathbb{R}_{+}^n$ are also unknown. This equality problem in the symmetric two-variable case (i.e., when $n=2$ and $p_1=p_2$, $q_1=q_2$) was solved under sixth-order regularity assumptions by Losonczi in 1999. The authors of this paper improved this result in 2023 by reaching the same conclusion assuming only first-order differentiability. In the nonsymmetric case, assuming third-order differentiability of $f$, $g$ and the first-order differentiability of at least three of the functions $p_1,\dots,p_n$, Gr\"unwald and P\'ales proved that \eq{0} holds if and only if there exist four constants $a,b,c,d\in\mathbb{R}$ with $ad\neq bc$ such that $$ cf+d>0,\qquad g=\frac{af+b}{cf+d},\qquad\mbox{and}\qquad q_\ell=(cf+d)p_\ell\qquad (\ell\in\{1,\dots,n\}). $$ The main goal of this paper is to establish the same conclusion under first-order differentiability.

In stark contrast to lower dimensions, we produce a plethora of ancient and immortal Ricci flows in real dimension $4$ with Einstein orbifolds as tangent flows at infinity. For instance, for any $k\in\mathbb{N}_0$, we obtain continuous families of non-isometric ancient Ricci flows on $\#k(\mathbb{S}^2\times \mathbb{S}^2)$ depending on a number of parameters growing linearly in $k$, and a family of half-PIC ancient Ricci flows on $\mathbb{CP}^2\#\mathbb{CP}^2$. The ancient/immortal dichotomy is determined by a notion of linear stability of orbifold singularities with respect to the expected way for them to appear along Ricci flow: by bubbling off Ricci-flat ALE metrics. We discuss the case of Ricci solitons orbifolds and motivate a conjecture that spherical and cylindrical solitons with orbifold singularities, which are unstable in our sense, should not appear along Ricci flow by bubbling off Ricci-flat ALE metrics.

The aim of this paper is to investigate the boundedness of periodic Fourier integral operators in Lebesgue spaces with variable exponent $L^{p(\cdot)}$ on the $n$-dimensional torus. We deal with operators of type $(\rho, \delta)$ which symbols belong to the H\"{o}rmander class $S^{m}_{\rho, \delta}(\mathbb{T}^{n}\times\mathbb{Z}^{n})$ for $0\leq\delta<\rho\leq1.$

Optimizing network utility in device-to-device networks is typically formulated as a non-convex optimization problem. This paper addresses the scenario where the optimization variables are from a bounded but continuous set, allowing each device to perform power control. The power at each link is optimized to maximize a desired network utility. Specifically, we consider the weighted-sum-rate. The state of the art benchmark for this problem is fractional programming with quadratic transform, known as FPLinQ. We propose a scalarization approach to transform the weighted-sum-rate, developing an iterative algorithm that depends on step sizes, a reference, and a direction vector. By employing the deep unfolding approach, we optimize these parameters by presenting the iterative algorithm as a finite sequence of steps, enabling it to be trained as a deep neural network. Numerical experiments demonstrate that the unfolded algorithm performs comparably to the benchmark in most cases while exhibiting lower complexity. Furthermore, the unfolded algorithm shows strong generalizability in terms of varying the number of users, the signal-to-noise ratio and arbitrary weights. The weighted-sum-rate maximizer can be integrated into a low-complexity fairness scheduler, updating priority weights via virtual queues and Lyapunov Drift Plus Penalty. This is demonstrated through experiments using proportional and max-min fairness.

Let $M$ be a compact, connected manifold of positive dimension and let $\mathcal G\leq\textrm{Homeo}(M)$ be \emph{locally approximating} in the sense that for all open $U\subseteq M$ compactly contained in a single Euclidean chart of $M$, the subgroup $\mathcal G[U]$ consisting of elements of $\mathcal G$ supported in $U$ is dense in the full group of homeomorphisms supported in $U$. We prove that $\mathcal G$ interprets first order arithmetic, as well as a first order predicate that encodes membership in finitely generated subgroups of $\mathcal G$. As a consequence, we show that if $\mathcal G$ is not finitely generated, then no group elementarily equivalent to $\mathcal G$ can be finitely generated. We show that many finitely generated locally approximating groups of homeomorphisms $\mathcal G$ of a manifold are prime models of their theories, and give conditions that guarantee any finitely presented group $G$ that is elementarily equivalent to $\mathcal G$ is isomorphic to $\mathcal G$. We thus recover some results of Lasserre about the model theory of Thompson's groups $F$ and $T$. Finally, we obtain several action rigidity result for locally approximating groups of homeomorphisms. If $\mathcal G$ acts in a locally approximating way on a compact, connected manifold $M$ then the dimension of $M$ is uniquely determined by the elementary equivalence class of $\mathcal G$. Moreover, if $\dim M\leq 3$ then $M$ is uniquely determined up to homeomorphism. In for general closed smooth manifolds, the homotopy type of $M$ is uniquely determined. In this way, we obtain a generalization of a well-known result of Rubin.

Brown constructed a series of threefold flips given by the GIT quotient of a hypersurface in C5. In this article, we classify threefold flips and flops which are the GIT quotients of complete intersections in C6. We also show that there are no more new examples as GIT quotients of complete intersections in Cn with n >= 7.

In this paper, we introduce a notion of clock moves for spanning trees in plane graphs. This enables us to develop a spanning tree model of an Alexander polynomial for a plane graph and prove the unimodal property of its associate coefficient sequence. In particular, this confirms the trapezoidal conjecture for planar singular knots and gives new insights to Fox's original conjecture on alternating knots.

A tuple $\underline{T}=(T_1, \dotsc, T_k)$ of operators on a Hilbert space $\mathcal H$ is said to be \textit{$q$-commuting with} $\|q\|=1$ or simply $q$-\textit{commuting} if there is a family of scalars $q=\{q_{ij} \in \mathbb C : |q_{ij}|=1, \ q_{ij}=q_{ji}^{-1}, \ 1 \leq i < j \leq k \}$ such that $T_i T_j =q_{ij}T_j T_i$ for $1 \leq i < j \leq k$. Moreover, if each $q_{ij}=-1$, then $\underline{T}$ is called an \textit{anti-commuting tuple}. A well-known result due to Holbrook \cite{Holbrook} states that a commuting $k$-tuple consisting of $2 \times 2$ scalar matrix contractions always dilates to a commuting $k$-tuple of unitaries for any $k\geq 1$. To find a generalization of this result for a $q$-commuting $k$-tuple of $2\times 2$ scalar matrix contractions, we first classify such tuples into three types upto similarity. Then we prove that a $q$-commuting tuple which is unitarily equivalent to any of these three types, admits a $\widetilde{q}$-unitary dilation, where $\widetilde q \subseteq q \cup \{1\}$. A special emphasis is given to the dilation of an anti-commuting tuple of $2 \times 2$ scalar matrix contractions.

In this work, we consider the target detection problem in a multistatic integrated sensing and communication (ISAC) scenario characterized by the cell-free MIMO communication network deployment, where multiple radio units (RUs) in the network cooperate with each other for the sensing task. By exploiting the angle resolution from multiple arrays deployed in the network and the delay resolution from the communication signals, i.e., orthogonal frequency division multiplexing (OFDM) signals, we formulate a cooperative sensing problem with coherent data fusion of multiple RUs' observations and propose a sparse Bayesian learning (SBL)-based method, where the global coordinates of target locations are directly detected. Intensive numerical results indicate promising target detection performance of the proposed SBL-based method. Additionally, a theoretical analysis of the considered cooperative multistatic sensing task is provided using the pairwise error probability (PEP) analysis, which can be used to provide design insights, e.g., illumination and beam patterns, for the considered problem.

Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of \textit{$D$-valued $B$-rational functions on $E$}, denoted by $Int^R_B(E,D)$, which naturally extends the concepts of integer-valued polynomials, defined as $ Int^R_B(E,D) \:=\lbrace f \in B(X);\; f(E)\subseteq D\rbrace.$ The notion of $Int^R_B(E,D)$ boils down to the usual notion of integer-valued rational functions when the subset $E$ is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from $Int^R(E,D)$ to $D$.

We introduce a novel relaxation strategy for denoising hyperbolic-valued data. The main challenge is here the non-convexity of the hyperbolic sheet. Instead of considering the denoising problem directly on the hyperbolic space, we exploit the Euclidean embedding and encode the hyperbolic sheet using a novel matrix representation. For denoising, we employ the Euclidean Tikhonov and total variation (TV) model, where we incorporate our matrix representation. The major contribution is then a convex relaxation of the variational ans\"atze allowing the utilization of well-established convex optimization procedures like the alternating directions method of multipliers (ADMM). The resulting denoisers are applied to a real-world Gaussian image processing task, where we simultaneously restore the pixelwise mean and standard deviation of a retina scan series.

We justify Prandtl equations and higher order Prandtl expansion from the hydrodynamic limit of the Boltzmann equations. Our fluid data is of the form $\text{shear flow}$, plus $\sqrt{\kappa}$ order term in analytic spaces in $(x_1, x_2)\in\mathbb T^2$ and Sobolev in $x_3\in\mathbb{R}_+$. This work is the first to rigorously justify the Prandtl equations from the hydrodynamic limits of the Boltzmann equations. The novelty lies in obtaining estimates for the linearized Boltzmann equation with a diffusive boundary condition around a Prandtl layer shear flow in analytic spaces. The key techniques involve delicate commutator estimates and the use of local conservation law.

A family of 4-dimensional lattices $L_k \subset \mathbb{R}^2$ is defined. Each lattice is defined by 2 quadratic extensions and has a \emph{finite} number of unit vectors, but the number of unit vectors in the family is \emph{unbounded}. $L_3$ is the Moser lattice.

The Stieltjes (or sometimes called the Cauchy) transform is a fundamental object associated with probability measures, corresponding to the generating function of the moments. In certain applications such as free probability it is essential to compute the inverses of the Stieltjes transform, which might be multivalued. This paper establishes conditions bounding the number of inverses based on properties of the measure which can be combined with contour integral-based root finding algorithms to rigorously compute all inverses.

We prove equivalence between nonnegative distributional solutions of the fractional heat equation and caloric functions, i.e., functions satisfying the mean value property with respect to the space-time isotropic $\alpha$-stable process. We also provide sufficient conditions for the boundary and exterior data under which the solutions are classical and we give off-diagonal estimates for the derivatives of the Dirichlet heat kernel and the lateral Poisson kernel, which might be of their own interest.

This note provides the first example of a nontrivial connected component of the space of symplectic structures standard at infinity in dimension four.

The Hamiltonian reduction $\mathcal N///T$ of the nilpotent cone in $\mathfrak{sl}_n$ by the torus of diagonal matrices is a Nakajima quiver variety which admits a symplectic resolution $\widetilde{\mathcal N///T}$, and the corresponding BFN Coulomb branch is the affine closure $\overline{T^*(G/U)}$ of the cotangent bundle of the base affine space. We construct a surjective map $\mathbb C\left[\overline{T^*(G/U)}^{T\times B/U}\right] \twoheadrightarrow H^*\left(\widetilde{\mathcal N /// T}\right)$ of graded algebras, which the Hikita conjecture predicts to be an isomorphism. Our map is inherited from a related case of the Hikita conjecture and factors through Kirwan surjectivity for quiver varieties. We conjecture that many other Hikita maps can be inherited from that of a related dual pair.

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen closed subset. The case of a Riemannian manifold with boundary with the boundary as the measurement set appears is a classical geometric inverse problem arising from Gel'fand's inverse boundary spectral problem. Examples of spaces satisfying our assumptions include some non-simple Riemannian manifolds, Euclidean domains with non-trivial topology, and metric trees.

We introduce a new family of oriented manifolds with boundaries called the forest biassociahedra and forest bimultiplihedra, generalizing the standard biassociahedra. They are defined as moduli spaces of ascending-descending biforests and are expected to act as parameter spaces for operations defined on Morse and Floer chains in the context of compact Lie group actions. We study the structure of their boundary, and derive some algebraic notions of ``$f$-bialgebras'', as well as related notions of bimodules, morphisms and categories. This allows us to state some conjectures describing compact Lie group actions on Morse and Floer chains, and on Fukaya categories.

A folklore result attributed to P\'olya states that there are $(1 + o(1))2^{\binom{n}{2}}/n!$ non-isomorphic graphs on $n$ vertices. Given two graphs $G$ and $H$, we say that $G$ is a unique subgraph of $H$ if $H$ contains exactly one subgraph isomorphic to $G$. For an $n$-vertex graph $H$, let $f(H)$ be the number of non-isomorphic unique subgraphs of $H$ divided by $2^{\binom{n}{2}}/n!$ and let $f(n)$ denote the maximum of $f(H)$ over all graphs $H$ on $n$ vertices. In 1975, Erd\H{o}s asked whether there exists $\delta>0$ such that $f(n)>\delta$ for all $n$ and offered $\$100$ for a proof and $\$25$ for a disproof, indicating he does not believe this to be true. We verify Erd\H{o}s' intuition by showing that $f(n)\rightarrow 0$ as $n$ tends to infinity, i.e. no graph on $n$ vertices contains a constant proportion of all graphs on $n$ vertices as unique subgraphs.

We establish one-to-one correspondences between maximal antichains in products of two finite linear orders and other mathematical objects, such as certain alignments of two strings, walks on a grid, lattice paths, words of two or three letters. Leaning on these correspondences, we gather what is known about the number of maximal antichains in products of two finite linear orders and we establish some new results.

Recent works by Altschuler and Parrilo and the authors have shown that it is possible to accelerate the convergence of gradient descent on smooth convex functions, even without momentum, just by picking special stepsizes. In this paper, we provide a general theory for composing stepsize schedules capturing all recent advances in this area and more. We propose three notions of ``composable'' stepsize schedules with elementary associated composition operations for combining them. From these operations, in addition to recovering recent works, we construct three highly optimized sequences of stepsize schedules. We first construct optimized stepsize schedules of every length generalizing the exponentially spaced silver stepsizes. We then construct highly optimized stepsizes schedules for minimizing final objective gap or gradient norm, improving on prior rates by constants and, more importantly, matching or beating the numerically computed minimax optimal schedules. We conjecture these schedules are in fact minimax (information theoretic) optimal. Several novel tertiary results follow from our theory including recovery of the recent dynamic gradient norm minimizing short stepsizes and extending them to objective gap minimization.

To implement the dynamics of a projected Hamiltonian or Lindbladian, the quantum Zeno effect is a fundamental quantum phenomenon that approximates the effective dynamic by intersecting the Hamiltonian or Lindblad evolution by any quantum operation that converges to the desired projected subspace. Unlike the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is limited to the order $1/n$. In this work, we improve the convergence rate using a multi-product formula to achieve any power of $1/n^{K+1}$, employing a modified Chernoff Lemma, a modified Dunford-Segal approximation, and the holomorphic functional calculus. We then briefly illustrate this scheme using the bosonic cat code, as well as a broad class of examples governed by the `Bang-Bang' method used to decouple systems from their environment.

Quantum simulation in its current state faces experimental overhead in terms of physical space and cooling. We propose boson sampling as an alternative compact synthetic platform performing at room temperature. Identifying the capability of estimating matrix permanents, we explore the applicability of boson sampling for tackling the dynamics of quantum systems without having access to information about the full state vector. By mapping the time-evolution unitary of a Hamiltonian onto an interferometer via continuous-variable gate decompositions, we present proof-of-principle results of localization characteristics of a single particle. We study the dynamics of one-dimensional tight-binding systems in the clean and quasiperiodic-disordered limits to observe Bloch oscillations and dynamical localization, and the delocalization-to-localization phase transition in the Aubry- Andre-Harper model respectively. Our computational results obtained using boson sampling are in complete agreement with the dynamical and static results of non-interacting tight-binding systems obtained using conventional numerical calculations. Additionally, our study highlights the role of number of sampling measurements or shots for simulation accuracy.

In this paper, we generalize the universal approximation property of single-hidden-layer feed-forward neural networks beyond the classical formulation over compact domains. More precisely, by assuming that the activation function is non-polynomial, we derive universal approximation results for neural networks within function spaces over non-compact subsets of a Euclidean space, e.g., weighted spaces, $L^p$-spaces, and (weighted) Sobolev spaces over unbounded domains, where the latter includes the approximation of the (weak) derivatives. Furthermore, we provide some dimension-independent rates for approximating a function with sufficiently regular and integrable Fourier transform by neural networks with non-polynomial activation function.

We develop a method for multifidelity Kolmogorov-Arnold networks (KANs), which use a low-fidelity model along with a small amount of high-fidelity data to train a model for the high-fidelity data accurately. Multifidelity KANs (MFKANs) reduce the amount of expensive high-fidelity data needed to accurately train a KAN by exploiting the correlations between the low- and high-fidelity data to give accurate and robust predictions in the absence of a large high-fidelity dataset. In addition, we show that multifidelity KANs can be used to increase the accuracy of physics-informed KANs (PIKANs), without the use of training data.

The goal of this note is to study integrable properties of a generating function of the HOMFLY-PT invariants of the Hopf link colored with different representations. We demonstrate that such a generating function is a $\tau$-function of the KP hierarchy. Furthermore, this Hopf generating function in the case of composite representations, which is a generating function of the 4-point functions in topological string (corresponding to the resolved conifold with branes on the four external legs), is a $\tau$-function of the universal character(UC) hierarchy put on the topological locus. We also briefly discuss a simple matrix model associated with the UC hierarchy.

Studying the interplay between the geometry of the loss landscape and the optimization trajectories of simple neural networks is a fundamental step for understanding their behavior in more complex settings. This paper reveals the presence of topological obstruction in the loss landscape of shallow ReLU neural networks trained using gradient flow. We discuss how the homogeneous nature of the ReLU activation function constrains the training trajectories to lie on a product of quadric hypersurfaces whose shape depends on the particular initialization of the network's parameters. When the neural network's output is a single scalar, we prove that these quadrics can have multiple connected components, limiting the set of reachable parameters during training. We analytically compute the number of these components and discuss the possibility of mapping one to the other through neuron rescaling and permutation. In this simple setting, we find that the non-connectedness results in a topological obstruction, which, depending on the initialization, can make the global optimum unreachable. We validate this result with numerical experiments.

Parametric quantile regression is illustrated for the one parameter new unit Rayleigh distribution called Median Based Unit Rayleigh distribution (MBUR) distribution. The estimation process using re-parameterized maximum likelihood function is highlighted with real dataset example. The inference and goodness of fit is also explored.

We propose a framework for topological quantum computation using newly discovered non-semisimple analogs of topological quantum field theories in 2+1 dimensions. These enhanced theories offer more powerful models for quantum computation. The conventional theory of Ising anyons, which is believed to describe excitations in the $\nu = 5/2$ fractional quantum Hall state, is not universal for quantum computation via braiding of quasiparticles. However, we show that the non-semisimple theory introduces new anyon types that extend the Ising framework. By adding just one new anyon type, universal quantum computation can be achieved through braiding alone. This result opens new avenues for realizing fault-tolerant quantum computing in topologically ordered systems.

Bid shading has become a standard practice in the digital advertising industry, in which most auctions for advertising (ad) opportunities are now of first price type. Given an ad opportunity, performing bid shading requires estimating not only the value of the opportunity but also the distribution of the highest bid from competitors (i.e. the competitive landscape). Since these two estimates tend to be very noisy in practice, first-price auction participants need a bid shading policy that is robust against relatively significant estimation errors. In this work, we provide a max-min formulation in which we maximize the surplus against an adversary that chooses a distribution both for the value and the competitive landscape, each from a Kullback-Leibler-based ambiguity set. As we demonstrate, the two ambiguity sets are essential to adjusting the shape of the bid-shading policy in a principled way so as to effectively cope with uncertainty. Our distributionally robust bid shading policy is efficient to compute and systematically outperforms its non-robust counterpart on real datasets provided by Yahoo DSP.

This paper deals with the long-term behaviour and incidence of a vaccine-preventable contact disease, under the assumption that both vaccine protection and immunity after recovery are not lifelong. The mathematical model is developed in a stochastic markovian framework. The evolution of the disease in a finite population is thus represented by a three-dimensional continuous-time Markov chain, which is versatile enough to be able to compensate for the loss of protection by including vaccination before the onset of the outbreak and also during the course of the epidemics.

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees in which each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable lineages associated with the leaf. For the rooted binary unlabeled trees, these integers equal 1. We address a variety of enumerative problems concerning rooted binary perfect phylogenies with sample size $s$: the rooted binary unlabeled trees in which a sample of size $s$ lineages is distributed across the leaves of an unlabeled tree with $n$ leaves, $1 \leq n \leq s$. The enumerations further characterize the rooted binary perfect phylogenies, which include the rooted binary unlabeled trees, and which can provide a set of structures useful for various biological contexts.

As electrical generation becomes more distributed and volatile, and loads become more uncertain, controllability of distributed energy resources (DERs), regardless of their ownership status, will be necessary for grid reliability. Grid operators lack direct control over end-users' grid interactions, such as energy usage, but incentives can influence behavior -- for example, an end-user that receives a grid-driven incentive may adjust their consumption or expose relevant control variables in response. A key challenge in studying such incentives is the lack of data about human behavior, which usually motivates strong assumptions, such as distributional assumptions on compliance or rational utility-maximization. In this paper, we propose a general incentive mechanism in the form of a constrained optimization problem -- our approach is distinguished from prior work by modeling human behavior (e.g., reactions to an incentive) as an arbitrary unknown function. We propose feedback-based optimization algorithms to solve this problem that each leverage different amounts of information and/or measurements. We show that each converges to an asymptotically stable incentive with (near)-optimality guarantees given mild assumptions on the problem. Finally, we evaluate our proposed techniques in voltage regulation simulations on standard test beds. We test a variety of settings, including those that break assumptions required for theoretical convergence (e.g., convexity, smoothness) to capture realistic settings. In this evaluation, our proposed algorithms are able to find near-optimal incentives even when the reaction to an incentive is modeled by a theoretically difficult (yet realistic) function.

Basement relief gravimetry is crucial in geophysics, especially for oil exploration and mineral prospecting. It involves solving an inverse problem to infer geological model parameters from observed data. The model represents basement relief with constant-density prisms, and the data reflect gravitational anomalies from these prisms. Inverse problems are often ill-posed, meaning small data changes can lead to large solution variations. To mitigate this, regularization techniques like Tikhonov's are used to stabilize solutions. This study compares regularization methods applied to gravimetric inversion, including Smoothness Constraints, Total Variation, Discrete Cosine Transform (DCT), and Discrete Wavelet Transform (DWT) using Daubechies D4 wavelets. Optimization, particularly with Genetic Algorithms (GA), is used to find prism depths that best match observed anomalies. GA, inspired by natural selection, selects the best solutions to minimize the objective function. The results, evaluated through fit metrics and error analysis, show the effectiveness of all regularization methods and GA, with the Smoothness constraint performing best in synthetic models. For the real data model, all methods performed similarly.

This paper proposes a computationally tractable algorithm for learning infinite-horizon average-reward linear mixture Markov decision processes (MDPs) under the Bellman optimality condition. Our algorithm for linear mixture MDPs achieves a nearly minimax optimal regret upper bound of $\widetilde{\mathcal{O}}(d\sqrt{\mathrm{sp}(v^*)T})$ over $T$ time steps where $\mathrm{sp}(v^*)$ is the span of the optimal bias function $v^*$ and $d$ is the dimension of the feature mapping. Our algorithm applies the recently developed technique of running value iteration on a discounted-reward MDP approximation with clipping by the span. We prove that the value iteration procedure, even with the clipping operation, converges. Moreover, we show that the associated variance term due to random transitions can be bounded even under clipping. Combined with the weighted ridge regression-based parameter estimation scheme, this leads to the nearly minimax optimal regret guarantee.

Aiming to accelerate the training of large deep neural models (DNN) in an energy-efficient way, an analog in-memory computing (AIMC) accelerator emerges as a solution with immense potential. In AIMC accelerators, trainable weights are kept in memory without the need to move from memory to processors during the training, reducing a bunch of overhead. However, although the in-memory feature enables efficient computation, it also constrains the use of data parallelism since copying weights from one AIMC to another is expensive. To enable parallel training using AIMC, we propose synchronous and asynchronous pipeline parallelism for AIMC accelerators inspired by the pipeline in digital domains. This paper provides a theoretical convergence guarantee for both synchronous and asynchronous pipelines in terms of both sampling and clock cycle complexity, which is non-trivial since the physical characteristic of AIMC accelerators leads to analog updates that suffer from asymmetric bias. The simulations of training DNN on real datasets verify the efficiency of pipeline training.

In this paper we construct a non-perturbative action of the higher spin symmetry algebra on the gravitational phase space. We introduce a symmetry algebroid $\mathcal{T}$ which allows us to include radiation in an algebraic framework. We show that $\mathcal{T}$ admits a non-linear realization on the asymptotic phase space generated by a Noether charge defined non-perturbatively for all spins. Besides, this Noether charge is conserved in the absence of radiation. Moreover, at non radiative cuts, the algebroid can be restricted to the wedge symmetry algebra studied in ArXiv:2409.12178. The key ingredient for our construction is to consider field and time dependent symmetry parameters constrained to evolve according to equations of motion dual to (a truncation of) the asymptotic Einstein's equations. This result then guarantees that the underlying symmetry algebra is also represented canonically.

The discrete cosine transform (DCT) is a central tool for image and video coding because it can be related to the Karhunen-Lo\`eve transform (KLT), which is the optimal transform in terms of retained transform coefficients and data decorrelation. In this paper, we introduce 16-, 32-, and 64-point low-complexity DCT approximations by minimizing individually the angle between the rows of the exact DCT matrix and the matrix induced by the approximate transforms. According to some classical figures of merit, the proposed transforms outperformed the approximations for the DCT already known in the literature. Fast algorithms were also developed for the low-complexity transforms, asserting a good balance between the performance and its computational cost. Practical applications in image encoding showed the relevance of the transforms in this context. In fact, the experiments showed that the proposed transforms had better results than the known approximations in the literature for the cases of 16, 32, and 64 blocklength.

In this paper, we study learning-augmented algorithms for the Bahncard problem. The Bahncard problem is a generalization of the ski-rental problem, where a traveler needs to irrevocably and repeatedly decide between a cheap short-term solution and an expensive long-term one with an unknown future. Even though the problem is canonical, only a primal-dual-based learning-augmented algorithm was explicitly designed for it. We develop a new learning-augmented algorithm, named PFSUM, that incorporates both history and short-term future to improve online decision making. We derive the competitive ratio of PFSUM as a function of the prediction error and conduct extensive experiments to show that PFSUM outperforms the primal-dual-based algorithm.

Data assimilation will be essential for the management and expansion of geological carbon storage operations. In traditional data assimilation approaches a fixed set of geological hyperparameters, such as mean and standard deviation of log-permeability, is often assumed. Such hyperparameters, however, may be highly uncertain in practical CO2 storage applications. In this study, we develop a hierarchical data assimilation framework for carbon storage that treats hyperparameters as uncertain variables characterized by hyperprior distributions. To deal with the computationally intractable likelihood function in hyperparameter estimation, we apply a likelihood-free (or simulation-based) inference algorithm, specifically sequential Monte Carlo-based approximate Bayesian computation (SMC-ABC), to draw independent posterior samples of hyperparameters given dynamic monitoring-well data. In the second step we use an ensemble smoother with multiple data assimilation (ESMDA) procedure to provide posterior realizations of grid-block permeability. To reduce computational costs, a 3D recurrent R-U-Net deep-learning surrogate model is applied for forward function evaluations. The accuracy of the surrogate model is established through comparisons to high-fidelity simulation results. A rejection sampling (RS) procedure for data assimilation is applied to provide reference posterior results. Detailed data assimilation results from SMC-ABC-ESMDA are compared to those from the reference RS method. These include marginal posterior distributions of hyperparameters, pairwise posterior samples, and history matching results for pressure and saturation at the monitoring location. Close agreement is achieved with 'converged' RS results, for two synthetic true models, in all quantities considered. Importantly, the SMC-ABC-ESMDA procedure provides speedup of 1-2 orders of magnitude relative to RS for the two cases.

Complex contagion models that involve contagion along higher-order structures, such as simplicial complexes and hypergraphs, yield new classes of mean-field models. Interestingly, the differential equations arising from many such models often exhibit a similar form, resulting in qualitatively comparable global bifurcation patterns. Motivated by this observation, we investigate a generalized mean-field-type model that provides a unified framework for analysing a range of different models. In particular, we derive analytical conditions for the emergence of different bifurcation regimes exhibited by three models of increasing complexity -- ranging from three- and four-body interactions to two connected populations with both pairwise and three-body interactions. For the first two cases, we give a complete characterisation of all possible outcomes, along with the corresponding conditions on network and epidemic parameters. In the third case, we demonstrate that multistability is possible despite only three-body interactions. Our results reveal that single population models with three-body interactions can only exhibit simple transcritical transitions or bistability, whereas with four-body interactions multistability with two distinct endemic steady states is possible. Surprisingly, the two-population model exhibits multistability via symmetry breaking despite three-body interactions only. Our work sheds light on the relationship between equation structure and model behaviour and makes the first step towards elucidating mechanisms by which different system behaviours arise, and how network and dynamic properties facilitate or hinder outcomes.

Synchronization is a fundamental dynamical state of interacting oscillators, observed in natural biological rhythms and in the brain. Global synchronization which occurs when non-linear or chaotic oscillators placed on the nodes of a network display the same dynamics as received great attention in network theory. Here we propose and investigate Global Topological Dirac Synchronization on higher-order networks such as cell and simplicial complexes. This is a state where oscillators associated to simplices and cells of arbitrary dimension, coupled by the Topological Dirac operator, operate at unison. By combining algebraic topology with non-linear dynamics and machine learning, we derive the topological conditions under which this state exists and the dynamical conditions under which it is stable. We provide evidence of 1-dimensional simplicial complexes (networks) and 2-dimensional simplicial and cell complexes where Global Topological Dirac Synchronization can be observed. Our results point out that Global Topological Dirac Synchronization is a possible dynamical state of simplicial and cell complexes that occur only in some specific network topologies and geometries, the latter ones being determined by the weights of the higher-order networks

In this paper, we consider a class of convex programming problems with linear equality constraints, which finds broad applications in machine learning and signal processing. We propose a new adaptive balanced augmented Lagrangian (ABAL) method for solving these problems. The proposed ABAL method adaptively selects the stepsize parameter and enjoys a low per-iteration complexity, involving only the computation of a proximal mapping of the objective function and the solution of a linear equation. These features make the proposed method well-suited to large-scale problems. We then custom-apply the ABAL method to solve the ISAC beamforming design problem, which is formulated as a nonlinear semidefinite program in a previous work. This customized application requires careful exploitation of the problem's special structure such as the property that all of its signal-to-interference-and-noise-ratio (SINR) constraints hold with equality at the solution and an efficient computation of the proximal mapping of the objective function. Simulation results demonstrate the efficiency of the proposed ABAL method.

Count data is prevalent in various fields like ecology, medical research, and genomics. In high-dimensional settings, where the number of features exceeds the sample size, feature selection becomes essential. While frequentist methods like Lasso have advanced in handling high-dimensional count data, Bayesian approaches remain under-explored with no theoretical results on prediction performance. This paper introduces a novel probabilistic machine learning framework for high-dimensional count data prediction. We propose a pseudo-Bayesian method that integrates a scaled Student prior to promote sparsity and uses an exponential weight aggregation procedure. A key contribution is a novel risk measure tailored to count data prediction, with theoretical guarantees for prediction risk using PAC-Bayesian bounds. Our results include non-asymptotic oracle inequalities, demonstrating rate-optimal prediction error without prior knowledge of sparsity. We implement this approach efficiently using Langevin Monte Carlo method. Simulations and a real data application highlight the strong performance of our method compared to the Lasso in various settings.

We develop a hybrid classical-quantum method for solving the Lorenz system. We use the forward Euler method to discretize the system in time, transforming it into a system of equations. This set of equations is solved using the Variational Quantum Linear Solver (VQLS) algorithm. We present numerical results comparing the hybrid method with the classical approach for solving the Lorenz system. The simulation results demonstrate that the VQLS method can effectively compute solutions comparable to classical methods. The method is easily extended to solving similar nonlinear differential equations.

In this work, we provide a global condition for contraction with respect to an invariant Riemannian metric on reductive homogeneous spaces. Using left-invariant frames, vector fields on the manifold are horizontally lifted to the ambient Lie group, where the Levi-Civita connection is globally characterized as a real matrix multiplication. By linearizing in these left-invariant frames, we characterize contraction using matrix measures on real square matrices, avoiding the use of local charts. Applying this global condition, we provide a necessary condition for a prescribed subset of the manifold to possibly admit a contracting system with respect to an invariant metric. Applied to the sphere, this condition implies that no closed hemisphere can be contained in a contraction region. Finally, we apply our results to compute reachable sets for an attitude control problem.

As large language models (LLMs) become increasingly deployed across various industries, concerns regarding their reliability, particularly due to hallucinations-outputs that are factually inaccurate or irrelevant to user input-have grown. Our research investigates the relationship between the training process and the emergence of hallucinations to address a key gap in existing research that focuses primarily on post hoc detection and mitigation strategies. Using models from the Pythia suite (70M-12B parameters) and several hallucination detection metrics, we analyze hallucination trends throughout training and explore LLM internal dynamics. We introduce SEnsitive Neuron Dropout (SeND), a novel training protocol designed to mitigate hallucinations by reducing variance during training. SeND achieves this by deterministically dropping neurons with significant variability on a dataset, referred to as Sensitive Neurons. In addition, we develop an unsupervised hallucination detection metric, Efficient EigenScore (EES), which approximates the traditional EigenScore in 2x speed. This efficient metric is integrated into our protocol, allowing SeND to be both computationally scalable and effective at reducing hallucinations. Our empirical evaluation demonstrates that our approach improves LLM reliability at test time by up to 40% compared to normal training while also providing an efficient method to improve factual accuracy when adapting LLMs to domains such as Wikipedia and Medical datasets.

The current work is a qualitative study that aims to explore the implementation of Twisted and Coiled Artificial Muscles (TCAMs) for actuating and replicating the bending motion of an octopus-like soft robot arm underwater. Additionally, it investigates the impact of hydrostatic and dynamic forces from steady-state fluid flow on the arm's motion. The artificial muscles are lightweight and low-cost actuators that generate a high power-to-weight ratio, producing tensile force up to 12,600 times their own weight, which is close to the functionality of biological muscles. The "extended" Cosserat theory of rods is employed to formulate a quasi-static continuum model of arm motion, where the arm's cross-section is not only capable of rigid rotation but also deforms within its plane. This planar deformation of the arm cross-section aligns with the biological behavior of the octopus arm, where the stiffness of the hydrostat is directly induced by the incompressibility of the tissues. In line with the main goal, a constitutive model is derived for the material of the octopus arm to capture its characteristic behavior.

As large-scale neural recordings become common, many neuroscientific investigations are focused on identifying functional connectivity from spatio-temporal measurements in two or more brain areas across multiple sessions. Spatial-temporal data in neural recordings can be represented as matrix-variate data, with time as the first dimension and space as the second. In this paper, we exploit the multiple matrix-variate Gaussian Graphical model to encode the common underlying spatial functional connectivity across multiple sessions of neural recordings. By effectively integrating information across multiple graphs, we develop a novel inferential framework that allows simultaneous testing to detect meaningful connectivity for a target edge subset of arbitrary size. Our test statistics are based on a group penalized regression approach and a high-dimensional Gaussian approximation technique. The validity of simultaneous testing is demonstrated theoretically under mild assumptions on sample size and non-stationary autoregressive temporal dependence. Our test is nearly optimal in achieving the testable region boundary. Additionally, our method involves only convex optimization and parametric bootstrap, making it computationally attractive. We demonstrate the efficacy of the new method through both simulations and an experimental study involving multiple local field potential (LFP) recordings in the Prefrontal Cortex (PFC) and visual area V4 during a memory-guided saccade task.

In Bayesian optimization, a black-box function is maximized via the use of a surrogate model. We apply distributed Thompson sampling, using a Gaussian process as a surrogate model, to approach the multi-agent Bayesian optimization problem. In our distributed Thompson sampling implementation, each agent receives sampled points from neighbors, where the communication network is encoded in a graph; each agent utilizes a Gaussian process to model the objective function. We demonstrate a theoretical bound on Bayesian Simple Regret, where the bound depends on the size of the largest complete subgraph of the communication graph. Unlike in batch Bayesian optimization, this bound is applicable in cases where the communication graph amongst agents is constrained. When compared to sequential Thompson sampling, our bound guarantees faster convergence with respect to time as long as there is a fully connected subgraph of at least two agents. We confirm the efficacy of our algorithm with numerical simulations on traditional optimization test functions, illustrating the significance of graph connectivity on improving regret convergence.

Satellite communications present challenging characteristics. Continuous end-to-end connectivity may not be available due to the large distances between satellites. Moreover, resources such as link capacity and buffer memory may be limited. Routing in satellite networks is therefore both complex and crucial to avoid packet losses and long delays. The Delay Tolerant Network (DTN) paradigm has emerged as an efficient solution for managing these challenging networks. Contact Graph Routing (CGR), a deterministic routing algorithm, is one of the most popular DTN algorithms. CGR is compatible with the ``store, carry, and forward" principle, whereby a node receives a message and stores it in its buffer until a transmission opportunity becomes available. However, CGR relies on simplified models to incorporate potential constraints in the route search. For instance, the linear volume assumption is often used to consider capacity constraints. Moreover, capacity management and buffer management are mostly performed during the forwarding phase, once an issue has occurred. In this paper, we propose to take measures before or during the route search in order to find routes that respect both contact-capacity limits and node-buffer limits. We introduce the contact splitting and edge pruning operations to effectively account for the routing constraints. This ensures that CGR outputs the optimal solution among the subset of valid solutions. The proposed approach can also be used to book resources to be used in case of issues during the forwarding step.

We present a neural network-enhanced column generation (CG) approach for a parallel machine scheduling problem. The proposed approach utilizes an encoder-decoder attention model, namely the transformer and pointer architectures, to develop job sequences with negative reduced cost and thus generate columns to add to the master problem. By training the neural network offline and using it in inference mode to predict negative reduced costs columns, we achieve significant computational time savings compared to dynamic programming (DP). Since the exact DP procedure is used to verify that no further columns with negative reduced cost can be identified at termination, the optimality guarantee of the original CG procedure is preserved. For small to medium-sized instances, our approach achieves an average 45% reduction in computation time compared to solving the subproblems with DP. Furthermore, the model generalizes not only to unseen, larger problem instances from the same probability distribution but also to instances from different probability distributions than those presented at training time. For large-sized instances, the proposed approach achieves an 80% improvement in the objective value in under 500 seconds, demonstrating both its scalability and efficiency.

Instrumental variables (IV) estimation is a fundamental method in econometrics and statistics for estimating causal effects in the presence of unobserved confounding. However, challenges such as untestable model assumptions and poor finite sample properties have undermined its reliability in practice. Viewing common issues in IV estimation as distributional uncertainties, we propose DRIVE, a distributionally robust framework of the classical IV estimation method. When the ambiguity set is based on a Wasserstein distance, DRIVE minimizes a square root ridge regularized variant of the two stage least squares (TSLS) objective. We develop a novel asymptotic theory for this regularized regression estimator based on the square root ridge, showing that it achieves consistency without requiring the regularization parameter to vanish. This result follows from a fundamental property of the square root ridge, which we call ``delayed shrinkage''. This novel property, which also holds for a class of generalized method of moments (GMM) estimators, ensures that the estimator is robust to distributional uncertainties that persist in large samples. We further derive the asymptotic distribution of Wasserstein DRIVE and propose data-driven procedures to select the regularization parameter based on theoretical results. Simulation studies confirm the superior finite sample performance of Wasserstein DRIVE. Thanks to its regularization and robustness properties, Wasserstein DRIVE could be preferable in practice, particularly when the practitioner is uncertain about model assumptions or distributional shifts in data.

We study large deviations and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate $\sqrt{t} / \log(t)$. We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate $\widetilde{\mathcal{O}}(t^{-1/2})$. Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate $\mathcal{O}(t^{-1})$, improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to $o(t^{-1/4})$.

We study the problem of finding optimal sparse, manifold-aligned counterfactual explanations for classifiers. Canonically, this can be formulated as an optimization problem with multiple non-convex components, including classifier loss functions and manifold alignment (or \emph{plausibility}) metrics. The added complexity of enforcing \emph{sparsity}, or shorter explanations, complicates the problem further. Existing methods often focus on specific models and plausibility measures, relying on convex $\ell_1$ regularizers to enforce sparsity. In this paper, we tackle the canonical formulation using the accelerated proximal gradient (APG) method, a simple yet efficient first-order procedure capable of handling smooth non-convex objectives and non-smooth $\ell_p$ (where $0 \leq p < 1$) regularizers. This enables our approach to seamlessly incorporate various classifiers and plausibility measures while producing sparser solutions. Our algorithm only requires differentiable data-manifold regularizers and supports box constraints for bounded feature ranges, ensuring the generated counterfactuals remain \emph{actionable}. Finally, experiments on real-world datasets demonstrate that our approach effectively produces sparse, manifold-aligned counterfactual explanations while maintaining proximity to the factual data and computational efficiency.

Multi-Output Regression (MOR) has been widely used in scientific data analysis for decision-making. Unlike traditional regression models, MOR aims to simultaneously predict multiple real-valued outputs given an input. However, the increasing dimensionality of the outputs poses significant challenges regarding interpretability and computational scalability for modern MOR applications. As a first step to address these challenges, this paper proposes a Sparse \& High-dimensional-Output REgression (SHORE) model by incorporating additional sparsity requirements to resolve the output interpretability, and then designs a computationally efficient two-stage optimization framework capable of solving SHORE with provable accuracy via compression on outputs. Theoretically, we show that the proposed framework is computationally scalable while maintaining the same order of training loss and prediction loss before-and-after compression under arbitrary or relatively weak sample set conditions. Empirically, numerical results further validate the theoretical findings, showcasing the efficiency and accuracy of the proposed framework.

We consider the variational principle for the Lagrangian 1-form structure for long-range models of Calogero-Moser (CM) type. The multiform variational principle involves variations with respect to both the field variables as well as the independent variables corresponding to deformations of the time-curves in a multi-time space. The ensuing generalised Euler-Lagrange (gEL) equations comprise a system of multi-time EL equations, as well as constraints from so-called `alien derivatives' and `corner equations' arising from how variations on different coordinate curves match up. The closure relation, i.e. closedness of the Lagrange 1-form on solutions of the EL system, guarantees the stationarity of the action functional under deformation of the time-curves, and hence the multidimensional consistency of the corresponding gEL system. Using this as an integrability criterion on the Lagrangian level, we apply the system to some ans\"atze on the kinetic form of the Lagrangian components, associated with models of CM type without specifying the potentials. We show that from this integrability criterion the general elliptic form of the three systems, Calogero-Moser, Ruijsenaars-Schneider, and Goldfish systems, can be derived. We extend the analysis to an associated Hamiltonian formalism, via Noether's theorem and by applying Legendre transformations. Thus, the multiform variational principle leads to a system of generalised Hamilton equations describing Hamiltonian commuting flows for the mentioned elliptic models.

We study the generalization capabilities of Group Convolutional Neural Networks (GCNNs) with ReLU activation function by deriving upper and lower bounds for their Vapnik-Chervonenkis (VC) dimension. Specifically, we analyze how factors such as the number of layers, weights, and input dimension affect the VC dimension. We further compare the derived bounds to those known for other types of neural networks. Our findings extend previous results on the VC dimension of continuous GCNNs with two layers, thereby providing new insights into the generalization properties of GCNNs, particularly regarding the dependence on the input resolution of the data.

The problem of measuring conditional dependence between two random phenomena arises when a third one (a confounder) has a potential influence on the amount of information shared by the original pair. A typical issue in this challenging problem is the inversion of ill-conditioned autocorrelation matrices. This paper presents a novel measure of conditional dependence based on the use of incomplete unbiased statistics of degree two, which allows to re-interpret independence as uncorrelatedness on a finite-dimensional feature space. This formulation enables to prune data according to the observations of the confounder itself, thus avoiding matrix inversions altogether. Moreover, the proposed approach is articulated as an extension of the Hilbert-Schmidt independence criterion, which becomes expressible through kernels that operate on 4-tuples of data.

This study explores the application of rectangular finite elements to model the stress-strain behavior of isotropic and orthotropic materials exhibiting negative Poisson's ratio, known as auxetic materials, under static shear conditions within linear elasticity. By employing the classical compatible shape functions for linear interpolation and the incompatible shape functions for quadratic interpolation within a displacement-based finite element framework, the research assesses the effectiveness of these approaches in capturing the mechanical response of auxetic materials. Additionally, the analytical expression for an incompatible rectangular finite element applicable to orthotropic materials is proposed. Hexachiral and re-entrant honeycomb structures, known for their auxetic behavior, are modeled as continuous media with homogenized properties using analytical expressions for their effective material constants. The findings reveal that while the classical shape functions may suffice for displacement modeling, they fall short in accurately predicting stress distributions in auxetic materials. In contrast, the incompatible shape functions demonstrate superior performance in providing appropriate stress and displacement predictions. This work underscores the relevance of using the incompatible rectangular finite elements in the modeling of advanced materials with a negative Poisson's ratio. It provides computationally efficient approaches for calculating auxetic honeycomb structures and their derived multilayer composites.

This paper presents a novel approach to solving convex optimization problems by leveraging the fact that, under certain regularity conditions, any set of primal or dual variables satisfying the Karush-Kuhn-Tucker (KKT) conditions is necessary and sufficient for optimality. Similar to Theory-Trained Neural Networks (TTNNs), the parameters of the convex optimization problem are input to the neural network, and the expected outputs are the optimal primal and dual variables. A choice for the loss function in this case is a loss, which we refer to as the KKT Loss, that measures how well the network's outputs satisfy the KKT conditions. We demonstrate the effectiveness of this approach using a linear program as an example. For this problem, we observe that minimizing the KKT Loss alone outperforms training the network with a weighted sum of the KKT Loss and a Data Loss (the mean-squared error between the ground truth optimal solutions and the network's output). Moreover, minimizing only the Data Loss yields inferior results compared to those obtained by minimizing the KKT Loss. While the approach is promising, the obtained primal and dual solutions are not sufficiently close to the ground truth optimal solutions. In the future, we aim to develop improved models to obtain solutions closer to the ground truth and extend the approach to other problem classes.

We modify the R\'enyi (1961) axioms for entropy to apply to negative (``signed") measures as arise, for example, in phase-space representations of quantum mechanics. We obtain two new measures of (lack of) information about a system -- which we propose as signed analogs to classical Shannon entropy and classical R\'enyi entropy, respectively. We show that signed R\'enyi entropy witnesses non-classicality of a system. Specifically, a measure has at least one negative component if and only if signed R\'enyi $\alpha$-entropy is negative for some $\alpha > 1$. The corresponding non-classicality test does not work with signed Shannon entropy. We next show that signed R\'enyi $2k$-entropy, when $k$ is a positive integer, is Schur-concave. (An example shows that signed Shannon entropy is not Schur-concave.) We then establish an abstract quantum H-theorem for signed measures. We prove that signed R\'enyi $2k$-entropy is non-decreasing under classical (``decohering") evolution of a signed measure, where the latter could be a Wigner function or other phase-space representation of a quantum system. (An example shows that signed Shannon entropy may be non-monotonic.) We also provide a characterization of the Second Law for signed R\'enyi $2$-entropy in terms of what we call eventual classicalization of evolution of a system. We conclude with an argument that signed R\'enyi $2$-entropy of the Wigner function is constant under Moyal bracket evolution.

We study agents acting in an unknown environment where the agent's goal is to find a robust policy. We consider robust policies as policies that achieve high cumulative rewards for all possible environments. To this end, we consider agents minimizing the maximum regret over different environment parameters, leading to the study of minimax regret. This research focuses on deriving information-theoretic bounds for minimax regret in Markov Decision Processes (MDPs) with a finite time horizon. Building on concepts from supervised learning, such as minimum excess risk (MER) and minimax excess risk, we use recent bounds on the Bayesian regret to derive minimax regret bounds. Specifically, we establish minimax theorems and use bounds on the Bayesian regret to perform minimax regret analysis using these minimax theorems. Our contributions include defining a suitable minimax regret in the context of MDPs, finding information-theoretic bounds for it, and applying these bounds in various scenarios.

A graph $G=(V,E)$ is defined as a star-$k$-PCG when it is possible to assign a positive real number weight $w$ to each vertex $V$, and define $k$ distinct intervals $I_1, I_2, \ldots I_k$, in such a way that there is an edge $uv$ in $E$ if and only if the sum of the weights of vertices $u$ and $v$ falls within the union of these intervals. The star-$k$-PCG class is connected to two significant categories of graphs, namely PCGs and multithreshold graphs. The star number of a graph $G$, is the smallest $k$ for which $G$ is a star-$k$-PCG. In this paper, we study the effects of various graph operations, such as the addition of twins, pendant vertices, universal vertices, or isolated vertices, on the star number of the graph resulting from these operations. As a direct application of our results, we determine the star number of lobster graphs and provide an upper bound for the star number of acyclic graphs.

It has been known for some time that generalised geometry provides a particularly elegant rewriting of the action and symmetries of 10-dimensional supergravity theories, up to the lowest nontrivial order in fermions. By exhibiting the full symmetry calculations in the second-order formalism, we show in the $\mathcal N=1$ case that this analysis can be upgraded to all orders in fermions and we obtain a strikingly simple form of the action as well as of the supersymmetry transformations, featuring overall only five higher-fermionic terms. Surprisingly, even after expressing the action in terms of classical (non-generalised geometric) variables one obtains a simplification of the usual formulae. This in particular confirms that generalised geometry provides the natural set of variables for studying (the massless level of) string theory. We also show how this new reformulation implies the compatibility of the Poisson-Lie T-duality with the equations of motion of the full supergravity theory.

Nonlinear systems of polynomial equations arise naturally in many applied settings, for example loglinear models on contingency tables and Gaussian graphical models. The solution sets to these systems over the reals are often positive dimensional spaces that in general may be very complicated yet have very nice local behavior almost everywhere. Standard methods in real algebraic geometry for describing positive dimensional real solution sets include cylindrical algebraic decomposition and numerical cell decomposition, both of which can be costly to compute in many practical applications. In this work we communicate recent progress towards a Monte Carlo framework for exploring such real solution sets. After describing how to construct probability distributions whose mass focuses on a variety of interest, we describe how Hamiltonian Monte Carlo methods can be used to sample points near the variety that may then be moved to the variety using endgames. We conclude by showcasing trial experiments using practical implementations of the method in the Bayesian engine Stan.

Regularization, whether explicit in terms of a penalty in the loss or implicit in the choice of algorithm, is a cornerstone of modern machine learning. Indeed, controlling the complexity of the model class is particularly important when data is scarce, noisy or contaminated, as it translates a statistical belief on the underlying structure of the data. This work investigates the question of how to choose the regularization norm $\lVert \cdot \rVert$ in the context of high-dimensional adversarial training for binary classification. To this end, we first derive an exact asymptotic description of the robust, regularized empirical risk minimizer for various types of adversarial attacks and regularization norms (including non-$\ell_p$ norms). We complement this analysis with a uniform convergence analysis, deriving bounds on the Rademacher Complexity for this class of problems. Leveraging our theoretical results, we quantitatively characterize the relationship between perturbation size and the optimal choice of $\lVert \cdot \rVert$, confirming the intuition that, in the data scarce regime, the type of regularization becomes increasingly important for adversarial training as perturbations grow in size.

We introduce LDAdam, a memory-efficient optimizer for training large models, that performs adaptive optimization steps within lower dimensional subspaces, while consistently exploring the full parameter space during training. This strategy keeps the optimizer's memory footprint to a fraction of the model size. LDAdam relies on a new projection-aware update rule for the optimizer states that allows for transitioning between subspaces, i.e., estimation of the statistics of the projected gradients. To mitigate the errors due to low-rank projection, LDAdam integrates a new generalized error feedback mechanism, which explicitly accounts for both gradient and optimizer state compression. We prove the convergence of LDAdam under standard assumptions, and show that LDAdam allows for accurate and efficient fine-tuning and pre-training of language models.

Line graph transformation has been widely studied in graph theory, where each node in a line graph corresponds to an edge in the original graph. This has inspired a series of graph neural networks (GNNs) applied to transformed line graphs, which have proven effective in various graph representation learning tasks. However, there is limited theoretical study on how line graph transformation affects the expressivity of GNN models. In this study, we focus on two types of graphs known to be challenging to the Weisfeiler-Leman (WL) tests: Cai-F\"urer-Immerman (CFI) graphs and strongly regular graphs, and show that applying line graph transformation helps exclude these challenging graph properties, thus potentially assist WL tests in distinguishing these graphs. We empirically validate our findings by conducting a series of experiments that compare the accuracy and efficiency of graph isomorphism tests and GNNs on both line-transformed and original graphs across these graph structure types.

This paper considers the problem for finding the $(\delta,\epsilon)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $\epsilon$ by a factor of $\epsilon^{-2/3}$.

We propose a scheme for the construction of deformed matrix geometries using Landau models. The level projection method cannot be applied straightforwardly to extract matrix geometries to the Landau models on deformed manifolds, as they do not generally exhibit degenerate energy levels (Landau levels). We overcome this problem by exploiting the idea of spectral flow. Taking a symmetric matrix geometry as a reference point of the spectral flow, we evolve the matrix geometry by deforming the Landau model. In this process, unitarity is automatically preserved. The explicit matrix realization of the coordinates is derived straightforwardly even for a non-perturbative deformation. We clarify the basic properties of the matrix geometries through the analysis of concrete models. The matrix geometries of an expanding two-sphere and ellipsoids are investigated using the non-relativistic and relativistic Landau models. The obtained ellipsoidal matrix geometries show behavior quantitatively different in each Landau level, but qualitatively similar to their classical counterpart. The difference between the ellipsoidal matrix geometry and the fuzzy ellipsoid is investigated numerically.

We explore how dynamic entry deterrence operates through feedback strategies in markets experiencing stochastic demand fluctuations. The incumbent firm, aware of its own cost structure, can deter a potential competitor by strategically adjusting prices. The potential entrant faces a one-time, irreversible decision to enter the market, incurring a fixed cost, with profits determined by market conditions and the incumbent's hidden type. Market demand follows a Chan-Karolyi-Longstaff-Sanders Brownian motion. If the demand is low, the threat of entry diminishes, making deterrence less advantageous. In equilibrium, a weak incumbent may be incentivized to reveal its type by raising prices. We derive an optimal equilibrium using path integral control, where the entrant enters once demand reaches a high enough level, and the weak incumbent mixes strategies between revealing itself when demand is sufficiently low.

Electric vehicles (EVs) play a significant role in enhancing the sustainability of transportation systems. However, their widespread adoption is hindered by inadequate public charging infrastructure, particularly to support long-distance travel. Identifying optimal charging station locations in large transportation networks presents a well-known NP-hard combinatorial optimization problem, as the search space grows exponentially with the number of potential charging station locations. This paper introduces a quantum search-based optimization algorithm designed to enhance the efficiency of solving this NP-hard problem for transportation networks. By leveraging quantum parallelism, amplitude amplification, and quantum phase estimation as a subroutine, the optimal solution is identified with a quadratic improvement in complexity compared to classical exact methods, such as branch and bound. The detailed design and complexity of a resource-efficient quantum circuit are discussed.

In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordstr\"om family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" $\mathfrak M_\mathrm{stab}$ of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordstr\"om family, such that any data in $\mathfrak M_\mathrm{stab}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordstr\"om family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordstr\"om family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordstr\"om black holes in the full nonlinear Einstein-Maxwell-scalar field model.

We provide a rigorous analysis of implicit regularization in an overparametrized tensor factorization problem beyond the lazy training regime. For matrix factorization problems, this phenomenon has been studied in a number of works. A particular challenge has been to design universal initialization strategies which provably lead to implicit regularization in gradient-descent methods. At the same time, it has been argued by Cohen et. al. 2016 that more general classes of neural networks can be captured by considering tensor factorizations. However, in the tensor case, implicit regularization has only been rigorously established for gradient flow or in the lazy training regime. In this paper, we prove the first tensor result of its kind for gradient descent rather than gradient flow. We focus on the tubal tensor product and the associated notion of low tubal rank, encouraged by the relevance of this model for image data. We establish that gradient descent in an overparametrized tensor factorization model with a small random initialization exhibits an implicit bias towards solutions of low tubal rank. Our theoretical findings are illustrated in an extensive set of numerical simulations show-casing the dynamics predicted by our theory as well as the crucial role of using a small random initialization.

We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a $\Lambda$-fold cup product that can produce a logical operator in the $\Lambda$-th level of the Clifford hierarchy on $\Lambda$ copies of the same quantum code, which we call the copy-cup gate. For any desired $\Lambda$, we can construct several families of quantum codes that support gates in the $\Lambda$-th level with various asymptotic code parameters.