We study $\mathbb{E}_n$-Koszul duality for pairs of algebras of the form $\mathrm{C}_{\bullet}(\Omega^{n}_*X;\Bbbk) \leftrightarrow \mathrm{C}^{\bullet}(X;\Bbbk)$, and the closely related question of $n$-affineness for Betti stacks. It was expected, but not known, that $\mathbb{E}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the $(\infty,n)$-category of iterated modules over $\mathrm{C}_{\bullet}(\Omega_*^{n+1}X;\Bbbk)$ is equivalent to the $(\infty,n)$-category of quasi-coherent sheaves of $(\infty,n-1)$-categories on $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$, where $\mathrm{cSpec}(\mathrm{C}^{\bullet}(X;\Bbbk))$ is the cospectrum of $\mathrm{C}^{\bullet}(X;\Bbbk)$. By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on $X$, $n\mathbf{LocSysCat}^{n-1}(X;\Bbbk)$. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of $\mathbb{E}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $\pi_{n+1}(X)$ is an obstruction to $n$-affineness.

As semantic communication (SemCom) gains increasing attention as a novel communication paradigm, ensuring the security of transmitted semantic information over open wireless channels becomes crucial. Existing secure SemCom solutions often lack explicit control over security. To address this, we propose a coding-enhanced jamming approach for secure SemCom over wiretap channels. This approach integrates deep joint source and channel coding (DeepJSCC) with neural network-based digital modulation, enabling controlled jamming through two-layer superposition coding. The outer constellation sequence encodes the source image, while the inner constellation sequence, derived from a secret image, acts as the jamming signal. By minimizing the mutual information between the outer and inner constellation sequences, the jamming effect is enhanced. The jamming signal is superposed on the outer constellation sequence, preventing the eavesdropper from recovering the source image. The power allocation coefficient (PAC) in the superposition coding can be adjusted to control system security. Experiments show that our approach matches existing methods in security while significantly improving reconstruction performance across varying channel signal-to-noise ratios (SNRs) and compression ratios.

Banyaga and Hurtubise defined the Morse-Bott-Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. In this paper, we unify the five conditions into only one degeneracy condition. This allows for a simpler definition of Morse-Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse-Smale function is quasi-isomorphic to the Morse-Smale-Witten chain complex. As a result, we obtain another proof of the Morse Homology Theorem.

Following our previous works on $C^*$-graph algebras and the associated Cuntz-Krieger graph families, in this paper we will try to have a look at the colored version of these structures and to see what a $C^*$-colored graph algebra might mean by employing some constructive examples very close to the toy example used in our previous works, and we also will try to study their graph theoretical properties as possible.

In this paper, by virtue of a determinantal formula for derivatives of the ratio between two differentiable functions, in view of the Fa\`a di Bruno formula, and with the help of several identities and closed-form formulas for the partial Bell polynomials $\operatorname{B}_{n,k}$, the author establishes thirteen Maclaurin series expansions of the functions \begin{align*} &\ln\frac{\operatorname{e}^x+1}{2}, && \ln\frac{\operatorname{e}^x-1}{x}, && \ln\cosh x, \\ &\ln\frac{\sinh x}{x}, && \biggl[\frac{\ln(1+x)}{x}\biggr]^r, && \biggl(\frac{\operatorname{e}^x-1}{x}\biggr)^r \end{align*} for $r=\pm\frac{1}{2}$ and $r\in\mathbb{R}$ in terms of the Dirichlet eta function $\eta(1-2k)$, the Riemann zeta function $\zeta(1-2k)$, and the Stirling numbers of the first and second kinds $s(n,k)$ and $S(n,k)$. presents four determinantal expressions and three recursive relations for the Bernoulli numbers $B_{2n}$. finds out three closed-form formulas for the Bernoulli numbers $B_{2n}$ and the generalized Bernoulli numbers $B_n^{(r)}$ in terms of the Stirling numbers of the second kind $S(n,k)$, and deduce two combinatorial identities for the Stirling numbers of the second kind $S(n,k)$. acquires two combinatorial identities, which can be regarded as diagonal recursive relations, involving the Stirling numbers of the first and second kinds $s(n,k)$ and $S(n,k)$. recovers an integral representation and a closed-form formula, and establish an alternative explicit and closed-form formula, for the Bernoulli numbers of the second kind $b_n$ in terms of the Stirling numbers of the first kind $s(n,k)$. obtains three identities connecting the Stirling numbers of the first and second kinds $s(n,k)$ and $S(n,k)$.

We show a construction for dense 3-uniform linear hypergraphs without $3\times 3$ grids, improving the lower bound on its Tur\'an number.

We investigate random partitions of complete graphs defined by Poissonian emsembles of Markov loops

In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom -- the H\"{o}lder divergence and the functional density power divergence (FDPD) -- based on their intersection and generalization. These divergence families include the density power divergence and the $\gamma$-divergence as special cases. First, we prove that the intersection of the H\"{o}lder divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001). Subsequently, motivated by the fact that H\"{o}lder's inequality is used in the proofs of nonnegativity for both the H\"{o}lder divergence and the FDPD, we define a generalized divergence family, referred to as the $\xi$-H\"{o}lder divergence. The nonnegativity of the $\xi$-H\"{o}lder divergence is established through a combination of the inequalities used to prove the nonnegativity of the H\"{o}lder divergence and the FDPD. Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the $\xi$-H\"{o}lder divergence. Finally, we prove that imposing the mathematical structure of the H\"{o}lder score on a composite scoring rule results in the $\xi$-H\"{o}lder divergence.

Nonlinear spectral problems arise across a range of fields, including mechanical vibrations, fluid-solid interactions, and photonic crystals. Discretizing infinite-dimensional nonlinear spectral problems often introduces significant computational challenges, particularly spectral pollution and invisibility, which can distort or obscure the true underlying spectrum. We present the first general, convergent computational method for computing the spectra and pseudospectra of nonlinear spectral problems. Our approach uses new results on nonlinear injection moduli and requires only minimal continuity assumptions: specifically, continuity with respect to the gap metric on operator graphs, making it applicable to a broad class of problems. We use the Solvability Complexity Index (SCI) hierarchy, which has recently been used to resolve the classical linear problem, to systematically classify the computational complexity of nonlinear spectral problems. Our results establish the optimality of the method and reveal that Hermiticity does not necessarily simplify the computational complexity of these nonlinear problems. Comprehensive examples -- including nonlinear shifts, Klein--Gordon equations, wave equations with acoustic boundary conditions, time-fractional beam equations, and biologically inspired delay differential equations -- demonstrate the robustness, accuracy, and broad applicability of our methodology.

A convergent power series solution is obtained for the SIR model, using an asymptotically motivated gauge function. For certain choices of model parameter values, the series converges over the full physical domain (i.e., for all positive time). Furthermore, the radius of convergence as a function of nondimensionalized initial susceptible and infected populations is obtained via a numerical root test.

We consider the robust version of a multi-commodity network flow problem. The robustness is defined with respect to the deletion, or failure, of edges. While the flow problem itself is a polynomially-sized linear program, its robust version is a saddle-point problem with discrete variables. We present two approaches for the solution of the robust network flow problem. One way is to formulate the problem as a bigger linear program. The other is to solve a multi-level optimization problem, where the linear programs appearing at the lower level can be solved by the dual simplex method with a warm start. We then consider the problem of robustifying the network. This is accomplished by optimally using a fixed budget for strengthening certain edges, i.e., increasing their capacity. This problem is solved by a sequence of linear programs at the upper level, while at the lower levels the mentioned dual simplex algorithm is employed.

We apply the resonance method to obtain large values of general exponential sums with positive coefficients. As applications, we show improved $\Omega$-bounds for Dirichlet and Piltz divisor problems, Gauss circle Problem, and error term for the mean square of the Riemann zeta function and the Dirichlet $L$-function.

We consider a proper parabolic subalgebra p of a simple Lie algebra g and the Inonu-Wigner contraction of p with respect to its decomposition into its standard Levi factor and its nilpotent radical : this is the Lie algebra which is isomorphic to p as a vector space, but where the nilpotent radical becomes an abelian ideal of this contraction. The study of the algebra of symmetric semi-invariants under the adjoint action associated with such a contraction was initiated in my paper entitled : Symmetric Semi-Invariants for some Inonu-Wigner Contractions-I, published in Transformation Groups, January 2025, wherein a lower bound for the formal character of this algebra was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when p is a maximal parabolic subalgebra in a classical simple Lie algebra g in type B, when the Levi subalgebra of p is associated with the set of all simple roots without a simple root of even index with Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for the algebra of symmetric semi-invariants associated with such a contraction and the polynomiality of such an algebra follows.

For any two partitions $\lambda$ and $\mu$ of a positive integer $N$, let $\chi_{\lambda}(\mu)$ denote the value of the irreducible character of the symmetric group $S_{N}$ associated with $\lambda$, evaluated at the conjugacy class of elements whose cycle type is determined by $\mu$. The quantity $Z_{t}(N)$ is defined as $$ Z_{t}(N):= \#\{(\lambda,\mu): \chi_{\lambda}(\mu) = 0 \quad \text{with $\lambda$ a $t$-core}\}. $$ We establish the bound $$ \max\limits_{1\leq t \leq N} Z_{t}(N) \geq c_{t}(N)p(N-t)\geq \frac{2\pi p(N)^{2}}{1.01e\sqrt{6N}\log N} \biggl(1+O(N^{-\frac{1}{2}}\log N)\biggr), $$ where $p(N)$ denotes the number of partitions of $N$. Also, we give lower bounds for $Z_{t}(N)$ in different ranges of $t$ and obtain a lower bound for the total number of zeros in the character table of $S_N$.

We prove the correctness of the AKS algorithm \cite{AKS} within the bounded arithmetic theory $T^{count}_2$ or, equivalently, the first-order consequence of the theory $VTC^0$ expanded by the smash function, which we denote by $VTC^0_2$. Our approach initially demonstrates the correctness within the theory $S^1_2 + iWPHP$ augmented by two algebraic axioms and then show that they are provable in $VTC^0_2$. The two axioms are: a generalized version of Fermat's Little Theorem and an axiom adding a new function symbol which injectively maps roots of polynomials over a definable finite field to numbers bounded by the degree of the given polynomial. To obtain our main result, we also give new formalizations of parts of number theory and algebra: $\bullet$ In $PV_1$: We formalize Legendre's Formula on the prime factorization of $n!$, key properties of the Combinatorial Number System and the existence of cyclotomic polynomials over the finite fields $Z/p$. $\bullet$ In $S^1_2$: We prove the inequality $lcm(1,\dots, 2n) \geq 2^n$. $\bullet$ In $VTC^0$: We verify the correctness of the Kung--Sieveking algorithm for polynomial division.

We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of non-intersecting paths. The starting point of our analysis is a formula for the correlation kernel due to Duits and Kuijlaars which involves the Christoffel-Darboux kernel of a particular family of non-Hermitian orthogonal polynomials. Our main results are split into two parts: the first part concerns the family of orthogonal polynomials, and the second concerns the behavior of the boundary of the so-called arctic curve. In the first half, we identify the orthogonal polynomials as a non-standard instance of little $q$-Jacobi polynomials and compute their large degree asymptotics in the $q \to 1$ regime. A consequence of this analysis is a proof that the zeros of the orthogonal polynomials accumulate on an arc of a circle and an asymptotic formula for the Christoffel-Darboux kernel. In the second half, we use these asymptotics to show that the boundary of the liquid region converges to the Airy process, in the sense of finite dimensional distributions, away from the boundary of the hexagon. At inflection points of the arctic curve, we show that we do not need to subtract/add a parabola to the Airy line ensemble, and this effect persists at distances which are $o(N^{-2/9})$ in the tangent direction.

We show that colorable hierarchically hyperbolic groups (HHGs) admit asymptotically CAT(0) metrics, that is, roughly, metrics where the CAT(0) inequality holds up to sublinear error in the size of the triangle. We use the asymptotically CAT(0) metrics to construct contractible simplicial complexes and compactifications that provide $\mathcal{Z}$-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit $\mathcal{Z}$-structures. As an application, we prove that many HHGs satisfy the Farrell--Jones Conjecture, including extra large-type Artin groups. To construct asymptotically CAT(0) metrics, we show that hulls of finitely many points in a colorable HHGs can be approximated by CAT(0) cube complexes in a way that adding a point to the finite set corresponds, up to finitely many hyperplanes deletions, to a convex embedding.

We completely determine the autotopism group of the (as of now) largest family of commutative semifields found by G\"olo\u{g}lu and K\"olsch. Since this family of semifields generally does not have large nuclei, this process is considerably harder than for families considered in preceding work. Our results show that all autotopisms are semilinear over the degree 2 subfield and that the autotopism group is always solvable. Using known connections, our results also completely determine the automorphism groups of the associated rank-metric codes and the collineation groups of the associated translation planes.

We employ a port-Hamiltonian approach to model nonlinear rigid multibody systems subject to both position and velocity constraints. Our formulation accommodates Cartesian and redundant coordinates, respectively, and captures kinematic as well as gyroscopic effects. The resulting equations take the form of nonlinear differential-algebraic equations that inherently preserve an energy balance. We show that the proposed class is closed under interconnection, and we provide several examples to illustrate the theory.

In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and $n$ additional vertices. In this note, we study how many ways a combinatorial triangulation can be drawn as geometric triangulation, that is, with straight-line segments, on a given point set in the plane. Our central contribution is that there exists a combinatorial triangulation with n vertices that can be drawn in at least $\Omega(1,31^n)$ ways on a set of n points as different geometric triangulations. We also show an upper bound on the number of drawings of a combinatorial triangulation on the so-called double chain point set.

Singular arcs emerge in the solutions of Optimal Control Problems (OCPs) when the optimal inputs on some finite time intervals cannot be directly obtained via the optimality conditions. Solving OCPs with singular arcs often requires tailored treatments, suitable for offline trajectory optimization. This approach can become increasingly impractical for online closed-loop implementations, especially for large-scale engineering problems. Recent development of Integrated Residual Methods (IRM) have indicated their suitability for handling singular arcs; the convergence of error measures in IRM automatically suppresses singular arc-induced fluctuations and leads to non-fluctuating solutions more suitable for practical problems. Through several examples, we demonstrate the advantages of solving OCPs with singular arcs using {IRM} under an economic model predictive control framework. In particular, the following observations are made: (i) IRM does not require special treatment for singular arcs, (ii) it solves the OCPs reliably with singular arc fluctuation suppressed, and (iii) the closed-loop results closely match the analytic optimal solutions.

We establish a central limit theorem and large deviations principle that characterises small noise fluctuations of the generalised Dean--Kawasaki stochastic PDE. The fluctuations agree to first order with fluctuations of certain interacting particle systems, such as the zero range process, about their hydrodynamic limits. Our main contribution is that we are able to consider stochastic PDEs on general $C^2$ bounded domains with Dirichlet boundary conditions. On the level of particles, the boundary condition corresponds to absorption or injection of particles at the boundary.

We resolve a question of Gillespie, Griffin, and Levinson that asks for a combinatorial bijection between two classes of trivalent trees, tournament trees and slide trees, that both naturally arise in the intersection theory of the moduli space $\overline{M}_{0,n+3}$ of stable genus zero curves with $n+3$ marked points. Each set of trees enumerates the same intersection product of certain pullbacks of $\psi$ classes under forgetting maps. We give an explicit combinatorial bijection between these two sets of trees using an insertion algorithm. We also classify the words that appear on the slide trees of caterpillar shape via pattern avoidance conditions.

We utilise the two principles of decoupling introduced in [arXiv:2407.16108] to prove decoupling for two types of surfaces exhibiting radial symmetry. The first type are surfaces of revolution in $\mathbb R^n$ generated by smooth surfaces in $\mathbb R^3$. The second type of surfaces are graphs of trivariate homogeneous smooth functions of nonzero degree.

Contraction metrics are crucial in control theory because they provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems. However, identifying these metrics for complex nonlinear systems remains an open challenge due to the lack of scalable and effective tools. This paper explores the approach of learning verifiable contraction metrics parametrized as neural networks (NNs) for discrete-time nonlinear dynamical systems. While prior works on formal verification of contraction metrics for general nonlinear systems have focused on convex optimization methods (e.g. linear matrix inequalities, etc) under the assumption of continuously differentiable dynamics, the growing prevalence of NN-based controllers, often utilizing ReLU activations, introduces challenges due to the non-smooth nature of the resulting closed-loop dynamics. To bridge this gap, we establish a new sufficient condition for establishing formal neural contraction metrics for general discrete-time nonlinear systems assuming only the continuity of the dynamics. We show that from a computational perspective, our sufficient condition can be efficiently verified using the state-of-the-art neural network verifier $\alpha,\!\beta$-CROWN, which scales up non-convex neural network verification via novel integration of symbolic linear bound propagation and branch-and-bound. Built upon our analysis tool, we further develop a learning method for synthesizing neural contraction metrics from sampled data. Finally, our approach is validated through the successful synthesis and verification of NN contraction metrics for various nonlinear examples.

Bifurcation is one of the major topics in the theory of dynamical systems. It characterizes the nature of qualitative changes in parametrized dynamical systems. In this work, we study combinatorial bifurcations within the framework of combinatorial multivector field theory--a young but already well-established theory providing a combinatorial model for continuous-time dynamical systems (or simply, flows). We introduce Conley-Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. The barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of Conley-Morse persistence barcode builds upon ideas from topological data analysis (TDA). Specifically, we consider a persistence module obtained from a zigzag filtration of topological pairs (formed by index pairs defining the Conley index) over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them with algorithms from TDA known for processing zigzag filtrations.

We investigate the global stability of large solutions to the compressible isentropic Navier-Stokes equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the strong solutions converge to an equilibrium state exponentially in the $L^2$-norm provided the density is essentially uniform-in-time bounded from above. Moreover, we obtain that the density converges to its equilibrium state exponentially in the $L^\infty$-norm if additionally the initial density is bounded away from zero. Furthermore, we derive that the vacuum states will not vanish for any time provided vacuum appears (even at a point) initially. This is the first result concerning the global stability for large strong solutions of compressible Navier-Stokes equations with vacuum in 3D general bounded domains.

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness $\varepsilon$, which scales with the minimum grid size. This reformulation extends the problem to a regular domain, incorporating boundary conditions via singular source terms. In this work, we analyze the convergence of a DDM approximation problem with transmission-type Neumann boundary conditions. We prove that the energy functional of the diffuse domain problem $\Gamma$--converges to the energy functional of the original problem as $\varepsilon \to 0$. Additionally, we show that the solution of the diffuse domain problem strongly converges in $H^1(\Omega)$, up to a subsequence, to the solution of the original problem, as $\varepsilon \to 0$.

This paper proposes critical pragmatism as a philosophy of sustainable mathematics education to bridge the gap between critical theory and the existing patchwork implementations. Combining existential sustainability as a holistic concept with pragmatic frameworks from the ethics in mathematics education literature creates a foundation enabling critical reflection and pragmatic implementation. We outline how their synthesis naturally leads to a three-stage implementation strategy: cultivating an ethical classroom culture, engaging with ethnomathematics, and tackling complex sustainability problems. Our critical pragmatic approach attempts to build a new philosophical perspective to equip teachers and students with the mathematical competencies, critical perspectives, and ethical grounding necessary to navigate and contribute to a sustainable future and to provide new analytic pathways.

Let $(M, g)$ be a closed oriented Riemannian surface, and let $SM$ be its unit tangent bundle. We show that the interior in the $\mathcal{C}^2$ topology of the set of smooth functions $\lambda:SM\to \mathbb{R}$ for which the thermostat $(M, g, \lambda)$ has no conjugate points is a subset of those functions for which the thermostat is projectively Anosov. Moreover, we prove that if a reversible thermostat is projectively Anosov, then its non-wandering set contains no conjugate points.

Stochastic Optimal Control (SOC) problems arise in systems influenced by uncertainty, such as autonomous robots or financial models. Traditional methods like dynamic programming are often intractable for high-dimensional, nonlinear systems due to the curse of dimensionality. This dissertation explores the path integral control framework as a scalable, sampling-based alternative. By reformulating SOC problems as expectations over stochastic trajectories, it enables efficient policy synthesis via Monte Carlo sampling and supports real-time implementation through GPU parallelization. We apply this framework to six classes of SOC problems: Chance-Constrained SOC, Stochastic Differential Games, Deceptive Control, Task Hierarchical Control, Risk Mitigation of Stealthy Attacks, and Discrete-Time LQR. A sample complexity analysis for the discrete-time case is also provided. These contributions establish a foundation for simulator-driven autonomy in complex, uncertain environments.

A permutiple is a natural number whose representation in some base is an integer multiple of a number whose representation has the same collection of digits. Previous efforts have made progress on finding such numbers using graph-theoretical and finite-state-machine constructions. These are the mother graph and the Hoey-Sloane machine (and its state graph). In this paper, we use the reflective symmetry of the mother graph as a starting point for understanding relationships between permutiple classes and how new classes can be determined from old. Such results are not only useful for finding new permutiples from old, they help us to see previous work through a new lens.

We prove that the span of normalized characters of subprincipal admissible modules over an affine Lie algebra of subprincipal admissible level $k$ is $SL_2(\mathbf{Z})$-invariant and find the explicit modular transformation formula.

Let $X$ be a smooth $p$-adic Stein space with free tangent sheaf. We use the notion of Hochschild cohomology for sheaves of Ind-Banach algebras developed in our previous work to study the Hochschild cohomology of the algebra of infinite order differential operators $\mathcal{D}_X$-cap. In particular, we show that the Hochschild cohomology complex of $\mathcal{D}_X$-cap is a strict complex of nuclear Fr\'echet spaces which is quasi-isomorphic to the de Rham complex of $X$. We then use this to compare the first Hochschild cohomology group of $\mathcal{D}_X$-cap with a wide array of Ext functors. Finally, we investigate the relation of the Hochschild cohomology of $\mathcal{D}_X$-cap with the deformation theory of $\mathcal{D}_X(X)$-cap. Assuming some finiteness conditions on the de Rham cohomology of $X$, we define explicit isomorphisms between the first Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of bounded outer derivations of $\mathcal{D}_X(X)$-cap, and between the second Hochschild cohomology group of $\mathcal{D}_X$-cap and the space of infinitesimal deformations of $\mathcal{D}_X(X)$-cap.

Let $\mathbb{F}$ be a normed field. In this work, we prove that every nil complete metric $\mathbb{F}$-algebra is nilpotent when $\mathbb{F}$ has characteristic zero. This result generalizes Grabiner's Theorem for Banach algebras, first proved in 1969. Furthermore, we show that a metric $\mathbb{F}$-algebra $\mathfrak{A}$ and its completion $C(\mathfrak{A})$ satisfy the same polynomial identities, and consequently, if $\mathsf{char}(\mathbb{F})=0$ and $C(\mathfrak{A})$ is nil, then $\mathfrak{A}$ is nilpotent. Our results allow us to resolve K\"othe's Problem affirmatively for complete metric algebras over normed fields of characteristic zero.

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0, x)=({\delta^{\nu+1}}{u_0}({\frac{x}{\delta}}),{\delta^\nu }{u_1}({\frac{x}{\delta}}))$ with $\nu\in\Bbb R$ and $(u_0, u_1)\in C_0^{\infty}(\Bbb R)$, which are a class of large initial data, we establish that when $\nu\le -\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of $(u_0, u_1)$ and cubic nonlinearity $F(u,{\partial u})$; when $\nu>-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $\nu=-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations.

Study of the KPZ universality class has seen the emergence of universal objects over the past decade which arrive as the scaling limit of the member models. One such object is the directed landscape, and it is known that exactly solvable last passage percolation (LPP) models converge to the directed landscape under the KPZ scaling. Large deviations of the directed landscape on the metric level were recently studied in arXiv:2405.14924, which also provides a general framework for establishing such large deviation principle (LDP). The main goal of the article is to employ and tweak that framework to establish a LDP for LPP models at the metric level without assuming exact solvability. We then use the LDP on the metric level to establish a LDP for geodesics in these models, providing a streamlined way to study large transversal fluctuations of geodesics in these models.

We study the continuous-time version of the empirical correlation coefficient between the paths of two possibly correlated Ornstein-Uhlenbeck processes, known as Yule's nonsense correlation for these paths. Using sharp tools from the analysis on Wiener chaos, we establish the asymptotic normality of the fluctuations of this correlation coefficient around its long-time limit, which is the mathematical correlation coefficient between the two processes. This asymptotic normality is quantified in Kolmogorov distance, which allows us to establish speeds of convergence in the Type-II error for two simple tests of independence of the paths, based on the empirical correlation, and based on its numerator. An application to independence of two observations of solutions to the stochastic heat equation is given, with excellent asymptotic power properties using merely a small number of the solutions' Fourier modes.

A fluid queue is a stochastic process which moves linearly with a rate that is determined by the state of a continuous-time Markov chain (CTMC). In this paper we construct an approximation to a fluid queue using a quasi birth-and-death process with rational arrival process components (QBD-RAP) and prove weak convergence via convergence of generators. The primary motivation for constructing the new approximation is to achieve a better approximation accuracy than existing methods while also ensuring that all approximations to probabilities have all probabilistic properties, such as non-negativity and bounded above by 1.

This paper investigates the existence of $m$-stiff configurations in the unit sphere $S^{d-1}$, which are spherical $(2m-1)$-designs that lie on $m$ parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer $m > 5$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $d$; (2) for each fixed integer $d > 10$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $m$. Furthermore, we provide a complete classification of the dimensions where $m$-stiff configurations exist for $m=2,3,4,5$. We also determine the non-existence (and the existence) of $m$-stiff configurations in $S^{d-1}$ for small $d$ ($3 \leq d \leq 120$) with arbitrary $m$, and also for small $m$ ($6 \leq m \leq 10$) with arbitrary $d$. Finally, we conjecture that there is no $m$-stiff configuration in $S^{d-1}$ for $(d,m)$ with $d\geq 3$ and $m\geq 6$.

In 1968, Paul Halmos initiated the research on density of the set of irreducible operators on a separable Hilbert space. Through the research, a long-standing unsolved problem inquires: is the set of irreducible operators dense in $B(H)$ with respect to the trace-class norm topology? Precisely, for each operator $T $ in $B(H)$ and every $\varepsilon >0$, is there a trace-class operator $K$ such that $T+K$ is irreducible and $\Vert K \Vert_1 < \varepsilon$? For $p>1$, to prove the $\Vert \cdot \Vert_p$-norm density of irreducible operators in $B(H)$, a type of Weyl-von Neumann theorem effects as a key technique. But the traditional method fails for the case $p=1$, where by $\Vert \cdot \Vert_p$-norm we denote the Schatten $p$-norm. In the current paper, for a large family of operators in $B(H)$, we give the above long-term problem an affirmative answer. The result is derived from a combination of techniques in both operator theory and operator algebras. Moreover, we discover that there is a strong connection between the problem and another related operator-theoretical problem related to type $\mathrm{II}_1$ von Neumann algebras.

It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fr\'echet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.

We study solutions to the $\alpha$-SQG equations, which interpolate between the incompressible Euler and surface quasi-geostrophic equations. We extend prior results on existence of bounded patches, proving propagation of $H^k$-regularity of the patch boundary, $k \ge 3$, for finite time for patches that are periodic in one spatial dimension. Such periodic patches also encompass layers, or two-sided fronts. As the authors have treated the Euler case in prior work, we now primarily focus on the range of $\alpha$ for which $\alpha$-SQG lies strictly between the Euler and SQG equations.

The celebrated theorem of Chung, Graham, and Wilson on quasirandom graphs implies that if the 4-cycle and edge counts in a graph $G$ are both close to their typical number in $\mathbb{G}(n,1/2),$ then this also holds for the counts of subgraphs isomorphic to $H$ for any $H$ of constant size. We aim to prove a similar statement where the notion of close is whether the given (signed) subgraph count can be used as a test between $\mathbb{G}(n,1/2)$ and a stochastic block model $\mathbb{SBM}.$ Quantitatively, this is related to approximately maximizing $H \longrightarrow |\Phi(H)|^{\frac{1}{|\mathsf{V}(H)|}},$ where $\Phi(H)$ is the Fourier coefficient of $\mathbb{SBM}$, indexed by subgraph $H.$ This formulation turns out to be equivalent to approximately maximizing the partition function of a spin model over alphabet equal to the community labels in $\mathbb{SBM}.$ We resolve the approximate maximization when $\mathbb{SBM}$ satisfies one of four conditions: 1) the probability of an edge between any two vertices in different communities is exactly $1/2$; 2) the probability of an edge between two vertices from any two communities is at least $1/2$ (this case is also covered in a recent work of Yu, Zadik, and Zhang); 3) the probability of belonging to any given community is at least $c$ for some universal constant $c>0$; 4) $\mathbb{SBM}$ has two communities. In each of these cases, we show that there is an approximate maximizer of $|\Phi(H)|^{\frac{1}{|\mathsf{V}(H)|}}$ in the set $\mathsf{A} = \{\text{stars, 4-cycle}\}.$ This implies that if there exists a constant-degree polynomial test distinguishing $\mathbb{G}(n,1/2)$ and $\mathbb{SBM},$ then the two distributions can also be distinguished via the signed count of some graph in $\mathsf{A}.$ We conjecture that the same holds true for distinguishing $\mathbb{G}(n,1/2)$ and any graphon if we also add triangles to $\mathsf{A}.$

Bilevel optimization is a fundamental tool in hierarchical decision-making and has been widely applied to machine learning tasks such as hyperparameter tuning, meta-learning, and continual learning. While significant progress has been made in bilevel optimization, existing methods predominantly focus on the {nonconvex-strongly convex, or the} nonconvex-PL settings, leaving the more general nonconvex-nonconvex framework underexplored. In this paper, we address this gap by developing an efficient gradient-based method inspired by the recently proposed Relaxed Gradient Flow (RXGF) framework with a continuous-time dynamic. In particular, we introduce a discretized variant of RXGF and formulate convex quadratic program subproblems with closed-form solutions. We provide a rigorous convergence analysis, demonstrating that under the existence of a KKT point and a regularity assumption {(lower-level gradient PL assumption)}, our method achieves an iteration complexity of $\mathcal{O}(1/\epsilon^{1.5})$ in terms of the squared norm of the KKT residual for the reformulated problem. Moreover, even in the absence of the regularity assumption, we establish an iteration complexity of $\mathcal{O}(1/\epsilon^{3})$ for the same metric. Through extensive numerical experiments on convex and nonconvex synthetic benchmarks and a hyper-data cleaning task, we illustrate the efficiency and scalability of our approach.

Let $A$ and $B$ be commutative Noetherian algebras over an arbitrary field $\Bbbk$ such that $A \otimes_\Bbbk B$ is Noetherian. We consider ideals $I$ and $J$ of $A$ and $B$, respectively, as well as nonzero finitely generated modules $L$ and $N$ over $A$ and $B$, respectively. In this paper, we investigate certain algebraic properties of the $A \otimes_\Bbbk B$-module $L\otimes_{\Bbbk} N$, which are often inherited from the properties of the $A$-module $L$ and the $B$-module $N$. Specifically, we provide characterizations for the Cohen-Macaulayness, generalized Cohen-Macaulayness, and sequentially Cohen-Macaulayness of $L\otimes_{\Bbbk} N$ with respect to the ideal $I \otimes_\Bbbk B + A \otimes_\Bbbk J$, in terms of the corresponding properties for $L$ and $N$ with respect to $I$ and $J$, respectively.

In his ICM report, Sullivan proposes the program of l-adic formalization of the concept of manifolds. In this program, he claims that smooth positive characteristic varieties should carry l-adic formal manifold structures. He also claims the existence of an abelianized Galois symmetry on l-adic formal manifold structures. This paper carries out this program, establishes the claims, and relates the abelianized Galois symmetry on l-adic formal manifold structures to the Galois symmetry of varieties. Meanwhile, we prove that simply-connected varieties are homotopically finite CW complexes in the l-adic sense.

We construct a surjective map from the set of conjugacy classes of depth-zero cuspidal enhanced L-parameters to that of isomorphism classes of depth-zero supercuspidal representations for simple adjoint groups, and check the bijectivity in various cases. We also prove that the Hiraga--Ichino--Ikeda conjecture on the formal degree of essentially square-integrable irreducible representations holds for this parametrization if it is bijective.

Consider a time-harmonic acoustic plane wave incident onto an elastic body with an unbounded periodic surface. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid air/fluid of constant mass density, while the elastic body is assumed to be isotropic and linear. By introducing the Dirichlet-to-Neumann (DtN) operators for acoustic and elastic waves simultaneously, the model is formulated as an acoustic-elastic interaction problem in periodic structures. Based on a duality argument, an a posteriori error estimate is derived for the associated truncated finite element approximation. The a posteriori error estimate consists of the finite element approximation error and the truncation error of two different DtN operators, where the latter decays exponentially with respect to the truncation parameter. Based on the a posteriori error, an adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem in periodic structures. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm.

The cyclic Nielsen realization problem for a closed, oriented manifold asks whether any mapping class of finite order can be represented by a homeomorphism of the same order. In this article, we resolve the smooth, metric, and complex cyclic Nielsen realization problem for certain "irreducible" mapping classes on the family of smooth 4-manifolds underlying del Pezzo surfaces. Both positive and negative examples of realizability are provided in various settings. Our techniques are varied, synthesizing results from reflection group theory and 4-manifold topology.

We establish a single-letter characterization of the fundamental distortion-rate-perception tradeoff with limited common randomness under the squared error distortion measure and the squared Wasserstein-2 perception measure. Moreover, it is shown that this single-letter characterization can be explicitly evaluated for the Gaussian source. Various notions of universal representation are also clarified.

It is known that the Dunkl-type fractional integral operator $I_\beta$ $(0 < \beta < 2\alpha + 2 =d_\alpha)$ is bounded from $L^p(\R,d\mu_\alpha)$ to $L^q (\R, d\mu_\alpha)$ when $1 < p < \frac{d_\alpha}{\beta}$ and $\frac{1}{p} - \frac{1}{q} = \frac{\beta}{d_\alpha}$. In \cite{spsa} , the authors introduced the generalized Dunkl-type fractional integral operator $T_\rho^\alpha$ and it's modified version $\tilde{T}_\rho^\alpha$ and extended the above boundedness results to the generalized Dunkl-type Morrey spaces and Dunkl-type $BMO_\phi$ spaces. In this paper we investigate the boundedness of generalized Dunkl-type fractional integral operators and it's modified version mainly on the Dunkl-type Campanato space.

The service rate region (SRR) has emerged as a critical performance metric for distributed systems that store data redundantly. It measures the system's ability to serve multiple users concurrently. Mathematically, the SRR is a polytope in R^k where each dimension corresponds to the service request rate of one of the k data objects. This paper focuses on systems employing a class of Maximum Distance Separable (MDS) codes. For each code in the class, we characterize the k axes intercept points of its SRR, and the smallest standard simplex that includes the SRR. We use these results to show that the SRR grows with the increasing number of systematic columns in the generator matrices. We establish a graph-theoretic framework associating this SRR problem with fractional matchings in quasi-uniform hypergraphs. Identifying the SRR polytope is equivalent to determining a particular image of the fractional-matching polytope. We introduce a notion of Greedy Matching and show that it is sufficient to focus on these matchings to characterize the SRR rather than the entire matching polytope. With these tools, we determine the SRR of a large subset of the considered class of codes. Our results generalize previous characterizations of systematic and non-systematic MDS-coded systems, offering a unified framework for analyzing service rate regions of codes.

In this paper we treat the Fourier coefficients of Siegel Eisenstein series of level $p$ with trivial or quadratic character, for an odd prime $p$. The Euler $p$-factor of the Fourier coefficient is called the ramified Siegel series. First we show that the ramified Siegel series attached to each cusp can be decomposed to $U(p)$-eigenfunctions explicitly, next we give recursion formulas of such $U(p)$-characteristic ramified Siegel series.

In this paper, we introduce a new bi-Lipschitz invariant for analytic function germs in two variables, enhancing the Henry-Parusinski invariant.

The weakly zero-divisor graph $W\Gamma(R)$ of a commutative ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two distinct vertices $x$, $y$ are adjacent if and only if there exist $w\in {\rm ann}(x)$ and $ z\in {\rm ann}(y)$ such that $wz =0$. In this paper, we determine the Sombor index for the weakly zero-divisor graph of the integers modulo ring $\mathbb{Z}_n$. Furthermore, we investigate the Sombor spectrum and establish bounds for the Sombor energy of the weakly zero-divisor graph of $\mathbb{Z}_n$.

Let $D$ be an oriented graph with skew adjacency matrix $S(D)$. Two oriented graphs $D$ and $C$ are said to share the same generalized skew spectrum if $S(D)$ and $S(C)$ have the same eigenvalues, and $J-S(D)$ and $J-S(C)$ also have the same eigenvalues, where $J$ is the all-ones matrix. Such graphs that are not isomorphic are generalized cospectral mates. We derive tight upper bounds on the number of non-isomorphic generalized cospectral mates an oriented graph can admit, based on arithmetic criteria involving the determinant of its skew-walk matrix. As a special case, we also provide a criterion for an oriented graph to be weakly determined by its generalized skew spectrum (WDGSS), that is, its only generalized cospectral mate is its transpose. These criteria relate directly to the controllability of graphs, a fundamental concept in the control of networked systems, thereby connecting spectral characterization of graphs to graph controllability.

We discuss the following question: given a finite BCK-algebra, what is the probability that two randomly selected elements commute? We call this probability the \textit{commuting degree} of a BCK-algebra. In a previous paper, the author gave sharp upper and lower bounds for the commuting degree of a BCK-algebra with order $n$. We expand on those results in this paper: we show that, for each $n\geq 3$, there is a BCK-algebra of order $n$ realizing each possible commuting degree and that the minimum commuting degree is achieved by a unique BCK-algebra of order $n$ Additionally, we show that every rational number in $(0,1]$ is the commuting degree of some finite BCK-algebra.

The Lost Notebook of Ramanujan contains a number of beautiful formulas, one of which can be found on its page 220. It involves an interesting function, which we denote as $\mathcal{F}_1(x)$. In this paper, we show that $\mathcal{F}_1(x)$ belongs to the category of period functions as it satisfies the period relations of Maass forms in the sense of Lewis and Zagier \cite{lz}. Hence, we refer to $\mathcal{F}_1(x)$ as the \emph{Ramanujan period function}. Moreover, one of the salient aspects of the Ramanujan period function $\mathcal{F}_1(x)$ that we found out is that it is a Hecke eigenfunction under the action of Hecke operators on the space of periods. We also establish that it naturally appears in a Kronecker limit formula of a certain zeta function, revealing its connections to various topics. Finally, we generalize $\mathcal{F}_1(x)$ to include a parameter $s,$ connecting our work to the broader theory of period functions developed by Bettin and Conrey \cite{bc} and Lewis and Zagier \cite{lz}. We emphasize that Ramanujan was the first to study this function, marking the beginning of the study of period functions.

Observing the special structure of the system and using the Poincar{\'{e}}-Sobolev inequality, we establish Liouville type theorems for the 3D steady tropical climate model under certain conditions on $u$, $v$, $\nabla \theta$. Our results extend and improve a Liouville type result of Cho-In-Yang (arXiv:2312.17441).

Identifying vertices that play a central role is a fundamental problem in network analysis. Although traditional centrality measures have been widely used for this purpose, the growing complexity of contemporary networks necessitates more sophisticated indicators. Forman curvature has recently emerged as a promising approach. In this paper, we define Forman curvature for multilayer networks, a class of complex networks characterized by multiple types of connections or layers between nodes, which are increasingly used to model intricate real-world phenomena. We establish the key properties of Forman curvature in the context of multilayer networks and demonstrate its utility for identifying vertices that hold central positions within these networks. Furthermore, we show that Forman curvature can also serve as an effective tool for the structural classification of entire multilayer networks.

This paper is concerned with the low Mach and Rossby number limits of $3$D compressible rotating Euler equations with ill-prepared initial data in the whole space. More precisely, the initial data is the sum of a $3$D part and a $2$D part. With the help of a suitable intermediate system, we perform this singular limit rigorously with the target system being a $2$D QG-type. This particularly gives an affirmative answer to the question raised by Ngo and Scrobogna [\emph{Discrete Contin. Dyn. Syst.}, 38 (2018), pp. 749-789]. As a by-product, our proof gives a rigorous justification from the $2$D inviscid rotating shallow water equations to the $2$D QG equations in whole space.

We determine the fields of values of the Isaacs' head characters of a finite solvable group.

In this paper we discuss how a general bilinear finite-dimensional closed quantum system with dispersed parameters can be steered between eigenstates. We show that, under suitable conditions on the separation of spectral gaps and the boundedness of parameter dispersion, rotating wave and adiabatic approximations can be employed in cascade to achieve population inversion between arbitrary eigenstates. We propose an explicit control law and test numerically the sharpness of the conditions on several examples.

In this paper the index of a family of critical points of the systole function on Teichm\"uller space is calculated. The members of this family are interesting in that their existence implies the existence of strata in the Thurston spine for which the systoles do not determine a basis for the homology of the surface. Previously, index calculations of critical points with this pathological feature were impossible, because the only known examples were in surfaces with huge genus. A related concept is that of a ``minimal filling subset'' of the systoles at the critical point. Such minimal filling sets are studied, as they relate to the dimension of the Thurston spine near the critical point. We find an example of a minimal filling set of simple closed geodesics in genus 5 with cardinality 8, that are presumably realised as systoles. More generally, we determine the smallest and largest cardinality of a minimal filling set related to a tesselation of a hyperbolic surface by regular, right-angled $m$-gons for $m \in \{ 5, 6, 7 \}$. For this, we use integer linear programming together with a hand-tailored symmetry breaking technique.

In this article, we study the continuous-discrete projection filter for the exponential-family manifolds with conjugate likehoods. We first derive the local projection error of the prediction step of the continuous-discrete projection filter. We then derive the exact Bayesian update algorithm for a class of discrete measurement processes with additive Gaussian noise. Lastly, we present a numerical simulation of the stochastic van der Pol filtering problem with a nonlinear measurement process. The proposed projection filter shows superior performance compared to several state-of-the-art parametric continuous-discrete filtering methods.

We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = \lambda K(x) |u|^{p-2} u \hspace{.1cm} \mbox { in } \hspace{.1cm} \mathbb{R}^N , \end{equation*} where $\lambda \in \mathbb{R}$, $p>1$, $K : \mathbb{R}^N \rightarrow \mathbb{R}$, $L : \mathbb{R}^N \rightarrow \mathbb{R}^+$ are locally integrable functions. The weight function $K$ is allowed to change sign, provided it remains positive on a set of nonzero measure. We establish the existence, regularity, and asymptotic behavior of the principal eigenfunctions. We also prove local and global antimaximum principles for a perturbed version of the problem.

In this paper, we introduce an asymmetric metric on the space of marked Euclidean triangles, and we prove several properties of this metric, including two equivalent definitions of this metric, one of them comparing ratios of functions of the edges, and the other one in terms of best Lipschitz maps. We give a description of the geodesics of this metric. We show that this metric is Finsler, and give a formula for its infinitesimal Finsler structure. We then generalise this study to the case of convex Euclidean polygons in the Euclidean plane and to surfaces equipped with singular Euclidean structures with an underlying fixed triangulation. After developing some elements of the theory of completeness and completion of asymmetric metrics which is adapted to our setting, we study the completeness of the metrics we introduce in this paper. These problems and the results obtained are motivated by Thurston's work developed in his paper Minimal stretch maps between hyperbolic surfaces. We provide an analogue of Thurston's theory in a Euclidean setting.

A family of explicit 15-stage Runge-Kutta methods of order 10 is derived.

In this paper, we study error exponents for a concatataned coding based class of DNA storage codes in which the number of reads performed can be variable. That is, the decoder can sequentially perform reads and choose whether to output the final decision or take more reads, and we are interested in minimizing the average number of reads performed rather than a fixed pre-specified value. We show that this flexibility leads to a considerable reduction in the error probability compared to a fixed number of reads, not only in terms of constants in the error exponent but also in the scaling laws. This is shown via an achievability result for a suitably-designed protocol, and in certain parameter regimes we additionally establish a matching converse that holds for all protocols within a broader concatenated coding based class.

In a previous work of the first authors, a non-holonomic model, generalising the micromorphic models and allowing for curvature (disclinations) to arise from the kinematic values, was presented. In the present paper, a generalisation of the classical models of Euler-Bernoulli and Timoshenko bending beams based on the mentioned work is proposed. The former is still composed of only one unidimensional scalar field, while the later introduces a third unidimensional scalar field, correcting the second order terms. The generalised Euler-Bernoulli beam is then shown to exhibit curvature (i.e. disclinations) linked to a third order derivative of the displacement, but no torsion (dislocations). Parallelly, the generalised Timoshenko beam is shown to exhibit both curvature and torsion, where the former is linked to the non-holonomy introduced in the generalisation. Lastly, using variational calculus, asymptotic values for the value taken by the curvature in static equilibrium are obtained when the second order contribution becomes negligible; along with an equation for the torsion in the generalised Timoshenko beam.

Due to increased energy demand and environmental concerns such as greenhouse gas emissions and natural resources depletion, optimizing energy and raw materials usage has recently drawn much attention. Achieving more synergy between different energy sectors and manufacturing processes could lead to substantial improvements. Energy hubs are already well-known solutions for studying multi-carrier energy systems. In the present study, a multi-carrier hub is defined as a geographic area where different processes take place to convert energy and material flows possibly consumed locally, stored, or exported. Hub boundaries correspond to an area small enough to neglect losses when energy and materials flows are exchanged between the processes. A general formulation is introduced to model such a hub. This formalism encompasses all the possible configurations without limitations or preconceptions regarding used carriers or processes' architecture. For this purpose, a new hub representation is proposed that allows optimization while considering all possible arrangements (parallel, serial, or a combination of them). It is implemented in a framework that allows the creation of several multi-carrier hubs and connecting some of them to exchange flows through dedicated networks. The formalism is based on linear programming (LP). The features of the developed framework are illustrated by a case study consisting of a network made of three hubs and considering heat, electricity, water, and hydrogen flows.The study aims to optimally schedule processes usage and materials and energy storage to minimize the cost of imported resources. The study is done in a short-term period. It shows the benefits of synergy between the different processes in the hub.

We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than $\pi$/2.

Linear upper bounds may be derived by imposing specific structural conditions on a generating set, such as additional constraints on ranks, eigenvalues, or the degree of the minimal polynomial of the generating matrices. This paper establishes a linear upper bound of \(3n-5\) for generating sets that contain a matrix whose minimal polynomial has a degree exceeding \(\frac{n}{2}\), where \(n\) denotes the order of the matrix. Compared to the bound provided in \cite[Theorem 3.1]{r2}, this result reduces the constraints on the Jordan canonical forms. Additionally, it is demonstrated that the bound \(\frac{7n}{2}-4\) holds when the generating set contains a matrix with a minimal polynomial of degree \(t\) satisfying \(2t\le n\le 3t-1\). The primary enhancements consist of quantitative bounds and reduced reliance on Jordan form structural constraints.

A piecewise continuous biharmonic problem in domains with corner points and a corresponding Schwarz type boundary value problem for monogenic functions in a commutative biharmonic algebra are considered. A method for reducing the problems to a system of integral equations is developed.

The elapsed-time model describes the behavior of interconnected neurons through the time since their last spike. It is an age-structured non-linear equation in which age corresponds to the elapsed time since the last discharge, and models many interesting dynamics depending on the type of interactions between neurons. We investigate the linearized stability of this equation by considering a discrete delay, which accounts for the possibility of a synaptic delay due to the time needed to transmit a nerve impulse from one neuron to the rest of the ensemble. We state a stability criterion that allows to determine if a steady state is linearly stable or unstable depending on the delay and the interaction between neurons. Our approach relies on the study of the asymptotic behavior of related Volterra-type integral equations in terms of theirs Laplace transforms. The analysis is complemented with numerical simulations illustrating the change of stability of a steady state in terms of the delay and the intensity of interconnections.

We establish sample-path large deviation principles for the centered cumulative functional of marked Poisson cluster processes in the Skorokhod space equipped with the M1 topology, under joint regular variation assumptions on the marks and the offspring distributions governing the propagation mechanism. These findings can also be interpreted as hidden regular variation of the cluster processes' functionals, extending the results in Dombry et al. (2022) to cluster processes with heavy-tailed characteristics, including mixed Binomial Poisson cluster processes and Hawkes processes. Notably, by restricting to the adequate subspace of measures on D([0, 1], R+), and applying the correct normalization and scaling to the paths of the centered cumulative functional, the limit measure concentrates on paths with multiple large jumps.

The complexity of a pair $(X,B)$ is an invariant that relates the dimension of $X$, the rank of the group of divisors, and the coefficients of $B$. If the complexity is less than one, then $X$ is a toric variety. We prove that if the complexity is less than two, then $X$ is a Fano type variety. Furthermore, if the complexity is less than 3/2, then $X$ admits a Calabi--Yau structure $(X,B)$ of complexity one and index at most two, and it admits a finite cover $Y \to X$ of degree at most 2, where $Y$ is a cluster type variety. In particular, if the complexity is one and the index is one, $(X,B)$ is cluster type. Finally, we establish a connection with the theory of $\mathbb{T}$-varieties. We prove that a variety of $\mathbb{T}$-complexity one admits a similar finite cover from a cluster type variety.

Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstra\ss-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics.

We establish certain oscillation estimates for weak solutions to nonlinear, anomalous phase transitions modeled on the nonlocal two-phase Stefan problem. The problem is singular in time, is scaling deficient and influenced by far-off effects. We study the the problem in a geometry adapted to the solution and obtain oscillation estimates in intrinsically scaled cylinders. Furthermore, via certain uniform estimates, we construct a continuous weak solution to the corresponding initial boundary value problem with a quantitative modulus of continuity.

The notion of graph cover, also known as locally bijective homomorphism, is a discretization of covering spaces known from general topology. It is a pair of incidence-preserving vertex- and edge-mappings between two graphs, the edge-component being bijective on the edge-neighborhoods of every vertex and its image. In line with the current trends in topological graph theory and its applications in mathematical physics, graphs are considered in the most relaxed form and as such they may contain multiple edges, loops and semi-edges. Nevertheless, simple graphs (binary structures without multiple edges, loops, or semi-edges) play an important role. It has been conjectured in [Bok et al.: List covering of regular multigraphs, Proceedings IWOCA 2022, LNCS 13270, pp. 228--242] that for every fixed graph $H$, deciding if a graph covers $H$ is either polynomial time solvable for arbitrary input graphs, or NP-complete for simple ones. A graph $A$ is called stronger than a graph $B$ if every simple graph that covers $A$ also covers $B$. This notion was defined and found useful for NP-hardness reductions for disconnected graphs in [Bok et al.: Computational complexity of covering disconnected multigraphs, Proceedings FCT 2022, LNCS 12867, pp. 85--99]. It was conjectured in [Kratochv\'{\i}l: Towards strong dichotomy of graphs covers, GROW 2022 - Book of open problems, p. 10, {\tt https://grow.famnit.upr.si/GROW-BOP.pdf}] that if $A$ has no semi-edges, then $A$ is stronger than $B$ if and only if $A$ covers $B$. We prove this conjecture for cubic one-vertex graphs, and we also justify it for all cubic graphs $A$ with at most 4 vertices.

Since the seminal work of Kenig and Pipher, the Dahlberg-Kenig-Pipher (DKP) condition on oscillations of the coefficient matrix became a standard threshold in the study of absolute continuity of the harmonic measure with respect to the Hausdorff measure on the boundary. It has been proved sufficient for absolute continuity in the domains with increasingly complex geometry, and known counterexamples show that in a certain sense it is necessary as well. In the present note, we introduce into the subject ideas from homogenization theory to exhibit a new class of operators for which the elliptic measure is well-behaved, featuring the coefficients violating the DKP condition, and on the contrary, oscillating so quickly, that the homogenization takes place.

This paper investigates an inverse source problem for general semilinear stochastic hyperbolic equations. Motivated by the challenges arising from both randomness and nonlinearity, we develop a globally convergent iterative regularization method that combines Carleman estimate with fixed-point iteration. Our approach enables the reconstruction of the unknown source function from partial lateral Cauchy data, without requiring a good initial guess. We establish a new Carleman estimate for stochastic hyperbolic equations and prove the convergence of the proposed method in weighted spaces. Furthermore, we design an efficient numerical algorithm that avoids solving backward stochastic partial differential equations and is robust to randomness in both the model and the data. Numerical experiments are provided to demonstrate the effectiveness of the method.

We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation in the presence of noise and deterministic forcing. The noise is space-dependent and statistically translation-invariant. We show that, for small forcing, solitons remain close to the family of traveling waves in a weighted Sobolev norm, with high probability. We study the effective dynamics of the soliton amplitude and position via their variational phase, for which we derive explicit modulation equations. The stability result holds on a time scale where the deterministic forcing induces significant amplitude modulation.

The propagation of nonlinear and dispersive waves in various materials can be described by the well-known Kadomtsev-Petviashvili (KP) equation, which is a (2+1)-dimensional partial differential equation. In this paper, we show that the KP equation can be used to describe the in-plane motion of compressible elastic solids with dispersion. Furthermore, a modified KP equation with cubic nonlinearity is obtained in the case of incompressible solids with dispersion. Then, several solutions of these partial differential equations are discussed and computed using a Fourier spectral method. In particular, both equations admit solitary wave solutions.

We establish boundary observability and control for the fractional heat equation over arbitrary time horizons $T > 0$, within the optimal range of fractional exponents $s \in (1/2, 1)$. Our approach introduces a novel synthesis of techniques from fractional partial differential equations and control theory, combining several key ingredients in an original and effective manner: 1. Boundary observability for low-frequency solutions of the fractional wave equation. We begin by analyzing the associated fractional wave equation. Using a fractional analogue of Pohozaev's identity, we establish a partial boundary observability result for the low-frequency solutions. The corresponding observability time horizon increases with the eigenmode frequency, reflecting the inherently slower propagation speed of the fractional waves. 2. Transmutation to the parabolic setting. Using transmutation techniques, we transfer the observability results from the wave setting to the parabolic one. This yields a frequency-dependent observability inequality for the fractional heat equation, which - via duality - enables control of its low-frequency components. 3. Frequency-wise iteration. Leveraging the dissipative nature of the fractional heat equation, we develop an iterative procedure to successively control the entire frequency spectrum of solutions. The condition $s \in (1/2, 1)$ is crucial in this analysis, as it guarantees sufficient decay of high-frequency components, enabling the convergence of the iteration. 4. Duality. By a duality argument, we derive boundary observability from the boundary controllability of the fractional heat equation. Remarkably, this type of boundary observability result is entirely new in the multi-dimensional setting and appears to be out of reach for existing methods. \end{itemize}

We present a formula that relates the Turaev coaction and the Goncharov-Brown coaction. Motivated by this relation, we introduce the reduced coaction equation. The skew-symmetric solutions to this equation form a Lie algebra under Ihara bracket.

We consider a network of identical, first-order linear systems, and investigate how replacing a subset of the systems composing the network with higher-order ones, either taken to be generic or specifically designed, may affect its controllability. After establishing a correspondence between state controllability in networks of first-order systems with output controllability in networks of higher-order systems, we show that adding higher-order dynamics may require significantly fewer subsystem modifications to achieve structural controllability, when compared to first-order heterogeneous subsystems. Furthermore, we characterize the topology of networks (which we call X-networks) in which the introduction of heterogeneous local dynamics is not necessary for structural output controllability, as the latter can be attained by suitable higher-order subsystems with homogeneous internal dynamics.

We propose an extension of the theory of parity sheaves, which allows for non-locally constant sheaves along strata. Our definition is tailored for proving the existence of (proper, quasihereditary, etc) stratifications of $\mathrm{Ext}$-algebras. We use this to study quiver Schur algebras $A(\alpha)$ for the cyclic quiver of length $2$. We find a polynomial quasihereditary structure on $A(\alpha)$ compatible with the categorified PBW basis of McNamara and Kleshchev-Muth, and sharpen their results to arbitrary characteristic. We also prove that semicuspidal algebras of $A(n\delta)$ are polynomial quasihereditary covers of semicuspidal algebras of the corresponding KLR algebra $R(n\delta)$, and compute them diagrammatically.

Based on a baryogenesis mechanism originating from the theory of causal fermion systems, we analyze its main geometric and analytic features in conformally flat spacetimes. An explicit formula is derived for the rate of baryogenesis in these spacetimes, which depends on the mass $m$ of the particles, the conformal factor $\Omega$ and a future directed timelike vector field $u$ (dubbed the regularizing vector field). Our analysis covers Friedmann-Lema{\^i}tre-Robertson-Walker, Milne and Milne-like spacetimes. It sets the ground for concrete, quantitative predictions for specific cosmological spacetimes.

By relating notions from quantum harmonic analysis and band-dominated operator theory, we prove that over any locally compact abelian group $G$, the operator algebra $\mathcal C_1$ from quantum harmonic analysis agrees with the intersection of band-dominated operators and Fourier band-dominated operators. As an application, we characterize the compactness of operators acting on $L^2(G)$ and compare it with previous results in the discrete case. In particular, our results can be seen as a generalization of the limit operator concept to the non-discrete world. Moreover, we briefly discuss property $A'$ for arbitrary locally compact abelian groups.

We use Majorana representations to study the subalgebras of the Griess algebra that have shape $(2B,3A,5A)$ and whose associated Miyamoto groups are isomorphic to $A_n$. We prove that these subalgebras exist only if $n\in \{5,6,8\}$. The case $n=5$ was already treated by Ivanov, Seress, McInroy, and Shpectorov. In case $n=6$ we prove that these algebras are all isomorphic and provide their precise description. In case $n=8$ we prove that these algebras do not arise from standard Majorana representations.

The null hypothesis of equality of distributions of functional data coming from $K$ samples is considered. The proposed test statistic is multivariate and its components are based on pairwise Cram\'{e}r von Mises comparisons of empirical characteristic functionals. The significance of the test statistic is evaluated via the novel multivariate permutation test, where the final single $p$-value is computed using the discrete optimal measure transport. The methodology is illustrated by real data on cumulative intraday returns of Bitcoin.

For a graph class $\mathcal G$ and a graph $H$, the four $\mathcal G$-covering numbers of $H$, namely global ${\rm cn}_{g}^{\mathcal{G}}(H)$, union ${\rm cn}_{u}^{\mathcal{G}}(H)$, local ${\rm cn}_{l}^{\mathcal{G}}(H)$, and folded ${\rm cn}_{f}^{\mathcal{G}}(H)$, each measure in a slightly different way how well $H$ can be covered with graphs from $\mathcal G$. For every $\mathcal G$ and $H$ it holds \[ {\rm cn}_{g}^{\mathcal{G}}(H) \geq {\rm cn}_{u}^{\mathcal{G}}(H) \geq {\rm cn}_{l}^{\mathcal{G}}(H) \geq {\rm cn}_{f}^{\mathcal{G}}(H) \] and in general each inequality can be arbitrarily far apart. We investigate structural properties of graph classes $\mathcal G$ and $\mathcal H$ such that for all graphs $H \in \mathcal{H}$, a larger $\mathcal G$-covering number of $H$ can be bounded in terms of a smaller $\mathcal G$-covering number of $H$. For example, we prove that if $\mathcal G$ is hereditary and the chromatic number of graphs in $\mathcal H$ is bounded, then there exists a function $f$ (called a binding function) such that for all $H \in \mathcal{H}$ it holds ${\rm cn}_{u}^{\mathcal{G}}(H) \leq f({\rm cn}_{g}^{\mathcal{G}}(H))$. For $\mathcal G$ we consider graph classes that are component-closed, hereditary, monotone, sparse, or of bounded chromatic number. For $\mathcal H$ we consider graph classes that are sparse, $M$-minor-free, of bounded chromatic number, or of bounded treewidth. For each combination and every pair of $\mathcal G$-covering numbers, we either give a binding function $f$ or provide an example of such $\mathcal{G},\mathcal{H}$ for which no binding function exists.

The method of range decreasing group homomorphisms can be applied to study various maps between mapping spaces, includin holomorphic maps, group homomorphisms, linear maps, semigroup homomorphisms, Lie algebra homomorphisms and algebra homomorphisms [Z1, Z2]. Previous studies on range decreasing group homomorphisms have primarily focused on specific subsets of mapping groups. In this paper, we provide a characterization of a general range decreasing group homomorphism applicable to the entire mapping group. As applications, we compute a particular class of homomorphisms between mapping groups and identify all range decreasing group homomorphisms defined on specific mapping groups.

We investigate the Cauchy problem for the spin-1 Gross-Pitaevskii(GP) equation, which is a model instrumental in characterizing the soliton dynamics within spinor Bose-Einstein condensates. Recently, Geng $etal.$ (Commun. Math. Phys. 382, 585-611 (2021)) reported the long-time asymptotic result with error $\mathcal{O}(\frac{\log t}t)$ for the spin-1 GP equation that only exists in the continuous spectrum. The main purpose of our work is to further generalize and improve Geng's work. Compared with the previous work, our asymptotic error accuracy has been improved from $\mathcal{O}(\frac{\log t}t)$ to $\mathcal{O}(t^{-3/4})$. More importantly, by establishing two matrix valued functions, we obtained effective asymptotic errors and successfully constructed asymptotic analysis of the spin-1 GP equation based on the characteristics of the spectral problem, including two cases: (i)coexistence of discrete and continuous spectrum; (ii)only continuous spectrum which considered by Geng's work with error $\mathcal{O}(\frac{\log t}t)$. For the case (i), the corresponding asymptotic approximations can be characterized with an $N$-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$. For the case (ii), the corresponding asymptotic approximations can be characterized with the leading term on the continuous spectrum and the residual error order $\mathcal{O}(t^{-3/4})$. Finally, our results confirm the soliton resolution conjecture for the spin-1 GP equation.

Water scarcity and the low quality of wastewater produced in industrial applications present significant challenges, particularly in managing fresh water intake and reusing residual quantities. These issues affect various industries, compelling plant owners and managers to optimise water resources within their process networks. To address this cross-sector business requirement, we propose a Decision Support System (DSS) designed to capture key network components, such as inlet streams, processes, and outlet streams. Data provided to the DSS are exploited by an optimisation module, which supports both network design and operational decisions. This module is coupled with a generic mixed-integer nonlinear programming (MINLP) model, which is linearised into a compact mixed-integer linear programming (MILP) formulation capable of delivering fast optimal solutions across various network designs and input parameterisations. Additionally, a Constraint Programming (CP) approach is incorporated to handle nonlinear expressions through straightforward modeling. This state-of-the-art generalised framework enables broad applicability across a wide range of real-world scenarios, setting it apart from the conventional reliance on customised solutions designed for specific use cases. The proposed framework was tested on 500 synthetic data instances inspired by historical data from three case studies. The obtained results confirm the validity, computational competence and practical impact of our approach both among their operational and network design phases, demonstrating significant improvements over current practices. Notably, the proposed approach achieved a 17.6% reduction in freshwater intake in a chemical industry case and facilitated the reuse of nearly 90% of wastewater in an oil refinery case.

We show the existence of odd fake $\mathbb{Q}$-homology quadrics, namely of minimal surfaces $S$ of general type which have the same $\mathbb{Q}$-homology as a smooth quadric $Q \cong (\mathbb{P}^1(\mathbb{C}))^2$, but have an odd intersection form on $ H^2(S, \mathbb{Z})/Tors(S)$, where $Tors(S)$ is the Torsion subgroup. Our examples are provided by a special 1-dimensional family of surfaces isogenous to a product of unmixed type.

We establish upper bounds for the expected Gaussian-smoothed $p$-Wasserstein distance between a probability measure $\mu$ and the corresponding empirical measure $\mu_N$, whenever $\mu$ has finite $q$-th moments for any $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also use a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive an upper bound for the classical $p$-Wasserstein distance. Although the latter upper bound is not optimal when $\mu$ has finite $q$-th moment with $q>p$, this bound does not require imposing such a moment condition on $\mu$, as it is usually done in the literature.

We consider unitary Shimura varieties at places where the totally real field ramifies over $\mbQ$. Our first result constructs comparison isomorphisms between absolute and relative local models in this context which relies on a reformulation of the Eisenstein condition of Rapoport--Zink and Rapoport--Smithling--Zhang. Our second result lifts this comparison to categories of $p$-divisible groups and, as a corollary, to various kinds of Rapoport--Zink spaces. This unifies multiple previously known results in this direction. Our third result and main application is to the arithmetic transfer conjecture of the third author. Using our statements about Rapoport--Zink spaces, we extend his previous proof from the unramified case to that of all $p$-adic local fields (for odd $p$). In general, our results have similar applications to other problems around the arithmetic of Shimura varieties as well, removing several ramification assumptions in the literature.

In this work we consider the $N$-body Hamiltonian describing the microscopic structure of a quantum gas of almost-bosonic anyons. This description includes both extended magnetic flux and spin-orbit/soft-disk interaction between the particles which are confined in a scalar trapping potential. We study a physically well-motivated ansatz for a sequence of trial states, consisting of Jastrow repulsive short-range correlations and a condensate, with sufficient variational freedom to approximate the ground state (and possibly also low-energy excited states) of the gas. In the limit $N \to \infty$, while taking the relative size of the anyons to zero and the total magnetic flux $2\pi\beta$ to remain finite, we rigorously derive the stationary Chern-Simons-Schr\"odinger/average-field-Pauli effective energy density functional for the condensate wave function. This includes a scalar self-interaction parameter $\gamma$ which depends both on $\beta$, the diluteness of the gas, and the spin-orbit coupling strength $g$, but becomes independent of these microscopic details for a particular value of the coupling $g=2$ in which supersymmetry is exhibited (on all scales, both microscopic and mesoscopic) with $\gamma=2\pi|\beta|$. Our findings confirm and clarify the predictions we have found in the physics literature.

This paper studies the inverse-closed subalgebras of the Roe algebra with coefficients of the type \(l^2(G, A)\). The coefficient \(A\) is chosen to be a non-commutative \(C^*\)-algebra, and the object of study is \(C^*(G, A)\) generated by the countable discrete group \(G\). By referring to the Sobolev-type algebra, the intersection of a family of Banach algebras is taken. It is proved that the intersection \(W_a^{\infty}(G, A)\) of Banach spaces is a spectrally invariant dense subalgebra of \(C^*(G, A)\), and a sufficient condition for this is that the group action of \(G\) has polynomial growth.

Shrinking target problems in the context of iterated function systems have received an increasing amount of interest in the past few years. The classical shrinking target problem concerns points returning infinitely many times to a sequence of shrinking balls. In the iterated function system context, the shrinking balls problem is only well tractable in the case of similarity maps, but the case of affine maps is more elusive due to many geometric-dynamical complications. In the current work, we push through these complications and compute the Hausdorff dimension of a set recurring to a shrinking target of geometric balls in some affine iterated function systems. For these results, we have pinpointed a representative class of affine iterated function systems, consisting of a pair of diagonal affine maps, that was introduced by Przytycki and Urba\'nski. The analysis splits into many sub-cases according to the type of the centre point of the targets, and the relative sizes of the targets and the contractions of the maps, illustrating the array of challenges of going beyond affine maps with nice projections. The proofs require heavy machinery from, and expand, the theory of Bernoulli convolutions.

Recently, G\'{o}rska, Lema\'{n}czyk, and de la Rue characterized the class of automorphisms disjoint from all ergodic automorphisms. Inspired by their work, we provide several characterizations of systems that are disjoint from all minimal systems. For a topological dynamical system $(X,T)$, it is disjoint from all minimal systems if and only if there exist minimal subsets $(M_i)_{i\in\mathbb{N}}$ of $X$ whose union is dense in $X$ and each of them is disjoint from $X$ (we also provide a measure-theoretical analogy of the result). For a semi-simple system $(X,T)$, it is disjoint from all minimal systems if and only if there exists a dense $G_{\delta}$ set $\Omega$ in $X \times X$ such that for every pair $(x_1,x_2) \in \Omega$, the subsystems $\overline{\mathcal{O}}(x_1,T)$ and $\overline{\mathcal{O}}(x_2,T)$ are disjoint. Furthermore, for a general system a characterization similar to the ergodic case is obtained.

It is shown that compact sets of complex matrices can always be brought, via similarity transformation, into a form where all matrix entries are bounded in absolute value by the joint spectral radius (JSR). The key tool for this is that every extremal norm of a matrix set admits an Auerbach basis; any such basis gives rise to a desired coordinate system. An immediate implication is that all diagonal entries - equivalently, all one-dimensional principal submatrices - are uniformly bounded above by the JSR. It is shown that the corresponding bounding property does not hold for higher dimensional principal submatrices. More precisely, we construct finite matrix sets for which, across the entire similarity orbit, the JSRs of all higher-dimensional principal submatrices exceed that of the original set. This shows that the bounding result does not extend to submatrices of dimension greater than one. The constructions rely on tools from the geometry of finite-dimensional Banach spaces, with projection constants of norms playing a key role. Additional bounds of the JSR of principal submatrices are obtained using John's ellipsoidal approximation and known estimates for projection constants.

In the short block length regime, pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities. However, in practice, the list size is limited due to the suboptimal scaling of the required area in hardware implementations. Automorphism ensemble decoding (AED) can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yielding a list of estimates from which the final estimate is selected. Yet, AED is limited to appropriately designed polar codes. Subcode ensemble decoding (ScED) was recently proposed for low-density parity-check codes and does not impose such design constraints. It uses multiple decodings in different subcodes, ensuring that the selected subcodes jointly cover the original code. We extend ScED to polar codes by expressing polar subcodes through suitable pre-transformations (PTs). To this end, we describe a framework classifying pre-transformations for pre-transformed polar codes based on their role in encoding and decoding. Within this framework, we propose a new type of PT enabling ScED for polar codes, analyze its properties, and discuss how to construct an efficient ensemble.

In this paper we show that Weyl-invariant commutator blueprints of type $(4, 4, 4)$ are faithful. As a consequence we answer a question of Tits from the late $1980$s about twin buildings. Moreover, we obtain the first example of a $2$-spherical Kac-Moody group over a finite field which is not finitely presented.

In this Part II of a two-part paper, we put forward secure network function computation, where in a directed acyclic network, a sink node is required to compute a target function of which the inputs are generated as source messages at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, is not allowed to obtain any information about a security function of the source messages. In Part I of the two-part paper, we have investigated securely computing linear functions with the wiretapper who can eavesdrop any edge subset up to a certain size r, referred to as the security level, where the security function is the identity function. The notion of this security is called source security. In the current paper, we consider another interesting model which is the same as the above one except that the security function is identical to the target function, i.e., we need to protect the information on the target function from being leaked to the wiretapper. The notion of this security is called target-function security. We first prove a non-trivial upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. In particular, when the security level r is equal to 0, the upper bound reduces to the computing capacity without security consideration. Further, from an algebraic point of view, we prove two equivalent conditions for target-function security and source security for the existence of the corresponding linear function-computing secure network codes. With them, for any linear function over a given finite field, we develop a code construction of linear secure network codes for target-function security and thus obtain a lower bound on the secure computing capacity; and also generalize the code construction developed in Part I for source security.

In this paper, we give upper estimates for the number and sum of eigenvalues below the bottom of the essential spectrum counting multiplicities of quantum waveguides in two dimensions. We consider both straight and curved waveguides of constant width, and the estimates are presented in terms of norms of the potential. For curved quantum waveguide, we assume that the waveguide is not self-intersecting and its curvature is a continuous and bounded function on R. The estimates are new, particularly for the case of curved quantum waveguides and this opens a window for their extension to different configurations such as waveguides with local defamations.

Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are simplices or whose vertices are in general position. We also obtain relatively tight asymptotic bounds for polytopes which do not contain lattice segments of large length.

The study of string C-group representations of rank at least $n/2$ for the symmetric group $S_n$ has gained a lot of attention in the last fifteen years. In a recent paper, Cameron et al. gave a list of permutation representation graphs of rank $r\geq n/2$ for $S_n$, having a fracture graph and a non-perfect split. They conjecture that these graphs are permutation representation graphs of string C-groups. In trying to prove this conjecture, we discovered two new techniques to glue two CPR graphs for symmetric groups together. We discuss the cases in which they yield new CPR graphs. By doing so, we invalidate the conjecture of Cameron et al. We believe our gluing techniques will be useful in the study of string C-group representations of high ranks for the symmetric groups.

The min-knapsack problem with compactness constraints extends the classical knapsack problem, in the case of ordered items, by introducing a restriction ensuring that they cannot be too far apart. This problem has applications in statistics, particularly in the detection of change-points in time series. In this paper, we propose a semidefinite programming approach for this problem, incorporating compactness in constraints or in objective. We study and compare the different relaxations, and argue that our method provides high-quality heuristics and tight bounds. In particular, the single hyperparameter of our penalized semidefinite models naturally balances the trade-off between compactness and accuracy of the computed solutions. Numerical experiments illustrate, on the hardest instances, the effectiveness and versatility of our approach compared to the existing mixed-integer programming formulation.

We consider scale-free percolation on a discrete torus $\mathbf{V}_N$ of size $N$. Conditionally on an i.i.d. sequence of Pareto weights $(W_i)_{i\in \mathbf{V}_N}$ with tail exponent $\tau-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{W_iW_j}{\|i-j\|^{\alpha}} \wedge 1$$ for some parameter $\alpha>0$. We focus on the (centred) Laplacian operator of this random graph and study its empirical spectral distribution. We explicitly identify the limiting distribution when $\alpha<1$ and $\tau>3$, in terms of the spectral distribution of some non-commutative unbounded operators.

This paper explores the algebraic characterization of directed graph substructures through principal minors of the Hermitian Laplacian matrix. By generalizing Bapat et al.'s nonsingular substructure theory and by defining substructures as vertex-edge pairs $(V',E')$ which allows edges to connect vertices outside $V'$, we establish a link between the principle minors and the topological properties of key substructures such as rootless trees and unicyclic graphs. Using the Cauchy-Binet formula, we decompose principal minors into sums of determinants of regular substructures. Specifically, we investigate how these algebraic invariants encode information about unicyclic substructures and their properties, contributing to the broader understanding of graph structures through the lens of Hermitian Laplacian matrix of algebraic graph theory.

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of $[0,1]$, IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given $(b_1,...b_\ell)\in\mathbb N^\ell$, there is no orthogonal vector $(a_1,\dots,a_\ell)\in\mathbb Z^\ell$ with some $|a_j|=1$ if and only if there is an increasing sequence of natural numbers $(n_k)_{k\in\mathbb N}$ with the property that for each $F\subseteq \{1,...,\ell\}$ there is a $\mu$-preserving transformation $T_F:[0,1]\rightarrow[0,1]$ ($\mu$ denotes the Lebesgue measure) such that for any measurable $A,B\subseteq [0,1]$, $$\lim_{k\rightarrow\infty}\mu(A\cap T_F^{-b_jn_k}B)=\begin{cases} \mu(A\cap B),\,\text{ if }j\in F,\\ \mu(A)\mu(B),\,\text{ if }j\not\in F. \end{cases}$$ We remark that this result has a natural extension to a wide class of families of sequences.

We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $$ \partial_tu-\text{div}_x \nabla_\xi f(\nabla u)=0 $$ in a space-time cylinder $\Omega_T=\Omega\times (0,T)$, subject to time-dependent boundary data $g\colon \partial_{\mathcal{P}}\Omega_T\to \mathbf{R}$ prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data $g$ along the lateral boundary $\partial\Omega\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\partial\Omega$ admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers.

We investigate the $R$-boundedness of parameter-dependent families of Poisson operators on the half-space $\mathbb R^n_+$ in various scales of function spaces. Applications concern maximal $L_q$-regularity for boundary value problems with dynamic boundary conditions.

This article exists first and foremost to contribute to a tribute to Patrick Cattiaux. One of the two authors has known Patrick Cattiaux for a very long time, and owes him a great deal. If we are to illustrate the adage that life is made up of chance, then what could be better than the meeting of two young people in the 80s, both of whom fell in love with the mathematics of randomness, and one of whom changed the other's life by letting him in on a secret: if you really believe in it, you can turn this passion into a profession. By another happy coincidence, this tribute comes at just the right time, as Michel Talagrand has been awarded the Abel prize. The temptation was therefore great to do a double. Following one of the many galleries opened up by mathematics, we shall first draw a link between the mathematics of Patrick Cattiaux and that of Michel Talagrand. Then we shall show how the abstract probabilistic material on the concentration of product measures thus revisited can be used to shed light on cut-off phenomena in our field of expertise, mathematical statistics. Nothing revolutionary here, as everyone knows the impact that Talagrand's work has had on the development of mathematical statistics since the late 90s, but we've chosen a very simple framework in which everything can be explained with minimal technicality, leaving the main ideas to the fore.

The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive integer $n$. More precisely, this doubling map defines a permutation $\sigma_{2,n}$ and each of $A$ and $B$ counts the number $C_2(n)$ of cycles of $\sigma_{2,n}$, hence $A=B$. In the first part of this note, we give a direct proof of this last equality. To do so, we consider and study a generalized $(k,n)$-perfect shuffle permutation $\sigma_{k,n}$, where we multiply by an integer $k\ge 2$ instead of $2$, and its number $C_k(n)$ of cycles. The second part of this note lists some of the many occurrences and applications of the doubling map and its generalizations in the literature: in mathematics (combinatorics of words, dynamical systems, number theory, correcting algorithms), but also in card-shuffling, juggling, bell-ringing, poetry, and music composition.

This work addresses control problems governed by a semilinear evolution equation with singular memory kernel $\kappa(t)=\alpha e^{-\beta t}\frac{t^{\nu-1}}{\Gamma(\nu)}$, where $\alpha>0, \beta\ge 0$, and $0<\nu<1$. We examine the existence of a mild solution and the approximate controllability of both linear and semilinear control systems. To this end, we introduce the concept of a resolvent family associated with the linear evolution equation with memory and develop some of its essential properties. Subsequently, we consider a linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, we derive sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a super-reflexive Banach space. Additionally, we present an approximate controllability result within the framework of a general Banach space. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory.

Gao and Xie (2021) conjectured that the inverse Kazhdan-Lusztig polynomial of any matroid is log-concave. Although the inverse Kazhdan-Lusztig polynomial may not always have only real roots, we conjecture that the Hadamard product of an inverse Kazhdan-Lusztig polynomial of degree $n$ and $(1+t)^n$ has only real roots. Using interlacing polynomials and multiplier sequences, we confirm this conjecture for paving matroids. This result allows us to confirm the log-concavity conjecture for these matroids by applying Newton's inequalities.

We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes $ X = (X_t)_{t \in [0, T]} $ and $ Y = (Y_t)_{t \in [0, T]}$, defined by the following McKean-Vlasov equations with common Brownian noise $B^0 = (B_t^0)_{t \in [0, T]}$: \begin{align} dX_t &= b(t, X_t, \mathcal{L}^1(X_t))dt + \sigma(t, X_t, \mathcal{L}^1(X_t))dB_t + \sigma^0(t, \mathcal{L}^1(X_t))dB^0_t, \\ dY_t &= \beta(t, Y_t, \mathcal{L}^1(Y_t))dt + \theta(t, Y_t, \mathcal{L}^1(Y_t))dB_t + \theta^0(t, \mathcal{L}^1(Y_t))dB^0_t, \end{align} where $\mathcal{L}^1(X_t)$ (respectively $\mathcal{L}^1(Y_t)$) denotes a version of the conditional distribution of $X_t$ (resp. $Y_t$) given $B^0$. These results extend those established for standard McKean-Vlasov equations in [Liu and Pag\`es, Ann. App. Prob. 2023] and [Liu and Pag\`es, Bernoulli 2022]. Under suitable conditions, for a (non-decreasing) convex functional $F$ on the path space with polynomial growth, we show $\mathbb{E}[F(X) \mid B^0] \leq \mathbb{E}[F(Y) \mid B^0]$ almost surely. Moreover, for a (non-decreasing) convex functional $G$ defined on the product space of paths and their marginal distributions, we establish \[ \mathbb{E} \Big[\,G\big(X, (\mathcal{L}^1(X_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big]\leq \mathbb{E} \Big[\,G\big(Y, (\mathcal{L}^1(Y_t))_{t\in[0, T]}\big)\,\Big| \, B^0\,\Big] \quad \text{almost surely}. \] Similar convex order results are also established for the corresponding particle system. We explore applications of these results to stochastic control problem - deducing in particular an associated comparison principle for Hamilton-Jacobi-Bellman equations with different coefficients - and to the interbank systemic risk model introduced by in [Carmona, Fouque and Sun, Comm. in Math. Sci. 2015].

We investigate a class of higher-order nonlinear dispersive equations posed on the circle, subject to additive forcing by a finite-dimensional control. Our main objective is to establish approximate controllability by using the controllability of the inviscid Burgers system, linearized around a suitably constructed trajectory. In contrast to earlier approaches based on Lie algebraic techniques, our method offers a more concise proof and sheds new light on the structure of the control. Although the approach necessitates a higher-dimensional control space, both the structure and dimension of the control remain uniform with respect to the order of the dispersive equation and the control time.

Continuing previous works on MacWilliams theory over codes and lattices, a generalization of the MacWilliams theory over $\mathbb{Z}_k$ for $m$ codes is established, and the complete weight enumerator MacWilliams identity also holds for codes over the finitely generated rings $\mathbb{Z}_k[\xi]$. In the context of lattices, the analogy of the MacWilliams identity associated with nu-function was conjectured by Sol\'{e} in 1995, and we present a new formula for nu-function over the lattices associated with a ternary code, which is rather different from the original conjecture. Furthermore, we provide many counterexamples to show that the Sol\'{e} conjecture never holds in the general case, except for the lattices associated with a binary code.

Professor Pierre Sabatier contributed much to the study of inverse problems in theory and practice. Two of these contributions were a focus on theory that actually supports practice, and the identification of well-posed aspects of inverse problems that may quite ill-posed. This paper illustrates these two themes in the context of Electrical Impedance Tomography (EIT), which is both very ill-posed and very practical. We show that for a highly constrained version of this inverse problem, in which a small elliptical inclusion in a homogeneous background is to be identified, optimization of the experimental design (that is, electrode locations) vastly improves the stability of the solution.

We investigate the properties of a class of piecewise-fractional maps arising from the introduction of an invariance under rescaling into convex quadratic maps. The subsequent maps are quasiconvex, and pseudoconvex on specific convex cones; they can be optimised via exact line search along admissible directions, and the iterates then inherit a bidimensional optimality property. We study the minimisation of such relaxed maps via coordinate descents with gradient-based rules, placing a special emphasis on coordinate directions verifying a maximum-alignment property in the reproducing kernel Hilbert spaces related to the underlying positive-semidefinite matrices. In this setting, we illustrate that accounting for the optimal rescaling of the iterates can in certain situations substantially accelerate the unconstrained minimisation of convex quadratic maps.

Let $D\subset\mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy--Szeg\"{o} projection $S_\omega$ defined with respect to a bounded, positive continuous multiple $\omega$ of induced Lebesgue measure, {maps $L^p(bD, \omega)$ to $L^p(bD, \omega)$ continuously} for any $1

The topic of multiple hypotheses testing now has a potpourri of novel theories and ubiquitous applications in diverse scientific fields. However, the universal utility of this field often hinders the possibility of having a generalized theory that accommodates every scenario. This tradeoff is better reflected through the lens of dependence, a central piece behind the theoretical and applied developments of multiple testing. Although omnipresent in many scientific avenues, the nature and extent of dependence vary substantially with the context and complexity of the particular scenario. Positive dependence is the norm in testing many treatments versus a single control or in spatial statistics. On the contrary, negative dependence arises naturally in tests based on split samples and in cyclical, ordered comparisons. In GWAS, the SNP markers are generally considered to be weakly dependent. Generalized familywise error rate (k-FWER) control has been one of the prominent frequentist approaches in simultaneous inference. However, the performances of k-FWER controlling procedures are yet unexplored under different dependencies. This paper revisits the classical testing problem of normal means in different correlated frameworks. We establish upper bounds on the generalized familywise error rates under each dependence, consequently giving rise to improved testing procedures. Towards this, we present improved probability inequalities, which are of independent theoretical interest

Since its introduction, the Potts model has gained widespread popularity across various fields due to its diverse applications. Even minor advancements in this model continue to captivate scientists worldwide, and small modifications often intrigue researchers from different disciplines. This paper investigates a one-dimensional \(q\)-state modified Potts model influenced by an external magnetic field. By leveraging the transfer matrix method, exact expressions are derived for key thermodynamic quantities, including free energy, entropy, magnetization, susceptibility, and specific heat capacity. Numerical analyses explore how these thermodynamic functions vary with relevant parameters, offering insights into the system's behavior. Additionally, the asymptotic properties of these quantities are examined in the limiting cases \(T \to 0\) and \(T \to \infty\). The findings contribute to a deeper understanding of the model's thermodynamic characteristics and highlight its potential applications across various disciplines.

We construct ample smooth strictly plurisubharmonic non-quadratic solutions to the Monge-Amp\`ere equation on either cylindrical type domains or the whole complex Euclidean space $\mathbb C^2$. Among these, the entire solutions defined on $\mathbb C^2$ induce flat Kahler metrics, as expected by a question of Calabi. In contrast, those on cylindrical domains produce a family of nowhere flat Kahler metrics. Beyond these smooth solutions, we also classify solutions that are radially symmetric in one variable, which exhibit various types of singularities. Finally, we explore analogous solutions to Donaldson's equation motivated by a result of He.

This paper addresses the unsourced/uncoordinated random access problem in an integrated sensing and communications (ISAC) system, with a focus on uplink multiple access code design. Recent theoretical advancements highlight that an ISAC system will be overwhelmed by the increasing number of active devices, driven by the growth of massive machine-type communication (mMTC). To meet the demands of future mMTC network, fundamental solutions are required that ensure robust capacity while maintaining favorable energy and spectral efficiency. One promising approach to support emerging massive connectivity is the development of systems based on the unsourced ISAC (UNISAC) framework. This paper proposes a spectrum-sharing compressive sensing-based UNISAC (SSCS-UNISAC) and offers insights into the practical design of UNISAC multiple access codes. In this framework, both communication signals (data transmission) and sensing signals (e.g., radar echoes) overlap within finite channel uses and are transmitted via the proposed UNISAC protocol. The proposed decoder exhibits robust performance, providing 20-30 dB capacity gains compared to conventional protocols such as TDMA and ALOHA. Numerical results validate the promising performance of the proposed scheme.

This work explores the channel estimation (CE) problem in uplink transmission for unsourced random access (URA) with a fluid antenna receiver. The additional spatial diversity in a fluid antenna system (FAS) addresses the needs of URA design in multiple-input and multiple-output (MIMO) systems. We present two CE strategies based on the activation of different FAS ports, namely alternate ports and partial ports CE. Both strategies facilitate the estimation of channel coefficients and angles of arrival (AoAs). Additionally, we discuss how to refine channel estimation by leveraging the sparsity of finite scatterers. Specifically, the proposed partial ports CE strategy is implemented using a regularized estimator, and we optimize the estimator's parameter to achieve the desired AoA precision and refinement. Extensive numerical results demonstrate the feasibility of the proposed strategies, and a comparison with a conventional receiver using half-wavelength antennas highlights the promising future of integrating URA and FAS.

We study the negative band number of braids, knots, and links using Birman, Ko, and Lee's left-canonical form of a braid. As applications, we characterize up to conjugacy strongly quasipositive braids and almost strongly quasipositive braids.

In $2003$, Pei and Wang introduced higher level analogs of the classical Cohen--Eisenstein series. In recent joint work with Beckwith, we found a weight $\frac{1}{2}$ sesquiharmonic preimage of their weight $\frac{3}{2}$ Eisenstein series under $\xi_{\frac{1}{2}}$ utilizing a construction from seminal work by Duke, Imamo\={g}lu and T\'{o}th. In further joint work with Beckwith, when restricting to prime level, we realized our preimage as a regularized Siegel theta lift and evaluated its (regularized) Fourier coefficients explicitly. This relied crucially on work by Bruinier, Funke and Imamo\={g}lu. In this paper, we extend both works to higher weights. That is, we provide a harmonic preimage of Pei and Wang's generalized Cohen--Eisenstein series under $\xi_{\frac{3}{2}-k}$, where $k > 1$. Furthermore, when restricting to prime level, we realize them as outputs of a regularized Shintani theta lift of a higher level holomorphic Eisenstein series, which builds on recent work by Alfes and Schwagenscheidt. Lastly, we evaluate the regularized Millson theta lift of a higher level Maass--Eisenstein series, which is known to be connected to the Shintani theta lift by a differential equation by earlier work of Alfes and Schwagenscheidt.

This paper is about topological rigidity of diagonal group actions on the homogeneous $\SL_4\big(\F(\!(t^{-1})\!)\big)/\SL_4(\F[t])$ where $\F$ is a finite field of characteristic $3$. We show that there is a non-closed relatively compact orbit of the diagonal group.

We observe that a particular first integral of the partially-averaged system in the secular theory of the three-body problem appears also as an important conserved quantity of integrable Kepler billiards. In this note we illustrate their common roots with the projective dynamics of the two-center problem. We then combine these two aspects to define a class of integrable billiard systems on surfaces of constant curvature.

This paper addresses the problem of robust fault detection filtering for linear time-varying (LTV) systems with non-Gaussian noise and additive faults. The conventional generalized likelihood ratio (GLR) method utilizes the Kalman filter, which may exhibit inadequate performance under non-Gaussian noise conditions. To mitigate this issue, a fault detection method employing the $H_{\infty}$ filter is proposed. The $H_{\infty}$ filter is first derived as the solution to a regularized least-squares (RLS) optimization problem, and the effect of faults on the output prediction error is then analyzed. The proposed approach using the $H_{\infty}$ filter demonstrates robustness in non-Gaussian noise environments and significantly improves fault detection performance compared to the original GLR method that employs the Kalman filter. The effectiveness of the proposed approach is illustrated using numerical examples.

We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where \(f\colon X\longrightarrow Y\) is twice continuously Fr\'echet-differentiable on Banach spaces and \(F\colon X\tto Y\) is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor \(u_{k+1}\), then incorporates second-order information in a Halley-type corrector step to obtain \(x_{k+1}\). Under metric regularity of the linearization at a reference solution and H\"older continuity of \(f''\), we prove that the iterates converge locally with order \(2+p\) (cubically when \(p=1\)). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart.

Let $X$ be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric $m$). We derive the variational formulas for the determinant of the Laplacian, ${\rm det}\,\Delta^m$, on $X$ under infinitesimal variations of the positions of the conical points and the conical angles (i. e. infinitesimal variations of $X$ in the class of polyhedra with the same number of vertices). Besides having an independent interest, this derivation may serve as a somewhat belated mathematical counterpart of the well-known heuristic calculation of ${\rm det}\,\Delta^m$ performed by Aurell and Salomonson in the 90-s.

Our research builds upon Halmos's foundational work on functional monadic Boolean algebras and our previous work on tense operators to develop three essential constructions, including the important concepts of fuzzy sets and powerset operators. These constructions have widespread applications across contemporary mathematical disciplines, including algebra, logic, and topology. The framework we present generates four covariant and two contravariant functors, establishing three adjoint situations.

We investigate the sharp material interface limit of the Darcy-Boussinesq model for convection in layered porous media with diffused material interfaces, which allow a gradual transition of material parameters between different layers. We demonstrate that as the thickness of these transition layers approaches zero, the conventional sharp interface model with interfacial boundary conditions, commonly adopted by the fluids community, is recovered under the assumption of constant porosity. Our results validate the widely used sharp interface model by bridging it with the more physically realistic case of diffused material interfaces. This limiting process is singular and involves a boundary layer in the velocity field. Our analysis requires del

In this work, in the THz UMa, extensive channel measurements are conducted and an accurate channel model is developed by combining ray-tracing, computer vision (CV), and statistical methods. Specifically, substantial channel measurement campaigns with distances up to 410~m are conducted at 220~GHz, with nanosecond-level absolute time synchronization. Based on the measurement results, the propagation phenomena are analyzed in detail and the channel characteristics are calculated and statistically modeled. Furthermore, a digital twin enabled channel model (DTECM) is proposed, which generates THz channel responses in a hybrid manner. Specifically, the dominant paths are generated deterministically by using the ray-tracing technique and CV methods. Apart from the path gains determined by ray-tracing, the additional foliage loss is accurately modeled based on foliage information extracted from panoramic pictures. To maintain a low computational complexity for the DTECM, non-dominant paths are then generated statistically. Numeric results reveal that compared to the traditional statistical channel models, the DTECM reduces the path loss modeling error from 14~dB to 4~dB, showing its great superiority. Furthermore, a preliminary link performance evaluation using the DTECM indicates that THz UMa is feasible, though requiring high antenna gains and coverage extension techniques to achieve high spectral efficiencies and wide coverage.

We provide a geometrical characterization of extremal negative dependence as a convex polytope in the simplex of multidimensional Bernoulli distributions, and we prove that it is an antichain that satisfies some minimality conditions with respect to the strongest negative dependence orders. We study the strongly Rayleigh property within this class and explicitly find a distribution that satisfies this property by maximizing the entropy. Furthermore, we construct a chain for the supermodular order starting from extremal negative dependence to independence by mixing the maximum entropy strongly Rayleigh distribution with independence.

In this work, we propose the joint use of a mixed penalty-interior point method and direct search, for addressing nonlinearly constrained derivative-free optimization problems. A merit function is considered, wherein the set of nonlinear inequality constraints is divided into two groups: one treated with a logarithmic barrier approach, and another, along with the equality constraints, addressed using a penalization term. This strategy, is adapted and incorporated into a direct search method, enabling the effective handling of general nonlinear constraints. Convergence to KKT-stationary points is established under continuous differentiability assumptions, without requiring any kind of convexity. Using CUTEst test problems, numerical experiments demonstrate the robustness, efficiency, and overall effectiveness of the proposed method, when compared with state-of-the-art solvers

Let $L$ be a finite extension of $\mathbb{Q}_p$. We calculate the dimension of $\text{Ext}^1$-groups of certain locally analytic representations of $\text{GL}_2(L)$ defined using coherent cohomology of Drinfeld curves. Furthermore, let $\rho_p$ be a $2$-dimensional continuous representation of $\text{Gal}(\bar L/L)$, which is de Rham with parallel Hodge-Tate weights $0,1$ and whose underlying Weil-Deligne representation is irreducible. We prove Breuil's locally analytic $\text{Ext}^1$ conjecture for such $\rho_p$. As an application, we show that the isomorphism class of the multiplicity space $\Pi^{\text{an}}_{\text{geo}}(\rho_p)$ of $\rho_p$ in the pro-\'etale cohomology of Drinfeld curves uniquely determines the isomorphism class of $\rho_p$.

In this article, we ask whether the Igusa zeta function of a restricted power series over $\mathbb{Q}_p$ can be determined solely from the terms of degree at most $D$. That is, we ask whether the truncated polynomial $f_D$, consisting of all terms of f of degree $\leq D$, yields the same Igusa zeta function as $f$ for sufficiently large $D$. Our main results include a counterexample already in the one-variable case, but also a positive result under the condition that $f$ is sufficiently non-degenerate with respect to its Newton polyhedron $\Gamma(f)$.

In this paper, we classify the Hopf hypersurfaces of the complex quadric $Q^m=SO_{m+2}/(SO_2SO_m)$ ($m\geq3$) with at most five distinct constant principal curvatures. We also classify the Hopf hypersurfaces of $Q^m$ ($m=3,4,5$) with constant principal curvatures. All these real hypersurfaces are open parts of homogeneous examples.

This paper provides a pedagogical introduction to classical sheaf theory and sheaf cohomology, followed by a research prospectus exploring potential applications to multi-agent artificial intelligence systems. The first section offers a comprehensive overview of fundamental sheaf-theoretic concepts-presheaves, sheaves, stalks, and cohomology-aimed at researchers in computer science and AI who may not have extensive background in algebraic topology. The second section presents a detailed research prospectus that outlines a roadmap for developing sheaf-theoretic approaches to model and analyze complex systems of interacting agents. We propose that sheaf theory's inherent local-to-global perspective may provide valuable mathematical tools for reasoning about how local agent behaviors collectively determine emergent system properties. The third section contains a literature review connecting sheaf theory with existing research in multi-agent systems, reinforcement learning, and economic modeling. This paper does not present a completed model but rather lays theoretical groundwork and identifies promising research directions that could bridge abstract mathematics with practical AI applications, potentially revealing new approaches to coordination and emergence in multi-agent systems.

In this paper, we address the problem of designing stochastic model predictive control (SMPC) schemes for linear systems affected by unbounded disturbances. The contribution of the paper is rooted in a measured-state initialization strategy. First, due to the nonzero probability of violating chance-constraints in the case of unbounded noise, we introduce ellipsoidal-based probabilistic reachable sets and we include constraint relaxations to recover recursive feasibility conditioned to the measured state. Second, we prove that the solution of this novel SMPC scheme guarantees closed-loop chance constraints satisfaction under minimum relaxation. Last, we demonstrate that, in expectation, the need of relaxing the constraints vanishes over time, which leads the closed-loop trajectories steered towards the unconstrained LQR invariant region. This novel SMPC scheme is proven to satisfy the recursive feasibility conditioned to the state realization, and its superiority with respect to open-loop initialization schemes is shown through numerical examples.

To determine the optimal locations for electric vehicle charging stations, optimisation models need to predict which charging stations users will select. We estimate discrete choice models to predict the usage of charging stations using only readily available information for charging network operators. Our parameter values are estimated from a unique, revealed preferences dataset of charging sessions in Montreal, Quebec. We find that user distance to stations, proximity to home areas, and the number of outlets at each station are significant factors for predicting station usage. Additionally, amenities near charging stations have a neutral effect overall, with some users demonstrating strong preference or aversion for these locations. High variability among the preferences of users highlight the importance of models which incorporate panel effects. Moreover, integrating mixed logit models within the optimization of charging station network design yields high-quality solutions, even when evaluated under other model specifications.

We generalize the theory of Widom factors to the $\mathbb C^n$ setting. We define Widom factors of compact subsets $K\subset \mathbb C^n$ associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets $K=K_1\times\cdots\times K_n$ of $\mathbb C^n$, where each $K_j$ is a non-polar compact subset of $\mathbb C$, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each $K_j$ is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions $w$; in particular, for the case $w\equiv 1$. Finally, we define the Mahler measure of a multivariate polynomial relative to $K\subset \mathbb C^n$ and obtain lower bounds for this quantity on product sets.

Poroelasticity describes the interaction of deformation and fluid flow in saturated porous media. A fully-mixed formulation of Biot's poroelasticity problem has the advantage of producing a better approximation of the Darcy velocity and stress field, as well as satisfying local mass and momentum conservation. In this work, we focus on a novel four-fields Virtual Element discretization of Biot's equations. The stress symmetry is strongly imposed in the definition of the discrete space, thus avoiding the use of an additional Lagrange multiplier. A complete a priori analysis is performed, showing the robustness of the proposed numerical method with respect to limiting material properties. The first order convergence of the lowest-order fully-discrete numerical method, which is obtained by coupling the spatial approximation with the backward Euler time-advancing scheme, is confirmed by a complete 3D numerical validation. A well known poroelasticity benchmark is also considered to assess the robustness properties and computational performance.

We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded number of inversions.

Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ tadpole Cartan matrix is modular, and they provided a proof for $r=2$. The $r=3$ case was recently resolved by Milas and Wang. We prove this conjecture for the next cases $r=4,5$. We also prove the modularity of some companion Nahm sums by establishing the corresponding Rogers--Ramanujan type identities. A key new ingredient in our proofs is some rank reduction formulas which allow us to decompose higher rank tadpole Nahm sums to mixed products of some lower rank Nahm-type sums and theta functions.

Burgers equation is a classic model, which arises in numerous applications. At its very core it is a simple conservation law, which serves as a toy model for various dynamics phenomena. In particular, it supports explicit heteroclinic solutions, both fronts and backs. Their stability has been studied in details. There has been substantial interest in considering dispersive and/or diffusive modifications, which present novel dynamical paradigms in such simple setting. More specificaly, the KdV-Burgers model has been showed to support unique fronts (not all of them monotone!) with fixed values at $\pm \infty$. Many articles, among which \cite{Pego}, \cite{NS1}, \cite{NS2}, have studied the question of stability of monotone (or close to monotone) fronts. In a breakthrough paper, \cite{BBHY}, the authors have extended these results in several different directions. They have considered a wider range of models. The fronts do not need to be monotone, but are subject of a spectral condition instead. Most importantly the method allows for large perturbations, as long as the heteroclinic conditions at $\pm \infty$ are met. That is, there is asymptotic attraction to the said fronts or equivalently the limit set consist of one point. The purpose of this paper is to extend the results of \cite{BBHY} by providing explicit algebraic rates of convergence as $t\to \infty$. We bootstrap these results from the results in \cite{BBHY} using additional energy estimates for two important examples namely KdV-Burgers and the fractional Burgers problem. These rates are likely not optimal, but we conjecture that they are algebraic nonetheless.

The orbifold/condensation completion procedure of defect topological quantum field theories can be seen as carrying out a lattice or state sum model construction internal to an ambient theory. In this paper, we propose a conceptual algebraic description of orbifolds/condensations for arbitrary tangential structures in terms of higher dagger structures and higher idempotents. In particular, we obtain (oriented) orbifold completion from (framed) condensation completion by using a general strictification procedure for higher dagger structures which we describe explicitly in low dimensions; we also discuss the spin and unoriented case. We provide several examples of higher dagger categories, such as those associated to state sum models, (orbifolds of) Landau--Ginzburg models, and truncated affine Rozansky--Witten models. We also explain how their higher dagger structures are naturally induced from rigid symmetric monoidal structures, recontextualizing and extending results from the literature.

It is shown that the existence of a local conserved charge supported by three neighboring sites, or its local version, Reshetikhin's condition, suffices to guarantee the existence of all higher conserved charges and hence the integrability of a quantum spin chain. This explains the ``coincidence'' that no counterexample is known to Grabowski and Mathieu's long-standing conjecture despite the folklore that the conservation of local charges of order higher than 4 imposes additional constraints not implied by the conservation of the three-local charge.

This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group $\mathrm{SL}(2,\mathbb{C})$ at an even level $k\in\mathbb{Z}_+$. Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The $*$-representation of the operator algebra is carried by the infinite dimensional Hilbert space $\mathcal{H}_{\vec{\lambda}}$ and closely connects to the infinite-dimensional $*$-representation of the quantum deformed Lorentz group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$, where $\mathbf{q}=\exp[\frac{2\pi i}{k}(1+b^2)]$ and $\widetilde{\mathbf{q}}=\exp[\frac{2\pi i}{k}(1+b^{-2})]$ with $|b|=1$. The quantum group $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$ also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a $m$-holed sphere $\Sigma_{0,m}$, the physical Hilbert space $\mathcal{H}_{phys}$ is identified by imposing the gauge invariance and the flatness constraint. The states in $\mathcal{H}_{phys}$ are the $\mathscr{U}_{\mathbf{q}}(sl_2)\otimes \mathscr{U}_{\widetilde{\mathbf{q}}}(sl_2)$-invariant linear functionals on a dense domain in $\mathcal{H}_{\vec{\lambda}}$. Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of $\Sigma_{0,m}$ are diagonalized.

This study presents a novel mathematical model derived from cohomology, leveraging the KEEL-proven theorem that establishes cohomology as tautological, generated by boundary classes of curves with fixed dual graphs. Simplicial complexes are constructed using skew-commutative graded algebra, and the structure theorem is applied to connect distinct homologies, enabling precise interpretations of the resulting geometric forms. The proposed model is utilized for protein structure analysis and prediction, with a specific application to the Flagellar Motor structure. This approach offers new insights into the geometric and algebraic foundations of biological macromolecular modeling, highlighting its potential for advancement in structural biology.

This work focuses on showing some arguments addressed to dismantle the extended idea about that social networks completely lacks of privacy properties. We consider the so-called active attacks to the privacy of social networks and the counterpart $(k,\ell)$-anonymity measure, which is used to quantify the privacy satisfied by a social network against active attacks. To this end, we make use of the graph theoretical concept of $k$-metric antidimensional graphs for which the case $k=1$ represents those graphs achieving the worst scenario in privacy whilst considering the $(k,\ell)$-anonymity measure. As a product of our investigation, we present a large number of computational results stating that social networks might not be as insecure as one often thinks. In particular, we develop a large number of experiments on random graphs which show that the number of $1$-metric antidimensional graphs is indeed ridiculously small with respect to the total number of graphs that can be considered. Moreover, we search on several real networks in order to check if they are $1$-metric antidimensional, and obtain that none of them are such. Along the way, we show some theoretical studies on the mathematical properties of the $k$-metric antidimensional graphs for any suitable $k\ge 1$. In addition, we also describe some operations on graphs that are $1$-metric antidimensional so that they get embedded into another larger graphs that are not such, in order to obscure their privacy properties against active attacks.

Supervised machine learning involves approximating an unknown functional relationship from a limited dataset of features and corresponding labels. The classical approach to feature-based machine learning typically relies on applying linear regression to standardized features, without considering their physical meaning. This may limit model explainability, particularly in scientific applications. This study proposes a physics-informed approach to feature-based machine learning that constructs non-linear feature maps informed by physical laws and dimensional analysis. These maps enhance model interpretability and, when physical laws are unknown, allow for the identification of relevant mechanisms through feature ranking. The method aims to improve both predictive performance in regression tasks and classification skill scores by integrating domain knowledge into the learning process, while also enabling the potential discovery of new physical equations within the context of explainable machine learning.

This paper studies the synthesis and mitigation of stealthy attacks in nonlinear cyber-physical systems (CPS). To quantify stealthiness, we employ the Kullback-Leibler (KL) divergence, a measure rooted in hypothesis testing and detection theory, which captures the trade-off between an attacker's desire to remain stealthy and her goal of degrading system performance. First, we synthesize the worst-case stealthy attack in nonlinear CPS using the path integral approach. Second, we consider how a controller can mitigate the impact of such stealthy attacks by formulating a minimax KL control problem, yielding a zero-sum game between the attacker and the controller. Again, we leverage a path integral-based solution that computes saddle-point policies for both players through Monte Carlo simulations. We validate our approach using unicycle navigation and cruise control problems, demonstrating how an attacker can covertly drive the system into unsafe regions, and how the controller can adapt her policy to combat the worst-case attacks.

In many practical situations, randomly assigning treatments to subjects is uncommon due to feasibility constraints. For example, economic aid programs and merit-based scholarships are often restricted to those meeting specific income or exam score thresholds. In these scenarios, traditional approaches to estimating treatment effects typically focus solely on observations near the cutoff point, thereby excluding a significant portion of the sample and potentially leading to information loss. Moreover, these methods generally achieve a non-parametric convergence rate. While some approaches, e.g., Mukherjee et al. (2021), attempt to tackle these issues, they commonly assume that treatment effects are constant across individuals, an assumption that is often unrealistic in practice. In this study, we propose a differencing and matching-based estimator of the average treatment effect on the treated (ATT) in the presence of heterogeneous treatment effects, utilizing all available observations. We establish the asymptotic normality of our estimator and illustrate its effectiveness through various synthetic and real data analyses. Additionally, we demonstrate that our method yields non-parametric estimates of the conditional average treatment effect (CATE) and individual treatment effect (ITE) as a byproduct.

In this work, we propose a joint design of encoding and encryption processes for KEMs like Kyber, without assuming the independence of the decoding noise entries. Our design features two techniques: ciphertext packing and lattice packing. First, we extend the Peikert-Vaikuntanathan-Waters (PVW) method to the Kyber: $\ell$ plaintexts are packed into a single ciphertext. This scheme is referred to as P$_\ell$-Kyber. We prove that the P$_\ell$-Kyber is IND-CCA secure under the M-LWE hardness assumption. We show that the decryption decoding noise entries across the $\ell$ plaintexts (also known as layers) are mutually independent. Second, we propose a cross-layer lattice encoding scheme for the P$_\ell$-Kyber, where every $\ell$ cross-layer information symbols are encoded to a lattice point. This way we obtain a \emph{coded} P$_\ell$-Kyber, where the decoding noise entries for each lattice point are mutually independent. Therefore, the decryption failure rate (DFR) analysis does not require the assumption of independence among the decryption decoding noise entries. Both DFR and communication cost (CER) are greatly decreased thanks to ciphertext packing and lattice packing. Finally, we demonstrate that with $\ell=24$ and Leech lattice encoder, the proposed coded P$_\ell$-KYBER1024 achieves DFR $<2^{-281}$ and CER $ = 4.6$, i.e., a decrease of CER by $90\%$ compared to KYBER1024.

Quick detection of transmittance changes in optical channel is crucial for secure communication. We demonstrate that pre-shared entanglement using two-mode squeezed vacuum states significantly reduces detection latency compared to classical and entanglement-augmented coherent-state probes. The change detection latency is inversely proportional to the quantum relative entropy (QRE), which goes to infinity in the absence of thermal noise, suggesting idealized instantaneous detection. However, in realistic scenarios, we show that QRE scales logarithmically with the inverse of the thermal noise mean photon number. We propose a receiver that achieves this scaling and quantify its performance gains over existing methods. Additionally, we explore the fundamental trade-off between communication capacity and change detection latency, highlighting how pre-shared entanglement enhances both.

Downsampling layers are crucial building blocks in CNN architectures, which help to increase the receptive field for learning high-level features and reduce the amount of memory/computation in the model. In this work, we study the generalization of the uniform downsampling layer for group equivariant architectures, e.g., G-CNNs. That is, we aim to downsample signals (feature maps) on general finite groups with anti-aliasing. This involves the following: (a) Given a finite group and a downsampling rate, we present an algorithm to form a suitable choice of subgroup. (b) Given a group and a subgroup, we study the notion of bandlimited-ness and propose how to perform anti-aliasing. Notably, our method generalizes the notion of downsampling based on classical sampling theory. When the signal is on a cyclic group, i.e., periodic, our method recovers the standard downsampling of an ideal low-pass filter followed by a subsampling operation. Finally, we conducted experiments on image classification tasks demonstrating that the proposed downsampling operation improves accuracy, better preserves equivariance, and reduces model size when incorporated into G-equivariant networks

We consider dynamical models given by rational ODE systems. Parameter estimation is an important and challenging task of recovering parameter values from observed data. Recently, a method based on differential algebra and rational interpolation was proposed to express parameter estimation in terms of polynomial system solving. Typically, polynomial system solving is a bottleneck, hence the choice of the polynomial solver is crucial. In this contribution, we compare two polynomial system solvers applied to parameter estimation: homotopy continuation solver from HomotopyContinuation.jl and our new implementation of a certified solver based on rational univariate representation (RUR) and real root isolation. We show how the new RUR solver can tackle examples that are out of reach for the homotopy methods and vice versa.

We consider the problem of recovering latent information from graphs under $\varepsilon$-edge local differential privacy where the presence of relationships/edges between two users/vertices remains confidential, even from the data curator. For the class of generalized random dot-product graphs, we show that a standard local differential privacy mechanism induces a specific geometric distortion in the latent positions. Leveraging this insight, we show that consistent recovery of the latent positions is achievable by appropriately adjusting the statistical inference procedure for the privatized graph. Furthermore, we prove that our procedure is nearly minimax-optimal under local edge differential privacy constraints. Lastly, we show that this framework allows for consistent recovery of geometric and topological information underlying the latent positions, as encoded in their persistence diagrams. Our results extend previous work from the private community detection literature to a substantially richer class of models and inferential tasks.

We study local, higher-spin conserved currents in integrable $2d$ sigma models that have been deformed via coupling to auxiliary fields. These currents generate integrability-preserving flows introduced by Smirnov and Zamolodchikov. For auxiliary field (AF) deformations of a free boson, we prove that local spin-$n$ currents exist for all $n$ and give recursion relations that characterize Smirnov-Zamolodchikov (SZ) flows driven by these currents. We then show how to construct spin-$2n$ currents in a unified class of auxiliary field sigma models with common structure -- including AF theories based on the principal chiral model (PCM), its non-Abelian T-dual, (bi-)Yang-Baxter deformations of the PCM, and symmetric space models -- for interaction functions of one variable, and describe SZ flows driven by any function of the stress tensor in these cases. Finally, we give perturbative solutions for spin-$3$ SZ flows in any member of our unified class of AF models with underlying $\mathfrak{su}(3)$ algebra. Part of our analysis shows that the class of AF deformations can be extended by allowing the interaction function to depend on a larger set of variables than has previously been considered.

In this paper, we establish a relation between the quantum corner VOA $q\widetilde{Y}_{L,0,N}[\Psi]$, which can be regarded as a generalization of quantum $W_N$ algebra, and Sergeev-Veselov super Macdonald polynomials. We demonstrate precisely that, under a specific map, the correlation functions of the currents of $q\widetilde{Y}_{L,0,N}[\Psi]$, coincide with the Sergeev-Veselov super Macdonald polynomials.

As state of the art neural networks (NNs) continue to grow in size, their resource-efficient implementation becomes ever more important. In this paper, we introduce a compression scheme that reduces the number of computations required for NN inference on reconfigurable hardware such as FPGAs. This is achieved by combining pruning via regularized training, weight sharing and linear computation coding (LCC). Contrary to common NN compression techniques, where the objective is to reduce the memory used for storing the weights of the NNs, our approach is optimized to reduce the number of additions required for inference in a hardware-friendly manner. The proposed scheme achieves competitive performance for simple multilayer perceptrons, as well as for large scale deep NNs such as ResNet-34.

We revisit the Markov Entropy Decomposition, a classical convex relaxation algorithm introduced by Poulin and Hastings to approximate the free energy in quantum spin lattices. We identify a sufficient condition for its convergence, namely the decay of the effective interaction. We prove that this condition is satisfied for systems in 1D at any temperature as well as in the high-temperature regime under a certain commutativity condition on the Hamiltonian. This yields polynomial and quasi-polynomial time approximation algorithms in these settings, respectively. Furthermore, the decay of the effective interaction implies the decay of the conditional mutual information for the Gibbs state of the system. We then use this fact to devise a rounding scheme that maps the solution of the convex relaxation to a global state and show that the scheme can be efficiently implemented on a quantum computer, thus proving efficiency of quantum Gibbs sampling under our assumption of decay of the effective interaction.

These lectures present a brief introduction to measurement theory for QFT in possibly curved spacetimes introduced by the author and R. Verch [Comm. Math. Phys. 378 (2020) 851-889]. Topics include: a brief introduction to algebraic QFT, measurement schemes in QFT, state updates, multiple measurements and the resolution of Sorkin's "impossible measurement" problem. Examples using suitable theories based on Green hyperbolic operators are given, and the interpretational significance of the framework is briefly considered. The basic style is to give details relating to QFT while taking for granted various facts from the theory of globally hyperbolic spacetimes.

We study how the application of injective morphisms affects the number $r$ of equal-letter runs in the Burrows-Wheeler Transform (BWT). This parameter has emerged as a key repetitiveness measure in compressed indexing. We focus on the notion of BWT-run sensitivity after application of an injective morphism. For binary alphabets, we characterize the class of morphisms that preserve the number of BWT-runs up to a bounded additive increase, by showing that it coincides with the known class of primitivity-preserving morphisms, which are those that map primitive words to primitive words. We further prove that deciding whether a given binary morphism has bounded BWT-run sensitivity is possible in polynomial time with respect to the total length of the images of the two letters. Additionally, we explore new structural and combinatorial properties of synchronizing and recognizable morphisms. These results establish new connections between BWT-based compressibility, code theory, and symbolic dynamics.

We present a unified free field realization of representations for the quantum toroidal algebra of $\mathfrak{gl}_1$ with arbitrary levels, constructed using six free boson fields. This realization arises from a specialized factorization of the structure function within the defining relations of the quantum toroidal algebra of $\mathfrak{gl}_1$. Utilizing this free field realization, we further develop intertwining operators for the algebra of $\mathfrak{gl}_1$.

The Sawada-Kotera (SK) equation is an integrable system characterized by a third-order Lax operator and is related to the modified Sawada-Kotera (mSK) equation through a Miura transformation. This work formulates the Riemann-Hilbert problem associated with the SK and mSK equations by using direct and inverse scattering transforms. The long-time asymptotic behaviors of the solutions to these equations are then analyzed via the Deift-Zhou steepest descent method for Riemann-Hilbert problems. It is shown that the asymptotic solutions of the SK and mSK equations are categorized into four distinct regions: the decay region, the dispersive wave region, the Painlev\'{e} region, and the rapid decay region. Notably, the Painlev\'{e} region is governed by the F-XVIII equation in the Painlev\'{e} classification of fourth-order ordinary differential equations, a fourth-order analogue of the Painlev\'{e} transcendents. This connection is established through the Riemann-Hilbert formulation in this work. Similar to the KdV equation, the SK equation exhibits a transition region between the dispersive wave and Painlev\'{e} regions, arising from the special values of the reflection coefficients at the origin. Finally, numerical comparisons demonstrate that the asymptotic solutions agree excellently with results from direct numerical simulations.

Accurate mobile device localization is critical for emerging 5G/6G applications such as autonomous vehicles and augmented reality. In this paper, we propose a unified localization method that integrates model-based and machine learning (ML)-based methods to reap their respective advantages by exploiting available map information. In order to avoid supervised learning, we generate training labels automatically via optimal transport (OT) by fusing geometric estimates with building layouts. Ray-tracing based simulations are carried out to demonstrate that the proposed method significantly improves positioning accuracy for both line-of-sight (LoS) users (compared to ML-based methods) and non-line-of-sight (NLoS) users (compared to model-based methods). Remarkably, the unified method is able to achieve competitive overall performance with the fully-supervised fingerprinting, while eliminating the need for cumbersome labeled data measurement and collection.

The Sum-of-Squares (SoS) hierarchy, also known as Lasserre hierarchy, has emerged as a promising tool in optimization. However, it remains unclear whether fixed-degree SoS proofs can be automated [O'Donnell (2017)]. Indeed, there are examples of polynomial systems with bounded coefficients that admit low-degree SoS proofs, but these proofs necessarily involve numbers with an exponential number of bits, implying that low-degree SoS proofs cannot always be found efficiently. A sufficient condition derived from the Nullstellensatz proof system [Raghavendra and Weitz (2017)] identifies cases where bit complexity issues can be circumvented. One of the main problems left open by Raghavendra and Weitz is proving any result for refutations, as their condition applies only to polynomial systems with a large set of solutions. In this work, we broaden the class of polynomial systems for which degree-$d$ SoS proofs can be automated. To achieve this, we develop a new criterion and we demonstrate how our criterion applies to polynomial systems beyond the scope of Raghavendra and Weitz's result. In particular, we establish a separation for instances arising from Constraint Satisfaction Problems (CSPs). Moreover, our result extends to refutations, establishing that polynomial-time refutation is possible for broad classes of polynomial time solvable constraint problems, highlighting a first advancement in this area.

The analytic extension of the Kerr spacetimes into the negative radial region contains closed causal curves for any non-zero rotation parameter $a$ and mass parameter $M$. Furthermore, the spacetimes become totally vicious when $|a|>M$, meaning that through every point there exists a closed timelike curve. Despite this, we prove that the Kerr spacetimes do not admit any closed null geodesics when $|a|\geq M$. This result generalises recent findings by one of the authors, which showed the nonexistence of closed causal geodesics in the case $|a|

We present new solutions to the Klein-Gordon equation for a scalar particle in a black string spacetime immersed in an anisotropic quintessence fluid surrounded by a cloud of strings, extending the analysis presented in our previous work. These novel solutions are dependent on the quintessence state parameter, $\alpha_{Q}$, and are now valid for a much larger domain of the radial coordinate. We investigate the cases when $\alpha_{Q} = 0,\,1/2,\,1$, encompassing both black hole and horizonless scenarios. We express the resulting radial wave functions using the confluent and biconfluent Heun functions, with special cases represented by Bessel functions. We derive restrictions on the allowed quantum energy levels by imposing constraints on the Heun parameters to ensure polynomial solutions. Furthermore, we investigate the emergence of "dark phases" associated with the radial wave function, focusing on the interesting case of $\alpha_{Q} = 1$. Our findings provide insights into the dynamics of scalar particles in this complex spacetime and the potential impact of dark energy on quantum systems.

We propose a method for constructing quantum error-correcting codes based on non-binary low-density parity-check codes with girth 16. In conventional constructions using circulant permutation matrices, the girth is upper-bounded by 12, which limits the suppression of harmful short cycles. Our construction employs affine permutation matrices and a randomized sequential selection procedure designed to eliminate short cycles, which are known to limit decoding performance. Joint belief propagation decoding is applied over depolarizing channels. Numerical experiments confirm that the proposed codes reduce the number of low-weight codewords in $C_X \setminus C_Z^\perp$ and $C_Z \setminus C_X^\perp$, and thus have the potential to suppress error floors. In addition, we obtain a significantly improved upper bound on the minimum distance, which we conjecture to be tight.