This paper constructs a graded-commutative, associative, differential Transverse Intersection Algebra TIA {on the torus (in any dimension) with its cubical decomposition by using a probabilistic wiggling interpretation. This structure agrees with the combinatorial graded intersection algebra (graded by codimension) defined by transversality on pairs of `cuboidal chains' which are in general position. In order to define an intersection of cuboids which are not necessarily in general position, the boundaries of the cuboids are considered to be `wiggled' by a distance small compared with the lattice parameter, according to a suitable probability distribution and then almost always the wiggled cuboids will be in general position, producing a transverse intersection with new probability distributions on the bounding sides. In order to make a closed theory, each geometric cuboid appears in an infinite number of forms with different probability distributions on the wiggled boundaries. The resulting structure is commutative, associative and satisfies the product rule with respect to the natural boundary operator deduced from the geometric boundary of the wiggled cuboids. This TIA can be viewed as a combinatorial analogue of differential forms in which the continuity of space has been replaced by a lattice with corrections to infinite order. See the comparison to Whitney forms at the end of the paper. For application to fluid algebra we also consider the same construction starting with the $2h$ cubical complex instead of the $h$ cubical complex. The adjoined higher order elements will be identical to those required in the $h$ cubical complex. The $d$-dimensional theory is a tensor product of $d$ copies of the one-dimensional theory.

This paper shows that on the Bergman space of the open unit disk, the slant Toeplitz operator $T_{p+\varphi}$ and $T_{p+\psi}$ commute if and only if $\varphi=\psi$ ,where $\varphi$ and $\psi$ are both bounded analytic functions, and $p$ is ananalytic polynomial.

In this paper we introduce and study \emph{rectangular torsion theories}, i.e.\ those torsion theories $(\C,\T,\F)$ with $\C$ a pointed category, where the canonical functor $\C\to \T\times\F$ is an equivalence of categories. In particular, we show that these are precisely the internal rectangular bands in the 2-category of pointed categories.

We study the topological properties of expanding invariant foliations of $C^{1+}$ diffeomorphisms, in the context of partially hyperbolic diffeomorphisms and laminations with $1$-dimensional center bundle. In this first version of the paper, we introduce a property we call *s-transversality* of a partially hyperbolic lamination with $1$-dimensional center bundle, which is robust under $C^1$ perturbations. We prove that under a weak expanding condition on the center bundle (called *some hyperbolicity*, or "SH"), any s-transverse partially hyperbolic lamination contains a disk tangent to the center-unstable direction (Theorem C). We obtain several corollaries, among them: if $f$ is a $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center, and the (strong) unstable foliation $W^{uu}$ of $f$ is minimal, then $W^{uu}$ is robustly minimal under $C^1$-small perturbations, provided that the stable and strong unstable bundles are not jointly integrable (Theorem B). Theorem B has applications in our upcoming work with Eskin, Potrie and Zhang, in which we prove that on ${\mathbb T}^3$, any $C^{1+}$ partially hyperbolic Anosov diffeomorphism with $1$-dimensional expanding center has a minimal strong unstable foliation, and has a unique $uu$-Gibbs measure provided that the stable and strong unstable bundles are not jointly integrable. In a future work, we address the density (in any $C^r$ topology) of minimality of strong unstable foliations for $C^{1+}$ partially hyperbolic diffeomorphisms with $1$-dimensional center and the SH property.

With the soaring demand for high-performing integrated circuits, 3D integrated circuits (ICs) have emerged as a promising alternative to traditional planar structures. Unlike existing 3D ICs that stack 2D layers, a full 3D IC features cubic circuit elements unrestricted by layers, offering greater design freedom. Design problems such as floorplanning, transistor sizing, and interconnect sizing are highly complex due to the 3D nature of the circuits and unavoidably require systematic approaches. We introduce geometric programming to solve these design optimization problems systematically and efficiently.

Forecasting atmospheric flows with traditional discretization methods, also called full order methods (e.g., finite element methods or finite volume methods), is computationally expensive. We propose to reduce the computational cost with a Reduced Order Model (ROM) that combines Extended Convolutional Autoencoders (E-CAE) and Reservoir Computing (RC). Thanks to an extended network depth, the E-CAE encodes the high-resolution data coming from the full order method into a compact latent representation and can decode it back into high-resolution with 75% lower reconstruction error than standard CAEs. The compressed data are fed to an RC network, which predicts their evolution. The advantage of RC networks is a reduced computational cost in the training phase compared to conventional predictive models. We assess our data-driven ROM through well-known 2D and 3D benchmarks for atmospheric flows. We show that our ROM accurately reconstructs and predicts the future system dynamics with errors below 6% in 2D and 8% in 3D, while significantly reducing the computational cost of a full-order simulation. Compared to other ROMs available in the literature, such as Dynamic Mode Decomposition and Proper Orthogonal Decomposition with Interpolation, our ROM is as efficient but more accurate. Thus, it is a promising alternative to high-dimensional atmospheric simulations.

We provide a formula for the third order free cumulants of products as entries. We apply this formula to find the third order free cumulants of various Random Matrix Ensambles including product of Ginibre Matrices and Wishart matrices, both in the Gaussian case.

We present an event-triggered boundary control scheme for a class of reaction-diffusion PDEs using operator learning and backstepping method. Our first-of-its-kind contribution aims at learning the backstepping kernels, which inherently induces the learning of the gains in the event trigger and the control law. The kernel functions in constructing the control law are approximated with neural operators (NOs) to improve the computational efficiency. Then, a dynamic event-triggering mechanism is designed, based on the plant and the continuous-in-time control law using kernels given by NOs,to determine the updating times of the actuation signal. In the resulting event-based closed-loop system, a strictly positive lower bound of the minimal dwell time is found, which is independent of initial conditions. As a result, the absence of a Zeno behavior is guaranteed. Besides, exponential convergence to zero of the L_2 norm of the reaction-diffusion PDE state and the dynamic variable in the event-triggering mechanism is proved via Lyapunov analysis. The effectiveness of the proposed method is illustrated by numerical simulation.

Multiparameter persistent homology is a generalization of classical persistent homology, a central and widely-used methodology from topological data analysis, which takes into account density estimation and is an effective tool for data analysis in the presence of noise. Similar to its classical single-parameter counterpart, however, it is challenging to compute and use in practice due to its complex algebraic construction. In this paper, we study a popular and tractable invariant for multiparameter persistent homology in a statistical setting: the multiparameter persistence landscape. We derive a functional central limit theorem for multiparameter persistence landscapes, from which we compute confidence bands, giving rise to one of the first statistical inference methodologies for multiparameter persistence landscapes. We provide an implementation of confidence bands and demonstrate their application in a machine learning task on synthetic data.

A $2-$dimensional mosaic floorplan is a partition of a rectangle by other rectangles with no empty rooms. These partitions (considered up to some deformations) are known to be in bijection with Baxter permutations. A $d$-floorplan is the generalisation of mosaic floorplans in higher dimensions, and a $d$-permutation is a $(d-1)$-tuple of permutations. Recently, in N. Bonichon and P.-J. Morel, {\it J. Integer Sequences} 25 (2022), Baxter $d$-permutations generalising the usual Baxter permutations were introduced. In this paper, we consider mosaic floorplans in arbitrary dimensions, and we construct a generating tree for $d$-floorplans, which generalises the known generating tree structure for $2$-floorplans. The corresponding labels and rewriting rules appear to be significantly more involved in higher dimensions. Moreover we give a bijection between the $2^{d-1}$-floorplans and $d$-permutations characterized by forbidden vincular patterns. Surprisingly, this set of $d$-permutations is strictly contained within the set of Baxter $d$-permutations.

We develop sample path large deviations for multivariate Hawkes processes with heavy-tailed mutual excitation rates. Our techniques rely on multivariate hidden regular variation, in conjunction with the cluster representation of Hawkes processes and a recent result on the tail asymptotics of the cluster sizes, to unravel the most likely configuration of (multiple) big jumps. Our proof hinges on establishing asymptotic equivalence between a suitably scaled multivariate Hawkes process and a coupled L\'evy process with multivariate hidden regular variation. Hence, along the way, we derive a sample-path large deviations principle for a class of multivariate heavy-tailed L\'evy processes that plays an auxiliary role in our analysis but is also of independent interest.

The aim of this paper is to study the class of quasicomplemented distributive nearlattices. We investigate $\alpha$-filters and $\alpha$-ideals in quasicomplemented distributive nearlattices and some results on ideals-congruence-kernels. Finally, we also study the notion of Stone distributive nearlattice and give a characterization by means $\sigma$-filters.

This paper focuses on the further development of the Lie bracket approximation approach for extremum seeking systems. Classical results in this area provide extremum seeking algorithms with exponential convergence rates for quadratic-like cost functions, and polynomial decay rates for cost functions of higher degrees. This paper proposes a novel control design approach that ensures the motion of the extremum seeking system along directions associated with higher-order Lie brackets, thereby ensuring exponential convergence for cost functions that are polynomial-like but with degree greater than two.

This work addresses an inverse problem for a semi-discrete parabolic equation, which consists of identifying the right-hand side of the equation based on solution measurements at an intermediate time and within a spatial subdomain. This result can be applied to establish a stability estimate for the spatially dependent potential function. Our approach relies on a novel semi-discrete Carleman estimate, whose parameter is constrained by the mesh size. As a consequence of the discrete terms arising in the Carleman inequality, this method naturally introduces an error term related to the solution's initial condition.

Interfacial Stokes flow can be efficiently computed using the Boundary Integral Equation method. In 3D, the fluid velocity at a target point is given by a 2D surface integral over all interfaces, thus reducing the dimension of the problem. A core challenge is that for target points near, but not on, an interface, the surface integral is near-singular and standard quadratures lose accuracy. This paper presents a method to accurately compute the near-singular integrals arising in elliptic boundary value problems in 3D. It is based on a local series approximation of the integrand about a base point on the surface, obtained by orthogonal projection of the target point onto the surface. The elementary functions in the resulting series approximation can be integrated to high accuracy in a neighborhood of the base point using a recursive algorithm. The remaining integral is evaluated numerically using a standard quadrature rule, chosen here to be the 4th order Trapezoidal rule. The method is reduced to the standard quadrature plus a correction, and is uniformly of 4th order. The method is applied to resolve Stokes flow past several ellipsoidal rigid bodies. We compare the error in the velocity near the bodies, and in the time and displacement of particles traveling around the bodies, computed with and without the corrections.

We develop Lie theory in the category $\text{Ver}_4^+$ over a field of characteristic 2, the simplest tensor category which is not Frobenius exact, as a continuation of arXiv:2406.10201. We provide a conceptual proof that an operadic Lie algebra in $\text{Ver}_4^+$ is a Lie algebra, i.e. satisfies the PBW theorem, exactly when its invariants form a usual Lie algebra. We then classify low-dimensional Lie algebras in $\text{Ver}_4^+$, construct elements in the center of $U(\mathfrak{gl}(X))$ for $X \in \text{Ver}_4^+$, and study representations of $\mathfrak{gl}(P)$, where $P$ is the indecomposable projective of $\text{Ver}_4^+$.

Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. We investigate the number and sizes of the $P_n\setminus S_n\ /\ P_n$ double cosets, showing that most double cosets have maximal size when $p$ is odd, or equivalently, that $P_n\cap P_n^x=1$ for most $x\in S_n$ when $n$ is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.

The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira and Invernizzi considered random geometric graphs on the $d$-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension $d$ go to infinity. They posed the question about the behavior when $d$ is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function.

Let $G$ be a finite solvable group with disconnected character degree graph $\Delta(G)$. Under these conditions, it follows from a result of P\'alfy that $\Delta(G)$ consists of two connected components. Another result of P\'alfy's gives an inequality relating the sizes of these two connected components. In this paper, we calculate the number of possible component size pairs that satisfy P\'alfy's inequality. Additionally, for a fixed positive integer $n$, the number of distinct graph orders for which exactly $n$ component size pairs satisfy P\'alfy's inequality is shown.

In this paper, we propose a randomized accelerated method for the minimization of a strongly convex function under linear constraints. The method is of Kaczmarz-type, i.e. it only uses a single linear equation in each iteration. To obtain acceleration we build on the fact that the Kaczmarz method is dual to a coordinate descent method. We use a recently proposed acceleration method for the randomized coordinate descent and transfer it to the primal space. This method inherits many of the attractive features of the accelerated coordinate descent method, including its worst-case convergence rates. A theoretical analysis of the convergence of the proposed method is given. Numerical experiments show that the proposed method is more efficient and faster than the existing methods for solving the same problem.

This article characterizes phase transitions in temperature within a specific space of H\"older continuous potentials, distinguished by their regularity and asymptotic behavior at zero. We also characterize the phase transitions in temperature that are robust within this space. Our results reveal a connection between phase transitions in temperature and ergodic optimization.

We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_\Omega F(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap \Omega), \] where $F$ is a density function satisfying some structural conditions. The onus of proof lies with the careful analysis of the ghost function, the gradient part in the Helmholtz-W\'eyl decomposition of a nonlinear flux that appears in the domain variation formula for $J(u)$. As an application we prove full regularity for a class of quasilinear Bernoulli type free boundary problems in $\R^3$.

We characterize covariant positive decomposable maps between unital C*-algebras in terms of a dilation theorem, which generalizes a seminal result by H. Scutaru from Rep. Math. Phys. 16 (1):79-87, 1979. As a case study, we provide a certain characterization of the operator sum representation of maps on $\mathbb{M}_n (\mathbb{C})$, covariant with respect to the maximal commutative subgroup of $\mathrm{U}(n)$. A connection to quantum dynamics is established by specifying sufficient and necessary conditions for covariance of D-divisible (decomposably divisible) quantum evolution families, recently introduced in J. Phys. A: Math. Theor. 56 (2023) 485202.

We construct a monotonicity formula for the free boundary problem of the form $\Delta u=\mu$, where $\mu$ is a Radon measure. It implies that the blow up limits of solutions are homogenous functions of degree one. The first formula is new even for classical Laplace operator. Our method of proof uses a careful application of the strong maximum principle.

We confirm a long standing conjecture in the case of rank 3 that MacPhersonians are homotopy equivalent to Grassmannians.

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$ is a quantitative measure of its $d$-dimensional rigidity, defined in terms of the eigenvalues of stiffness matrices associated with different embeddings of the graph into $\mathbb{R}^d$. For a function $a:V\to \mathbb{N}$, we denote by $G^{(a)}$ the $a$-blow-up of $G$, that is, the graph obtained from $G$ by replacing every vertex $v\in V$ with an independent set of size $a(v)$. We determine a relation between the stiffness matrix eigenvalues of $G^{(a)}$ and the eigenvalues of certain weighted stiffness matrices associated with the original graph $G$. This resolves, as a special case, a conjecture of Lew, Nevo, Peled and Raz on the stiffness eigenvalues of balanced blow-ups of the complete graph. As an application, we obtain a lower bound on the $d$-dimensional algebraic connectivity of complete bipartite graphs. More precisely, we prove the following: Let $K_{n,m}$ be the complete bipartite graph with sides of size $n$ and $m$ respectively. Then, for every $d\ge 1$ there exists $c_d>0$ such that, for all $n,m\ge d+1$ with $n+m\ge \binom{d+2}{2}$, $a_d(K_{n,m})\ge c_d\cdot \min\{n,m\}$. This bound is tight up to the multiplicative constant. In the special case $d=2$, $n=m=3$, we obtain the improved bound $a_2(K_{3,3})\ge 2(1-\lambda)$, where $\lambda\approx 0.6903845$ is the unique positive real root of the polynomial $176 x^4-200 x^3+47 x^2+18 x-9$, which we conjecture to be tight.

This paper investigates the cuspidal spectrum of the quotient of the real Lie group $G= SU(n,1)$ and a principal congruence subgroup $\Gamma(m)$ for $m\geq 3$, focusing on the multiplicities of integrable discrete series representations. Using the Selberg trace formula, we derive an explicit formula for the multiplicity $m(\Gamma(m), \pi_\tau)$ of a representation $\pi_\tau$ of integrable discrete series of $G$ within $L^2(\Gamma(m) \backslash G)$. The formula involves the Harish-Chandra parameter $\tau$, the discriminant $D_\ell$ of the imaginary quadratic field $\ell$ over which $G$ is defined and special values of the Dirichlet $L$-function $L_\ell$ associated to $\ell$. We apply these results on the one hand to compute the cuspidal cohomology of locally symmetric spaces $\Gamma(m) \backslash G / K$, where $K$ is a maximal compact subgroup of $G$. On the other hand we use them to reprove a known rationality result involving the values of $L_\ell$ at odd positive integers and make them more explicit. This work extends previous studies on real and quaternionic hyperbolic spaces to the complex hyperbolic case, contributing to the understanding of the spectrum of $\mathbb{R}$-rank one algebraic groups.

We consider the problem of assigning tasks efficiently to a set of workers that can exhaust themselves as a result of processing tasks. If a worker is exhausted, it will take a longer time to recover. To model efficiency of workers with exhaustion, we use a continuous-time Markov chain (CTMC). By taking samples from the internal states of the workers, the source assigns tasks to the workers when they are found to be in their efficient states. We consider two different settings where (i) the source can assign tasks to the workers only when they are in their most efficient state, and (ii) it can assign tasks to workers when they are also moderately efficient in spite of a potentially reduced success probability. In the former case, we find the optimal policy to be a threshold-based sampling policy where the thresholds depend on the workers' recovery and exhaustion rates. In the latter case, we solve a non-convex sum-of-ratios problem using a branch-and-bound approach which performs well compared with the globally optimal solution.

We consider a scattering problem generated by the Sturm-Liouville equation on a tree which consists of an equilateral compact subtree and a half-infinite lead attached to its root. We assume that the potential on the lead is identically zero while the potentials on the finite edges are real. We show how to find the shape of the tree using the S-function of the scattering problem and the eigenvalues of the operators associated with the compact tree.

This paper calculates transient distributions of a special class of Markov processes with continuous state space and in continuous time, up to an explicit error bound. We approximate specific queues on R with one-sided L\'evy input, such as the M/G/1 workload process, with a finite-state Markov chain. The transient distribution of the original process is approximated by a distribution with a density which is piecewise constant on the state space. Easy-to-calculate error bounds for the difference between the approximated and actual transient distributions are provided in the Wasserstein distance. Our method is fast: to achieve a practically useful error bound, it usually requires only a few seconds or at most minutes of computation time.

In this article, a new approach for 3D motion planning, applicable to aerial vehicles, is proposed to connect an initial and final configuration subject to pitch rate and yaw rate constraints. The motion planning problem for a curvature-constrained vehicle over the surface of a sphere is identified as an intermediary problem to be solved, and it is the focus of this paper. In this article, the optimal path candidates for a vehicle with a minimum turning radius $r$ moving over a unit sphere are derived using a phase portrait approach. We show that the optimal path is $CGC$ or concatenations of $C$ segments through simple proofs, where $C = L, R$ denotes a turn of radius $r$ and $G$ denotes a great circular arc. We generalize the previous result of optimal paths being $CGC$ and $CCC$ paths for $r \in \left(0, \frac{1}{2} \right]\bigcup\{\frac{1}{\sqrt{2}}\}$ to $r \leq \frac{\sqrt{3}}{2}$ to account for vehicles with a larger $r$. We show that the optimal path is $CGC, CCCC,$ for $r \leq \frac{1}{\sqrt{2}},$ and $CGC, CC_\pi C, CCCCC$ for $r \leq \frac{\sqrt{3}}{2}.$ Additionally, we analytically construct all candidate paths and provide the code in a publicly accessible repository.

In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a geometric interpretation of the BCFW recurrence. Arkani-Hamed and Trnka conjectured that each way of iterating the BCFW recurrence gives a "triangulation" or "tiling" of the m=4 amplituhedron. In this article we prove the BCFW tiling conjecture of Arkani-Hamed and Trnka. We also prove the cluster adjacency conjecture for BCFW tiles of the amplituhedron, which says that facets of tiles are cut out by collections of compatible cluster variables for the Grassmannian Gr(4,n). Moreover we show that each BCFW tile is the subset of the Grassmannian where certain cluster variables have particular signs.

Let $F$ be a $p$-adic field. In this article, we consider representations of split special orthogonal groups $\mathrm{SO}_{2n+1}(F)$ and symplectic groups $\mathrm{Sp}_{2n}(F)$ of rank $n$. We denote by $\pi_1 \times \ldots \times \pi_r \rtimes \pi$ the normalized parabolically induced representation of either. Now let $u_i$ be essentially Speh representations and $\pi$ a representation of Arthur type. We prove that the parabolic induction $u_1 \times \ldots \times u_r \rtimes \pi$ is irreducible if and only if $u_i \times u_j$, $u_i \times u_j^\vee$ and $u_i \rtimes \pi$ are irreducible for all choices of $i\neq j$. If $u_i$ are Speh representations, we determine the composition series of the above parabolically induced representation. Through this, we are able to produce a new collection of unitary representations.

In the papers Ziegler(2001) and Goldstein(2012) it was previously shown that any subset of the Boolean cube $ S \subset \{0,1\}^n $ for $ n \leq 9 $ can be partitioned into $n+1$ parts of smaller diameter, i.e., the Borsuk conjecture holds for such subsets. In this paper, it is shown that this is also true for $ n=10 $; however, the complexity of the computational verification increases significantly. In order to perform the computations in a reasonable time, several heuristics were developed to reduce the search tree. The SAT solver $\textbf{kissat}$ was used to cut off the search branches.

We obtain improved local well-posedness results for the Lorentzian timelike minimal surface equation. In dimension $d=3$, for a surface of arbitrary co-dimension, we show a gain of $1/3$ derivative regularity compared to a generic equation of this type. The first result of this kind, going substantially beyond a general Strichartz threshold for quasilinear hyperbolic equations, was shown for minimal surfaces of co-dimension one with a gain of $1/4$ regularity by Ai-Ifrim-Tataru. We use a geometric formulation of the problem, relying on its parametric representation. The natural dynamic variables in this formulation are the parametrizing map, the induced metric and the second fundamental form of the immersion. The main geometric observation used in this paper is the Gauss (and Ricci) equation, dictating that the Riemann curvature of the induced metric (and the curvature of the normal bundle) can be expressed as the wedge product of the second fundamental form with itself. The second fundamental form, in turn, satisfies a wave equation with respect to the induced metric. Exploiting the problem's diffeomorphism freedom, stemming from the non-uniqueness of the parametrization, we introduce a new gauge - a choice of a coordinate system on the parametrizing manifold - in which the metric recovers the full regularity of its Riemann curvature, including the crucial $L^1L^\infty$ estimate for the first derivatives of the metric. Analysis of minimal surfaces with co-dimension bigger than one requires that we impose and take advantage of an additional special gauge on the normal bundle of the surface. The proof also uses both the additional structure contained in the wedge product and the Strichartz estimates with losses developed earlier in the context of a well-posedness theory for quasilinear hyperbolic equations.

Multiple works regarding convergence analysis of Markov chains have led to spectral gap decomposition formulas of the form \[ \mathrm{Gap}(S) \geq c_0 \left[\inf_z \mathrm{Gap}(Q_z)\right] \mathrm{Gap}(\bar{S}), \] where $c_0$ is a constant, $\mathrm{Gap}$ denotes the right spectral gap of a reversible Markov operator, $S$ is the Markov transition kernel (Mtk) of interest, $\bar{S}$ is an idealized or simplified version of $S$, and $\{Q_z\}$ is a collection of Mtks characterizing the differences between $S$ and $\bar{S}$. This type of relationship has been established in various contexts, including: 1. decomposition of Markov chains based on a finite cover of the state space, 2. hybrid Gibbs samplers, and 3. spectral independence and localization schemes. We show that multiple key decomposition results across these domains can be connected within a unified framework, rooted in a simple sandwich structure of $S$. Within the general framework, we establish new instances of spectral gap decomposition for hybrid hit-and-run samplers and hybrid data augmentation algorithms with two intractable conditional distributions. Additionally, we explore several other properties of the sandwich structure, and derive extensions of the spectral gap decomposition formula.

Consider a robot that walks along a knot once on a knot diagram and switches every undercrossing it meets, stopping when it comes back to the starting position. We show that such a robot always unknots the knot. In fact, we prove that the robot produces an ascending diagram, and we provide a purely combinatorial proof that every ascending or descending knot diagram with C crossings can be transformed into the zero-crossing unknot diagram using at most (7C+1)C Reidemeister moves. Moreover, we show that an ascending or descending knot diagram can always be transformed into a zero-crossing unknot diagram using Reidemeister moves that do not increase the number of crossings.

As a medium for cold data storage, DNA stands out as it promises significant gains in storage capacity and lifetime. However, it comes with its own data processing challenges to overcome. Constrained codes over the DNA alphabet $\{A,T,G,C\}$ have been used to design DNA sequences that are free of long homopolymers to increase stability, yet effective error detection and error correction are required to achieve reliability in data retrieval. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely DNA LOCO (D-LOCO) codes, with error detection. In this paper, we equip our D-LOCO codes with error correction for substitution errors via syndrome-like decoding, designated as residue decoding. We only use D-LOCO codewords of indices divisible by a suitable redundancy metric $R(m) > 0$, where $m$ is the code length, for error correction. We provide the community with a construction of constrained codes forbidding runs of length higher than fixed $\ell \in \{1,2,3\}$ and $GC$-content in $\big [0.5-\frac{1}{2K},0.5+\frac{1}{2K}\big ]$ that correct $K$ segmented substitution errors, one per codeword. We call the proposed codes error-correction (EC) D-LOCO codes. We also give a list-decoding procedure with near-quadratic time-complexity in $m$ to correct double-substitution errors within EC D-LOCO codewords, which has $> 98.20\%$ average success rate. The redundancy metric is projected to require $2\log_2(m)+O(1)$-bit allocation for a length-$m$ codeword. Hence, our EC D-LOCO codes are projected to be capacity-approaching with respect to the error-free constrained system.

The $p$-angular distance was first introduced by Maligranda in 2006, while the skew $p$-angular distance was first introduced by Rooin in 2018. In this paper, we shall introduce a new geometric constant named Maligranda-Rooin constant in Banach spaces to compare $p$-angular distance and skew $p$-angular distance. We denote the Maligranda-Rooin constant as $\mathcal{M} \mathcal{R}_p(\mathcal{X})$. First, the upper and lower bounds for the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant is given. Next, it's shown that, a normed linear space is an inner space if and only if $\mathcal{M} \mathcal{R}_p(\mathcal{X})=1$. Moreover, an equivalent form of this new constant is established. By means of the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, we carry out the quantification of the characterization of uniform nonsquareness. Finally, we study the relationship between the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, uniform convexity, uniform smooth and normal structure.

We show (Theorem 3) that the symplectic reduction of the spatial $n$-body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions. Each stratum is realized as coadjoint orbit in the dual of the Lie algebra of the linear symplectic group $Sp(2n-2)$. The planar stratum arises as the frontier upon taking the closure of the spatial stratum. We reduce by going to center-of-mass coordinates to reduce by translations and boosts and then performing symplectic reduction with respect to the orthogonal group $O(3)$. The theorem is a special case of a general theorem (Theorem 2) which holds for the $n$-body problem in any dimension $d$. This theorem follows largely from a ``Poisson reduction'' theorem, Theorem 1. We achieve our reduction theorems by combining the Howe dual pair perspective of reduction espoused by Lerman-Montgomery-Sjamaar with a normal form arising from a symplectic singular value decomposition due to Xu. We begin the paper by showing how Poisson reduction by the Galilean group rewrites Newton's equations for the $n$-body problem as a Lax pair. In section 6.4 we show that this Lax pair representation of the $n$-body equations is equivalent to the Albouy-Chenciner representation in terms of symmetric matrices.

In this paper, we introduce cosine Thurston maps. In particular, we construct postsingularly finite topological cosine maps and focus on such maps with strictly preperiodic critical points. We use the techniques of Hubbard, Schleicher, and Shishikura to prove that, subject to a condition on the critical points, a postsingularly finite topological cosine map with strictly preperiodic critical points is combinatorially equivalent to $C_\lambda(z) = \lambda \cos z$ for a unique $\lambda \in \mathbb{C}^*$ if only if it has no degenerate Levy cycle.

Let $\mu$ be a nondegenerate probability measure with finite entropy on a countable group $G \leq \mathrm{Homeo}_+(S^1)$ of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on $S^1$. We prove that the circle $S^1$ endowed with its unique $\mu$-stationary probability measure is not the Poisson boundary of $(G,\mu)$. When $G$ is Thompson's group $T$ and $\mu$ is finitely supported, this answers a question posed by B. Deroin [Ergodic Theory Dynam. Systems, 2013] and A. Navas [Proceedings of the International Congress of Mathematicians, 2018].

This paper investigates the global existence of classical solutions to the isentropic compressible Navier-Stokes equations with slip boundary condition in a three-dimensional (3D) exterior domain. It is shown that the classical solutions with large initial energy and vacuum exist globally in time when the adiabatic exponent $\gamma>1$ is sufficiently close to 1 (near-isothermal regime). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183-200). To the best of our knowledge, we establish the first result on the global existence of large-energy solutions with vacuum to the compressible Navier-Stokes equations with slip boundary condition in a 3D exterior domain, which improves previous related works where either small initial energy is required or boundary effects are ignored.

Learning the governing equations from time-series data has gained increasing attention due to its potential to extract useful dynamics from real-world data. Despite significant progress, it becomes challenging in the presence of noise, especially when derivatives need to be calculated. To reduce the effect of noise, we propose a method that simultaneously fits both the derivative and trajectory from noisy time-series data. Our approach formulates derivative estimation as an inverse problem involving integral operators within the forward model, and estimates the derivative function by solving a regularization problem in a vector-valued reproducing kernel Hilbert space (vRKHS). We derive an integral-form representer theorem, which enables the computation of the regularized solution by solving a finite-dimensional problem and facilitates efficiently estimating the optimal regularization parameter. By embedding the dynamics within a vRKHS and utilizing the fitted derivative and trajectory, we can recover the underlying dynamics from noisy data by solving a linear regularization problem. Several numerical experiments are conducted to validate the effectiveness and efficiency of our method.

Let $ A(G) $ be the adjacency matrix of a simple graph $ G $, and let $\chi(G)$ denote its chromatic number. For $ p > 0 $, we define the positive and negative $ p $-energies of $ G $ as $$ \mathcal{E}_p^+(G) = \sum_{\lambda_i > 0} \lambda_i^p(A(G)), \quad \mathcal{E}_p^-(G) = \sum_{\lambda_i < 0} |\lambda_i(A(G))|^p, $$ where $ \lambda_1(A(G)) \geq \cdots \geq \lambda_n(A(G)) $ are the eigenvalues of $ A(G) $. We first prove that $$ \chi(G) \geq 1 + \max \left\{ \frac{\mathcal{E}_p^+(G)}{\mathcal{E}_p^-(G)}, \frac{\mathcal{E}_p^-(G)}{\mathcal{E}_p^+(G)} \right\} $$ holds for all $ 0 < p < 1 $. This result has already been established for $ p = 0 $ and $ p = 2 $, and it holds trivially for $ p = 1 $. Furthermore, we demonstrate that for certain graphs, non-integer values of $p$ yield sharper lower bounds on $\chi(G)$ than existing spectral bounds. Finally, we conjecture that the same inequality continues to hold for all $ 1 < p < 2 $.

Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, and the nonlinear Schr\"{o}dinger equations, the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the definition of the discrete variational derivative, based on which, a novel semi-analytical multiple invariants-preserving integrator for the conservative partial differential equations is constructed by projection technique. The proposed integrators are shown to have the same order of accuracy as the underlying integrators. For applications, some concrete mass-momentum-energy-preserving integrators are derived for the KdV equation.

We consider the asymptotic behavior of the multidimensional Laplace's integral. On the asymptotic analysis for this integral, the so-called Laplace's method or Laplace's approximation introduced by P.S. Laplace (1812) is well known. It has been developed in various forms over many years of study. In this paper, we derive a new formula for the higher-order asymptotic expansion of the multidimensional Laplace's integral, with an error estimate, which generalizes previous results.

In this paper, we prove exponential tail bounds for canonical (or degenerate) $U$-statistics and $U$-processes under exponential-type tail assumptions on the kernels. Most of the existing results in the relevant literature often assume bounded kernels or obtain sub-optimal tail behavior under unbounded kernels. We obtain sharp rates and optimal tail behavior under sub-Weibull kernel functions. Some examples from nonparametric and semiparametric statistics literature are considered.

Recently the author presented a new approach to solving the coefficient problems for various classes of holomorphic functions $f(z) = \sum\limits_0^\infty c_n z^n$, not necessarily univalent. This approach is based on lifting the given polynomial coefficient functionals $J(f) = J(c_{m_1}, \dots, c_{m_s}), 2 < c_{m_1} < \dots < c_{m_s} < \infty$, onto the Bers fiber space over universal Teichmuller space and applying the analytic and geometric features of Teichm\"{u}ller spaces, especially the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. In this paper, we extend this approach to more general classes of functions. In particular, this provides a strengthening of de Branges' theorem solving the Bieberbach conjecture.

This paper focuses on the Hopf bifurcation in an activator-inhibitor system without diffusion which can be modeled as a delay differential equation. The main result of this paper is the existence of the Poincar\'e-Lindstedt series to all orders for the bifurcating periodic solutions. The model has a non-linearity which is non-polynomial, and yet this allows us to exploit the use of Fourier-Taylor series to develop order-by-order calculations that lead to linear recurrence equations for the coefficients of the Poincar\'e-Lindstedt series. As applications, we implement the computation of the coefficients of these series for any finite order, and use a pseudo-arclength continuation to compute branches of periodic solutions.

The truncated Euler--Maruyama (EM) method, developed by Mao (2015), is used to solve multi-dimensional nonlinear stochastic differential equations (SDEs). However, its convergence rate is suboptimal due to an unnecessary infinitesimal factor. The primary goal of this paper is to demonstrate the optimal convergence of the truncated EM method without infinitesimal factors. Besides, the logarithmic truncated EM method has not been studied in multi-dimensional cases, which is the other goal of this paper. We will show the optimal strong convergence order of the positivity-preserving logarithmic truncated EM method for solving multi-dimensional SDEs with positive solutions. Numerical examples are given to support our theoretical conclusions.

In flow control, finite energy may be injected to push out material trapped in the attractor and to eliminate stagnation and circulate the flow. To describe such phenomena and to give a lower bound on the energy required, we generalize the existing concepts of chain recurrence. In fact, this paper introduces concepts of ``coarse chain recurrences'' and Morse graphs with finite errors. Using these concepts, we describe toy models of escape from attracting basins and elimination of stagnation by controls using finite energy, persistence of recurrent points, and singular limit behaviors where energy injections go to zero. Furthermore, we construct filtrations associated with dynamical systems, which indicate the persistence of circulations.

Structural stability of piecewise M\"obius transformations (PMTs) is investigated from several angles. A result about structural stability restricted to the space of PMTs is obtained using hyperbolic features for the component functions and the pre-singularities set, allowing a holomorphic motion. Is it defined and analyzed for PMTs the analogous concept of J-stabilty for rational maps, finding some relations with the general structural stability. The notions of hyperbolic and expansive PMTs are defined, showing that they are not equivalent and none of them implies structural stability. Combining the previous results and analyzes, sufficient conditions are given for structural stability. Finally, an example of structural stability in the complexified tent maps family is shown.

For a locally free sheaf $\mathcal{E}$ on a smooth projective curve, we can define the punctual Quot scheme which parametrizes torsion quotients of $\mathcal{E}$ of length $n$ supported at a fixed point. It is known that the punctual Quot scheme is a normal projective variety with canonical Gorenstein singularities. In this note, we show that the punctual Quot scheme is a $\mathbb{Q}$-factorial Fano variety of Picard number one.

We concern the long-time behavior of mild solutions to the spatially homogeneous Boltzmann--Fermi--Dirac equation with moderately soft potential. Based on the well-posedness results in [X-G. Lu, J. Stat. Phys., 105, (2001), 353-388], we prove that the mild solution decays algebraically to the Fermi--Dirac statistics with an explicit rate. Under the framework of the level set analysis by De Giorgi, we derive an $L^\infty$ estimate which is uniform with respect to the quantum parameter $\varepsilon$. All quantitative estimates are independent of $\varepsilon$, which implies that they also hold in the classical limit, i.e., the Boltzmann equation.

The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and Clifford analysis. This paper explores the fundamental properties of these monogenic functions, focusing on the Cauchy-Pompeiu integral formula and Taylor series expansion in hypercomplex subspaces, among which the non-commutativity and especially non-associativity of multiplications demand full considerations. The theory presented herein provides a robust framework for understanding monogenic functions in the context of real alternative $\ast$-algebras, shedding light on the interplay between algebraic structures and hypercomplex analysis.

This paper addresses the reconstruction of a potential coefficient in an elliptic problem from distributed observations within the Bayesian framework. In such problems, the selection of an appropriate prior distribution is crucial, particularly when the function to be inferred exhibits sharp discontinuities, as traditional Gaussian priors often prove inadequate. To tackle this challenge, we develop the topological prior (TP), a new prior constructed using persistent homology. The proposed prior utilizes persistent pairs to characterize and record the topological variations of the functions under reconstruction, thereby encoding prior information about the structure and discontinuities of the function. The TP prior, however, only exists in a discretized formulation, which leads to the absence of a well-defined posterior measure in function spaces. To resolve this issue, we propose a TP-Gaussian hybrid prior, where the TP component detects sharp discontinuities in the function, while the Gaussian distribution acts as a reference measure, ensuring a well-defined posterior measure in the function space. The proposed TP prior demonstrates effects similar to the classical total variation (TV) prior but offers greater flexibility and broader applicability due to three key advantages. First, it is defined on a general topological space, making it easily adaptable to a wider range of applications. Second, the persistent distance captures richer topological information compared to the discrete TV prior. Third, it incorporates more adjustable parameters, providing enhanced flexibility to achieve robust numerical results. These features make the TP prior a powerful tool for addressing inverse problems involving functions with sharp discontinuities.

In this article, we describe the theoretical foundations of the Macaulay2 package ConnectionMatrices and explain how to use it. For a left ideal in the Weyl algebra that is of finite holonomic rank, we implement the computation of the encoded system of linear PDEs in connection form with respect to an elimination term order that depends on a chosen positive weight vector. We also implement the gauge transformation for carrying out a change of basis over the field of rational functions. We demonstrate all implemented algorithms with examples.

The isomorphism problem of regular Higman-Thompson groups was solved in arXiv:1006.1759, via embedding it into the Leavitt algebra. In this paper, we will expand these results to embed the Higman-Thompson groups of unfolding trees of directed graphs into the Leavitt path algebra. This embedding allows us to show that any isomorphism of rooted Leavitt path algebras induces an isomorphism between Higman-Thompson groups.

Erd\H{o}s proved an upper bound on the number of edges in an $n$-vertex non-Hamiltonian graph with given minimum degree and showed sharpness via two members of a particular graph family. F\"{u}redi, Kostochka and Luo showed that these two graphs play the same role when ``number of edges'' is replaced by ``number of t-stars,'' and that two members of a more general graph family maximize the number of edges among non-$k$-edge-Hamiltonian graphs. In this paper we generalize their former result from Hamiltonicity to related properties (traceability, Hamiltonian-connectedness, $k$-edge Hamiltonicity, $k$-Hamiltonicity) and their latter result from edges to $t$-stars. We identify a family of extremal graphs for each property that is forbidden. This problem without the minimum degree condition was also open; here we conjecture a complete description of the extremal family for each property, and prove the characterization in some cases. Finally, using a different family of extremal graphs, we find the maximum number of $t$-stars in non-$k$-connected graphs.

The integrated sensing and communication (ISAC) technology has been extensively researched to enhance communication rates and radar sensing capabilities. Additionally, a new technology known as fluid antenna system (FAS) has recently been proposed to obtain higher communication rates for future wireless networks by dynamically altering the antenna position to obtain a more favorable channel condition. The application of the FAS technology in ISAC scenarios holds significant research potential. In this paper, we investigate a FAS-assisted multiple-input multiple-output (MIMO) ISAC system for maximizing the radar sensing signal-clutter-noise ratio (SCNR) under communication signal-to-interference-plus-noise ratio (SINR) and antenna position constraints. We devise an iterative algorithm that tackles the optimization problem by maximizing a lower bound of SCNR with respect to the transmit precoding matrix and the antenna position. By addressing the non-convexity of the problem through this iterative approach, our method significantly improves the SCNR. Our simulation results demonstrate that the proposed scheme achieves a higher SCNR compared to the baselines.

This paper presents a novel approach to solve capacitated facility location problems (FLP) that encompass various resource allocation problems. FLPs are a class of NP-hard combinatorial optimization problems, involving optimal placement and assignment of a small number of facilities over a large number of demand points, with each facility subject to upper and lower bounds on its resource utilization (e.g., the number of demand points it can serve). To address the challenges posed by inequality constraints and the combinatorial nature of the solution space, we reformulate the problem as a dynamic control design problem, enabling structured constraint handling and enhanced solution efficiency. Our method integrates a Control Barrier Function (CBF) and Control Lyapunov Function (CLF)-based framework with a maximum-entropy principle-based framework to ensure feasibility, optimality, and improved exploration of solutions. Numerical experiments demonstrate that this approach significantly enhances computational efficiency, yielding better solutions and showing negligible growth in computation time with problem size as compared to existing solvers. These results highlight the potential of control-theoretic and entropy-based methods for large-scale facility location problems.

We give formulas for computing efficiently the generalized Kloosterman sums appearing in the Hardy-Ramanujan-Rademacher expansions of the Fourier coefficients of general eta-quotients given by Sussman and Chern, as well as explicit bounds for the tails of these series.

We describe the derived McKay correspondence for real reflection groups of rank $3$ in terms of a maximal resolution of the logarithmic pair consisting of the quotient variety and the discriminant divisor with coefficient $\frac{1}{2}$. As an application, we verify a conjecture by Polishchuk and Van den Bergh on the existence of a certain semiorthgonal decomposition of the equivariant derived category into the derived categories of affine spaces for any real reflection group of rank $3$.

We examine cyclic, non-well-founded and well-founded derivations in the provability logic $\mathsf{GLP}$. While allowing cyclic derivations does not change the system, the non-well-founded and well-founded derivations we consider define the same proper infinitary extension of $\mathsf{GLP}$. We establish that this extension is strongly algebraic and neighbourhood complete with respect to both local and global semantic consequence relations. In fact, these completeness results are proved for generalizations of global and local consequence relations, which we call global-local. In addition, we prove strong local neighbourhood completeness for the original system $\mathsf{GLP}$ (with ordinary derivations only).

In this work, we derive a new expression for the diagonal matrix elements of irreducible representations of the direct product group $S_r\times S_s$ using Grime's hook fusion procedure for symmetric groups, which simplifies the fusion procedure by reducing the number of auxiliary parameters needed. By extending this approach to the product group setting, we provide a method for constructing a complete set of orthogonal primitive idempotents.

In this paper, we present sufficient conditions for asymptotic stability and exponential stability of a class of impulsive neutral differential equations with discrete and distributed delays. Our approaches are based on the method using fixed point theory, which do not resort to any Lyapunov functions or Lyapunov functionals. Our conditions do not require the differentiability of delays, nor do they ask for a fixed sign on the coefficient functions. Our results improve some previous ones in the literature. Examples are given to illustrate our main results.

Discrete latent factor models (DLFMs) are widely used in various domains such as machine learning, economics, neuroscience, psychology, etc. Currently, fitting a DLFM to some dataset relies on a customized solver for individual models, which requires lots of effort to implement and is limited to the targeted specific instance of DLFMs. In this paper, we propose a generic framework based on CVXPY, which allows users to specify and solve the fitting problem of a wide range of DLFMs, including both regression and classification models, within a very short script. Our framework is flexible and inherently supports the integration of regularization terms and constraints on the DLFM parameters and latent factors, such that the users can easily prototype the DLFM structure according to their dataset and application scenario. We introduce our open-source Python implementation and illustrate the framework in several examples.

In this expository note, recent results of Kishimoto and Matsushita on triangulated manifolds are linked to the classical criterion on the normal Stiefel-Whitney classes for existence of an embedding of a smooth closed manifold into Euclidean space of given dimension. We also look back at Atiyah's K-theoretic condition for the existence of a smooth embedding.

In this paper, we consider maps from pseudo-Hermitian manifolds to K\"{a}hler manifolds and introduce partial energy functionals for these maps. First, we obtain a foliated Lichnerowicz type result on general pseudo-Hermitian manifolds, which generalizes a related result on Sasakian manifolds in \cite{SSZ2013holomorphic}. Next, we investigate critical maps of the partial energy functionals, which are referred to as $\overline{\partial }_{b}$-harmonic maps and $\partial _{b}$-harmonic maps. We give a foliated result for both $\overline{\partial }_{b}$- and $\partial _{b}$-harmonic maps, generalizing a foliated result of Petit \cite{Pet2002harmonic} for harmonic maps. Then we are able to generalize Siu's holomorphicity result for harmonic maps \cite{Siu1980rigid} to the case for $\overline{\partial }_{b}$- and $\partial _{b}$-harmonic maps.

We present a probabilistic proof of the mean-field limit and propagation of chaos of a classical N-particle system in three dimensions with Coulomb interaction force of the form $f^N(q)=\pm\frac{q}{|q|^3}$ and $N$-dependent cut-off at $|q|>N^{-\frac{5}{12}+\sigma}$ where $\sigma>0$ can be chosen arbitrarily small. This cut-off size is much smaller than the typical distance to the nearest neighbour. In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Poisson system. The proof is based on a Gronwall estimate for the maximal distance between the exact microscopic dynamics and the approximate mean-field dynamics. Thus our result leads to a derivation of the Vlasov-Poisson equation from the microscopic $N$-particle dynamics with force term arbitrary close to the physically relevant Coulomb force.

Convex Hull (CH) pricing, used in US electricity markets and raising interest in Europe, is a pricing rule designed to handle markets with non-convexities such as startup costs and minimum up and down times. In such markets, the market operator makes side payments to generators to cover lost opportunity costs, and CH prices minimize the total "lost opportunity costs", which include both actual losses and missed profit opportunities. These prices can also be obtained by solving a (partial) Lagrangian dual of the original mixed-integer program, where power balance constraints are dualized. Computing CH prices then amounts to minimizing a sum of nonsmooth convex objective functions, where each term depends only on a single generator. The subgradient of each of those terms can be obtained independently by solving smaller mixed-integer programs. In this work, we benchmark a large panel of first-order methods to solve the above dual CH pricing problem. We test several dual methods, most of which not previously considered for CH pricing, namely a proximal variant of the bundle level method, subgradient methods with three different stepsize strategies, two recent parameter-free methods and an accelerated gradient method combined with smoothing. We compare those methods on two representative sets of real-world large-scale instances and complement the comparison with a (Dantzig-Wolfe) primal column generation method shown to be efficient at computing CH prices, for reference. Our numerical experiments show that the bundle proximal level method and two variants of the subgradient method perform the best among all dual methods and compare favorably with the Dantzig-Wolfe primal method.

In this paper, we design a controller for an interconnected system where a linear Stochastic Differential Equation (SDE) is actuated through a linear parabolic heat equation. These dynamics arise in various applications, such as coupled heat transfer systems and chemical reaction processes that are subject to disturbances. Our goal is to develop a computational method for approximating the controller that minimizes a quadratic cost associated with the state of the SDE component. To achieve this, we first perform a change of variables to shift the actuation inside the PDE domain and reformulate the system as a linear Stochastic Partial Differential Equation (SPDE). We use a spectral approximation of the Laplacian operator to discretize the coupled dynamics into a finite-dimensional SDE and compute the optimal control for this approximated system. The resulting control serves as an approximation of the optimal control for the original system. We then establish the convergence of the approximated optimal control and the corresponding closed-loop dynamics to their infinite-dimensional counterparts. Numerical simulations are provided to illustrate the effectiveness of our approach.

In this paper, we develop a general mathematical framework for analyzing electostatics within multi-layer metamaterial structures. The multi-layer structure can be designed by nesting complementary negative and regular materials together, and it can be easily achieved by truncating bulk metallic material in a specific configuration. Using layer potentials and symmetrization techniques, we establish the perturbation formula in terms of Neumann-Poincar\'e (NP) operator for general multi-layered medium, and obtain the spectral properties of the NP operator, which demonstrates that the number of plasmon modes increases with the number of layers. Based on Fourier series, we present an exact matrix representation of the NP operator in an apparently unsymmetrical structure, exemplified by multi-layer confocal ellipses. By highly intricate and delicate analysis, we establish a handy algebraic framework for studying the splitting of the plasmon modes within multi-layer structures. Moreover, the asymptotic profiles of the plasmon modes are also obtained. This framework helps reveal the effects of material truncation and rotational symmetry breaking on the splitting of the plasmon modes, thereby inducing desired resonances and enabling the realization of customized applications.

In this paper we carry out a computational study of a novel microscopic follow-the-leader model for traffic flow on road networks. We assume that each driver has its own origin and destination, and wants to complete its journey in minimal time. We also assume that each driver is able to take rational decisions at junctions and can change route while moving depending on the traffic conditions. The main novelty of the model is that vehicles can automatically and anonymously share information about their position, destination, and planned path when they are close to each other within a certain distance. The pieces of information acquired during the journey are used to optimize the route itself. In the limit case of a infinite communication range, we recover the classical Reactive User Equilibrium and Dynamic User Equilibrium.

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure, according to certain weights assigned to each cycle length. The proof is based on uniform control of the characteristic polynomial using results from the singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which has already appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian Holomorphic Chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.

We propose and study a fully implicit finite volume scheme for the pressureless Euler-Poisson-Boltzmann equations on the one dimensional torus. Especially, we design a consistent and dissipative discretization of the force term which yields an unconditional energy decay. In addition, we establish a discrete analogue of the modulated energy estimate around constant states with a small velocity. Numerical experiments are carried to illustrate our theoretical results and to assess the accuracy of our scheme. A test case of the literature is also illustrated.

The digital transformation of our society is a constant challenge, as data is generated in almost every digital interaction. To use data effectively, it must be of high quality. This raises the question: what exactly is data quality? A systematic literature review of the existing literature shows that data quality is a multifaceted concept, characterized by a number of quality dimensions. However, the definitions of data quality vary widely. We used feature-oriented domain analysis to specify a taxonomy of data quality definitions and to classify the existing definitions. This allows us to identify research gaps and future topics.

We establish a shape-derivative formula for the Dirichlet-to-Neumann operator on a compact manifold. Then, we apply this formula to obtain the well-posedness in H 1 under a specific Rayleigh-Taylor condition to an equation describing cell protrusions. This equation is a generalisation of the theoretical part of [9] to any -2D as well as 3D-surfaces.

We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of $\mathrm{SL}_n(\mathbb{Z})$ that are not finitely presented for all $n \geq 6$.

Recently, the high-resolution ordinary differential equation (ODE) framework, which retains higher-order terms, has been proposed to analyze gradient-based optimization algorithms. Through this framework, the term $\nabla^2 f(X_t)\dot{X_t}$, known as the gradient-correction term, was found to be essential for reducing oscillations and accelerating the convergence rate of function values. Despite the importance of this term, simply adding it to the low-resolution ODE may sometimes lead to a slower convergence rate. To fully understand this phenomenon, we propose a generalized perturbed ODE and analyze the role of the gradient and gradient-correction perturbation terms under both continuous-time and discrete-time settings. We demonstrate that while the gradient-correction perturbation is essential for obtaining accelerations, it can hinder the convergence rate of function values in certain cases. However, this adverse effect can be mitigated by involving an additional gradient perturbation term. Moreover, by conducting a comprehensive analysis, we derive proper choices of perturbation parameters. Numerical experiments are also provided to validate our theoretical findings.

In this paper, we consider critical points of the horizontal energy $E_{\HH}(f)$ for a smooth map $f$ between two Riemannian foliations. These critical points are referred to as horizontally harmonic maps. In particular, if the maps are foliated, they become transversally harmonic maps. By utilizing the stress-energy tensor, we establish some monotonicity formulas for horizontally harmonic maps from Euclidean spaces, the quotients $K_{m}$ of Heisenberg groups and also for transversally harmonic maps from Riemannian foliations with appropriate curvature pinching conditions. Finally, we give Jin-type theorems for either horizontally harmonic maps or transversally harmonic maps under some asymptotic conditions at infinity.

Let $X$ be a smooth projective curve over a field $k$ with an action of a finite group $G$. A well-known result of Chevalley and Weil describes the $k[G]$-module structure of cohomologies of $X$ in the case when the characteristic of $k$ does not divide $\# G$. It is unlikely that such a formula can be derived in the general case, since the representation theory of groups with non-cyclic $p$-Sylow subgroups is wild in characteristic $p$. The goal of this article is to show that when $G$ has a cyclic $p$-Sylow subgroup, the $G$-structure of the de Rham cohomology of $X$ is completely determined by the ramification data. In principle, this leads to new formulas in the spirit of Chevalley and Weil for such curves. We provide such an explicit description of the de Rham cohomology in the cases when $G = \mathbb Z/p^n$ and when the $p$-Sylow subgroup of $G$ is normal of order $p$.

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large matrix, with as few matrix-matrix multiplications as possible. More precisely, let $ \Pi_{2^{m}}^* $ represent the set of polynomials computable with $m$ matrix-matrix multiplications, but with an arbitrary number of matrix additions and scaling operations. We characterize this set through a tabular parameterization. By deriving equivalence transformations of the tabular representation, we establish new methods that can be used to construct elements of $ \Pi_{2^{m}}^* $ and determine general properties of the set. The transformations allow us to eliminate variables and prove that the dimension is bounded by $m^2$. Numerical simulations suggest that this is a sharp bound. Consequently, we have identified a parameterization, which, to our knowledge, is the first minimal parameterization. Furthermore, we conduct a study using computational tools from algebraic geometry to determine the largest degree $d$ such that all polynomials of that degree belong to $ \Pi_{2^{m}}^* $, or its closure. In many cases, the computational setup is constructive in the sense that it can also be used to determine a specific evaluation scheme for a given polynomial.

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erd\H{o}s-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.

In this paper, we sharpen results obtained by the author in 2023. The new results reduce the Mathieu Conjecture on $SU(N)$ (formulated for all compact connected Lie groups by O. Mathieu in 1997) to a conjecture involving only functions on $\mathbb{R}^n\times (S^1)^m$ with $n,m$ non-negative integers instead of involving functions on $\mathbb{R}^n\times (S^1\setminus\{1\})^m$. The proofs rely on a more recent work of the author (2024) and a specific $KAK$ decomposition. Finally, with these results we can also improve the results on the groups $Sp(N)$ and $G_2$ in the latter paper, since they relied on the construction introduced in the 2023 paper.

In this work, we investigate the arithmetic properties of $p_{1,7^k}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are multiples of $7^k$. By constructing generating functions for $p_{1,7^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type infinite family of congruences modulo powers of $7$.

In fully-implicit two-phase pore-network models, developing a well-converged scheme remains a major challenge, primarily due to the discontinuities in the phase conductivities. This paper addresses these numerical issues by proposing a generalized flux function that establishes a continuous flux expression for two-phase flows by introducing an additional throat variable $\Theta$. Two approaches for expressing this additional throat variable are introduced: the first applies regularization strategies, while the second constructs an additional residual constraint equation. It is shown that this approach significantly improves accuracy and ensures the temporal convergence, as demonstrated through various numerical examples.

Empirical likelihood serves as a powerful tool for constructing confidence intervals in nonparametric regression and regression discontinuity designs (RDD). The original empirical likelihood framework can be naturally extended to these settings using local linear smoothers, with Wilks' theorem holding only when an undersmoothed bandwidth is selected. However, the generalization of bias-corrected versions of empirical likelihood under more realistic conditions is non-trivial and has remained an open challenge in the literature. This paper provides a satisfactory solution by proposing a novel approach, referred to as robust empirical likelihood, designed for nonparametric regression and RDD. The core idea is to construct robust weights which simultaneously achieve bias correction and account for the additional variability introduced by the estimated bias, thereby enabling valid confidence interval construction without extra estimation steps involved. We demonstrate that the Wilks' phenomenon still holds under weaker conditions in nonparametric regression, sharp and fuzzy RDD settings. Extensive simulation studies confirm the effectiveness of our proposed approach, showing superior performance over existing methods in terms of coverage probabilities and interval lengths. Moreover, the proposed procedure exhibits robustness to bandwidth selection, making it a flexible and reliable tool for empirical analyses. The practical usefulness is further illustrated through applications to two real datasets.

This paper investigates the dynamical behavior of periodic solutions for a class of second-order non-autonomous differential equations. First, based on the Lyapunov-Schmidt reduction method for finite-dimensional functions, the corresponding bifurcation function is constructed, and it is proven that the system possesses at least one T-periodic solution. Second, a two-timing method is employed to perform perturbation analysis on the original equation. By separating the fast and slow time scales, an explicit expression for the approximate T-periodic solution is derived. Furthermore, for the stability of the system under parametric excitation, the bifurcation characteristics near the first instability tongue are revealed through perturbation expansion and eigenvalue analysis. Additionally, the Ince-Strutt stability diagram is plotted to illustrate the stability boundaries.

Many ecological population models consider taxis as the directed movement of animals in response to a stimulus. The taxis is named direct if the animals are guided by the density gradient of some other population or indirect if they are guided by the density of a chemical secreted by individuals of the other population. Let $u$ and $v$ denote the densities of two populations and $w$ the density of the chemical secreted by individuals in the $v$ population. We consider a bounded, open set $\Omega \subset \mathbb{R}^N$ with regular boundary and prove that for the space dimension $N\leq 2$ the solution to the Lotka-Volterra competition model with repulsive indirect taxis and homogeneous Neumann boundary conditions $$u_t - d_u\Delta u = \chi \nabla \cdot u \nabla w +\mu_1u(1-u-a_1v)\,,$$ $$ v_t - d_v\Delta v = \mu_2v(1-v-a_2u)\,,$$ $$\varepsilon ( w_t - d_w\Delta w )= v- w\, , $$ converges to the solution of repulsive direct-taxis model: $$ u_t - d_u\Delta u = \chi \nabla \cdot u \nabla v +\mu_1u(1-u-a_1v)\,,$$ $$ v_t - d_v\Delta v = \mu_2v(1-v-a_2u)\,$$ when $\varepsilon\longrightarrow 0$. For space dimension $N\geq 3$ we use the compactness argument to show that the result holds in some weak sense. A similar result is also proved for a typical prey-predator model with prey taxis and logistic growth of predators.

Given a graph $G$ and an integer $d\ge 0$, its $d$-defective chromatic number $\chi^d(G)$ is the smallest size of a partition of the vertices into parts inducing subgraphs with maximum degree at most $d$. Guo, Kang and Zwaneveld recently studied the relationship between the $d$-defective chromatic number of the $(d+1)$-fold (clique) blowup $G\boxtimes K_{d+1}$ of a graph $G$ and its ordinary chromatic number, and conjectured that $\chi(G)=\chi^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. In this note we disprove this conjecture by constructing graphs $G$ of arbitrarily large chromatic number such that $\chi(G)\ge \frac{30}{29}\chi^d(G\boxtimes K_{d+1})$ for infinitely many $d$. On the positive side, we show that the conjecture holds with a constant factor correction, namely $\chi^d(G\boxtimes K_{d+1})\le \chi(G)\le 2\chi^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$.

We consider the free boundary problem for a liquid drop of nearly spherical shape with capillarity, and we study the existence of nontrivial (i.e., non spherical) rotating traveling profiles bifurcating from the spherical shape, where the bifurcation parameter is the angular velocity. We prove that every eigenvalue of the linearized problem is a bifurcation point, extending the known result for simple eigenvalues to the general case of eigenvalues of any multiplicity. We also obtain a lower bound on the number of bifurcating solutions. The proof is based on the Hamiltonian structure of the problem and on the variational argument of constrained critical points for traveling waves of Craig and Nicholls (2000, SIAM J. Math. Anal. 32, 323-359), adapted to the nearly spherical geometry; in particular, the role of the action functional is played here by the angular momentum with respect to the rotation axis. Moreover, the bifurcation equation presents a 2-dimensional degeneration, related to some symmetries of the physical problem. This additional difficulty is overcome thanks to a crucial transversality property, obtained by using the Hamiltonian structure and the prime integrals corresponding to those symmetries by Noether theorem, which are the fluid mass and the component along the rotation axis of the velocity of the fluid barycenter.

The goal of this paper is to propose a new approach to asymptotic analysis of the finite predictor for stationary sequences. It produces the exact asymptotics of the relative prediction error and the partial correlation coefficients. The assumptions are analytic in nature and applicable to processes with long range dependence. The ARIMA type process driven by the fractional Gaussian noise (fGn), which previously remained elusive, serves as our study case.

We use height arguments to prove two results about the dynamical Mordell-Lang problem. We are more interested in the positive characteristic case due to our original purpose. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov exponent of any iteration is not an integer. (ii) Let $f\times g:X\times C\rightarrow X\times C$ be an endomorphism in which $f$ and $g$ are endomorphisms of a projective variety $X$ and a curve $C$, respectively. If the degree of $g$ is greater than the first dynamical degree of $f$, then the return sets of the system $(X\times C,f\times g)$ have the same form as the return sets of the system $(X,f)$. Using the second result, we deal with the case of split endomorphisms of products of curves, for which the degrees of the factors are pairwise distinct. In the cases that the height argument cannot be applied, we find examples which show that the return set can be very complicated -- more complicated than experts once imagine -- even for endomorphisms of tori of zero entropy.

Shape optimization is commonly applied in engineering to optimize shapes with respect to an objective functional relying on PDE solutions. In this paper, we view shape optimization as optimization on Riemannian shape manifolds. We consider so-called outer metrics on the diffeomorphism group to solve PDE-constrained shape optimization problems efficiently. Commonly, the numerical solution of such problems relies on the Riemannian version of the steepest descent method. One key difference between this version and the standard method is that iterates are updated via geodesics or retractions. Due to the lack of explicit expressions for geodesics, for most of the previously proposed metrics, very limited progress has been made in this direction. Leveraging the existence of explicit expressions for the geodesic equations associated to the outer metrics on the diffeomorphism group, we aim to study the viability of using such equations in the context of PDE-constrained shape optimization. However, solving geodesic equations is computationally challenging and often restrictive. Therefore, this paper discusses potential numerical approaches to simplify the numerical burden of using geodesics, making the proposed method computationally competitive with previously established methods.

In 2022, Huh-Matherne-Meszaros-St. Dizier showed that normalized Schur polynomials are Lorentzian, thereby yielding their continuous (resp. discrete) log-concavity on the positive orthant (resp. on their support, in type-$A$ root directions). A reinterpretation of this result is that the characters of finite-dimensional simple representations of $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. In the same paper, these authors also showed that shifted characters of Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. In this work we extend these results to a larger family of modules that subsumes both of the above: we show that shifted characters of all parabolic Verma modules over $\mathfrak{sl}_{n+1}(\mathbb{C})$ are denormalized Lorentzian. The proof involves certain graphs on $[n+1]$; more strongly, we explain why the character (i.e., generating function) of the Kostant partition function of any loopless multigraph on $[n+1]$ is Lorentzian after shifting and normalizing. In contrast, we show that a larger universal family of highest weight modules, the higher order Verma modules, do not have discretely log-concave characters. Finally, we extend all of these results to parabolic (i.e. "first order") and higher order Verma modules over the semisimple Lie algebras $\oplus_{t=1}^T \mathfrak{sl}_{n_t+1}(\mathbb{C})$.

Copositive matrices and copositive polynomials are objects from optimization. We connect these to the geometry of Feynman integrals in physics. The integral is guaranteed to converge if its kinematic parameters lie in the copositive cone. P\'olya's method makes this manifest. We study the copositive cone for the second Symanzik polynomial of any Feynman graph. Its algebraic boundary is described by Landau discriminants.

We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient. More precisely, we essentially assume that the drift coefficient is piecewise Lipschitz continuous with an exceptional set $\Theta\subset \mathbb{R}^d$ that is an orientable $C^4$-hypersurface of positive reach, the diffusion coefficient is assumed to be Lipschitz continuous and, in a neighborhood of $\Theta$, both coefficients are bounded and the diffusion coefficient has a non-degenerate portion orthogonal to $\Theta$. In recent years, a number of results have been proven in the literature for strong approximation of such SDEs and, in particular, the performance of the Euler-Maruyama scheme was studied. For $d=1$ and finite $\Theta$ it was shown that the Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\geq 1$ as in the classical case of Lipschitz continuous coefficients. For $d>1$, it was only known so far, that the Euler-Maruyama scheme achieves an $L_2$-error rate of at least $1/4-$ if, additionally, the coefficients $\mu$ and $\sigma$ are globally bounded. In this article, we prove that in the above setting the Euler-Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2-$ for all $d\in\mathbb{N}$ and all $p\geq 1$. The proof of this result is based on the well-known approach of transforming such an SDE into an SDE with globally Lipschitz continuous coefficients, a new It\^{o} formula for a class of functions which are not globally $C^2$ and a detailed analysis of the expected total time that the actual position of the time-continuous Euler-Maruyama scheme and its position at the preceding time point on the underlying grid are on 'different sides' of the hypersurface $\Theta$.

We consider functional ellipsoids in the sense defined by Ivanov and Nasz\'odi and we study the problem of constructing a decomposition of the identity similar to the one given by Fritz John in his fundamental theorem.

Let $A$ be a commutative ring with identity. A proper submodule $N$ of $A$-module $M$ is said to be prime submodule if $ax \in N$ where $a \in A, x \in M$, implies $x \in N$ or $aM \subseteq N$. A proper submodule $N \subset M$ is said to be weakly prime submodule if $0 \neq ax \in N$ where $a \in A, x \in M$, then either $x \in N$ or $aM \subseteq N$. The notion of weakly prime submodule was introduced by Atani and Farzalipour \cite{atani2007weakly}. The purpose of this paper is to study the form of prime and weakly prime submodules of duplication of the $A$-module $M$ along the ideal $I$ (denoted by $M \bowtie I$), introduced and studied by E. M. Bouba, N. Mahdou and M. Tamekkante. A number of results concerning prime and weakly prime submodules on amalgamated duplication and examples are given.

In this paper, we study the appearance of a spanning subdivision of a clique in graphs satisfying certain pseudorandom conditions. Specifically, we show the following three results. Firstly, that there are constants $C>0$ and $c\in (0,1]$ such that, whenever $d/\lambda\ge C$, every $(n,d,\lambda)$-graph contains a spanning subdivision of $K_t$ for all $2\le t \le \min\{cd,c\sqrt{\frac{n}{\log n}}\}$. Secondly, that there are constants $C>0$ and $c\in (0,1]$ such that, whenever $d/\lambda\ge C\log^3n$, every $(n,d,\lambda)$-graph contains a spanning nearly-balanced subdivision of $K_t$ for all $2\le t \le \min\{cd,c\sqrt{\frac{n}{\log^3n}}\}$. Finally, we show that for every $\mu>0$, there are constants $c,\varepsilon\in (0,1]$ and $n_0\in \mathbb N$ such that, whenever $n\ge n_0$, every $n$-vertex graph with minimum degree at least $\mu n$ and no bipartite holes of size $\varepsilon n$ contains a spanning nearly-balanced subdivision of $K_t$ for all $2\le t \le c\sqrt{n}$.

We investigate the maximum number of limit cycles bifurcating from the period annulus of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic piecewise smooth polynomials. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. When the switching line is $x=0$ or $y=0$, we obtain the sharp bounds of the number of limit cycles for the perturbed systems by using the first order averaging method. Our results indicate that non-smooth systems can have more limit cycles than smooth ones, and the switching lines play an important role in the dynamics of non-smooth systems.

We analyze community recovery in the planted partition model (PPM) in regimes where the number of communities is arbitrarily large. We examine the three standard recovery regimes: exact recovery, almost exact recovery, and weak recovery. When communities vary in size, traditional accuracy- or alignment-based metrics become unsuitable for assessing the correctness of a predicted partition. To address this, we redefine these recovery regimes using the correlation coefficient, a more versatile metric for comparing partitions. We then demonstrate that \emph{Diamond Percolation}, an algorithm based on common-neighbors, successfully recovers communities under mild assumptions on edge probabilities, with minimal restrictions on the number and sizes of communities. As a key application, we consider the case where community sizes follow a power-law distribution, a characteristic frequently found in real-world networks. To the best of our knowledge, we provide the first recovery results for such unbalanced partitions.

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers $A$ and $B$ have natural density $1$, then their product set $A \cdot B := \{ab : a \in A, b \in B\}$ also has natural density $1$. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set $A$ of density 1 such that $A\cdot A$ has a ''large'' complement.

In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schr\"odinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schr\"odinger system, we present a complete classification of minimal-mass blowup solutions. Specifically, we demonstrate that all such solutions must be either solitons or their pseudo-conformal transformations. To prove this result, we develop a modulation analysis that accounts for multi-component interactions to overcome the multiply phase transformations caused by the multi-component. A long time Strichartz estimate for vector-valued solutions is established to solve the difficulty posed by the Galilean transformations and spatial translation, where a new vector-valued bilinear estimate is proven to address the challenges caused by the coupled nonlinear interaction. For the infinitely coupled focusing nonlinear Schr\"odinger system when the mass is slightly above the scattering threshold in \cite{CGHY}, we show that scattering is the only dynamical behavior of the solutions to the infinitely coupled system.

Reliability-based design optimization (RBDO) is a methodology for designing structures under the consideration for uncertainty with the assumption that the input distribution is completely known. In practical engineering, the number of input data is often limited, which can damage the validity of the optimal results obtained by RBDO. Confidence-based design optimization (CBDO) has been proposed to account for the uncertainty of the input distribution. However, this approach faces challenges, computational cost and accuracy when dealing with highly nonlinear performance constraints. In this paper, we consider the compliance minimization problem of truss structures with uncertain external forces. Armed with the advanced risk measure, conditional Value-at-Risk (CVaR), we formulate a bi-objective optimization problem for the worst-case expected value and the worst-case CVaR of compliance, which allows us to account for the tail risk of performance functions not addressed in CBDO. Employing kernel density estimation for estimation of the input distribution allows us to eliminate the need for modeling the input distribution. We show that this problem reduces to a second-order cone programming when assigning either uniform kernel or triangular kernel. Finally, through numerical experiments, we obtain the Pareto front for the bi-objective optimization problem of the worst-case expected value and CVaR of compliance of truss structures, and confirm the changes in the Pareto solutions.

Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $\psi_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $\psi_p(G)\leq\psi_p(C_{p^nm})$, where $C_{p^nm}$ is the cyclic group of order $p^nm$, and the equality holds if and only if $G$ is $p$-nilpotent of a particular type. A generalization of this result is also presented.

In the radiative Vlasov-Maxwell equations, the Lorentz force is modified by the addition of radiation reaction forces. The radiation forces produce damping of particle energy but the forces are no longer divergence-free in momentum space, which has an effect of concentration to zero momentum. We prove unconditional global regularity of solutions for a class of radiative Vlasov-Maxwell equations with large initial data.

The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $\Omega$, $u=f$ on $\partial\Omega$, from the Dirichlet to Neumann (DN) map $\Lambda_a$. We show that this can be achieved, when the coefficient $a$ has H\"older continuous first order derivatives and the exponents satisfy $1<p\neq q<\infty$. Our reconstruction method relies on a careful analysis of the asymptotic behavior of the solution $u$ to the double phase problem with small or large Dirichlet datum $f$ (depending on the ordering of $p$ and $q$) as well as the related DN map $\Lambda_a$. As is common for inverse boundary value problems, we need a sufficiently rich family of special solutions to a related partial differential equation, which is independent of the coefficient one aims to reconstruct (in our case to the $p$-Laplace equation). We construct such families of solutions by a suitable linearization technique.

An $SL_k$-tiling is a bi-infinite array of integers having all adjacent $k\times k$ minors equal to one and all adjacent $(k+1)\times (k+1)$ minors equal to zero. Introduced and studied by Bergeron and Reutenauer, $SL_k$-tilings generalize the notion of Conway-Coxeter frieze patterns in the case $k=2$. In a recent paper, Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and $SL_2$-tilings. We extend this result to higher $k$ by constructing a bijection between $SL_k$-tilings and certain pairs of bi-infinite strips of vectors in $\mathbb{Z}^k$ called paths. The key ingredient in the proof is the connection to Pl\"ucker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings.

Let $p\geq5$ be a prime number and let $K$ be an imaginary quadratic field where $p$ is unramified. Under mild technical assumptions, in this paper we prove the non-existence of non-trivial finite $\Lambda$-submodules of Pontryagin duals of signed Selmer groups of a $p$-supersingular rational elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$, where $\Lambda$ is the corresponding Iwasawa algebra. In particular, we work under the assumption that our plus/minus Selmer groups have $\Lambda$-corank $1$, so they are not $\Lambda$-cotorsion. Our main theorem extends to the supersinular case analogous non-existence results by Bertolini in the ordinary setting; furthermore, since we cover the case where $p$ is inert in $K$, we refine previous results of Hatley-Lei-Vigni, which deal with $p$-supersingular elliptic curves under the assumption that $p$ splits in $K$.

We develop high-order flux splitting schemes for the one- and two-dimensional Euler equations of gas dynamics. The proposed schemes are high-order extensions of the existing first-order flux splitting schemes introduced in [ E. F. Toro, M. E. V\'azquez-Cend\'on, Comput. \& Fluids, 70 (2012), pp. 1--12], where the Euler equations of gas dynamics are split into two subsystems: the advection and pressure systems. In this paper, we formulate the TV splitting within the semi-discrete framework to extend it to higher orders of accuracy for the first time. The second-order extension is obtained by using piecewise linear interpolant to reconstruct the one-sided point values of the unknowns. The third- and fifth-order schemes are developed using the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework, which is particularly effective in handling multidimensional problems and provides a more straightforward approach to constructing higher-order WENO schemes. These extensions significantly improve the resolution of discontinuities and the accuracy of numerical solutions, as demonstrated by a series of numerical experiments of both the one- and two-dimensional Euler equations of gas dynamics.

We consider the (universal) local isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, any pole order, and allowing for twisted/ramified formal normal forms at each pole. This covers the general case, and we particularly study the fundamental groups of the spaces of admissible deformations of irregular types/classes, in the viewpoint of (twisted/nonsplit) reflections cosets.

Poisson's equation is fundamental to the study of Markov chains, and arises in connection with martingale representations and central limit theorems for additive functionals, perturbation theory for stationary distributions, and average reward Markov decision process problems. In this paper, we develop a new probabilistic representation for the solution of Poisson's equation, and use Lyapunov functions to bound this solution representation explicitly. In contrast to most prior work on this problem, our bounds are computable. Our contribution is closely connected to recent work of Herve and Ledoux (2025), in which they focus their study on a special class of Harris chains satisfying a particular small set condition. However, our theory covers general Harris chains, and often provides a tighter bound. In addition to the new bound and representation, we also develop a computable uniform bound on marginal expectations for Harris chains, and a computable bound on the potential kernel representation of the solution to Poisson's equation.

Green and Schroll give an easy criterion for a monomial algebra $A$ to be quasi-hereditary with respect to some partial order $\leq_A$. A natural follow-up question is under which conditions a monomial quasi-hereditary algebra $(A, \leq_A)$ admits an exact Borel subalgebra in the sense of K\"onig. In this article, we show that it always admits a Reedy decomposition consisting of an exact Borel subalgebra $B$, which has a basis given by paths, and a dual subalgebra. Moreover, we give an explicit description of $B$ and show that it is the unique exact Borel subalgebra of $A$ with a basis given by paths. Additionally, we give a criterion for when $B$ is regular, using a criterion by Conde.

Given two quasi-hereditary algebras, their tensor product is quasi-hereditary. In this article, we show that given two exact Borel subalgebras for these quasi-hereditary algebras, their tensor product is an exact Borel subalgebra. Moreover, we describe in which cases the tensor product of two regular exact Borel subalgebras is again regular. Additionally, we investigate tensor algebras of generalised species of quasi-hereditary algebras and exact Borel subalgebras thereof.

Given a finite set $S$ of points in $\mathbb{R}^d$, which we regard as the locations of voters on a $d$-dimensional political `spectrum', two candidates (Alice and Bob) select one point in $\mathbb{R}^d$ each, in an attempt to get as many votes as possible. Alice goes first and Bob goes second, and then each voter simply votes for the candidate closer to them in terms of Euclidean distance. If a voter's distance from the two candidates is the same, they vote for nobody. We give a geometric characterization of the sets $S$ for which each candidate wins, assuming that Alice wins if they get an equal number of votes. We also show that, if not all the voters lie on a single line, then, whenever Alice has a winning strategy, there is a unique winning point for her. We also provide an algorithm which decides whether Alice has a winning point, and determines the location of that point, both in finite (in fact polynomial) time.

Jones introduced a method to produce unoriented links from elements of the Thompson's group $F$, and proved that any link can be produced by this construction. In this paper, we attempt to investigate the relations between conjugacy classes of the group $F$ and the links being constructed. For each unoriented link $L$, we find a sequence of elements of $F$ from distinct conjugacy classes which yield $L$ via Jones's construction. We also show that a sequence of $2$-bridge links can be constructed from elements in the conjugacy class of $x_0$ (resp. $x_1$).

In this paper, we want to study the asymptotic behavior of the first $p$-Laplacian eigenvalue, with Robin boundary conditions, with negative boundary parameter. In particular, we prove that the limit of the eigenfunctions is a viscosity solution for the infinity Laplacian eigenvalue problem.

MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85\%, which is much higher than previously reported results.

In this paper, we propose an adaptive proximal inexact gradient (APIG) framework for solving a class of nonsmooth composite optimization problems involving function and gradient errors. Unlike existing inexact proximal gradient methods, the proposed framework introduces a new line search condition that jointly adapts to function and gradient errors, enabling adaptive stepsize selection while maintaining theoretical guarantees. Specifically, we prove that the proposed framework achieves an $\epsilon$-stationary point within $\mathcal{O}(\epsilon^{-2})$ iterations for nonconvex objectives and an $\epsilon$-optimal solution within $\mathcal{O}(\epsilon^{-1})$ iterations for convex cases, matching the best-known complexity in this context. We then custom-apply the APIG framework to an important signal processing problem: the joint beamforming and compression problem (JBCP) with per-antenna power constraints (PAPCs) in cooperative cellular networks. This customized application requires careful exploitation of the problem's special structure such as the tightness of the semidefinite relaxation (SDR) and the differentiability of the dual. Numerical experiments demonstrate the superior performance of our custom-application over state-of-the-art benchmarks for the JBCP.

We present a novel approach for simulating acoustic (pressure) wave propagation across different media separated by a diffuse interface through the use of a weak compressibility formulation. Our method builds on our previous work on an entropy-stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes/Cahn-Hilliard system \cite{manzanero2020entropyNSCH}, and incorporates a modified weak compressibility formulation that allows different sound speeds in each phase. We validate our method through numerical experiments, demonstrating spectral convergence for acoustic transmission and reflection coefficients in one dimension and for the angle defined by Snell's law in two dimensions. Special attention is given to quantifying the modeling errors introduced by the width of the diffuse interface. Our results show that the method successfully captures the behavior of acoustic waves across interfaces, allowing exponential convergence in transmitted waves. The transmitted angles in two dimensions are accurately captured for air-water conditions, up to the critical angle of $13^\circ$. This work represents a step forward in modeling acoustic propagation in incompressible multiphase systems, with potential applications to marine aeroacoustics.

Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to their syndrome-measurement circuit's low operational depth. Numerous studies advocate for the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor while decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of the \emph{hypergraph-product} and \emph{lifted-product} codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets, which are subgraphs formed by stabilizers, and (2) classical trapping sets, which originate from the classical codes utilized in the construction of hypergraph-product and lifted-product codes. The dynamics of stabilizer-induced trapping sets is examined and a straightforward modification of iterative decoders is proposed to circumvent these trapping sets. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent classical trapping sets in both hypergraph product and lifted product codes, from decoders capable of avoiding their trapping set in the parent classical LDPC code. When decoders that can avoid stabilizer-induced trapping sets are run in parallel with those that can mitigate the effect of classical TS, the logical error rate improves significantly in the error-floor region.

We study the direct enstrophy cascade at small spatial scales in statistically stationary forced-dissipated 2D Navier-Stokes equations subject to the Coriolis force in the $\beta$-plane approximation. We provide sufficient conditions inspired by [6,63] to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function's asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. Our result indicates that at small scales, the enstrophy flux from larger to smaller scales is not affected by the Coriolis force, confirming experimental and numerical observations. To the best of our knowledge, this is the first mathematically rigorous study of the above equations.

An alternative method to invert the Radon transforms without the use of Courand-Hilbert's identities has been proposed and developed independently from the space dimension. For the universal representation of inverse Radon transform, we study the consequences of inhomogeneity of outset function without the restrictions on the angular Radon coordinates. We show that this inhomogeneity yields a natural evidence for the presence of the extra contributions in the case of the full angular region. In addition, if the outset function is well-localized in the space, we demonstrate that the corresponding boundary conditions and the angular restrictions should be applied for both the direct and inverse Radon transforms. Besides, we relate the angular restrictions on the Radon variable to the boundary exclusion of outset function and its Radon image.

A hereditarily atomic von Neumann algebra $A$ is a $W^*$ product of matrix algebras, regarded as the underlying function algebra of a quantum set. Projections in $A\overline{\otimes}A^{\circ}$ are interpreted as quantum binary relations on $A$, with the supremum of all $p\otimes (1-p)$ representing quantum inequality. We prove that the symmetrized weak$^*$-closed linear span of all such quantum-inequality projections is precisely the symmetric summand of the joint kernel of multiplication and opposite multiplication, a result valid without the symmetrization qualification for plain matrix algebras. The proof exploits the symmetries of the spaces involved under the compact unitary group of $A$, and related results include a classification of those von Neumann algebras (hereditarily atomic or not) for which the unitary group operates jointly continuously with respect to the weak$^*$ topology.

The entanglement of curves within a 3-periodic box provides a model for complicated space-filling entangled structures occurring in biological materials and structural chemistry. Quantifying the complexity of the entanglement within these models enhances the characterisation of these structures. In this paper, we introduce a new measure of entanglement complexity through the untangling number, reminiscent of the unknotting number in knot theory. The untangling number quantifies the minimum distance between a given 3-periodic structure and its least tangled version, called ground state, through a sequence of operations in a diagrammatic representation of the structure. For entanglements that consist of only infinite open curves, we show that the generic ground states of these structures are crystallographic rod packings, well-known in structural chemistry.

We introduce a property of convex cones, being "well-clipped", that is inspired by the work of several complex algebraic geometers on the Morrison-Kawamata cone conjecture. That property is satisfied by movable cones of divisors on various complex projective varieties of Calabi-Yau type, such as abelian varieties and projective hyperk\"ahler manifolds. The property of being well-clipped has the advantage to descend under taking invariants by a finite group action, and to be stable by direct sums. In the class of well-clipped cones, we also provide a simple characterization of those cones that admit a rational polyhedral fundamental domain under some natural group action. We use this framework to prove the movable cone conjecture for finite quotients of various projective varieties of Calabi-Yau type, notably products of projective primitive symplectic varieties, abelian varieties, and smooth rational surfaces underlying klt Calabi-Yau pairs. This entails Enriques manifolds. We deduce that such finite quotients admit finitely many unmarked small $\mathbb{Q}$-factorial modifications, and that the nef cone conjecture holds for them.

This paper studies the measure-preserving homeomorphisms on compact groups and proposes new methods for constructing measure-preserving homeomorphisms on direct products of compact groups and non-commutative compact groups. On the direct product of compact groups, we construct measure-preserving homeomorphisms using the method of integration. In particular, by applying this method to the \(n\)-dimensional torus \({\mathbb{T}}^{n}\), we can construct many new examples of measure-preserving homeomorphisms. We completely characterize the measure-preserving homeomorphisms on the two-dimensional torus where one coordinate is a translation depending on the other coordinate, and generalize this result to the \(n\)-dimensional torus. For non-commutative compact groups, we generalize the concept of the normalizer subgroup \(N\left( H\right)\) of the subgroup \(H\) to the normalizer subset \({E}_{K}( P)\) from the subset \(K\) to the subset \(P\) of the group of measure-preserving homeomorphisms. We prove that if \(\mu\) is the unique \(K\)-invariant measure, then the elements in \({E}_{K}\left( P\right)\) also preserve \(\mu\). In some non-commutative compact groups the normalizer subset \({E}_{G}\left( {\mathrm{{AF}}\left( G\right) }\right)\) can give non-affine homeomorphisms that preserve the Haar measure. Finally, we prove that when \(G\) is a finite cyclic group and a \(n\)-dimensional torus, then \(\mathrm{{AF}}\left( G\right)= N\left( G\right) = {E}_{G}\left( {\mathrm{{AF}}\left( G\right) }\right)\).

In this paper we focus on the Cahn-Hilliard equation with dynamic boundary conditions, by adding two hyperbolic relaxation terms to the system. We verify that the energy of the total system is decreasing with time. By adding two stabilization terms, we have constructed a first-order temporal accuracy numerical scheme, which is linear and energy stable. Then we prove that the scheme is of first-order in time by the error estimates. At last we carry out enough numerical results to validate the the temporal convergence and the energy stability of such scheme. Moreover, we have present the differences of the numerical results with and without the hyperbolic terms, which show that the hyperbolic terms can help the total energy decreasing slowly.

We study the asymptotic behavior of a nonlinear PDE model for ant trail formation, which was introduced in [3]. We establish the existence of a compact global attractor and prove the nonlinear instability of the homogeneous steady state under an inviscid instability condition. We also provide a dimensional lower bound on the attractor. Alternatively, we prove that if the interaction parameter is sufficiently small, the homogeneous steady state is globally asymptotically stable.

Recent results in quantitative homogenisation of the wave equation with rapidly oscillating coefficients are discussed from the operator-theoretic perspective, which views the solution as the result of applying the operator of hyperbolic dynamics, i.e. the unitary group of a self-adjoint operator on a suitable Hilbert space. A prototype one-dimensional example of utilising the framework of Ryzhov boundary triples is analysed, where operator-norm resolvent estimates for the problem of classical moderate-contrast homogenisation are obtained. By an appropriate "dilation" procedure, these are shown to upgrade to second-order (and more generally, higher-order) estimates for the resolvent and the unitary group describing the evolution for the related wave equation.

We study a generalization of Moreau-Yosida regularization that is adapted to the geometry of Banach spaces where the dual space is uniformly convex with modulus of convexity of power type. Important properties for regularized convex functions are given, in particular strong monotonicity of the subdifferential of their convex conjugate and H\"older-continuity of their gradient.

Finiteness and duality of cohomology of families of $(\varphi,\Gamma)$-modules were proved by Kedlaya-Pottharst-Xiao. In this paper, we study solid locally analytic representations introduced by Rodrigues Jacinto-Rodr\'iguez Camargo in terms of analytic stacks and 6-functor formalisms, which are developed by Clausen-Scholze, Heyer-Mann, respectively. By using this, we will provide a generalization of the result of Kedlaya-Pottharst-Xiao, giving a new proof for cases already proved there.

Proper scoring rules have been a subject of growing interest in recent years, not only as tools for evaluation of probabilistic forecasts but also as methods for estimating probability distributions. In this article, we review the mathematical foundations of proper scoring rules including general characterization results and important families of scoring rules. We discuss their role in statistics and machine learning for estimation and forecast evaluation. Furthermore, we comment on interesting developments of their usage in applications.

Voiculescu's notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitarily invariant. In this work, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local unitary matrices. Assuming the existence of the \emph{tensor distribution} limit described by tuples of permutations, we show that an independent family of local unitary invariant random matrices satisfies asymptotically a novel form of freeness, which we term \emph{tensor freeness}. It can be defined via the vanishing of mixed \emph{tensor free cumulants}, allowing the joint tensor distribution of tensor free elements to be described in terms of that of individual elements. We present several applications of these results in the context of random matrices with a tensor product structure, such as partial transpositions of (local) unitarily invariant random matrices and tensor embeddings of random matrices. Furthermore, we propose a tensor free version of the central limit theorem, which extends and recovers several previous results for tensor products of free variables.

Due to their wide appearance in environmental settings as well as industrial and medical applications, the Stokes-Darcy problems with different sets of interface conditions establish an active research area in the community of mathematical modelers and computational scientists. For numerical simulation of such coupled problems in applications, robust and efficient computational algorithms are needed. In this work, we consider a generalization of the Beavers-Joseph interface condition recently developed using homogenization and boundary layer theory. This extension is applicable not only for the parallel flows to the fluid-porous interface as its predecessor, but also for arbitrary flow directions. To solve the Stokes-Darcy problem with these generalized interface conditions efficiently, we develop and analyze a Robin-Robin domain decomposition method using Fourier analysis to identify optimal weights in the Robin interface conditions. We study efficiency and robustness of the proposed method and provide numerical simulations which confirm the obtained theoretical results.

Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in G/P$ is opposite to the Weyl chambers determined by $F$, we define a projection $\Phi(F, x) \in F$ which is continuous and $G$-equivariant. Furthermore, if $q \geq 3$, we exhibit a $G$-equivariant continuous map defined on an open subset of full measure of the space of $q$-tuples of $(G/P)^q$ with image in $G/K$. When $G$ is the orientation preserving isometries of real hyperbolic space and $q = 3$, we recover the geometric barycenter of the corresponding ideal triangle. All our proofs are constructive.

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs. Then seaweed subalgebras $\mathfrak q\subset\mathfrak g$ were defined by Panyushev for any reductive $\mathfrak g$. A few years later Joseph generalised this notion to the setting of (untwisted) affine Kac--Moody algebras $\widehat{\mathfrak g}$. Furthermore, he proved that the index of such a seaweed can be computed by the same formula that had been known for $\mathfrak g$. In this paper, we construct graphs that help to understand the index of a seaweed $\mathfrak q\subset\widehat{\mathfrak g}$, where $\widehat{\mathfrak g}$ is of affine type A or C.

Let $m$ be a positive integer and $\mathcal{C}$ be a collection of closed subspaces in $L^2(\mathbb{R})$. Given the measurements $\mathcal{F}_Y=\left\lbrace \left\lbrace y_k^1 \right\rbrace_{k\in \mathbb{Z}},\ldots, \left\lbrace y_k^m \right\rbrace_{k\in \mathbb{Z}} \right\rbrace \subset \ell^2(\mathbb{Z})$ of unknown functions $\mathcal{F}=\left\{f_1, \ldots,f_m \right\} \subset L^2( \mathbb{R})$, in this paper we study the problem of finding an optimal space $S$ in $\mathcal{C}$ that is ``closest" to the measurements $\mathcal{F}_Y$ of $\mathcal{F}$. Since the class of finitely generated shift-invariant spaces (FSISs) is popularly used for modelling signals, we assume $\mathcal{C}$ consists of FSISs. We will be considering three cases. In the first case, $\mathcal{C}$ consists of FSISs without any assumption on extra invariance. In the second case, we assume $\mathcal{C}$ consists of extra invariant FSISs, and in the third case, we assume $\mathcal{C}$ has translation-invariant FSISs. In all three cases, we prove the existence of an optimal space.

We study the regularity of entropy solutions for quasilinear parabolic equations with anisotropic degeneracy and stochastic forcing. Building on previous works, we establish space-time regularity under a non-degeneracy condition that does not require an assumption on the derivative of the symbol of the corresponding kinetic equation, a restriction imposed in earlier studies. This allows us to obtain regularity results for certain equations not accounted for by prior theory, albeit with reduced regularity exponents. Our approach uses a kinetic formulation with two transport equations, one of second order and one of first order, leveraging a form of "parabolic regularity" inherent in these equations that was not utilized in previous studies.

A {\em hole} is an induced cycle of length at least 4, a $k$-hole is a hole of length $k$, and an {\em odd hole} is a hole of odd length. Let $\ell\ge 2$ be an integer. Let ${\cal A}_{\ell}$ be the family of graphs of girth at least $2\ell$ and having no odd holes of length at least $2\ell+3$, let ${\cal B}_{\ell}$ be the triangle-free graphs which have no 5-holes and no odd holes of length at least $2\ell+3$, and let ${\cal G}_{\ell}$ be the family of graphs of girth $2\ell+1$ and have no odd hole of length at least $2\ell+5$. Chudnovsky {\em et al.} \cite{CSS2016} proved that every graph in ${\cal A}_{2}$ is 58000-colorable, and every graph in ${\cal B}_{\ell}$ is $(\ell+1)4^{\ell-1}$-colorable. Lan and liu \cite{LL2023} showed that for $\ell\geq3$, every graph in ${\cal G}_{\ell}$ is 4-colorable. It is not known whether there exists a small constant $c$ such that graphs of ${\cal G}_2$ are $c$-colorable. In this paper, we show that every graph in ${\cal G}_2$ is 1456-colorable, and every graph in ${\cal A}_{3}$ is 4-colorable. We also show that every 7-hole free graph in ${\cal B}_{\ell}$ is $(12\ell+8)$-colorable.

We construct a parametrized version of scissors congruence $K$-theory of manifolds, which in particular gives a topologized version of the scissors congruence $K$-theory of oriented manifolds, and we describe this spectrum as mediating between the cobordism category and usual algebraic $K$-theory of spaces. We show that on $\pi_0$, the scissors congruence $K$-theory of oriented manifolds agrees with a version of the cobordism category where we allow free boundaries.

In this paper, we investigated the Fourier partial sums with respect to general orthonormal systems when the function $f$ belongs to some differentiable class of functions

Reconstruction problems lie at the very heart of both mathematics and science, posing the enigmatic challenge: \emph{How does one resurrect a hidden structure from the shards of incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the profound insights of the Vaisman Atiyah--Molino framework. Our method renders the reconstruction problem computationally tractable while exhibiting exceptional robustness in the presence of noise. Central to our theory is the Hantjies tensor -- a curvature-like invariant that precisely quantifies noise propagation and enables error-bounded reconstructions. This synthesis of differential geometry, integral analysis, and algebraic topology not only resolves long-standing ambiguities in inverse problems but also paves the way for transformative applications across a broad spectrum of scientific disciplines.

We describe an algorithm which computes the ring laws for Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithm uses an isomorphism of Illusie in order to compute in an adequate polynomial ring. We also give an implementation of the algorithm in SageMath, which turns out to be faster that Finotti's algorithm, which was until now the most efficient one for these operations.

This paper focuses on nonparametric statistical inference of the hazard rate function of discrete distributions based on $\delta$-record data. We derive the explicit expression of the maximum likelihood estimator and determine its exact distribution, as well as some important characteristics such as its bias and mean squared error. We then discuss the construction of confidence intervals and goodness-of-fit tests. The performance of our proposals is evaluated using simulation methods. Applications to real data are given, as well. The estimation of the hazard rate function based on usual records has been studied in the literature, although many procedures require several samples of records. In contrast, our approach relies on a single sequence of $\delta$-records, simplifying the experimental design and increasing the applicability of the methods.

The de Bruijn identity states that Fisher information is the half of the derivative of Shannon differential entropy along heat flow. In the same spirit, in this paper we introduce a generalized version of Fisher information, named as the R\'enyi--Fisher information, which is the half of the derivative of R\'enyi information along heat flow. Based on this R\'enyi--Fisher information, we establish sharp R\'enyi-entropic isoperimetric inequalities, which generalize the classic entropic isoperimetric inequality to the R\'enyi setting. Utilizing these isoperimetric inequalities, we extend the classical Cram\'er--Rao inequality from Fisher information to R\'enyi--Fisher information. Lastly, we use these generalized Cram\'er--Rao inequalities to determine the signs of derivatives of entropy along heat flow, strengthening existing results on the complete monotonicity of entropy.

Heterogeneity in federated learning (FL) is a critical and challenging aspect that significantly impacts model performance and convergence. In this paper, we propose a novel framework by formulating heterogeneous FL as a hierarchical optimization problem. This new framework captures both local and global training process through a bilevel formulation and is capable of the following: (i) addressing client heterogeneity through a personalized learning framework; (ii) capturing pre-training process on server's side; (iii) updating global model through nonstandard aggregation; (iv) allowing for nonidentical local steps; and (v) capturing clients' local constraints. We design and analyze an implicit zeroth-order FL method (ZO-HFL), provided with nonasymptotic convergence guarantees for both the server-agent and the individual client-agents, and asymptotic guarantees for both the server-agent and client-agents in an almost sure sense. Notably, our method does not rely on standard assumptions in heterogeneous FL, such as the bounded gradient dissimilarity condition. We implement our method on image classification tasks and compare with other methods under different heterogeneous settings.

In harmonic analysis, the studies of inequalities of classical operators (= singular, maximal, Riesz potentials etc.) in various function spaces have a very important place. The maturation of many topics in the field of harmonic analysis, as a result of various needs and developments to respond to the problems of the time, has also led to the emergence of many studies and works on these topics. In [3], under some conditions, the boundedness of Riesz potential on homogeneous variable exponent Herz-Morrey-Hardy spaces has been given. Inspired by the work of [3], in this work, by the atomic decompositions, we obtain the boundedness of commutators generated by the Riesz potentials on homogeneous variable exponent Herz-Morrey-Hardy spaces.

We investigate the hyperbolic decomposition of the Dirichlet norm and distance between autoregressive moving average (ARMA) models. Beginning with the K\"ahler information geometry of linear systems in the Hardy space and weighted Hardy spaces, we demonstrate that the Dirichlet norm and distance of ARMA models, corresponding to the mutual information between the past and future, are decomposed into functions of the hyperbolic distance between the poles and zeros of the ARMA models.

We develop an axiomatic framework for fractal analysis and fractal number theory grounded in hierarchies of definability. Central to this approach is a sequence of formal systems F_n, each corresponding to a definability level S_n contained in R of constructively accessible mathematical objects. This structure refines classical analysis by replacing uncountable global constructs with countable, syntactically constrained approximations. The axioms formalize: - A hierarchy of definability levels S_n, indexed by syntactic and ordinal complexity; - Fractal topologies and the induced notions of continuity, compactness, and differentiability; - Layered integration and differentiation with explicit convergence and definability bounds; - Arithmetic and function spaces over the stratified continuum R_{S_n}, which is a subset of R. This framework synthesizes constructive mathematics, proof-theoretic stratification, and fractal geometric intuition into a unified, finitistically structured model. Key results include the definability-based classification of real numbers (e.g., algebraic, computable, Liouville), a stratified fundamental theorem of calculus with syntactic error bounds, and compatibility with base systems such as RCA_0 and ACA_0. The framework enables constructive approximation and syntactic regularization of classical analysis, with applications to proof assistants, computable mathematics, and foundational studies of the continuum.

In this paper, we prove the $W$-entropy formula and its monotonicity for the heat flow on RCD$(K, N)$ and RCD$(K, n, N)$ spaces $(X, d, \mu)$, where $K\in \mathbb{R}$, $n$ is the geometric dimension of $(X, d, \mu)$ and $N\geq n$. We also prove the $K$-concavity of the Shannon entropy power on RCD$(K, N)$ spaces. As an application, we derive the Shannon entropy isoperimetric inequality and the Stam logarithmic Sobolev inequality on RCD$(0, N)$ spaces with maximal volume growth condition. Finally, we prove the rigidity theorem for the Stam logarithmic Sobolev inequality with sharp constant on noncollapsing RCD$(0, N)$ spaces.

The Chen-Ng\^o Conjecture predicts that the Hitchin morphism from the moduli stack of $G$-Higgs bundles on a smooth projective variety surjects onto the space of spectral data. The conjecture is known to hold for the group $GL_n$ and any surface, and for the group $GL_2$ and any smooth projective variety. We prove the Chen-Ng\^o Conjecture for any reductive group when the variety is a ruled surface or (a blowup of) a nonisotrivial elliptic fibration with reduced fibers. Furthermore, if the group is a classical group, i.e. $G \in \{SL_n,SO_n,Sp_{2n}\}$, then we prove the Hitchin morphism restricted to the Dolbeault moduli space of semiharmonic $G$-Higgs bundles surjects onto the space of spectral data.

The Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry constructs a mirror space and a superpotential from the data of a Lagrangian torus fibration on a K\"ahler manifold with effective first Chern class. For K\"ahler manifolds whose first Chern class is not nef, the SYZ construction is further complicated by the presence of additional holomorphic discs with non-positive Maslov index. In this paper, we study SYZ mirror symmetry for two of the simplest toric examples: the non-Fano Hirzebruch surfaces F_3 and F_4. For F_3, we determine the SYZ mirror associated to generic perturbations of the complex structure, and demonstrate that the SYZ mirror depends on the choice of perturbation. For F_4, we determine the SYZ mirror for a specific perturbation of complex structure, where the mirror superpotential is an explicit infinite Laurent series. Finally, we relate this superpotential to those arising from other perturbations of F_4 via a scattering diagram.

It is proved that any smooth manifold $\mathcal M$ of dimension $m$ admits a smooth polynomially convex embedding into $\mathbb C^n$ when $n\geq \lfloor 5m/4\rfloor$. Further, such embeddings are dense in the space of smooth maps from $\mathcal M$ into $\mathbb C^n$ in the $\mathcal C^3$-topology. The components of any such embedding give smooth generators of the algebra of complex-valued continuous functions on $\mathcal M$. A key ingredient of the proof is a coordinate-free description of certain notions of (non)degeneracy, as defined by Webster and Coffman, for CR-singularities of order one of an embedded real manifold in $\mathbb C^n$. The main result is obtained by inductively perturbing each stratum of degeneracy to produce a global polynomially convex embedding.

Let $H$ be a subdigraph of a digraph $D$. An ear of $H$ in $D$ is a path or a cycle in $D$ whose ends lie in $H$ but whose internal vertices do not. An \emph{ear decomposition} of a strong digraph $D$ is a nested sequence $(D_0,D_1,\ldots , D_k)$ of strong subdigraphs of $D$ such that: 1) $D_0$ is a cycle, 2) $D_{i+1} = D_i\cup P_i$, where $P_i$ is an ear of $D_i$ in $D$, for every $i\in \{0,1,\ldots,k-1\}$, and 3) $D_k=D$. In this work, the $\mathcal{LE}_i$ is defined as the family of strong digraphs, with an ear decomposition such that every ear has a length of at least $i\geq 1$. It is proved that Seymour's second Neighborhood Conjecture and the Laborde, Payan, and Soung conjecture, are true in the family $\mathcal{LE}_2$, and the Small quasi-kernel conjecture is true for digraphs in $\mathcal{LE}_3$. Also, some sufficient conditions for a strong nonseparable digraph in $\mathcal{LE}_2$ with a kernel to imply that the previous (following) subdigraph in the ear decomposition has a kernel too, are presented. It is proved that digraphs in $\mathcal{LE}_2$ have a chromatic number at most 3, and a dichromatic number 2 or 3. Finally, the oriented chromatic number of asymmetrical digraphs in $\mathcal{LE}_3$ is bounded by 6, and it is shown that the oriented chromatic number of asymmetrical digraphs in $\mathcal{LE}_2$ is not bounded.

Anderson modules form a generalization of Drinfeld modules and are commonly understood as the counterpart of abelian varieties but with function field coefficients. In an attempt to study their ``motivic theory'', two objects of semilinear algebra are attached to an Anderson module: its motive and its dual motive. While the former is better suited to follow the analogy with Grothendieck motives, the latter has proven much useful in the study of transcendence questions in positive characteristic. Despite sharing similar definitions, the relationship between motives and dual motives has remained nebulous. Over perfect fields, it was only proved recently by the second author that the finite generation of the motive is equivalent to the finite generation of the dual motive, answering a long-standing open question in function field arithmetic (the ``abelian equals $A$-finite'' theorem). This work constructs a perfect pairing among the motive and the dual motive of an Anderson module, with values in a module of differentials, thus answering a question raised by Hartl and Juschka. Our construction involves taking the residue of certain formal power series in the Frobenius endomorphism. Although it may seem peculiar, this pairing is natural and compatible with base change. It also comes with several new consequences in function field arithmetic; for example, we generalize the ``abelian equals A-finite'' theorem to a large class of algebras, including fields, perfect algebras and noetherian regular domains.

Given parameters $c>0, \delta\ne0$ and a sequence $(X_n)$ of real-valued, integrable, independent and identically $F$-distributed random variables, we characterise distributions $F$ such that $(N_n-cM_n)$ is a martingale, where $N_n$ denotes the number of observations $X_k$ among $X_1,\ldots,X_n$ such that $X_k>M_{k-1}+\delta$, called $\delta$-records, and $M_k=\max\{X_1,\ldots, X_k\}$. The problem is recast as $1-F(x+\delta)=c\int_{x}^{\infty}(1-F)(t)dt$, for $x\in T$, with $F(T)=1$. Unlike standard functional equations, where the equality must hold for all $x$ in a fixed set, our problem involves a domain that depends on $F$ itself, introducing complexity but allowing for more possibilities of solutions. We find the explicit expressions of all solutions when $\delta < 0$ and, when $\delta > 0$, for distributions with bounded support. In the unbounded support case, we focus attention on continuous and lattice distributions. In the continuous setting, with support $\mathbb{R}_+$, we reduce the problem to a delay differential equation, showing that, besides particular cases of the exponential distribution, mixtures of exponential and gamma distributions and many others are solutions as well. The lattice case, with support $\mathbb{Z}_+$ is treated analogously and reduced to the study of a difference equation. Analogous results are obtained; in particular, mixtures of geometric and negative binomial distributions are found to solve the problem.

We develop a novel source coding strategy for sampling and monitoring of a Wiener process. For the encoding process, we employ a four level ``quantization'' scheme, which employs monotone function thresholds as opposed to fixed constant thresholds. Leveraging the hitting times of the Wiener process with these thresholds, we devise a sampling and encoding strategy which does not incur any quantization errors. We give analytical expressions for the mean squared error (MSE) and find the optimal source code lengths to minimize the MSE under this monotone function threshold scheme, subject to a sampling rate constraint.

Let $K_q(n,r)$ denote the minimum size of a $q$-ary covering code of word length $n$ and covering radius $r$. In other words, $K_q(n,r)$ is the minimum size of a set of $q$-ary codewords of length $n$ such that the Hamming balls of radius $r$ around the codewords cover the Hamming space $\{0,\ldots,q-1\}^n$. The special case $K_3(n,1)$ is often referred to as the football pool problem, as it is equivalent to finding a set of forecasts on $n$ football matches that is guaranteed to contain a forecast with at most one wrong outcome. In this paper, we build and expand upon the work of Gijswijt (2005), who introduced a semidefinite programming lower bound on $K_q(n,r)$ via matrix cuts. We develop techniques that strengthen this bound, by introducing new semidefinite constraints inspired by Lasserre's hierarchy for 0-1 programs and symmetry reduction methods, and a more powerful objective function. The techniques lead to sharper lower bounds, setting new records across a broad range of values of $q$, $n$, and $r$.

The extension from toric varieties to quantum toric stacks allows for the study of moduli spaces of toric objects with fixed combinatorial structures, as we now consider general finitely generated subgroups of $\mathbb{R}^n$ as "lattices." This paper aims to construct a moduli space that encompasses all such moduli spaces for a given dimension of the ambient space. To achieve this, we adapt the construction of the secondary fan within the quantum framework. This approach provides a description of wall-crossings between different moduli spaces, analogous to those observed in LVMB manifolds.

In the Edge Inference (EI) paradigm, where a Deep Neural Network (DNN) is split across the transceivers to wirelessly communicate goal-defined features in solving a computational task, the wireless medium has been commonly treated as a source of noise. In this paper, motivated by the emerging technologies of Reconfigurable Intelligent Surfaces (RISs) and Stacked Intelligent Metasurfaces (SIM) that offer programmable propagation of wireless signals, either through controllable reflections or diffractions, we optimize the RIS/SIM-enabled smart wireless environment as a means of over-the-air computing, resembling the operations of DNN layers. We propose a framework of Metasurfaces-Integrated Neural Networks (MINNs) for EI, presenting its modeling, training through a backpropagation variation for fading channels, and deployment aspects. The overall end-to-end DNN architecture is general enough to admit RIS and SIM devices, through controllable reconfiguration before each transmission or fixed configurations after training, while both channel-aware and channel-agnostic transceivers are considered. Our numerical evaluation showcases metasurfaces to be instrumental in performing image classification under link budgets that impede conventional communications or metasurface-free systems. It is demonstrated that our MINN framework can significantly simplify EI requirements, achieving near-optimal performance with $50~$dB lower testing signal-to-noise ratio compared to training, even without transceiver channel knowledge.

Compressed sensing with subsampled unitary matrices benefits from \emph{optimized} sampling schemes, which feature improved theoretical guarantees and empirical performance relative to uniform subsampling. We provide, in a first of its kind in compressed sensing, theoretical guarantees showing that the error caused by the measurement noise vanishes with an increasing number of measurements for optimized sampling schemes, assuming that the noise is Gaussian. We moreover provide similar guarantees for measurements sampled with-replacement with arbitrary probability weights. All our results hold on prior sets contained in a union of low-dimensional subspaces. Finally, we demonstrate that this denoising behavior appears in empirical experiments with a rate that closely matches our theoretical guarantees when the prior set is the range of a generative ReLU neural network and when it is the set of sparse vectors.

In this paper, we use a machine learning approach to predict the stationary distributions of the number of customers in a single-staiton multi server system. We consider two systems, the first is $c$ homogeneous servers, namely the $GI/GI/c$ queue. The second is a two-heterogeneous server system, namely the $GI/GI_i/2$ queue. We train a neural network for these queueing models, using the first four inter-arrival and service time moments. We demonstrate empirically that using the fifth moment and beyond does not increase accuracy. Compared to existing methods, we show that in terms of the stationary distribution and the mean value of the number of customers in a $GI/GI/c$ queue, we are state-of-the-art. Further, we are the only ones to predict the stationary distribution of the number of customers in the system in a $GI/GI_i/2$ queue. We conduct a thorough performance evaluation to assert that our model is accurate. In most cases, we demonstrate that our error is less than 5\%. Finally, we show that making inferences is very fast, where 5000 inferences can be made in parallel within a fraction of a second.

Coffee leaf rust (CLR) severely affects coffee production worldwide, leading to reduced yields and economic losses. To reduce the cost of control, small-scale farmers often only apply control measures once a noticeable level of infection is reached. In this work, we develop mathematical models to better understand CLR dynamics and impulsive biocontrol with threshold-based interventions. We first use ordinary and impulsive differential equations to describe disease spread and the application of control measures once a certain infection level is detected. These models help determine when and how often interventions should occur. To capture the early stages of the disease and the chance that it might die out by itself, we then use a continuous-time Markov chain approach. This stochastic model allows us to estimate the probability that the pathogen fails to establish, thereby avoiding serious outbreaks.

Double-bracket quantum imaginary-time evolution (DB-QITE) is a quantum algorithm which coherently implements steps in the Riemannian steepest-descent direction for the energy cost function. DB-QITE is derived from Brockett's double-bracket flow which exhibits saddle points where gradients vanish. In this work, we perform numerical simulations of DB-QITE and describe signatures of transitioning through the vicinity of such saddle points. We provide an explicit gate count analysis using quantum compilation programmed in Qrisp.

We study infinite distance limits in the complex structure moduli space of Type IIB compactifications on Calabi--Yau threefolds, in light of the Emergent String Conjecture. We focus on the so-called type II limits, which, based on the asymptotic behaviour of the physical couplings in the low-energy effective theory, are candidates for emergent string limits. However, due to the absence of Type IIB branes of suitable dimensionality, the emergence of a unique critical string accompanied by a tower of Kaluza--Klein states has so far remained elusive. By considering a broad class of type II$_b$ limits, corresponding to so-called Tyurin degenerations, and studying the asymptotic behaviour of four-dimensional EFT strings in this geometry, we argue that the worldsheet theory of the latter describes a unique critical heterotic string on $T^2\times\mathrm{K3}$ with a gauge bundle whose rank depends on $b$. In addition, we establish the presence of an infinite tower of BPS particles arising from wrapped D3-branes by identifying a suitable set of special Lagrangian 3-cycles in the geometry. The associated BPS invariants are conjectured to be counted by generalisations of modular forms. As a consistency check, we further show that in special cases mirror symmetry identifies the EFT strings with the well-understood emergent string limits in the K\"ahler moduli space of Type IIA compactifications on K3-fibred Calabi--Yau threefolds. Finally, we discuss the implications of the Emergent String Conjecture for type II limits which do not correspond to Tyurin degenerations, and predict new constraints on the possible geometries of type II degenerations which resemble those arising in the Kulikov classification of degenerations of K3 surfaces.

We consider a heat equation with memory which is defined on a bounded domain in $\mathbb{R}^d$ and is driven by $m$ control inputs acting on the interior of the domain. Our objective is to numerically construct a state feedback controller for this equation such that, for each initial state, the solution of the closed-loop system decays exponentially to zero with a decay rate larger than a given rate $\omega>0$, i.e. we want to solve the $\omega$-stabilization problem for the heat equation with memory. We first show that the spectrum of the state operator $A$ associated with this equation has an accumulation point at $-\omega_0<0$. Given a $\omega\in(0,\omega_0)$, we show that the $\omega$-stabilization problem for the heat equation with memory is solvable provided certain verifiable conditions on the control operator $B$ associated with this equation hold. We then consider an appropriate LQR problem for the heat equation with memory. For each $n\in\mathbb{N}$, we construct finite-dimensional approximations $A_n$ and $B_n$ of $A$ and $B$, respectively, and then show that by solving a corresponding approximation of the LQR problem a feedback operator $K_n$ can be computed such that all the eigenvalues of $A_n + B_n K_n$ have real part less than $-\omega$. We prove that $K_n$ for $n$ sufficiently large solves the $\omega$-stabilization problem for the heat equation with memory. A crucial and nontrivial step in our proof is establishing the uniform (in $n$) stabilizability of the pair $(A_n+\omega I, B_n)$. We have validated our theoretical results numerically using two examples: an 1D example on a unit interval and a 2D example on a square domain.

The problem of incompressible fluid mixing arises in numerous engineering applications and has been well-studied over the years, yet many open questions remain. This paper aims to address the question "what do efficient flow fields for mixing look like, and how do they behave?" We approach this question by developing a framework which is inspired by the dynamic and geometric approach to optimal mass transport. Specifically, we formulate the fluid mixing problem as an optimal control problem where the dynamics are given by the continuity equation together with an incompressibility constraint. We show that within this framework, the set of reachable fluid configurations can formally be endowed with the structure of an infinite-dimensional Riemannian manifold, with a metric which is induced by the control effort, and that flow fields which are maximally efficient at mixing correspond to geodesics in this Riemannian space.

It is long known that quantum Calogero models feature intertwining operators, which increase or decrease the coupling constant by an integer amount, for any fixed number of particles. We name these as ``horizontal'' and construct new ``vertical'' intertwiners, which \emph{change the number of interacting particles} for a fixed but integer value of the coupling constant. The emerging new structure of a grid of intertwiners exists only in the algebraically integrable situation (integer coupling) and allows one to obtain each Liouville charge from the free power sum in the particle momenta by iterated intertwining either horizontally or vertically. We present recursion formul\ae\ for the new intertwiners as a factorization problem for partial differential operators and prove their existence for small values of particle number and coupling. As a byproduct, a new basis of non-symmetric Liouville integrals appears, algebraically related to the standard symmetric one.

We develop a novel optimization framework with distribution-based fairness constraints for efficiently producing demographically blind, explainable models across a wide range of fairness levels. This is accomplished through post-processing, avoiding the need for retraining. Our framework, which is based on stochastic gradient descent, can be applied to a wide range of model types, with a particular emphasis on the post-processing of gradient-boosted decision trees. Additionally, we design a broad class of interpretable global bias metrics compatible with our method by building on previous work. We empirically test our methodology on a variety of datasets and compare it to other methods.

Quantum computing (QC) emulators, which simulate quantum algorithms on classical hardware, are indispensable platforms for testing quantum algorithms before scalable quantum computers become widely available. A critical challenge in QC emulation is managing numerical errors from finite arithmetic precision, especially truncation errors in resource-efficient fixed-point arithmetic. Despite its importance, systematic studies quantifying how truncation errors impact quantum algorithm accuracy are limited. In this paper, we propose a rigorous quantitative framework analyzing truncation error propagation in fixed-point QC emulation, focusing on Grover's quantum search algorithm. First, we introduce a simplified two-value amplitude representation of quantum states during Grover's iterations and prove its theoretical validity. Using this representation, we derive explicit mathematical expressions characterizing truncation error accumulation across quantum gate operations. We quantify the overall emulation error by the $\ell_2$ distance between ideal and emulated probability distributions, obtaining asymptotic bounds scaling as $O(2^{n-f})$, where $n$ is the number of qubits and $f$ is fractional-bit precision. Extensive numerical simulations and empirical experiments on a practical fixed-point QC emulator confirm that observed errors precisely match our theoretical predictions. Finally, we provide a closed-form formula to determine the minimal fractional-bit precision required to achieve a specified error threshold, offering clear guidelines for emulator designers balancing accuracy and resource utilization.

While unmanned aerial vehicles (UAVs) with flexible mobility are envisioned to enhance physical layer security in wireless communications, the efficient security design that adapts to such high network dynamics is rather challenging. The conventional approaches extended from optimization perspectives are usually quite involved, especially when jointly considering factors in different scales such as deployment and transmission in UAV-related scenarios. In this paper, we address the UAV-enabled multi-user secure communications by proposing a deep graph reinforcement learning framework. Specifically, we reinterpret the security beamforming as a graph neural network (GNN) learning task, where mutual interference among users is managed through the message-passing mechanism. Then, the UAV deployment is obtained through soft actor-critic reinforcement learning, where the GNN-based security beamforming is exploited to guide the deployment strategy update. Simulation results demonstrate that the proposed approach achieves near-optimal security performance and significantly enhances the efficiency of strategy determination. Moreover, the deep graph reinforcement learning framework offers a scalable solution, adaptable to various network scenarios and configurations, establishing a robust basis for information security in UAV-enabled communications.

This paper develops a model-based framework for continuous-time policy evaluation (CTPE) in reinforcement learning, incorporating both Brownian and L\'evy noise to model stochastic dynamics influenced by rare and extreme events. Our approach formulates the policy evaluation problem as solving a partial integro-differential equation (PIDE) for the value function with unknown coefficients. A key challenge in this setting is accurately recovering the unknown coefficients in the stochastic dynamics, particularly when driven by L\'evy processes with heavy tail effects. To address this, we propose a robust numerical approach that effectively handles both unbiased and censored trajectory datasets. This method combines maximum likelihood estimation with an iterative tail correction mechanism, improving the stability and accuracy of coefficient recovery. Additionally, we establish a theoretical bound for the policy evaluation error based on coefficient recovery error. Through numerical experiments, we demonstrate the effectiveness and robustness of our method in recovering heavy-tailed L\'evy dynamics and verify the theoretical error analysis in policy evaluation.

We study different online optimization problems in the random-order model. There is a finite set of bins with known capacity and a finite set of items arriving in a random order. Upon arrival of an item, its size and its value for each of the bins is revealed and it has to be decided immediately and irrevocably to which bin the item is assigned, or to not assign the item at all. In this setting, an algorithm is $\alpha$-competitive if the total value of all items assigned to the bins is at least an $\alpha$-fraction of the total value of an optimal assignment that knows all items beforehand. We give an algorithm that is $\alpha$-competitive with $\alpha = (1-\ln(2))/2 \approx 1/6.52$ improving upon the previous best algorithm with $\alpha \approx 1/6.99$ for the generalized assignment problem and the previous best algorithm with $\alpha \approx 1/6.65$ for the integral knapsack problem. We then study the fractional knapsack problem where we have a single bin and it is also allowed to pack items fractionally. For that case, we obtain an algorithm that is $\alpha$-competitive with $\alpha = 1/e \approx 1/2.71$ improving on the previous best algorithm with $\alpha = 1/4.39$. We further show that this competitive ratio is the best-possible for deterministic algorithms in this model.

Global variants of four-dimensional gauge theories are specified by their spectrum of genuine Wilson-'t Hooft line operators. The choice of global variant has significant consequences when spacetime is taken to be $\mathbb{R}^3 \times S^1$. We focus on $\mathcal N=1^*$ theories, which are closely connected to twisted elliptic Calogero-Moser systems. We establish, on general grounds, how this gauge-theoretic topological data manifests itself on the integrable system side by introducing a notion of global variants for complex many-body integrable systems associated with Lie algebras. Focusing on $\mathcal N=1^*$ theories of type $A$ and $B_2$, we elucidate the implications for the structure of gapped vacua, the emergent (generalized) symmetries realized in each vacuum, and the action of spontaneously broken modular invariance.

The preservation of endangered languages is a widely discussed issue nowadays. Languages represent essential cultural heritage and can provide valuable botanical, biological, and geographical information. Therefore, it is necessary to develop efficient measures to preserve and revitalize endangered languages. However, the language shift process is complex and requires an interdisciplinary approach, including mathematical modeling techniques. This paper develops a new mathematical model that extends previous works on this topic. We introduce the factor of ethnic identity, which is a proxy for a more complex nexus of variables involved in an individual's self-identity and/or a group's identity. This proxy is socially constructed rather than solely inherited, shaped by community-determined factors, with language both indexing and creating the identity. In our model, we divide speakers into groups depending on with which language they identify themselves with. Moreover, every group includes monolinguals and bilinguals. The proposed model naturally allows us to consider cases of language coexistence and describe a broader class of linguistic situations. For example, the simulation results show that our model can result in cyclic language dynamics, drawing a parallel to cell population models. In this way, the proposed mathematical model can serve as a useful tool for developing efficient measures for language preservation and revitalization.

Scaled Relative Graphs (SRGs) provide a novel graphical frequency domain method for the analysis of nonlinear systems. In this paper, we use the restriction of the SRG to particular input spaces to compute frequency-dependent gain bounds for incrementally stable nonlinear systems. This leads to a nonlinear (NL) generalization of the Bode diagram, where the sinusoidal, harmonic, and subharmonic inputs are considered separately. When applied to the analysis of the NL loop transfer and sensitivity, we define a notion of bandwidth for both the open-loop and closed-loop, compatible with the LTI definitions. We illustrate the power of our method on the analysis of a DC motor with a parasitic nonlinearity, verifying our results in simulations.

Non-collocated vibration absorption (NCVA) concept using delayed resonator for in-situ tuning is analyzed and experimentally validated. There are two critical contributions of this work. One is on the scalable analytical pathway for verifying the concept of resonant substructure as the basis of the ideal vibration absorption. The second is to experimentally validate the spatial and spectral tunability of NCVA structures for the first time. For both novelties arbitrarily large dimensions of interconnected mass-spring-damper chains are considered. Following the state of the art on NCVA, control synthesis is performed over the resonant substructure comprising the delayed resonator and a part of the primary structure involved in the vibration absorption. The experimental validation of the proposed NCVA concept is performed on a mechatronic setup with three interconnected cart-bodies. Based on the spectral analysis, an excitation frequency is selected for which a stable vibration suppression can be achieved sequentially for all the three bodies, one collocated and two non-collocated. The experimental results closely match the simulations for complete vibration suppression at the targeted bodies, and thus validating the crucial spatial tunability characteristic as well as the traditional spectral tuning.

Probabilistic generative models based on measure transport, such as diffusion and flow-based models, are often formulated in the language of Markovian stochastic dynamics, where the choice of the underlying process impacts both algorithmic design choices and theoretical analysis. In this paper, we aim to establish a rigorous mathematical foundation for denoising Markov models, a broad class of generative models that postulate a forward process transitioning from the target distribution to a simple, easy-to-sample distribution, alongside a backward process particularly constructed to enable efficient sampling in the reverse direction. Leveraging deep connections with nonequilibrium statistical mechanics and generalized Doob's $h$-transform, we propose a minimal set of assumptions that ensure: (1) explicit construction of the backward generator, (2) a unified variational objective directly minimizing the measure transport discrepancy, and (3) adaptations of the classical score-matching approach across diverse dynamics. Our framework unifies existing formulations of continuous and discrete diffusion models, identifies the most general form of denoising Markov models under certain regularity assumptions on forward generators, and provides a systematic recipe for designing denoising Markov models driven by arbitrary L\'evy-type processes. We illustrate the versatility and practical effectiveness of our approach through novel denoising Markov models employing geometric Brownian motion and jump processes as forward dynamics, highlighting the framework's potential flexibility and capability in modeling complex distributions.