This paper addresses the non-preemptive single-machine scheduling problem with heterogeneous release and processing times, where minimizing total waiting time is NP-hard. We introduce a novel waiting-time formula that preserves global information, which derive a specific structural condition that transforms the problem into a tractable form. Leveraging this formula, we develop a structure-driven analytical framework that uses inductive reasoning to characterize the feasible solution space. This space enables the identification of key job movements that theoretically minimize total waiting time, followed by working backward from the optimality conditions to schedule the remaining jobs. Based on this framework, we design a provably exact algorithm with $\mathcal{O}(n^9)$ time complexity. Numerical experiments validate both the optimality and computational efficiency of our method, demonstrating substantial performance gains over Gurobi. These results highlight the algorithm's potential as a powerful tool for solving a class of otherwise intractable scheduling problems.

We define the tricategory of algebraic conformal nets, defects, sectors and intertwiners where algebraic refers to the absence of a topology on the relevant algebras and modules. We aim at making these definitions satisfying from a categorical point of view, and in a few cases, e.g., for defects, we propose more general definitions than what can be found in the literature. This allows us in particular to propose an implementation of solitons as defects. Then we focus on the locally constant case where we provide an identification of the tricategory of locally constant algebraic conformal nets, defects, sectors and intertwiners with the tricategory of commutative algebras, algebras, bimodules and bimodule homomorphisms.

The optimal strategies to catch a randomly walking cat in various environments are presented. All games have a player that opens a box at step $i$. If the cat is in this box the player wins, if not, the cat moves randomly to an adjacent box, or in case the box is open to the outside, the cat may escape. If the cat is not escaping, the next step of the game is played. In case the cat has doors to escape, the optimal strategy is determined that minimizes the escape chances of the cat. If there are no doors to the outside, the strategies are calculated that minimize the game duration. The environments are 1) a one-dimensional array of up to 9 boxes in a line 2) this line is connected to a ring and 3) a $2 \times m$ grid (with $2\le m\le 4$) of boxes. In cases 1) and 3) the boxes may or may not have exits to the outside. Numerical proofs for the optimality of the presented strategies are outlined. Also a formula is presented for the average number of steps it takes a randomly walking cat to exit a $2\times m$ grid when no player is involved. This formula is only based on Fibonacci numbers if $m$ is even and Lucas numbers if $m$ is odd.

We study the geometric aspects of the magic trick called Gozinta Boxes. We generalize Gozinta Boxes to other dimensions, and we show that in three and higher dimensions, the maximum number of boxes is 3, and in two dimensions, the maximum is 4. We discuss other properties of Gozinta Boxes and provide a plethora of examples.

We derive an asymptotic formula for the sum $$ H = \sum_{0<\gamma_k\leqslant T,\, 1\leqslant k\leqslant m}h(a_1\gamma_1+a_2\gamma_2+\cdots + a_m\gamma_m), $$ where $a_1, a_2, \ldots, a_m$ are integers whose sum equals zero, $\gamma_1, \ldots, \gamma_m$ independently run through the imaginary parts of the non-trivial zeros of the Riemann zeta function, each zero occuring in the sum the number of times of its multiplicity, and the function $h$ belongs to some special class.

The author studies an average version of Brun--Titchmarsh theorem with large moduli. Using Maynard's recent breakthrough on the Bombieri--Friedlander--Iwaniec type triple convolution estimates, we refine the previous result of Baker and Harman (1996). As an application, we improve a result of Baker and Harman (1998) on shifted primes with a large prime factor, showing that the largest prime factor of $p - 1$ is larger than $p^{0.679}$ for infinitely many primes $p$.

Amdeberhan, Sellers, and Singh introduced the function $a_c(n)$ that counts the number of generalized cubic partitions of $n$, which are partitions of $n$ whose even parts may appear in $c\geq 1$ different colors. Recently, Dockery obtained via modular forms the following isolated congruences modulo $7$ and $11$ for $a_c(n)$, namely $a_5(49n+31)\equiv 0\pmod{7}$ and $a_9(121n+36)\equiv 0\pmod{11}$ for $n\geq 0$. We prove in this short note a generalization of these congruences by employing a result of Ahlgren on the coefficients of a certain product of powers of Euler's product.

Generalized Cunningham chains are sets of the form $\{f^n(z)\}_{n\ge0}$ where all its elements are prime numbers and $f$ is a linear polynomial with integer coefficients. We generalize this definition further to include starting terms that are not prime, and we obtain the bound of $\ell(z)< z$ if $z$ is big enough, where $\ell(z)$ is the size of the generalized Cunningham chain. Unlike a direct generalization of previous results, which require $z$ to have a prime factor that does not divide the leading term of $f$, this result is only dependent on the size of $z$ and not on its prime factorization.

This paper explicitly constructs a family of weight-shifting operators mapping from the space of weight-k smooth automorphic functions, A_k(Gamma), to the space of weight-t automorphic functions, A_t(Gamma). These operators are defined as integral transforms whose kernel, K, is the product of the covariant factor P given in Definition 1.2 and a function, F, which is invariant under the diagonal action of SL(2, R). The key assumption in the operator's construction is the spectral condition posited in Hypothesis 1.1, namely that the kernel K is an eigenfunction of the weight-t hyperbolic Laplacian Delta_t with respect to the first variable. This partial differential equation reduces to an ordinary differential equation for the invariant function F, which is shown to be equivalent to the Hypergeometric Differential Equation (HDE). The main result of this paper, presented in Theorem 2.1, is the explicit determination of the HDE parameters (a,b,c) as functions of the initial automorphic data (k, t, q) and the operator's spectral parameter lambda_K. This work presents an analytic methodology for the explicit construction of an intertwining operator that maps a Laplacian eigenspace to its corresponding eigenspace, demonstrating how the kernel is determined from the automorphic data via the Hypergeometric Differential Equation.

We formulate Shintani's invariant in terms of the cyclic quantum dilogarithm. Building on earlier results that expressed Shintani's invariant using the $q$-Pochhammer symbol, we show how the cyclic quantum dilogarithm naturally arises in this context, providing new perspectives on the arithmetic significance of Shintani's construction. The present note is an announcement; a full account with complete proofs will appear in a forthcoming paper.

We study whether symplectic quandle colorings can reveal causal structure encoded by "sky links" - i.e. links consisting of spheres of all light rays through two points in the space of all light rays of a spacetime. Building on the known limitations of the Alexander-Conway polynomial, we compare the connected sum of two Hopf links (which represents all causally unrelated situations) with the first two Allen-Swenberg links (that are the only known examples when this polynomial does not work). For each diagram we report both the quandle counting invariant (total number of colorings) and an enhanced version that records how many distinct colors appear in each coloring. In our tests over small finite fields, plain counts often agree across examples, but the enhanced invariant consistently separates the Hopf case from the Allen-Swenberg family, and becomes more discriminating as the field grows. A simple transfer step suggests that this effect persists along the whole sequence. These results point to enhanced symplectic quandle colorings as a practical, computable indication of causality that classical polynomials alone may miss; the first examples of this kind were discovered by Jain.

This work aims to initiate a discussion on finding solutions to non-homoge\-neous differential equations in terms of generalized functions. For simplicity, we conduct the analysis within the specific context of the stationary Klein-Gordon equation with a point-like source, identifying a generalized function that solves such an equation and aligns with the solution obtained through the Fourier approach with dimensional regularization. In addition to being regular at the source singularity, a notable advantage of our solution is its presentation as a single expression, eliminating the need for piecewise definitions. The arguments presented here are applicable to a broader range of situations, offering a novel approach to addressing divergences in field theories using generalized functions. Moreover, we anticipate that the approach introduced in this work could provide a new method for handling Green functions regularized at coincident points, thereby simplifying the renorma\-lization process in a wide range of theories.

We study $\eta$-correction terms in the Kauffman bracket skein algebra of the once-punctured torus $K_t(\Sigma_{1,1})$. While the Frohman-Gelca product-to-sum rule is explicit on the closed torus, punctures introduce correction terms in the ideal $(\eta)$ whose structure has resisted systematic description beyond low-determinant cases. We give a closed form for the family $P_n=(1,0)\cdot T_n((1,2))$ (determinant $2$): the correction $\epsilon_n$ has an explicit Chebyshev expansion with coefficients that factor as geometric sums in $t^{\pm4}$ governed by parity. Tracking creation and cancellation of $\eta$ through the Chebyshev recurrence, and using diffeomorphism invariance, the formula transports to all products $C_1\cdot T_k(C_2)$ with $|\det(C_1,C_2)|=2$. We further treat the \emph{primitive maximal-thread} regime, where one Frohman-Gelca summand is fully threaded and the other is simple or doubly covered. In this case we obtain a closed form for the discrepancy: an $\eta$-linear cascade with Chebyshev $S$-coefficients that lowers the thread degree by two at each step. Equivalently, in this maximal-thread regime we solve the specialized Wang-Wong recursion in closed form; the coefficients match their fast algorithm term-by-term (up to the $T_0$ normalization) and subsume Cho's $|\det|=2$ case. The resulting rules give compact, scalable formulas for symbolic multiplication in $K_t(\Sigma_{1,1})$

We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary obtained by cutting the closed domain surface of the Morse function at the levels of regular values. We consider Morse functions having a bounded number of critical points and one single local minimum. We find a small set of Morse functions which are close enough to any other Morse function in the sense that they share the same characterizing surfaces with boundary.

We compare different periodic point invariants for families of maps parameterized over a compact manifold. Malkiewich and Ponto showed that, in the case of a single map, the Fuller trace is equivalent to the collection of Reidmeister traces of iterates. In this paper, we show that, in contrast to the case of a single map, the fiberwise Fuller trace is a strictly more sensitive invariant than the collection of fiberwise Reidemeister traces of iterates. This resolves a conjecture of Malkiewich and Ponto.

We study stationary configurations of compressible barotropic fluids lying inside an incompressible fluid and acted upon by a constant gravitational field. Without gravity, it is a simple matter to construct solutions consisting of perfectly spherical bubbles of compressible fluid, but with gravity the stratification of the compressible fluid's density prevents the existence of solutions with such simple geometries. We construct smooth solutions with nontrivial gravity and nearly spherical, axisymmetric geometry by means of a bifurcation argument in an appropriate weighted Sobolev space, defined in terms of the Legendre operator. Along the way, we develop a number of essential linear and nonlinear functional analytic tools for this scale of spaces, which are of independent interest.

This paper proposes the theoretical grounds for emulating the Dynamic System Optimum with desired arrival times on regional networks, using aggregated traffic dynamics based on the Macroscopic Fundamental Diagram. We used a projection gradient-based solution method that avoids the need to compute approximations of marginal travel times. We demonstrate the application of our framework on 8-region networks and show that our approach yields improved solutions compared to classical approximation methods in the literature, such as the Method of Successive Averages and the gap-based approach.

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha m)\right|^2 \gg y\log y$$ whenever $y \ll x^{1/3-\varepsilon}.$ This answers a question of Ramaré by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} \Lambda(m) \mathrm{e}(\alpha m)\right| \gg x^{1/6 - \varepsilon}.$$

We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata.

For an integer $m\geq 2$, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb{Z}/2 \mathbb{Z}\times\mathbb{Z}/2^m \mathbb{Z}$, as the Galois group of the maximal unramified $2$-extension (resp. pro-$2$-extension) of certain number fields of $2$-power degree (resp. cyclotomic $\mathbb Z_2$-extensions). Furthermore, we show up some new techniques for studying Greenberg's conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields $F=\mathbb{Q}(\sqrt{\eta q rs})$, the real biquadratic fields $K=\mathbb{Q}(\sqrt{\eta q},\sqrt{rs})$, with $\eta\in\{1,2\}$, and the Fröhlich multiquadratic fields of the form $\mathbb F=\mathbb{Q}(\sqrt{q }, \sqrt {r}, \sqrt{s})$, where $q$, $r$ and $s$ are odd primes numbers.

Controlling instability in plasma is one of the central challenges in fusion energy research. Among the various sources of instability, kinetic effects play a significant role. In this work, we aim to suppress the instability induced by kinetic effects by designing an external electric field. However, rather than directly solving the full kinetic Vlasov-Poisson system, we focus on a reduced-order model, specifically the moment-based system, to capture the underlying dynamics. This approach is motivated by the desire to reduce the computational cost associated with repeatedly solving the high-dimensional kinetic equations during the optimization of the electric field. Additionally, moment-based data is more readily accessible in practice, making a moment-based control framework more adaptable to data-driven scenarios. We investigate the effectiveness of moment-based control both analytically and numerically, by comparing it to control based on the full kinetic model.

Charles L. Dodgson, also known as Lewis Carroll, in his book "Pillow problems" from 1893 asked for the likelihood of a random triangle to be obtuse. Clearly, the answer to Dodgson's question depends strongly on the assumed random distribution. In this article, we show nevertheless that there are certain fundamental limitations imposed by the geometry of Euclidean space. Specifically, we give universal lower bounds for how improbable obtuse triangles can be, if drawn from a distribution in $\mathbb{R}^d$. We prove that planar obtuse triangles cannot be less likely than 1/3, and construct a distribution for which the probability is 4/9. Analogous results are provided in three and higher dimensions, where obtuse triangles can be increasingly less likely. Sharpness of the lower bounds are left as open problems.

This paper demonstrates the stability of the global regularity for a class of pseudo-differential operators under lower-order perturbations. We establish that if an operator has a globally hypoelliptic symbol, its global regularity (in the sense of Schwartz functions and tempered distributions) is preserved when perturbed by operators of sufficiently lower order. This result applies in particular to operators within the Shubin and SG classes. Furthermore, we discuss why this stability result does not hold in the standard Hörmander classes.

Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods are classical approaches for the numerical integration of functions $f$ over $[0,1]^d$. While QMC methods can achieve faster convergence rates than MC in moderate dimensions, their tractability in high dimensions typically relies on additional structure -- such as low effective dimension or carefully chosen coordinate weights -- since worst-case error bounds grow prohibitively large as $d$ increases. In this work, we study the construction of high-dimensional QMC point sets via combinatorial discrepancy, extending the recent QMC method of Bansal and Jiang. We establish error bounds for these constructions in weighted function spaces, and for functions with low effective dimension in both the superposition and truncation sense. We also present numerical experiments to empirically assess the performance of these constructions.

Two approaches to the Painlevé I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between both formalisms is built which gives the identification of these setups. In particular, this provides new expressions for the Lax matrices and Hamiltonians.

We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under which the convex hull of the feasible region of the linearized problem is SOC-representable. While the proposed formulation may be of exponential size in general, we identify additional structural conditions that guarantee the existence of a polynomial-size SOC-representable formulation, which can be constructed in polynomial time. Under these conditions, the optimal value of the nonconvex quadratic program coincides with that of a polynomial-size second-order cone program. Our results serve as a starting point for bridging the gap between the Boolean quadric polytope of sparse problems and its continuous counterpart.

We study the dissipativity of linear infinite-dimensional systems with respect to a prescribed quadratic supply rate functional. We characterize this property via an operator inequality that also yields the system's dissipation rate. We also derive implications for the linear-quadratic optimal control problem on the nonnegative real half-line.

In this paper, we study the centraliser of $\mathfrak{osp}(1|2)$, denoted the total angular momentum algebra (TAMA), in the Weyl Clifford algebra. The TAMA extends the angular momentum algebra (AMA), which arises as the centraliser of $\mathfrak{sl}(2)$ and admits a diagrammatic presentation via the crossing relation described by Feigin and Hakobyan. Using Young symmetrisers we construct an analogue relation for the even subalgebra of the TAMA. We prove that for rank $4$ and $5$ these relations generate a presentation for the even subalgebra of the TAMA.

Rowland found a matrix product formula for generating functions counting binomial coefficients by their $p$-adic valuations. A natural generalization of binomial coefficients was introduced by Knuth and Wilf defined by a sequence $C$. We obtain analogous matrix product formulas counting these $C$-nomial coefficients when $C$ is a strong divisibility sequence. Surprisingly, the matrices are universal in the sense that they are independent of $C$. We further extend this product to $C$-multinomial coefficients.

Let $\mathcal{F}(\textbf{k},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\textbf{k}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,\pi)$ weighted by powers of central $L$-values $L(1/2,\pi)^r$, where $\pi$ runs through $\mathcal{F}(\textbf{k},\mathfrak{q})$. For $r=1,2,3$, we show that the resulting distributions $W_r$ match with predictions from Random Matrix Theory. For general $r \geq 1$, we also formulate a conjectural formula for $W_r$ based on the ``recipe'' method.

This work begins the process of using the decomposition of the diagonal as a tool for studying the rationality of invariant fields of finite groups $G$. Our ground field must be characteristic 0 because of the use we make of Bertini theorems. The steps we take are, first, defining and studying an "open" version of Chow zero. Second, we use this to translate our study to that of a Chow group of $G$ Galois extensions. We prove a "Sylow" property and thereby yield a connection between the invariants of $G$ and that of its Sylow subgroups. In particular, we show that if $G$ is a finite group with $p$ Sylow subgroup $P$, $V$ is a faithful $G$ module, and $F(V)^P$ has nontrivial unramified cohomology, then $F(V)^G$ is not retract rational. Finally, we prove Sylow type theorems for decomposition of the diagonal and the centers of generic division algebras.

Let $M$ be a closed orientable hypersurface of dimension $n$, with nonwhere vanishing mean curvature $H$, immersed into a Riemannian Spin$^c$ manifold $\mathcal Z$ carrying a parallel spinor field. The first eigenvalue $\lambda_1(\not\hspace{-0.1cm}D)$ (with the least absolute value) of the induced Dirac operator $\not\hspace{-0.1cm}D$ of $M$ satisfies the Spin$^c$ Bär inequality \begin{eqnarray*} \lambda_1^2 (\not\hspace{-0.1cm}D) \leq \frac{n^2}{4 \ \mathrm{vol}(M)}\int_M H^2 dV, \end{eqnarray*} where $\mathrm{vol}(M)$ is the volume of $M$ and $dV$ is the volume form of the manifold $M$. In this paper, we classify hypersurfaces $M$ that satisfy the equality case in the Spin$^c$ Bär inequality when $\mathcal Z = (0,+\infty) \times P$ is the cone over a Riemannian Spin$^c$ manifold $P$ carrying a real Killing spinor, under two conditions: one being a Ricci condition on $\mathcal Z$, and the second one the curvature of the auxiliary line bundle associated with the Spin$^c$ structure on $\mathcal Z$. More precisely, we prove that $M$ are the slices $\{s\} \times P$, where $s \in (0,+\infty)$. In the special case, when $\mathcal Z=\mathbb R^{n+1}$, i.e., the cone over the sphere, which is a Spin manifold with a parallel spinor, the classification result was previously obtained by Hijazi and Montiel.

A three-dimensional convex body is said to have Rupert's property if its copy can be passed through a straight hole inside that body. In this work we construct a polyhedron which is provably not Rupert, thus we disprove a conjecture from 2017. We also find a polyhedron that is Rupert but not locally Rupert.

In this work, we extend the sensitivity-based rank condition (SERC) test for local observability to another class of systems, namely smooth and nonsmooth differential-algebraic equation (DAE) systems of index-1. The newly introduced test for DAEs, which we call the lexicographic SERC (L-SERC) observability test, utilizes the theory of lexicographic differentiation to compute sensitivity information. Moreover, the newly introduced L-SERC observability test is useful in the context of partial observability as it can judge which states are observable and which are not. Additionally, we introduce a novel sensitivity-based extended Kalman filter (S-EKF) algorithm for state estimation, applicable to both smooth and nonsmooth DAE systems. Finally, we apply the newly developed S-EKF to estimate the states of a wind turbine power system model.

A classical result states that if $f(z)$ is a polynomial of degree at most $n$ with nonnegative coefficients, then $f(z)$ has no zeros in the sector $|\arg(z)| < \frac{\pi}{n}$ of the complex plane, and the bound $\frac{\pi}{n}$ is tight. Motivated by the Shapiro--Shapiro conjecture and related problems in real Schubert calculus, we generalize this result to Wronskians of polynomials. Namely, let $f_1(z), \dots, f_k(z)$ be linearly independent polynomials of degree at most $n$ whose coefficient matrix has all nonnegative $k\times k$ minors (that is, the polynomials span an element of the totally nonnegative Grassmannian in the sense of Lusztig and Postnikov). We show that the Wronskian polynomial $\operatorname{Wr}(f_1, \dots, f_k)$ has no complex zeros in the sector $|\arg(z)| < \frac{\pi}{n}$ (independent of $k$), and the bound $\frac{\pi}{n}$ is tight. Our proof uses classical results of Gantmakher and Krein (1950) and Obreschkoff (1923) on sign variation.

We call an oriented odd cycle alternating if it has exactly one vertex whose in-degree and out-degree are both positive. In this paper, we investigate whether certain graphs admit an orientation that avoids alternating odd cycles as subgraphs, or one in which all their shortest odd cycles become alternating. Our focus is on topologically $\chi$-chromatic graphs, that is, graphs for which the topological method yields a sharp lower bound on the chromatic number. We present results for several graph families, including Kneser graphs, Schrijver graphs, and generalized Mycielski graphs.

In this work, we establish global gradient estimates to solutions of quasilinear elliptic models in non-divergence form with general degeneracy law and a Hamiltonian term, given by $$ -\Psi(x, |\nabla u|)\Delta_p^{\mathrm{N}}u(x)+\mathscr{H}(x,\nabla u)=f(x) \quad \mathrm{in} \quad \Omega, \quad \mathrm{for} \,\,\,1<p< \infty, $$ under suitable assumptions on the data of the problem. Particularly, our results are relevant for a class of quasi-linear models with Hamiltonian terms. Additionally, we address non-degeneracy estimates for such solutions and present a couple of applications.

Unlike conventional imaging modalities, such as magnetic resonance imaging, which are often well described by a linear regression framework, coherent imaging systems follow a significantly more complex model. In these systems, the task is to recover the unknown signal or image $\mathbf{x}_o \in \mathbb{R}^n$ from observations $\mathbf{y}_1, \ldots, \mathbf{y}_L \in \mathbb{R}^m$ of the form \[ \mathbf{y}_l = A_l X_o \mathbf{w}_l + \mathbf{z}_l, \quad l = 1, \ldots, L, \] where $X_o = \mathrm{diag}(\mathbf{x}_o)$ is an $n \times n$ diagonal matrix, $\mathbf{w}_1, \ldots, \mathbf{w}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,I_n)$ represent speckle noise, and $\mathbf{z}_1, \ldots, \mathbf{z}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,\sigma_z^2 I_m)$ denote additive noise. The matrices $A_1, \ldots, A_L$ are known forward operators determined by the imaging system. Our goal is to characterize the minimax risk of estimating $\mathbf{x}_o$, in high-dimensional settings where $m$ could be even less than $n$. Motivated by insights from sparse regression, we note that the structure of $\mathbf{x}_o$ plays a central role in the estimation error. Here, we adopt a general notion of structure better suited to coherent imaging: we assume that $\mathbf{x}_o$ lies in a signal class $\mathcal{C}_k$ whose Minkowski dimension is bounded by $k \ll n$. We show that, when $A_1,\ldots,A_L$ are independent $m \times n$ Gaussian matrices, the minimax mean squared error (MSE) scales as \[ \frac{\max\{\sigma_z^4,\, m^2,\, n^2\}\, k \log n}{m^2 n L}. \]

This article explores Ricci-Yamabe solitons on a specific class of 4-dimensional Walker manifolds. Walker manifolds, characterized by the existence of a parallel null distribution, find applications in general relativity and are fundamental objects of geometric study. We consider a particular pseudo-Riemannian metric, which depends on the smooth functions $f_1$, $f_2$, $f_3$. The main objective is to determine the conditions under which this manifold admits a Ricci-Yamabe soliton. We will explicitly calculate the components of the Ricci tensor, the scalar curvature, and the components of the Hessian Perelman potential. Solving the resulting system of partial differential equations, we will identify the constraints on the functions f1, f2, f3 and the vector field X for the existence of such solitons. Specific examples and their geometric properties will also be discussed.

Sign Gradient Descent (SignGD) is a simple yet robust optimization method, widely used in machine learning for its resilience to gradient noise and compatibility with low-precision computations. While its empirical performance is well established, its theoretical understanding remains limited. In this work, we revisit SignGD from a continuous-time perspective, showing that it arises as an Euler discretization of a norm-constrained gradient flow. This viewpoint reveals a trust-region interpretation and connects SignGD to a broader class of methods defined by different norm constraints, such as normalized gradient descent and greedy coordinate descent. We further study the discontinuous nature of the underlying dynamics using Filippov's differential inclusion framework, which allows us to derive new algorithmic variants, such as the convex-combination sliding update for the $\ell_1$-constrained flow, that faithfully approximate Filippov solutions even at discontinuity points. While we do not provide convergence guarantees for these variants, we demonstrate that they preserve descent properties and perform well empirically. We also introduce an accelerated version of SignGD based on a momentum-augmented discretization of the sign-gradient flow, and show its effectiveness in practice. Finally, we establish provable convergence guarantees for standard SignGD in the setting of strongly convex optimization. Our results provide new geometric, algorithmic, and analytical insights into SignGD and its norm-constrained extensions.

For each integer $N>1$, we define a notion of complexity $c(N)$, which is a nonnegative integer and carries some information on the shape of the Ford fundamental domain of the congruence subgroup $\Gamma_0(N)$. The property that $c(N)=0$ first appeared as a technical assumption in a paper by Anke Pohl, which is closely related to a conjecture of Don Zagier on the "reduction theory" of $\Gamma_0(N)$. In this paper, we give a complete classification of integers $N>1$ with $c(N)=0$, and we also show that $c(N)$ goes to infinity if both the number of distinct prime factors of $N$ and the smallest prime factor of $N$ go to infinity.

It has been well known since Gauss that the principality of an ideal in a real quadratic field $K$ is equivalent to the solvability of a certain generalized Pell equations. In this paper, we combine this classical result with Srinivasan's conditions for the existence of well-rounded ideals in $K$ to obtain necessary and sufficient criteria for a real quadratic field to have principal well-rounded (PWR) ideals. Using these criteria, we prove that there are infinitely many real quadratic fields that have PWR ideals. Moreover, these ideals are pairwise non-similar. We then construct new algorithms that produce these PWR ideals, especially when the field discriminant is large. Our algorithms run in sub-exponential time theoretically; however, they are very fast in practice by employing some commonly used probabilistic algorithms for testing squarefreeness. Finally, we briefly consider criteria for the existence of prime PWR ideals and show that there are infinitely many real quadratic fields that have prime PWR ideals.

Have you ever played or watched a game of pool? If so, you have already seen a billiard system in action. In mathematics and physics, a billiard system describes a ball that moves in straight lines and bounces off walls. Despite these simple rules, billiard systems can produce remarkably rich behaviors: some table shapes generate regular, periodic patterns, while others give rise to complete chaos. Scientists also study what happens when we shrink the ball down to the size of an electron to a world where quantum effects take over and the familiar reflection rules no longer apply. In this article, we discuss billiard systems in their many forms and show how such a simple setup can reveal fundamental insights into the behavior of nature at both classical and quantum scales.

We prove that for any non-trivial knot K, infinitely many r-surgeries K(r) along K have a unique surgery description along a knot. Moreover, we show that for any hyperbolic L-space knot K and infinitely many integer slopes n, the manifold K(n) has a unique surgery description. Here we say a 3-manifold M has a unique surgery description along a knot in S^3 if there is a unique pair (K,r) of a knot K and a slope r such that M is orientation-preservingly diffeomorphic to K(r). This generalises the notion of characterising slopes. Conversely, we provide new families of manifolds with several distinct surgery descriptions along knots. More precisely, we construct for every non-zero integer m a knot K_m such that for any integer n, the manifold K_m(m+1/n) can also be obtained by surgery on another knot.

Chemical reaction networks in living cells maintain precise control over thousands of metabolites despite operating far from equilibrium under constant perturbations. While mass action kinetics accurately describe the underlying dynamics, the resulting nonlinear differential equations are difficult to analyze and control, particularly for large networks. We propose a simplified model where reaction quotients (the ratios that measure how far reactions are from equilibrium) evolve exponentially toward their equilibrium values when viewed on a logarithmic scale. This principle leads to linear dynamics in log-space, providing several key advantages: analytical solutions exist for arbitrary network topologies, thermodynamic constraints are automatically satisfied through the relationship between reaction quotients and Gibbs free energy, conservation laws decouple from reaction quotient dynamics simplifying both analysis and control design, and external energy sources couple linearly to the dynamics, unifying diverse biological regulatory mechanisms.

In the early 2000s, Frigerio, Martelli, and Petronio studied $3$-manifolds of smallest combinatorial complexity that admit hyperbolic structures. As part of this work they defined and studied the class $M_{g,k}$ of smallest complexity manifolds having $k$ torus cusps and connected totally geodesic boundary a surface of genus $g$. In this paper, we provide a complete classification of the manifolds in $M_{k,k}$ and $M_{k+1,k}$, which are the cases when the genus $g$ is as small as possible. In addition to classifying manifolds in $M_{k,k}$, $M_{k+1,k}$, we describe their isometry groups as well as a relationship between these two sets via Dehn filling on small slopes. Finally, we give a description of important commensurability invariants of the manifolds in $M_{k,k}$.

We prove a linear--independence principle for finite families of translates in irreducible, square--integrable modulo the center (SI/Z) representations of step--two nilpotent Lie groups with one--dimensional center. Fix a one--parameter subgroup $\exp(\mathbb R X)$ and a noncentral element $Y$ (not assumed to commute with $X$). For every nonzero smooth vector $f$, any finite collection $\{\pi(\exp(x_k X))f\}_{k=1}^m$ of translates along $\exp(\mathbb R X)$, together with the single translate $\pi(\exp Y)f$, is linearly independent. The proof combines Schwartz regularity of matrix coefficients in the noncentral variables with an ergodic log--product estimate on a compact torus generated by commutator phases. Specializing to the Schrödinger representation of the Heisenberg group $\mathbb H_d$, we obtain an HRT--type statement for $(m,1)$ configurations (on--line direction with zero $X_{k_0}$--component and rogue $\exp X_{k_0}$) for all nonzero Schwartz windows, without spacing

Optimization-based controllers often lack regularity guarantees, such as Lipschitz continuity, when multiple constraints are present. When used to control a dynamical system, these conditions are essential to ensure the existence and uniqueness of the system's trajectory. Here we propose a general method to convert a Quadratic Program (QP) into a Second-Order Cone Problem (SOCP), which is shown to be Lipschitz continuous. Key features of our approach are that (i) the regularity of the resulting formulation does not depend on the structural properties of the constraints, such as the linear independence of their gradients; and (ii) it admits a closed-form solution, which is not available for general QPs with multiple constraints, enabling faster computation. We support our method with rigorous analysis and examples.

Rudyak's conjecture states that for any degree one map $f:M\to N$ between oriented closed manifolds there is the inequality $\cat (M)\ge \cat(N)$ for the Lusternik-Shnirelmann category. We prove the Rudyak's conjecture for $ n$-dimensional simply connected manifolds for $n\le 8$.

Mathematical models of complex social systems can enrich social scientific theory, inform interventions, and shape policy. From voting behavior to economic inequality and urban development, such models influence decisions that affect millions of lives. Thus, it is especially important to formulate and present them with transparency, reproducibility, and humility. Modeling in social domains, however, is often uniquely challenging. Unlike in physics or engineering, researchers often lack controlled experiments or abundant, clean data. Observational data is sparse, noisy, partial, and missing in systematic ways. In such an environment, how can we build models that can inform science and decision-making in transparent and responsible ways?

We identify two nontrivial errors in the proof of the main results of the work \emph{generalized baby Mandelbrot sets adorned with halos in families of rational maps} by Jang, So, and Marotta. We correct these errors, showing that the main results remain true.

This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.

Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S \subseteq V(G)$, denoted $\Lambda^k(S)$, is the set of vertices that are adjacent to at least $k$ vertices in $S$. The $k$-th binding number $\beta^k(G)$ is defined as the minimum ratio $|\Lambda^k(S)|/|S|$ over all subsets $S \subseteq V(G)$ with $|S| \ge k$ and $\Lambda^k(S) \ne V(G)$. This parameter generalizes the classical binding number introduced by Woodall. Andersen showed that the condition $\beta^1(G) \ge 1$ does not guarantee the existence of a $1$-factor in $G$, while Barát et al. proved that $\beta^2(G) \ge 1$ suffices for the existence of a $2$-factor. In this paper, we extend this result to general $k \ge 2$ by showing that any graph $G$ with even $k|V(G)|$ and $\beta^k(G) \ge 1$ contains a $k$-factor. Moreover, if $G$ is additionally a split graph of even order, then it admits a $(k+1)$-factor. We also prove that any graph $G$ with $\beta^k(G) \ge 1$ contains at least $k-1$ disjoint perfect or near-perfect matchings. Finally, for any bipartite graph $G$ with bipartition $(X, Y)$, we introduce an analogue of the $k$-th binding number and show that, under the condition $\beta^k(G, X) \ge 1$, the graph admits $k$ disjoint matchings, each covering $X$.

We prove group existence and structure theorems in a general setting of tame topological theories. More precisely, we identify a linear/non-linear dividing line -- called topological 1-basedness -- among the class of t-minimal theories with the independent neighborhood property. This is a wide class including all visceral theories, as well as all dense weakly o-minimal and C-minimal theories (even those where exchange fails). Now assume $\mathcal M$ is highly saturated and t-minimal with the independent neighborhood property. We show that if $\mathcal M$ is non-trivial and topologically 1-based, it admits a type-definable abelian group $(G,+)$ with $G$ an open subset of $M$. Moreover, we can ensure that $G$ is a topological group with the subspace topology inherited from $M$; and in this case, we show that the induced structure on $G$ satisfies an appropriate topological analog of the Hrushovski-Pillay classification of 1-based stable groups.

We study Borel polychromatic colorings of grid graphs arising from free Borel actions of $\mathbb{Z}^d$. A polychromatic coloring is one in which every unit $d$-dimensional cube sees all available colors. In the classical setting, every grid admits a $2^d$-polychromatic coloring, while in the Borel setting this fails. Our main result shows that every free $\mathbb{Z}^d$-action admits a Borel $(2^d-1)$-polychromatic coloring. This result is sharp: any action where the generators act ergodically does not admit a Borel $2^d$-polychromatic coloring. We conclude with open directions for extending the theory beyond cube tilings and for exploring the dependence of Borel polychromatic numbers on the underlying action.

While single-purchase choice models have been widely studied in assortment optimization, customers in modern retail and e-commerce environments often purchase multiple items across distinct product categories, exhibiting both substitution and complementarity. We consider the cross-category assortment optimization problem where retailers jointly determine assortments across categories to maximize expected revenue. Most prior work on the topic either overlooks complementarity or proposes models that lead to intractable optimization problems, despite being based on the multinomial logit (MNL) choice model. We propose a sequential multi-purchase choice model for cross-category choice that incorporates complementarity through a Markovian transition structure across categories, while allowing general Random Utility Maximization (RUM)-based choice models to capture the within-category substitution. We develop an Expectation-Maximization algorithm for estimation, and a polynomial-time algorithm for unconstrained assortment optimization that yields the optimal solution when the within-category substitution follows a Markov chain choice model. Furthermore, we introduce an empirical metric to quantify complementarity strength across product categories and conduct extensive numerical experiments on both synthetic data and a large-scale transaction-level dataset from a major US grocery store. Our model yields improvements in predictive accuracy, model fit, and expected revenue in setting with complementarity, and it reveals intuitive market structures such as brand-loyal cross-category purchasing. Overall, we believe that our model provides a theoretically-grounded and practical framework for modeling complementarity and making better cross-category assortment decisions.

Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along Möbius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions.

This paper investigates the nonparametric estimation of a circular regression function in an errors-in-variables framework. Two settings are studied, depending on whether the covariates are circular or linear. Adaptive estimators are constructed and their theoretical performance is assessed through convergence rates over Sobolev and Hölder smoothness classes. Numerical experiments on simulated and real datasets illustrate the practical relevance of the methodology.

Extremely large-scale reconfigurable intelligent surface (XL-RIS) can effectively overcome severe fading and provide higher communication performance. However, current research on XL-RIS overlooks the discrete phase-shift characteristics of RIS in practical systems, which will result in significant performance this http URL this paper, we investigate near-field communication schemes assisted by XL-RIS with discrete phase this http URL, we propose a hierarchical beam training method to obtain the user channel state information (CSI), and develop the jointly optimized codebook construction (JOCC) method and separately optimized codebook construction (SOCC) method for base station (BS) precoding and XL-RIS phase shifts, respectively. With JOCC, the most superior beam training performance can be this http URL SOCC, higher performance than the single-antenna BS codebook can be obtained at a similar this http URL, we propose a flexible multiuser interference management (IM) method that is simple to solve. The IM method uses adaptive gain matrix approximation to take into account user fairness and can be solved in closed-form iterations. In addition, we extend the proposed method to a hybrid precoding design. Simulation results demonstrate that the proposed multi-resolution codebook construction method can obtain more accurate beam patterns and user CSI, and the proposed IM method obtains superior performance over the benchmark methods.

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where $H(\lambda_1,\ldots,\lambda_k)$ is an explicit constant that is best possible. The proof combines several ingredients, including a lower bound estimate on the measure of sums of linear transformations of compact sets in $\mathbb{R}^d$, a variant of Freiman's theorem tuned specifically to sums of dilates and the analysis of what we call lattice density, which succinctly captures how a subset of $\mathbb{Z}^d$ is arranged relative to a given flag of lattices. As an application, we revisit the study of sums of linear transformations of finite sets, in particular proving an asymptotically best possible lower bound for sums of two linear transformations.

We show that the star graph defined as the Cayley graph of ${\mathfrak S}_{n+1}$ generated by the star transpositions is an ${\mathfrak S}_n$-cover of the complete graph $K_{n+1}$, which is known to have fine spectral properties. In the case $n = 3$, the star graph also has fine geometric properties: it embeds into the honeycomb lattice and has a spectrum computable via both representation theory and an explicit Fourier formula. Intermediate covers correspond to the cube and truncated tetrahedron, offering a new interpretation of their integral spectra.

We present an algorithm to determine the Galois group of an irreducible monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree at most five. Following work of Conrad, Dummit, and Stauduhar this comes down to answering two questions: Is a given integer a square? and Does a given polynomial have an integral root? Since these are both easily addressed with a calculator, our algorithm amounts to Galois theory by calculator. For example, we have an implementation at this http URL. In an appendix we present a simplified version of our algorithm, suitable for a handheld calculator, in case $f(x) = x^n + px + q$.

We study half-line discrete Schrödinger operators and their rank-one perturbations. We establish certain continuity and stability properties of the Fourier transform of the associated spectral measures. Using these results, we construct a sparse potential whose essential spectrum contains an open interval, and show that for every rank-one perturbation the corresponding spectral measure is non-Rajchman. This resolves a question posed in [24].

Let $\mathscr{M}$ be a conditional oriented matroid. We define a graded algebra $\widehat{\mathscr{VG}}_\mathscr{M}$ with vector space dimension given by the number of covectors in $\mathscr{M}$ which admits a distinguished filtration indexed by the poset $\mathscr{L}(\mathscr{M})$ of flats of $\mathscr{M}$. The subquotients of this filtration are isomorphic to graded Varchenko-Gelfand rings of contractions of $\mathscr{M}$, so we call $\widehat{\mathscr{VG}}_\mathscr{M}$ the {\em graded big Varchenko-Gelfand ring of $\mathscr{M}$.} We describe a no broken circuit type basis of $\widehat{\mathscr{VG}}_\mathscr{M}$ and study its equivariant structure under the action of $\mathrm{Aut}(\mathscr{M})$. Our key technique is the orbit harmonics deformation which encodes $\widehat{\mathscr{VG}}_\mathscr{M}$ (as well as the classical Varchenko-Gelfand ring) in terms of a locus of points.

We provide a combinatorial description of morphisms in the coherent sheaf category ${\rm coh}\mbox{-}\mathbb{X}(p,q)$ over weighted projective line of type $(p,q)$ via a marked annulus. This leads to a geometric realization of exceptional sequences in ${\rm coh}\mbox{-}\mathbb{X}(p,q)$. As applications, we present a classification of complete exceptional sequences, an effective method for enlarging exceptional sequences, and a new proof of the transitivity of the braid group action on complete exceptional sequences. Besides, we offer a combinatorial description of tilting bundles via lattice paths and count the number of tilting sheaves in ${\rm coh}\mbox{-}\mathbb{X}(p,q)$, up to the Auslander-Reiten translation.

Self-efficacy is a significant construct in education due to its predictive relationship with achievement. Existing measures of assessment-related self-efficacy concentrate on students' beliefs about content-specific tasks but omit beliefs around assessment-taking. This research aimed to develop and test the Measure of Assessment Self-Efficacy (MASE), designed to assess two types of efficacy beliefs related to assessment (i.e., 'comprehension and execution' and 'emotional regulation') in two scenarios (i.e., a low-stakes online quiz and a high-stakes final exam). Results from confirmatory factor analysis in Study 1 (N = 301) supported the hypothesised two-factor measurement models for both assessment scenarios. In Study 2, results from MGCFA (N = 277) confirmed these models were invariant over time and provided evidence for the scales' validity. Study 3 demonstrated the exam-related MASE was invariant across cohorts of students (Ns = 277; 329). Potential uses of the developed scales in educational research are discussed.

This study addresses the rebalancing of empty modular transit pods between scheduled service trips in fixed-route bus systems. A two-stage hierarchical optimization framework is proposed. The first stage determines the minimum fleet size and initial vehicle assignments by solving a maximum matching problem on a bipartite graph, using a GPU-accelerated push-relabel algorithm. The second stage formulates detailed routing as a series of minimum-cost flow problems on time-space networks. To manage memory usage in large instances, a capped-interval heuristic limits the size of each network by dividing long scheduling intervals into subintervals. Computational experiments on the Manhattan bus network show that the proposed method achieves performance comparable to the full-scale time-space network formulation in terms of objective value, while enabling the solution of instances that are otherwise intractable due to memory limitations.

This article celebrates the 40th anniversary of Dr. Ivan Dmitrievich Remizov, a mathematician who made a number of important contributions to the theory of one-parameter operator semigroups -- a branch of functional analysis which has applications to differential equations, mathematical physics, random processes, control theory, and quantum mechanics. Born on December 7, 1984, Dr. Remizov obtained his Ph.D. in 2018 from Moscow State University and has made substantial contributions, particularly in the study of Chernoff approximations of one-parameter semigroups of operators. This article reviews his academic background, research achievements, and his impact on the mathematical community.

We prove that every separable Banach space admits an isometric embedding into $\linf$ intersecting $\cseq$ only at the origin. Moreover, we prove that any finite-or countable-dimensional, or more generally separable, subspace of $(\ell_\infty \setminus \cseq)\cup \{0\}$ can be extended to a subspace containing an isometric copy of an arbitrary separable Banach space, while still avoiding $\cseq$.

In this paper, we first endow the space of decorated planar rooted forests with a coproduct that equips it with the structure of a bialgebra and further a Moerdijk Hopf algebra. We also present a combinatorial description of this coproduct, and further give an explicit formulation of its dual coproducts through the newly defined notion of forest-representable matrices. By viewing the Moerdijk Hopf algebra within the framework of operated algebras, we introduce the notion of a multiple cocycle Hopf algebra, incorporating a symmetric Hochschild 1-cocycle condition. We then show that the antipode of this Hopf algebra is a Rota-Baxter operator on Moerdijk Hopf algebras. Furthermore, we investigate the universal properties of cocycle Hopf algebras. As an application, we construct the initial object in the category of free cocycle Hopf algebras on undecorated planar rooted forests, which coincides with the well-known Moerdijk Hopf algebra.

Let G be an automorphism group of a nontrivial t-(k^2,k,\lambda) design. In this paper, we prove that if G is block-transitive, then the socle of G cannot be a finite simple exceptional group of Lie type.

We study cancellation in sums of Hecke eigenvalues over irreducible quadratic polynomials over short intervals. In particular, we look at an average over bases of Hecke forms of weight $k$ in the range $\vert k-K\vert<K^\theta$ where $1/3<\theta<1$. We see that when averaged over this family such sums admit square root cancellation. The key new arithmetic input for such a result is a bound on sums of Kloosterman sums over irreducible quadratic polynomials. Then using work of Nelson, we relate such sums to the mass equidistribution conjecture for modular forms on compact arithmetic surfaces, and we show that almost all forms satisfy the mass equidistribution conjecture. Furthermore, such forms will satisfy the conjecture with an effective convergence rate of $k^{-\delta}$ for any $\delta<1/2$.

Jim Pitman's (1999) celebrated coagulation-fragmentation duality for the PD($\alpha$,$\theta$) family of laws of Pitman and Marc Yor (1997) has resisted generalization beyond its canonical setting. We resolve this by introducing a novel, four-part coupled process built upon the Poisson Hierarchical Indian Buffet Process (PHIBP), a framework we recently developed for modeling microbiome species sampling across multiple ($J \ge 1$) groups. This provides a tractable generalization of the duality to processes driven by arbitrary subordinators, yielding explicit laws for both single-group and the previously uncharacterised multi-group structured partitions. The static, fixed-time partitions are revealed to be a single projection of an inherently dynamic, four-component coupled system. This new construction simultaneously defines: (i) the fine-grained partition, (ii) its coagulation operator, (iii) a forward-in-time system of coupled, time-homogeneous fragmentation processes in the sense of Jean Bertoin (2006), and (iv) a dual, backward-in-time structured coalescent. All four components are governed by the same underlying compositional structure, which yields their exact compound Poisson representations and, for the coalescent, drives simultaneous, across-group merger events. The chief result is a general coagulation-fragmentation duality that generalizes Pitman's duality in two fundamental directions. First, it holds for arbitrary driving subordinators, defining a vast new family of single-group dualities. Second, it operates in the previously uncharacterised multi-group ($J$-group) setting. This work establishes a unified and tractable framework for a rich class of partition-valued dynamics, revealing structural connections between the theories of fragmentation and coalescence.

In this paper, hybrid and block-pulse functions are used to approximate the solution of a class of Fredholm integro-differential equations that was first studied by Hemeda. By employing suitable approximations, the equation has been converted into a system of algebraic equations that can be solved with classical methods. Finally, the method is explained with illustrative examples and results are compared to the results obtained by Hemeda's method to show the usefulness and efficiency of the block-pulse and hybrid functions approach.

The $g$-fan $\Sigma(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $\Sigma(A)$ is convex, we call $A$ $g$-convex. In this case, the $g$-polytope ${\rm P}(A)$ of $A$ is a reflexive polytope. Thus, in each dimension, there are only finitely many isomorphism classes of fans that can be realized as $g$-fans of $g$-convex algebras. An important problem is to classify such fans for a fixed dimension $d$. In this paper, we give a complete answer for the case $d=3$: we prove that there are precisely 61 convex $g$-fans of dimension 3 up to isomorphism. Our method is based on the decomposition of fans into the $2^3$ orthants in the real Grothendieck group of $A$, together with a detailed analysis of possible sequences of $g$-vectors arising from iterated mutations.

This letter presents a closed-form characterization of the joint distribution of first arrival time (FAT) and first arrival position (FAP) in diffusion-based molecular communication (MC) systems with drift. Prior studies have investigated FAT modeling via inverse Gaussian distributions [1] and applied FAT statistics for parameter estimation and synchronization tasks [2], [3], while more recent work has characterized FAP for spatial channel analysis [4]. In contrast, we derive an explicit joint probability density function (PDF) under constant drift and isotropic diffusion in arbitrary spatial dimensions. Our result reveals a nontrivial coupling between arrival time and lateral position, generalizing known inverse Gaussian models. We further compute the Fisher information matrix (FIM) with respect to key channel parameters, showing that the joint observation enables estimation of lateral drift and improves sensitivity to the diffusion coefficient -- capabilities not achievable with time-only or position-only models. This joint framework enhances the modeling and inference capabilities for molecular communication channels where spatial randomness itself carries non-negligible information.

We develop an information-theoretic framework for bounding the expected supremum and tail probabilities of stochastic processes, offering a simpler and sharper alternative to classical chaining and slicing arguments for generalization bounds. The key idea is a lifting argument that produces information-theoretic analogues of empirical process bounds, such as Dudley's entropy integral. The lifting introduces symmetry, yielding sharp bounds even when the classical Dudley integral is loose. As a by-product, we obtain a concise proof of the majorizing measure theorem, providing explicit constants. The information-theoretic approach provides a soft version of classical localized complexity bounds in generalization theory, but is more concise and does not require the slicing argument. We apply this approach to empirical risk minimization over Sobolev ellipsoids and weak $\ell_q$ balls, obtaining sharper convergence rates or extensions to settings not covered by classical methods.

This paper investigates the existence and properties of spherical $5$-designs of minimal type. We focus on two cases: tight spherical $5$-designs and antipodal spherical $4$-distance $5$-designs. We prove that a tight spherical $5$-design is of minimal type if and only if it possesses a specific $Q$-polynomial coherent configuration structure. For tight spherical $5$-designs in $\mathbb{R}^d$ of minimal type, we demonstrate that half of the derived code forms an equiangular tight frames (ETF) with parameters $(d-1, \frac{(d-1)(d+1)}{3})$. This provides a sufficient condition for constructing such ETFs from maximal ETFs with parameters $(d, \frac{d(d+1)}{2})$. Moreover, we establish that tight spherical $5$-designs of minimal type cannot exist if the dimension $d$ satisfies a certain arithmetic condition, which holds for infinitely many values of $d$, including $d=119$ and $527$. For antipodal spherical $4$-distance $5$-designs, we utilize valency theory to derive necessary conditions for certain special types of antipodal spherical $4$-distance $5$-designs to be of minimal type.

We study the twisted de Rham complex associated with a holomorphic function on a Kähler manifold whose critical point set is compact. We prove the $E_1$-degeneration of the Hodge-to-de Rham spectral sequence. It is a generalization of Barannikov-Kontsevich Theorem.

In this article, we establish the $L^p$-Heisenberg-Pauli-Weyl uncertainty inequalities on the Laguerre hypergroup $\mathbb{K}$, the natural setting for radial analysis on the Heisenberg group. For $1 \leq p < 2$, under the condition $b > Q(1/p - 1/2)$, and for $2 \leq p < \infty$, with $0 < a < Q/p$ and $0 < b < 4$, we derive $L^p$-HPW uncertainty inequalities and as a consequence, we obtain a refined $L^2$-HPW inequality on $\mathbb{K}$, valid for all $a, b > 0$, improving upon the earlier result of Atef (2013) which required $a, b \geq 1$. Our proofs rely on the Fourier-Laguerre transform, dilation and rescaling invariance, and Hausdorff-Young and Plancherel inequalities, thus avoiding heat kernel methods. These results extend Xiao's Euclidean $L^p$-HPW uncertainty inequalities (2022) and parallel recent developments on nilpotent Lie groups, thereby providing a complete $L^p$-framework for uncertainty inequalities on the Laguerre hypergroup.

We present the first known efficient decoding algorithm for correcting multiple insertion-deletion errors in Helberg codes and their non-binary generalizations, extending a known algorithm for correcting multiple deletion errors.

In this paper, we introduce the concept of Inhomogeneous sub-self-similar (ISSS) sets, building upon the foundations laid by Falconer (Trans. Amer. Math. Soc. 347 (1995) 3121-3129) in the study of sub-self-similar sets and drawing inspiration from Barnsley's work on inhomogeneous self-similar sets (Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 24). We explore a range of examples of ISSS sets and elucidate a method to construct ISSS sets. We also investigate the upper and lower box dimensions of ISSS sets and discuss the continuity of the Hausdorff dimension.

In this work, we propose a new class of explicit Runge-Kutta schemes for backward stochastic differential equations (BSDEs). By extending the classical Butcher theory from ordinary differential equations to the BSDE setting, we systematically derive order conditions for schemes with arbitrary numbers of stages and orders. Numerical experiments are conducted to demonstrate and validate the theoretical findings.

We construct a general class of correspondences on hyperelliptic Riemann surfaces of arbitrary genus that combine finitely many Fuchsian genus zero orbifold groups and Blaschke products. As an intermediate step, we first construct analytic combinations of these objects as partially defined maps on the Riemann sphere. We then give an algebraic characterization of these analytic combinations in terms of hyperelliptic involutions and meromorphic maps on compact Riemann surfaces. These involutions and meromorphic maps, in turn, give rise to the desired correspondences. The moduli space of such correspondences can be identified with a product of Teichmüller spaces and Blaschke spaces. The explicit description of the correspondences then allows us to construct a dynamically natural injection of this product space into appropriate Hurwitz spaces.

Integrated sensing and communication (ISAC) plays a crucial role in 6G, to enable innovative applications such as drone surveillance, urban air mobility, and low-altitude logistics. However, the hybrid ISAC signal, which comprises deterministic pilot and random data payload components, poses challenges for target detection due to two reasons: 1) these two components cause coupled shifts in both the mean and variance of the received signal, and 2) the random data payloads are typically unknown to the sensing receiver in the bistatic setting. Unfortunately, these challenges could not be tackled by existing target detection algorithms. In this paper, a generalized likelihood ratio test (GLRT)-based detector is derived, by leveraging the known deterministic pilots and the statistical characteristics of the unknown random data payloads. Due to the analytical intractability of exact performance characterization, we perform an asymptotic analysis for the false alarm probability and detection probability of the proposed detector. The results highlight a critical trade-off: both deterministic and random components improve detection reliability, but the latter also brings statistical uncertainty that hinders detection performance. Simulations validate the theoretical findings and demonstrate the effectiveness of the proposed detector, which highlights the necessity of designing a dedicated detector to fully exploited the signaling resources assigned to random data payloads.

Let $G$ be a dense graph with good expansion properties and not too close to being bipartite. Let $\boldsymbol d$ be a graphical degree sequence. Under very weak conditions, we find the number of subgraphs of $G$ with degree sequence $\boldsymbol d$ to arbitrary precision. The average degree can be any power of $n$ and the variation in degrees can be very large. The method uses an explicit bound on the tail of the cumulant generating function found by the first author. As a first application, we prove that there is an asymptotic expansion for the number of regular graphs and find several terms explicitly. We believe that this is the first combinatorial application of the Fourier inversion method for which the integral outside the dominant regions cannot be bounded by the integral of the absolute value, and we give a general method for dealing with that situation.

In this paper, we investigate the well-posedness of quadratic backward stochastic differential equations driven by G-Brownian motion (referred to as G-BSDEs) with double mean reflections. By employing a representation of the solution via G-BMO martingale techniques, along with fixed point arguments, the Skorokhod problem, the backward Skorokhod problem, and the {\theta}-method, we establish existence and uniqueness results for such G-BSDEs under both bounded and unbounded terminal conditions.

This paper is devoted to the regularity of Navier-Stokes (NS) equations with additive white noise on two-dimensional torus $\mathbb T^2$. Under the conditions that the external force $f(x)$ belongs to the phase space $ H$ and the noise intensity function $h(x)$ satisfies $\|\nabla h\|_{L^\infty} \leq \sqrt \pi \nu \lambda_1$, where $ \nu $ is the kinematic viscosity of the fluid and $\lambda_1$ is the first eigenvalue of the Stokes operator, it was proved that the random NS equations possess a tempered $(H,H^2)$-random attractor whose (box-counting) fractal dimension in $H^2$ is finite. This was achieved by establishing, first, an $H^2$ bounded absorbing set and, second, an $(H,H^2)$-smoothing effect of the system which lifts the compactness and finite-dimensionality of the attractor in $H$ to that in $H^2$. Since the force $f$ belongs only to $H$, the $H^2$-regularity of solutions as well as the $H^2$-bounded absorbing set was constructed by an indirect approach of estimating the $H^2$-distance between the solution of the random NS equations and that of the corresponding deterministic equations.

We solve three optimal recovery problems for operators on classes of $L$-space (which is a semilinear metric space with two additional axioms that connect the metric with the algebraic operations) valued functions that are defined by a majorant of their modulus of continuity. Consideration of $L$-spaces valued functions allows to treat multi- and fuzzy-valued functions, as well as random processes and other non-real valued functions in a unified manner.

In this paper, we study the asymptotic limit, as $\varepsilon\to 0$, of solutions to a vector-valued Allen-Cahn equation $$ \partial_t u = \Delta u - \frac{1}{\varepsilon^2} \partial_u F(u), $$ where $u: \Omega \subset \mathbb{R}^m \to \mathbb{R}^n$ and $F(u): \mathbb{R}^n \to \mathbb{R}$ is a nonnegative radial function which vanishes precisely on two concentric spheres. This equation, proposed and studied by Bronsard and Stoth [Trans. Amer. Math. Soc. 1998] for the case $n=2$, serves as a typical example for a general reaction-diffusion equation introduced by Rubinstein, Sternberg, and Keller to model chemical reactions and diffusions as well as phase transitions. We establish that the sharp interface limit is a two-phase flow system: (i) The interface evolves by mean curvature flow; (ii) Within the bulk phase regions, the solution follows the harmonic map heat flow into $\mathbb{S}^{n-1}$; (iii) Across the interface, the $\mathbb{S}^{n-1}$-valued vectors on the two sides satisfy a mixed boundary condition. Furthermore, we rigorously justify this limit using the matched asymptotic expansion method. First, we employ the idea of ``quasi-minimal connecting orbits'' developed in Fei, Lin, Wang, and Zhang [Invent. Math. 2023] to construct approximated solutions up to arbitrary order. Second, we derive a uniform spectral lower bound for the linearized operator around the approximate solution, which relies on a novel application of the boundary condition. To achieve this, we introduce a suitable decomposition which can reduce the problem to spectral analysis of two scalar one-dimensional linear operators and some singular product estimates.

Wi-Fi plays a crucial role in connecting electronic devices and providing communication services in everyday life. Recently, there has been a growing demand for services that require low-latency communication, such as real-time applications. The latest amendments to Wi-Fi, IEEE 802.11bn, are being developed to address these demands with technologies such as the multiple access point coordination (MAPC). In this paper, we demonstrate that coordinated TDMA (Co-TDMA), one of the MAPC techniques, effectively reduces the latency of transmitting time-sensitive traffic. In particular, we focus on worst-case latency and jitter, which are key metrics for evaluating the performance of real-time applications. We first introduce a Co-TDMA scheduling strategy. We then investigate how this scheduling strategy impacts latency under varying levels of network congestion and traffic volume characteristics. Finally, we validate our findings through system-level simulations. Our simulation results demonstrate that Co-TDMA effectively mitigates jitter and worst-case latency for LL traffic, with the latter exhibiting an improvement of approximately 24%.

In the 1970s Thurston introduced a technique known as ``jiggling'' which brings any triangulation into general position (a stronger version of transversality) by subdividing and perturbing. This result is now known as Thurston's jiggling lemma. In this paper we provide an alternative, more conceptual proof of the lemma. In particular we also prove the generalization to manifolds, whose proof had previously only been sketched.

Gradient-flow (GF) viewpoints unify and illuminate optimization algorithms. Yet most GF analyses focus on unconstrained settings. We develop a geometry-respecting framework for constrained problems by (i) reparameterizing feasible sets with maps whose Jacobians vanish on the boundary (orthant/box) or are rank \(n{-}1\) (simplex), (ii) deriving flows in the parameter space which induce feasible primal dynamics, (iii) discretizing with A-stable implicit schemes solvable by robust inner loops (Modified Gauss-Newton or a KL-prox (negative-entropy) inner solver), and (iv) proving that stationarity of the dynamics implies KKT-with complementary slackness arising from a simple kinematic mechanism (''null speed'' or ''constant dual speed with vanishing Jacobian''). We also give a Stiefel-manifold treatment where Riemannian stationarity coincides with KKT. These results yield efficient, geometry-respecting algorithms for each constraint class. We include a brief A-stability discussion and present numerical tests (NNLS, simplex- and box-constrained least squares, orthogonality) demonstrating stability, accuracy, and runtime efficiency of the implicit schemes.

Decision-makers often encounter uncertainty, and the distribution of uncertain parameters plays a crucial role in making reliable decisions. However, complete information is rarely available. The sample average approximation (SAA) approach utilizes historical data to address this, but struggles with insufficient data. Conversely, moment-based distributionally robust optimization (DRO) effectively employs partial distributional information but can yield conservative solutions even with ample data. To bridge these approaches, we propose a novel method called harmonizing optimization (HO), which integrates SAA and DRO by adaptively adjusting the weights of data and information based on sample size N. This allows HO to amplify data effects in large samples while emphasizing information in smaller ones. More importantly, HO performs well across varying data sizes without needing to classify them as large or small. We provide practical methods for determining these weights and demonstrate that HO offers finite-sample performance guarantees, proving asymptotic optimality when the weight of information follows a 1/\sqrt{N}-rate. In addition, HO can be applied to enhance scenario reduction, improving approximation quality and reducing completion time by retaining critical information from reduced scenarios. Numerical results show significant advantages of HO in solution quality compared to Wasserstein-based DRO, and highlight its effectiveness in scenario reduction.

We study the monodromy representation of the hypergeometric system $\mathcal{F}_{C}^{p,m}(a,B)$ in $m$ variables of rank $p^m$ with parameters $a$ and $B$. This system can be regarded as a multi-variable model of the generalized hypergeometric equation of rank $p$. We construct $m+1$ loops which generate the fundamental group of the complement of the singular locus of $\mathcal{F}_{C}^{p,m}(a,B)$, and we show that they satisfy certain relations as elements of the fundamental group. We produce circuit matrices along these loops with respect to a fundamental system of solutions to $\mathcal{F}_C^{p,m}(a,B)$ under certain non-integrality conditions on parameters $a$ and $B$.

95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the interval \[ \left[x, x+x^{1-{1\over 250}}\right] \qquad \hbox{for $x$ sufficiently large}. \] More recently Baker, Harman, Pintz proved existence of primes in the interval \[ \left[x, x+ x^{1-{19\over 40}}\right] \qquad \hbox{for $x$ sufficiently large}. \] In the present article I will, to the extent possible, make some of these statements effective. Specifically, among other things, I shall show that \[ \forall n \geq 4, \qquad\forall x \geq \exp(\exp(33)), \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]; \] \[ \forall n \geq 156, \qquad\forall x \geq [155^{155}]^{n/(n-155)} , \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] Furthermore \[ \forall n \geq 185, \qquad\forall x \geq 1, \qquad \hbox{there are primes in the interval} \left[x, x+ x^{1-{1\over n}}\right]. \] In particular this last observation makes both the Hoheisel and Heilbronn results fully explicit and effective. This (relatively) specific observation can be extended and generalized in various manners.

We consider random genus-0 hyperbolic surfaces $\mathcal{S}_n$ with $n + 1$ punctures, sampled according to the Weil-Petersson measure. We show that, after rescaling the metric by $n^{-1/4}$, the surface $\mathcal{S}_n$ converges in distribution to the Brownian sphere - a random compact metric space homeomorphic to the 2-sphere, exhibiting fractal geometry and appearing as a universal scaling limit in various models of random planar maps. Without rescaling the metric, we establish a local Benjamini--Schramm convergence of $\mathcal{S}_n$ to a random infinite-volume hyperbolic surface with countably many punctures, homeomorphic to $\mathbb{R}^2 \setminus \mathbb{Z}^2$. Our proofs mirror techniques from the theory of random planar maps. In particular, we develop an encoding of punctured hyperbolic surfaces via a family of plane trees with continuous labels, akin to Schaeffer's bijection. This encoding stems from the Epstein-Penner decomposition and, through a series of transformations, reduces to a model of single-type Galton--Watson trees, enabling the application of known invariance principles.

Hoffman's bound is a well-known eigenvalue bound on the chromatic number of a graph. By interpreting this bound as a parameter, we show multiple applications of colorings attaining the bound (Hoffman colorings) for several notions of graph regularity: regular, (co-)edge-regular, and strongly regular. For strongly regular graphs, we prove that Hoffman colorability implies pseudo-geometricity, and we strengthen Haemers' finiteness result on strongly regular graphs with a bounded chromatic number by considering the Hoffman bound instead of the chromatic number. Furthermore, by using Hoffman colorings we show that a sufficient condition for non-unique vector colorability shown by Godsil, Roberson, Rooney, Šámal and Varvitsiotis [European J. Combin. 79, 2019] can be relaxed in the setting of strongly regular graphs. Lastly, using Hoffman colorings we derive several new characterizations of the mentioned graph regularity notions.

In this paper, we consider the isotropic-nematic phase transition with anisotropic elasticity governed by the Landau-de Gennes dynamics of liquid crystals. For $-\frac{3}{2}< L<0,$ we rigorously justify the limit from the Landau-de Gennes flow to a sharp interface system characterized by a two-phase flow: The interface evolves via motion by mean curvature; In the isotropic region, $Q=0$; In the nematic region, $Q=s_+(nn-\frac{1}{3}I)$ with $n\in \mathbb{S}^2$ and $s_+>0$, where the alignment vector field $n$ satisfies $$(2s_+^2\partial_t n+h)\times n=0$$ and $h=-\frac{\delta E(n,\nabla n)}{\delta n}$ with $E(n,\nabla n)$ denoting the Oseen-Frank energy; On the interface, the strong anchoring condition $n=\nu$ is satisfied. This result rigorously verifies a claim made by de Gennes [Mol. Cryst. Liq. Cryst. 1971] regarding the surface tension strength of isotropic-nematic interfaces in dynamical settings. Furthermore, we rigorously justify this limit using the method of matched asymptotic expansions. First, we employ the idea of ``quasi-minimal connecting orbits'' developed by Fei-Lin-Wang-Zhang [this http URL. 2023] to construct approximated solutions up to arbitrary order. Second, we derive a uniform spectral lower bound for the linearized operator around the approximate solution. To achieve this, we introduce a suitable basis decomposition and a coordinate transformation to reduce the problem to spectral analysis of two scalar one-dimensional linear operators and some singular product estimates. To address the difficulties arising from anisotropic elasticity and the strong anchoring boundary condition, we introduce a div-curl decomposition and, when estimating the cross terms, combine these with the anisotropic elastic terms to close the energy estimates.

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3\alpha\}$, a transition between long- and short-range regimes at a crossover value $\alpha_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the first of a series of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. In this paper, we introduce our non-perturbative real-space renormalization group method and apply this method to analyze the HD regime $d>\min\{6,3\alpha\}$. In particular, we compute the tail of the cluster volume and establish the superprocess scaling limits of the model, which transition between super-Levy and super-Brownian behavior when $\alpha=2$. All our results hold unconditionally for $d> 3\alpha$, without any perturbative assumptions on the model; beyond this regime, when $d> 6$ and $\alpha \geq d/3$, they hold under the assumption that appropriate two-point function estimates hold as provided for spread-out models by the lace expansion. Our results on scaling limits also hold (with possible slowly-varying corrections to scaling) in the critical-dimensional regime with $d=3\alpha<6$ subject to a marginal-triviality condition we call the hydrodynamic condition; this condition is verified in the third paper in this series, in which we also compute the precise logarithmic corrections to mean-field scaling when $d=3\alpha<6$.

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3\alpha\}$, a transition between long- and short-range regimes at a crossover value $\alpha_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the second of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. We focus on the long-range low-dimensional (LR-LD) regime $d/3<\alpha<\alpha_c(d)$, where the model is below its upper critical dimension. Since computing $\alpha_c(d)$ for $2<d<6$ appears to be beyond the scope of current techniques, we give an axiomatic definition of the LR regime which we prove holds for $\alpha <1$. Using this, we prove up-to-constants estimates for the critical and slightly subcritical two-point function in the LR regime and for the volume tail and $k$-point function in the LR-LD regime. We deduce that the critical exponents satisfy the identities \[ \eta = 2-\alpha, \qquad \gamma = (2-\eta)\nu, \qquad \text{ and } \qquad \Delta = \nu d_f \] in the LR regime (if $\gamma$, $\nu$, or $\Delta$ is well-defined) and that $\delta$ and $d_f$ follow the hyperscaling identities \[ \delta = \frac{d+\alpha}{d-\alpha} \qquad \text{ and } \qquad d_f = \frac{d+\alpha}{2} \] in the LR-LD regime. Our results are suggestive of conformal invariance in the LR-LD regime, with the critical $k$-point function matching an explicit Möbius-covariant function up-to-constants.

In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model is conjectured to exhibit eight qualitatively different second-order critical behaviours, with a transition between mean-field and low-dimensional regimes when $d=\min\{6,3\alpha\}$, a transition between long- and short-range regimes at a crossover value $\alpha_c(d)$, and with various logarithmic corrections at the boundaries between these regimes. This is the second of three papers developing a rigorous theory of the model's critical behavior in five of these eight regimes, including all long-range (LR) and high-dimensional (HD) regimes. Here, we analyze the model at its upper critical dimension $d=3\alpha<6$. We prove the hydrodynamic condition holds, which allows us to apply our first paper's RG analysis to deduce that the model has the same superprocess scaling limits as in high dimension, after accounting for slowly varying corrections to scaling. We then compute the precise logarithmic corrections to scaling by analyzing the RG flow to second order. Our results yield in particular that for $d=3\alpha < 6$ the critical volume tail is \[ \mathbb{P}_{\beta_c}(|K|\geq n) \sim C \frac{(\log n)^{1/4}}{\sqrt{n}} \] as $n\to \infty$, while the critical two- and three-point functions are \[ \mathbb{P}_{\beta_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+\alpha} \; \text{ and } \; \mathbb{P}_{\beta_c}(x\leftrightarrow y \leftrightarrow z) \asymp \sqrt{\frac{\|x-y\|^{-d+\alpha}\|y-z\|^{-d+\alpha}\|z-x\|^{-d+\alpha}}{\log(1+\min\{\|x-y\|,\|y-z\|,\|z-x\|\})}}. \] These logarithmic corrections match those in hierarchical percolation but differ from those conjectured for nearest-neighbour percolation on $\mathbb{Z}^6$.

We exhibit automorphisms of a certain K3 surface in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with an isolated fixed point at which the induced action on the stalk of the structure sheaf is arbitrarily close to the identity. This implies that the multiplicities of these automorphisms at the fixed point can be arbitrarily large. As another application, we show that the intersection multiplicity of two isomorphic curves at a point can be arbitrarily large on this K3 surface.

We establish a rigorous framework for the Zakharov system on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$ ($m,n\geq 1$), which models the nonlinear coupling between optical and acoustic modes in confined geometries such as optical fibers. Our analysis reveals that the sharp \textit{shell-type Strichartz estimate} for $\mathbb{R}^2 \times \mathbb{T}$ is globally valid in time and exhibits no derivative loss via the measure estimate of semi-algebraic sets, unlike the periodic case studied in \cite{MR4665720}. In addition, we demonstrate that such an estimate fails on the product space $\mathbb{R} \times \mathbb{T}^2$ by constructing a counter-example. Moreover, we derive analogues of these shell-type estimates in other dimensions, both in the waveguide and Euclidean settings. As a direct application, we establish, for the first time, a local well-posedness theory for the partially periodic Zakharov system. To summarize, we compare shell-type Strichartz estimates in different settings (the Euclidean, the periodic, and the waveguide). Numerical verification on $\mathbb{R}^2\times\mathbb{T}$ reveals a uniform $L^4$-spacetime bound, while $\mathbb{R}\times\mathbb{T}^2$ exhibits sublinear growth, quantitatively confirming the theoretical dichotomy between geometries with different dimensional confinement. These findings advance the understanding of dispersive effects in hybrid geometries and provide mathematical foundations for efficient waveguide design and signal transmission. Finally, for the Euclidean case, we establish well-posedness theory for supercritical nonlinear Schrödinger equation (NLS) with \textit{strip-type} frequency-restricted initial data, revealing a trade-off between dispersion and confinement, which is of independent mathematical interest. This provides a deterministic analogue to random data theory of NLS.

In 1980 Lovász introduced the concept of a double circuit in a matroid. The 2nd, 3rd and 4th authors recently generalised this notion to $k$-fold circuits (for any natural number $k$) and proved foundational results about these $k$-fold circuits. In this article we use $k$-fold circuits to derive new results on the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. These results include analysing 2-sums, showing sufficient conditions for the $k$-fold circuit property to hold for $k$-fold $\mathcal{R}_d$-circuits, and giving an extension of Whiteley's coning lemma. The last of these allows us to reduce the problem of determining if a graph $G$ with a vertex $v$ of sufficiently high degree is independent in $\mathcal{R}_d$ to that of verifying matroidal properties of $G-v$ in $\mathcal{R}_{d-1}$.

In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d \mu\right) (-\Delta)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and $0<s<1$. Under suitable assumptions on $h(x)$, we first prove the existence of ground state solutions by the mountain-pass theorem for $p>4$; then we verify the existence of ground state sign-changing solutions based on the method of Nehari manifold for $p>6$. Finally, we establish the multiplicity of nontrivial weak solutions.

We give an independent combinatorial proof of Nogin's Theorem concerning the minimum distance of the Grassmann codes using a special decomposition of the Grassmannians. We use the same idea to also compute the second minimum weight of the Grassmann codes.

Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $\ell$-error-correcting pairs ($\ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $\ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $\ell$-ECPs, which can correct up to $\ell$ errors in polynomial time, with $\ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results.

As a continuation to \cite{MRW} where the Poincaré and log-Sobolev inequalities were studied for the sticky-reflected Brownian motion on Riemannian manifolds with boundary, this paper establishes the super and weak Poincaré inequalities for more general sticky-reflected diffusion processes. As applications, the convergence rate and uniform integrability of the associated diffusion semigroups are characterized. The main results are illustrated by concrete examples.

The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author.

We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero-Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero-Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also the value at zero point, we identify them with renormalized Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function.

This paper investigates the asymptotic behavior of a forward-backward-forward (FBF) type differential equation and its discrete counterpart for solving quasimonotone variational inequalities (VIs). Building on recent continuous-time dynamical system frameworks for VIs, we extend these methods to accommodate quasimonotone operators. We establish weak and strong convergence under significantly relaxed conditions, without requiring strong pseudomonotonicity or sequential weak-to-weak continuity. Additionally, we prove ergodic convergence of the continuous trajectories, offering further insight into the long-term stability of the system. In the discrete setting, we propose a novel Bregman-type algorithm that incorporates a nonmonotone adaptive step-size rule based on the golden ratio technique. A key contribution of this work is demonstrating that the proposed method ensures strong convergence under the assumption of uniform continuity of the operator, thereby relaxing the standard Lipschitz continuity requirement prevalent in existing methods. Numerical experiments, including infinite-dimensional and non-Lipschitz cases, are presented to illustrate the improved convergence and broader applicability of the proposed approach.

These notes are based on a mini-course given at the ACOTCA conference 2025. The goal is to present full proofs of the first two key results regarding hypercyclic Toeplitz operators, in a way that is accessible to beginners.

We study the Hsu-Robbins-Erdös theorem to the case when all moments under sub-linear expectations exist, but the moment generating function does not, viz, we suppose that $C_{\mathbb{v}}\left\{\exp\left\{\left(\log^{+}|X|\right)^{\alpha}\right\}\right\}<\infty$ for some $\alpha>1$. We also investigate multi-index version.

We provide a criterion for a vertex operator superalgebra homomorphism from an affine vertex algebra to another vertex superalgebra to be conformal, and an additional criterion that guarantees that this homomorphism is surjective. This situation is applied to W-algebras and W-superalgebras and we list all cases where our criterion applies. This gives many new examples of W-algebras that collapse to affine vertex algebras or are conformal extensions. In particular, we provide many examples of simple W-algebras at non-admissible levels that collapse to admissible level affine vertex algebras.

Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit.

In this short note, we point out a mistake in this http URL's proof of his version of the universal approximation theorem which has been widely cited. This mistake might not be easily fixable along the idea of his proof and it also leads to an interesting question in measure theory.

We consider a large mass limit of the non-local isoperimetric problem with a repulsive Yukawa potential in two space dimensions. In this limit, the non-local term concentrates on the boundary, resulting in the existence of a critical regime in which the perimeter and the non-local terms cancel each other out to leading order. We show that under appropriate scaling assumptions the next-order $\Gamma$-limit of the energy with respect to the $L^1$ convergence of the rescaled sets is given by a weighted sum of the perimeter and Euler's elastica functional, where the latter is understood via the lower-semicontinuous relaxation and is evaluated on the system of boundary curves. As a consequence, we prove that in the considered regime the energy minimizers always exist and converge to either disks or annuli, depending on the relative strength of the elastica term.

For coprime $p,q\in\mathbb{Z}_{\geq 2}$, the triplet vertex operator algebra $W_{p,q}$ is a non-simple extension of the universal Virasoro vertex operator algebra of central charge $c_{p,q}=1-\frac{6(p-q)^2}{pq}$, and it is a basic example of a vertex operator algebra appearing in logarithmic conformal field theory. Here, we give a new construction of $W_{p,q}$ different from the original screening operator definition of Feigin-Gainutdinov-Semikhatov-Tipunin. Using our earlier work on the tensor category structure of modules for the Virasoro algebra at central charge $c_{p,q}$, we show that the simple modules appearing in the decomposition of $W_{p,q}$ as a module for the Virasoro algebra have $PSL_2$-fusion rules and generate a symmetric tensor category equivalent to $\mathrm{Rep}\,PSL_2$. Then we use the theory of commutative algebras in braided tensor categories to construct $W_{p,q}$ as an appropriate non-simple modification of the canonical algebra in the Deligne tensor product of $\mathrm{Rep}\,PSL_2$ with this Virasoro subcategory. As a consequence, we show that the automorphism group of $W_{p,q}$ is $PSL_2(\mathbb{C})$. We also define a braided tensor category $\mathcal{O}_{c_{p,q}}^0$ consisting of modules for the Virasoro algebra at central charge $c_{p,q}$ that induce to untwisted modules of $W_{p,q}$. We show that $\mathcal{O}_{c_{p,q}}^0$ tensor embeds into the $PSL_2(\mathbb{C})$-equivariantization of the category of $W_{p,q}$-modules and is closed under contragredient modules. We conjecture that $\mathcal{O}_{c_{p,q}}^0$ has enough projective objects and is the correct category of Virasoro modules for constructing logarithmic minimal models in conformal field theory.

We investigate the relationship between variants of the uniformization property for ladder system colorings and fragments of Martin's Axiom. The well-known forcing properties of having precaliber $\aleph_1$ and being $\sigma$-centered correspond to uncountable refinement and countable decomposition into centered subsets, respectively, and the associated forcing axioms have been widely studied. In this paper, we focus on a forcing axiom for the property corresponding to stationary refinement, namely the stationary precaliber $\aleph_1$ property. Analogously, we observe that ladder system coloring uniformization also admits both stationary refinement and countable decomposition variants. We discuss the interaction between these uniformization properties and various forcing axioms. Through this analysis, we obtain as a main result the separation between the forcing axioms for stationary precaliber $\aleph_1$ and for $\sigma$-linked posets.

We analyze a semi-implicit finite volume scheme for the Gray--Scott system, a model for pattern formation in chemical and biological media. We prove unconditional well-posedness of the fully discrete problem and establish qualitative properties, including positivity and boundedness of the numerical solution. A convergence result is obtained by compactness arguments, showing that the discrete approximations converge strongly to a weak solution of the continuous system. Under additional regularity assumptions, we further derive a priori error estimates in the $L^2$ norm. Numerical experiments validate the theoretical analysis, confirm a rate of convergence of order 1, and illustrate the ability of the scheme to capture classical Gray--Scott patterns.

We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small degree.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. A strictly-Perron polynomial is the minimal polynomial of a Perron number $\lambda$ such that $\lambda$ is neither a Pisot number, an anti-Pisot number, nor a Salem number. For any natural number $n\ge 2$, we prove that there exist infinitely many monogenic strictly-Perron polynomials of degree $n$.

Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of length two (e.g. $\mathbb{Z}/p^2\mathbb{Z}$ and $\mathbb F_q[t]/\langle t^2\rangle$). We study the power map $g\mapsto g^L$, where $L$ is a positive integer. We consider $L$ to be coprime to $p$ (an odd prime), the characteristic of the residue field $k$ of $\mathscr O_2$. We classify all the elements in the image, whose mod-$\mathfrak m$ reduction in $\mathrm{GL}_n(k)$ are either regular semisimple or cyclic, where $\mathfrak m$ is the unique maximal ideal of $\mathscr O_2$. Our main tool is a Hensel lifting for polynomial equations over $\mathrm{M}_n(\mathscr O_2)$, which we establish in this work. A central contribution of this work is the construction of canonical forms for certain natural classes of matrices over $\mathscr O_2$. As applications, we derive explicit generating functions for the probabilities that a random element of $\mathrm{GL}_n(\mathscr O_2)$ is regular semisimple, $L$-power regular semisimple, compatible cyclic, or $L$-power compatible cyclic.

A {\bf $\mathbf{k}$-majority coloring} of a digraph $D=(V,A)$ is a coloring of $V$ with $k$ colors so that each vertex $v\in V$ has at least as many out-neighbours of color different from its own color as it has out-neighbours with the same color as itself. Majority colorings have received much attention in the last years and many interesting open problems remain. Inspired by this and the fact that digraphs can be modelled via 2-edge-colored graphs we study several problems concerning vertex partitions of 2-edge-colored graphs. In particular we study vertex partitions with the property that for each $c=1,2$ every vertex $v$ has least as many edges of colour $c$ to vertices outside the set it belongs to as it has to vertices inside its own set. We call such a vertex partition with $k$ sets a {\bf $\mathbf{k}$-majority partition. Among other things we show that every 2-edge-coloured graph has a 4-majority partition and that it is NP-complete to decide whether a 2-edge-coloured graph has a 3-majority partition. We also apply probabilistic tools to show that every $2$-edge-colored graph $G$ of minimum color-degree $\delta$ and maximum degree $\Delta \le \frac{e^{\delta/18}}{9\delta}-2$ has a balanced majority $3$-partition.

The paper is devoted to tests for uniformity based on sum-functions of overlapping spacings, where the order of spacings can diverge to infinity as the sample size increases. In particular, it is shown that the asymptotic local power of these tests depends significantly on the asymptotic properties of the counterpart statistics based on disjoint spacings.

In this paper, we focus on the problem of phase retrieval from intensity measurements of the Short-Time Linear Canonical Transform (STLCT). Specifically, we show that the STLCT allows for the unique recovery of any square-integrable function through phaseless STLCT sampling on rectangular square-root lattices. When turning to the uniform lattices, we establish counterexamples about the STLCT phase retrieval problems in L2(R). Nevertheless, for functions in band-limited function spaces, phase retrieval results on uniform lattices can still be accomplished.

Let $G = G_{1} \times G_{2}$ be a product of two locally compact, second countable groups and $\mu \in \mathrm{Prob}(G)$ be of the form $\mu = \mu_{1} \times \mu_{2}$, where $\mu_{i} \in \mathrm{Prob}(G_{i})$. Let $(B,\nu_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\mathcal{M}$ with \[ \mathcal{N} \subseteq \mathcal{M} \subseteq \mathcal{N} \,\bar{\otimes}\, L^{\infty}(B,\nu) \] splits as a tensor product of the form $\mathcal{N}\bar{\otimes}L^{\infty}(C,\nu_C)$, where $(C,\nu_C)$ is a $(G,\mu)$-boundary. Here, $\mathcal{N}$ is a tracial von Neumann algebra on which $G$ acts trace-preservingly. This generalizes the Intermediate Factor Theorem proved by Bader--Shalom (\cite[Theorem~1.9]{BS06}) in the measurable setup. In addition, we give various other examples of the splitting phenomenon associated with $W^{*}$-dynamics. We also show that certain assumptions are necessary for the intermediate algebras to split, and ideals in the ambient tensor product algebra obstruct the splitting phenomenon. We also use the Master theorem from \cite{glasner2023intermediate} to resolve the second part of \cite[Problem~5.2]{jiangskalski} in the affirmative.

We study positivity-preserving properties for the elastic flow of non-compact, complete curves in Euclidean space. Despite the fact that the canonical elastic energy is infinite in this context, we extend our recent work based on the adapted elastic energy to derive nontrivial optimal thresholds for maintaining planar embeddedness and graphicality, respectively. We also obtain a new Li--Yau type inequality for complete planar curves.

We consider the initial-boundary value problem for the incompressible two-dimensional micropolar fluid model with angular viscosity in the upper half-plane. This model describes the motion of viscous fluids with microstructure. The global well-posedness of strong solutions for this problem with positive angular viscosity can be established via the standard energy method, as presented in the classical monograph [Łkaszewicz, {\it Micropolar fluids: Theory and applications.} Birkhäuser, 1999]. Corresponding results for the zero angular viscosity case were established recently in [Liu, Wang, {\it Commun. Math. Sci.} 16 (2018), no. 8, 2147-2165]. However, the link between the positive angular viscosity model (the full diffusive system) and the zero angular viscosity model (the partially diffusive system) via the vanishing diffusion limit remains unknown. In this work, we first construct Prandtl-type boundary layer profiles. We then provide a rigorous justification for the vanishing angular viscosity limit of global strong solutions, without imposing smallness assumptions on the initial data. Our analysis reveals the emergence of a strong boundary layer in the angular velocity field (micro-rotation velocity of the fluid particles) during this vanishing viscosity process. Moreover, we also obtain the optimal $L^\infty$ convergence rate as the angular viscosity tends to zero. Our approach combines anisotropic Sobolev spaces with careful energy estimates to address the nonlinear interaction between the velocity and angular velocity fields.

This paper introduces a framework for the categorification of the Le-Murakami-Ohtsuki (LMO) invariant of 3-manifolds, defining a new invariant, the LMO Spectrum, via factorization homology. The theoretical foundation of this framework is the main algebraic result of this paper (Theorem A): a proof that the algebra of Jacobi diagrams, A_Jac, possesses the structure of a homotopy E3-algebra. This algebraic structure is shown to be a consequence of 3-dimensional geometry, originating from a "Tetrahedron Principle" where the fundamental IHX relation is interpreted as an algebraic manifestation of the tetrahedron's combinatorial properties. This geometric perspective is used to provide a rigorous proof of the "Principle of Decomposability" that underpins the Goussarov-Habiro clasper calculus, thereby supplying a mathematical justification for the consistency of this geometric surgery theory. To establish a computational basis, the framework is built from the first principles of factorization homology, from which a universal surgery formula is derived via the excision axiom (Theorem B) independently of any conjectural models. The utility of this axiomatically-grounded theory is demonstrated by constructing a new observable, the "H_1-decorated LMO invariant." This invariant is defined in accordance with the principles of TQFT, where the evaluation of a closed diagram (a numerical invariant) is the trace of the corresponding open diagram (an operator). As the main applied result of this paper (Theorem C), it is proven that this invariant distinguishes the lens spaces L(156, 5) and L(156, 29), a pair known to be indistinguishable by the classical LMO invariant. This result validates the proposed framework and establishes the LMO Spectrum as a new tool in quantum topology that unifies the geometric intuition of clasper theory with the algebraic rigor of modern homotopy theory.

We introduce the notion of intersective polynomials over the Gaussian integers, and prove that given any intersective polynomial $p(x)$ over the Gaussian integers, every subset $A$ of the Gaussian integers of positive upper density contains two distinct elements such that their difference is $p(z)$ for some Gaussian integer $z$. Moreover, we also obtain a quantitative version of this result. The proof is motivated by an argument due to Lucier, and the Fourier-free proof of the Furstenberg--Sárközy theorem over the integers by Green, Tao and Ziegler.

We introduce the concept of soliton solutions of integrable nonlinear partial differential equations and point out that the inverse spectral method represents the rigorous mathematical formalism to construct such solutions. We work with the travelling waves of the KdV, mKdV and nonlinear Schrödinger (NLS) equations and derive a pedagogic method to find their soliton solutions. The travelling wave of the KdV equation leads directly to the well known bell type KdV soliton while the mKdV equation needs some additional consideration in respect of this. The travelling waves of a generalized mKdV and NLS equations are obtained in terms of $sn(u,m)$, the so called Jacobi elliptic sine function. The choice $m=1$ provides a constraint on the parameters of the equations and gives their kink and anti-kink soliton solutions. We further show that by expressing the travelling waves of these equations in terms of Jacobi elliptic cosine functions, $cn(u,m)$, it is possible to construct bell-type soliton solutions.

We introduce a framework for stochastic differential equations (SDEs) with interaction on compact, connected, $d$-dimensional manifolds. For SDEs whose drift and diffusion coefficients may depend on both the state variable and the empirical distribution, we establish existence and uniqueness of strong solutions under general regularity assumptions. We study the associated measure valued process on the Wasserstein space over the manifold, deriving an explicit Itô Wiener decomposition. We prove Malliavin differentiability of the solution and, using directional derivatives in the Wasserstein space, establish smooth dependence of the solution on the measure component for a class of coefficients.

Research in Economics and Game theory has necessitated results on Carathéodory-type selections. In particular, one has to obtain Carathéodory type-selections from correspondences that need not be continuous (neither lower-semicontinuous nor upper-semicontinuous). We provide new theorems on Carathéodory type-selections that include as corollaries the results in Kim-Prikry-Yannelis \cite{KPY:87}. We also, obtain new random fixed-point theorems, random maximal elements, random (Nash) equilibrium and Bayesian equilibrium extending and generalizing theorems of Browder \cite{Browder:68}, Fan \cite{Fan:52} and Nash \cite{Nash}, among others.

We present an online-adaptive hyperreduced reduced basis element method for model order reduction of parameterized, component-based nonlinear systems. The method, in the offline phase, prepares a library of hyperreduced archetype components of various fidelity levels and, in the online phase, assembles the target system using instantiated components whose fidelity is adaptively selected to satisfy a user-prescribed system-level error tolerance. To achieve this, we introduce a hierarchical error estimation framework that compares solutions at successive fidelity levels and drives a local refinement strategy based on component-wise error indicators. We also provide an efficient estimator for the system-level error to ensure that the adaptive strategy meets the desired accuracy. Component-wise hyperreduction is performed using an empirical quadrature procedure, with the training accuracy guided by the Brezzi--Rappaz--Raviart theorem. The proposed method is demonstrated on a family of nonlinear thermal fin systems comprising up to 225 components and 68 parameters. Numerical results show that the hyperreduced basis element model achieves O(100) computational reduction at 1% error level relative to the truth finite-element model. In addition, the adaptive refinement strategy provides more effective error control than uniform refinement by selectively enriching components with higher local errors.

Chung and Graham~[J. London Math. Soc., 1983] claimed that there exists an $n$-vertex graph $G$ containing all $n$-vertex trees as subgraphs that has at most \( \frac{5}{2}n \log_2 n + O(n)\) edges. We identify an error in their proof. This error can be corrected by adding more edges, which increases the number of edges to \( e(G) \le \frac{7}{2}n \log_2 n + O(n). \) Moreover, we further improve this by showing that there exists such an $n$-vertex graph with at most \( \left(5- \frac{1}{3}\right)n \log_3 n + O(n) \leq 2.945 \cdot n\log_2(n)\) edges. This is the first improvement of the bound since Chung and Graham's pioneering work four decades ago.

In this paper we take a look at compactly generated weak Hausdorff spaces equipped with an action of a compact Lie group $G$ together with their colimits and homotopy colimits. In particular, we investigate relations between (homotopy) colimits and mapping spaces, and consequently, homotopy groups.

For $\alpha>0$ and $\sigma > 0$, we consider the following probability distribution on $\alpha\mathbb N_0$: $\pi_{\alpha,\sigma} = \exp \big(- \frac{\sigma}{{\alpha}^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\frac{\sigma}{{\alpha}^2}\big)^n {\delta}_{\alpha n}$, where $\delta_y$ denotes the Dirac measure with mass at $y$. For $\alpha=1$, $\pi_{1,\sigma}$ is the Poisson distribution with parameter $\sigma$. Furthermore, the centered probability distribution $\tilde \pi_{\alpha,\sigma} = \exp \big(- \frac{\sigma}{{\alpha}^2}\big) \sum_{n=0}^{\infty} \frac{1}{n!} \big(\frac{\sigma}{{\alpha}^2}\big)^n {\delta}_{\alpha n-\sigma/\alpha}$ weakly converges to $\mu_\sigma$ as $\alpha\to0$. Here $\mu_\sigma$ is the Gaussian distribution with mean zero and variance $\sigma$. Let $(c_n)_{n=0}^\infty$ be the monic polynomial sequence that is orthogonal with respect to the measure $\mu_{\alpha,\sigma}$. In particular, for $\alpha=1$, $(c_n)_{n=0}^\infty$ is a sequence of Charlier polynomials. Let $\mathbb F_\sigma(\mathbb C)$ denote the Bargmann space of all entire functions $f(z)=\sum_{n=0}^\infty f_nz^n$ with $f_n \in \mathbb C$ satisfying $ \sum_{n=0}^{\infty} {| f_n |}^2 \, n! \, \sigma^n < \infty$. The generalized Segal--Bargmann transform associated with the measure $\pi_{\alpha,\sigma}$ is a unitary operator $\mathcal S:L^2(\alpha\mathbb N_0,\pi_{\alpha,\sigma})\to \mathbb F_\sigma(\mathbb C)$ that satisfies $(\mathcal Sc_n)(z)=z^n$ for $n\in\mathbb N_0$. We present some new results related to the operator $\mathcal S$. In particular, we observe how the study of $\mathcal S$ naturally leads to the normal ordering in the Weyl algebra.

I prove connectedness of the moduli space $\mathcal M_n$ of $SU(2)$ instantons on $S^3\times S^1$ with charge $n$.

The widely used Heun algorithm for the numerical integration of stochastic differential equations (SDEs) is critically re-examined. We discuss and evaluate several alternative implementations, motivated by the fact that the standard Heun scheme is constructed from a low-order integrator. The convergence, stability, and equilibrium properties of these alternatives are assessed through extensive numerical simulations. Our results confirm that the standard Heun scheme remains a benchmark integration algorithm for SDEs due to its robust performance. As a byproduct of this analysis, we also disprove a previous claim in the literature regarding the strong convergence of the Heun scheme.

We study stable Sp-decompositions of the cokernel of the Johnson homomorphism. Continuing the work of Conant in 2016, which identified the 1-loop part of the Johnson cokernel as the Enomoto-Satoh obstruction, we study the 2-loop part. Using the corresponding 2-loop trace map, we capture all the components of the Johnson cokernels in degree 6 that cannot be detected by the Enomoto-Satoh trace.

Multi-stage stochastic optimization is a well-known quantitative tool for decision-making under uncertainty. It is broadly used in financial and investment planning, inventory control, and also natural disaster risk management. Theoretical solutions of multi-stage stochastic programs can be found explicitly only in very exceptional cases due to their variational form and interdependency of uncertainty in time. Nevertheless, numerical solutions are often inaccurate, as they rely on Monte-Carlo sampling, which requires the Law of Large Numbers to hold for the approximation quality. In this article, we introduce a new approximation scheme, which computes and groups together stage-wise optimal quantizers of conditional Fréchet distributions for optimal weighting of value functions in the dynamic programming. We consider optimality of scenario quantization methods in the sense of minimal Kantorovich-Wasserstein distance at each stage of the scenario tree. By this, we bound the approximation error with convergence guarantees. We also provide global solution guarantees under convexity and monotonicity conditions on the value function. We apply the developed methods to the governmental budget allocation problem for risk management of flood events in Austria. For this, we propose an extremely efficient way to approximate optimal quantizers for conditional Fréchet distributions. Our approach allows to enhance the overall efficiency of dynamic programming via the use of different parameter estimation methods for different groups of quantizers. The groups are distinguished by a particular risk threshold and are able to differentiate between higher- and lower-impact flood events.

We prove an extension of the Furstenberg set theorem to families of graphs satisfying a transversality condition. We apply the result to derive bounds on $L^{p}$-norms of Fourier transforms of fractal measures supported on plane curves.

We introduce TROOP, a tree-based Riccati optimistic online planner, that is designed to generate near-optimal control laws for discrete-time switched linear systems with switched quadratic costs. The key challenge that we address is balancing computational resources against control performance, which is important as constructing near-optimal inputs often requires substantial amount of computations. TROOP addresses this trade-off by adopting an online best-first search strategy inspired by A*, allowing for efficient estimates of the optimal value function. The control laws obtained guarantee both near-optimality and stability properties for the closed-loop system. These properties depend on the planning depth, which determines how far into the future the algorithm explores and is closely related to the amount of computations. TROOP thus strikes a balance between computational efficiency and control performance, which is illustrated by numerical simulations on an example.

We prove that for any reductive group $G$ of adjoint type cuspidal automorphic twisted D-modules have non-vanishing quantum Whittaker coefficients. The argument provides a microlocal interpretation of quantum Whittaker coefficients for any $\check{\Lambda}^+$-valued divisor under some hypothesis on singular support.

In this paper we study the number of four-rich points defined by pencils of certain algebraic objects. Our main result concerns the number of four-rich points defined by four sheaves of planes; under certain non-degeneracy conditions, we prove that four sheaves of $n$ planes in $\mathbb P^3$ determine at most $O(n^{8/3})$ four-rich points. We prove this using the four dimensional Elekes-Szabó theorem. Using the same method, we prove an upper bound on the number of four-rich points determined by four sets of concentric spheres in $\mathbb C^3$. Furthermore, using the same technique with the 3-d Elekes-Szabó theorem, one can prove upper bounds on four-rich points determined by various configurations of lines/circles in the plane $\mathbb C^2$; we give one such example, involving two pencils of lines and two pencils of concentric circles in $\mathbb C^2$.

In this paper, we derive explicit reconstruction formulas for two common measurement geometries: a plane and a sphere. The problem is formulated as inverting the forward operator $R^a$, which maps the initial source to the measured wave data. Our first result pertains to planar observation surfaces. By extending the domain of $R^a$ to tempered distributions, we provide a complete characterization of its range and establish that the inverse operator $(R^a)^{-1}$ is uniquely defined and "almost" continuous in the distributional topology. Our second result addresses the case of a spherical observation geometry. Here, with the operator acting on $L^2$ spaces, we derive a stable reconstruction formula of the filtered backprojection type.

We give an introduction to our results on cluster structures for schemes of $(G,c)$-bands emphasizing their connections with seminal works of Frenkel and Reshetikhin in the 90's. In particular we construct using $(G,c)$-bands a discrete analogue of the difference Miura transformation of the loop group $LG$, and we show that it calculates the $q$-characters of the finite-dimensional representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of the same $A$, $D$, $E$ type as $G$, thus verifying a conjecture of Frenkel and Reshetikhin.

We give a survey on Auslander-Gorenstein algebras with a focus on finite-dimensional algebras. We put an emphasis on recent classification results for special classes of algebras and the newly discovered interactions of the Auslander-Reiten bijection with other well studied bijections in the literature.

Objective: Evaluate and compare multiple mechanics-based and traditional regularization strategies within a variational image registration framework for quasi-static ultrasound elastography. Methods:We reformulate a previously proposed momentum-equation-based post-processing method (SPREME) as a regularization term directly integrated into an image registration energy functional. Four regularization types are implemented and compared: a strain magnitude ($\R_\epsilon$), a strain magnitude with incompressibility constraint ($R_{\epsilon i}$), and a momentum-based regularization under plane strain ($R_{P\epsilon}$) and plane stress ($R_{P\sigma}$) assumptions. Each is evaluated in a variational framework solved via Gauss-Newton optimization. Data:Registration performance is assessed using synthetic ultrasound image sequences generated from 2D and 3D finite element simulations, as well as experimental phantom data. Comparisons are based on displacement and strain field errors, strain contrast, and contrast-to-noise ratio (CNR). Results: Momentum-based regularization, particularly under plane stress assumptions ($R_{P\sigma}$), achieved the lowest strain errors and highest strain contrast across both single-frame and accumulated measurements, even when the underlying tissue deformation violated 2D assumptions. In contrast, strain magnitude regularization with an incompressibility constraint ($R_{\epsilon i}$) produced unstable results in 3D and accumulated displacement scenarios. Conclusions: Mechanics-based regularization that incorporates momentum conservation outperforms strain-based techniques in elastographic image registration, particularly when applied directly in the optimization framework. This approach improves robustness to noise and model mismatch, offering a promising direction for future displacement-based inverse imaging methods.

We investigate the time-periodic solutions to the nonlinear wave and beam equations and uncover their intricate, fractal-like structure. In particular, we identify a new class of large-energy solutions with complex mode compositions and propose a systematic framework for their analysis. A Floquet stability study reveals that this class contains solutions that are linearly stable, suggesting that they may play a significant role in the nonlinear dynamics of the systems.

We present a new method for approximating real-valued functions on ${\mathbb R}^+$ by linear combinations of exponential functions with complex coefficients. The approach is based on a multi-point Padé approximation of the Laplace transform and employs a highly efficient continued fraction technique to construct the corresponding rational approximant. We demonstrate the accuracy of this method through a variety of examples, including the Gaussian function, probability density functions of the lognormal and Gompertz-Makeham distributions, the hockey stick and unit step functions, as well as a function arising in the approximation of the gamma and Barnes $G$-functions.

The problem of designing adaptive stepsize sequences for the gradient descent method applied to convex and locally smooth functions is studied. We take an adaptive control perspective and design update rules for the stepsize that make use of both past (measured) and future (predicted) information. We show that Lyapunov analysis can guide in the systematic design of adaptive parameters striking a balance between convergence rates and robustness to computational errors or inexact gradient information. Theoretical and numerical results indicate that closed-loop adaptation guided by system theory is a promising approach for designing new classes of adaptive optimization algorithms with improved convergence properties.

We introduce a new generalization of Euler's $\varphi$-function associated with a system of polynomials of several variables. We reprove by a short direct approach certain known related identities, and study some other special cases that do not appear in the literature. We also discuss the unitary analogues of these functions.

In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have been introduced and studied various topologies on the fundamental group from the beginning of the 21st century onwards. In this paper by reviewing these topologies, using the concept of subgroup topology, we are going to compare these topologies in order to present some results on topologized fundamental groups.

In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of solutions of an algebraic equation. The existence of multiple stationary solutions is established through countably many saddle-node bifurcations at critical shear speeds for the algebraic equation. We then establish similar local bifurcations for the stationary solutions; more precisely, (i) for each critical shear speed, there is a unique stationary solution, the stationary solution disappears for smaller shear speed, and two stationary solutions nearby bifurcate for larger shear speed; (ii) more importantly, under a generic condition, there is exactly one zero eigenvalue for the linearization of the shear flow at the critical stationary solution, and the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution. These results extend and are motivated by the works of Dorn and Liu [{\em J. Differential Equations {\bf 253} (2012), 3184-3210}] and of Jiao, et. al. [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}].

In this manuscript, we study a nonlinear model of tumor growth, described by a coupled hyperbolic-elliptic system of partial differential equations. In this model, the compressible flow of tumor cells is modeled by a transport equation for the cell density, which takes into account transport via a background flow (given by a potential solving a Brinkman-type equation), and which has a source term modeling cell growth and death. In this manuscript, we show that for sufficiently large viscosity, the tumor growth system admits nontrivial global strong solutions for positive initial data having a gradient with sufficiently small norm. This illustrates the regularizing effects of the source term representing tumor cell growth and death on the resulting transport dynamics of the equation. Furthermore, we characterize the long-time behavior of global strong solutions to the tumor growth system using a level-set analysis, in which we analyze how level sets evolve as they are transported by the flow, in terms of expansion/contraction and accretion/depletion of cells. While there has been past work on global existence of weak solutions for this tumor growth system, this manuscript opens the study of well-posedness in terms of more regular strong/classical solutions, which exist globally in time.

We study the solutions of a McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the mean field limit of a system of $N$ interacting excitatory neurons with $N$ large. Each neuron spikes randomly with rate depending on its membrane potential. At each spiking time, the neuron potential is reset to the value $\bar v$, its adaptation variable is incremented by $\bar w$ and all other neurons receive an additional amount $J/N$ of potential after some delay where $J$ is the connection strength. Between jumps, the neurons drift according to some two-dimensional ordinary differential equation with explosive behavior. We prove the existence and uniqueness of solutions of a heuristically derived mean-field limit of the system when $N\to\infty$. We then study the existence of stationary distributions and provide several properties (regularity, tail decay, etc.) based on a Doeblin estimate using a Lyapunov function. Numerical simulations are provided to assess the hypotheses underlying the results.

We construct an explicit family of finite-area, infinite-genus translation surfaces whose vertical translation flow is strongly mixing. This provides a positive answer to a question posed by Lindsey and Treviño~\cite{LT}

We show that any finitely presented group with an index two subgroup is realized as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilize after adding a single copy of the complex projective plane.

In this paper, we introduce the group version of a Lie-Leibniz triple. We define a Lie group-rack triple whose tangent structure is a Lie-Leibniz triple. This triple is a generalization of an augmented Lie rack. We show that any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple by generalizing the integration procedure of an augmented Leibniz algebra into an augmented Lie rack.

In this review paper, we demonstrate that several classes of point processes in a locally compact Polish space $X$ appear as the joint spectral measure of a rigorously defined particle density of a representation of the canonical anticommutation relations (CAR) or the canonical commutation relations (CCR) in a Fock space. For these representations of the CAR/CCR, the vacuum state on the corresponding $*$-algebra is quasi-free. The classes of point process that arise in such a way include determinantal and permanental point processes.

A practical approach is proposed to construct short presentations for Euclidean crystallographic groups in terms of generators and relations. For our purposes a short presentation is the one with a small number of short relators for a given generating set. The connection is emphasized between relators of a group presentation and cycles in the associated Cayley graph. It is shown by examples that a short presentation is usually the one where relators correspond to strong rings in the Cayley graph and therefore provide a natural upper bound for their size. Presentations are computed for vertex-transitive groups which act with trivial vertex stabilizers on a number of high-symmetry 2-, 3- and 4-periodic graphs. Higher-dimensional as well as subperiodic examples are also considered. Relations are explored between geodesics in periodic graphs and corresponding cycles in their quotients.

In present work we deal with the class $\mathcal{C}=\mathcal{C}_1\cup \mathcal{C}_2$ where $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$) is formed by all separable Uniform algebras (respectively, separable commutative C$^*$-algebras) with no compact elements. For a given algebra $A$ in $\mathcal{C}_1$ (respectively, $A$ in $\mathcal{C}_2$) we show that $A$ is isometrically isomorphic as algebra (respectively, as C$^*$-algebra) to a subalgebra $M$ of $\ell_{\infty}$ with $M\subset (\ell_{\infty}\setminus c_0)\cup\{0\}.$ Under the additional assumption that $A$ is non-unital we verify that there exists a copy of $M(A)$ (the multipliers algebra of $A$ which is non-separable) inside $(\ell_{\infty}\setminus c_0)\cup\{0\}$. For an infinitely generated abelian C$^*$-algebra $B,$ we study the least cardinality possible of a system of generators ($gen_{C^*}(B)$). In fact we deduce that $gen_{C^*}(B)$ coincides with the smallest cardinal number $n$ such that an embedding of $\Delta(B)$ (= the spectrum of $B$) in $\mathbb{R}^n$ exists - The finitely generated version of this result was proved by Nagisa. In addition, we introduce new concepts of algebrability in terms of $gen_{C^*}(B)$ ($(C^*)$-genalgebrability) and its natural variations. From our methods we infer that there is $^*$-isomorphic copy of $\ell_{\infty}$ in $(\ell_{\infty}\setminus c_0)\cup\{0\}$. In particular, $(\ell_{\infty}\setminus c_0)\cup\{0\}$ contains a copy of every separable Banach space. Moreover, all the positive answers of this work holds if we replace the set $(\ell_{\infty}\setminus c_0)\cup\{0\}$ with $(B(\ell_2(\mathbb{N}))\setminus K(\ell_2(\mathbb{N}))\cup \{ 0\}.$

We give two new constructions of the harmonic algebra of a lattice polytope $P$, a bigraded algebra whose character is the $q$-Ehrhart series of $P$ defined by Reiner and Rhoades. First, we show that the harmonic algebra is the associated graded algebra of the semigroup algebra of $P$ with respect to a certain natural filtration, clarifying it's relationship with the more classical semigroup algebra. We then give a geometric interpretation of the harmonic algebra as a quotient of the ring of global sections of a certain family of line bundles on the blowup of the toric variety associated to $P$ at a generic point. Using this connection to toric geometry we resolve one the main conjectures of Reiner and Rhoades by showing that the harmonic algebra is not finitely generated in general.

In this paper, we propose Stoch-IDENT, a novel method for identifying Stochastic Partial Differential Equations (SPDEs) from observational data. Our method can handle linear and nonlinear high-order SPDEs driven by time-dependent Wiener processes with both additive or multiplicative structures. Theoretically, we establish a rigorous connection between the spectral properties of the solution's mean and covariance and the identifiability of the underlying SPDEs. Our analysis covers key classes of equations, including linear SPDEs with constant coefficients, as well as parabolic and hyperbolic types, generalizing the theory of identification of deterministic PDEs. Algorithmically, the drift term is identified using a sample mean generalization of Robust-IDENT (He et al., 2023). For the diffusion term, we develop a new greedy algorithm, Quadratic Subspace Pursuit (QSP), which can address general sparse regression problems with quadratic measurements. We validate Stoch-IDENT extensively on various SPDEs, demonstrating its effectiveness through quantitative and qualitative evaluations.

The goal of this paper is to provide a new approach to address the $L^p-$boundedness of bilinear rough singular integral operators. This approach relies on local Fourier series expansion of input functions leading to trilinear estimates with desired decay in the frequency parameter. This approach departs from the existing methods of the wavelet decomposition of the multiplier employed in the work of Grafakos, He and Honzík and in a series of subsequent papers in the context of bilinear rough singular integrals. With this new approach, we provide a new and self contained proof of $L^p-$boundedness of bilinear rough singular integral operators in dimension one for the optimal range of exponents. Furthermore, this approach allows us to prove sharp $L^p-$estimates for maximally truncated bilinear rough singular integrals when the kernel is supported away from the diagonal in the plane.

Graphlets are small subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence, gives a topological description of the surrounding of the analyzed vertex. In this article, we study properties and uniqueness of graphlet degree sequences. The information given by graphlets up to size (n-1) is utilized graphs having certain type of asymmetric vertex-deleted subgraphs. Moreover, we show a reconstruction of trees from their (<= n-1)-graphlet degree sequences, which is much easier compared to the standard reconstruction from vertex-deleted subgraphs.

In this paper we consider the numerical approximation of the incompressible surface Navier--Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\ell \geq 2$. In the semidiscrete continuous-in-time setting we are able to prove a stability estimate that mimics a corresponding result for the continuous problem. Some numerical results, including a convergence experiment, demonstrate the practicality and accuracy of the proposed method.

In this work we achieve a full understanding of the so-called entanglement of radicals, showing that over any field there are extremely few additive relations among radicals. Our results complete a famous theorem by Kneser from 1975 on the linear independence of radicals and solve a problem discussed by Lenstra in 2006.

Naidu classified pairs of finite groups and 3-cocycles that lead to equivalent Dijkgraaf-Witten theories for 3-manifolds. We establish analogous equivalences for arithmetic Dijkgraaf-Witten theory over totally imaginary number fields F containing n-th roots of unity, where n is invertible on X subset spec O_F. For the full ring of integers X = spec O_F, we give examples with quadratic fields and the quaternion group Q_8 where these equivalences fail, but also identify sufficient conditions under which they still hold.

We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which the Griffiths bundle extends amply. In particular the Griffiths bundle is semiample. We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture of Prokhorov--Shokurov. The semiampleness results (and the construction of the Baily--Borel compactifications) crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Kollár on the geometry of minimal lc centers to verify the extra conditions.

We rigorously establish the existence of dark-bright solitons as traveling wave solutions to a one dimensional defocusing Gross-Pitaevskii system, a widely used model for describing mixtures of Bose-Einstein condensates and nonlinear optical systems. These solitons are shown to exhibit symmetry and radial monotonicity in modulus, and to propagate at subsonic speed. Our method relies on minimizing an energy functional subject to two constraints: the mass of the bright component and a modified momentum of the dark component. The compactness of minimizing sequences is obtained via a concentration-compactness argument, which requires some novel estimates based on symmetric decreasing rearrangements.

The classical dimer model is concerned with the (weighted) enumeration of perfect matchings of a graph. An $n$-dimer cover is a multiset of edges that can be realized as the disjoint union of $n$ individual matchings. For a probability measure recently defined by Douglas, Kenyon, and Shi, which we call the $M_n$-dimer model, we study random $n$-dimer covers on bipartite graphs with matrix edge weights and produce formulas for local edge statistics and correlations. We also classify local moves that can be used to simplify the analysis of such graphs.

In this work, we study the Kuelbs-Steadman-2 space (KS-2 space), a Hilbert space constructed via the Henstock-Kurzweil integral, which allows handling non-absolutely integrable functions. We present the construction of the KS-2 space over measurable subsets of $\mathbb{R}^d$ and explore its functional properties with particular focus on integral operators associated with symmetric kernels. A Mercer-type representation theorem is established for such kernels in a KS-2 space, leading to the characterization of the associated Reproducing Kernel Hilbert Spaces (RKHS). As an application, we derive asymptotic upper and lower bounds for the covering numbers of the embedding of the RKHS into the KS-2 space, highlighting how the Fourier coefficients decay rate of the kernels influences the estimates.

This paper studies the nonsmooth and nonconvex composite optimization on Riemannian manifolds. We propose a new inexact manifold proximal linear algorithm (IManPL) that incorporates both high- and low-accuracy conditions. At each iteration, IManPL solves convex subproblems on tangent spaces inexactly, guided by two adaptive stopping criteria. We establish convergence guarantees and show that IManPL achieves the best iteration complexity for solving the nonsmooth manifold composite optimization. Numerical experiments on the sparse spectral clustering (SSC) and the sparse principal component analysis (SPCA) demonstrate that our methods outperform existing approaches, including the manifold proximal linear algorithm and manifold proximal gradient algorithm.

We show how the Husain-Kuchař model can be understood as a Carrollian limit of the Holst term in the context of background-independent field theories described in terms of coframes and spin connections. We also discuss the footprint of the Carrollian symmetry in the Hamiltonian formulation of the Husain-Kuchař action.

We present a distributed algorithm and implementation of the variational quantum eigensolver (VQE), termed distributed VQE (DVQE). DVQE, provided as an open-source Python package, enables the execution of parameterized quantum circuits across multiple logical quantum processing units (QPUs) in a distributed fashion. This approach addresses key hardware limitations of near-term quantum devices, including restricted qubit counts and limited circuit depth. Distributed ansatz circuits are constructed to preserve the quantum state fidelity of their monolithic counterparts, allowing consistent energy estimation while distributing the computational load. To improve the convergence and robustness of the optimization loop for identifying the variational parameters of the DVQE ansatz circuit, we use the ADAM optimizer in combination with metaheuristic initialization strategies, which outperform random initialization across various test cases. The complete DVQE pipeline is implemented in a modular Python package that accepts QUBO problems as input and supports monolithic and distributed execution modes. The framework leverages Qiskit to construct and simulate distributed circuits, and includes an internal greedy algorithm for automatic qubit allocation across multiple QPUs. Simulation results on QUBO benchmarks confirm the correctness of the approach, paving the way for real QPU deployment and further exploration of distributed quantum optimization. \textbf{The simulator is publicly available on \href{this https URL}{GitHub} under a package named raiselab, with a collection of tutorial examples.}

We introduce a unified framework for learning the spatio-temporal dynamics of vector valued functions by combining operator valued reproducing kernel Hilbert spaces (OV-RKHS) with kernel based Koopman operator methods. The approach enables nonparametric and data driven estimation of complex time evolving vector fields while preserving both spatial and temporal structure. We establish representer theorems for time dependent OV-RKHS interpolation, derive Sobolev type approximation bounds for smooth vector fields, and provide spectral convergence guarantees for kernel Koopman operator approximations. This framework supports efficient reduced order modeling and long term prediction of high dimensional nonlinear systems, offering theoretically grounded tools for forecasting, control, and uncertainty quantification in spatio-temporal machine learning.

Defects in conformal field theories (CFTs) play a key role in critical phenomena by modifying scaling behaviors and generating new universality classes. We introduce Parisi-Sourlas (PS) supersymmetry in the presence of extended operators and demonstrate that any $p$-dimensional defect in a CFT$_d$ can be uplifted to a defect in a PS-supersymmetric CFT$_{d+2}$. Surprisingly, there are actually two distinct uplifted defects--of dimensions $p$ and $p+2$--which reduce to the original one. We show how this reduction works for correlators with insertions both in the bulk and on the defect. As a byproduct, we find new relations between defect conformal blocks in dimensions $d$ and $d+2$. We further show that the reduction of the $p$-dimensional defect implies and extend a "global symmetry reduction" previously considered in the literature. Finally, we provide various examples of these uplifts, including perturbative computations in epsilon expansion of the uplift of the Ising magnetic line defect, as well as exact computations of observables in the four-dimensional uplift of minimal models with boundaries.

We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a~full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a~direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an~odd number of twists in each twist region.

Parametric Feynman integrals with the regions of integration defined by some polynomials are considered in this paper. It is shown that integrals with irregular integration regions can be converted to standard parametric integrals, for which a reduction method is known. An application of this method to the analytic calculation of three-point energy correlators is presented. In principle, this method applies to more general event shapes and even jet observables.

We study the problem of reconfiguring odd matchings, that is, matchings that cover all but a single vertex. Our reconfiguration operation is a so-called flip where the unmatched vertex of the first matching gets matched, while consequently another vertex becomes unmatched. We consider two distinct settings: the geometric setting, in which the vertices are points embedded in the plane and all occurring odd matchings are crossing-free, and a combinatorial setting, in which we consider odd matchings in general graphs. For the latter setting, we provide a complete polynomial time checkable characterization of graphs in which any two odd matchings can be reconfigured into each another. This complements the previously known result that the flip graph is always connected in the geometric setting [Aichholzer, Brötzner, Perz, and Schnider. Flips in odd matchings]. In the combinatorial setting, we prove that the diameter of the flip graph, if connected, is linear in the number of vertices. Furthermore, we establish that deciding whether there exists a flip sequence of length $k$ transforming one given matching into another is NP-complete in both the combinatorial and the geometric settings. To prove the latter, we introduce a framework that allows us to transform partial order types into general position with only polynomial overhead. Finally, we demonstrate that when parameterized by the flip distance $k$, the problem is fixed-parameter tractable (FPT) in the geometric setting when restricted to convex point sets.

A main open question in contemporary AI research is quantifying the forms of reasoning neural networks can perform when perfectly trained. This paper answers this by interpreting reasoning tasks as circuit emulation, where the gates define the type of reasoning; e.g. Boolean gates for predicate logic, tropical circuits for dynamic programming, arithmetic and analytic gates for symbolic mathematical representation, and hybrids thereof for deeper reasoning; e.g. higher-order logic. We present a systematic meta-algorithm that converts essentially any circuit into a feedforward neural network (NN) with ReLU activations by iteratively replacing each gate with a canonical ReLU MLP emulator. We show that, on any digital computer, our construction emulates the circuit exactly--no approximation, no rounding, modular overflow included--demonstrating that no reasoning task lies beyond the reach of neural networks. The number of neurons in the resulting network (parametric complexity) scales with the circuit's complexity, and the network's computational graph (structure) mirrors that of the emulated circuit. This formalizes the folklore that NNs networks trade algorithmic run-time (circuit runtime) for space complexity (number of neurons). We derive a range of applications of our main result, from emulating shortest-path algorithms on graphs with cubic--size NNs, to simulating stopped Turing machines with roughly quadratically--large NNs, and even the emulation of randomized Boolean circuits. Lastly, we demonstrate that our result is strictly more powerful than a classical universal approximation theorem: any universal function approximator can be encoded as a circuit and directly emulated by a NN.

We consider the orbits of a discrete Painlevé equation over finite fields and show that the number of points in such orbits satisfy the Hasse bound. The orbits turn out to lie on algebraic curves, whose defining polynomials are given explicitly. Moreover, these curves are shown to have genus less than or equal to one, which contrasts sharply with the case of discrete Painlevé equations over $\mathbb{C}$, whose generic solutions are believed to be more transcendental than elliptic functions.

In this paper, we propose a new definition of the trace-map bordism within the Atiyah-Segal framework for Witten-type TQFTs constructed from the topological twist of mass-gapped theories. We demonstrate that these Witten-type TQFTs are unitary under this revised definition and conjecture the self-consistency of the modified bordism category.

In this paper, we present a general framework for testing relevant hypotheses in functional time series. Our unified approach covers one-sample, two-sample, and change point problems under contaminated observations with arbitrary sampling schemes. By employing B-spline estimators and the self-normalization technique, we propose nuisance-parameter-free testing procedures, obviating the need for additional procedures such as estimating long-run covariance or measurement-error variance functions. A key challenge arises from related nonparametric statistics may not be tight, complicating the joint weak convergence for the test statistics and self-normalizers, particularly in sparse scenarios. To address this, we leverage a Gaussian approximation in a diverging-dimension regime to derive a pivotal approximate distribution. Then, we develop consistent decision rules, provide sufficient conditions ensuring non-degeneracy, and establish phase transition boundaries from sparse to dense. We also examine the multiple change point scenario and extend the theory when one obtains consistent estimates of the change points. The choice of self-normalizers is further discussed, including the recently developed range-adjusted self-normalizer. Extensive numerical experiments support the proposed theory, and we illustrate our methodologies using the this http URL implied volatility and traffic volume datasets.

We present a unified framework combining statistical physics, coding theory, and algebraic topology for efficient multi-class image classification. High-dimensional feature vectors from a frozen MobileNetV2 backbone are interpreted as spins on a sparse Multi-Edge Type quasi-cyclic LDPC (MET-QC-LDPC) graph, forming a Random-Bond Ising Model (RBIM). We operate this RBIM at its Nishimori temperature, $\beta_N$, where the smallest eigenvalue of the Bethe-Hessian matrix vanishes, maximizing class separability. Our theoretical contribution establishes a correspondence between local trapping sets in the code's graph and topological invariants (Betti numbers, bordism classes) of the feature manifold. A practical algorithm estimates $\beta_N$ efficiently with a quadratic interpolant and Newton correction, achieving a six-fold speed-up over bisection. Guided by topology, we design spherical and toroidal MET-QC-LDPC graph ensembles, using permanent bounds to suppress harmful trapping sets. This compresses 1280-dimensional features to 32 or 64 dimensions for ImageNet-10 and -100 subsets. Despite massive compression (40x fewer parameters), we achieve 98.7% accuracy on ImageNet-10 and 82.7% on ImageNet-100, demonstrating that topology-guided graph design yields highly efficient, physics-inspired embeddings with state-of-the-art performance.

In the (1-dimensional) bin packing problem, we are asked to pack all the given items into bins, each of capacity one, so that the number of non-empty bins is minimized. Zhu~[Chaos, Solitons \& Fractals 2016] proposed an approximation algorithm $MM$ that sorts the item sequence in a non-increasing order by size at the beginning, and then repeatedly packs, into the current single open bin, first as many of the largest items in the remaining sequence as possible and then as many of the smallest items in the remaining sequence as possible. In this paper we prove that the asymptotic approximation ratio of $MM$ is at most 1.5. Next, focusing on the fact that $MM$ is at the intersection of two algorithm classes, max-min algorithms and 1-bounded space algorithms, we comprehensively analyze the theoretical performance bounds of each subclass derived from the two classes. Our results include a lower bound of 1.25 for the intersection of the two classes. Furthermore, we extend the theoretical analysis over algorithm classes to the cardinality constrained bin packing problem.

The rapid expansion of sixth-generation (6G) wireless networks and the Internet of Things (IoT) has catalyzed the evolution from centralized cloud intelligence towards decentralized edge general intelligence. However, traditional edge intelligence methods, characterized by static models and limited cognitive autonomy, fail to address the dynamic, heterogeneous, and resource-constrained scenarios inherent to emerging edge networks. Agentic artificial intelligence (Agentic AI) emerges as a transformative solution, enabling edge systems to autonomously perceive multimodal environments, reason contextually, and adapt proactively through continuous perception-reasoning-action loops. In this context, the agentification of edge intelligence serves as a key paradigm shift, where distributed entities evolve into autonomous agents capable of collaboration and continual adaptation. This paper presents a comprehensive survey dedicated to Agentic AI and agentification frameworks tailored explicitly for edge general intelligence. First, we systematically introduce foundational concepts and clarify distinctions from traditional edge intelligence paradigms. Second, we analyze important enabling technologies, including compact model compression, energy-aware computing strategies, robust connectivity frameworks, and advanced knowledge representation and reasoning mechanisms. Third, we provide representative case studies demonstrating Agentic AI's capabilities in low-altitude economy networks, intent-driven networking, vehicular networks, and human-centric service provisioning, supported by numerical evaluations. Furthermore, we identify current research challenges, review emerging open-source platforms, and highlight promising future research directions to guide robust, scalable, and trustworthy Agentic AI deployments for next-generation edge environments.

We analyze the privacy guarantees of the Laplace mechanism releasing the histogram of a dataset through the lens of pointwise maximal leakage (PML). While differential privacy is commonly used to quantify the privacy loss, it is a context-free definition that does not depend on the data distribution. In contrast, PML enables a more refined analysis by incorporating assumptions about the data distribution. We show that when the probability of each histogram bin is bounded away from zero, stronger privacy protection can be achieved for a fixed level of noise. Our results demonstrate the advantage of context-aware privacy measures and show that incorporating assumptions about the data can improve privacy-utility tradeoffs.

Reliable predictions from systems biology models require knowing whether parameters can be estimated from available data, and with what certainty. Identifiability analysis reveals whether parameters are learnable in principle (structural identifiability) and in practice (practical identifiability). We introduce the core ideas using linear models, highlighting how experimental design and output sensitivity shape identifiability. In nonlinear models, identifiability can vary with parameter values, motivating global and simulation-based approaches. We summarise computational methods for assessing identifiability noting that weakly identifiable parameters can undermine predictions beyond the calibration dataset. Strategies to improve identifiability include measuring different outputs, refining model structure, and adding prior knowledge. Far from a technical afterthought, identifiability determines the limits of inference and prediction. Recognising and addressing identifiability is essential for building models that are not only well-fitted to data, but also capable of delivering predictions with robust, quantifiable uncertainty.

We develop a resurgence analysis for large central charge (large $C$) expansions in two-dimensional CFTs using the Coulomb gas formalism. Through the exact Borel-Laplace representations of the conformal blocks $I_1(C,z)$ and $I_2(C,z)$ associated with the four-point correlation function $ \langle \phi_{2,1}(0)\phi_{2,1}(z,\bar{z})\phi_{2,1}(1)\phi_{2,1}(\infty)\rangle$, we demonstrate that $I_1(C,z)$ participates in the Stokes phenomenon of $I_2(C,z)$ (and vice versa), and establish that monodromy in $z$ arises from alien calculus in the Borel plane variable $\zeta$ (Borel dual to $C$). From a given conformal block, resurgence theory thus enables us to discover other internal operators (conformal blocks). This approach establishes a non-perturbative connection between conformal blocks, shedding light on the resurgence phenomena in more general quantum field theories.

Privacy regulations require the erasure of data from deep learning models. This is a significant challenge that is amplified in Federated Learning, where data remains on clients, making full retraining or coordinated updates often infeasible. This work introduces an efficient Federated Unlearning framework based on information theory, modeling leakage as a parameter estimation problem. Our method uses second-order Hessian information to identify and selectively reset only the parameters most sensitive to the data being forgotten, followed by minimal federated retraining. This model-agnostic approach supports categorical and client unlearning without requiring server access to raw client data after initial information aggregation. Evaluations on benchmark datasets demonstrate strong privacy (MIA success near random, categorical knowledge erased) and high performance (Normalized Accuracy against re-trained benchmarks of $\approx$ 0.9), while aiming for increased efficiency over complete retraining. Furthermore, in a targeted backdoor attack scenario, our framework effectively neutralizes the malicious trigger, restoring model integrity. This offers a practical solution for data forgetting in FL.

Single-Pixel Imaging enables reconstructing objects using a single detector through sequential illuminations with structured light patterns. We propose a bilevel optimisation method for learning task-specific, binary illumination patterns, optimised for applications like single-pixel fluorescence microscopy. We address the non-differentiable nature of binary pattern optimisation using the Straight-Through Estimator and leveraging a Total Deep Variation regulariser in the bilevel formulation. We demonstrate our method on the CytoImageNet microscopy dataset and show that learned patterns achieve superior reconstruction performance compared to baseline methods, especially in highly undersampled regimes.

Empirical science needs to be based on facts and claims that can be reproduced. This calls for replicating the studies that proclaim the claims, but practice in most fields still fails to implement this idea. When such studies emerged in the past decade, the results were generally disappointing. There have been an overwhelming number of papers addressing the ``reproducibility crisis'' in the last 20 years. Nevertheless, terminology is not yet settled, and there is no consensus about when a replication should be called successful. This paper intends to clarify such issues. A fundamental problem in empirical science is that usual claims only state that effects are non-zero, and such statements are scientifically void. An effect must have a \emph{relevant} size to become a reasonable item of knowledge. Therefore, estimation of an effect, with an indication of precision, forms a substantial scientific task, whereas testing it against zero does not. A relevant effect is one that is shown to exceed a relevance threshold. This paradigm has implications for the judgement on replication success. A further issue is the unavoidable variability between studies, called heterogeneity in meta-analysis. Therefore, it is of little value, again, to test for zero difference between an original effect and its replication, but exceedance of a corresponding relevance threshold should be tested. In order to estimate the degree of heterogeneity, more than one replication is needed, and an appropriate indication of the precision of an estimated effect requires such an estimate. These insights, which are discussed in the paper, show the complexity of obtaining solid scientific results, implying the need for a strategy to make replication happen.

Large-scale fault-tolerant quantum computers of the future will likely be modular by necessity or by design. Modularity is inevitable if the substrate cannot support the desired error-correction code due to its planar geometry or manufacturing constraints resulting in a limited number of logical qubits per module. Even if the computer is compact enough there may be functional requirements to distribute the quantum computation substrate over distant regions of varying scales. In both cases, matter-based quantum information, such as spins, ions or neutral atoms, is the most conveniently transmitted or mediated by photonic interconnects. To avoid long algorithm execution times and reduce errors, each module of a universal quantum computer should be dynamically interconnected with as many other modules as possible. This task relies on an optical switching network providing any-to-any or sufficiently high simultaneous connectivity. In this work we construct several novel and decentralized switching schemes based on the properties of the Generalized Mach-Zehnder Interferometer (GMZI) that are more economic and less noisy compared to commonly considered alternatives while achieving the same functionality.

This work considers positive energy theorems in asymptotically, locally AdS spacetimes. Particular attention is given to spacetimes where conformal infinity has compact, Einstein cross-sections admitting Killing or parallel spinors; a positive energy theorem is derived for such spacetimes in terms of geometric data intrinsic to the cross-section. This is followed by the first complete proofs of the BPS inequalities in (the bosonic sectors of) 4D and 5D minimal, gauged supergravity, including with magnetic fields. The BPS inequalities are proven for asymptotically AdS spacetimes, but also generalised to the aforementioned class of asymptotically, locally AdS spacetimes.

The Goal-oriented Communication (GoC) paradigm breaks the separation between communication and the content of the data, tailoring communication decisions to the specific needs of the receiver and targeting application performance. While recent studies show impressive encoding performance in point-to-point scenarios, the multi-node distributed scenario is still almost unexplored. Moreover, the few studies to investigate this consider a centralized collision-free approach, where a central scheduler decides the transmission order of the nodes. In this work, we address the Goal-oriented Multiple Access (GoMA) problem, in which multiple intelligent agents must coordinate to share a wireless channel and avoid mutual interference. We propose a theoretical framework for the analysis and optimization of distributed GoMA, serving as a first step towards its complete characterization. We prove that the problem is non-convex and may admit multiple Nash Equilibrium (NE) solutions. We provide a characterization of each node's best response to others' strategies and propose an optimization approach that provably reaches one such NE, outperforming centralized approaches by up to 100% while also reducing energy consumption. We also design a distributed learning algorithm that operates with limited feedback and no prior knowledge.

Recurrent neural networks (RNNs) have become increasingly popular in information processing tasks involving time series and temporal data. A fundamental property of RNNs is their ability to create reliable input/output responses, often linked to how the network handles its memory of the information it processed. Various notions have been proposed to conceptualize the behavior of memory in RNNs, including steady states, echo states, state forgetting, input forgetting, and fading memory. Although these notions are often used interchangeably, their precise relationships remain unclear. This work aims to unify these notions in a common language, derive new implications and equivalences between them, and provide alternative proofs to some existing results. By clarifying the relationships between these concepts, this research contributes to a deeper understanding of RNNs and their temporal information processing capabilities.

Polarimetry, which is the ability to measure the scattering response of the environment across orthogonal polarizations, is fundamental to enhancing wireless communication and radar system performance. In this paper, we utilize the Zak-OTFS modulation to enable instantaneous polarimetry within a single transmission frame. We transmit a Zak-OTFS carrier waveform and a spread carrier waveform mutually unbiased to it simultaneously over orthogonal polarizations. The mutual unbiasedness of the two waveforms enables the receiver to estimate the full polarimetric response of the scattering environment from a single received frame. Unlike existing methods for instantaneous polarimetry with computational complexity quadratic in the time-bandwidth product, the proposed method enables instantaneous polarimetry at complexity that is only sublinear in the time-bandwidth product. Via numerical simulations, we show ideal polarimetric target detection and parameter estimation results with the proposed method, with improvements in performance and computational complexity over comparable baselines.

We study the extension of Presburger arithmetic by the class of sub-polynomial Hardy field functions, and show the majority of these extensions to be undecidable. More precisely, we show that the theory $\mathrm{Th}(\mathbb{Z}; <, +, \lfloor f \rceil)$, where $f$ is a Hardy field function and $\lfloor \cdot \rceil$ the nearest integer operator, is undecidable when $f$ grows polynomially faster than $x$. Further, we show that when $f$ grows sub-linearly quickly, but still as fast as some polynomial, the theory $\mathrm{Th}(\mathbb{Z}; <, +, \lfloor f \rceil)$ is undecidable.

We derive a new family of quantum error-correcting codes. The main technical tool used to do so is the physically intuitive concept of condensation, which is employed to create new domain walls between the quantum double of $\Z_4$ and an instance of the doubled semion phase. Specifically, we provide explicit constructions, first at the lattice-level and then subsequently at the macroscopic logical-level. To provide intuition, we provide a series of explicit examples using the derived topological interfaces. We discuss the code's utility in the burgeoning area of quantum error-correction with an emphasis on the interplay between logical error rates and decoding. We conclude by outlining how such codes' representation and design can be automated. We expect our results, which provide explicit step-by-step instructions in the form of algorithms, to pave the path for new higher-algebraic-dimensional codes to be discovered and implemented in configurations that take advantage of various hardware's distinct strengths.

The notion of an individual random sequence goes back to von Mises. We describe the evolution of this notion, especially the use of martingales (suggested by Ville), and the development of algorithmic information theory in 1960s and 1970s (Solomonov, Kolmogorov, Martin-Lof, Levin, Chaitin, Schnorr and others). We conclude with some remarks about the use of the algorithmic information theory in the foundations of probability theory.

This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

We informally review a few PDEs for which the Monge-Kantorovich distance between pairs of solutions, possibly with some judicious cost function, decays: heat equation, Fokker-Planck equation, heat equation with varying coefficients, fractional heat equation with varying coefficients, homogeneous Boltzmann equation for Maxwell molecules, and some nonlinear integro-differential equations arising in neurosciences. We always use the same method, that consists in building a coupling between two solutions. This amounts to solve a well-chosen PDE posed on the Euclidian square of the physical space, i.e. doubling the variables. Finally, although the above method fails, we recall a simple idea to treat the case of the porous media equation. We also introduce another method based on the dual Monge-Kantorovich problem.

We study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on the densities of pollinators. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interaction graph and the quantities of resources exchanged between species. Our model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, we show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.

We solve the completion problem of $3\times3$ upper triangular operator matrix acting on a direct sum of Banach spaces and hence generalize the famous result of Han, Lee, Lee (Proc. Amer. Math. Soc. 128 (1) (2000), 119-123) to a greater dimension of a matrix. Our main tools are Harte's ghost of an index theorem and Banach spaces embeddings. We overcome the lack of orthogonality in Banach spaces by exploiting decomposition properties of inner regular operators, and of Fredholm regular operators when needed. Finally, we provide some necessity results related to the invertibility of $n\times n$ upper triangular operators, $n>3$.

We study the distribution of resonances for discrete Hamiltonians of the form $H_0+V$ near the thresholds of the spectrum of $H_0$. Here, the unperturbed operator $H_0$ is a multichannel Laplace type operator on $\ell^2(\mathbb Z; \mathbb C^N) \cong \ell^2(\mathbb Z)\otimes \mathbb C^N$ and $V$ is a non-selfadjoint compact perturbation. We compute the exact number of resonances and give a precise description on their location in clusters around some special points in the complex plane.

Both evaluating the service quality of a public transport system and understanding how passengers choose between modes or routes is imperative for public transport operators, providers of competing mobility services and policy makers. However, the literature does not offer consensus on how either of these tasks should be performed, which can lead to inconsistent or counter-intuitive results. This paper provides a formal treatment on how fundamental elements of public transport systems (route sets, timetables and line plans) can be evaluated consistently, and how travelers distribute over routes. Our main insight is that evaluation and routing are two sides of the same coin: by solving an appropriate optimization model one obtains both the quality of the route set, timetable or line plan (the optimal objective value), and the distribution of the travelers over the routes (the optimal solution itself). The practical relevance of the new framework is demonstrated with several applications that are validated with real data from the Dutch and Swiss railway networks. The measures and route choice models developed in this paper enable planners to create better line plans and to effectively analyze timetables for inefficiencies. The framework also reveals: (i) the importance of using the right model for the right stage of planning, (ii) that it is not always necessary for public transport planners to accurately model travel behavior, especially for high-level planning, and (iii) that combining models in an inconsistent way can have significant negative consequences that are avoided with the new framework.

This paper studies the dynamics of mean curvature flow as it approaches a cylindrical singularity. We proved that the rescaled mean curvature flow converging to a smooth generalized cylinder can be written as a graph over the cylinder in a ball of radius $K\sqrt{t}$, and a normal form of the asymptotics. Using the normal form, we can define the nondegeneracy of cylindrical singularities, and we show that nondegenerate cylindrical singularities are isolated in space, have a mean convex neighborhood, and are type-I.

In this paper, we prove that transnormal functions are isoparametric functions on Finsler space forms (N(c), F) under certain conditions, which generalize Theorem B given by Q.M. Wang in Riemannian case. Next, we discuss the relationship between umbilic hypersurfaces and isoparametric hypersurfaces in (N(c), F). Further more, we construct a locally isoparametric function by a distance function and give a global isoparametric function on a standard Finsler sphere.

This paper presents a sizing procedure for a rocket family capable of fulfilling multiple missions, considering the commonalities between the vehicles. The procedure aims to take full advantage of sharing a common part across multiple rockets whose payload capability differs entirely, ultimately leading to cost savings in designing a rocket family. As the foundation of the proposed rocket family design method, an integrated sizing method with trajectory optimization for a single rocket is first formulated as a single optimal control problem. This formulation can find the optimal sizing along with trajectory results in a tractable manner. Building upon this formulation, the proposed rocket family design method is developed to 1) determine the feasible design space of the rocket family design problem (i.e., commonality check), and 2) if a feasible design space is determined to exist, minimize the cost function within that feasible space by solving an optimization problem in which the optimal control problem is embedded as a subproblem. A case study is carried out on a rocket family composed of expendable and reusable launchers to demonstrate the novelty of the proposed procedure.

We study quantum $q$-series invariants of 3-manifolds $\widehat{Z}_\sigma$ of Gukov-Pei-Putrov-Vafa, using techniques from the theory of normal surface singularities such as splice diagrams. We show that the (suitably normalized) sum of all $\widehat{Z}_\sigma$ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal abelian cover. We use these ideas to find simple formulas for $\widehat{Z}_\sigma$ invariants of Seifert manifolds. Applications include a better understanding of the vanishing of the $q$-series $\widehat{Z}_\sigma$. Additionally, we study moduli spaces of flat $\operatorname{SL}_2(\mathbb{C})$ connections on Seifert manifolds and their relation to spectra of surface singularities, extending a result of Boden and Curtis for Brieskorn spheres to Seifert rational homology spheres with 3 singular fibers and to Seifert homology spheres with any number of fibers.

We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,\mathcal{B},\mu)$ with $\mu$ a $\sigma$-finite measure and $\mu(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known from results by Aaronson that in general the asymptotic behaviour of the Birkhoff sums $S_Ng(x):= \sum_{j=1}^N\, (g\circ T^{j-1})(x)$ strongly depends on the point $x\in X$, and that there exists no sequence $(d_N)$ for which $S_Ng(x)/d_N \to 1$ for $\mu$-almost every $x\in X$. In this paper we consider the case $g\not\in L^1(X,\mu)$ assuming that there exists $E\in\mathcal{B}$ with $\mu(E)<\infty$ and $\int_E g\,\mathrm{d}\mu=\infty$ and continue the investigation initiated in previous work by the authors. We show that for transformations $T$ with strong mixing assumptions for the induced map on a finite measure set, the almost sure asymptotic behaviour of $S_Ng(x)$ for an unbounded observable $g$ may be obtained using two methods, adding a number of summands depending on $x$ to $S_Ng$ and trimming. The obtained sums are then asymptotic to a scalar multiple of $N$. The results are applied to a couple of non-regular continued fraction algorithms, the backward (or Rényi type) continued fraction and the even-integer continued fraction algorithms, to obtain the almost sure asymptotic behaviour of the sums of the digits of the algorithms.

For a profinite group $G$ we describe an abelian group $W_G(R; M)$ of $G$-typical Witt vectors with coefficients in an $R$-module $M$ (where $R$ is a commutative ring). This simultaneously generalises the ring $W_G(R)$ of Dress and Siebeneicher and the Witt vectors with coefficients $W(R; M)$ of Dotto, Krause, Nikolaus and Patchkoria, both of which extend the usual Witt vectors of a ring. We use this new variant of Witt vectors to give a purely algebraic description of the zeroth equivariant stable homotopy groups of the Hill-Hopkins-Ravenel norm $N_{\{e\}}^G(X)$ of a connective spectrum $X$, for any finite group $G$. Our construction is reasonably analogous to the constructions of previous variants of Witt vectors, and as such is amenable to fairly explicit concrete computations.

We propose an implicit Discontinuous Galerkin (DG) discretization for incompressible two-phase flows using an artificial compressibility formulation. The conservative level set (CLS) method is employed in combination with a reinitialization procedure to capture the moving interface. A projection method based on the L-stable TR-BDF2 method is adopted for the time discretization of the Navier-Stokes equations and of the level set method. Adaptive Mesh Refinement (AMR) is employed to enhance the resolution in correspondence of the interface between the two fluids. The effectiveness of the proposed approach is shown in a number of classical benchmarks. A specific analysis on the influence of different choices of the mixture viscosity is also carried out.

Let $K\subset \mathbb{R}^n$, $n\geq 3$, be a convex body. A point $p$ the interior of $K$ is said to be a Larman point of $K$ if for every hyperplane $\Pi$ passing through $p$ the section $\Pi\cap K$ has a $(n-2)$-plane of symmetry. If $p$ is a Larman point of $K$ and, in addition, for every section $\Pi\cap K$, $p$ is in the corresponding $(n-2)$-plane of symmetry, then we call $p$ a revolution point of $K$. We conjecture that if $K$ contains a Larman point which is not a revolution point, then $K$ is either an ellipsoid or a body of revolution. This generalizes a conjecture of K. Bezdek for convex bodies in $\mathbb{R}^3$ to $n \geq 4$. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if $K \subset \mathbb{R}^n$ is a strictly convex origin symmetric body that contains a revolution point $p$ which is not the origin, then $K$ is a body of revolution. This generalizes the False Axis of Revolution Theorem. We also show that if $p$ is a Larman point of $K \subset \mathbb{R}^3$ and there exists a line $L$ such that $p\notin L$ and, for every plane $\Pi$ passing through $p$, the line of symmetry of the section $\Pi \cap K$ intersects $L$, then $K$ is a body of revolution (in some cases, we conclude that $K$ is a sphere). We obtain a similar result for projections of $K$. Additionally, for $K \subset \mathbb{R}^n$, $n \geq 4$, we show that if every hyperplane section or projection of $K$ is a body of revolution and $K$ has a unique diameter $D$, then $K$ is a body of revolution with axis $D$.

We study an abstract family of asymptotically degenerating variational problems. Those are natural generalisations of families of problems emerging upon application of a rescaled Floquet-Bloch-Gelfand transform to resolvent problems for high-contrast elliptic PDEs with highly oscillatory periodic coefficients. An asymptotic analysis of these models leads us to a hierarchy of approximation results with uniform operator-type error estimates under various assumptions, satisfied by specific examples. We provide approximations for the resolvents in terms of a certain `bivariate' operator which appears an abstract generalisation of the two-scale limit operators for highly oscillatory high-contrast PDEs. The resulting approximating self-adjoint operator, providing tight operator error estimates, is the bivariate operator sandwiched by a connecting operator which for a broad class of periodic problems specialises to a new two-scale version of the classical Whittaker-Shannon interpolation. An explicit description of the limit spectrum in the abstract setting is provided, and new tight error estimates on the distance between the original and limit spectra are established. Our generic approach allows us to readily consider a wide class of asymptotically degenerating problems including but also going beyond high-contrast highly oscillatory PDEs. The obtained results are illustrated by various examples.

We show that curve enumeration invariants of complex threefolds with nef anti-canonical bundle are determined by their values on local curves. This implies the MNOP conjecture of Maulik, Nekrasov, Okounkov, and Pandharipande relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants, for all complex threefolds with nef anti-canonical bundle (in particular, all Calabi--Yau threefolds) and primary insertions (no descendents), given its known validity for local curves due to Bryan, Okounkov, and Pandharipande. The main new technical ingredient in our work is a generic transversality result for holomorphic curves in complex manifolds. Due to the rigidity of complex structures, this result is necessarily weaker than the corresponding generic transversality property for holomorphic curves in almost complex manifolds. Despite this weaker nature, it is enough to obtain our main result by following the proof of the Gopakumar--Vafa integrality conjecture by Ionel and Parker.

In the present paper, we formulate two versions of Frank--Wolfe algorithm or conditional gradient method to solve the DC optimization problem with an adaptive step size. The DC objective function consists of two components; the first is thought to be differentiable with a continuous Lipschitz gradient, while the second is only thought to be convex. The second version is based on the first and employs finite differences to approximate the gradient of the first component of the objective function. In contrast to past formulations that used the curvature/Lipschitz-type constant of the objective function, the step size computed does not require any constant associated with the components. For the first version, we established that the algorithm is well-defined of the algorithm and that every limit point of the generated sequence is a stationary point of the problem. We also introduce the class of weak-star-convex functions and show that, despite the fact that these functions are non-convex in general, the rate of convergence of the first version of the algorithm to minimize these functions is ${\cal O}(1/k)$. The finite difference used to approximate the gradient in the second version of the Frank-Wolfe algorithm is computed with the step-size adaptively updated using two previous iterations. Unlike previous applications of finite difference in the Frank-Wolfe algorithm, which provided approximate gradients with absolute error, the one used here provides us with a relative error, simplifying the algorithm analysis. In this case, we show that all limit points of the generated sequence for the second version of the Frank-Wolfe algorithm are stationary points for the problem under consideration, and we establish that the rate of convergence for the duality gap is ${\cal O}(1/\sqrt{k})$.

This study explores strategic decision-making in professional golf's Stroke Play format through a computational lens. We develop a Markov Decision Process (MDP) model-specifically, a stochastic shortest path formulation-to optimize a golfer's strategy on any given course, incorporating both course layout and player skill data. While MDPs have been widely used in sports analytics, applying them to golf presents significant scalability challenges due to the curse of dimensionality. Our primary objective is not to predict player performance with high precision, but rather to demonstrate that an exact, data-driven MDP approach is computationally tractable on full scale, real-world instances. We show that, with careful problem structuring, low-level coding, and efficient memory management, it is possible to solve such large-scale models without resorting to heuristics or Q-learning approximations, as used in existing approaches. To illustrate the model's potential, we show how one can use PGA Tour data and aerial course imagery to simulate strategic outcomes and analyze how different skill profiles influence performance. In particular, we assess the relative impact of driving and putting, challenging the popular adage "Drive for show, putt for dough." These results support the value of our methodology as a robust proof of concept and a foundation for future enhancements. All code and analyses (in R and C++) are made available as open-source resources to support reproducibility and further research.

We give a generators and relations presentation for the full monoidal subcategory of representations of the quantum orthogonal group generated by the quantum exterior powers of the defining representation.

We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of finite subsets of $I$. As a special case, using the Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the imaginary sorts of the ring of rational adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning from monadic second order imaginaries to first-order reducts, and for proving a certain ``one-way'' model-theoretic orthogonality within the adelic imaginaries.

We use a dynamic programming approach to construct management strategies for a hydropower plant with a dam and a continuously adjustable unit. Along the way, we estimate unknown variables via simple models using historical data and forecasts. Our suggested scheme achieves on average 97.1 % of the theoretical maximum using small computational effort. We also apply our scheme to a Run-of-River hydropower plant and compare the strategies and results to the much more involved PDE-based optimal switching method studied earlier by the authors in (Optimization and Engineering (2021): 1-25); this comparison shows that our simple approach may be preferable if the underlying data is sufficiently rich.

This paper introduces deep super ReLU networks (DSRNs) as a method for approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$ for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily of ReLU DNNs, and several layers of the square of ReLU added at the end to smooth the networks output. This approach retains the advantages of ReLU DNNs, leading to the straightforward training. The paper also proves the optimality of DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs, and obtains the generalization error in Sobolev spaces via an estimate of the pseudo-dimension of higher-order derivatives of DNNs.

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.

We show that the nearby cycles functor for the $p$-adic Hecke stack at parahoric level is perverse t-exact, by developing a theory of Wakimoto filtrations at Iwahori level, and that it lifts to the $\mathbb{E}_1$-center. We apply these tools to construct the Arkhipov-Bezrukavnikov functor for $p$-adic affine flag varieties at Iwahori level, and prove that it is an equivalence for all classical groups and also exceptional groups of type $E_6$ and $E_7$.

Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts each eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as Weyl interlacing, Cauchy interlacing, Dirichlet--Neumann bracketing and so on. We prove a sharp version of the interlacing inequalities for ``finite-dimensional perturbations in boundary conditions'', expressed as bounds on the spectral shift between two self-adjoint extensions of a fixed symmetric operator with finite and equal defect numbers. The bounds are given in terms of the Duistermaat index, a topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the self-adjoint extensions being compared, while the third corresponds to the Friedrichs extension, which acts as a reference point. Along the way several auxiliary results are established, including one-sided continuity properties of the Duistermaat triple index, smoothness of the (abstract) Cauchy data space without unique continuation-type assumptions, and a formula for the Morse index of an extension of a non-negative symmetric operator.

A stochastic process $X$ becomes occupied when it is enlarged with its occupation flow $\mathcal{O}$, which tracks the time spent by the path at each level. Crucially, the occupied process $(\mathcal{O},X)$ enjoys a Markov structure when $X$ is Markov. We develop a novel Itô calculus for occupied processes that lies midway between Dupire's functional Itô calculus and the classical version. We derive a surprisingly simple Itô formula and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where $\mathcal{O}$ plays the role of time. The space variable, given by the current value of $X$, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods. In the financial applications, we demonstrate that occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price and manage derivatives books with a single numerical solver. We then explore avenues in financial modeling where volatility is driven by the occupied process. In particular, we propose the local occupied volatility (LOV) model which not only calibrates to European vanilla options but also offers the flexibility to capture stylized facts of volatility and fit other instruments. We also present an extension of forward variance models that leverages the entire forward occupation surface.

In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with $U-V\in\mathcal{S}_{2}(\hilh)$ (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\|f(U)-f(V)\|_2\leq \|f\|_{\lip(\cir)}\|U-V\|_2$ for all Lipschitz functions $f:\cir\to\C$. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is a strict contraction, which significantly extends the class of functions from results known earlier. More specifically, for each $p\in(1,\infty)$ and for every pair of contractions $(T_0,T_1)$ with $\|T_0\|<1$, there exists a constant $d_{f, p,T_0}>0$ such that $\|f(T_1)-f(T_0)\|_p\leq d_{f,p, T_0}\|T_1-T_0\|_p$ for all Lipschitz functions on $\cir$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations.

We introduce the notion of generalised Gorenstein spin structure on a curve and we give an explicit description of the associated section ring for curves of genus two with ample canonical bundle, obtaining five different formats.

The study of extensions realizing affine datum is specialized to central extensions in varieties with a difference term which leads to generalizations of several classical theorems on central extensions from group theory. We establish a 1-dimensional Hochschild-Serre sequence for a central extension equipped with affine datum. This is used to develop a Schur-Hopf formula which characterizes the $2^{\mathrm{nd}}$-cohomology group of regular datum in terms of the transgression map and commutators in free presentations. We prove, assuming the existence of an idempotent, the existence of covers and provide a cohomological characterization of perfect algebras. The class of varieties with a difference term contain all varieties of algebras with modular congruence lattices; for example, any variety of groups with multiple operators in the parlance of P.J. Higgins or algebras of Loday-type - analogous results recently established for these algebras can be recovered by specialization.

This paper is devoted to the study of source reconstruction algorithms for coupled systems of heat equations, with either constant or spatially dependent coupling terms, where internal measurements are available from a reduced number of observed states. Two classes of systems are considered. The first comprises parabolic equations with constant zero-order coupling terms (through a matrix potential term or via the diffusion matrix). The second type considers parabolic equations coupled by a matrix potential that depends on spatial variables, which leads to the analysis of a non-self-adjoint operator. In all configurations, the source is assumed to be of separate variables, the temporal part is a known scalar function, and the spatial dependence is an unknown vector field. Several numerical examples using the finite element method in 1D and 2D are presented to show the reconstruction of space-dependent sources.

Bent functions are balanced by restricting their domains to vectors with either even or odd Hamming weights, which ensures an equal number of pre-images for both, 0 and 1. Using the previous fact, we can construct bent functions on two affine spaces. Additionally, we employ the Maiorana-McFarland secondary construction to obtain bent functions. Furthermore, the functions produced are balanced when we restrict their domain to vectors with even Hamming weights. We also conduct specific comparisons of the Walsh-Hadamard transform in relation to both the original bent functions and the newly generated ones. With all the necessary tools, we present an algorithm that leverages the Maiorana-McFarland secondary construction approach repeatedly to generate bent functions of any dimension.

We propose a continuous approach to computing the pseudospectra of linear operators with compact or compact-plus-scalar resolvent, following a 'solve-then-discretize' strategy. Instead of taking a finite section approach or using a finite-dimensional matrix to approximate the operator of interest, the new method employs an operator analogue of the Lanczos process to work with operators and functions directly. The method is shown to be free of spectral pollution and spectral invisibility, fully adaptive, and nearly optimal in accuracy. The advantages of the method are demonstrated by extensive numerical examples and comparison with the traditional method.

We explain how the (shifted) Ratios Conjecture for $L(s,\chi)$ would extend a randomization argument of Harper from a conductor-limited range to an unlimited range of ``beyond square-root cancellation'' for character twists of the Liouville function. As a corollary, the Liouville function would have nontrivial cancellation in arithmetic progressions of modulus just exceeding the well-known square-root barrier. Morally, the paper passes from random matrices to random multiplicative functions.

We show that the Yoneda embedding extends to an $(\infty,2)$-natural transformation. Furthermore, as such, it is uniquely determined by its value at the trivial $\infty$-category. We also study the naturality of the Yoneda lemma in its arguments, showing that it is an isomorphism of $(\infty,2)$-natural transformations.

We construct a $p$-adic Rankin-Selberg $L$-function associated to the product of two families of modular forms, where the first is an ordinary (Hida) family, and the second an arbitrary universal-deformation family (without any ordinarity condition at $p$). This gives a function on a 4-dimensional base space - strictly larger than the ordinary eigenvariety, which is 3-dimensional in this case. We prove our $p$-adic $L$-function interpolates all critical values of the Rankin-Selberg $L$-functions for the classical specialisations of our family, and derive a functional equation for our $p$-adic $L$-function.

Let $(\mathcal{H}_k, \mathcal{H}_{\ell})$ be a pair of Hilbert function spaces with kernels $k, \ell$. In a 2005 paper, Shimorin showed that a certain factorization condition on $(k, \ell)$ yields a commutant lifting theorem for multipliers $\mathcal{H}_k\to\mathcal{H}_{\ell}$, thus unifying and extending previous results due to Ball-Trent-Vinnikov and Volberg-Treil. Our main result is a strong converse to Shimorin's theorem for a large class of holomorphic pairs $(k, \ell),$ which leads to a full characterization of the complete Pick property for such pairs. We also present a short alternative proof of sufficiency for Shimorin's condition. Finally, we establish necessary conditions for abstract pairs $(k, \ell)$ to satisfy the complete Pick property, further generalizing Shimorin's work with proofs that are new even in the single-kernel case $k=\ell.$ Our approach differs from Shimorin's in that we do not work with the Nevanlinna-Pick problem directly; instead, we are able to extract vital information for $(k, \ell)$ through Carathéodory-Fejér interpolation.

Neural networks have become standard tools in many areas, yet many important statistical questions remain open. This paper studies the question of how much data are needed to train a ReLU feed-forward neural network. Our theoretical and empirical results suggest that the generalization error of ReLU feed-forward neural networks scales at the rate $1/\sqrt{n}$ in the sample size $n$ rather than the usual "parametric rate" $1/n$. Thus, broadly speaking, our results underpin the common belief that neural networks need "many" training samples.

Considering supercritical Bernoulli percolation on $\mathbb{Z}^d$, Garet and Marchand [GM09] proved a diffusive concentration for the graph distance. In this paper, we sharpen this result by establishing the subdiffusive concentration inequality, which revisits the sublinear bound of the variance proved by Dembin [Dem22] as a consequence. Our approach is inspired by similar work in First-passage percolation [BR08, DHS14], combined with new tools to address the challenge posed by the infinite weight of the model. These tools, including the notion of effective radius and its properties, enable a simple one-step renormalization process as a systematic means of managing the effects of resampling edges.

The spinor-helicity formalism in particle physics gives rise to natural subvarieties in the product of two Grassmannians. These include two-step flag varieties for subspaces of complementary dimension. Taking Hadamard products leads to Mandelstam varieties. We study these varieties through the lens of combinatorics and commutative algebra, and we explore their tropicalization, positive geometry, and scattering correspondence.

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M \subset H$, there exists a reflection mapping the hypersection of $K_1$ defined by $\mathrm{aff}\{p_1, M\}$ onto the hypersection of $K_2$ defined by $\mathrm{aff}\{p_2, M\}$, then there exists a reflection which maps $K_1$ onto $K_2$.

Given a random sample from a density function supported on a manifold $M$, a new method for the estimating highest density regions of the underlying population is introduced. The new proposal is based on the empirical version of the opening operator from mathematical morphology combined with a preliminary estimator of the density function. This results in an estimator that is easy-to-compute since it simply consists of a list of centers and a radius $r$ that are adequately selected from the data. The new estimator is shown to be consistent and its convergence rates in terms of the Hausdorff distance are provided. All consistency results are established uniformly on the level of the set and for any Riemannian manifold $M$ satisfying mild assumptions. The applicability of the procedure is shown by some illustrative examples.

In their study of the giant component in inhomogeneous random graphs, Bollobás, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that the tree structures underlying two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic, a notion introduced by Grebík and Rocha. A different class of branching processes was introduced by Hladký, Nachmias, and Tran in relation to uniform spanning trees in finite graphs approximating a given connected graphon. We prove that that the tree structures of two such branching processes have the same distributions if and only if the corresponding graphons are fractionally isomorphic up to scalar multiple. Combined with a recent result of Archer and Shalev, this implies that if uniform spanning trees of two dense graphs have a similar local structure, they have a similar scaling limit. As a side result we give a characterization of fractional isomorphism for graphs as well as graphons in terms of their connected components.

Radial basis functions (RBFs) play an important role in function interpolation, in particular in an arbitrary set of interpolation nodes. The accuracy of the interpolation depends on a parameter called the shape parameter. There are many approaches in literature on how to appropriately choose it as to increase the accuracy of interpolation while avoiding instability issues. However, finding the optimal shape parameter value in general remains a challenge. In this work, we present a novel approach to determine the shape parameter in RBFs. First, we construct an optimisation problem to obtain a shape parameter that leads to an interpolation matrix with bounded condition number, then, we introduce a data-driven method that controls the condition of the interpolation matrix to avoid numerically unstable interpolations, while keeping a very good accuracy. In addition, a fall-back procedure is proposed to enforce a strict upper bound on the condition number, as well as a learning strategy to improve the performance of the data-driven method by learning from previously run simulations. We present numerical test cases to assess the performance of the proposed methods in interpolation tasks and in a RBF based finite difference (RBF-FD) method, in one and two-space dimensions.

Driven by the surmounting interest for dedicated infrastructure in cislunar space, this work considers the satellite constellation design for cislunar space situational awareness (CSSA). We propose a mixed-integer linear programming (MILP)-based formulation that simultaneously tackles the constellation design and sensor-tasking subproblems surrounding CSSA. Our approach generates constellation designs that provide coverage with considerations for the field-of-view of observers. We propose a time-expanded p-Median problem (TE-p-MP) which considers the optimal placement of p space-based observers into discretized locations based on orbital slots along libration point orbits, simultaneously with observer pointing directions across discretized time. We further develop a Lagrangian method for the TE-p-MP, where a relaxed problem with an analytical solution is derived, and customized heuristics leveraging the orbital structure of candidate observer locations are devised. The performance of the proposed formulation is demonstrated with several case studies for CSSA constellations monitoring the cislunar Cone of Shame and a periodic time-varying transit window for low-energy transfers located in the Earth-Moon L2 neck region. The proposed problem formulation, along with the Lagrangian method, is demonstrated to enable a fast assessment of near-optimal CSSA constellations, equipping decision-makers with a critical technique for exploring the design trade space.

We propose and study a new quasi-interpolation method on spheres featuring the following two-phase construction and analysis. In Phase I, we analyze and characterize a large family of zonal kernels (e.g., the spherical version of Poisson kernel, Gaussian, compactly-supported radial kernels), so that the underlying spherical convolution operators (upon the introduction of a scaling parameter) attains a high-order of approximation to target functions. In Phase II, we discretize the spherical integrals utilizing quadrature rules of optimal order to produce the final quasi-interpolants. Numerical experiments demonstrate that the new quasi-interpolation algorithm is robust and amenable to integrated as well as distributed ways of implementation. Moreover, the underlying error-analysis shows that by fine-tuning the scaling parameter in the radial kernels employed, the resulting quasi-interpolants achieve a well-balanced trade-off between approximation and sampling errors.

We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a stability theory a la Bourgain-Brezis-Mironescu as $s \to 1$, answering a question raised by Brazke--Schikorra--Yung. Part of the results are new even for $p=q$.

Let $ V$ be a braided tensor category and $ C$ a tensor category equipped with a braided tensor functor $G:V\to Z(C)$. For any exact indecomposable $C$-module category $M$, we explicitly construct a right adjoint of the action functor $\rho:Z^V(C)\to C^*_{M}$ afforded by $M$. Here $Z^V(C)$ is the Müger's centralizer of the subcategory $G(V)$ inside the center $Z^V(C)$, also known as the relative center. The construction is parallel to the one presented by K. Shimizu, but using instead the relative coend end. This adjunction turns out to be monadic, thus inducing Hopf monads $T_{V}: C\to C$, such that there is a monoidal equivalence of categories $ C_{T_{V}}\simeq Z^V(C).$ If $\bar{\rho}: C^*_{ M}\to Z^V(C)$ is the right adjoint of $\rho,$ then $\bar{\rho}(Id_{M})$ is the braided commutative algebra constructed in [R. Laugwitz and C. Walton. Braided commutative algebras over quantized enveloping algebras, Transform. Groups 26(3) (2021), 957--993]. As a consequence of our construction of these algebras, in terms of the right adjoint to $\rho$, we can provide a recipe to compute them when $C=Rep(H\# T)$ is the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H\# T$ obtained by bosonization, and choosing an arbitrary $Rep(H\# T)$-module category $M$. We show an explicit example in the case of Taft algebras.

In this paper, we consider the subcritical half-wave equation in one dimension. Let $R_k(t,x)$, $k=1,2$, represent two-solitary wave solutions of the half-wave equation, each with different translations $x_1,x_2$. We prove that if the relative distance $x_2-x_1$ between the two solitary waves is large enough, then the sum of $R_k(t)$ is weakly stable. Our proof relies on an energy method and the local mass monotonicity property. Unlike the single-solitary wave or NLS cases, the interactions between different waves are significantly stronger here. To establish the local mass monotonicity property, as well as to analyze non-local effects on localization functions and non-local operator $D$, we utilize the Carlderón estimate and the integral representation formula of the half-wave operator.

This paper presents a novel boundary-optimized fast Fourier extension algorithm for efficient approximation of non-periodic functions. The proposed methodology constructs periodic extensions through strategic utilization of boundary interval data, which is subsequently combined with original function samples to form an extended periodic representation. We develop a parameter optimization framework that preserves superalgebraic convergence while requiring only a few boundary node deployment, resulting in computational complexity marginally exceeding that of standard FFT implementations. Furthermore, we present an improved version of the algorithm tailored for functions exhibiting boundary oscillations. This variant employs grid refinement near the boundaries, which reduces the resolution constant to approximately one-fourth of that in conventional approaches. Comprehensive numerical experiments confirm the efficiency and accuracy of the proposed methods and establish practical guidelines for parameter selection.

This article investigates Dehornoy's monomial representations for structure groups and Coxeter-like groups of a set-theoretic solution to the Yang-Baxter equation. Using the brace structure of these two groups and the language of cycle sets, we relate the irreducibility of monomial representations to the indecomposability of the solutions. Furthermore, in the case of an indecomposable solution, we show how to obtain these representations by induction from explicit one-dimensional representations.

The mean squared error and regularized versions of it are standard loss functions in supervised machine learning. However, calculating these losses for large data sets can be computationally demanding. Modifying an approach of J. Dick and M. Feischl [Journal of Complexity 67 (2021)], we present algorithms to reduce extensive data sets to a smaller size using rank-1 lattices. Rank-1 lattices are quasi-Monte Carlo (QMC) point sets that are, if carefully chosen, well-distributed in a multidimensional unit cube. The compression strategy in the preprocessing step assigns every lattice point a pair of weights depending on the original data and responses, representing its relative importance. As a result, the compressed data makes iterative loss calculations in optimization steps much faster. We analyze the errors of our QMC data compression algorithms and the cost of the preprocessing step for functions whose Fourier coefficients decay sufficiently fast so that they lie in certain Wiener algebras or Korobov spaces. In particular, we prove that our approach can lead to arbitrary high convergence rates as long as the functions are sufficiently smooth.

Gaiotto, Moore, and Neitzke predicted that the hyperkähler Ooguri-Vafa space $\mathcal{M}^{\rm ov}$ should provide a local model for Hitchin moduli spaces near the discriminant locus. To this end, Tulli identified $\mathcal{M}^{\rm ov}$ with a certain space of framed Higgs bundles with an irregular singularity. We extend this result by identifying the Ooguri-Vafa holomorphic symplectic form with a regularized version of the Atiyah-Bott form on the associated space of framed connections. We also prove the analogous statement for the corresponding semiflat forms. Finally, restricting to the Hitchin section, we identify a regularized version of Hitchin's $L^2$-metric with the Ooguri-Vafa metric.

In this paper, we prove a conjecture proposed by Schnell concerning the Kodaira dimensions of algebraic fiber spaces. As a consequence, we establish the equivalence between the non-vanishing conjecture and its generalized form, as proposed by Campana and Peternell.

Given a local $\otimes$-triangulated category, and a fiber sequence $y\to 1 \to x$, one may ask if there is always a nonzero object $z$ such that either $z\otimes f$ or $z\otimes g$ is $\otimes$-nilpotent. The claim that this property holds for all local $\otimes$-triangulated categories is equivalent to Balmer's "nerves of steel conjecture" from arXiv:2001.00284. In the present paper, we will see how this property can fail if the category we start with is not rigid, discuss a large class of categories where the property holds, and ultimately prove that the nerves of steel conjecture is equivalent to a stronger form of this property.

We present new expressions for the $k$-generalized Fibonacci numbers, say $F_k(n)$. They satisfy the recurrence $F_k(n) = F_k(n-1) +\dots+F_k(n-k)$. Explicit expressions for the roots of the auxiliary (or characteristic) polynomial are presented, using Fuss-Catalan numbers. Properties of the roots are enumerated. We quantify the accuracy of asymptotic approximations for $F_k(n)$ for $n\gg1$. Our results subsume and extend some results published by previous authors. We also present a basis (or `fundamental solutions') to solve the above recurrence for arbitrary initial conditions. We comment on the use of generating functions and multinomial sums for the $k$-generalized Fibonacci numbers and related sequences. We note that the resulting multinomial sums are Dickson polynomials of the second kind in several variables. We also present what may be a new identity for companion matrices.

Identification capacity has been established as a relevant performance metric for various goal-/task-oriented applications, where the receiver may be interested in only a particular message that represents an event or a task. For example, in olfactory molecular communications (MCs), odors or pheromones, which are often a mixture of various molecule types, may signal nearby danger, food, or a mate. In this paper, we examine the identification capacity with deterministic encoder for the discrete affine Poisson channel which can be used to model MC systems with molecule counting receivers. We establish lower and upper bounds on the identification capacity in terms of features of the affinity matrix between the released molecules and receptors at the receiver. As a key finding, we show that even when the number of receptor types scales sub-linearly in the number of molecule types $N,$ the number of reliably identifiable messages can grow super-exponentially with the rank of the affinity matrix, $T,$ i.e., $\sim 2^{(T \log T)R},$ where $R$ denotes the coding rate. We further derive lower and upper bounds on $R,$ and show that the proposed capacity theorem includes several known results in the literature as its special cases.

We derive a necessary and sufficient condition for a homeomorphism with the shadowing property to be topologically transitive: to have an invariant subset $A$, dense in the non-wandering set, where the barycenter property holds. To elucidate its dynamical nature, we compare this condition with other properties known to be sufficient for an Anosov diffeomorphism to be topologically transitive. We also describe the $C^1$ interior of the set of diffeomorphisms which comply with this condition, discuss examples with a variety of dynamics and present some applications of interest.

We prove new results concerning the topology and Hodge theory of singular varieties. A common theme is that concrete conditions on the complexity of the singularities, from a number of different perspectives, are closely related to the symmetries of the Hodge-Du Bois diamond. We relate this to the theory of rational homology manifolds, and characterize these among low-dimensional varieties with rational singularities.

Refined canonical stable Grothendieck polynomials were introduced by Hwang, Jang, Kim, Song, and Song. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan, Pappe, Poh, and Schilling. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden--Greene's jeu de taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieck polynomials in terms of biflagged tableaux, which naturally appear in the characterization of the image of the uncrowding map.

It is known that standard stochastic Galerkin methods face challenges when solving partial differential equations (PDEs) with random inputs. These challenges are typically attributed to the large number of required physical basis functions and stochastic basis functions. Therefore, it becomes crucial to select effective basis functions to properly reduce the dimensionality of both the physical and stochastic approximation spaces. In this study, our focus is on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC). We delve into the low-rank approximation of the quasimatrix, whose columns represent the coefficients in the gPC expansions of the solution. We conduct an investigation into the singular value decomposition (SVD) of this quasimatrix, proposing a strategy to identify the rank required for a desired accuracy. Subsequently, we introduce both a simultaneous low-rank projection approach and an alternating low-rank projection approach to compute the low-rank approximation of the solution for PDEs with random inputs. Numerical results demonstrate the efficiency of our proposed methods for both diffusion and Helmholtz problems.

We discuss a general method to prove quantitative improvements on correlation inequalities and apply it to arm estimates for Bernoulli bond percolation on the square lattice. Our first result is that the two-arm exponent is strictly larger than twice the one-arm exponent and can be seen as a quantitative improvement on the Harris-FKG inequality. This answers a question of Garban and Steif, which was motivated by the study of exceptional times in dynamical percolation. Our second result is that the monochromatic arm exponents are strictly larger than their polychromatic versions, and can be seen as a quantitative improvement on Reimer's main lemma. This second result is not new and was already proved by Beffara and Nolin using a different argument.

Answering a conjecture by S. Kobayashi, in 1986, K. Sekigawa and L. Vanhecke proved that an almost hermitian manifold whose local geodesic symmetries preserve the Kähler 2-form is a locally symmetric hermitian space. In the present paper, we relax the hermitean hypothesis by only requiring the manifold to be symplectic. In other words, we study the symplectic manifolds equipped with a symplectic connection whose geodesic symmetries are (local) symplectomorphisms. We call ``S-type'' these affine symplectic manifolds.

We investigate the depth of the tensor product of finitely generated modules over local rings. One of the main ingredients of our approach is a lifting construction introduced by Huneke, Jorgensen, and Wiegand. We recover a result of Celikbas, Sadeghi, and Takahashi for local complete intersection rings. Additionally, we provide a negative answer to a question they asked and establish a corresponding lower bound. We derive a result on the depth of the tensor product of certain modules over local complete $\mathcal{TE}$ rings. Some general conditions on the existence of hulls and approximations are also studied.

We present an adaptive trust-region method for unconstrained optimization that allows inexact solutions to the trust-region subproblems. Our method is a simple variant of the classical trust-region method of \citet{sorensen1982newton}. The method achieves the best possible convergence bound up to an additive log factor, for finding an $\epsilon$-approximate stationary point, i.e., $O( \Delta_f L^{1/2} \epsilon^{-3/2}) + \tilde{O}(1)$ iterations where $L$ is the Lipschitz constant of the Hessian, $\Delta_f$ is the optimality gap, and $\epsilon$ is the termination tolerance for the gradient norm. This improves over existing trust-region methods whose worst-case bound is at least a factor of $L$ worse. We compare our performance with state-of-the-art trust-region (TRU) and cubic regularization (ARC) methods from the GALAHAD library on the CUTEst benchmark set on problems with more than 100 variables. We use fewer function, gradient, and Hessian evaluations than these methods. For instance, our algorithm's median number of gradient evaluations is $23$ compared to $36$ for TRU and $29$ for ARC. Compared to the conference version of this paper \cite{hamad2022consistently}, our revised method includes several practical enhancements. These modifications dramatically improved performance, including an order of magnitude reduction in the shifted geometric mean of wall-clock times. We also show it suffices for the second derivatives to be locally Lipschitz to guarantee that either the minimum gradient norm converges to zero or the objective value tends towards negative infinity, even when the iterates diverge.

Let $C_\ell/\mathbb Q$ denote the curve with affine model $y^\ell=x(x^\ell-1)$, where $\ell\geq 3$ is prime. In this paper we study the limiting distributions of the normalized $L$-polynomials of the curves by computing their Sato-Tate groups and distributions. We also provide results for the number of points on the curves over finite fields, including a formula in terms of Jacobi sums when the field $\mathbb F_q$ satisfies $q\equiv 1 \pmod{\ell^2}$.

We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength perturbations approximately obey a system of dispersive nonlinear wave equations. Computational experiments demonstrate that solutions of the 1D Euler equations agree well with this dispersive model, with solutions consisting mainly of solitary waves. Shock formation seems to be avoided for moderate-amplitude initial data, while shock formation occurs for larger initial data. We investigate the threshold for transition between these behaviors, validating a previously-proposed criterion based on further computational experiments. These results support the existence of large-time non-breaking solutions to the 1D compressible Euler equations, as hypothesized in previous works.

The robustness of fault detection algorithms against uncertainty is crucial in the real-world industrial environment. Recently, a new probabilistic design scheme called distributionally robust fault detection (DRFD) has emerged and received immense interest. Despite its robustness against unknown distributions in practice, current DRFD focuses on the overall detectability of all possible faults rather than the detectability of critical faults that are a priori known. Henceforth, a new DRFD trade-off design scheme is put forward in this work by utilizing prior fault information. The key contribution includes a novel distributional robustness metric of detecting a known fault and a new relaxed distributionally robust chance constraint that ensures robust detectability. Then, a new DRFD design problem of fault detection under unknown probability distributions is proposed, and this offers a flexible balance between the robustness of detecting known critical faults and the overall detectability against all possible faults. To solve the resulting semi-infinite chance-constrained problem, we first reformulate it to a finite-dimensional problem characterized by bilinear matrix inequalities (BMIs). Subsequently, a customized solution algorithm is developed, which includes a sequential optimization procedure and an initialization strategy. Finally, case studies on a simulated three-tank system and a real-world battery cell are carried out to showcase the usefulness of our DRFD method.

The famous Goldbach conjecture remains open for nearly three centuries. Recently Goldbach graphs are introduced to relate the problem with the literature of Graph Theory. It is shown that the connectedness of the graphs is equivalent to the affirmative answer of the conjecture. Some modified version of the graphs, say, near Goldbach graphs are shown to be Hamiltonian for small number of vertices. In this context, we introduce a class of graphs, namely, prime multiple missing graphs such that near Goldbach graphs are finite intersections of these graphs. We study these graphs for primes 3,5 and in general for any odd prime p. We prove that these graphs are connected with diameter at most 3 and Hamiltonian for even (>2) vertices. Next the intersection of prime multiple missing graphs for primes 3 and 5 are studied. We prove that these graphs are connected with diameter at most 4 and they are also Hamiltonian for even (>2) vertices. We observe that the diameters of finite Goldbach graphs and near Goldbach graphs are bounded by 5 (up to 10000 vertices). We believe further study on these graphs with big data analysis will help to understand structures of near Goldbach graphs.

Hybridizable \(H(\textrm{div})\)-conforming finite elements for symmetric tensors on simplices with barycentric refinement are developed in this work for arbitrary dimensions and any polynomial order. By employing barycentric refinement and an intrinsic tangential-normal (\(t\)-\(n\)) decomposition, novel basis functions are constructed to redistribute degrees of freedom while preserving \(H(\textrm{div})\)-conformity and symmetry, and ensuring inf-sup stability. These hybridizable elements enhance computational flexibility and efficiency, with applications to mixed finite element methods for linear elasticity.

Consider the nonlinear stochastic heat equation $$ \frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ \sigma(u (t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$ in the space variable. When $\sigma(0)=0$, the well-posedness of the solution and its Hölder continuity have been proved by Hu et al. \cite{HHLNT2017}. In this paper, we study the asymptotic properties of the temporal gradient $u(t+\varepsilon, x)-u(t, x)$ at any fixed $t \ge 0$ and $x\in \mathbb R$, as $\varepsilon\downarrow 0$. As applications, we deduce Khintchine's law of iterated logarithm, Chung's law of iterated logarithm, and a result on the $q$-variations of the temporal process $\{u(t, x)\}_{t \ge 0}$, where $x\in \mathbb R$ is fixed.

We consider Banach spaces $X$ that can be linearly lifted into their Lipschitz-free spaces $\mathcal{F}(X)$ and, for a group $G$ acting on $X$ by linear isometries, we study the possible existence of $G$-equivariant linear liftings. In particular, we prove that such lifting exists when $G$ is compact in the strong operator topology, or an increasing union of such groups and $\mathcal{F}(X)$ is complemented in its bidual by an equivariant projection. As an example of application, we define and study a complex version of the Lipschitz-free space $\mathcal{F}(X)$ when $X$ is a subset of a complex Banach space stable under the action of the circle group.

Log-canonical and $F$-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form $p^a + x^b \in {\bf Z}_p [[ x ]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a + x^b \in {\bf F}_p [[ t, x ]]$. We also discuss a few other sporadic examples.

We study sublevel set and superlevel set persistent homology on discrete functions through the perspective of finite ordered sets of both linearly ordered and cyclically ordered domains. Finite ordered sets also serve as the codomain of our functions making all arguments finite and discrete. We prove duality of filtrations of sublevel sets and superlevel sets that undergirths a range of duality results of sublevel set persistent homology without the need to invoke complications of continuous functions or classical Morse theory. We show that Morse-like behavior can be achieved for flat extrema without assuming genericity. Additionally, we show that with inversion of order, one can compute sublevel set persistence from superlevel set persistence, and vice versa via a duality result that does not require the boundary to be treated as a special case. Furthermore, we discuss aspects of barcode construction rules, surgery of circular and linearly ordered sets, as well as surgery on auxiliary structures such as box snakes, which segment the ordered set by extrema and monotones.

We introduce a new mathematical model to explore the dynamic relationship between prescribed burning and bushfire occurrence, formulated as a system of ordinary differential equations. The model admits a unique steady-state, and its stability is shown to depend critically on the policy framework governing prescribed burning. In particular, reactive policies, where prescribed burning is increased in response to bushfire events, can lead to system instability unless augmented with appropriate feedback control. Conversely, proactive policies that implement moderate, consistent prescribed burning are more effective at stabilizing the system and mitigating the frequency of bushfires. Additionally, higher vegetation regeneration rates contribute positively to equilibrium stability. While this model does not capture the full complexity of fire management, it highlights the potential risks of emotionally driven, reactive responses and underscores the value of preventive, stability-focused strategies. To the best of our knowledge, this is the first mathematical analysis of preventive burning strategies and their long-term impact on bushfire mitigation.

This paper addresses the critical problem of co-optimizing the optimal locations for orbital depots and the sequence of in-space servicing for a satellite constellation. While most traditional studies used network optimization for this problem, assuming a fixed set of discretized nodes in the network (i.e., a limited number of depot location candidates), this work is unique in that it develops a method to optimize the depot location in continuous space. The problem is formulated as mixed-integer nonlinear programming, and we propose a solution methodology that iteratively solves two decoupled problems: one using mixed-integer linear programming and the other using nonlinear programming with an analytic transfer solution. To demonstrate the effectiveness of our approach, we apply this methodology to a case study involving a GPS satellite constellation. Numerical experiments confirm the stability of our proposed solutions.

We introduce a new class of sparse sequences that are ergodic and pointwise universally $L^2$-good for ergodic averages. That is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions. These sequences are generated randomly as return or hitting times in systems exhibiting a rapid correlation decay. This can be seen as a natural variant of Bourgain's Return Times Theorem. As an example, we obtain that for any $a\in (0,1/2)$, the sequence $\left\{n\in\mathbb{N}:\ 2^ny\mod{1}\in (0,n^{-a})\right\}$ is ergodic and pointwise universally $L^2$-good for Lebesgue almost every $y\in [0,1]$. Our approach builds on techniques developed by Frantzikinakis, Lesigne, and Wierdl in their study of sequences generated by independent random variables, which we adapt to the non-independent case.

A new way of encoding a non-self-adjoint Jacobi matrix $J$ by a spectral measure of $|J|$ together with a phase function was described by Pushnitski--\v Stampach in the bounded case. We present another perspective on this correspondence, based on Weyl functions instead of moments, which simplifies some proofs and generalizes the correspondence to the unbounded case. In particular, we find a bijection between proper Jacobi matrices with positive off-diagonal elements, and a class of spectral data. We prove that this mapping is continuous in a suitable sense. To prove injectivity of the map, we prove a local Borg--Marchenko theorem for unbounded non-self-adjoint Jacobi matrices in this class that may be of independent interest.

In this paper we explore the algebraic structure and combinatorial properties of eulerian magnitude homology. First, we analyze the diagonality conditions of eulerian magnitude homology, providing a characterization of complete graphs. Then, we construct the regular magnitude-path spectral sequence as the spectral sequence of the (filtered) injective nerve of the reachability category, and explore its consequences. Among others, we show that such spectral sequence converges to the complex of injective words on a digraph, and yields characterization results for the regular path homology of diagonal directed graphs.

Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the Fermat's factorization algorithm [1 p.6,3 p.1470], implementation technique which I call the Fermat's factorization method (like the title, translated into English, of the reference document [2] published in Italian) and which is, among the implementation techniques [1 pp.6-8,2,3 pp.1470-1471] of the Fermat's factorization algorithm, the one with which a smaller iterations number occurs to identify the factors, trivial or non-trivial, of n (except for the circumstance in which two factors, trivial or non-trivial, of n are so close to each other that they are identified at the 1st iteration with each of the implementation techniques of the Fermat's factorization algorithm). In fact, through the way in which the Euler's function [4] is applied to the Fermat's factorization method, we arrive at this new factorization algorithm with which we obtain the certain reduction in the iterations number (except for the cases in which two factors of n are so close to each other that they are identified at the 1st iteration with the Fermat's factorization method) compared to the iterations number that occurs with the Fermat's factorization method. Furthermore, in this article I represent the hypotheses field according to which it will eventually be possible to further reduce the iterations number. Finally and always in relation to this new factorization algorithm, in this article I represent in detail the limit iterations number, which is smaller than the iterations number that occurs to reach the condition x - y = 1 which characterizes the pair of trivial factors of n, beyond which it is no longer possible for pairs of non-trivial factors of n to occur.

We develop analysis of Hölder continuous mappings with applications to geometry and topology of the Heisenberg groups. We cover the theory of distributional Jacobians of Hölder continuous mappings and pullbacks of differential forms under Hölder continuous mappings. That includes versions of the change of variables formula and the Stokes theorem for Hölder continuous mappings. The main applications are in the setting of the Heisenberg groups, where we provide a simple proof of a generalization of the Gromov non-embedding theorem, and new results about the Hölder homotopy groups of the Heisenberg groups.

In this work, we extend the celebrated result of Avila--Forni~\cite{avila2007weak} on the weak mixing property of interval exchange transformations to the setting of linear involutions, which naturally arise from the study of vertical foliations on half-translation surfaces. Using recent advances on the Kontsevich--Zorich cocycle for quadratic differentials~\cite{belldiagonal, gutierrez2019classification, trevino2013non}, we establish that, for every dynamically irreducible generalized permutation, the associated linear involution is weakly mixing for almost every admissible parameter.

We study the propagation of energy in one-dimensional anharmonic chains subject to a periodic, localized forcing. For the purely harmonic case, forcing frequencies outside the linear spectrum produce exponentially localized responses, preventing equi-distribution of energy per degree of freedom. We extend this result to anharmonic perturbations with bounded second derivatives and boundary dissipation, proving that for small perturbations and non-resonant forcing, the dynamics converges to a periodic stationary state with energy exponentially localized uniformly in the system size. The perturbed periodic state is described by a convergent power type expansion in the strength of the anharmonicity. This excludes chaoticity induced by anharmonicity, independently of the size of the system. Our perturbative scheme can also be applied in higher dimensions.

We introduce symmetric cone games (SCGs), a broad class of multi-player games where each player's strategy lies in a generalized simplex (the trace-one slice of a symmetric cone). This framework unifies a wide spectrum of settings, including normal-form games (simplex strategies), quantum games (density matrices), and continuous games with ball-constrained strategies. It also captures several structured machine learning and optimization problems, such as distance metric learning and Fermat-Weber facility location, as two-player zero-sum SCGs. To compute approximate Nash equilibria in two-player zero-sum SCGs, we propose a single online learning algorithm: Optimistic Symmetric Cone Multiplicative Weights Updates (OSCMWU). Unlike prior methods tailored to specific geometries, OSCMWU provides closed-form, projection-free updates over any symmetric cone and achieves an optimal $\tilde{\mathcal{O}}(1/\epsilon)$ iteration complexity for computing $\epsilon$-saddle points. Our analysis builds on the Optimistic Follow-the-Regularized-Leader framework and hinges on a key technical contribution: We prove that the symmetric cone negative entropy is strongly convex with respect to the trace-one norm. This result extends known results for the simplex and spectraplex to all symmetric cones, and may be of independent interest.

This work explores the rich structure of spherically symmetric time-periodic solutions of the cubic conformal wave equation on $\mathbb{S}^{3}$. We discover that the families of solutions bifurcating from the eigenmodes of the linearised equation form patterns similar to the ones observed for the cubic wave equation. Alongside the Galerkin approaches, we study them using the new method based on the Padé approximants. To do so, we provide a rigorous perturbative construction of solutions. Due to the conformal symmetry, the solutions presented in this work serve as examples of large time-periodic solutions of the conformally coupled scalar field on the anti-de Sitter background.

The projection filter is one of the approximations to the solution of the optimal filtering problem. It approximates the filtering density by projecting the dynamics of the square-root filtering density onto the tangent space of the square-root parametric density manifold. While the projection filters for exponential and mixture families with continuous measurement processes have been well studied, the continuous-discrete projection filtering algorithm for non-conjugate priors has received less attention. In this paper, we introduce a simple Riemannian optimization method to be used for the Bayesian update step in the continuous-discrete projection filter for exponential families. Specifically, we show that the Bayesian update can be formulated as an optimization problem of $\alpha$-Rényi divergence, where the corresponding Riemannian gradient can be easily computed. We demonstrate the effectiveness of the proposed method via two highly non-Gaussian Bayesian update problems.

We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.

We study the Takagi-van der Waerden functions $f_r (x)$, a well-known class of continuous but nowhere differentiable functions, from probabilistic point of view. As an application of elephant random walks remembering the very recent past (ERWVRP, a.k.a. symmetric correlated random walks), we obtain precise estimates for the oscillations of $f_r (x)$. We also establish a result on the necessary and sufficient condition for localization of the ERWVRP with variable step length, which can be applied to obtain a complete description of the differentiability properties of the Takagi-van der Waerden class functions.

In this paper, we explore a modified Leslie-Gower type predator-prey model with Holling I functional response and addictive Allee effect in prey. It is shown that the highest codimension of a nilpotent cusp 4, and the model can undergo degenerate Bogdanov-Takens bifurcation of codimension 4. Besides, when the model has a center-type equilibrium, we show that the model can admit five limit cycles bifurcated from Hopf bifurcation. Our results indicate that addictive Allee effect can induce not only richer dynamics and bifurcations, but also the coextinction of both populations with some positive initial densities. Finally, numerical simulations, including three limit cycles and four limit cycles, are presented to illustrate the theoretical results.

In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player shares the same feedback form with the strategy of the social equilibrium, we say the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations(FBSDEs). By employing the elliptic master equations, a class of distribution-dependent elliptic PDEs , we provide a representation for the Nash equilibrium. We prove the Yamada-Watanabe theorem and show the weak uniqueness for infinite-time FBSDEs. And we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.

The article presents four reasons why the elliptic genus is the most general characteristic class that admits a generalization to singular spaces. We prove that the elliptic characteristic class (with an additional factor) is essentially the only characteristic class invariant under certain modifications, such as the Atiyah flop, Grassmannian flops, and modifications of Bott-Samelson this http URL result confirms and extends Totaro's result concerning the cobordism ring modulo classical flops. However, our approach is based on local calculus in equivariant cohomology.

The basic question of delineating those statistical problems that are solvable without making any assumptions on the underlying data distribution has long animated statistics and learning theory. This paper characterizes when a convex M-estimation or stochastic optimization problem is solvable in such an assumption-free setting, providing a precise dividing line between solvable and unsolvable problems. The conditions we identify show, perhaps surprisingly, that Lipschitz continuity of the loss being minimized is not necessary for distribution free minimization, and they are also distinct from classical characterizations of learnability in machine learning.

An analysis of the Markov tree is presented. Markov triplets, {x,R,z}, are the positive integer solutions to the Diophantine equation x2 + R2 + z2 = 3xRz. Inspired by patterns of the Fibonacci and Pell triplets in Region 1 and Region 2 of the tree, an investigation of interior regions of the Markov tree finds generating functions and sequence functions for all triplets of all regions. These sequence functions lead to the discovery of a Pell equation for the Markov region numbers along the edges of all regions. Analysis of this Pell equation leads to the resolution of the Uniqueness Conjecture. Further analysis using these sequence functions finds palindromic repeat cycles of the last digits of region numbers along the edges of all regions. Then, since all Markov numbers are the sum of the squares of two integers and again inspired by the patterns of the two unique squares which sum to form the region numbers of certain Fibonacci triplets in Region 1, an investigation of interior regions of the Markov tree finds generating functions and sequence functions for the two special square terms which sum to form the region numbers of the triplets along the edges of all regions. Further analysis using these sequence functions finds palindromic repeat cycles of the last digits of these two special square terms for all regions.

The vivid contrast between two competing algorithms for solving Abel's equation $g(\theta(x)) = g(x) + 1$, given $\theta(x)$, is easily sketched. EJ is faster and more efficient, but ML evaluates a limit characterizing the principal solution $g(x)$ directly. EJ finds $g(x)+\delta$, where $\delta$ is possibly nonzero but independent of $x$. If we were to know an exact expression for $\delta$, then the "intrinsicality" of ML would be subsumed by EJ. Filling this gap in our knowledge is the aim of this paper.

The first-order linear positivity preserving schemes in time are available for the time dependent Poisson-Nernst-Planck (PNP) equations, second-order linear ones are still challenging. In this paper, we propose the first- and second-order exponential time differencing schemes with the finite difference spatial discretization for PNP equations, based on the Slotboom transformation of the Nernst-Planck equation. The proposed schemes are linear and preserve the mass conservation and positivity preservation of ion concentration at full discrete level without any constraints on the time step size. The corresponding energy stability analysis is also presented, demonstrating that the second-order scheme can dissipate the modified energy. Extensive numerical results are carried out to support the theoretical findings and showcase the performance of the proposed schemes.

In this paper, we construct a partial group \(\mathcal{P}(F)\) that represents the "partial symmetry" inherent in a subset \(F\) of \(d\)-dimensional Euclidean space. In cases where \(F\) is not connected, \(\mathcal{P}(F)\) captures more detailed information than the conventional symmetry group \(G(F)\). To establish a stronger connection between \(\mathcal{P}(F)\) and \(F\), we introduce a novel definition of partial group action. Furthermore, to characterize \(\mathcal{P}(F)\) in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.

We show that Schur classes of ample vector bundles on smooth projective varieties satisfy Hodge-Riemann relations on $H^{p,q}$ under the assumption that $H^{p-2,q-2}$ vanishes. More generally, we study Hodge-Riemann polynomials, which are partially symmetric polynomials that produce cohomology classes satisfying the Hodge-Riemann property when evaluated at Chern roots of ample vector bundles. In the case of line bundles and in bidegree $(1,1)$, these are precisely the nonzero dually Lorentzian polynomials. We prove various properties of Hodge-Riemann polynomials, confirming predictions and answering questions of Ross and Toma. As an application, we show that the derivative sequence of any product of Schur polynomials is Schur log-concave, confirming conjectures of Ross and Wu.

We study a mathematical model of voting contest with $m$ voters and $n$ candidates, with each voter ranking the candidates in order of preference, without ties. A Condorcet winner is a candidate who gets more than $m/2$ votes in pairwise contest with every other candidate. An ``impartial culture'' setting is the case when each voter chooses his/her candidate preference list uniformly at random from all $n!$ preferences, and does it independently of all other voters. For impartial culture case, Robert May and Lisa Sauermann showed that when $m=2k-1$ is fixed ($k=2$ and $k>2$ respectively), and $n$ grows indefinitely, the probability of a Condorcet winner is small, of order $n^{-(k-1)/k}$. We show that when $m$ grows indefinitely and $m\gg n^4$, the probability of a Condercet winner is of order n^{-1}\log^{1/2}n \cdot \log\log n, thus converges to zero.

Based on the $\phi(\Delta)$-operator introduced by Kim \cite{Kim2}, where $\phi$ is a Bernstein function, we establish weighted $L_q(L_p)$-estimates for solutions to the fractional evolution equation $$ \partial_t^\alpha w(t,x) = \phi(\Delta) w(t,x) + h(t,x), \quad w(0,\cdot) = w_0, \quad t > 0, \; x \in \mathbb{R}^d, $$ where $\partial_t^\alpha$ denotes the Caputo derivative of order $\alpha \in (0,1)$. Specifically, for all $p, q \in (1, \infty)$ and $\gamma \in \mathbb{R}$, we prove that $$ \left\| \partial_t^\alpha w \right\|_{L_q(0,T,\mu_2dt; H^{\phi,\gamma}_p(\mu_1))} + \left\| \phi(\Delta) w \right\|_{L_q(0,T,\mu_2dt; H^{\phi,\gamma}_p(\mu_1))} \leq C \left( \left\| h \right\|_{L_q(0,T,\mu_2dt; H^{\phi,\gamma}_p(\mu_1))} + \left\| w_0 \right\|_{N_{\alpha,p,\phi}} \right), $$ where $\mu_1(x) \in A_p(\mathbb{R}^d)$ and $\mu_2(t) \in A_q(\mathbb{R})$ are Muckenhoupt weights, and $N_{\alpha,p,\phi}$ is a suitably defined Banach space capturing the initial data $w_0$. For instance, when $\mu_2 \equiv 1$ and $\alpha q > 1$, we have $N_{\alpha,p,\phi} \sim B^{\phi,\gamma+2-\frac{2}{\alpha q}}_{p,q}(\mu_1)$. Our approach relies on tools from harmonic analysis, including the Fefferman--Stein inequality, Hardy--Littlewood maximal estimates in weighted mixed-norm spaces, and sharp function techniques for estimating solution operators. These results extend earlier work by Han and Kim \cite[\emph{J.~Differ.~Equ.}]{Han,Kim1} and complement those of Dong \cite[\emph{Calc.~Var.~Partial~Differ.~Equ.}]{Dong}.

We investigate the competing mechanisms of localisation in one-dimensional block disordered subwavelength resonator systems subject to non-reciprocal damping, induced by an imaginary gauge potential. Using a symmetrisation approach to enable the adaptation of tools from Hermitian systems, we derive the limiting spectral distribution of these systems as the number of blocks goes to infinity and characterise their spectral properties in terms of the spectral properties of their constituent blocks. By employing a transfer matrix approach, we then clarify, in terms of Lyapunov exponents, the competition between the edge localisation due to imaginary gauge potentials and the bulk localisation due to disorder. In particular, we demonstrate how the disorder acts as insulation against the non-Hermitian skin effect, preventing edge localisation for small imaginary gauge potentials.

We study the Frank-Wolfe algorithm for constrained optimization problems with relatively smooth objectives. Building upon our previous work, we propose a fully adaptive variant of the Frank-Wolfe method that dynamically adjusts the step size. Our method does not require prior knowledge of the function parameters and guarantees convergence using only local information. We establish a linear convergence rate under relative strong convexity and provide a detailed theoretical analysis of the proposed adaptive step-size rule. Furthermore, we demonstrate how relative smoothness and strong convexity naturally arise in the setting of centralized distributed optimization. Under a variance-type assumption on the gradients, we show that the global objective becomes relatively strongly convex with respect to the Bregman divergence generated by a local function. This structure allows us to apply our adaptive Frank-Wolfe algorithm, leading to provable acceleration due to an improved relative condition number.

Earth observation satellites (EOS) play a pivotal role in capturing and analyzing planetary phenomena, ranging from natural disasters to societal development. The EOS scheduling problem (EOSSP), which optimizes the schedule of EOS, is often solved with respect to nadir-directional EOS systems, thus restricting the observation time of targets and, consequently, the effectiveness of each EOS. This paper leverages state-of-the-art constellation reconfigurability to develop the reconfigurable EOS scheduling problem (REOSSP), wherein EOS are assumed to be maneuverable, forming a more optimal constellation configuration at multiple opportunities during a schedule. This paper develops a novel mixed-integer linear programming formulation for the REOSSP to optimally solve the scheduling problem for given parameters. Additionally, since the REOSSP can be computationally expensive for large-scale problems, a rolling horizon procedure (RHP) solution method is developed. The performance of the REOSSP is benchmarked against the EOSSP, which serves as a baseline, through a set of random instances where problem characteristics are varied and a case study in which Hurricane Sandy is used to demonstrate realistic performance. These experiments demonstrate the value of constellation reconfigurability in its application to the EOSSP, yielding solutions that improve performance, while the RHP enhances computational runtime for large-scale REOSSP instances.

We study the differential smoothness of Sklyanin algebras in three and four variables. We show that all non-degenerate three-dimensional cases are differentially smooth, while none of the four-dimensional Sklyanin algebras admit a connected integrable differential calculus of suitable dimension.

We give a new proof of the operator extension of the strong subadditivity of von Neumann entropy $\rho_{AB} \otimes \sigma_{C}^{-1} \leq \rho_{A} \otimes \sigma_{BC}^{-1}$ by identifying the mathematical structure behind it as Connes' theory of spatial derivatives. This immediately generalizes the inequality to arbitrary inclusions of von Neumann algebras. In the case of standard representations, it reduces to the monotonicity of the relative modular operator.

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to all orders on the lower bound and $L_2$ on the upper, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. Our theory achieves the desired form but may require slightly more smoothness on the kernel than the regular $>d/2$ assumption. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

In this paper, we focus on the problem of optimal portfolio-consumption policies in a multi-asset financial market, where the n risky assets follow Exponential Ornstein-Uhlenbeck processes, along with one risk-free bond. The investor's preferences are modeled using Constant Relative Risk Aversion utility with state-dependent stochastic discounting. The problem can be formulated as a high-dimensional stochastic optimal control problem, wherein the associated value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation, which constitutes a necessary condition for optimality. We apply a variable separation technique to transform the HJB equation to a system of ordinary differential equations (ODEs). Then a class of hybrid numerical approaches that integrate exponential Rosenbrock-type methods with Runge-Kutta methods is proposed to solve the ODE system. More importantly, we establish a rigorous verification theorem that provides sufficient conditions for the existence of value function and admissible optimal control, which can be verified numerically. A series of experiments are performed, demonstrating that our proposed method outperforms the conventional grid-based method in both accuracy and computational cost. Furthermore, the numerically derived optimal policy achieves superior performance over all other considered admissible policies.

The focus of the paper is on the maximal dimension of affine subspaces of nilpotent $n \times n $ matrices with fixed rank. In particular we obtain two results in the "border" cases rank equal to $n-1$ and rank equal to $1$.

Rényi divergences play a pivotal role in information theory, statistics, and machine learning. While several estimators of these divergences have been proposed in the literature with their consistency properties established and minimax convergence rates quantified, existing accounts of probabilistic bounds governing the estimation error are relatively underdeveloped. Here, we make progress in this regard by establishing exponential deviation inequalities for smoothed plug-in estimators and neural estimators by relating the error to an appropriate empirical process and leveraging tools from empirical process theory. In particular, our approach does not require the underlying distributions to be compactly supported or have densities bounded away from zero, an assumption prevalent in existing results. The deviation inequality also leads to a one-sided concentration bound from the expectation, which is useful in random-coding arguments over continuous alphabets in information theory with potential applications to physical-layer security. As another concrete application, we consider a hypothesis testing framework for auditing Rényi differential privacy using the neural estimator as a test statistic and obtain non-asymptotic performance guarantees for such a test.

We propose a motivic version of T. Hausel and M. Thaddeus' Topological Mirror Symmetry for character stacks associated with arbitrary semisimple groups, which is an analogue of F. Loeser and D. Wyss' result for Chow motives of moduli spaces of Higgs bundles. As first steps towards it, we generalize A. Mellit's cell decomposition to arbitrary connected and reductive groups. We use it to describe all automorphisms on the associated character stacks. Then we show that the Weil pairing induces a duality between cells that interchanges automorphisms by connected components. As a toy example, we show that these results imply our conjecture for the special linear group of rank two.

In this paper, we describe explicit algebraic equations of cyclic gonal curves reflecting the action of the normalizer of a tame cyclic $k$-gonal automorphism. This completes the known situation obtained by Wootton for the case when $k$ is a prime integer.

We examine sets $\mathscr A$ of natural numbers having the property that for some real number $p\in (0,2)$, one has the subconvex bound $$\int_0^1 \Bigl| \sum_{n\in \mathscr A\cap [1,N]}e(n\alpha)\Bigr|^p\, {\rm d}\alpha \ll N^{-1}|\mathscr A\cap [1,N]|^p.$$ We show that exponential sums over such sets satisfy inequalities analogous to Weyl's inequality, and in many circumstances of the same strength as classical versions of Weyl's bound. We also examine equidistribution of polynomials modulo $1$ in which the summands are restricted to these subconvex $L^p$-sets. In addition, we describe applications to problems involving character sums and averages of arithmetic functions.

In this paper, we present two infinite-dimensional Kolmogorov theorems based on nonresonant frequencies of Bourgain's Diophantine type or even weaker conditions. To be more precise, under a nondegenerate condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with the universally prescribed frequency that is independent of any spectral asymptotics. As an application, we prove that for a class of perturbed networks with weakly coupled oscillators described by \[\frac{{{{\rm d}^2}{x_n}}}{{{\rm d}{t^2}}} + V'\left( {x_n} \right) = \varepsilon_n {W'\left( {x_{n + 1} - {x_n}} \right) - \varepsilon_{n-1}W'\left( {{x_n} - {x_{n - 1}}} \right)} ,\;\;n \in \mathbb{Z},\] or even for more general perturbed networks, almost periodic breathers with frequency-preserving do exist, provided that the local potential $ V $ and the coupling potential $ W $ satisfy certain assumptions. In particular, this yields the first frequency-preserving version of the Aubry-MacKay conjecture [8,21].

In the first Heisenberg group, we study entire, locally Sobolev intrinsic graphs that are stable for the sub-Riemannian area. We show that, under appropriate integrability conditions for the derivatives, the intrinsic graph must be an intrinsic plane, i.e., a coset of a two dimensional subgroup. This result extends \cite{arXiv:1809.04586} beyond the Lipschitz class.

We introduce the concept of representations of a matched pair of left-symmetric algebras. Special attentions are paid to the cohomology, infinitesimal deformation and abelian extension theory. Finally, we construct the Wells type exact sequences for automorphisms of abelian extension of a matched pair of left-symmetric algebras.

Starting from any proper action of any locally compact quantum group on any discrete quantum space, we show that its equivariant representation theory yields a concrete unitary 2-category of finite type Hilbert bimodules over the discrete quantum space, from which the quantum group and its action may be completely reconstructed as in a previous article by the author. In particular, this shows that any locally compact quantum group acting properly on a discrete quantum space must be an algebraic quantum group.

This chapter describes selected reference-type multi-criteria decision-making (MCDM) methods that rank alternatives by comparing them with one or more reference solutions derived from an alternatives-criteria matrix (ACM). After explaining the idea of constructing positive ideal, negative ideal and/or average reference solutions, the chapter details the algorithmic steps of each method, illustrating them with a common ACM example. The 9 methods covered are: Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Gray/Grey Relational Analysis (GRA), VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), Evaluation Based on Distance from Average Solution (EDAS), Multi-attributive Border Approximation Area Comparison (MABAC), Combinative Distance-based Assessment (CODAS), Proximity Indexed Value (PIV), Measurement of Alternatives and Ranking According to Compromise Solution (MARCOS) and Preference Ranking on the Basis of Ideal-average Distance (PROBID). The advantages (e.g., computational simplicity) and limitations (e.g., susceptibility to rank reversal) of each method are discussed. A consolidated summary highlights how the different treatments of reference solutions can ultimately drive variations in the ranking of alternatives, underscoring the value of applying several methods in practice. By studying this chapter, readers can (1) describe the principles and steps of each reference-type method, (2) implement them on an ACM, and (3) choose an appropriate reference-type method for their decision-making problems.

We continue the study of Lebesgue-type parameters for various greedy algorithms in quasi-Banach spaces. First, we introduce a parameter that can be used with the quasi-greedy parameter to obtain the exact growth of the Lebesgue parameter for strong partially greedy bases. Second, we establish a new upper bound for the Lebesgue parameter for semi-greedy bases using the quasi-greedy and the squeeze symmetry parameters. Finally, we answer several open questions regarding the optimal power in various bounds proved in [F. Albiac, J. L. Ansorena, and P. M. Berná, New parameters and Lebesgue-type estimates in greedy approximation, Forum Math. Sigma 10 (2022), 1-39].

We study variance bounds under symmetry constraints in classical, free, and Boolean probability. Starting from the classical fact that for a Bernoulli random variable $X$ and an independent $Y$ with $X+Y$ symmetric about $0$, one must have $\mathrm{Var}(Y)\geq pq$, we establish exact analogues in the free and Boolean settings. Using projections as the noncommutative counterpart of Bernoulli variables, we prove that the same bound persists for freely and Boolean independent symmetrizers. These results not only unify variance inequalities across different probabilistic frameworks but also have potential applications in quantum information and noncommutative stochastic modeling.

An Anosov flow on a smooth three-manifold $M$ gives rise to a Liouville structure on $\mathbb{R} \times M$ by a construction of Mitsumatsu. In a recent paper, Cieliebak, Lazarev, Massoni and Moreno ask whether an embedded torus $\Sigma \subseteq M$ transverse to an Anosov flow gives rise to a Lagrangian in $\mathbb{R} \times M$. We show that the answer to this question is in general negative by finding a topological obstruction related to the foliations induced on the torus by the weak stable and unstable bundles of the flow. Going in the opposite direction, we show that the answer is positive if the induced foliations do not admit parallel compact leaves.

Acquiring perfect channel state information (CSI) introduces substantial challenges in cell-free massive MIMO (CF-mMIMO) systems, primarily due to the large dimensionality of channel parameters, especially under ultra-reliable low-latency communication (uRLLC) constraints. Furthermore, the impact of imperfect CSI on the average achievable rate within the finite blocklength regime remains largely unexplored. Motivated by this gap, this paper proposes a novel analytical framework that provides a closed-form expression for the average achievable rate with imperfect CSI in the Laplace domain. We demonstrate analytically that both the channel dispersion and the expected channel capacity can be expressed explicitly in terms of the Laplace transform of the large-scale fading component. Numerical simulations confirm that the derived expressions match closely with Monte Carlo simulations, verifying their accuracy. Furthermore, we theoretically show that although imperfect CSI degrades performance in the finite blocklength regime, the inherent characteristics of CF-mMIMO architecture effectively mitigates this loss.

Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis remains limited, especially regarding convergence guarantees. This work develops a general optimisation framework for energy minimisation problems arising from linear self-adjoint elliptic PDEs, formulated over nonlinear but analytically tractable approximation spaces. The framework accommodates a natural split between linear and nonlinear parameters and supports hybrid optimisation strategies: linear variables are updated via linear solves or steepest descent, while nonlinear variables are handled using constrained projected descent. We establish both local and global convergence of the resulting algorithm under modular structural assumptions on the discrete energy functional, including differentiability, boundedness, regularity, and directional convexity. These assumptions are stated in an abstract form, allowing the framework to apply to a broad class of nonlinear approximation manifolds. In a companion paper [Magueresse, Badia (2025, arXiv:2508.17705)], we introduce a concrete instance of such a space based on overlapping free-knot tensor-product B-splines, which satisfies the required assumptions and enables geometrically adaptive solvers with rigorous convergence guarantees.

Accurately solving PDEs with localised features requires refined meshes that adapt to the solution. Traditional numerical methods, such as finite elements, are linear in nature and often ineffective for such problems, as the mesh is not tailored to the solution. Adaptive strategies, such as $h$- and $p$-refinement, improve efficiency by sequentially refining the mesh based on a posteriori error estimates. However, these methods are geometrically rigid -- limited to specific refinement rules -- and require solving the problem on a sequence of adaptive meshes, which can be computationally expensive. Moreover, the design of effective a posteriori error estimates is problem-dependent and non-trivial. In this work, we study a specific nonlinear approximation scheme based on overlapping tensor-product free-knot B-spline patches, where knot positions act as nonlinear parameters controlling the geometry of the discretisation. We analyse the corresponding energy minimisation problem for linear, self-adjoint elliptic PDEs, showing that, under a mild mesh size condition, the discrete energy satisfies the structural properties required for the local and global convergence of the constrained optimisation scheme developed in our companion work [Magueresse, Badia (2025, arXiv:2508.17687)]. This establishes a direct connection between the two analyses: the adaptive free-knot B-spline space considered here fits into the abstract framework, ensuring convergence of projected gradient descent for the joint optimisation of knot positions and coefficients. Numerical experiments illustrate the method's efficiency and its ability to capture localised features with significantly fewer degrees of freedom than standard finite element discretisations.

We consider inhomogeneous square random matrices of size $N$ with independent entries of mean 0 and finite variance. We assume that the variance profile of this matrix is doubly stochastic and has a band-like structure with an appropriately defined bandwidth $W$. We prove that when the entries have a bounded density and a subgaussian tail, then the empirical spectral measure for the eigenvalues of the matrix converges to the circular law as $N$ tends to infinity whenever $W\geq N^{1/2+c}$ for any $c>0$. In the special case of block band matrices the density assumption is not needed and the moment condition is relaxed. This establishes the circular law limit throughout the entire delocalization regime in 1-d: $W\geq N^{1/2+c}$ and extends the previous thresholds for the circular law limit with exponent $\frac{5}{6},\frac{8}{9},\frac{33}{34}$ in $N$. The main technical input is a new lower bound on the small-ish singular values via Green function estimates and a new lower bound on the least singular value with fewer moment conditions.

In this work we consider a growing random graph sequence where a new vertex is less likely to join to an existing vertex with high degree and more likely to join to a vertex with low degree. In contrast to the well studied \emph{preferential attachment random graphs} \cite{BarAlb99}, we call such a sequence a \emph{de-preferential attachment random graph model}. We consider two types of models, namely, \emph{inverse de-preferential}, where the attachment probabilities are inversely proportional to the degree and \emph{linear de-preferential}, where the attachment probabilities are proportional to $c-$degree, where $c > 0$ is a constant. For the case when each new vertex comes with exactly one half-edge we show that the degree of a fixed vertex is asymptotically of the order $\sqrt{\log n}$ for the inverse de-preferential case and of the order $\log n$ for the linear case. These show that compared to preferential attachment, the degree of a fixed vertex grows to infinity at a much slower rate for these models. We also show that in both cases limiting degree distributions have exponential tails. In fact we show that for the inverse de-preferential model the tail of the limiting degree distribution is faster than exponential while that for the linear de-preferential model is exactly the $\mbox{Geometric}\left(\frac{1}{2}\right)$ distribution. For the case when each new vertex comes with $m > 1$ half-edges, we show that similar asymptotic results hold for fixed vertex degree in both inverse and linear de-preferential models. Our proofs make use of the martingale approach as well as embedding to certain continuous time age dependent branching processes.

For a class of groups $G$ over a field $\mathbb{F}$, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product ${\rm SL}(2,\mathbb{R}) \ltimes \mathrm{Sym}^n(\mathbb{R}^2)$. Furthermore, for affine groups of the form ${\rm GL}(n,\mathbb{R}) \ltimes \mathbb{R}^n$, we show that if $x \in {\rm GL}(n,\mathbb{R})$ is rational, then $(x,v)$ is rational for every $v \in \mathbb{R}^n$.

In this article, we propose an adapted sequential Monte Carlo approximate Bayesian computation (SMC-ABC) algorithm for network inference in coupled stochastic differential equations (SDEs) used for multivariate time series modeling. Our approach is motivated by neuroscience, specifically the challenge of estimating brain connectivity before and during epileptic seizures. To this end, we make four key contributions. First, we introduce a 6N-dimensional SDE to model the activity of N coupled neuronal populations, extending the (single-population) stochastic Jansen and Rit neural mass model used to describe human electroencephalography (EEG) rhythms, particularly epileptic activity. Second, we construct a reliable and efficient numerical splitting scheme for the model simulation. Third, we apply the proposed adapted SMC-ABC algorithm to the neural mass model and validate it on different types of simulated data. Compared to standard SMC-ABC, our approach significantly reduces computational cost by requiring fewer model simulations to reach the desired posterior region, thanks to the inclusion of binary parameters describing the presence or absence of coupling directions. Finally, we apply our method to real multi-channel EEG data, uncovering potential similarities in patients' brain activities across different epileptic seizures, as well as differences between pre-seizure and seizure periods.

We consider the problem of learning classification trees that are robust to distribution shifts between training and testing/deployment data. This problem arises frequently in high stakes settings such as public health and social work where data is often collected using self-reported surveys which are highly sensitive to e.g., the framing of the questions, the time when and place where the survey is conducted, and the level of comfort the interviewee has in sharing information with the interviewer. We propose a method for learning optimal robust classification trees based on mixed-integer robust optimization technology. In particular, we demonstrate that the problem of learning an optimal robust tree can be cast as a single-stage mixed-integer robust optimization problem with a highly nonlinear and discontinuous objective. We reformulate this problem equivalently as a two-stage linear robust optimization problem for which we devise a tailored solution procedure based on constraint generation. We evaluate the performance of our approach on numerous publicly available datasets, and compare the performance to a regularized, non-robust optimal tree. We show an increase of up to 12.48% in worst-case accuracy and of up to 4.85% in average-case accuracy across several datasets and distribution shifts from using our robust solution in comparison to the non-robust one.

We perform a quantitative assessment of different strategies to compute the contribution due to surface tension in incompressible two-phase flows using a conservative level set (CLS) method. More specifically, we compare classical approaches, such as the direct computation of the curvature from the level set or the Laplace-Beltrami operator, with an evolution equation for the mean curvature recently proposed in literature. We consider the test case of a static bubble, for which an exact solution for the pressure jump across the interface is available, and the test case of an oscillating bubble, showing pros and cons of the different approaches.

We present a quantitative assessment of the impact of high-order mappings on the simulation of flows over complex orography. Curved boundaries were not used in early numerical methods, whereas they are employed to an increasing extent in state of the art computational fluid dynamics codes, in combination with high-order methods, such as the Finite Element Method and the Spectral Element Method. Here we consider a specific Discontinuous Galerkin (DG) method implemented in the framework of the this http URL library, which natively supports high-order mappings. A number of numerical experiments based on classical benchmarks over idealized orographic profiles demonstrate the positive impact of curved boundaries on the accuracy of the results, with no significantly adverse effect on the computational cost of the simulation. These findings are also supported by results of the application of this approach to non-smooth and realistic orographic profiles.

We study the expressive power of deep polynomial neural networks through the geometry of their neurovariety. We introduce the notion of the activation degree threshold of a network architecture to express when the dimension of the neurovariety achieves its theoretical maximum. We prove the existence of the activation degree threshold for all polynomial neural networks without width-one bottlenecks and demonstrate a universal upper bound that is quadratic in the width of largest size. In doing so, we prove the high activation degree conjecture of Kileel, Trager, and Bruna. Certain structured architectures have exceptional activation degree thresholds, making them especially expressive in the sense of their neurovariety dimension. In this direction, we prove that polynomial neural networks with equi-width architectures are maximally expressive by showing their activation degree threshold is one.

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, $\rho_1$ and $\rho_2$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator $(\rho_1 - \rho_2)$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.

Motivated by a recent method for approximate solution of Fredholm equations of the first kind, we develop a corresponding method for a class of Fredholm equations of the \emph{second kind}. In particular, we consider the class of equations for which the solution is a probability measure. The approach centres around specifying a functional whose gradient flow admits a minimizer corresponding to a regularized version of the solution of the underlying equation and using a mean-field particle system to approximately simulate that flow. Theoretical support for the method is presented, along with some illustrative numerical results.

This article delves into the concept of quantum contextuality, specifically focusing on proofs of the Kochen-Specker theorem obtained by assigning Pauli observables to hypergraph vertices satisfying a given commutation relation. The abstract structure composed of this hypergraph and the graph of anticommutations is named a hypergram. Its labelings with Pauli observables generalize the well-known magic sets. A first result is that all these correct quantum labelings of a given hypergram inherently possess the same degree of contextuality. Then we provide a necessary and sufficient condition for the existence of such quantum labelings and an efficient algorithm to find one of them. We finally attach to each assignable hypergram an abstract notion of contextuality degree. By presenting the study of observable-based Kochen-Specker proofs from the perspectives of graphs and matrices, this abstraction opens the way to new methods to search for original contextual configurations.

This study discusses the current state of FSO technology, as well as global trends and developments in the industrial ecosystem to identify obstacles to the full realization of optical space-to-ground communication networks. Additionally, link performance and network availability trade-off studies are presented, comparing overall system performance between portable and large OGS networks in conjunction with a constellation of small low Earth orbit (LEO) satellites. The paper provides an up-to-date overview and critical analysis of the FSO industry and assesses the feasibility of low-cost portable terminals as an alternative to larger high-capacity OGS systems. This initiative aims to better inform optical communications stakeholders, including governments, academic institutions, satellite operators, manufacturers, and communication service providers

Reliable Bayesian predictive inference has long been an open problem under unidentified transformation models, since the Markov Chain Monte Carlo (MCMC) chains of posterior predictive distribution (PPD) values are generally poorly mixed. We address the poorly mixed PPD value chains under unidentified transformation models through an adaptive scheme for prior adjustment. Specifically, we originate a conception of sufficient informativeness, which explicitly quantifies the information level provided by nonparametric priors, and assesses MCMC mixing by comparison with the within-chain MCMC variance. We formulate the prior information level by a set of hyperparameters induced from the nonparametric prior elicitation with an analytic expression, which is guaranteed by asymptotic theory for the posterior variance under unidentified transformation models. The analytic prior information level consequently drives a hyperparameter tuning procedure to achieve MCMC mixing. The proposed method is general enough to cover various data domains through a multiplicative error working model. Comprehensive simulations and real-world data analysis demonstrate that our method successfully achieves MCMC mixing and outperforms state-of-the-art competitors in predictive capability.

Starting with an explicit framework for designing logical Clifford circuits for CSS codes, we construct logical gates for Hypergraph Product Codes. We first derive symplectic matrices for CNOT, CZ, Phase, and Hadamard operators, which together generate the Clifford group. This enables us to design explicit transformations that result in targeted logical gates for arbitrary codes in this family. As a concrete example, we give logical circuits for the $[[18,2,3]]$ toric code.

In the context of building electrification, the operation of distributed energy resources integrating multiple energy carriers (electricity, heat, mobility) poses a significant challenge due to the nonlinear device dynamics, uncertainty, and computational issues. As such, energy management systems seek to decide the power dispatch in the best way possible. The objective is to minimize and balance operative costs (energy bills or asset degradation) with user requirements (mobility, heating, etc.). Current energy management uses empirical battery ageing models outside of their specific fitting conditions, resulting in inaccuracies and poor performance. Moreover, the link to thermal systems is also overlooked. This paper presents an ageing-aware day-ahead algorithm for electrified buildings that incorporates physics-based battery ageing models. The models distinguish between energy storage systems and make explicit the trade-off between grid cost and battery degradation. The proposed day-ahead algorithm can either cut down on grid costs or extend battery lifetime (electric vehicle or stationary battery packs). Moreover, it exploits the differences between cathode chemistries improving grid costs by 25% when using LFP cells, with respect to NMC cells. Finally, the performance using aged batteries is also enhanced with 35% grid cost observed savings, when passing from new to aged batteries in the summer.

The constrained zonotope is a polytopic set representation widely used for set-based analysis and control of dynamic systems. This paper develops methods to formulate and solve optimization problems for dynamic systems in real time using constrained zonotope reachability analysis. An alternating direction method of multipliers (ADMM) algorithm is presented that makes efficient use of the constrained zonotope structure. To increase the efficiency of the ADMM iterations, reachability calculations are presented that increase the sparsity of the matrices used to define a constrained zonotope when compared to typical methods. The developed methods are used to formulate and solve predictive control, state estimation, and safety verification problems. Numerical results show that optimization times using the proposed approach are competitive with state-of-the-art QP solvers and conventional problem formulations. A combined set-valued state estimation and moving horizon estimation algorithm is presented and experimentally demonstrated in the context of robot localization.

Maximizing revenue for grid-scale battery energy storage systems in continuous intraday electricity markets requires strategies that are able to seize trading opportunities as soon as new information arrives. This paper introduces and evaluates an automated high-frequency trading strategy for battery energy storage systems trading on the intraday market for power while explicitly considering the dynamics of the limit order book, market rules, and technical parameters. The standard rolling intrinsic strategy is adapted for continuous intraday electricity markets and solved using a dynamic programming approximation that is two to three orders of magnitude faster than an exact mixed-integer linear programming solution. A detailed backtest over a full year of German order book data demonstrates that the proposed dynamic programming formulation does not reduce trading profits and enables the policy to react to every relevant order book update, enabling realistic rapid backtesting. Our results show the significant revenue potential of high-frequency trading: our policy earns 58% more than when re-optimizing only once every hour and 14% more than when re-optimizing once per minute, highlighting that profits critically depend on trading speed. Furthermore, we leverage the speed of our algorithm to train a parametric extension of the rolling intrinsic, increasing yearly revenue by 8.4% out of sample.

Since the advent of the ``Neural Ordinary Differential Equation (Neural ODE)'' paper, learning ODEs with deep learning has been applied to system identification, time-series forecasting, and related areas. Exploiting the diffeomorphic nature of ODE solution maps, neural ODEs has also enabled their use in generative modeling. Despite the rich potential to incorporate various kinds of physical information, training Neural ODEs remains challenging in practice. This study demonstrates, through the simplest one-dimensional linear model, why training Neural ODEs is difficult. We then propose a new stabilization method and provide an analytical convergence analysis. The insights and techniques presented here serve as a concise tutorial for researchers beginning work on Neural ODEs.

In this paper, we establish a numerical method for simulation of wall-bounded incompressible turbulent flows by integrating the technology of random vortex method with the core idea of Large Eddy Simulation (LES). Specifically, we utilize the filtering function in LES, interpreted as spatial averaging, along with the integral representation theorem for parabolic equations, to achieve a closure numerical scheme which may be used for calculating solutions of Navier-Stokes equations. This approach circumvents the challenge associated with handling the non-locally integrable 3-dimensional integral kernel in the random vortex method and facilitates the computation of numerical solutions for flow systems via Monte-Carlo method. Comprehensive numerical simulations, along with systematic comparisons against other numerical approaches, are conducted for both turbulent and laminar flows in unbounded and wall-bounded domains, considering two- and three-dimensional cases. These results collectively demonstrate the validity and effectiveness of the proposed method.

Large eddy simulation (LES) has become a central technique for simulating turbulent flows in engineering and applied sciences, offering a compromise between accuracy and computational cost by resolving large scale motions and modeling the effects of smaller, unresolved scales through a subgrid scale (SGS) model. The fidelity and robustness of LES depends critically on the SGS model, particularly in coarse simulations where much of the turbulence spectrum remains unresolved. In this work, we extend the tau-orthogonal (TO) method, a data-driven SGS modeling framework, to three-dimensional turbulent flows. The method reformulates the high-dimensional SGS closure problem as a low-dimensional prediction task focused on scale-aware quantities of interest (QoIs). We extend the model to incorporate QoI-state dependence and temporal correlations by combining regularized least-squares regression with a multivariate Gaussian residual model. This yields a simple yet effective stochastic time-series prediction model (the LRS model), with orders-of-magnitude fewer parameters than typical deep learning approaches which try to directly learn the high-dimensional SGS closure. We demonstrate the effectiveness of the TO LRS model in three-dimensional forced isotropic turbulence and turbulent channel flow. The model achieves accurate long-term QoI distributions, robust performance across hyperparameter settings, and good reproduction of key flow features such as kinetic energy spectra and coherent structures, despite being trained solely on QoI trajectories. Comparisons against classical SGS models, including Smagorinsky and WALE formulations, highlight the TO LRS model's balance of accuracy and computational efficiency.

In $3+1$ dimensions, we study the stability of Kasner solutions for the Einstein-Maxwell-scalar field-Vlasov system. This system incorporates gravity, electromagnetic, weak and strong interactions for the initial stage of our universe. Due to the presence of the Vlasov field, various new challenges arise. By observing detailed mathematical structures and designing new delicate arguments, we identify a new strong sub-critical regime and prove the nonlinear stability with Kasner exponents lying in this full regime. This extends the result of Fournodavlos-Rodnianski-Speck [8] from the Einstein-scalar field system to the physically more complex system with the Vlasov field.

The dominant paradigm in modern neural networks relies on simple, monotonically-increasing activation functions like ReLU. While effective, this paradigm necessitates large, massively-parameterized models to approximate complex functions. In this paper, we introduce the Periodic Linear Unit (PLU), a learnable sine-wave based activation with periodic non-monotonicity. PLU is designed for maximum expressive power and numerical stability, achieved through its formulation and a paired innovation we term Repulsive Reparameterization, which prevents the activation from collapsing into a non-expressive linear function. We demonstrate that a minimal MLP with only two PLU neurons can solve the spiral classification task, a feat impossible for equivalent networks using standard activations. This suggests a paradigm shift from networks as piecewise Taylor-like approximators to powerful Fourier-like function synthesizers, achieving exponential gains in parameter efficiency by placing intelligence in the neuron itself.

This paper introduces a nonstandard finite difference (NSFD) approach to a reaction-diffusion SEIQR epidemiological model, which captures the spatiotemporal dynamics of infectious disease transmission. Formulated as a system of semilinear parabolic partial differential equations (PDEs), the model extends classical compartmental models by incorporating spatial diffusion to account for population movement and spatial heterogeneity. The proposed NSFD discretization is designed to preserve the continuous model's essential qualitative features, such as positivity, boundedness, and stability, which are often compromised by standard finite difference methods. We rigorously analyze the model's well-posedness, construct a structure-preserving NSFD scheme for the PDE system, and study its convergence and local truncation error. Numerical simulations validate the theoretical findings and demonstrate the scheme's effectiveness in preserving biologically consistent dynamics.