We introduce the CLUSTER algorithm (\textbf{c}oordinate-\textbf{l}evel \textbf{u}pdate \textbf{s}trategy for \textbf{t}rust-region step \textbf{e}valuation \textbf{r}efinement) for local derivative-free optimization problems where there is a cost to changing each parameter (or clusters of parameters). For example, this type of cost model is appropriate for optimizing robot-controlled laboratory experiments, in which a robot may incur a separate motion for each parameter cluster to be adjusted. We build off of a class of quadratic-interpolation optimization algorithms by Powell and Conn that are known to perform well for twice-differentiable objectives (e.g. low-noise experiments), and show that the CLUSTER variants improve performance on a variety of test problems (including an optics laboratory experiment) by around 50$\%$, and greatly outperform common competing algorithms for laboratory optimization (Bayesian optimization and Nelder--Mead). We also adapt the convergence proof of the Conn algorithm to obtain a similar convergence guarantee for CLUSTER-Conn.

WepresentatwolevelSchwarzmethodasanalternativetoAlgebraicMultigridmethod(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poission equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes eqaution. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

The \emph{minimum positive codegree} $δ^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Turán density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $δ^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<γ^+(F_1, F_2) < \min\{γ^+(F_1), γ^+(F_2)\}. $$

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$--exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le β\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

In this paper, a braid is regarded as a dynamical system of points in the plane. The states of this dynamical system are given by Delaunay triangulations. This construction makes it possible to define an abstract groupoid $\overset{abc}{\mathcal{G}^{4}_{n+3}}$, which gives a representation of the colored braid groupoid $\text{ColB}(n)$. We define homomorphisms ${f}_{n+3}:\overset{abc}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{Q})$ and ${f}'_{n+3}:\overset{abc}{\mathcal{G}^{4}_{n+3}} \rightarrow\text{GL}_{2n+1}(\mathbb{C})$, and describe an algorithm for computing the resulting invariants.

We introduce the notion of a coarsely minimal Reeb flow, generalizing the notion of minimal geodesic flow, and prove the following rigidity theorem: That a coarsely minimal Reeb flow satisfying a divergence property is orbitally equivalent to the geodesic flow of a Riemannian metric of negative sectional curvature. Without the divergence assumption, we obtain an orbital semi-equivalence. This extends a rigidity result for geodesic flows of negatively curved Riemannian metrics which is due to Gromov. We use Floer homology and Morse's hyperbolic `stability' Lemma.

We prove that a lacunary hyperbolic group $G = \varinjlim G_i$ with sufficient generics is selfless in the sense of Amrutam--Gao--Kunnawalkam Elayavalli--Patchell, provided the hyperbolicity constants $δ_i$ and injectivity radii $r_i$ satisfy $δ_i(\log r_i)^{7} = o(r_i)$. The proof replaces the acylindricity-based machinery of that work with a direct geodesic $n$-gon criterion due to Arzhantseva, which applies in any $δ$-hyperbolic space. As a consequence, combined with rapid decay, $G$ is $C^*$-selfless. The condition is mild: torsion-free Tarski monsters, Jacobson's mixed-identity-free elementary amenable groups and Gromov monster groups satisfy it for appropriate parameter choices. The amenable examples are selfless but cannot be $C^*$-selfless, providing examples that separate these properties. Finally we remark that the Gromov monster group examples provide a potential avenue to a non-exact $C^*$-algebra with strict comparison.

We prove that every graph admits a linked, componental, rooted tree-cut decomposition of finite adhesion that displays all undominated edge-ends. As a first application, we deduce that this tree-cut decomposition also displays the edge-degrees of all undominated edge-ends. For locally finite graphs $-$ where every end is an undominated edge-end $-$ this yields a linked tree-cut decomposition of finite adhesion into $\textit{finite}$ parts that displays all ends and their edge-degrees. As a second application, this latter tree-cut decomposition yields short, unified deductions of Thomassen's theorem on boundary-linked finite partitions, and of Bruhn and Stein's characterisation of Eulerian locally finite graphs in terms of even ends.

We report some work in progress on the relationship between étale and quasicoherent cohomological dimensions.

Torsion pairs, and in particular t-structures, play a central role in the study of triangulated categories. Specifically, t-structures induced by silting (or tilting) objects often admit desirable properties with strong connections to derived equivalences. In this paper, using the correspondence of Saorín-Šťovíček between cohereditary cotorsion pairs in Frobenius exact categories and t-structures in their stable categories, we construct a family of t-structures in the $Q$-shaped derived category of Holm and Jorgensen, arising from admissible partitions of $Q$. We give an explicit description of the associated cotorsion pairs inside the Frobenius exact category of the bifibrant objects, and we identify the corresponding co-aisles by certain homological vanishing conditions. Such t-structures are proved to be induced by a silting object, that can be completely determined by the combinatorics of $Q$. Finally, we illustrate our results by recovering well-known equivalences in the $Q$-shaped setting, while also providing examples where the combinatorial conditions fail (e.g. cyclic quivers), showing that such categories may admit no non-trivial t-structures, revealing phenomena analogous to those observed by Linckelmann in stable module categories.

We establish a direct high-probability last-iterate guarantee for the standard same-sample two-point Gaussian zeroth-order stochastic-gradient method applied to smooth, strongly convex stochastic optimization. At each iteration, the method draws a fresh Gaussian direction, evaluates the objective at two symmetric perturbations using the same stochastic sample, and takes a norm-normalized stochastic-approximation step. Assuming unbiased stochastic gradients and a conditional exponential-moment bound on the squared norm of the stochastic-gradient noise, we prove that, whenever \(d\ge16\log(6T/δ)\), \[ f(\bx_T)-f(\bx^*) = \widetilde{\mathcal O}\!\left(\frac{d}{T}\right) \] with probability at least \(1-δ\), up to fixed problem parameters and logarithmic factors. The confidence dependence is therefore logarithmic rather than polynomial in \(1/δ\). The analysis is direct: it neither invokes Markov's inequality to convert an expectation bound nor truncates the noise. We are not aware of a prior direct high-probability last-iterate result at this zeroth-order scale for the same-sample Gaussian recursion under conditional sub-Gaussian stochastic-gradient noise. The proof combines a uniform weighted scan for Gaussian angles with an angle-enlarged product-martingale boundary that controls the signed suffix-product term arising from the unrolled stochastic recursion.

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

We study interactions between simplex faces of lattice polytopes and quadratic generation of toric ideals. We prove that, under a mild condition on edges, if the toric ideal of a lattice polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. In particular, if the toric ideal of a $(0,1)$-polytope is generated by quadratic binomials, then every clique of its 1-skeleton is the vertex set of a face. For $(0,1)$-polytopes satisfying condition (E), we characterize this clique-face property in terms of divisibility by quadratic monomials appearing in quadratic binomials of the toric ideal; as a consequence, such toric ideals have no indispensable monomials of degree $\ge 3$. We apply these results to edge polytopes and cut polytopes, for which the clique-face property is equivalent to quadratic generation. Finally, motivated by conjectures on quadratic toric ideals, we verify the clique-face property for simple polytopes, matroid independence polytopes, and matroid base polytopes, and discuss stable set polytopes.

We introduce the $G$-Daugavet property ($G$-DPr, for short) for Banach spaces endowed with an action of a group $G$ by surjective linear isometries. This notion provides a common framework for the classical Daugavet property and the alternative Daugavet property, which correspond respectively to the trivial action and to the scalar action of $S_{\mathbb{K}}$. We establish several characterizations of the $G$-DPr in terms of $G$-slices and closed convex $G$-invariant hulls, recovering the usual slice descriptions of the DPr and the aDPr as particular cases. We show that the presence of a group action leads to new behavior in Daugavet theory. In particular, the $G$-DPr may hold on classical reflexive spaces in sharp contrast with the classical Daugavet property. We relate this phenomenon to convex transitivity, almost transitivity and finite-dimensional rotation problems. We also prove group-action versions of the classical characterizations for $L^1(μ, X)$- and $C(K,X)$-spaces. The paper also studies group separable determination, $G$-versions of numerical radius and numerical index, and connections between the $G$-DPr and strong Radon-Nikodým and SCD operators. Finally, we introduce a parameter which measures how far the $G$-DPr is from the classical DPr in a quantitative manner. As a consequence of these results, we obtain conditions under which the $G$-DPr recovers several classical implications, including the failure of the RNP for both $X$ and $X^*$, the presence of copies of $\ell_1$ and the failure of the unit ball to be an SCD set.

We reconcile privacy protection and rate-double-robust inference. The privacy of individuals is protected by a local privacy mechanism: injecting noise into their sensitive data, revealing only the noisy data for inference. Hence, privacy protection hinders inference. In contrast, the inference of a target parameter is rate-double-robust when the large-sample bias of an estimator of the parameter is characterised by a trade-off between the estimation errors of two other, nuisance, parameters. Hence, rate-double-robustness facilitates inference. Our starting point of reconciliation is a class of rate-double-robust target parameters indexed linearly by an infinite-dimensional and nonlinearly by a low-dimensional regression. Among others, this includes causal parameters. To infer these targets privately, we show how suitable privacy mechanisms transfer the semiparametric properties of the sensitive-data model to the private setting. Rate-double-robustness is transferred, enabling locally-private, unbiased and semiparametrically efficient inference of our target parameters. Finally, we transform general nonparametric nuisance estimators into private ones, which inherit convergence properties of their nonprivate counterparts. For parametric nuisance models, we develop a private method-of-moments estimator and its large-sample inference theory.

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. First, we show that for arrangements of $d<9$ lines the intersection lattice determines the exponent data considered here. Then we list six distinct Ziegler pair with $d=10$. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions.

A c-direct category is a small category equipped with an ordinal degree function such that every morphism is level or degree-raising. Every c-direct category is c-Reedy. The c-Reedy model structure on any functor category from a c-direct category to a model category coincides with the projective model structure. In this framework, a realization functor is a colimit-preserving functor satisfying some mild homotopical conditions from the category of presheaves on a c-direct category with cofibrant representables to a model category. We prove that any two such realization functors are weakly equivalent on cofibrant presheaves. For categories of cubes, we prove that thick categories have cofibrant representables. As an application, we introduce the $\varepsilon$-branching space of an $\mathcal A$-set for any thick category of cubes $\mathcal A$. It is obtained as a coend over a c-direct category with cofibrant representables constructed from $\mathcal A$. We prove that, on free $\mathcal A$-sets generated by precubical sets, this new definition coincides with the earlier one. We prove that, for cofibrant $\mathcal A$-sets, the resulting space is independent of $\varepsilon$ up to homotopy.

We define a combinatorial property of a projective Fraisse category which we call the \emph{approximate Ramsey property}. Let $F$ be a continuum, $G$ a closed subgroup of the homeomorphism group of $F$, and $\mathbb{F}$ the limit of projective Fraisse category $\mathcal{F}$ such that $\textrm{Aut}(\mathbb{F})$ is dense in $G$. We prove that $\mathcal{F}$ has the approximate Ramsey property if and only if $G$ is extremely amenable. We prove that the group of homeomorphisms of the universal pseudo-solenoid has non-metrizable universal minimal flow.

We develop the theory of equilibrium states via specification properties for a wide class of continuous flows on complete separable metric spaces. An important motivating example is geodesic flow over negatively curved manifolds without pinching assumptions and geodesic flow over CAT(-1) spaces. Since our phase space is non-compact, we need to establish all the basic definitions and results to make this theory work, including a suitable notion of topological pressure and fundamental results such as the variational principle. We introduce a notion of strong positive recurrence in this setting and use it as a criterion to prove the existence and uniqueness of an equilibrium state.

We prove that every $K_5$-free $n$-vertex graph with sublinear independence number can be made bipartite by removing at most $n^2(1/18+o(1))$ edges, where the constant $1/18$ is best possible. The proof method is related to extensions of Turán Theorem in edge-weighted settings, and part of the proof uses flag algebra.

This paper investigates the vanishing and non-vanishing of Betti and Bass numbers for non-finitely generated modules. We prove that for \(d\)-dimensional Cohen--Macaulay local rings, every non-zero \(\mathfrak{m}\)-torsion module satisfies \(β_d(M)\neq 0\), and we establish the Betti number behavior of the injective hull \(E_R(k)\). We study Tor-rigidity for \(H^d_{\mathfrak{m}}(R)\). We also provide partial positive answers to Schoutens' question on whether the vanishing of sufficiently high Betti numbers of a big Cohen--Macaulay algebra forces the Cohen--Macaulay property of \(R\). For the absolute integral closure \(R^+\), we establish both Tor and Ext results. On the Tor side, we prove that \(\operatorname{Tor}_i^R(R^+,k)=0\) for some \(i>0\) implies regularity in a series cases including quotient singularities. On the Ext side, we prove that \(\operatorname{Ext}^i_R(k,R^+)=0\) for some \(i\geq d\) forces regularity for Gorenstein domains of prime characteristic, and we obtain analogous results for graded normal domains of dimension \(2\) and also for quotient and isolated singularities in any dimension.

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA), corresponding to a BKM Cartan matrix $A$ filled by negative integers. Let $P^+\subset \mathfrak{h}^*$ the classical dominant integral cone (wherein pairings are non-negative). The non-integrable simple highest weight modules $L(μ)$'s widely studied were broadly those by Naito ([Trans. Amer. Soc., 1995]), for $μ$'s dot-linked to $P^+$-translates of sums $- \sum_{j\in J}α_j$ of mutually orthogonal and imaginary simple roots $α_j$'s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$'s, and characters of $L(ρ)$ for Weyl vector $ρ$ in negative type-$A$. These needed a family of ``integrable'' $L(μ)$'s for $μ$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $μ(α_i^{\vee})\leq 0$ therein are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(μ)$ contain ``Chevalley-Serre relations'' $f_i^{\frac{2}{A_{ii}}{μ(α_i^{\vee})}+1}L(μ)_μ=0$; which differ from relations in $L(λ)$ for all $λ\in P^+$, and are seemingly unstudied earlier (also by Naito). This paper initiates the study in rank-2, of the module structures and maximal vectors (or Verma embeddings) in the Verma covers $M(μ)$ of $L(μ)$'s for $μ\in P^{\pm}$. In this, our goal is to explore in weight spaces of those Verma covers, the strictness (or otherwise, an uniform equality) of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$'s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on the unique solution case in Lemma 3.1 from that paper.

Nonlinear geotechnical analysis often involves complex geometries, staged construction, local failure, and mesh-dependent stress and plastic strain responses. This study develops a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis and implements it in ABAQUS through the user element subroutine. The proposed method combines Wachspress interpolation with cell-based strain smoothing, in which the smoothed strain--displacement matrix is evaluated by boundary integration over polygonal smoothing subcells. This formulation avoids direct calculation of shape-function derivatives inside polygonal elements and enables standard polygonal meshes and hybrid quadtree meshes with hanging nodes to be handled in a unified framework. Nonlinear geomaterial behavior is incorporated through incremental elasto-plastic constitutive updates, including the Mohr--Coulomb model and the Duncan--Chang model. Several benchmark and engineering examples, including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, are presented for verification. The results show that the proposed method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety, while providing improved mesh flexibility and computational efficiency for nonlinear geotechnical analysis.

In this paper, we study multi-stage stochastic programming (MSP) problems with nonsmooth composite objectives. Tailored to their intrinsic stage-wise and scenario-wise structure, we develop a single-loop minorized dual decomposition method, in which each iteration constructs a minorized problem and its restricted Wolfe dual, and then performs \textit{one iteration} of the symmetric Gauss--Seidel based inexact alternating direction method of multipliers on the resulting dual problem to generate the next iterate. A key feature of the proposed optimization framework is that the resulting updates preserve the stage-wise and scenario-wise decomposable structure of the MSP problem and are suitable for parallel implementation. We establish global convergence of the generated iterates for the three-stage case and further establish the corresponding global convergence theorem for the general multi-stage setting. Numerical experiments illustrate the computational viability of the proposed framework and its favorable scaling behavior with respect to the stage-wise and scenario-wise structure.

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We report on recent results concerning minimal $G$-invariant characters of $N$ (which are the sums of the characters on each orbit of the action of $G$ by conjugation on $\text{Irr}(N)$) and their influence on the structure of $N$, as well as their relationship to the $G$-conjugacy classes of $N$.

In 1984, Frankl and Füredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2^{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $Ω(r^{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from $\{0,2,\ldots,2k\}$.

The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

$q$-Matroids, a $q$-analogue of classical matroids have attracted a lot of attention over the last decade, yet their enumeration remains largely unexplored. In this paper, we study the number of $q$-matroids, paving and sparse-paving $q$-matroids defined on a fixed ground space and with prescribed rank. We derive new lower bounds using constructions from constant-dimension codes and improve existing estimates. On the upper bound side, we develop two approaches: a combinatorial method based on controlling the number of dependent hyperplanes for paving $q$-matroids, and an entropy-based counting argument applicable to classes of $q$-matroids closed under contraction. These techniques yield explicit upper bounds on the logarithmic number of $q$-matroids with fixed rank and ground space. Finally, we analyze the asymptotic behavior of these bounds, and identify gaps between lower and upper estimates, leading to conjectures on the true asymptotic growth.

In this note we investigate the Fitting ideals associated to the Tjurina ideal of a non quasi-homogeneous plane curve singularity. Special properties occur when the difference between Milnor number and Tjurina number is at most 2.