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.

$q$-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.

In this article, we introduce a novel data-dependent Shepard interpolation method inspired by the adaptive strategies proposed in [2]. In this case, as Shepard interpolation does not produce oscillations, our approach has the core objective of reducing the smearing near jump discontinuities in the data in one and two dimensions. While the original work in [2] focuses in on Radial Basis Function (RBF) interpolation, we extend these ideas to the Shepard framework by incorporating a data-dependent adaptation mechanism. Specifically, we modify the classical Shepard interpolation by adaptively adjusting the influence weights based on local smoothness indicators that modify the shape parameter. These indicators, similar to those used in [2], are designed to detect discontinuities: for grid-based data, we use squared undivided second-order differences, and for scattered data, we employ squared least-squares approximations of the Laplacian scaled by the square of the mean local separation of stencil points. The resulting data-dependent weighting scheme forces the kernels close to a discontinuity to behave like a local delta function, effectively reducing the smearing of the discontinuities introduced by the classical Shepard approach. We establish the theoretical foundation of the method, including the properties of the new interpolation and we theoretically prove that the reduction of the smearing of discontinuities is possible. Numerical experiments in one and two dimensions confirm that the proposed data-dependent Shepard interpolation significantly reduces the smearing of jump discontinuities while maintaining high accuracy in smooth regions.

We continue recent work of Chao, Gabel, Larson, and Nasr in using cyclotomic polynomials for dice relabeling. In their work, one idea they expand on is finding pairs of dice with different number of sides which maintain the sum frequency of two normal dice. We continue this idea in this paper by studying pairs of dice where the number of sides of each is a different perfect square (which we call "square-sided" dice). We additionally provide conjectures offering ideas for future exploration.

In this paper, based on recent results for the prescribed Hermitian-Yang-Mills (HYM) tensor and its twisted variants by Fan-Wang-Yang-Yau, we provide a dynamical construction of Hermitian-Einstein metrics on stable holomorphic vector bundles and its extension to Higgs bundles. Additionally, in the appendix, we use the heat flow method to give a new proof of the existence and uniqueness of solutions to the twisted prescribed HYM tensor equation, as well as its generalization to Higgs bundles.

We consider large-scale noisy derivative-free optimization (DFO) problems in which only function values are available and gradient or subgradient information cannot be reliably estimated. Matrix-adaptation evolution strategies (MAES) and their limited-memory variants are among the most robust DFO methods under noise; however, their performance may deteriorate when the noise level is large. In such regimes, sorting and selection may misidentify informative sampled points, making the recombination step less reliable and weakening the scaling information used by affine or matrix-adaptation mechanisms. This can substantially reduce the efficiency of MAES-type methods, especially in high-dimensional settings. To address this limitation, we propose a DFO method that replaces the full affine-scaling matrix with a diagonal approximation constructed from conjugacy-type conditions. The proposed mechanism does not attempt to estimate gradients, subgradients, or interpolation models, nor does it learn dense covariance information from noisy rankings. Instead, it uses consecutive normalized recombination displacements in a conservative diagonal update, thereby limiting the influence of unreliable selection information while preserving the derivative-free structure of the underlying evolutionary framework. As a result, the method is computationally cheaper than full matrix-adaptation schemes and limited-memory affine-scaling variants, while providing a stable scaling mechanism in noisy environments. Numerical experiments on noisy benchmark problems show that the proposed method is competitive with, and often more efficient than, MAES-type baselines, particularly when the noise level is large and ranking-based selection becomes unreliable.

The toughness of a graph is a crucial parameter for characterizing its structural properties. The toughness of a non-complete graph $G$ is defined as $τ(G) = \min \{ \dfrac{|S|}{c(G - S)} : S \subseteq V(G), c(G-S) > 1 \}$, where $c(G)$ denotes the number of components of $G$. We define $τ(K_n) = \infty$. A graph $G$ is said to be $τ$-tough if $|S| \ge τ\cdot c(G-S)$ for every vertex cut $S$ of $G$. Let $k \ge 3$ be an integer. For $\frac{1}{k-η}$-tough graphs with $η\in \{0, 1\}$, Liu, Fan and Shu \cite{a34} derived sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Jia and Lu \cite{a24}, for the case $\frac{1}{k-1} \leq τ(G) < \frac{1}{k-2}$, established sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Motivated by these results, in this paper, we further investigate sufficient conditions for the existence of a spanning $k$-tree when $\frac{1}{k} \leq τ(G) < \frac{1}{k-1}$. Specifically, for a connected $\frac{t}{t(k-1)+1}$-tough graph of sufficiently large order $n$ (where $t \ge 1$ is an integer), we provide sufficient conditions for the existence of a spanning $k$-tree in terms of the spectral radius and the signless Laplacian spectral radius. Furthermore, we establish a lower bound on the size (number of edges) to guarantee the existence of a spanning $k$-tree.

We prove a Littlewood-Paley formula for the Hardy space of Dirichlet series $\mathscr{H}^p$ with $1\leq p<\infty$ in terms of almost every vertical limit function. This significantly strengthens previous results, which hold either only as an average over the vertical limit functions or under additional assumptions of uniform convergence. As part of our approach, we obtain a Hardy-Stein identity for the derivative of the $p$-mean of almost every vertical limit. We further show that the mean counting function exists for any $f$ in $\mathscr{H}^p$ in terms of almost all of its vertical limit functions. This is done by establishing a version of Jensen's formula in this setting. In the process, we also deduce ergodic versions of Fatou's lemma and the monotone and dominated convergence theorems for the Kronecker flow.

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(Ω;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(Ω)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $Ω=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $Ω=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $Ω$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

We study unit groups of weighted Leavitt path algebras. Let $K$ be a field of characteristic $0$ and let $(E,ω)$ be a finite connected weighted graph. We prove that $L_K(E,ω)^\times$ is abelian if and only if $L_K(E,ω)$ is a domain. Equivalently, $L_K(E,ω)^\times$ contains no non-cyclic free subgroup if and only if $L_K(E,ω)$ is a domain.

Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker $x^{2}+r^{2}=2n$, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.

We investigate the spectral stability of small-amplitude, periodic, traveling-wave solutions of the modified Camassa-Holm equation with cubic nonlinearities. More precisely, we analyze the $L^2(\mr)$-spectrum of the associated linearized operator in a neighborhood of the origin in the spectral plane. Inspired by a recently novel method based on Kato's perturbation theory [Berti et al, Full description of Benjamin-Feir instability of Stokes waves in deep water, \textit{Invent. Math.}, 230 (2022), 651-711.], we provide a complete description of the spectrum near the origin of the linearized operator--an integro-differential operator with periodic coefficients--and thus prove that such waves are not subject to modulational instability. Moreover, a spectral analysis reveals a remarkable threshold phenomenon: such waves with wave number $k^2\leq 3$ exhibit spectral stability, while instability emerges when $k^2>3$.

In 1929, O. Zariski found a pair of complex plane algebraic curves of the same degree and with the same collection of singularities, but embedded into the plane in a topologically different way. Accordingly, such curves belong to different components of the equisingular family. This phenomenon has been intensively studied till now. In this note, we propose a different insight on this subject: Two curves $C',C''\subset\PP^2$ form an {\it anti-Zariski pair}, if $(\PP^2,C')$ and $(\PP^2,C'')$ are homeomorhic, but $C'$ and $C''$ belong to different components of the equisingular family. We exhibit examples of anti-Zariski pairs and discuss related issues.

In this paper, we investigate the tree-independence number of graph classes that do not contain $K_{1,d}$ as an induced subgraph. Dallard et al. conjectured that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. Our main contribution towards this conjecture is showing that the conjecture holds for outerstring graphs. Additionally we give linear and quadratic bounds for the tree-independence number of various $K_{1,d}$-free graph classes, sharpening previous bounds. Finally, we bound the tree-independence number of $K_{2,d}$-free graphs additionally forbidding holes of length at least $5$.

The theory of fiber bundles over small categories is developed, viewing them as locally constant functors to the category of small categories. The Grothendieck construction yields a total category equipped with a projection that is a bifibration. We show that, up to natural isomorphism, every such bundle admits a constant fiber, and that the monodromy gives a representation of the fundamental groupoid in the automorphism group of the fiber, which allows the classification of fiber bundles up to isomorphism. The gauge group of the bundle is proved to be isomorphic to the centralizer of the monodromy subgroup. We then give a precise analysis of sections and (lax) fixed points of the fiber bundle. Beat points for functors are introduced, and it is proved that every fiber bundle with some finiteness and acyclic conditions admits a minimal core, using a rigidity lemma for finite acyclic categories. These concepts are illustrated with explicit examples.

We take up the study, initiated by Fred Cohen, of the Cartan--Leray spectral sequence for Euclidean configuration spaces, establishing a decomposition as a direct sum of atomic spectral sequences. As an immediate consequence, we recover a difficult theorem of Arone--Mahowald on the vanishing of Goodwillie derivatives of the identity.

Drones are increasingly used in agriculture, where tight margins demand efficient planning. Current optimization tools suffer from exponential runtimes as problem sizes grow, necessitating practical heuristics for daily operations. This paper presents an operational framework and benchmarking analysis for drone spraying operations. We evaluate the trade-offs between facility siting methods and tiered routing parameters. For facility siting, comparing a Mixed-Integer Program (MIP) baseline against a $p$-Median heuristic shows that the heuristic reduces runtime by three orders of magnitude, from over 97 seconds to under 1.2 seconds, with only a 4\% reduction in serviced field area. For route planning, a tiered problem decomposition approach partitioning the target area into 6 to 8 spatial clusters reduces computation time by an order of magnitude with minimal degradation in serviced area. This framework achieves minute-scale planning on commodity hardware, demonstrating operational relevance. Future research will incorporate weather modeling, integrated optimization of facility location and routing, and validation across diverse field geometries.