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.

This work concerns regularity properties of weak solutions to elliptic equations in divergence form -div(a$\nabla$u) = div F , under low regularity assumptions on both the coefficient a and the source term F . We introduce generalized Morrey and Campanato spaces extending the classical definitions by replacing uniform boundedness requirements with suitable integrability conditions. Within this framework, we establish regularity estimates for the gradient of weak solutions in these generalized spaces. As applications, we recover classical H{ö}lder and Lebesgue estimates and derive fractional Sobolev regularity results. In particular, the proposed approach yields fractional Sobolev estimates in situations where the coefficient may be discontinuous and the gradient of the solution is not expected to be locally bounded.

We propose a conservative adaptive rank method for the 1D1V Wigner-Poisson system. The method targets a central challenge in deterministic quantum kinetic simulations: reducing the cost of phase-space evolution while preserving the macroscopic invariants needed for physical fidelity. The scheme combines a sampling-based adaptive rank Wigner-Poisson update [7] with a conservative macroscopic correction. A conservative density-momentum solve provides local macroscopic updates, a Fermi-Dirac-type reconstruction transfers them to the kinetic solution, and a global quadratic moment correction enforces the discrete total energy constraint at the kinetic level. Unlike Maxwell-Boltzmann-type corrections commonly used in classical kinetic settings, the reconstruction uses a Fermi-Dirac-type form motivated by the model's quantum-statistical structure. The corrected state is incorporated into an ACA SVD representation, allowing the numerical rank to adapt to the phase-space complexity generated by the nonlocal Wigner operator and self-consistent Poisson field. Numerical experiments for the two-stream instability, strong Landau damping, and bump-on tail instability show that the method captures benchmark Wigner-Poisson dynamics for several values of the quantum parameter H, maintains bounded adaptive ranks, and preserves the specified global discrete invariants with conservation errors near machine precision. We also compare this formulation, which uses local density-momentum correction plus global total energy correction, with a related globally conservative formulation for mass, momentum, and energy [8]. The two approaches produce nearly identical phase-space and diagnostic results for the periodic benchmark test considered here, indicating that both correction strategies are compatible with adaptive rank compression for Wigner-Poisson dynamics in the tested 1D1V periodic setting.

Expanding products of modal logics are bimodal logics obtained from the combination of a `horizontal component' logic and a `vertical component' logic, lying between the fusion and the Cartesian product of the two logics. Gabelaia et al. showed that expanding products are often decidable when the first component is Noetherian, although their methods are semantical and do not yield complete axiomatisations. They do, however, propose a candidate, dubbed the expanding commutator of the two logics and known to be complete in many `non-Noetherian' cases. In this paper, we consider various expanding products of modal logics whose vertical component is $\sf GL$. We show that the standard axiomatisation is complete when the horizontal component is either $ {\sf K4}$ or $ {\sf GL} $, but incomplete when it is ${\sf Grz}$ or any logic between ${\sf K4.3}$ and ${\sf Grz.3}$, thus yielding a partial solution to a question posed by Gabelaia et al. more than two decades ago.

Inspired by the model theory of difference fields in characteristic zero, a class of automorphisms of an algebraic variety, here called compound fundamental isotrivial, is introduced. These are algebraic dynamical systems that are built up via a finite sequence of equivariant fibrations from (possibly nonautonomous) algebraic dynamics which trivialise after base extension over themselves. Every wild automorphism of an abelian variety is compound fundamental isotrivial. Conversely, it is shown that the only irreducible projective varieties admitting a wild automorphism that is compound fundamental isotrivial are the abelian varieties. That is, the wild automorphism conjecture of Reichstein, Rogalski, and Zhang is here proven for compound fundamental isotrivial dynamics. Along the way, a counterexample to the naive generalisation of the conjecture to the nonautonomous setting of $σ$-varieties is provided.

We characterize the pointwise notion of almost equicontinuity for families of real-valued measurable functions on subsets of $\mathbb R^n$ of finite measure. The characterization is given by means of an integral truncated translation condition. We also provide examples showing that the finite measure assumption and the truncation are essential.

In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(Ω)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $β\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.

Let $G = \mathbf{GL}_d(\mathbb{F})$ be the general linear group over a field of cardinal $q$, and let $\mathbb{k}$ be a field of positive characteristic which does not divide $q(q-1)$. Building on the works of Cline, Parshall, and Scott, we show how to compute Ext-groups between $\mathbb{k}G$-modules using the quantum Schur algebra. The main novelty is our ability to compute these Ext-groups in higher degree than what was done before. More precisely, let $\ell$ be the order of $q$ in $\mathbb{k}$. In previous work, this method enabled the computation of the cohomology groups $H^*(\mathbf{GL}_d,M)$ in degree $*\leq \ell-1$. We show that for a lot of modules $M$, we can compute these cohomology groups in higher degree, with an example where we can compute until degree $3(\ell-1)$. We also show some new result on Ext-groups between modules over the quantum Schur algebra along the way.

In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(ρ_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $ρ_0$ has a positive lower bound. Furthermore, if $ρ_0 \in C^2$ and $||ρ_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.

For a complex single variable polynomial $f$ of degree $d$, let $K$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K$ has an invariant component $K^*$ on which $f$ acts as a degree $d_*<d$ map. This is a simplest instance of holomorphic polynomial-like renormalization (Douady-Hubbard). One can associate a certain Cantor-like subset $G'$ of the circle with $K^*$; it is defined as the set of arguments of all smooth or broken rays to $K^*$. We will describe a role the Hausdorff dimension of $G'$ and the respective Hausdorff measure play in geometry of $K^*$. In particular, we give upper and lower bounds on the modulus of renormalization in terms of the Hausdorff measure of $K^*$.

We study properties of graphs (or rather graph sequences) saying that some restricted count of subgraphs is approximatively what is expected in a random graph. It has been shown by several authors that many such properties characterize quasi-random graphs, but there are also some exceptions. We continue here the line of investigation in Janson and Sós (2013), and introduce some new versions of these properties, in order to better understand why many of these properties are quasi-random, and to understand the structure of the exceptions that are not. A new feature in the proofs is a simple decomposition of the subspace of symmetric functions in $L^2([0,1]^m)$ into subspaces that are irreducible for the action of measure-preserving transformations of $[0,1]$; this simplifies some arguments and gives structure to others.

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur positivity of symmetric functions naturally associated with set partitions, with respect to two different notions of descent. In the first case, the Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings. In the second case, the Schur functions correspond to two-rows Young diagrams, and the coefficients are partial sums of associated Bell numbers. A key ingredient of our approach in the second case is the notion of a removable singleton, defined algebraically and shown to admit an equivalent combinatorial interpretation via jeu-de-taquin rectification of skew tableaux. As an application, we establish Schur positivity for various classes of symmetric functions indexed by non-crossing partitions and partitions with a given number of parts. We provide an explicit combinatorial description of the tableaux that contribute to the Schur expansion, and we connects the obtained coefficients to some known integer sequences.

We observe that the canonical map \(KK(A, \prod_{n \in \mathbb{N}} B_n) \to \prod_{n \in \mathbb{N}} KK(A,B_n)\) is an isomorphism of abelian groups whenever \(A\) enjoys the Universal Coefficient Theorem and \(B_n\) are unital, simple and purely infinite C*-algebras. This clarifies an aspect of previous work of Dadarlat--Eilers and Tikuisis--White--Winter.

This paper aims to explore the formation mechanism of probability distribution in situations where the differences among random experiments are distinguishable, and these differences continue to evolve along with the dynamic changes in conditions and their mechanisms of action. To this end, we are motivated to devise a new theoretical system -- theory of uncertain probability (TUP) with Kolmogorov's system and nonlinear theories as special cases. TUP develops a novel model that integrates probability and uncertainty as well as the known and unknown to more accurately depict numerous typical random phenomena under more realistic assumptions, and thus provides appropriate tools for greater variety of real needs. It also allows for pioneering interpretation of the causal mechanisms underlying many important distributional characteristics and incorporation of pathwise property to distribution model.

For a regular semisimple Higgs bundle with a smooth spectral curve, we prove that, over the \etale\ locus, the Atiyah class of the underlying bundle is induced by the Atiyah class of the spectral line bundle and takes values in the centralizer of the Higgs field. Further, when the discriminant is reduced, we construct a logarithmic refinement across the branch divisor: the Atiyah class extends as a class with logarithmic poles and values in a natural regularized centralizer sheaf.

Let $f$ be a transcendental entire function, $V$ be a simply connected Fatou component of $f,$ and $U$ be a Fatou component with $f(U)\subset V.$ There is a natural way to associate $f|_U$ to an inner function, namely a function $g_f:=ψ^{-1}\circ f\circφ,$ where $φ:\mathbb{D}\to U$ and $ψ:\mathbb{D}\to V$ are Riemann maps. Inner functions have been used as a tool in the study of the iterates of transcendental entire, and more recently meromorphic, functions. However, there are only a few examples where associated inner functions have been calculated explicitly, with the case where $f$ has infinite degree in $U$ being the least well understood and more complicated. In this paper, we introduce a general method for calculating associated inner functions to a wide class of entire functions arising as `lifts'. In particular, if $f$ is a lift of a transcendental entire function $h,$ we show that an inner function associated to $f|_U$ can be obtained by relating it to an inner function associated to $h|_G,$ where $G$ is the Fatou component that lifts to $U.$ This result significantly generalises the main part of a theorem by Evdoridou, Rempe and Sixmith, and can be applied to several functions that have been studied so far. In both finite- and infinite-degree settings, the results hold for forward-invariant Fatou components as well as for wandering domains.