Papers

We show that if $(S,g,\xi_i,\eta_i,\Phi_i)$ is a quasi-regular $3$-Sasakian orbifold of dimension $7$, then its quotient $Z$ by one of the Reeb vector fields inherits an $SU(3)$-structure parametrized by two real parameters. We compute its torsion and show that it is an LT-structure. Moreover, for certain values of the parameters, we can provide a nearly Kähler structure on $Z$ appearing as a special case of our construction. We give several examples in the regular, quasi-regular and orbifold cases.

Let $N\geq2$ and let $\Omega\subset\R^N$ be a connected Lipschitz domain, possibly unbounded, symmetric with respect to the origin, and such that $0<\gammaN(\Omega)<1$. We assume that the Gaussian Sobolev embedding $H^1(\Omega,\gammaN)\hookrightarrow L^2(\Omega,\gammaN)$ is compact; a sufficient condition is the existence of a bounded Gaussian Sobolev extension operator from $\Omega$ to $\R^N$. Denote by \[ 0=\mu_0(\Omega)<\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots \] the Neumann eigenvalues of the positive Ornstein–Uhlenbeck operator $-\Delta+x\cdot\nabla$ in $\Omega$. We prove the sharp reciprocal-sum inequality \[ \sum_{k=1}^{N}\frac{1}{\mu_k(\Omega)} \geq \frac{N}{\mu_1(B_R)}, \] where $B_R$ is the Euclidean ball centred at the origin and satisfying $\gammaN(B_R)=\gammaN(\Omega)$. Equality holds if and only if $\Omega=B_R$. The proof combines a coupled $N$-dimensional Ritz argument with a Gaussian raywise rearrangement. The angular imbalance is encoded by a symmetric trace-free matrix, whose contribution is controlled by a finite-dimensional convexity inequality.

We study the problem of sequential change detection over a general class of probability distributions ($\mathcal P$), where both the pre-change and post-change distributions are unknown and belong to $\mathcal P$. We do not assume a pre-specified partition of $\mathcal P$ into pre- and post-change families. We propose a general class of sequential change detectors obtained by aggregating point-null e-processes over possible changepoints and taking an infimum over candidate no-change distributions. The weights in the aggregation scheme determine whether they attain average run length (ARL) control and probability-of-false-alarm (PFA) control. Under suitable assumptions, we prove that our methods achieve first-order asymptotically optimal detection delay. Concrete examples include sub-Gaussian and bounded mean changes, Gaussian mean changes with unknown variance, as well as changes in Markov transition matrices.

We show that every countable group $G$ that is mixed-identity-free (MIF) and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. We also present a purely dynamical criterion that implies this result. Our criterion recovers several of the existing results on primitivity, including those involving acylindrically hyperbolic groups. Furthermore, our criterion also applies (positively) to a plethora of new examples, such as Thompson-like groups, commensurator groups of hyperbolic groups, some Kac-Moody groups, and many more.

The Nahm transform for 4-dimensional flat hyperkahler tori is an isometry between the moduli space of anti-self-dual (ASD) instantons on a torus $T^4$ and the moduli space of ASD instantons on the dual torus $\hat{T^4}$ parametrising flat line bundles on $T^4$. This paper studies a generalised Nahm transform on an 8-dimensional torus with a Spin(7) structure. I construct instanton bundles with Dirac kernels respectively in positive and negative chiralities, demonstrating that the usual Nahm transform is not well-defined. I then define a notion of asymptotic holonomy for instantons twisted by a high power $k \gg 1$ of an instanton line bundle, and I show that this asymptotic holonomy reduces to Spin(7) to second order in $k$. Finally, I provide examples for which the asymptotic holonomy is $\mathfrak{u}(1)^4$, and thus not Spin(7).

Motivated by Erdős's conjecture on the Turán number of degenerate bipartite graphs, Bradač, Janzer, Sudakov and Tomon proved that $ \ex(n,T \Box P)=\Theta_{T,P}(n^{3/2})$ for every nontrivial tree $T$ and every nontrivial path $P$, and conjectured that the same order of magnitude holds for the Cartesian product of any two nontrivial trees. We prove their conjecture. More generally, for every integer $r\ge2$, we introduce a class of bipartite $r$-degenerate graphs, called $r$-star-flip graphs, that are obtained from a seed tree by a sequence of local vertex-duplication operations. We prove that every fixed $r$-star-flip graph $H$ satisfies $\ex(n,H)=O_H(n^{2-1/r})$. Every Cartesian product of two trees is a $2$-star-flip graph, while the star-flip class also contains graphs that do not arise as such products. As a further application, our framework yields a new proof of Füredi's theorem: if $H$ is a fixed bipartite graph in which at most one vertex in one colour class has degree greater than $r$, then $\ex(n,H)=O_H(n^{2-1/r})$. The key ingredient is a conditional-resampling procedure that extends the tree branching random walk on the seed tree to a random homomorphism of the entire star-flip graph, while preserving the branching-random-walk distribution on every live tree.

This paper develops bootstrap methods for likelihood-based inference in autoregressive conditional duration (ACD) models, where the sample size is endogenously determined by durations observed over a fixed time span. This feature fundamentally shapes the asymptotic framework, particularly so when the durations do not have finite expectation. Building on recent limit theory for heavy-tailed and integrated ACD processes, we analyze recursive bootstrap schemes that either fix the time span (yielding a random sample size) or fix the number of durations (yielding a random time span). We establish a bootstrap theory for ACD models that links naturally to renewal theory with random sample sizes. For the fixedcount bootstrap, we prove first-order validity in the finite-mean and boundary cases and characterize the random limiting bootstrap distribution in the infinite-mean case. Although classical bootstrap consistency can fail when the durations have infinite expectation, we argue that the bootstrap remains valid and yields asymptotically normal t-statistics. Monte Carlo evidence shows that the proposed methods have good finite-sample properties in both finite- and infinite-mean settings, and are robust to distributional misspecification relative to the exponential likelihood. We conclude with an empirical application to cryptocurrency ETFs.

We study boundary contributions to the holomorphic Lefschetz formula for strict self-maps of compact complex manifolds with a simple normal crossings divisor. At a normally non-resonant boundary fixed point, the local term reduces to the fixed-point problem on the boundary stratum and vanishes in the de Rham specialisation. For resonant points satisfying a normal admissibility condition, finite flat rescaling and relative duality recover the local trace as the coefficient at the excess normal order of a holomorphic family of relative residue traces; in the balanced case, this coefficient is the trace of a class on the special fibre. The numerical coefficient is independent of the admissible presentation. In a coupled family, the de Rham boundary index is the product of the normal contact order and the tangential complete-intersection multiplicity. Boundary contributions to the fixed-point formula for the complement are supported on the resonant fixed points.

Computing numerical approximations of matrix functions frequently relies on the efficient evaluation of high-degree matrix polynomials. Although computational bounds are historically governed by the Paterson–Stockmeyer (PS) method, recent theoretical developments have demonstrated the viability of evaluation schemes that eliminate two matrix products ($2M$). Existing literature documents stable instances of this $2M$ reduction only for isolated cases, such as specific degrees of Taylor approximations for the matrix exponential and the matrix logarithm. However, a generalized approach for arbitrary polynomials remains unestablished. To address this limitation, this work presents a software-driven procedure that extends these computational savings to polynomials of degrees $m \in \{18, 21, 24, 26, 27, 28\}$ and all $m \ge 30$, requiring primarily a non-zero leading coefficient. Since the underlying evaluation coefficients must be determined by solving systems of nonlinear equations (SNEs), selecting a numerically stable solution set is critical. We introduce an automated verification routine designed to filter and validate robust coefficient sets for floating-point execution. The primary contribution is a MATLAB implementation leveraging variable precision arithmetic to handle the underlying SNEs, verify stability, and project precision bounds. Numerical experiments involving various matrix functions verify that the developed implementation preserves or, in some instances, enhances the numerical accuracy of the classic PS method, while systematically achieving the theoretical reduction of $2M$.

In many applications, graphs are observed repeatedly, where the task is to observe and track weak localized structural perturbations under noise, such as vertex-coordinate displacements or edge-weight/attribute changes on a fixed vertex set. One-shot graph distances can miss these localized perturbations or fluctuate too strongly to support reliable monitoring, especially in large graphs where signal dilutes with scale. We propose Robust Order-aware Spectral Amplification (ROSA), a distance-amplification operator that integrates metric evaluations along an order-aware edge-removal filtration. We prove that ROSA cannot reduce the included base distance and conditionally increases it whenever a filtered step exposes additional signal; separately, we prove that it can strictly improve an inverse coefficient of variation stability score (IS$^2$), defined as the mean of a noisy graph distance divided by its standard deviation, under an explicit sufficient condition that can be checked empirically. Experiments on synthetic graphs with localized edits under three noise models, varying graph densities, and three algorithmically selected edit sites show that ROSA can often approximately double the IS$^2$ of the base spectral distance. We show where ROSA works and stops working on real-world use cases including mitochondrial networks, fMRI correlation matrices, tissue networks, protein conformer graphs, and retinal vasculature multiplex graphs. The empirical results across graph, operator, and noise settings are consistent with the theoretical amplification conditions and boundary constructions. Evaluated with empirically estimated operative quantities, the bound agrees with the observed gain in direction and approximate scale.

We determine the capacity of a family of $q$-ary sticky-insertion channels. Fix $q\geq2$ and $d\geq1$, and let $\lambda$ be the unique positive solution of $\lambda^d = (q-1) (\lambda^{d-1} + \cdots + \lambda + 1 )$. We prove that, for every repetition law supported on $1+d\mathbb{Z}_{\geq0}$ and satisfying a coefficientwise-domination criterion with domination constant $\gamma\geq\lambda^{-d}$, the Shannon capacity equals the zero-error capacity, both being $\log_2\lambda$ bits per symbol. We also exhibit explicit repetition laws satisfying these conditions, one of which is given by the weighted Fuss–Catalan numbers. To the best of our knowledge, these are the first known cases of nontrivial repeat channels whose Shannon capacity has been determined exactly.

Let $N\in\{2,3\}$, $0<\mu _1\leq\mu _2$, and $0<\beta<\mu _1$. We prove that the equal-frequency two-component cubic Schrödinger system \[ -\Delta u+u=\mu _1u^3+\beta uv^2, \qquad -\Delta v+v=\mu _2v^3+\beta u^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the synchronized state constructed from the unique positive radial solution of $-\Delta w+w=w^3$ in $\mathbb{R}^N$. This settles Sirakov's equal-frequency uniqueness conjecture throughout the weak-coupling range. The main difficulty in the proof is to exclude radial solutions for which the ratio of the normalized components is nonconstant. After normalization, the two components satisfy scalar equations with a common potential. We construct a weighted Pohozaev functional for the system together with a correction term and prove that both the corrected functional and the associated weighted functional are strictly positive. Combining these sign properties with a radial flux identity and an auxiliary quotient associated with the component ratio forces synchronization.

Semantic communication is an emerging paradigm that can preserve the meaning of data during transmission. However, human users are often interested in specific semantic content based on their intent, and users' intent is often not considered in current semantic coding design. Moreover, most of the existing semantic models are fine-tuned using specific datasets, which limits their generalization capability. Furthermore, how to prioritize semantically important features in large-scale multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) systems remains largely unexplored. To address the aforementioned challenges, in this paper, we propose a generalized query-oriented image semantic coding (QO-ISC) framework. In the proposed framework, the transmitter extracts features which are relevant to the user's query and the receiver reconstructs an image based on those features. We use a pretrained large artificial intelligence (AI) model (LAM) to enhance general feature representations. We develop a semantic-aware hybrid beamforming (SA-HBF) algorithm to prioritize semantically important features for large-scale MIMO-OFDM system. When evaluated on unseen object categories within the dataset, simulation results show that our proposed generalized QO-ISC framework achieves better performance than the traditional codec and two state-of-the-art semantic coding schemes.

The most general class of multipoint inhomogeneous boundary-value problems for systems of linear ordinary differential equations of arbitrary order $r\geq 1$ is investigated, whose solutions belong to a given Sobolev space $W_p^{n+r}$, where $n\geq 0$ and $1\leq p\leq \infty$. The boundary conditions in these problems contain Caputo derivatives of fractional or integer orders, which may exceed the order of the differential system equations. Constructive sufficient conditions are established under which the solutions of these problems are continuous with respect to a parameter from an abstract metric space in the Sobolev space $W_p^{n+r}$.

We consider optimization problems in the space of measures. As a regularization term, the problem includes the transport distance to a given prior measure. For the derivation of second-order optimality conditions of no-gap type, the theory of weak-$\star$ second subderivatives is used which will lead to an equivalence with quadratic growth under additional assumptions on the smooth part of the objective and on the Kantorovich potential, i.e., the solution of the dual transport problem. Further, the weak-$\star$ second subderivative is calculated and weak-$\star$ epidifferentiability is proven. Finally, the results are applied to optimal control problems in measure space.

In this paper, the corner Poisson structure of four-dimensional Palatini-Cartan gravity is derived. Building on the classical description of gravity on manifolds with boundary, specifically on the boundary constraint algebra, a pre-Dirac structure on the space of corner fields is obtained together with a reduction procedure that yields a maximal Dirac structure, identified as the graph of a Poisson bivector field, on the reduced space of corner fields. It is further shown that this Poisson structure admits an equivalent affine Poisson description, which naturally exhibits the reduced corner theory as a $BF$-like theory and leads to a BF$^2$V formulation. This provides the basis for a unified framework for the bulk, boundary, and corner structures of Palatini-Cartan gravity.

We introduce Topoformer, a lightweight and scalable framework for graph representation learning that encodes topological structure into attention-friendly sequences. At the core of our method is Topo-Scan, a novel module that decomposes a graph into a short, ordered sequence of topological tokens by slicing over node or edge filtrations. These sequences capture multi-scale structural patterns, from local motifs to global organization, and are processed by a Transformer to produce expressive graph-level embeddings. Unlike traditional persistent homology pipelines, Topo-Scan is parallelizable, avoids costly diagram computations, and integrates seamlessly with standard deep learning architectures. We provide theoretical guarantees on the stability of our topological encodings and demonstrate state-of-the-art performance across graph classification and molecular property prediction benchmarks. Our results show that Topoformer matches or exceeds strong GNN and topology-based baselines while offering predictable and efficient compute. This work opens a new path for parallelizable and unifying approaches to graph representation learning that integrate topological inductive biases into attention frameworks.

We introduce and solve the Interacting Quantum Symmetric Exclusion Process (IQSEP), a family of models describing the stochastic quantum hopping of charged particles along the edges of a lattice, with hopping amplitudes that depend on the occupations of neighbouring sites. In the absence of interactions, they reduce to the standard quantum simple symmetric exclusion process, exhibiting coherent diffusive transport. For interactions of order one, they capture incoherent diffusive transport and its fluctuations, characterized by density-dependent diffusivity and mobility, making contact with the macroscopic fluctuation theory. By rescaling the interaction strength appropriately with the lattice mesh, we define a mesoscopic scaling regime that retains a finite coherence length in the continuous thermodynamic limit. This regime interpolates between coherent behavior at small length scales and incoherent behavior at large scales. The resulting scaling theory accounts for fluctuations of quantum coherences in interacting diffusive systems, going beyond the scope of standard fluctuating hydrodynamics.

We investigate the geometric classification of complex left-symmetric algebras and left-symmetric superalgebras in low dimensions. Building on the algebraic classifications of Bai and Zhang for 3-dimensional left-symmetric algebras and for 2- and 3-dimensional left-symmetric superalgebras, we determine all degenerations and non-degenerations among their isomorphism classes. We prove that the variety of 3-dimensional left-symmetric algebras has dimension 10 and decomposes into 31 irreducible components, ten of which are rigid. For left-symmetric superalgebras, we describe the irreducible components, determine their dimensions, and identify the rigid superalgebras in the varieties of dimensions (1,1), (1,2), and (2,1).

Let $k>t\ge 1$ be integers and set $d=k-t$. A $k$-uniform hypergraph $\mathcal F$ is called $t$-intersecting if any two edges intersect in at least $t$ vertices, and is called $t$-critical if its minimum $t$-transversal has size $k$. Frankl proved that, for $k\ge d^4$,$|\mathcal F|\le \binom{k+d}{d},$ with equality only for the complete $k$-graph on $k+d$ vertices, and conjectured that the same conclusion should hold when $k>c d^2$ for some constant $c$. In this paper we confirm this conjecture for $c=30$. The proof relies on Frankl's fixed-edge decomposition and Füredi's pseudo-sunflower method.

This paper investigates the approximation properties of linear Kantorovich-type sampling operators in the setting of general Banach lattices $X$ on the torus $\mathbb{T}$ and the real line $\mathbb{R}$. Under a natural assumption on the uniform boundedness of the Steklov averaging operators, we establish direct and inverse approximation estimates, as well as strong converse inequalities. Our framework does not require the underlying spaces to be translation-invariant, thereby covering a wide variety of classes and extending the estimates for Kantorovich-type operators previously known primarily for Lebesgue spaces $L_p$.

We study partition functions $p_{\delta,g}(n)$ counting partitions into parts congruent to $0$ or $\pm g \pmod\delta$. Using generalized Dedekind eta functions and Rankin-Cohen brackets, we derive infinite families of Euler-type recurrences involving divisor sums and Fourier coefficients of cusp forms. We also obtain an explicit recurrence for $\delta=5$, which, as a corollary, gives a Ramanujan-type congruence. As a corollary of our method of proof, we obtain a Rademacher-type formula involving Kloosterman sums and Bessel functions.

The chromatic threshold, originating in a question of Erdős and Simonovits, asks when a linear minimum-degree condition forces bounded chromatic number in H-free graphs. Motivated by a question of Thomassen, the homomorphism threshold asks for the stronger conclusion that every such graph admits a homomorphism to an H-free graph of bounded order. Since the work of Goddard and Lyle determined the clique case, exact homomorphism thresholds for individual non-complete forbidden graphs have remained unknown. In this paper, we extend the clique case to a larger family of forbidden graphs, determining the homomorphism threshold exactly for every graph in this family.

This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We establish the existence of kinetic solutions for continuous obstacles and further derive $L^1$-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin–Wentzell large deviation principle in $L^1(0,T;L^1(\mathbb{T}^N))$. Unlike existing large-deviation arguments based on control-uniform penalization of the obstacle, we work directly with the reflected skeleton equation and retain the possibly singular Radon reflection measure throughout, while a viscous approximation provides the spatial $H^1$-regularity required for compactness before passing to the inviscid limit. Our results provide a hyperbolic counterpart to the large deviation theory for parabolic reflected SPDEs developed by Matoussi, Sabbagh, and Zhang (2021).

Algebraic multigrid (AMG) methods are robust and efficient black-box preconditioners for linear systems arising from low-order discretizations of elliptic partial differential equations, but their performance often deteriorates for high-order methods. We introduce an algebraic-geometric multilevel preconditioner for continuous lagrangian finite element discretizations. The method automatically constructs prolongation operators and a Galerkin hierarchy using only the coordinates of finite element support points, without requiring a prescribed mesh hierarchy or domain decomposition. The support-point are recursively partitioned by an R-tree algorithm based on axis-aligned bounding boxes, producing a hierarchy of agglomerates. In contrast to classical AMG, which typically uses piecewise-constant aggregate coarse spaces, our method embeds local discontinuous polynomial spaces of degree $p'>0$, defined on the agglomerate boxes, through continuous nodal interpolation. This yields a high-order extension of the Nicolaides coarse space. A two-level analysis quantifies how the coarse polynomial degree offsets the effects of large agglomerates and high-order fine discretizations, under uniform box-regularity, interpolation-stability, and stable-decomposition assumptions. Numerical experiments in two and three dimensions show that the resulting V-cycle, used as a conjugate-gradient preconditioner, maintains stable iteration counts and remains effective in regimes where standard AMG deteriorates.

This paper studies iterated Hopf Ore extensions (IHOEs) over a field $\Bbbk$ of characteristic zero, a significant class of connected Hopf algebras with finite Gelfand-Kirillov dimension. We establish two fundamental structural properties for arbitrary IHOEs of $\Bbbk$. First, the class of IHOEs of $\Bbbk$ is closed under Hopf subalgebras and quotient Hopf algebras. Second, every one-sided coideal subalgebra of an IHOE of $\Bbbk$ is an iterated Ore extension of $\Bbbk$. These structural facts are built upon the thin replacement machinery for PBW generating systems originating from Kharchenko's work. We further prove a general inheritance theorem for PBW generating systems, which serves as a systematic perturbation method to produce PBW generating systems for subalgebras from those of the ambient algebra under mild hypotheses. Based on these theoretical advances, we develop an explicit combinatorial algorithm for classifying all one-sided coideal subalgebras of arbitrary IHOEs of $\Bbbk$. As practical illustrations, we explicitly classify all right coideal subalgebras of noncocommutative connected Hopf algebras of GK-dimension three.

We define a new class of superalgebras, called tensor product degenerate spin affine Hecke superalgebras, and study their structure theory. We establish an isomorphism that relates the tensor product degenerate spin affine Hecke superalgebras to the tensor product degenerate affine Hecke-Clifford superalgebras. The relationship between these superalgebras is a tensor product analogue of the relationship between the degenerate spin affine Hecke superalgebras and degenerate affine Hecke-Clifford superalgebras.

Let $g\in H(\mathbb D)$, the generalized Hilbert operator $\mathcal H_g$ is defined by \[ \mathcal H_g(f)(z)=\int_0^1 f(t)g'(tz)dt,\ \ z\in \mathbb D\, \ \ f \in H(\mathbb D). \] Let $\mathcal R_p=\mathcal H(H^p)$ be the range of the classical Hilbert operator on Hardy space, equipped with the pullback norm, and let $(\mathcal R_p,H^p)$ denote the Hadamard multiplier space. For $1<p<\infty$, we prove the exact multiplier characterization \[ \mathcal H_g:H^{p}\longrightarrow H^{p} \ \ \text{is bounded} \quad\Longleftrightarrow\quad g'\in(\mathcal R_p,H^p), \] and an equivalent Hilbert-matrix bilinear criterion $\mathfrak B_p(g)<\infty$. We identify the multiplier space completely when $1<p\le2$: \[ (\mathcal R_p,H^p)=H\left(p,\infty,\frac1{p'}\right). \] For $p>2$, we prove that the multiplier space $(\mathcal R_p,H^p)$ is strictly contained in $H\left(p,\infty,\frac1{p'}\right)$. This shows that \(g\in \Lambda(p,1/p)\) does not imply that $\mathcal H_g$ is bounded on \(H^p\), giving a negative answer to the conjecture posed by Galanopoulos, Girela, Peláez and Siskakis. In addition, we locate two previously known sufficient classes inside the multiplier space. This allow us obtain a complete coefficient characterization of $\mathcal H_g$ on $H^{p}$ for $g \in H(\mathbb D) $ with nonnegative decreasing Taylor coefficients. We then study the structure of $(\mathcal R_p,H^p)$. % It turns out that $(\mathcal{R}_p,H^p)$ contains all polynomials as well as Cauchy transforms. We show that the multiplier spaces $(\mathcal{R}_p,H^p)$ form a strictly increasing family with respect to the exponent $p$.

We study parametric classes of almost stochastic dominance on general Polish spaces as order relations for probability distributions with a parameter $\gamma \in [0,1]$. Larger values of $\gamma$ correspond to weaker order relations: $\gamma=0$ gives classical stochastic dominance $\le_{st}$, whereas $\gamma=1$ gives a complete preorder based on comparison of expectations of a fixed increasing function $g$. It is well known that $X \le_{st} Y$ can be characterized by the existence of a solution to an optimal transport problem with $\mathrm{OT}_c(X,Y)=0$ for a suitable cost function $c$. We generalize this idea so that the best possible parameter $\gamma$ for almost stochastic dominance can be determined from the solution of an optimal transport problem. Using a generalization of the classical Kantorovich–Rubinstein duality theorem to quasi-pseudo-metrics, we derive a dual characterization of the order in terms of expectation comparisons for a parametric class of test functions. Consequently, our relations are always transitive, in contrast to some other recent approaches to almost stochastic dominance based on optimal transport. A natural multivariate approach to almost stochastic dominance, based on classes of test functions with bounds on partial derivatives, was recently introduced by Müller et al. (2025). We show that this approach is a special case of our framework and derive the best possible parameters $\gamma$ for examples considered there, as well as for other examples from the literature. We also prove a robustness result showing that, under small perturbations of the distributions in a Wasserstein-type metric related to the optimal transport problem, the best possible $\gamma$ increases only slightly.

We prove that ordinary Hilbert-Kunz multiplicity can be transcendental. More precisely, over every uncountable algebraically closed field of characteristic $p>2$, there exists a normal standard graded domain $S$ and an $S_+$-primary homogeneous ideal $I\subseteq S$ such that $e_{\rm HK}(IS_{S_+})$ is transcendental.