We study the vanishing viscosity limit for the incompressible Navier-Stokes equations with anisotropic viscosity in bounded domains, analyzing certain classes of symmetric flows: plane parallel, pipe parallel and circularly symmetric. By anisotropic viscosity, it is meant here that the viscosity coefficient in the direction normal to the wall is different than that in the direction tangential to the wall. Using boundary layer theory and semigroup techniques, we establish the validity of the vanishing viscosity limit in the energy norm, that is, in $L^2$ in space uniformly in time, for all three classes of flows, with explicit convergence rates. We further obtain higher-order estimates under suitable assumptions on the anisotropic viscosity coefficients. In particular, we consider both the case in which the tangential viscosity coefficient goes to zero faster than the normal one and, conversely, the case when the normal coefficient vanishes faster then the tangential one. Our results extend previous works on isotropic viscosity and provide new examples where the vanishing viscosity limit can be rigorously justified in the anisotropic setting.

We investigate the impact of large synaptic delays on the emergence of periodic dynamics in inhibitory neuronal networks, within the framework of the NNLIF model. Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficitently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Ces{à}ro mean toward a limit function determined solely by these pseudo-equilibria.

In response to the growing demand for sustainable and efficient urban deliveries, this study introduces a two-echelon vehicle sharing and repositioning problem for containerized delivery operations within a hyperconnected urban logistics system. We leverage a Physical Internet (PI)-enabled three-tier logistic hub network, comprising gateway, local, and access hubs, to facilitate efficient flows. By adopting containerized delivery, vehicles can rapidly swap standardized modular containers at hubs to reduce handling time. Moreover, inspired by the PI concept of open resource sharing, we determine optimal service routes within a two-echelon structure that jointly utilizes heterogeneous vehicle fleets and enables dynamic vehicle relocation across hubs. We formulate this problem as a multi-period integer program that integrates path-based service vehicle planning with arc-based container routing. To address real-world large-scale instances, we propose a decomposition-based heuristic with capacity-aware flow assignment, which partitions the problem into subproblems structured by echelons and regions. A case study on the Atlanta metropolitan area demonstrates the effectiveness of the proposed model and solution approach. Experimental results show that the two-echelon hyperconnected delivery system reduces CO2-eq emissions by 45.0% and total costs by 16.8% at full market share compared to a traditional single-echelon alternative, while enabling vehicle repositioning further lowers costs by up to 17.7%.

The classical Frenet frame is defined by a concrete construction from the tangent, principal normal, and binormal vectors of a regular space curve. However, this construction breaks down at singular points and at points where the curvature vanishes. Motivated by this observation, we reconsider the Frenet frame from an axiomatic viewpoint and identify the fundamental properties that characterize it independently of its classical construction. Based on the theory of frontals on the unit sphere and Legendre duality, we introduce a generalized Frenet frame for singular space curves. Furthermore, we introduce the notion of a frame sequence, which gives rise to an integer-indexed family of Frenet frames together with the corresponding curvatures and torsions. This viewpoint provides a unified framework encompassing both the Frenet and Bishop frames of space curves and the evolute-involute correspondence for spherical frontals. Moreover, explicit recursive formulas are derived, revealing that the curvatures and torsions at each level encode, respectively, the magnitude and rotational behavior of the invariants arising at the preceding level.

Menger's theorem is one of the most fundamental results in graph theory. It states that if a graph $G$ does not contain $k$ disjoint paths between two given sets $X$ and $Y$ of vertices in $G$, then there is a set of at most $k-1$ vertices that intersects every path between $X$ and $Y$. Nguyen, Scott, and Seymour gave a counterexample to the conjectured natural coarse variant in which the paths are required to be pairwise at distance at least $d$, and, conversely, there is a set of at most $k-1$ bounded-radius balls intersecting every path between $X$ and $Y$. In other words, the coarse Menger property does not hold in general. We prove that graphs whose cycles space is generated by cycles of bounded length do have the coarse Menger property. As a corollary, we show that many natural graphs and geodesic metric spaces have the coarse Menger property. These include hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded face size, and complete Riemannian planes.

For a manifold $\overline{M}$ with boundary $\partial M$ and interior $M$, we introduce and study a weakening of the concept of projective compactness for torsion-free linear connections on $M$, which we call projective pre-compactness. Via the Levi-Civita connection, this concept applies to pseudo-Riemannian metrics on $M$. This is motivated by scattering theory and general relativity (GR), via asymptotic forms of metrics used in these areas. In the general setting of a projectively pre-compact connection $\nabla$ we show that, assuming weak asymptotic conditions on the Ricci curvature, there is an induced projective structure on the boundary. Under a slightly stronger condition on Ricci, we show that the standard tractor bundle and its normal tractor connection arise naturally on this boundary structure. The key ingredient to this is that $\nabla$ admits a smooth extension to the boundary as a linear connection on the tensor product of Melrose's b-tangent bundle with a density bundle, which then restricts to the boundary tractor bundle. A projectively pre-compact pseudo-Riemannian metric (satisfying the conditions on the Ricci curvature) is then shown to induce a holonomy reduction of the boundary projective structure to an indefinite orthogonal group. This endows the boundary with a decomposition into so-called curved orbits, which are either open or embedded hypersurfaces, representing space-like, time-like and light-like infinities in a GR context. We introduce and study a new asymptotic form for such metrics which is available near any boundary point and relate it to an asymptotic form used in general relativity, which is only available near boundary points in the open curved orbits. We show that, in that region, projective pre-compactness essentially is equivalent to the asymptotic form from GR, and projective compactness is equivalent to vanishing of the mass aspect.

Many fractals $G$ admit a harmonic immersion into $\R^n$, i.e. an immersion which minimises a natural energy under fixed boundary conditions; we look for harmonic immersions of the Sierpinski gasket into the hyperbolic plane. We show that, given any three points $\tilde A$, $\tilde B$, $\tilde C$ in the hyperbolic plane there is a harmonic map bringing the three points $A$, $B$, $C$ of the boundary of the gasket to $\tilde A$, $\tilde B$, $\tilde C$ respectively. Moreover, if the points $\tilde A$, $\tilde B$, $\tilde C$ are sufficiently close in the hyperbolic distance, then the harmonic map is unique and depends differentiably on $\tilde A$, $\tilde B$, $\tilde C$. Lastly, we show that, if the harmonic map $ϕ$ is injective, then it brings geodesics of the gasket $G$ into geodesics of $ϕ(G)$.

In this paper, we develop a framework based on differential forms that enables us to deal with the Hall problem on domains in any dimension $n\ge2$. In the case of smooth bounded domains of $R^n$, we prove local existence of mild solutions in subcritical spaces.

In this study, the set of orthomorphisms of a finite group G of order 3M with a normal subgroup of order M has been partitioned into classes. Additionally, a number of relationships between the orders of these classes have been established. Ultimately, this approach has been used to theoretically determine all 3840 orthomorphisms of alternating groups on four symbols by just counting 30 orthomorphisms, a problem that had been unsolved since 1992.

We study spectral intertwining operators between spectral Eisenstein series $\operatorname{Eis}_{P^\vee}$, $\operatorname{Eis}_{Q^\vee}$ for two parabolic subgroups $P, Q$ of a $p$-adic reductive group $G$ with the same Levi subgroup $M$, inspired by the analogy with the classical intertwining operators between parabolic induced representations of $p$-adic reductive groups. In particular, we construct the normalized (canonical) intertwining operator that satisfies the transitivity and an unnormalized intertwining operator that is adjoint to a rational section of an analog of the Bruhat-Mackey's filtration. Moreover, the normalized and unnormalized intertwining operators differ by a ratio of L-functions, analogously to the Langlands conjecture about classical ones up to units. Finally we prove that the spectral intertwining operators correspond to classical ones up to units under any conjectural categorical local Langlands correspondence.

In this paper, we provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.

In this paper, we introduce the power mean transform $P_λ(T)$ of an operator $T$ on a Hilbert space, which is a convex combination of some classical operator transforms such as the mean transform $M(T)$, the Aluthge transform $Δ(T)$, and the Duggal transform $T^D$. In particular, when $T$ is invertible, this transform coincides with the induced Aluthge transform $Δ_{\mathsf{m}_{f}}(T)$ recently defined by Yamazaki \cite{yamazaki-laa-2021} with $f(x)=(λ+(1-λ)\sqrt{x})^2$ for $x\in(0,\infty)$ and $λ\in(0,1)$. We study basic properties of $P_λ(T)$ including its spectrum, norm and numerical radius. Moreover, we use the power mean transform to give new characterizations of normal, quasinormal and binormal operators. The questions of Golla et al. \cite{yamazaki-laa-2023} and some new results on the Duggal transform are also mentioned. We obtain a result close to the recent one of Osaka and Yamazaki \cite[Theorem 3.3]{yamazaki-tams-2025} on the iteration of the induced Aluthge transform for centered operators. Finally, we describe the form of bijective maps commuting with the power mean transform of the product of matrices.

Linear-multi-parametric optimization problems are a widely studied class of optimization problems. The objective function in such a problem is affine linear dependent on a parameter vector, and the goal is to compute a set of solutions that contains an optimal solution for every fixed parameter vector. However, this is known to be computationally challenging: The underlying non-parametric problem might be NP-hard, and, in addition, optimal solution sets might have exponential cardinality. Parametric approximation aims at providing polynomial-time algorithms that overcome these challenges. Instead of computing an optimal solution set, the goal is to compute an approximation set that contains only an approximate solution for every fixed parameter vector. Several new parametric approximation algorithms have been developed in recent literature. However, all of these share a common set of assumptions, which limits the class of parametric optimization problems that can be approximated. Namely, they do not allow negative parameter dependencies and have their parameter sets fixed to the positive orthant. We present a new adaptive approximation (and, also, exact) algorithm that can be applied to a wider class of linear-multi-parametric optimization problems. Our algorithm builds upon existing algorithms from both the fields of parametric and multi-objective optimization and generalizes these algorithms. In addition, we provide structural results for the transformation of parameter sets, and demonstrate that, for linear-multi-parametric maximization problems, the assumption of non-negative optimal objective values over the whole parameter set is not sufficient to ensure approximability.

We study an infinite-horizon optimal stopping problem for a normally reflected two-dimensional diffusion in the positive quadrant with nonsmooth max-type reward \(G(x_1,x_2)=x_1\vee αx_2\). The paper develops a conditional measure-theoretic framework for the associated reflected obstacle problem. The main innovation is to show that the stopping gain \(Γ=c+rG-\mathcal LG\) is a signed measure, not a function: the kink of \(G\) generates an explicit negative surface measure on \(Δ=\{x_1=αx_2\}\). We then prove that the correct potential representation uses the resolvent of the reflected diffusion killed on first entry into the stopping set, rather than the unrestricted reflected resolvent. Under explicit monotonicity, regularity, and measure-superharmonicity assumptions, we derive an epigraph representation, a continuation-side boundary-trace condition, and a candidate verification theorem. The framework clarifies hidden regularity and uniqueness assumptions in multidimensional nonsmooth optimal stopping.

In this article, we study the image of the Hitchin morphism for some classical groups over an algebraic surface. The Hitchin morphism is a map from the moduli stack of $G$-Higgs bundles $\mathscr{M}_{X,G}$ to the Hitchin base $\mathscr{A}_{X,G}$, where $X$ is a smooth projective variety. In general, this morphism is not surjective when the dimension of $X$ is greater than one. Chen and Ng{ô} showed that the Hitchin morphism factors through a closed subscheme $\mathscr{B}_{X,G}$ of the Hitchin base, which is called the spectral base. They conjectured that the image of the Hitchin morphism is exactly the spectral base. When $X$ is a smooth projective surface, we prove that this conjecture holds for the special linear algebraic group of odd rank. We also confirm this conjecture for the classical groups ${\rm SL}_n$ and ${\rm Sp}_{2n}$ when $X$ is a product of smooth curves.

In this paper, we study the Schrödinger-type operator associated with geometric stable processes on $\mathbb{R}^{d}$, especially the differentiability of spectral function. Let $\mathcal{H}$ be the generator of the geometric stable process and $μ$ a smooth measure on $\mathbb{R}^{d}$. Then the spectral function $C(θ)$ is defined as $C(θ) = -\inf σ(-\mathcal{H} - θμ)$, where $σ(\mathcal{A})$ denotes the spectrum of $\mathcal{A}$ and $θ$ is a real parameter. Since the geometric stable process exhibits severe local singularities in its Lévy measure, its transition semigroup lacks ultracontractivity, which invalidates classical methods for proving the differentiability. To overcome this obstacle, we use the compact embedding of the extended Dirichlet space into $L^2(μ)$. As a primary application of this differentiability, we establish a large deviation principle for a positive continuous additive functional associated with the smooth measure $μ$.

We investigate data-driven finite-volume discretizations of the linear advection equation in one dimension. Neural networks for use as numerical advection schemes are constructed adhering to first principles of numerical analysis, allowing us to examine how normalization, training data, and architectural choices influence stability, accuracy, and shape preservation. (i) We show that reconstruction based solely on cell averages leads to a multi-valued learning problem, explaining limited generalization when training data includes widely different curvature regimes. (ii) Numerical stability and good generalization can be achieved by enforcing semilinearity (Lin and Rood 1998) through local stencil-scale normalization, which ensures invariance under affine transformations of the inputs. (iii) A new data-driven flux limiter is introduced that outperforms the classical 'OSTVD3' (Arora and Roe, 1997) scheme in shape preservation by introducing mild antidiffusion in near-linear regimes, while higher-order reconstruction in non-monotonic regions provides limited benefit. (iv) We show that training on polynomial profiles yields stable, high-order accurate discretizations, with the polynomial degree controlling the formal order of accuracy. Together, these results illustrate how the representational, architectural, and training choices govern the stability and generalization of data-driven finite-volume schemes for linear advection.

It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schrödinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+αLu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $α$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-Pólya idea to the spectral problem at hand.

The purpose of this note is to study spectral properties of the horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution. We show that the horizontal Laplacian is unitarily equivalent to a twisted Laplacian acting on the space of sections of a certain infinite-rank flat vector bundle over the base manifold of the Riemannian submersion. We give an application of this interpretation to the asymptotic behavior of the scaled first nonzero eigenvalue of the canonical variations introduced by Berard-Bergery and Bourguignon. Our approach enables us to compare the horizontal Laplacian with the usual Laplacian on a Riemannian covering over the base manifold, and, when the holonomy group is infinite and amenable, we prove a coincidence of the essential spectrum, which strengthen, in our special setup, a result due to Kordyukov in the context of geometric analysis on foliated manifolds.

This work introduces and investigates the function \( \mathcal{V}(G) = \frac{\text{Simp}(G)}{|L(G)|} \), where \( \text{Simp}(G) \) denotes the number of simple subgroups and \( |L(G)| \) the total number of subgroups of a finite group \( G \). The function \( \mathcal{V}(G) \), defined on the interval \( [0,1] \), represents the proportion of simple subgroups relative to the total number of subgroups. It serves as a tool for analyzing structural patterns in finite groups, particularly in p-groups and other families.

We develop a new combinatorial large sieve method for sets with bounded algebraic multiplicities. The method exploits algebraic splitting modulo many small primes: local congruence branching produces many modular collisions, while global bounded-multiplicity hypotheses force these collisions to be rare. As a first application, we prove that every Sidon subset $A\subset\{1^2,\ldots,N^2\}$ satisfies \[ |A| \le N\exp\left( -c\frac{\log N}{\log\log N} \right) \] for some absolute constant $c>0$. This gives the first super-polylogarithmic saving for a classical problem of Alon and Erdős. As a second application, we establish new upper bounds for two grid-distance problems. We show that the largest subset of $[N]^2$ with no repeated distance has size at most $N\exp\left(-c\log N/\log\log N\right)$, giving the first progress in over thirty years on a problem of Erdős and Guy. The same method also gives a super-polylogarithmic saving for subsets of $[N]^2$ with no isosceles triangles, a problem recently popularized by Ellenberg and by the PatternBoost work of Charton, Ellenberg, Wagner, and Williamson. We then develop an entropic version of the method. This gives bounds for $B_2[g]$-sets in the squares and for analogous bounded-multiplicity problems associated with norm forms over arbitrary number fields. Moreover, we prove the first nontrivial bounds for $B_3[g]$-sets in the cubes and $B_4[g]$-sets in the fourth powers.

We study the Dirichlet problem for the $p$-Poisson equation in the metric measure space setting equipped with a doubling measure and supporting a $(p,p)$-Poincaré inequality. We prove the existence of the solutions by using a variational approach. We prove the stability and uniqueness of the solutions, when the space is also geodesic.

Tanaka's formula is a classical identity for Brownian motion, and Tsukada (2018) extended it to Lévy processes not necessarily symmetric. From a potential-theoretic point of view, this formula shows that the invariant function for the process killed upon hitting a singleton can be decomposed into the sum of a martingale part and a local time. In this paper, we generalize this singleton setting and derive a Tanaka-type formula for a compact set $B$. To this end, we introduce the equilibrium measure, defined as the rescaled limit of the $q$-capacity measures, and show that the invariant function for the process killed upon hitting $B$ can be represented as the integral, with respect to the equilibrium measure, of the invariant functions associated with processes killed upon hitting singletons, up to an additive constant called the Robin constant. Moreover, when $B$ is an interval, we obtain explicit representations of the equilibrium measure, the Robin constant, and the martingale part for recurrent stable processes as well as for recurrent spectrally negative Lévy processes. Finally, we discuss how an analogous Tanaka-type formula can also be established for transient Lévy processes.

The rapid growth of AI workloads is substantially increasing data center electricity demand and carbon emissions, motivating the development of carbon-aware scheduling methods. However, effective scheduling is challenging because renewable generation and AI workloads are subject to forecast errors, while training and inference workloads exhibit heterogeneity in computational characteristics. This paper proposes a contextual robust optimization framework for AI data center operation. The proposed model explicitly captures the heterogeneous computational characteristics of AI training and inference workloads. To deal with renewable generation and workload forecast errors, we develop loss-based uncertainty learning models that directly map contextual features to covariate-dependent uncertainty sets. The resulting contextual joint chance-constrained scheduling problem is reformulated into a tractable robust optimization problem, and a calibration algorithm is developed to provide finite-sample probabilistic feasibility guarantees for multiple joint chance constraints. Numerical experiments based on real-world AI workload traces and renewable generation data show that the proposed method reduces operating costs by an average of 5.57% compared to benchmark methods while maintaining reliable feasibility and strong computational scalability.

Superhedging supermartingales, introduced by the authors in previous work, are non-probabilistic processes defined via subadditive outer integrals that carry a purely financial interpretation in terms of superhedging cost. Building on the Leinert-König theory of non-lattice integration, the present paper establishes several results that are classical in probability theory but whose non-probabilistic proofs require fundamentally new arguments: (i) a tower inequality for the conditional outer integral \overlineσ_j applied at stopping times, reducing to equality when the integrand is conditionally integrable; (ii) three versions of Doob's optional stopping theorem, organised by the class of supermartingale and the range of the stopping times; and (iii) Dubins' upcrossing inequality in both finite- and infinite-time horizons. A key structural result, property (K)-a.e., identifies conditions under which the two superhedging operators \overlineσ_j and \overline{I}_j coincide on non-negative functions, extending the scope of all preceding results to the positive operator \overline{I}_j. None of the proofs invoke classical measure-theoretic tools; in particular, (classical) integrability and measurability are not assumed. The analogues of classical stochastic results acquire a purely financial interpretation and, in this way, gain depth and generality by providing a context that is independent of any a priori probabilistic structure.

For every integer $g\ge 2$, we determine exactly which integers occur in the greedy 3-sumfree sequence that starts with $1$, $g$, and $g+1$. This gives a direct proof of a conjecture of Bosma, Bruin, Fokkink, Grube, Reuijl, and Tromp. We also obtain an explicit eventual periodic description, including both the preperiod and the repeating block.

The study of principal symmetric ideals (PSIs) in ambient polynomial rings was complicated by the combinatorial instability of minimal generators for ideal powers. We resolve this instability in the two variable case by translating the problem into the arithmetic geometry of symmetric affine plane curves. By working topdown within the Dedekind domain of a symmetric coordinate ring, we establish a precise geometric dictionary for PSIs. We prove that the prime factorization of a PSI is strictly determined by the $S_2$-orbits of its symmetric intersection locus, and that ramification corresponds exactly to tangential intersections, which are detected globally by a novel Symmetric Discriminant ideal. Crucially, we demonstrate that the ideal class of any PSI is a $2$-torsion element in the Ideal Class Group. This establishes that the powers of a PSI exhibit strict periodicity, alternating between being principal and requiring exactly two generators. Finally, we localize this arithmetic obstruction to the axis of symmetry, culminating in a Parity Criterion that determines principality based on intersection multiplicities along the diagonal.

In recent work of Dombos, the set of integer partitions of $n$ wherein the parts are either divisible by 4 or congruent to $\pm 1 \pmod{6}$ arose in a natural way. In this work, we will denote the function which counts the number of such partitions of $n$ by $dp(n)$. Using elementary generating function manipulations and classical $q$--series results, we prove several congruences satisfied by $dp(n)$. As an example, we prove that, for all $α\geq 1$ and all $n \geq 0$, \begin{equation*} dp \left( 3^{2α+ 1}n + \frac{7 \cdot 9^α+ 1}{4} \right) \equiv 0 \pmod{3}. \end{equation*}

Let $L =-Δ+V$ be a Schrödinger operator on $\mathbb{R}^n$, $n \geq 3$, with the potential $V$ being nonnegative and belonging to the reverse Hölder class $RH_q$ for some $q >n/2$. For $0< α<2$, the Lipschitz space $Λ_L^α(\mathbb{R}^n)$ adapted to $L$ is defined as the space of all measurable functions $f$ on $\mathbb{R}^n$ such that \[ \|f\|_{Λ_L^α}:= \|ρ(\cdot)^{-α}f(\cdot)\|_{L^\infty}+ \sup_{z \in \mathbb{R}^n \backslash \{0\}} \frac{\|f(\cdot + z) + f(\cdot -z) -2 f(\cdot)\|_{L^\infty}}{|z|^α} <\infty, \] where $ρ$ is the critical radius function related to $L$. In this paper, we provide characterizations of $Λ^α_L(\mathbb{R}^n)$ in terms of Littlewood-Paley-type decompositions and Carleson measures, for $0< α< 2 -(n /q)$.

We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of $\mathbb{P}^n$. If $X$ is a complete intersection of dimension $\geq 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of L{ó}pez and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.