Papers

In this paper we construct the first example of a non-locally trivial $\mathrm{W}^*$-bundle whose fibres are all isomorphic to some fixed $\mathrm{II}_1$ factor. This is achieved by introducing a notion of uniformly having spectral gap for $\mathrm{W}^*$-bundles. For bundles with fixed factorial fibres, the negation of having this uniform spectral gap property provides an obstruction for being locally trivial. This results in seemingly elementary examples of $\mathrm{W}^*$-bundles whose fibres are all isomorphic to some fixed factor but that are not locally trivial, even over spaces with covering dimension equal to zero.

Let $S$ and $T$ be disjoint sets with $|S|=i$ and $|T|=r-1$ for $2\le i\le r-1$, and let $B_i^{(r)}$ be the $r$-graph on $S\cup T$ whose edges are the $r$-subsets containing $S$ or $T$. We study the deficit $q_{r,i}:=1-\pi(B_i^{(r)})$ in its Turán density. Balogh, Bohman, Bollobás, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\to\infty$, for every fixed integer $a\ge1$, $q_{r,a+1}=\Theta_a(r^{-a})$, and for every fixed integer $b\ge2$, $q_{r,r-b}=\Theta_b(r^{-b}\log r)$.

This paper develops a fully discrete cumulative distribution transform (CDT) for atomic probability measures on the real line. The transform is defined through monotone quantile maps and admits explicit linear-time algorithms for both forward transformation and inverse reconstruction based solely on cumulative mass matching. Unlike the classical continuous setting, deterministic transport between atomic measures cannot generally split masses, so exact reconstruction may fail at finite resolution. We establish a precise cumulative-mass compatibility criterion for exact finite-resolution recovery and prove weak convergence of reconstructed measures under reference refinement. Several structural properties of the discrete CDT are derived, including translation, composition, and scaling laws, and the framework is extended to a discrete signed cumulative distribution transform with thresholded stabilization near zero crossings. By avoiding continuous interpolation, the proposed framework provides a simple fixed-reference transport representation for discrete data. Numerical examples illustrate translation linearization, compatibility-controlled reconstruction, refinement consistency, and stabilization of the signed transform.

For a given Coxeter group, we study distributive lattices called middle orders refining the weak order and refined by the Bruhat order. In type $A$, we construct such lattices indexed by binary trees using a direct bijection between permutations and lower sets of a certain partition of the root poset into rectangles. When the binary tree is a left-comb tree, we recover the middle order defined by Bouvel, Ferrari, and Tenner (2025). We study combinatorial properties of these lattices, and show they are the only distributive lattices between the weak and Bruhat orders in type $A$. For general Coxeter groups, we study middle orders on parabolic quotients and use these to generalize our construction in type $A$ to other Weyl groups, obtaining so-called “minuscule middle orders”. We show that they are a subset of sorting orders defined by Armstrong (2009), and we give conjectural descriptions of all middle orders that are not minuscule.

We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak–Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of our ZOMA framework is a multi-level unification, along the following directions: (i) estimator - our framework adopts a hybrid zeroth-order estimator, which accommodates, among others, both coordinate-wise and randomized uniform smoothing estimators; (ii) bias correction - our framework subsumes a wide range of bias-correction strategies, including gradient tracking (GT), exact diffusion (ED), and EXTRA and (iii) acceleration - our framework facilitates a broad class of acceleration techniques, including zeroth-order versions of STORM, PAGE, and L2S. The general nature of ZOMA leads to many novel decentralized zeroth-order minimax methods and allows us to establish unified convergence guarantees, matching the performance of state-of-the-art centralized zeroth-order minimax methods, while providing benefits, such as linear speed-up in the number of users. The unified framework also provides a systematic way to assess algorithmic suitability by specializing the convergence rates to specific problem structures and method designs. We validate the performance of the proposed algorithms via numerical simulations.

We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ with real-valued potentials in $\ell^\infty(\mathbb{Z})$ and show that the generic spectral type is pure point. More specifically, we show that for a generic bounded potential, the essential spectrum of the associated Schrödinger operator is a Cantor set and has zero weight with respect to all spectral measures.

Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely eliminating inner iterative loops. In both regimes, per-iteration complexity scales linearly with dataset size, and we provide a rank-sufficiency certificate for global optimality verification. Extensive experiments across a range of problem sizes demonstrate that our regularization-free first- and second-order solvers achieve faster convergence and superior performance over existing state-of-the-art low-rank OT solvers.

Let $X \subset \P^5_{\C}$ be a smooth cubic fourfold.The Kuznetsov component $\sA_X$ is contained in the derived category $D^b(X)$ and the transcendental motive $t(X)$ is contained in the category of Chow motives $\sM_{rat}(\C))$. If $X$ and $Y$ are Fourier -Mukai partners and hence the categories $\sA_X$ and $\sA_Y$ are equivalent, then their transcendental motives $t(X)$ and $t(Y)$ are isomorphic. The aim of this note is to consider families of special cubic fourfolds $X$ with their FM-partners $Y$ and to give an explicit description of the isomorphism between the transcendental motives, in the case $X$ and $Y$ are rational and when they are conjecturally irrational. We also prove that ,for special cubic fourfolds $X $ in countably many Hassett divisors, with a symplectic automorphism of order 3, there exists another special cubic fourfold $Y$, an equivalence of categories $\sA^G_X \simeq \sA_{Y}$, where $\sA^G_X$ is the equivariant Kuznetsov component, and an isomorphism $t(X) \simeq t(Y)$.

Lyapunov feedback laws can be constructed for performing orbital transfer maneuvers based on Gauss Variational Equations, or GVEs, while satisfying specified state and control constraints. These state and control constraints are enforced using barrier functions and saturation, respectively, while the reference governor, employed as a convergence governor, is used to avoid getting stuck at spurious equilibria that may be created by barrier functions. In this article, these Lyapunov feedback laws are extended to rendezvous maneuvers where not only five orbital elements are matched to prescribed target values, but also the true anomaly matches its time dependent target value. The modification involves altering the commanded semi major axis with an outer loop feedback law, also designed using Lyapunov techniques. We illustrate the resulting safe closed loop rendezvous maneuvers in simulations for conventional thrust based propulsion and Lorentz force-based propulsion. In the latter case, only the current through the tether is controlled, subject to current limits, to accomplish prescribed orbital transfer and rendezvous maneuvers.

The integral representation of completely monotone functions of several real variables as Laplace or Stieltjes-Fantappié transforms of positive measures opens a Hilbert space path toward their finite-point interpolation by simpler functions. We combine, within a non-commutative Radon transform framework, the matrix pencil realization of the positive semi-definite Hankel kernel associated with the sampling of a completely monotone function with Weyl's operational calculus and Fantappiè's analytic calculus. The interpolation is achieved by finitely determined entire or rational functions, respectively, which are directionally completely monotone. In our relaxation scheme, the original positive measure is approximated by a sequence of specific Wigner distributions, which can also be regarded as analytic functionals. Throughout the interpolation process, tight bounds are enforced on the modulus or the real part of the holomorphic extension to the underlying tube domain.

We study Dirichlet Green's functions associated with second-order parabolic systems with rapidly oscillating periodic coefficients that are symmetric and independent of time. For bounded $C^{1,1}$ domains, we obtain a sharp zeroth-order convergence estimate from the oscillating Green's function to its homogenized counterpart, with the optimal rate $O(\varepsilon)$ and Gaussian off-diagonal decay. For bounded $C^{2,1}$ domains, we also prove a first-order expansion for the spatial gradient in terms of Dirichlet correctors, with an $O(\varepsilon)$ error up to a logarithmic factor. In this time-independent symmetric setting, these results improve the convergence rates established by Geng in [Calc. Var. Partial Differ. Equ., 62(6), 2023] for parabolic systems with time-dependent periodic coefficient matrices.

In this chapter we review some results describing the high-frequency eigenmodes of the Laplacian on compact manifolds, or Euclidean domains, for which the geodesic flow is chaotic. We focus on the macroscopic distribution of these eigenmodes, which is described by the concept of semiclassical measure. The main result on the question is the Quantum Ergodicity theorem, originally due to Schnirelman. We provide the detailed proof of this theorem, including the adjustments necessary to treat the case of manifolds with boundary. We also discuss the Quantum Unique Ergodicity conjecture, and some progress towards this conjecture for strongly chaotic (Anosov) systems. In particular, we describe the constraints on admissible semiclassical measures, in terms of their Kolmogorov-Sinai entropy, as well as more recent delocalization results.

In this paper, we propose two fully decoupled, linear and structure-preserving block-centered finite difference schemes for the classical Keller-Segel chemotaxis system on staggered non-uniform spatial grids. Both novel schemes are second-order accurate in space; one is first-order accurate in time, while the other achieves second-order temporal accuracy. Moreover, we show that the schemes preserve several inherent physical laws at the discrete level: (i) the positivity of both the cell density and the chemoattractant concentration; (ii) the conservation of total cell mass; and (iii) a discrete energy dissipation property for the first-order scheme. In particular, the temporally first-order scheme unconditionally preserves positivity, mass conservation, and energy dissipation, whereas the second-order scheme ensures positivity under a sufficient (but not necessary) time-step condition. The proposed methods yield more accurate and efficient simulations of chemotactic dynamics, especially in the presence of rapid blow-up phenomena, on specified non-uniform spatial grids. Numerical experiments are conducted to validate the theoretical findings and to illustrate the accuracy and reliability of the proposed schemes.

Freedman and Krushkal introduced a triple torsion linking form for rational homology $3$-spheres and used it to obstruct locally flat embeddings in $S^4$. For every odd prime $p$, we identify their triple torsion form, computed with parameter $t=p$ on rational homology $3$-spheres whose first homology has exponent $p$, with the mod-$p$ triple cup product under torsion-linking duality. For algebraically split $\pm p$-framed surgery links, this gives a signed formula in terms of Milnor's integral length-three invariants $\bar\mu_{ijk}$, with the framing-sign factor dictated by torsion-linking duality. We then use Borromean band-sums to realize arbitrary mod-$p$ triple cup tensors on rational homology $3$-spheres with $H_1\cong(\mathbb Z/p)^6$ and fixed hyperbolic ordinary torsion linking form. Finally, using the classical spinor/Klein model for the split six-dimensional quadratic space, we classify the tensors with no dual null Hantzsche pair. This produces, for every odd prime $p$, a rational homology $3$-sphere with hyperbolic ordinary torsion linking form but with no locally flat embedding in $S^4$, and indeed no locally flat embedding in any integer homology $4$-sphere.

For integers $k\ge 1$ and $n\ge k+1$, let $\operatorname{ex}^{\mathrm{HIST}}_k(n)$ denote the maximum number of edges in a $k$-connected graph of order $n$ which contains no homeomorphically irreducible spanning tree (or briefly HIST). We determine these extremal numbers for $k=1$ and $k=2$. More precisely, we prove that $\operatorname{ex}^{\mathrm{HIST}}_1(n)=\binom{n-2}{2}+2$ for $n\ge 9$, with $L_n$ as the unique extremal graph, and that $\operatorname{ex}^{\mathrm{HIST}}_2(n)=\binom{n-3}{2}+4$ for $n\ge 13$, with $B_n$ as the unique extremal graph. This provides a Turán-type extremal result for spanning trees with no vertices of degree two.

We solve exactly the relativistic finite-difference equation for the quantum three-dimensional ring-shaped Quesne oscillator potential. Our investigation is based on a finite-difference version of relativistic quantum mechanics. So-called relativistic configurational r-space is a key concept here. We show that the radial wavefunctions and angular wavefunctions are expressed through the continuous dual Hahn polynomials and Jacobi polynomials, respectively. A discrete energy spectrum has been found. The radial wave functions and energy spectrum have the correct nonrelativistic limit. We also build a dynamical symmetry group SU (1, 1) for the radial part of the equation of motion, which allows us to find the energy spectrum purely algebraically.

We describe the anlogue of the Einsidler, Kaydev, Polo and Sarig on Intrinsic Ergodicity in the simple setting of expanding maps and, in particular, $\beta$-transformations.

Let $\mathbb F$ be either $\mathbb R$ or $\mathbb C$, and let $L$ be a nonempty extremally disconnected compact Hausdorff space. We show that $(X,C(L,\mathbb F))$ has the Bishop–Phelps–Bollobás property with $\eta(\varepsilon)=\varepsilon^2/2$ for every Banach space $X$ over $\mathbb F$. For each fixed $L$ and either scalar field, this bound is optimal uniformly over $X$.

We prove Sobolev-type inequality and local volume noncollapsing with optimal exponents for compact Kähler manifolds of uniformly bounded $q$-Nash entropy.

We characterize complex-elliptic operators $\mathbb A(D)$ through a hierarchy of overdeterminacy ($\ell$-vanishing) quantifying the structural twisting of their symbols. This framework yields the optimal dimensional estimate for $BV^{\mathbb A}$-functions: a measure ${\mathbb A} u$ cannot concentrate on sets of dimension below $n-1$. Consequently, the jump part of ${\mathbb A} u$ is characterized as an $(n-1)$-dimensional surface measure with density given by the symbol and the two-sided traces. Building on this dimensional bound, we prove that measures satisfying $\frac{{\mathbb A} u}{|{\mathbb A} u|} \in \mathrm{span}\{P_0\}$ precisely decompose into finite sums of one-dimensional $BV$ profiles. Ultimately, these results reveal that complex-ellipticity strictly enforces a plane-wave structure on tangent measures.

In this note we propose an effective method to estimate the dimension of the Feigenbaum attractor for the period doubling phenomenon. In particular, we will describe a way to convert the highly accurate estimates for $g$ into better estimates on $\dim(X)$.

Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous a posteriori upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two-sided error enclosures. In this paper, we derive computable a posteriori lower bounds for PINN errors in ordinary differential equations on suitable certified state-space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural-network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time-invariant and time-varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed-residual finite-probe certificate based on coordinate unit vectors. We also formulate a certificate-informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post-training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.

The main purpose of this note is to give a proof of the fact that the Toda brackets \ $\langle\bar{\nu},\sigma,\bar{\nu}\rangle$ and $\langle\nu,\eta, \bar{\sigma}\rangle$ are not trivial. This is an affirmative answer of M.~Mahowald's Conjecture (J. Mukai, Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs 13(2008), 355–383). The second purpose is to determine the relations including $\bar{\nu}_6\omega_{14}$ in $\pi^6_{30}$ and $\bar{\nu}_7\omega_{15}$ in $\pi^7_{31}$. To this end, we provide relations between the Toda bracket and the $J$-homomorphism, and between the Toda bracket and the generalized $P$-homomorphism.

We prove two conjectural $q$-congruences proposed by Guo. The first is Conjecture 7.2 in Guo's work on $q$-analogues of two “divergent” Ramanujan-type supercongruences; it asserts a square-cyclotomic congruence for a truncated $q$-analogue of a Ramanujan-type sum when $n\equiv1\pmod4$. The second is Conjecture 4.1 in Guo's extension of Van Hamme's $(A.2)$ supercongruence; it gives divisibility modulo $[n]$ for a family of truncated basic hypergeometric sums with a parameter $s$. The proof of the first result relies on a known Watson-transformation congruence obtained by Guo. The proof of the second result is based on period decomposition at primitive roots of unity and a reflection cancellation inside residue blocks.

We study the topology of isometric classes of tensorial $H$-structures for closed and connected subgroups $H\leqslant\mathrm{SO}(n)$. The map assigning an $H$-structure its induced metric admits a global trivialization; consequently, the space of $H$-structures is homotopy equivalent to any fixed isometric class. On flat tori, these classes are mapping spaces into $\mathrm{SO}(n)/H$. We give examples with infinitely many connected components for $\mathrm{G}_2$, $\mathrm{Spin}(7)$, and almost Hermitian structures, determine the number of components for $\mathrm{SU}(m)$-structures on flat tori, and study the corresponding moduli spaces. For closed manifolds of dimension $n>2$, the intrinsic torsion energy on the full space of $H$-structures has infimum zero on every nonempty path component, and its critical points are precisely the torsion-free structures. In fixed isometric classes, we contrast the concentration phenomena underlying finite-time singularities with cohomological classes admitting positive energy lower bounds and harmonic representatives. Finally, we establish a lifting principle for metric-dependent flows, derive an evolution identity for isometric flows, and extend harmonic-flow estimates to general $H$-structures.

A fundamental concept in linear algebra and its applications to network analysis is the Perron–Frobenius (PF) theorem, which underpins eigenvector-based centrality measures such as eigenvector centrality, PageRank, and hubs and authorities. By invoking the PF theorem, we know for strongly connected networks with positive edge weights that the eigenvector corresponding to the largest eigenvalue of the weight matrix yields a well-defined centrality measure (namely, eigenvector centrality). Traditional formulations of the PF theorem and associated centrality measures assume that networks have real-valued weights. However, many networks in areas such as quantum information, quantum chemistry, electrodynamics, and machine learning have complex-valued edge weights. In this paper, we study generalizations of the PF theorem to complex-valued matrices, establish connections between these generalizations, and propose generalized eigenvector-based centrality measures to analyzing node importances in networks with complex edge weights. We also prove results about the existence of complex-weighted networks that satisfy generalized PF properties and calculate associated centrality measures for several examples, which we draw from application areas such as electron transport, circuit analysis, mathematical chemistry, and communication networks.

We develop a framework for comparing probability measures on finite posets via hockey-stick domination, an order relation defined through interval-counting test functions. The theory introduces poset integrals, derivatives, power functions and the associated moment functionals, all of which are invariant under poset isomorphisms. We prove that hockey-stick domination admits an exact quantitative characterization: whenever $\mu$ is dominated by $\nu$ in the hockey-stick order, the corresponding Zolotarev-type distance is equal to one half of the second-order poset moment of $\nu-\mu$. We further develop a constructive theory for generating such domination relations. In particular, we show that hockey-stick domination is preserved under direct products, disjoint unions, ordinal sums, and suitable ideal restrictions, yielding natural families of examples on chains, Boolean posets, rectangular lattices, rooted trees, and Young diagrams.

Let ${\mathcal C} ^{\rm rt}$ be the class of rooted trees and $\langle {\mathcal C} ^{\rm rt}\rangle _{\dot{\cup }\Pi}$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\mathbb X}= \dot{\bigcup} _{i<n}\prod _{j<m_i}{\mathbb X}_i^j$, where ${\mathbb X}_i^j$ are rooted trees. Defining ${\mathcal T}=\mathop{\rm Th} ({\mathbb X})$, ${\mathcal T} _i ^j=\mathop{\rm Th}({\mathbb X}_i^j)$, for $i<n$ and $j<m_i$, and $\kappa = \prod _{i<n}\prod _{j<m_i}I({\mathcal T} _i^j)$, we have (a) Vaught's conjecture is true for ${\mathcal T}$: $I({\mathcal T})=\kappa $, if $\kappa\in \{ 1,\omega ,{\mathfrak{c}}\}$, and, otherwise, $I({\mathcal T}) \in [3,\omega)$; (b) ${\mathbb Y} \equiv {\mathbb X}$ iff $\;{\mathbb Y} \cong \dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb Y} _i^j$, where ${\mathbb Y}_i^j\equiv {\mathbb X}_i^j$, for $i<n$ and $j<m_i$; (c) ${\mathbb E}\preccurlyeq {\mathbb X}$ iff $\;{\mathbb E} =\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb E}_i^j$, where ${\mathbb E}_i^j\preccurlyeq {\mathbb X}_i^j$, for $i<n$ and $j<m_i$; (d) ${\mathcal T}$ is atomic iff $\;{\mathcal T} _i^j$, for $i<n$ and $j<m_i$, are atomic; then $\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb A}_i^j$ is a countable atomic model of ${\mathcal T}$, where ${\mathbb A}_i^j$ is a countable atomic model of ${\mathcal T} _i^j$, for $i<n$ and $j<m_i$; (e) ${\mathcal T}$ is small iff $\;{\mathcal T} _i^j$, for $i<n$ and $j<m_i$, are small; then $\dot{\bigcup}_{i<n}\prod _{j<m_i}{\mathbb S}_i^j$ is a countably saturated model of ${\mathcal T}$, where ${\mathbb S}_i^j$ is a countably saturated model of ${\mathcal T}_i^j$, for $i<n$ and $j<m_i$.

This manuscript investigates the spectral geometry of Riemannian submersions whose fibers have a basic mean curvature. By restricting the Laplace–Beltrami operator to the space of basic functions, we reduce the spectral problem on $M$ to the spectral problem for a weighted Laplacian on the base manifold, where the weight is determined by the fiber-volume function $S$. We derive a summation formula for the reciprocal of the basic Dirichlet eigenvalues (Basel-type series). Furthermore, using the framework of Supersymmetric Quantum Mechanics (SUSYQM), we establish a supersym\-me\-tric duality relating the basic Dirichlet and Neumann spectra under the trans\-for\-ma\-tion $S \mapsto 1/S$.

A model for predicting the residual stress gradient in a thin film segment is developed on the basis of the theory of dislocation pile-ups. The initial shear stress within the film is relaxed via the formation of a pile-up of screw dislocations against the impenetrable film-substrate interface. Plastic strain is related to the dislocation density, leading to a fundamental equation, which links the residual stress to this density. The distribution of dislocations within the pile-up for an arbitrary, non-uniform residual stress profile is derived analytically by applying the force balance condition. This results in a singular integro-differential equation for the residual stress profile. The equation is solved numerically by a collocation method for various initial stress distributions: constant, linear, parabolic, and exponential functions. The solutions demonstrate that the established residual stress profile strongly depends on the film segment's thickness-to-width ratio and the initial stress distribution. As this ratio increases, stress relaxation becomes more effective away from the film-substrate interface. In all cases, equilibrium requires a pile-up containing dislocations with both positive and negative Burgers vectors. The total number of dislocations and their density distribution vary significantly with the initial stress profile. This model provides a critical step towards more complex models of residual stress formation in constrained material systems, specifically, thin films.