Papers

We prove a quantitative propagation of chaos estimate and a central limit theorem for the particle system associated with a class of degenerate kinetic McKean–Vlasov SDEs with external drifts and singular interaction kernels in Kato's class. In particular, the interaction kernel can be in the mixed $L^q_tL^{p_v}_vL^{p_x}_x$-space, where $\frac2q+\frac{3d}{p_x}+\frac d{p_v}<1$. For the associated $N$-particle system, we obtain a path-space relative entropy bound of order $k/N$ for the first $k$ particles, assuming only entropic chaoticity of the initial data. The key ingredients are kinetic Krylov–Khasminskii estimates and a conditional Hilbert-space subgaussian estimate for empirical interaction fields. For the CLT, we also prove a Berry–Esseen-type bound for finite-dimensional projections.

The logarithmic double ramification cycle is the virtual fundamental class of the locus where a line bundle on a family of curves is fiberwise trivial. We construct a K-theoretic logarithmic double ramification class and prove a product formula and a \(\mathrm{GL}_r(\mathbb Z)\)-invariance property. We also give an explicit formula for this class in terms of a Grothendieck polynomial via a novel $K$-theoretic Thom–Porteous formula for vector bundles on algebraic stacks.

Column generation (CG) is central to many large-scale optimization algorithms, including branch-price-and-cut methods for vehicle routing problems, but unstable dual solutions can substantially slow its convergence. Existing deep dual-optimal inequalities can reduce this instability by restricting the dual space. Their construction, however, typically relies on problem-specific exchange arguments that are difficult to establish for routing problems with capacity limits, time windows, and other resource constraints. We introduce learned pairwise deep dual-optimal inequalities (L-PDDOIs), a learning framework that predicts pairwise orderings between dual variables and incorporates their primal counterparts directly into the master problem. To construct training labels, the framework samples optimal dual solutions and selects pairwise order relations that hold simultaneously on a sufficiently large common subset of the samples. A classifier then assigns a score to each candidate relation. Because conflicts and redundancies among the predicted relations can impair performance, graph-based postprocessing filters and compresses the candidate set before deployment. We further introduce a recovery procedure that selectively relaxes learned inequalities and provides a certificate when the baseline CG bound has been restored. On the main test sets for the capacitated vehicle routing problem and the vehicle routing problem with time windows, direct deployment of L-PDDOIs reduces the geometric mean root CG time by 89.7% and 93.9%, respectively, while incurring mean bound losses of only 1.3% and 0.5%. The recovery procedure retains corresponding time reductions of 54.8% and 83.1%, respectively, while guaranteeing no loss in the CG bound.

We give a characterization of linear isometries with finite corank between two arbitrary unital C$^*$-algebras in terms of Jordan $^*$-homomorphisms and “finite-dimensional remainders”. Also, the corresponding results are given for linear isometries and linear order embeddings between self-adjoint parts of C$^*$-algebras. In the finite-dimensional case, we give new examples of linear isometries between matrix algebras.

A discrete formulation of the multiphase moist compressible Euler equations is presented for which the energy and tracer variance conserving dynamics is coupled to the non-equilibrium moist phase exchanges in a thermodynamically consistent manner. A neural network is then trained to learn the mass exchanges between vapour, liquid and ice phases in atmospheric convection. The network is trained on convection resolving output from a regional configuration of the LFRic model with a multi-moment microphysics parameterisation (CASIM). The loss function for learning these phase exchanges is formulated under the assumptions of thermal and mechanical equilibrium (same temperature and pressure for all phases), and chemical dis-equilibrium (different Gibbs free energies for all phases). The network outputs determine the exchanges of vapour, liquid and ice so as to conserve mass, and the resulting change in entropy is determined from the network outputs so as to conserve energy. The neural network is implemented in a thermodynamically consistent manner within the discontinuous Galerkin moist compressible Euler model in order to simulate the formation of three-phase clouds for convection at sub-km resolution. The results are compared to those from a physics based representation of three-phase moist processes at thermodynamic equilibrium, as well as to an idealised configuration of LFRic model.

This is the third part in a series of papers developing a new framework to study the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\geq 2$. Parts I and II introduced an intrinsic frequency function for $T$ – planar frequency – and used its monotonicity properties, among other things, to establish that ${\mathcal H}^{n-2}$-a.e. branch point is a rapid-decay branch point where the planar frequency is at least $1 + \alpha$. This paper analyses branch points of planar frequency $\neq 2$. It establishes: (1) higher order asymptotics: at ${\mathcal H}^{n-2}$-a.e. such point, the current admits an expansion of finite order $>1$, with precise decay estimates for the remainder term; (2) branch set decomposition: the set of such branch points locally decomposes into finitely many pairwise disjoint, locally $n-2$ rectifiable sets (of locally finite measure); (3) topological control: near any branch point satisfying a specific planar-frequency criterion, the support of $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, \mu}$ parametrization. (Classical complex algebraic examples show that when this frequency criterion fails, the current need not be locally homeomorphic to an $n$-disk). The work here (as well as in parts I & II) avoids the use of center manifolds – a technically demanding foundational component of the classical Almgren framework – and uses instead intrinsic geometric arguments based on the monotonicity formula for planar frequency. In part IV, a center manifold is utilised to analyse planar frequency 2 points, where the center manifold becomes necessary and geometrically canonical, satisfying additional simplifying properties. Reduced reliance on center manifolds in our framekwork is necessitated by the structural results it establishes for $T$.

This article investigates positive supersolutions of the fully nonlinear elliptic system \[ \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)+|\nabla u|^{q} \geq\lambda_{1}f_{1}(v)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0},\\ -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)+|\nabla v|^{q} \geq\lambda_{2}f_{2}(u)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0}, \end{cases} \] where $q>1,\lambda_{1},\lambda_{2}>0,$ and the nonlinearities exhibit power-type behaviour either near the origin or at infinity. Introducing the effective dimension $\widetilde n_{+}=\frac{\lambda}{\Lambda}(n-1)+1$ associated to extremal Pucci operator, we identify the critical exponent $q_{c}=\frac{\widetilde n_{+}}{\widetilde n_{+}-1},$ which governs the qualitative behaviour of positive supersolutions. Using this framework, we establish sharp Liouville-type nonexistence theorems in exterior domains and determine optimal nonexistence regions through the interaction between the gradient exponent $q,$ the effective dimension $\widetilde n_{+}$ and the nonlinear couplings. In the prototype case $f_{1}(t)=t^{p_{1}},$ $f_{2}(t)=t^{p_{2}}$ the obtained conditions are shown to be optimal. The analysis is carried out under the natural regularity assumption $u,v\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus B_{R_0}),$ for $p>n,$ which is the regularity available for fully nonlinear uniformly elliptic equations, rather than within a classical $C^{2}$ framework. Our results provide the fully nonlinear Pucci analogue of the Liouville theory for semilinear elliptic systems involving nonlinear gradient terms and reveal the fundamental role of the effective dimension in determining the critical nonexistence thresholds.

We investigate a radially symmetric initial-boundary value problem whose separation of variables leads to a regular self-adjoint Sturm-Liouville problem with explicitly determined coefficients and a positive weight function. The resulting Sturm-Liouville operator is defined by a special geometry-induced coefficient pair that gives rise to a nonstandard weighted spectral structure. The associated radial operator possesses a discrete real spectrum and a complete orthonormal system of eigenfunctions in the corresponding weighted Hilbert space, yielding an exact spectral representation of the evolution problem. An exact transcendental spectral equation governing the eigenvalues is derived. To analyze the high-frequency regime, the Liouville-Green transformation is applied directly to the radial Sturm-Liouville equation. This yields explicit asymptotic formulas for the eigenvalues and eigenfunctions together with corresponding Liouville-Green quasimodes. Their weighted orthogonality properties and asymptotic completeness are established within the spectral framework of the exact operator. The resulting asymptotic spectral data are used to construct approximate solutions of the original boundary-value problem and to obtain error estimates for the spectral reconstruction. Numerical computations of the exact Sturm-Liouville spectrum show excellent agreement with the Liouville-Green predictions, thereby validating the asymptotic quantization formula and confirming the accuracy of the proposed spectral approximation.

We study multidimensional semilinear stochastic evolution equations driven by multiplicative noise and subject to rough initial data. The drift and diffusion coefficients are assumed to take values in negative fractional order spaces and may exhibit temporal singularities at the initial time. Our main results establish well-posedness and pathwise spatio-temporal regularity of the solutions when these coefficients are globally Lipschitz, and we prove the existence, uniqueness, and pathwise regularity of maximal local solutions when the coefficients are locally Lipschitz. The local Lipschitz condition is formulated with respect to a specific time-weighted norm. This enables the application of our theoretical results to linear stochastic partial differential equations, as well as to a broad class of models with non-globally Lipschitz nonlinearities. These include the stochastic Burgers, Allen–Cahn, Fisher–KPP, and Burgers–Fisher equations, alongside the Ginzburg–Landau system and models featuring exponential growth, such as the stochastic solid fuel ignition model.

We prove that there is a density-one set of positive integers for which the products of all distinct consecutive blocks are distinct, answering a question of Erdős and Graham. The construction retains or rejects entire interiors of consecutive-prime gaps and is therefore stable under passage to longer prefixes. Uniform affine-curve estimates control the witnesses with no previous rejected gap, and a scale-contracting forest controls every remaining witness unconditionally.

We study the Shield number of the stable marriage problem, defined as the minimum, over all preference profiles of size n, of the maximum number of blocking pairs attainable by a complete matching. We establish the universal quadratic lower bound $\sigma(n) \ge \lceil n(n-2)/6 \rceil$ using a rearrangement inequality and show that this bound is the strongest consequence obtainable from uniform first-moment arguments. We further prove a sharp min-rearrangement inequality and construct an explicit cyclic Hollow-Shell instance attaining $\lfloor (n-1)^2/4 \rfloor$ blocking pairs. We also establish sufficient Hall-type conditions for biclique realizability, prove structural limitations of uniform averaging and two-parameter linear assignment objectives, and provide complete proofs for all unconditional results. Computer-assisted verification is used only for explicitly identified computational statements and is clearly distinguished from mathematical proofs.

We study congruences and binary reducibility in completely additive ternary $\Gamma$-semirings through their two-sorted atomic cores. For an odd $m\geq3$, an abelian group $G$ of exponent dividing $m-1$, and $u,v\in G$, we construct a two-branch symmetric $m$-ary band $F_m(G;u,v)$. We prove \[ \Con(F_m(G;u,v))\cong\operatorname{Sub}(G)\times B_2 \] and classify every carrier–index congruence pair: $(\theta,\phi)$ is typed exactly when $\theta$ is an ordinary congruence and $\phi$ refines $\theta$. We also give a quotient-by-quotient reducibility criterion. In the finite case, $F_m(G;u,v)$ is quotient-critically irreducible exactly when $G$ is a cyclic $p$-group and $u-v$ has order $p$. The specialization $G=\mathbb Z_2$ yields a four-point family $H_m$ extending the irreducible ternary example of Devillet and Mathonet. Its congruence lattice is $B_3$, its alternating core has exactly $36$ typed congruence pairs, and its automorphism group is $C_2$. Its seven proper quotients are reducible and form five isomorphism types; in arity five their exact reduction counts are determined. The powerset lift of $H_5$ is a $16$-element atom-total complete atomic Boolean ternary $\Gamma$-semiring with no atomic binary collapse, whereas each proper diagonal strong atom-saturated quotient has one. Its parity congruence is recovered on atoms by one explicitly specified depth-one polynomial inequation.

We introduce a uniform reduction for derivative sums of balanced gamma quotients. For exponent data $(a_i,e_i)$ satisfying $\sum_i e_i a_i=0$, the translation-dependent gamma prefactor is governed by the characteristic power sums $\chi_m=\sum_i e_i a_i^m$ through $\log C(u)=\sum_{m\ge2}(-1)^m\chi_m\zeta(m)u^m/m$. This separates the universal gamma contribution from a hypergeometric coefficient-extraction problem and organizes three exponential families. For $\chi_m=2-2^m$, diagonal and symmetric specializations of a four-parameter Wilf–Zeilberger identity prove Conjectures 4.2 and 4.3 of Zhi-Wei Sun. For $\chi_m=3-3^m$, exact span certificates in the coefficient spaces of Au's Example IV prove corrected forms of Conjectures 4.4 and 4.5. For $\chi_m=4^m-10\cdot2^m+16$, a half-integer specialization of Au's first $1/\pi^4$ construction, combined with the diagonal transformation in his Example VI, proves Conjecture 4.6 through weight eleven. The transformed sides reduce to ordinary multiple zeta values, and every computer-assisted acceptance test is exact: explicit rational WZ certificates and separately implemented MZV certificate checkers use no numerical recognition, PSLQ, or conjectural MZV dimensions. We also identify four errors in the printed statements of Conjectures 4.2–4.5.

Derivative-free optimization (DFO) addresses unconstrained problems $\min_{\x\in\RR^n} f(\x)$ where $f$ is accessed only through a zeroth-order oracle. Model-based trust-region methods construct underdetermined quadratic interpolation models from $\mathcal{O}(n)$ points and solve a KKT system to determine model parameters, costing $\mathcal{O}(m^3)$ operations and limiting parallel scalability. It is shown that the KKT matrix for the minimum Frobenius norm updating model depends entirely on inner products of shifted coordinates. Reflecting the interpolation set across a single coordinate axis preserves these inner products and changes only one row and column of the KKT matrix, inducing a rank-at-most-two perturbation whose inverse update via the Sherman-Morrison-Woodbury formula costs $\mathcal{O}(n^2)$ when $m=\mathcal{O}(n)$. The reflection is an isometry in centered Euclidean trust regions and preserves the poisedness constant of the interpolation set; together with standard fully linear model-management assumptions this supports first-order global convergence. The mechanism is embedded in a master-worker parallel algorithm with a Truncated Conjugate Gradient subproblem solver. Numerical results on 530 benchmark problems compare performance against an established DFO solver.

Let $F(n)$ be the largest size of a set $A\subseteq[1,n]$ such that $a\nmid bc$ whenever $a,b,c\in A$ and $a\notin\{b,c\}$, with $b$ and $c$ allowed to coincide. We prove \[ F(n)=\pi(n)+\left(\frac{27}{2}+o(1)\right)\frac{n^{2/3}}{(\log n)^2}. \] This determines the second-order constant conjectured by Erdős; the upper bound keeps the leading constants in his multiplicative basis, while the lower bound packs scale-separated prime triples by proper edge-colourings.

Let A = F_q[T] with q a power of an odd prime p, let [n] = T^(q^n) - T, and let rho be the Carlitz module. A monic prime P of A is a c-Wieferich prime (to base 1) if rho_P(1) = 1 mod P^2. Thakur suggested in 2015, on the basis of limited data and of proofs in degrees 2 and 3, that in odd characteristic every c-Wieferich prime has degree divisible by p; the question was restated as open in 2024, and Bamunoba and Bergstrom, after extensive computations, expressed the belief that the statement holds in odd characteristic. We show that it is false: an explicit irreducible c-Wieferich prime of degree 5 over F_{19^3} is exhibited, with 19 not dividing 5. We further give a closed form for the resulting common factor of [5] and M_5: it equals mu(T^q - T) for an explicit quintic mu with coefficients in the prime field F_19, squarefree of degree 5*19^3, and it divides gcd([5], M_5); we conjecture equality, and verify it for the part of low degree over the prime field. Degree 5 is the least possible degree of such a counterexample, and exhaustive computations show that no counterexample exists over the prime fields F_p in a substantial range of degrees and characteristics. The proof that degree 5 is minimal, and the method by which the example was found, appear in a companion paper.

We present a quantum-inspired tensor-network framework for solving advection-diffusion-reaction (ADR) partial differential equations. Discretized solution fields are encoded as matrix product states (MPS), while differential operators are represented as matrix product operators (MPOs). Time integration is performed entirely in tensor-network form using explicit Euler updates with controlled truncation. The method is evaluated on one- and two-dimensional ADR problems and compared with high-accuracy Runge-Kutta reference solutions. Numerical results show that the proposed representation remains compact, stable, and accurate across a range of dynamical regimes. The solver captures both local solution profiles and global observables while maintaining small bond dimensions throughout the simulation. These results highlight the potential of tensor networks as efficient structure-preserving tools for PDE simulation in multiple spatial dimensions.

The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.

We study the deterministic query complexity of minimizing a convex Lipschitz function over a $d$-dimensional Euclidean ball using only exact function values. At accuracy $\Theta(d^{-1/2})$, the previously applicable lower bound was $\Omega(d)$, inherited from the stronger full first-order oracle, while an upper bound from Protasov's value-only method requires $O(d^2\log^2 d)$ evaluations. By providing a lower bound of $\Omega(\,\frac{d^2}{\log(d+1)})$ on the oracle complexity in this setting, we thereby close this gap dating back to 1996, up to polylogarithmic factors. Furthermore, we are able to lift this result to the mixed-integer setting: Mixed-integer convex optimization with $d$ continuous and $n$ discrete variables using function values requires $\tilde{\Omega}(d^2\cdot 2^n)$ queries.

We introduce the Zariski frame of any presentably symmetric monoidal $\infty$-category. This allows us to unify several spectral theories arising in higher algebra. The Zariski frame is coherent whenever the category is compactly generated, and the associated spectral space recovers both the classical Zariski spectrum of a commutative ring and the Hochster dual of the Balmer spectrum of a commutative $2$-ring. Moreover, the smashing frame of any stable presentably symmetric monoidal $\infty$-category can be identified with the Zariski frame of its category of dualizable modules. This construction is based on the principle that ideals in a symmetric monoidal $\infty$-category should be understood as monomorphisms into the unit object. In suitable contexts, this notion recovers the kinds of ideals appearing in the preceding examples, including thick ideals and smashing ideals, and it also accommodates the smashing ideals of non-stable $\infty$-categories. We also study the problem of forming quotients by ideals, which is subtle in the setting of higher algebra. To address this, we introduce two properties of pointed $\infty$-categories, called $\Sigma$-triviality and $\Sigma$-exactness. These conditions ensure that quotienting by ideals behaves well. As an application, we construct quotients of $\mathbb{E}_\infty$-semirings.

Constant domain intuitionistic logic admits a complete semantics in presheaf toposes, by interpreting sorts as constant presheaves and predicates as arbitrary sub-presheaves. The goal of this note is to point out how this fits in topos theory, replacing constant presheaves with objects that are covert and Hausdorff when considered as discrete locales. We call these objects "CD" and we show that they form a Boolean pretopos in any topos.

The existence, several properties, and constructions of Generalized Weighing-Hadamard (GWH) matrices over finite fields are addressed in this work. We study the subset of invertible GWH matrices and show that it forms a group under matrix multiplication. Besides that, we introduce a strong notion of equivalence between such matrices, defined via orthogonal transformations, and further prove that the corresponding quotient group by the subgroup of orthogonal matrices is abelian. Finally, we discuss some applications of these matrices in coding theory

This work extends several fundamental results previously established for rings of characteristic $p$ to the broader class of rings that admit a contracting endomorphism. Specifically, let $\phi: R \to R$ be such an endomorphism, and denote by $^{\phi}R$ the ring $R$ endowed with the left $R$-module structure induced by $\phi$. We employ the functor $\text{Hom}_R(^{\phi}R,-)$ to derive new characterizations of regular rings. Furthermore, we examine the behavior of the functor $(-) \otimes_R {^{\phi}}R$ on local cohomology modules and injective modules.

The intrinsic singularity of the tempered fractional Laplacian (TFL) remains a major challenge in developing numerical methods that are simultaneously accurate, efficient, and easy to implement. We develop a weighted integral-regularized finite difference (WIRFD) method that regularizes the singular integrand via a multidimensional Taylor expansion incorporating a smooth window function. The resulting integral is decomposed into a regularized term, which is discretized by a punctured trapezoidal rule, and a directly evaluated correction term. For the multidimensional TFL operator, we derive an $O(h^{4-\alpha})$ truncation error bound in the $l^{\infty}$-norm for $\alpha\in(0,2)$ and $u\in C^s(\mathbb{R}^d)$ with $s\geq 8$ by introducing a smooth auxiliary function together with the aliasing formula. For the one-dimensional TFL equation, we establish stability in both the $l^2$- and $l^{\infty}$-norms and optimal $O(h^{4-\alpha})$ convergence for $\alpha\in[1,2)$ based on the strict diagonal dominance of the discrete matrix and a lower bound for its minimum eigenvalue. The Toeplitz structure of the discrete matrix enables FFT-based matrix-vector multiplication, and the resulting linear systems are solved efficiently by a preconditioned conjugate gradient (PCG) method. Numerical experiments corroborate the theoretical results, demonstrating the accuracy, efficiency, and robustness of the proposed method.

There exists an exceptional quotient of braid groups $\text{Br}_4 \twoheadrightarrow \text{Br}_3$ that is related to many interesting constructions in algebra, topology, and geometry. This quotient map also descends to a quotient of "truncated" braid groups $\text{Br}_4(d) \twoheadrightarrow \text{Br}_3(d)$, which have an added torsion relation on their half twist generators. In this article, we find a presentation for the kernel of this truncated quotient map that takes the form of what we deem a "primitive" Burnside group. We give a few finiteness results on these primitive Burnside groups. Our methods are purely group theoretic, but we comment on an interpretation involving Lefschetz fibrations at the end.

We establish a general criterion ensuring the absence of infinite connected components at criticality for a broad class of two-dimensional dependent percolation models. Our assumptions include monotonicity, stationarity and ergodicity, positive association, a quantitative decoupling condition, continuity in the percolation parameter, and trivial behavior at the extremes. As application, we show how our criterion recovers known results for classical dependent percolation models, such as Gaussian level-set percolation and Boolean percolation.

We view metrics through their isometric embeddigns $f_g:(M^n,g)\rightarrow (\mb{S}^{\tn},\tg)$ and their deformations. If $M$ carries a metric $g$ of constant scalar curvature $s_g$ and Ricci tensor $r_g \leq 0$, and if this $M$ does not carry scalar flat metrics other than Ricci flat ones, then $M$ is not a manifold of Kazdan-Warner (KW) type I, and if the space of Ricci flat metrics is not empty, $M$ is a manifold of KW type II, while otherwise, $M$ is of KW type III. If $M$ carries a metric $g'$ of nontrivial scalar curvature $s_{g'}\geq 0$, and an Einstein metric $g_{-}$ such that $r_{g_{-}}<0$, then $M$ must carry both, scalar flat non Ricci flat and Ricci flat metrics, and if orientable, it is spinnable. No such manifold exists if $n\leq 3$, and if $r_{g'}$ is assumed further to be positive, no such manifold exists if $n\leq 4$, and in these dimensions, $M^{n}$ can admit Einstein metrics of scalar curvature of at most one sign. If $M$ has a contractible universal cover and carries no Ricci flat metrics at all, $M$ is of KW type III. Based on these resulst, we find the KW type and sigma invariant of several manifolds $M=X/\Gamma_M$ with model geometry $({\rm Isom}(X,g),X)$ of Thurston. Notably, we show that an $M^n=\mb{H}^n/\Gamma_M$ of hyperbolic model $(\mb{H}^n,g_{\mb{H}^n})$ is of KW type III, that if $n\geq 3$ its $\Gamma_M$ invariant hyperbolic metric $g_M$ and class realize its sigma invariant, and that the space of hyperbolic metrics on $M$ is path connected and consists of isotopic deformations $f_{g_t}$ of $f_{g_0}:=f_{g_M}$ of equal volume metrics $g_t$ of constant sectional curvature $-1$, with $(M,g_t)$ isometric to $(M,g_M)$ for all $t$, while $3$d nil, solv, $\widetilde{\mb{P}\mb{S}\mb{L}}(2,\mb{R})$ and $\mb{R}\times \mb{H}^2$ manifolds are all of KW type III also, but have vanishing nonachievable sigma invariant.

We prove a novel characterization of the exponential distribution via a linear combination of order statistics. The theorem established herein generalizes the previous characterization theorem given in Milošević and Obradović (2016a). The proof relies on properties of Stirling numbers of the second kind, which are derived for this purpose and may be of independent interest.

We study the existence of equilibria for state constrained, convex generalized Nash equilibrium problems (GNEPs) coupled with hyperbolic partial differential equations (PDEs). Analogous problems have been addressed for state constrained GNEPs coupled with elliptic and parabolic PDEs, respectively, but the problematic regularity of the set-valued constraint maps has been a barrier for development of an existence theory in the hyperbolic case. This is mainly due to compactness issues with the strategy-to-state maps. Beyond existence, we also provide first-order optimality conditions for a class of fairly general linear hyperbolic PDEs and show its relevance for applications such as the wave equation, advertising dynamics, and the linearized isothermal Euler system on networks.

We develop a geometric framework in which the token dynamics of a transformer are modeled by a system of interacting particles on a Riemannian manifold $\mathcal M$, the attention mechanism being encoded by a time-independent two-body interaction law, that is, a section of the pullback bundle $\pi_2^{*}(T\mathcal M)$ over $\mathcal M\times\mathcal M$. Within this framework we isolate two features that a family of interaction laws must possess in order to model language: it must realize generic nonlocal and nonreciprocal forces, and it must parametrize vector fields on a high-dimensional manifold efficiently. We show that both features are achieved simultaneously in a transformer model. Our main theorem produces a finitely parametrized family of interaction laws that is universal: it realizes an arbitrary prescribed attention digraph. Moreover, we show that the cost of realizing a given attention digraph is governed by two combinatorial invariants of the digraph, namely its biclique cover number, which we identify with the least number of hubs in a hub extension, and its hub-chromatic index.