Papers

While traditional hub capacity planning models optimize effectively for quantitative inputs, they often fail to digest qualitative business context. We propose a novel framework where a large language model (LLM) agent iteratively proposes hub capacity decisions guided by natural-language business context descriptions. The key mechanism is a chain-of-thought reasoning protocol: the LLM constructs a structured decision table that maps each contextual item to specific capacity adjustments based on the implied direction and magnitude of changes. The new capacity decision is then validated through a feedback loop with an optimization model, which provides routing-based performance metrics to guide the agent's selection. On a real-world 13-hub freight network in the southeastern US, our framework achieves a 2.8% optimality gap relative to the hidden ground-truth, a significant improvement over the 11.0% gap produced by the traditional optimization model without textual business inputs. This demonstrates that LLMs can serve as a contextual bridge, integrating qualitative business insights into Operations Research workflows.

We prove that any closed orientable hypersurface in a contact manifold of dimension five or greater is isotopic to a robustly non-convex hypersurface via an arbitrarily $C^0$-small isotopy. This strengthens a recent result of the first author and yields a strong counterpart to the groundbreaking density theorem of Honda-Huang and Giroux. This is proven by combining a new convexity obstruction via heteroclinics and recent advances in robust heterodimensional dynamics due to Li-Turaev to produce a robust deconvexifying plug, which is a local and robust convexity obstruction.

We develop a morphological representation theory for operators acting in the frequency domain of $L^2(\mathbb{R}^n)$. Equipping the space with the Fourier inf-semilattice order $\leq_{\!\mathcal{F}}$ (spectral modulus inequality with phase equality), convolution becomes a morphological erosion and the adjoint dilation is the Wiener inverse filter. A key observation is that spatial translation-invariance is vacuous in the spectral order, replaced structurally by positive homogeneity of the operator on spectral moduli. Under this hypothesis we prove the Morphological Representation Theorem for the Fourier modulus lattice $\hat{L}$: any increasing, USC, positively homogeneous $\Psi:\hat{L}\to\hat{L}$ decomposes exactly as a supremum of max-times erosions $\hat{f}/\psi$, indexed by a minimal morphological basis. Three structural corollaries follow: the spectral modulus map $|\cdot|:L^2(\mathbb{R}^n)\to\hat{L}$ is an idempotent cross-lattice projection (the morphological analogue of ReLU) that makes depth non-trivial; wavelet scattering coefficients form the canonical dictionary for the basis via Littlewood–Paley density; and spatial pooling is a Fourier erosion, so that pool-then-unpool is the ideal band-pass opening, U-Net skip connections are Fourier top-hat transforms, and strided pooling preserves the morphological structure if and only if the Nyquist condition holds. The framework carries a natural $\mathbb{C}^*$-group morphology structure, positioning mathematical morphology as the constructive operator-theoretic complement of spectral bias, scattering, and tropical geometry theories.

Cooperative perception enables vehicles and infrastructure to exchange sensor data via Vehicle-to-Everything (V2X) communication, extending sensing coverage beyond occlusions and mitigating blind spots. While critical for autonomous driving and safety, practical deployments often rely on bandwidth-efficient late fusion. Recently, intermediate fusion has emerged as a promising approach for an optimal bandwidth-accuracy trade-off. However, in dense urban environments, cumulative bandwidth demands can overwhelm network capacity, potentially compromising safety-critical Cooperative Intelligent Transport Systems (C-ITS) functions. To alleviate these problems, this paper proposes Hierarchical Multi-Scale Semantic-Aware Cooperative Perception (HMS-SCP), a robust noise-resilient and bandwidth-efficient framework for task-oriented semantic communication in cooperative perception. HMS-SCP employs a spatial importance predictor to identify task-relevant grid elements at each scale, which are then directly mapped into complex-valued symbols for Joint Source-Channel Coding (JSCC). Unlike prior methods that rely on high-dimensional symbol projections for robustness, HMS-SCP exploits structural semantic redundancy across multiple scales to enhance resilience against channel noise, while maintaining an ultra-low symbol rate. This design significantly reduces bandwidth consumption and mitigates network congestion in high-density vehicular environments. Extensive evaluations on the simulated OPV2V and real-world DAIR-V2X datasets demonstrate that HMS-SCP effectively prevents performance collapse under severe Rayleigh fading and extreme compression ratio, maintaining high-confidence far-field detection with a real-time latency of below 16~ms, well within the safety-critical thresholds for dynamic V2X environments.

The Cartwright–McMullen theorem (1978) establishes the uniqueness of the Riemann–Liouville family of fractional integral operators on a compact interval $[0,1]$ under a natural set of axioms: inclusion of the classical integral, a semigroup property, positivity, and continuity. By unpacking the proof into its constituent steps and understanding its structure clearly, we provide improvements and extensions of the original result. Firstly, by using some properties of topological groups, we demonstrate that the positivity axiom can be removed with almost no effect on the result. Secondly, we consider a version of the theorem on the whole real line $\mathbb{R}$, with constant of integration $-\infty$ instead of $0$, which was mentioned without proof in the 1978 paper. The theorem can be extended to this setting, but it is not trivial to do so: the theory of Fréchet spaces must be used, and we introduced a new shift-commutativity axiom to get the density result that we need to extend the proof to spaces of functions and distributions on $\mathbb{R}$ with left-bounded support.

We show that for certain classes of matrices over quaternions, a more restrictive version of breakthrough numerical radius inequalities obtained by Kittaneh [\textit{{Stud. Math.}, 2005}] hold.

This paper presents an information-gain-guided reinforcement-learning sensor-selection framework for single-vessel tracking in heterogeneous maritime sensor networks. The proposed approach is motivated by information-theoretic sensor management: instead of activating all sensors or repeatedly performing computationally expensive online expected-information-gain evaluation, a learned policy selects one tracking-relevant sensor at each decision epoch. A Bayesian sequential Monte Carlo tracker estimates the vessel state from noisy measurements and provides a belief representation for scheduling under nonlinear and non-Gaussian conditions. A Proximal Policy Optimization agent selects one of five sensors in a georeferenced simulation of the CMMI Smart Marina testbed at Ayia Napa Marina, Cyprus. The policy is trained on the testbed's actual five-sensor configuration. The agent observes belief-state, detection-history, coverage, sensor-geometry, and realized-information-gain features. The reward is defined as a realized-information-gain term gated by an observability mask. Final-test simulations compare the proposed framework with random single-sensor selection, always-on sensing using all sensors simultaneously, and the expected-information-gain sensor-selection baseline proposed in our previous work. Results show that the learned policy achieves tracking performance close to always-on sensing while activating only one sensor per decision time step and avoiding the computationally expensive online entropy search required by expected-information-gain selection. Additional zero-shot evaluation without retraining on ten moderately perturbed versions of actual layout configuration showed broadly stable tracking, with any increase in positional tracking error remaining below 1 meter across all perturbations.

The multipole decomposition of the single-site density matrix provides a symmetry-adapted representation of local electronic degrees of freedom. Conventional, so-called fixed-shell, formulations do not span the full local single-particle operator space, as only operators mapping within the same orbital $l$ are resolved. Here we construct a complete orthogonal basis of real Hermitian multipole operators for the local density matrix by extending the existing formulation to inter-shell operators. We revisit the multipole decomposition as a decomposition of the operator space by $\mathrm{SO}(3)$ by first decomposing the orbital and spin operator spaces. Then by coupling them we arrive at the spin-$\frac{1}{2}$ local single-particle operator space, staying consistent with the existing fixed-shell formulations. We then classify the multipoles by parity and time-reversal symmetry, which allows for unique identification of multipole moments of the density matrix that contribute to expectation values of fully symmetry-resolved observables. As an application, we analyze the two enantiomers of chiral trigonal tellurium by computing the electric toroidal monopole moment selected by symmetry.

According to a theorem by F.~Dyson, every continuous function $f: S^2 \to {\mathbb R}$ takes the same value at the vertices of a square inscribed into a great circle of $S^2$. We give a proof of this statement based on the theory of characteristic classes and indicate other potential applications of this approach.

After giving a number of properties of continued fractions of polynomial type, in particular focusing on convergence properties and Bauer-Muir-Apéry acceleration techniques, we give a large list of continued fractions, both for specific real numbers, and for special functions, some extracted from a number of different sources, but most others being probably new. In addition to providing such a list, one of our main additions is to include the exact speed of convergence of these continued fractions (sometimes only up to a multiplicative constant), which is almost always omitted in the literature.

In this paper, picture fuzzy multisets were studied together with their associated properties. We also introduced the concept of picture fuzzy multigroups and established some of their algebraic properties.

This paper presents a polygonal cell-based smoothed finite element method (CS-FEM) for two-dimensional steady-state and transient heat-conduction analysis. In the proposed formulation, Wachspress shape functions are employed to construct the temperature approximation over general polygonal elements, and the smoothed temperature gradient is evaluated through boundary integration over cell-based smoothing domains. The resulting formulation is implemented in ABAQUS through the user-defined element (UEL) interface, enabling heat-conduction analysis using polygonal meshes within a commercial finite element environment. Several numerical examples, including a linear patch test, steady-state benchmark problems, and transient heat-conduction problems with different boundary conditions, are investigated to verify the accuracy, convergence behavior, and robustness of the proposed method. The numerical results show good agreement with analytical or reference solutions. Compared with conventional FEM using triangular and quadrilateral elements, the proposed polygonal CS-FEM exhibits favorable accuracy and convergence performance, while providing greater flexibility in mesh generation for complex geometries. The proposed framework therefore offers an accurate and robust numerical approach for steady-state and transient heat-conduction analysis.

We recast the physics discussions in the paper of Dieter Van den Bleeken \cite{svan2012bps} within the context of wall-crossing structure à la Kontsevich and Soibelman \cite{kontsevich2014wall}. In particular, we compare the Hesse flow given in \cite{svan2012bps} and the attractor flow on the base of the complex integrable system, and show that both can be used in the formalism of wall-crossing structure. We also propose the notions of dual Hesse flow and dual attractor flow, and show that under the rotation of the $\mathbb{Z}$-affine structure, the Hesse flow can be transformed into the dual attractor flow, while the attractor flow into the dual Hesse flow. This suggests its possible use in Mirror Symmetry.

We present a short "proof from the Book" that $321$-avoiding permutations without double deficiencies are counted by the Motzkin numbers. Although this result was first proved by Rubey and Stump, our proof gives a much simpler and more transparent description of the same bijection.

This article extends the definition of the Noncommutative Foliation Invariant (NCFI) for foliations of odd codimension and computes certain key examples for both even and odd codimension cases: fibrational foliations, irrational Kronecker, rational Kronecker (both the vertical and the horzontal foliation using a flat connection) and weighted Hopf/orbifold cases. We also prove some more general results along the way. More concretely, we compute the Noncommutative Foliation Invariant (NCFI) for several basic families of foliated manifolds. In even codimension the invariant is the Chern–Connes pairing of Connes' transverse fundamental cyclic cocycle with the \(K_0\)-class associated with the transverse geometric module. In odd codimension the transverse cocycle has odd degree, so a numerical pairing also requires a specified odd \(K_1\)-class; the odd-codimensional value is therefore an invariant of the foliation together with this chosen odd-favourable structure.

We give generators and presentations for various local restrictions of Steinberg modules over fields and relate them to partial Borel–Serre compactifications of locally symmetric spaces in the case of number fields, extending the existing theory of partial modular symbols for $\mathrm{GL}_2(\Q)$. Along the way, we clarify the relationship between “circuit” and “Bykovskii”-type presentations for such modules. In an appendix, we relate the existence of such presentations to Koszulity properties of the Steinberg $\mathrm{VB}$-algebra.

For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison–Reiman data with a nonsingular $M$-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular $M$-matrix assumption is structural. In the larger completely-$\mathcal{S}$ class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai–Dieker question has a positive answer in the Harrison–Reiman $M$-matrix class and a negative answer in a natural completely-$\mathcal{S}$ extension.

This paper focuses on computing the compositional inverses of permutation polynomials of the form $x+\gamma \operatorname{Tr}_{q}^{q^{n}}(H(x))$ over finite fields via the local method. We explicitly construct compositional inverses for four families of permutation polynomials of this type over $\mathbb{F}_{q^2}$, another four families over $\mathbb{F}_{q^3}$, and one general family over $\mathbb{F}_{q^n}$. The closed-form inverse expressions derived in this work supplement the theory of trace permutation polynomials.

In this paper, we investigate the compositional inverses of permutation polynomials of the form \[ F(x)=h(\psi(x))\varphi(x)+g(\psi(x)) \in \mathbb{F}_{q^n}[x], \] where \(\psi(x),\varphi(x) \in \mathbb{F}_{q^n}[x]\) are additive polynomials, \(h(x), g(x) \in \mathbb{F}_{q^n}[x]\) satisfy $ h(\psi(\mathbb{F}_{q^n})) \subseteq \mathbb{F}_q^*, $ and there exists a polynomial \(\bar{\psi}(x) \in \mathbb{F}_{q^n}[x]\) such that $ \bar{\psi}(F(x)) = \psi(x). $

This paper presents an accelerated solver for the unit commitment problem in large-scale power systems. The approach is based on the concurrent execution of GPU- and CPU-based optimization solvers on a single machine, with the solver that converges first terminating the other to minimize overall runtime. This strategy effectively harnesses the complementary strengths of different solvers. Convergence is further accelerated through a systematic and aggressive presolve approach. Numerical experiments on a 6,049-bus system with millions of decision variables and constraints demonstrate speedups ranging from 2.14x to 5.61x, reducing the maximum runtime from 42.12 minutes to 5.77 minutes across 45 test cases. These results highlight the scalability and computational efficiency of the proposed GPU-CPU concurrent solver framework.

In emerging urban mobility and logistics applications, such as advanced air mobility, electric vehicle charging, and shared service systems, agents with heterogeneous valuations choose among multiple outcomes while sharing congested resources. Existing congestion game and resource-allocation models, however, typically restrict either the structure of heterogeneous outcome preferences or the form of congestion externalities. We introduce a nonatomic congestion game framework in which agents have multidimensional heterogeneous valuations over a finite set of outcomes, with the population represented by a measure over the valuation space. We show that, although there is a continuum of heterogeneous agents, both Nash equilibria and social optima admit finite-dimensional characterizations. Nash equilibria are characterized by aggregate flows and outcome-level threshold vectors, while social optima admit a semi-discrete optimal transport formulation characterized by aggregate flows and strategy-level Kantorovich dual potentials. For separable utilities and latency functions $\ell_e$ such that $x\mapsto x\ell_e(x)$ is convex for every resource $e$, the social welfare problem becomes a convex optimization problem, and we characterize social optima through marginal congestion costs and capacity-dual prices. Our results provide a geometric characterization of congestion games with multidimensional heterogeneous valuations and connect nonatomic congestion games with semi-discrete optimal transport.

By a convergent set is meant a set of stochastic matrices where every infinite product of matrices from every compact subset converges to a rank one matrix. Well-known examples include the set of all scrambling matrices, the set of all stochastic matrices with all diagonal entries positive and a rooted graph, the set of all Sarymsakov matrices, and the set of doubly stochastic matrices with positive diagonal entries and a weakly connected graph. It is known that every infinite product from each compact set of every convergent set converges to its limit exponentially fast, but not much is known about the rate of convergence when not all matrices involved are scrambling matrices. This paper deals with bounding the rate of convergence in convergent sets using submultiplicative seminorms. It is shown that only in some convergent sets all matrices are contractions in the same seminorm, and in particular that this method cannot be used to determine the convergence rate for the class of matrices with positive diagonal entries and a rooted graph. As a second contribution, it is shown that for every compact convergent set and every submultiplicative seminorm, there is a finite number $k$ such that all products of $k$ matrices from the set are contractions in the seminorm. Finally, several open questions are posed for future research.

Any Pythagorean triple $\{a,b,c\}$ such that $a^{2}+b^{2}=c^{2}$ satisfies an elegant relation between its partitions into three parts, namely $p(a,3)+p(b,3)=p(c,3)$. While this property follows from elementary analytic methods, we give the first combinatorial proof of this relation.

We derive new exact formulae for elliptic genera in two spacetime dimensions with $\mathcal{N}\ge 2$ supercharges, as well as their one- and zero-dimensional counterparts (for Witten indices and matrix-model partition functions). Our results are written as a discrete sum over geometric/combinatorial structures of crystals introduced previously by the authors. The contribution from each crystal may be expressed in terms of the boundary data of a finite substructure of the crystal called the molecules. Our results provide vast generalisations of the celebrated Nekrasov partition function enumerated by the Young diagrams, and uncover new combinatorics underlying the crystal melting. We also analyse the thermodynamic limit of crystals arising from two-dimensional $\mathcal{N}=(0,2)$ theories associated with toric Calabi-Yau fourfolds, where a projection of the Calabi-Yau geometry emerges in the limit.

Observer-quotient security studies interactive cryptographic systems whose security depends on what an admissible observer can distinguish across transcripts, leakage traces, and hidden implementation continuations. The paper defines observer-indexed experiments with session identifiers, adaptive schedulers, oracle forwarding, simulators, ideal quotient functionalities, and nonuniform environments, and proves a real/ideal emulation theorem in which sequential morphism defects add, parallel defects obey a product-TV bound, and adaptive observer choice is absorbed by an explicit wrapper construction. The resulting advantage bound is indexed by $\delta_{\mathrm{obs},t}$, $\delta_{K,t}$, $\delta_{\mathrm{post},t}$, $\delta_{\mathrm{sim},t}$, $\eta_t$, and the residual floor $\rho_T(\mathbb E Z_T)$. The framework is instantiated for IND-CPA encryption with timing leakage, deterministic encryption with entropy ledgers, and finite-state side-channel refinement under transcript, timing, cache, power, EM, and profiled observers. The optimization/control component identifies hidden continuations with observability kernels, treats sensor redesign as quotient refinement, and converts dissipativity, PL-type rates, and ISS residual bounds into concrete reductions in distinguishing advantage. Ancillary code and synthetic data reproduce the finite-state leakage audit and LTI observer-design benchmark.

How much information is required to represent a dynamical system in terms of an interaction structure and an evolution rule? We address this question using algorithmic information theory. We introduce Representational Complexity, the excess description length of a structure-plus-rule model relative to the shortest possible description of the dynamics itself. This intrinsic description defines a universal lower bound: no exact structural representation can be more concise. If arbitrary rules are allowed, graphs, hypergraphs, and other formalisms can all reach this bound by shifting information between structure and dynamics, so expressiveness alone cannot distinguish them. Meaningful differences arise only when scientific modeling restricts the admissible structures and rules. Within this setting, we identify conditions under which graph and hypergraph descriptions are informationally equivalent, and show how graph-preferred, hypergraph-preferred, and mixed regimes can emerge when those conditions are relaxed. Because Kolmogorov complexity is not computable, we complement the formal results with explicit description-length estimates. Our framework reframes the choice of network representation as a question of informational cost and mechanistic transparency rather than universal expressive power.

We develop a top-down counterpart of the minimal proper-time formulation of quantum field theory previously introduced as an effective bottom-up framework. Starting from a deterministic pre-geometric substrate of causally ordered events, we show how coarse-graining over microscopic histories leads, at low energies, to an effective Nambu-like quantum dynamics. The elementary deterministic update is identified with the minimal proper-time step, while the growth of coarse-grained equivalence classes controls both the ultraviolet dissipative correction and the scale dependence of the effective quantization strength, encoded in a running Planck constant. In this way, the proper-time cutoff kernel of the bottom-up formulation acquires a microscopic interpretation as the inverse growth of unresolved deterministic histories. In the infrared limit, the dissipative term vanishes and standard unitary quantum field theory is recovered. The same coarse-grained structure also provides a natural setting for an emergent relativistic spacetime geometry, compatible in the macroscopic limit with Einstein gravity. The resulting picture suggests a common deterministic origin for minimal-scale structure, quantum behavior, and relativistic spacetime.

A cone structure on a complex manifold $M$ is a closed submanifold $\mathcal C\subset \mathbb P TM$ of the projectivized tangent bundle of $M$ that is submersive over $M$. So this defines a set $\mathcal C_x$ of distinguished directions in each point $x\in M$. A conic connection on $\mathcal C$ is then a family of unparametrized curves on $M$ that comprises exactly one curve through each point $x\in M$ in each direction in $\mathcal C_x\subset \mathbb PT_xM$. This can be encoded by a line subbundle $\mathcal F\subset T\mathcal C$. The subclass of characteristic conic connections is defined by the vanishing of a simple invariant, called characteristic torsion. For those, one has a much more subtle and slightly mysterious invariant called the cubic torsion. The first aim of this article is to provide a new approach to the cubic torsion, which also leads to a geometric condition characterizing its vanishing. We then specialize to the case of isotrivial cone structures, for which the fibers of $\mathcal C$ are assumed to be of some fixed type. Such a structure induces a first-order $G$-structure on $M$ whose structure group is the projective automorphism group of the model fiber. Moreover, any connection $\gamma$ on the associated $G$-structure induces a conic connection $\mathcal F^\gamma$ on $\mathcal C$. Assuming that the model fiber is homogeneous, we study the relation between the torsion and curvature of a connection $\gamma$ and the characteristic and cubic torsion of $\mathcal F^\gamma$. As an application we discuss cone structures of subadjoint type, showing in particular that there are such structures admitting conic connections with vanishing characteristic and cubic torsion that are not locally flat.

We study the disjoint separators problem in graphs, an analogue of the famous disjoint paths problem. Given a graph $G$ and four pairwise disjoint subsets of vertices $S_r$, $T_r$, $S_b$, $T_b$, we ask whether there exist an $(S_r,T_r)$-separator and an $(S_b,T_b)$-separator which are disjoint. This is equivalent to coloring the vertices in red or blue, with $S_r \cup T_r$ in red and $S_b \cup T_b$ in blue, such that there is no red $(S_r,T_r)$-path and no blue $(S_b,T_b)$-path. On the one hand, we show that the disjoint separators problem is NP-complete. We actually exhibit several NP-complete restrictions of this problem, including planar graphs of bounded maximum degree, and graphs of bounded maximum degree when $|S_r|=|T_r|=|S_b|=|T_b|=1$. On the other hand, these hardness results turn out to be quite tight, as we provide a structural characterization and a polynomial-time algorithm for planar graphs when $|S_r|=|T_r|=|S_b|=|T_b|=1$. This has an interesting consequence about the popular board game Hex: for the generalized game that may be played on any board, our result characterizes the planar boards on which draws are impossible, thus extending the well-known result about impossibility of draws on the standard commercialized board.

We introduce the notion of very weak suitable solutions for the Navier–Stokes equations. Here, the velocity and the pressure satisfy minimal conditions that make sense of the local energy inequality in the distributional setting. A well-known but still relevant question is to find sufficient conditions ensuring that very-weak solutions are in fact Leray solutions. Exploiting the local energy inequality within the general framework of local Morrey spaces, we establish such conditions. Local Morrey spaces provide a general framework that contains other useful functional settings in the theoretical analysis of the Navier–Stokes equations, such as Lebesgue, Lorentz, homogeneous Morrey, and parabolic Morrey spaces. As a by-product, we also derive some sufficient conditions to study the uniqueness and regularity of the resulting Leray solutions.