Papers

This paper investigates the well-posedness and hidden regularity of boundary-degenerate hyperbolic equations in a two-dimensional setting. The shape design method is employed, which approximates the original degenerate problem by a family of uniformly elliptic problems on truncated subdomains and passes to the limit through uniform estimates. Within this framework, the existence and uniqueness of weak solutions are established in suitable weighted Sobolev spaces. A hidden regularity estimate for the conormal derivative on the nondegenerate portion of the boundary is obtained, providing a uniform bound in terms of the natural weighted energy norms of the initial data and source term. This estimate yields the boundary trace information essential for observability and controllability of degenerate hyperbolic systems.

We study a dynamic coalition-formation process in the tradition of Konishi and Ray (2003): players repeatedly form and dissolve binding agreements, evaluate states by discounted long-term expected payoffs, and hold self-confirming beliefs about the process. States and payoff sharing follow Heitzig and Kornek (2018): a state is a hierarchy of nested agreements, and the members of a new agreement share the surplus it generates, measured against the state without that agreement. All payoff assumptions are structural. We prove that every grand state ever reached is absorbing, and that every absorbing state is grand, for every discount factor. A grand state is actually reached, almost surely, for small discount factors, for three players, and, at every discount factor, whenever distributional stakes are smaller than each player s share of the efficiency gain. Otherwise the process can fail only by cycling for ever among non-grand states. We give exact necessary conditions on such a cycle, decidable for a fixed candidate cycle by linear programming, and exhibit, under an earlier and weaker notion of profitability, a four-player payoff structure whose only closed class is a cycle of two pairs forming and dissolving alternately. Under the present definition, and under either termination rule, no equilibrium traverses a cycle on a fixed schedule: somebody always reaches a state they would rather not leave, and the axioms give them the floor. Whether arrival can fail by cycling at random is open. The axioms are not merely postulated: we exhibit a bargaining game proposal, amendment by substitutes voted on by their own signatories, final unanimity, and an arbitrarily small delay on failure whose equilibria satisfy them as the delay vanishes, and which settles each period in its first round.

Deep level Deligne-Lusztig representations, as natural extensions of the classical Deligne-Lusztig representations, have recently seen various applications in the Langlands program. In this note, we address their distinction problem for symmetric pairs over finite fields, yielding a generalization of a classical result by Lusztig.

In this work, we compute the first integral homology, or abelianization, of the congruence subgroups $\Gamma(A, \mathfrak{m}_A), \Gamma_1(A, \mathfrak{m}_A)$, and $\Gamma_0(A, \mathfrak{m}_A)$ for a local ring $A$ with maximal ideal $\mathfrak{m}_A$, showing that $H_1(\Gamma(A, \mathfrak{m}_A), \mathbb{Z})$ is isomorphic to the additive group of $\mathfrak{sl}_2(\mathfrak{m}_A/\mathfrak{m}_A^2)$. We then use these results to determine the structure of the groups $H_1(\Gamma(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$, $H_1(\Gamma_1(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$ and $H_1(\Gamma_0(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a Dedekind domain of arithmetic type, not totally imaginary, $|S| \geq 2$, and $\mathfrak{p}$ is a nonzero prime ideal. The computations are given in terms of the residue field $\kappa(\mathfrak{p})$ and the known $H_1(\mathrm{SL}_2(\mathcal{O}_{K, S}), \mathbb{Z})$. As a consequence, we also obtain the torsion subgroup of their second integral cohomology. These results will be of paramount importance for a forthcoming work concerning $H_2(\mathrm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$.

For finite-domain constraint satisfaction problems (CSPs), the polymorphism clone provides an algebraic framework for studying the associated constraint language. A central question in clone theory is whether a clone is finitely related, that is, whether it is determined by the operations preserving a finite set of relations. Here, we study this question through the action algebra of a clone on its own $n$-ary part. We prove that a clone $\mathcal{C}$ on a finite domain $C$ is finitely related if and only if its action algebra $\widetilde{\mathcal{C}} = \mathcal{C} \curvearrowright \mathcal{C}^{(n)}$ is finitely related, provided that $n \geq |C|$. We also apply natural clone isomorphisms to obtain a finite-relatedness criterion for free algebras: a finite algebra is finitely related if and only if its free algebra of sufficiently large finite rank is finitely related.

In this article, we introduce a novel class of mappings identified by their property of contracting the hypotenuse of a right-angled triangle in the setting of metric spaces. We establish sufficient conditions ensuring both the existence and uniqueness of fixed points. A geometric analysis, complemented by illustrative diagrams, is provided to differentiate these mappings from other familiar contraction types, namely perimeter and area contractions, supported by examples. Furthermore, we investigate the Ulam-Hyers stability of the associated fixed point equation, thereby strengthening the robustness of the theoretical framework. Finally, the derived results are applied to demonstrate the existence of solutions for a nonhomogeneous linear parabolic partial differential equation (PDE).

Intelligent channel state information (CSI) feedback is essential for realizing the high capacity and spectral efficiency goals of future 6G systems, yet existing deep learning solutions face a trade-off between model generalization and scenario-specific performance. Large neural networks generalize well but incur high computational and tuning costs, while small models excel in particular environments but require repetitive costly end-to-end training for each base station (BS). To address these challenges, we introduce a model repository-based deployment framework in which a centralized AI data center maintains a catalog of scene-specific CSI models. The repository is enhanced with a Learnware-based framework, where each model is associated with a specification including semantic part (network architecture parameters) and statistical part (codeboo-fingerprint embeddings of training-data distributions). A BS submits only its local statistical specifications to retrieve the most relevant pre-trained model, enhancing data privacy by avoiding raw CSI transmission and drastically reducing retrieval latency and communication overhead. We further develop a data-driven search strategy that matches codebook fingerprints to model performance, achieving over 90% selection accuracy. In simulations, our scheme yields 18.8% and 57.7% performance improvements over the General Model in LOS and NLOS scenarios, respectively while reducing local fine-tuning by up to 1000 samples and 100 epochs. This Learnware-based approach minimizes redundant training, maximizes model reuse, and supports rapid,privacy-enhancing deployment of CSI feedback models.

Single-particle cryo-electron microscopy images a macromolecule as many noisy tomographic projections of its electrostatic potential. We reconstruct the protein backbone directly from such projections, as an atomic point cloud, without the intermediate step of reconstructing the 3D electrostatic potential map. We formulate this as an indirect shape matching problem: a point-cloud template of the backbone is deformed until its simulated projections agree with the data, with the structure observed only through the imaging operator. The deformation is computed via a gradient flow on a Lie group, and we derive the resulting framework in a general geometric setting before adapting it for single-particle cryo-electron microscopy. On synthetic data, we recover single- and multichain proteins and capture conformational transitions.

Synchronizing nonlinear dynamical networks typically requires solving matrix inequalities or detailed system models, which fail for large networks. This paper offers a simple fix : a purely graph-theoretic framework using only a single Lipschitz-like bound on the dynamics. Coupling strengths are computed directly from the digraph, bypassing inequality solvers entirely. The method succeeds where existing approaches encounter infeasibility due to connectivity patterns. It examines only $n-1$ directed paths per strongly connected component versus $\frac{n(n-1)}{2}$ undirected paths before, achieving $O(n^3)$ complexity. Results show network connectivity can be exploited to synchronize a large class of nonlinear dynamical networks.

We study the set of degrees of $\mathbb{Z}/m$-equivariant maps from spheres to complex Stiefel manifolds, motivated by the work of Astey–Gitler–Micha–Pastor. Under a suitable arithmetic condition, this set is determined using results of James, Atiyah–Todd, and Adams–Walker. Our approach is homotopy-theoretic.

Let $J_G$ denote the binomial edge ideal of a finite graph $G$ in the polynomial ring $S$. We determine all triples $(n,t,d)$ with $n\geq 3$ for which there exists a finite connected graph $G$ on $n$ vertices with ${\mathrm{depth}}(S/J_{G})=t$ and $\dim(S/J_G)=d$.

For every field $k$, we construct a commutative coherent $k$-algebra admitting a Gorenstein projective module that is not Gorenstein flat.

We give computer-assisted proofs that every principal minor of each of the \(70\times70\) and \(143\times143\) Fourier matrices is nonzero. A lifting theorem of Caragea, Lee, Malikiosis, and Pfander reduces the two assertions to the nonvanishing of all principal minors of the Fourier matrix of order \(10\) in characteristic \(7\), and of order \(11\) in characteristic \(13\), respectively. We realize primitive roots in \(\mathbb F_{7^4}\) and \(\mathbb F_{13^{10}}\) and evaluate all \(2^{10}\) and \(2^{11}\) principal determinants by exact, division-free arithmetic. None vanishes. The lifting theorem in fact yields the stronger conclusions that every \(10\)-principal minor of the order-\(70\) matrix and every \(11\)-principal minor of the order-\(143\) matrix is nonzero. Self-contained standard-library verifiers for the finite-field calculations accompany the paper.

Let $\delta>0$ be fixed, let $m = \lceil(1+\delta)n\rceil$, and let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope $G_m = \Gamma(B_1^m)$ we prove $$ \p\left\{d_{\mathrm{BM}}(G_m,B_1^n) \ge c_\delta n^{5/8}(\log n)^{-1/8}\right\} \ge 1-Ce^{-cn}. $$ Consequently, for all sufficiently large $n$, $$ R_1(n)\ge c n^{5/8}(\log n)^{-1/8}. $$ The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.

For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In his 1977 formulation of the problem, Erdős discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n^{2/3}-7$ has a relabelling with at most three common edges. Consequently, \[ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, \] and hence \[ f(n,4)=2n-o(n). \] Thus we resolve Erdős's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n^{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.

We show that the integral $2$-adic Tate module of an elliptic curve over a complete discretely valued field of odd residue characteristic is determined by its $2$-torsion representation, together with the square class of $c$ and local information about the pairwise differences of the roots of $f$ in a model $E\colon y^2=cf(x)$, where $f$ is monic of degree $3$. The proof uses explicit halving formulae to determine the Galois action on the full tower of $2$-power torsion.

The quantum approximate optimization algorithm (QAOA) targets NP-hard maximum-likelihood (ML) detection in multiple-input multiple-output (MIMO) systems. Existing $M$-ary quadrature amplitude modulation (M-QAM) detectors design angles by expected Ising energy: online per instance, warm-started, or ramped, while train-once designs remain B/QPSK-only or block-local, leaving M-QAM without a size-scalable benchmark. Their infinite-size spin-glass theory assumes independent disorder, matching the retained covariances at B/QPSK but not M-QAM's correlated couplings and fields. We develop a correlated infinite-size multi-species spin-glass framework whose covariance-matched evaluators make that energy an offline objective with a size-scalable benchmark. In addition, the ML rate, the exponential rate of sampling the ML string, is for the first time exploited for QAOA angle design in MIMO detection. The energy evaluator is $q$-free at $O(p\,4^p)$ cost while the ML rate transfers angles from a fixed $q_{\rm ref}$-qubit reference. Tests reach 4096-QAM, 128 antennas, $p=30$ and per-symbol SNR 0-45 dB. In simulations, ML rates fall as a power law $r_0\,p^{-\alpha}$, with larger exponents for the sampling design, which tracks exact ML at $5\times5$ 16-QAM (0-20 dB) and $3\times3$ 64-QAM (8-28 dB) while its bit-error rate (BER) advantage widens with SNR to two orders of magnitude. The approach points toward near-optimum decoding on deeper noiseless fault-tolerant quantum (FTQ) circuits.

We study singular conformal metrics \(g=Fg_0\) near codimension-two corners, where \(F\) is a finite positive sum of monomial singularities. The competing orders are encoded by the homogeneous Newton support function \(\mathcal H(p,q)=\max_j(pa_j+qb_j)\). We prove the sharp projective accessibility criterion \(\mathcal H(p,q)<2\min\{p,q\}\), and deduce that the corner lies at finite metric distance exactly when \(\max_j(a_j+b_j)<2\). At every accessible corner, all normal approaches over a fixed corner point determine one canonical completion point. The induced boundary metric is locally bi-Lipschitz to a snowflake of the smooth corner metric, yielding an explicit Hausdorff-dimension formula. For rational exponent data, the Newton decomposition is also realized by compatible rooted and monoidal modifications. The metric conclusions themselves require only positivity, bounded coefficients, and uniform ellipticity.

We study the case where the tangent bundle of a Kähler surface splits orthogonally into two integrable complex-line subbundles. This amounts to the presence of a unit-length closed anti-self-dual 2-form, leading in turn to a curvature condition. Therefore, such a decomposition need not exist, even locally, in contrast with general Riemannian four-manifolds which, under the assumption of real-analyticity, were shown by Grant and Vickers [11] to always admit, locally, two mutually orthogonal integrable rank-two distributions. We exhibit local coordinates naturally adapted to a decomposition as above in a Kähler surface, which may be used for simple constructions of examples. We also provide a characterization of the Kähler case within the wider class of almost-Kähler surfaces arising from the coordinates just mentioned. The characterization involves a system of four first-order quasilinear partial differential equations imposed on four unknown functions of four variables and, using Cartan's test, we prove the system's local solvability. Finally, we show that, for such a decomposition in a Kähler surface, one of the summands is holomorphic if and only if the other one has totally geodesic leaves, and describe a general construction of examples with this last property.

We develop a uniform framework for “big” categorification of representation categories of towers of finite classical groups and wreath product groups. We construct actions of symmetric products of Heisenberg categories, quantum in the finite classical group case and degenerate in the wreath product case. These actions lead to categorical actions of suitable symmetric products of Kac–Moody 2-categories, and hence to actions of large Lie algebras on Grothendieck groups. In characteristic zero, the resulting actions control the full graded centers of the group algebras through diagrammatic central elements, and the associated colored weight functions separate all irreducible ordinary characters. We also obtain modular block descriptions for wreath product groups through categorification.

It is shown that the trivial extension of the Xu–Snashall algebra is a symmetric counterexample to the Snashall–Solberg conjecture, which states that the Hochschild cohomology ring of a finite-dimensional algebra modulo nilpotence is a finitely generated algebra. To the best of our knowledge, this is the first known selfinjective counterexample.

Mobile robots that operate in side by side with humans and critical facilities must reach their goals at low cost, despite often unknown true traversal costs of the map apriori and imperfect actuation. Planners that solve the underlying stochastic shortest path problem exactly, such as value iteration, require computation that grows with the diameter of the map, whereas Dijkstra's algorithm is fast but is usually considered inexact once transitions are stochastic. This study shows that Dijkstra's algorithm can remain an exact planning engine under a condition that is much weaker than the causality condition often invoked in the literature, namely nonnegativity of a reduced cost defined on the determinized map. Building on this characterization, an online learner DORA (Dijkstra Oracle Reduced-cost Algorithm) is proposed for robot navigation that calls a shortest path oracle a fixed number of times per episode, never estimates a transition kernel, and adds a logarithmic survival weight when the probability of contact with a dynamic obstacle must stay within a budget. In the numerical experiments involving three other benchmarks that cover grid world navigation, directional drilling, and drone surveillance, the learner matches optimistic value iteration that is given the true transition kernel while performing 4.5 to 19.3 times less planner work, reduces contacts during learning by a factor of seventeen relative to determinize and replan, and keeps the contact rate within budgets that span two orders of magnitude. These results indicate that shortest path search supports safe and efficient online navigation and path planning tasks.

Motivated by the equivalent definitions of positive definite (resp. positive semidefinite) matrices over real and complex fields, we give four (resp. five) inequivalent definitions for these matrices over finite fields. Our starting point is the recent definition due to Cooper–Hanna–Whitlatch (RMJ. Math., 2024) of positive elements in finite fields. We also use this definition to study totally positive matrices over finite fields. For all of these cases, we give explicit enumeration formulae or give bounds. Most of our formulas are new, but we summarize results from the existing literature for completeness. For positive semidefinite matrices of type 5 and totally positive matrices, we give structural formulas using the rationality of the Weil zeta function, i.e. Dwork's theorem, and conjecture a quasipolynomial-type formula.

This paper establishes a fundamental connection between quasilinear potential theory and convex geometric analysis by investigating the interplay between the quasilinear Laplace operator and quermassintegrals. We introduce a quasilinear heat dispersion law for convex conductors and prove that, among all convex conductors of a fixed mean width, the closed ball is a unique maximizer of this dispersion. By characterizing the quasilinear heat loss of a convex conductor explicitly in terms of its quermassintegrals, we demonstrate not only a formal equivalence between the isocapacitary and isoperimetric inequalities in the setting of mathematical physics but also that, among all convex conductors of a fixed mean width, the closed ball is a unique maximizer of this loss. These results provide a novel bridge between the metric properties of convex conductors and the variational analysis of quasilinear elliptic operators, offering a unified perspective on sharp geometric inequalities and their extremal cases.

We argue for the universality of quantum theory using a dynamical consistency argument, within a specific Hamiltonian setting. We analyse two different types of coupling between simple quantum harmonic oscillators. Each illustrates an aspect of the free and interacting quantum fields and shows the inadequacy of semiclassical models. In particular, we establish that requiring the canonical algebra to be preserved under joint unitary dynamics rules out specific hybrid classical-quantum models. We apply our reasoning to the gravitational field in the linear regime, coupled to the quantised electromagnetic field and, separately, to quantised matter. We conclude with a comparison to DeWitt's analysis of quantum measurement, in which the apparatus, if classical, must be at least stochastic to preserve the Heisenberg Uncertainty Principle. We also note that stochastic models are inconsistent with the strict version of conservation principles, even if they comply with a probabilistic (on average) conservation.

For a real transcendental number $\xi$, let $\omega_n^*(\xi)$ denote the supremum of all $\omega$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $\leq n$ satisfying $|\xi-\alpha|\leq H(\alpha)^{-\omega -1}$, where $H(\alpha)$ is the naive height of the minimal polynomial of $\alpha$. A celebrated result of Wirsing gives the uniform lower bound $\omega_n^*(\xi)\geq\frac{n+1}{2}$, which was improved significantly in a recent work of Poëls to $\frac{n}{2-\log 2}$. In this paper, we establish a $p$-adic counterpart of Poëls's result. Let $p$ be a prime and $\xi\in\Qp$ be transcendental. Let $\omega_{n,p}^*(\xi)$ be the supremum of all real numbers $\omega$ for which there exist infinitely many algebraic numbers $\alpha \in \Qp$ of degree $\leq n$ such that $|\xi-\alpha|_p\leq H(\alpha)^{-\omega -1}$. We show that $\omega^*_{n,p}(\xi)\geq\frac{n}{2-\log 2}-1$. This improves the known lower bounds in the $p$-adic setting, namely the analogue of Wirsing's theorem, due to Morrison and Teulié.

Impulsive noise (IN), characterized by its high power and non-Gaussian distribution, poses a critical challenge in modern orthogonal frequency-division multiplexing (OFDM) systems, driven by the proliferation of electronic devices. Current IN mitigation techniques rely heavily on time-domain processing. These methods are applied before the discrete Fourier transform (DFT), introducing additional complexity, failing to align with OFDM's inherent frequency-domain processing flow, and risking the destruction of subcarrier orthogonality due to imperfect IN subtraction. To address these limitations, we propose a frequency-domain, block-based framework for mitigating IN. The statistical representation of IN in the frequency domain is first derived using a transformed Gaussian mixture model. Based on this model, we develop an optimal receiver that leverages perfect noise state information (NSI), thereby identifying scenarios in which NSI is critical. We then propose an unsupervised block-based expectation-maximization (EM) framework for NSI estimation and develop three variants for evaluation. These include a simple symbol-by-symbol variance-updated EM, a sequence-based transition-updated EM, and a MAP-based EM that exploits a sparsity-promoting prior to automatically prune the number of states. Our frequency-domain design operates after the DFT, seamlessly integrates with the OFDM processing chain, preserves subcarrier orthogonality, and leverages the known IN block structure to achieve substantial performance gains without the immense complexity of time-domain impulse reconstruction.

We develop the representation theory of imaginary Verma modules for twisted quantum affine algebras and construct the corresponding Kashiwara algebras. The twisted case presents substantial new difficulties compared with the untwisted setting: the PBW root vectors have nontrivial orbit structure, the imaginary root spaces occur with multiplicities, and roots of unity enter essentially into the defining commutation relations. Our first main result is an explicit PBW-type basis for twisted quantum imaginary Verma modules associated with the natural imaginary partition of the affine root system. This provides a precise compatibility between the twisted quantum and classical theories that is not immediate from the standard PBW theory. We then determine the structure and irreducibility of the twisted quantum imaginary Verma modules. We prove that the Heisenberg submodule generated by the imaginary root vectors is irreducible precisely at nonzero central charge, and establish the corresponding irreducibility criterion for the reduced twisted imaginary Verma modules at zero central charge. The second main part of the paper introduces the Kashiwara algebra in the twisted setting. The required current formula for the generators differs essentially from the untwisted formula and is needed to construct the Omega operators. We derive the resulting Omega-operator commutation relations, including the root-of-unity factors specific to the twisted cases. These relations lead to a new presentation of the Kashiwara algebra associated with the reduced twisted imaginary Verma modules. We prove that the negative current algebra is a simple module over this Kashiwara algebra and construct a symmetric non-degenerate bilinear form characterized by the Omega operators.

We explicitly determine the first Galois obstruction in the cokernel of the Johnson homomorphism of the mapping class group of a surface of genus $g$, for every genus $g \ge 2$. It is described as a sum of two terms which are considerably different in character. One lies in the kernel of the Enomoto-Satoh trace map, whereas the other belongs to a certain ideal which vanishes upon passage to the closed surface case.

This work proposes a semi-analytical theory for improving orbit design of the GARATÉA-L Brazilian lunar mission. The dynamical model incorporates the lunar gravitational potential up to degree 12 and order 3 harmonics and a third-body perturbation model accounting for the eccentricity and inclination of the Earth's orbit with respect to the Moon. Through Hori's method, short- and medium-period terms are eliminated from the Hamiltonian to derive a first-order theory in mean orbital elements. It is demonstrated that nominal frozen orbits designed under simplified circular and equatorial assumptions for the motion of the third body are not maintained in this more realistic dynamical environment, leading to significant eccentricity growth and potential collisions of the probe with the lunar surface. The inclusion of the third body's inclination and eccentricity breaks the system's axial symmetry, introducing the longitude of the ascending node into the double-averaged variational equations and rendering stationary frozen orbits practically unattainable. Consequently, the problem is addressed through a parametric study around the simplified model's frozen conditions to identify “low-drift" states where eccentricity variations are highly decreased. A new nominal orbit is proposed for GARATÉA-L. Numerical validation against high-fidelity propagation and JPL ephemeris confirms that this new configuration maintains the “low-drift" behaviour throughout the mission, ensuring orbital safety and avoiding the collision predicted by simpler models.