Papers

Baste et al. (2020) conjectured that every regular graph of positive degree has domination number at most its edge domination number, the least size of a maximal matching. Combining published bounds settles the inequality for every degree at least nine. A reduction proves the inequality whenever one endpoint of each edge of a minimum maximal matching can be chosen to form a dominating set, and the Lovász Local Lemma shows such a choice exists for every degree at least seven, newly closing degrees seven and eight and leaving degrees three through six open. The reduction settles each open degree up to a bounded number of vertices, forty-eight for cubic graphs. At fifty vertices, however, the reduction meets an explicit cubic graph it cannot settle, though the inequality holds there too. The inequality cannot be tightened, since infinitely many cubic graphs have equal domination and edge domination numbers. The cubic case stays open, and even linear arguments from the local structure cannot close it. The middle degrees stay open beyond the graphs already settled.

Complex systems are often organised into hierarchies whose internal interactions are stronger or faster than interactions across levels. When markets are treated as genuinely multilevel systems it becomes natural to represent them as systems with hierarchical causality. Here we show that reflexivity can be formulated within such a discrete hierarchical causal system; but one in which a higher-level actor state restricts the lower-level transition kernels that remain admissible. Then event dynamics can be separated from calendar embeddings: a set-valued actor-conditioned correspondence can be used to define the admissible family of event kernels, while joint state and waiting-time laws can be used to determine compatible timing to then natural demonstrate reflexivity. A selected event-state law need not determine a unique calendar embedding. Locally, uniqueness of the joint event or timing specification requires uniqueness of both the admissible event kernel and its compatible timing law. Reflexivity is thus the endogenous closure of a hierarchical constraint loop, while timing and projection can generate calendar-time memory or causal ambiguity even for Markov event dynamics.

High-altitude platform station (HAPS) serves as a promising enabler for wide-area connectivity of low-power wireless devices, particularly in remote and underserved regions. However, the strong line-of-sight characteristics of HAPS links increase the risk of eavesdropping, while the limited energy budget of ground devices remains a major operational constraint. In this work, we propose a secure wireless information and energy harvesting framework for HAPS-based networks in the presence of spatially distributed eavesdroppers. The proposed system integrates friendly jamming and power transfer nodes equipped with null-steering capability antennas, such that they not only degrade the reception quality at eavesdroppers but also act as additional radio-frequency energy sources for legitimate users. A time-switching wireless information and power transfer architecture is adopted at the user side. Under this framework, we derive tractable expressions for the joint rate-energy coverage and the average secrecy rate using stochastic geometry tools. Numerical and Monte Carlo results validate the developed analysis and reveal key design trade-offs among the time allocation factor, null-steering-zone radius around each user, and jammer transmit power. In particular, the results show that properly coordinated null-steering jamming can simultaneously support secure communication and adequate wireless power transfer, while an appropriate choice of system parameters, such as time allocation factor and jamming power, is required to balance harvested energy, communication reliability, and secrecy performance.

We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.

The Jin–Xin relaxation of a nonlinear hyperbolic conservation law introduces a relaxation parameter that controls the width of the internal layer resolving a shock; the discontinuity of the limiting conservation law emerges only in the singular limit as this width vanishes. Physics-informed neural networks (PINNs) use smooth network approximations and are therefore not well suited to this limit, while relaxation PINNs with a fixed parameter resolve only a single scale and cannot follow the multiscale transition toward the limiting solution. We propose the Jin–Xin relaxation gradual convergence method (JXRGCM), which treats the relaxation parameter as a continuation variable, annealing it to zero along a schedule and warm-starting each stage from the previous one, so that the approximation follows the relaxation profile through progressively sharper scales. Under the sub-characteristic condition we establish a stability estimate whose constant is independent of the relaxation parameter; combined with the relaxation limit, it yields for scalar conservation laws an $L^2$ convergence rate of $\mathcal{O}(\varepsilon^{1/4})$ toward the entropy solution. Numerical experiments on the Burgers equation, the shallow-water dam-break problem, and the Sod shock tube show that JXRGCM improves shock and rarefaction resolution compared with fixed-parameter relaxation PINNs and other physics-informed approaches.

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniformly bounded circumdisk radii, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a nonempty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.

This work presents a concise, unified stability theory of high-order decoupled \lan{implicit-explicit linear multistep (IELM)} methods based on the well-known consistent splitting technique for the incompressible Navier-Stokes equation. With the help of the recent semi-generating function approach and the global discrete energy analysis, one can establish the unconditional stability of a consistent splitting IELM method with respect to the $\ell^{\infty}(H^1)\cap \ell^{2}(H^2)$ norm if the associated implicit-explicit controllability intensity is larger than $\sqrt{2}/2$, a constant determined by the Stokes pressure estimate. It is shown that the $\beta$-parameterized GBDF-$\rmk$ ($2\le \rmk\le5$) schemes and $\gamma$-parameterized SIELM-$\rmk$ ($2\le \rmk\le9$) schemes can fulfill this requirement of implicit-explicit controllability intensity by choosing proper parameters so that they can theoretically maintain the unconditional stability of the associated consistent splitting IELM methods. Numerical experiments are also included to support our theory.

Let $(v,p)$ be a smooth stationary Navier–Stokes solution in $\mathbb R^3$ with $v(x)\to0$ as $|x|\to\infty$. We prove that $v\equiv0$ if its vorticity $\omega:=\nabla \times v$ belongs to $L^{s,\infty}(\mathbb R^3)$ for some $9/5\le s\le1.87$. This gives a global weak-Lorentz vorticity criterion beyond $9/5$ requiring neither smallness nor an additional Fubini-type hypothesis. The finite Dirichlet integral is recovered from the vorticity hypothesis rather than assumed. In particular, $|\omega(x)|=O(|x|^{-\alpha})$ implies triviality for $\alpha\ge300/187$, crossing the $|x|^{-5/3}$ vorticity-decay scale. The proof combines this automatic finite-energy upgrade with a new variable-denominator Bernoulli–vorticity quotient identity. Its completed-square form gives nonnegative defect terms, and localizing the concavity of the denominator creates a positive measure on a Bernoulli level surface. A pressure-weighted convex extension of this identity yields a quantitative normalized-vorticity trace and quotient-gradient control. These estimates are then coupled to the Bernoulli-gradient identity through a nonlinear feedback estimate.

The queen's graph $Q_n$ has the squares of the $n\times n$ chessboard as vertices, with adjacency defined by a common row, column, or diagonal. We give an explicit set of $82$ queens on $Q_{163}$. In centered coordinates, all queen coordinates are odd, every odd row and odd column is occupied exactly once, and the occupied rows, columns, and diagonals satisfy the conditions for a type A $1$-cover in the terminology of Ostergard and Weakley. A direct independent verification checks every one of the $163^2=26{,}569$ board squares and finds none uncovered. Hence $\gamma(Q_{163})\leq82$. The Finozhenok-Weakley lower bound $\gamma(Q_n)\geq\lceil n/2\rceil$, valid here, gives the matching inequality and therefore $\gamma(Q_{163})=82$. For this cover, the parameters defined by Ostergard and Weakley are $e=16$, $f=15$, and $u=24$; the complete difference- and sum-diagonal multisets are displayed in the paper. It consequently also supplies an explicit finite input to their amplification theorem for type A covers, giving $\gamma(Q_N)\leq(17/33)N+O(1)$. This last coefficient improves both the earlier type A coefficient $69/133$ and the subsequent general coefficient $101/195$ of Burger and Mynhardt. The coordinates and a complete standard-library Python verifier are included.

We study the stars at infinity in the Thurston boundary of Teichmüller space for the Thurston and Teichmüller metrics. For the Thurston metric, we prove that for every finite-type surface, the star of a projective measured lamination is exactly its zero set, extending a theorem of Liu-Shi from closed surfaces. Moreover, the based star agrees with the star. For the Teichmüller metric, we show that both the based star and the star of a projective measured lamination coincide with its two-step zero set. This set can be strictly larger than the zero set, disproving a conjecture of Karlsson.

We investigate geometric structures defined by 4-forms, encompassing important classes of geometric structures with torsion including almost Hermitian, $\mathrm{G}_2$, $\mathrm{Spin}(7)$, and almost quaternion/hyper Hermitian. Within this unified framework, the underlying 4-form naturally determines geometric data that define a string algebroid over the manifold. In particular, it gives rise to a canonical notion of instanton, thereby linking these geometries to gauge-theoretic structures in generalized geometry and heterotic string theory. We further prove that these data induce generalized Ricci-flat metrics. Finally, we examine the main examples in the literature and construct several explicit ones.

We propose a holonomy-based framework for discretizing curvature on graphs equipped with local symmetric positive-definite metrics. Each vertex carries a fibre metric \(g_i\), and each directed edge carries a reversible metric-compatible transport \(F_{ij}\). The ordered product around an oriented triangular loop \(\mathcal C\) gives a holonomy \(H_{\mathcal C}\), whose normalized logarithm \(\Omega_{\mathcal C}=-s_{\mathcal C}^{-1}\operatorname{Log}(H_{\mathcal C})\) is used as a finite-loop curvature observation. Thus the construction discretizes the geometric principle that infinitesimal holonomy is controlled by curvature, rather than treating holonomy as a heuristic feature. Since \(\Omega_{\mathcal C}\) lies in the \(g_i\)-orthogonal Lie algebra, it is not itself a velocity of an SPD metric. We therefore introduce two aggregation mechanisms: a commutator with a symmetric response matrix, producing symmetric Ricci-type metric responses, and an incidence-aware covariant divergence of curvature-induced edge fluxes, reflecting the relation between trace and covariant divergence. The resulting responses are locally orthogonal-gauge equivariant and can drive exponential updates that preserve positive definiteness. We also give a reversible metric-compatible parametrization of edge transports, allowing orthogonal edge factors, loop scales, weights, and response matrices to be learned while respecting the graph geometry. Known-geometry calibrations on the unit sphere test the holonomy–curvature relation, curvature preservation under nontrivial local metric representations, and the empirical recovery of edge transports from local observations.

Let $N=2^k-1$, and let $S_{t,k}$ consist of the pairs $0\le a,b<N$ such that $a+b\equiv t\pmod N$ and $\wt(a)+\wt(b)<k$. The Tu–Deng bound $|S_{t,k}|\le 2^{k-1}$ has recently been proved. In this paper, we give a combinatorial proof of the equality criterion. If the $k$-bit cyclic word of $t$ has $Z$ zeros and $g_1,\ldots,g_Z$ are the numbers of ones between successive zeros, then $$|S_{t,k}|=2^{k-1} \quad\Longleftrightarrow\quad g_i\ge Z-1\quad(1\le i\le Z).$$ Moreover, we recover the resulting enumeration of the equality parameters. Beyond equality, if $R\ge Z\ge2$, where $R$ is the number of ones, then every nonequality parameter satisfies $$2^{k-1}-|S_{t,k}|\ge 2^{R-Z+1},$$ and we classify all cases in which this bound is attained. For $R<Z$ we obtain a congruence for $|S_{t,k}|$ and a lower bound for the deficit in terms of the number of cyclic runs of ones.

In this note, we show that invertible composition operators on S(R) are never generalized hyperbolic and, when the symbol has a fixed point, the corresponding operator fails to have the positive shadowing property. We then establish sufficient conditions on the symbol under which the associated composition operator is positively topologically expansive. In particular, we obtain a complete characterization for affine symbols and prove expansivity for broad classes of odd-degree polynomial symbols, while showing that polynomial symbols of even degree cannot generate topologically expansive operators. Our results reveal a strong connection between the dynamics of the symbol and the linear dynamics of the induced composition operator, and provide new examples and obstructions for topological expansivity in locally convex spaces.

In this paper, we give a counterexample to the Bruzzo–Graña Otero conjecture on the curve semistability for Higgs bundles. Let $\Sigma$ be a very general smooth plane quintic curve and let $X=\Sigma^{(2)}$ be its second symmetric product. Starting from the tautological bundle $F=\mathcal{O}_\Sigma(1)^{[2]}$, we construct a rank four Higgs bundle $\mathcal{E}_s=(E_s,\theta_s)$ on $X$. Then, for every smooth projective curve $C$ and every morphism $f:C\to X$, the pullback Higgs bundle $f^*\mathcal{E}_s=(f^*E_s,f^*\theta_s)$ is semistable. However, on the other hand, $\mathrm{det}(E_s)\cong\mathcal{O}_X$ and $\int_Xc_2(E_s)=10$. Consequently, the discriminant of $E_s$ does not vanish, and $\mathcal{E}_s$ provides the desired counterexample.

We consider the $1$-equivariant harmonic map heat flow (HMHF) from $\mathbb R^2$ to $\mathbb S^2$ and the radial energy-critical nonlinear heat equation (NLH) in dimensions $d=3,4,5$. We prove that every finite-energy solution of (HMHF) and every $\dot H^1$-bounded solution of (NLH) has at most one bubble: every finite-time blow-up has exactly one bubble, whereas every global solution has either no bubble or one bubble at infinite time. Starting from the soliton resolution, we exclude bubble trees by modulation analysis, deriving a contradiction from the relative dynamics of the two innermost scales. This energy method does not rely on maximum principle and, in particular, applies without restriction on the bubble signs.

We propose and analyze an IMEX-SDIRK2 mean-reverting concurrent-correction scalar auxiliary variable (mr-ccSAV) method for the forced two-dimensional periodic Navier–Stokes equations in vorticity form. The viscous term is treated by Alexander's SDIRK2 method and advection explicitly. Each stage requires two elliptic solves with the same shifted Laplacian and the solution of either a cubic or a linear scalar algebraic equation. For initial vorticity in $\dot L^s(\Omega)$, $s>2$, a stage solution exists for every positive time step; uniqueness is established separately under an explicit small-step condition. The principal result is a direct, unconditional uniform-in-time enstrophy bound for arbitrary positive time steps. For persistently bounded forcing, this estimate is absorbing: the influence of the initial data decays, and the forcing contribution does not accumulate in time. Under additional regularity, uniformly bounded step sizes, and bounded neighboring step ratios, we also establish uniform-in-time $H^1$ and $H^2$ vorticity bounds without a small-step condition. For smooth solutions, the method converges optimally at second order. Numerical experiments confirm its accuracy, long-time robustness, and effectiveness of a companion embedded time-step selector.

We present a derivation of the integral kernel of a metaplectic representation operator in the Bargmann-Fock model based solely on its defining intertwining property with the Heisenberg representation. Expressing a metaplectic operator as an integral operator transforms the corresponding infinitesimal intertwining identities into a system of first-order partial differential equations satisfied by its kernel. We show that this system determines the kernel (up to a unit scalar) which is precisely the classical Gaussian associated with the symplectic transformation. A similar result is obtained for the Schrödinger model.

We give a local newform theory for the metaplectic group of rank 1 over a non-archimedean local field of characteristic 0 with odd residual characteristic. It was suggested by some results of Kohnen and Ueda on the classical theory of modular forms of half-integral weight.

A central question in higher index theory and operator algebras is whether the Roe algebra and the quasi-local algebra associated with a metric space of bounded geometry coincide, or at least have the same $K$-theory. In this paper, we focus on a sparse metric space $X$. We prove the following three main results: (1) For a block-diagonal operator $T$ with uniformly bounded block-rank, $T$ is quasi-local if and only if it is in the Roe algebra. (2) In general, we discover a ghost block-diagonal projection which is quasi-local but not in the Roe algebra. (3) For a sequence of expander graphs with sufficiently large girth, the inclusion of the uniform Roe algebra into the uniform quasi-local algebra induces a non-surjective map on their $K_0$-groups. This yields the first known $K$-theoretic distinction between the uniform Roe algebra and the uniform quasi-local algebra.

In this paper, we show how to adapt a derivation of Hartree equations due to Knowles and Pickl in the case where the one-particle hamiltonian is an Anderson-Hermite operator in dimension 1 and 2 or a formal Anderson-Hermite operator in dimension 3. By proving energy estimates for solution of the limit equation, uniform in the noise in bounded sets, we achieve a double limit. Namely, we show that we can commute the mean-field limit with a limit in noise.

In this paper, we propose an extension of the well-known Robinson implicit function theorem for generalized equations from the upper Lipschitzian case to the upper Hölderian case. Explicit exponents dependence between the generalized equation and its linearization is determined. Applications to ball constrained least squares problems, including linear least squares and separable nonlinear least squares, are studied. In particular, we establish that the solution mapping of ball constrained linear least squares under linear perturbation is locally upper Hölder continuous with exponent $1/3$, which is of independent interest. Ultilizing the upper Hölderian version of the implicit function theorem, we show the local upper Hölderian of the solution mapping of parametric ball constrained linear least squares.

We study the Marshall quotient \[ M(n)=M(\mathbb Z/n\mathbb Z) \] obtained from the ring of integers modulo $n$ by quotienting by the square classes of non-zero-divisors. Using elementary arithmetic of square classes modulo prime powers and the Chinese Remainder Theorem, we give an explicit description of these quotients and classify several of their structural properties. We determine when the quotient relation is arithmetically elementary, when $M(n)$ is hyperbolic, when it can be real reduced, and when it can be formally real. We also analyze the subset of invertible classes together with zero, proving exactly when it is a submultiring, when it is a hyperfield, and when it is hyperbolic. The results provide a finite family of test examples for questions connecting multirings, special hyperfields, real semigroups, and abstract quadratic-form theory.

We classify Satake diagrams for general linear and orthosymplectic non-twisted affine Lie superalgebras and prove that each of them generates a family of proper spherical subalgebras. Furthermore, we demonstrate that every such family contains subalgebras with matrix invariants, which are viewed as classical analogs of K-matrices solving supersymmetric Reflection equation.

Many nonlinear and variational problems are posed fibrewise over a probability space, whereas the dynamically relevant objects are measurable sections coupled by measure-preserving transformations. We study structural obstructions created by this fibre-to-section lifting. First, for a global continuously differentiable potential on a real Hilbert space, absence of influence between closed subspaces is reciprocal; for finite orthogonal decompositions this yields an additive decomposition over dependency components. Second, finite Koopman memory channels are identifiable exactly under a sharp orbit-separation condition. Resolving the classical Hilbert-space potentiality criterion channel by channel then gives a shifted-adjoint balance between forward and reverse memories. At the global finite-difference level, if a section map with arbitrary finite dynamical memory is the gradient of a global continuously differentiable potential, its genuinely active lags are invariant under time reversal. Thus globally variational memory is reversal-complete, and one-sided memory collapses to present-state locality. Continuous proximity operators inherit the same rigidity through their convex-potential representation; a discontinuous proximal selection shows that continuity cannot generally be removed. Finally, under pointwise nondegeneracy of two coherent branches, vanishing branchpasting residual is equivalent to asymptotic invariance of the branch labels. For finitely generated ergodic probability-preserving actions, strong ergodicity is therefore the exact threshold for relative compactness of all branch-pasting approximate-zero sequences. We also record a strong-compactness criterion for full measurable-selection lifts and a nonconvex Wasserstein minimizing-step illustration.

For every Gorenstein cubic $\mathbb Z$-order, we prove an identity equating Yun's zeta function, defined by counting finite-index submodules of the trace dual, with the explicit overorder zeta function introduced in our previous work on Beyond Endoscopy for $\mathrm{GL}_3$. This is posed as Conjecture A in an early draft of Deng-Espinosa and Lee subsequently proved the functional equation of the overorder zeta function, making the Deng–Espinosa isolation of the trivial representation fully unconditional. His argument computes the local factors explicitly. Our proof is independent of Lee's and does not evaluate the individual cubic overorder factors: it matches natural decompositions of the two sides and concludes by induction. As an application, we give a short, uniform evaluation of the local $\mathrm{GL}_3 $ Kloosterman Dirichlet series in the Poisson-summation argument of Deng–Espinosa. The direct local analysis of the Kloosterman series occupies nearly ninety pages in Deng–Espinosa while our treatment here replaces its case-by-case enumeration with a short uniform proof.

The Barzilai-Borwein (BB) method is an efficient gradient-based approach for unconstrained optimization that approximates spectral information of the Hessian matrix to capture curvature at low computational cost. In this paper, we extend the BB stepsize strategy to composite nonconvex optimization problems consisting of a smooth nonconvex term and a proper closed convex term, and propose an adaptive Barzilai-Borwein proximal gradient method for nonconvex optimization (AdaBBNC). The proposed method incorporates a flexible BB-based curvature estimate into the proximal gradient framework to enhance adaptability in nonconvex settings. Under mild assumptions, we establish that AdaBBNC achieves the optimal iteration complexity of $\mathcal{O}(\epsilon^{-2})$ for finding an $\epsilon$-stationary point, without requiring any prior knowledge of the global Lipschitz constant. Numerical experiments demonstrate the effectiveness and robustness of the proposed method. Compared with recent parameter-free and line-search-free adaptive proximal gradient methods, AdaBBNC exhibits more aggressive yet stable behavior in ill-conditioned optimization problems.

In this article, we prove two conjectures of Burness and Fusari [Timothy Burness and Marco Fusari, On derangements in simple permutation groups, Forum Math. Sigma 13 (2025)] concerning the powers and products of conjugacy classes of derangements in the symmetric and alternating groups: (1) We show that there exist two conjugacy classes $C$ and $D$ of derangements in $S_n$ such that $S_n=C^2\cup CD$, and (2) We show that there exists a conjugacy class $C$ of derangements in $A_n$ such that $C^2=A_n$, whenever $n\equiv 3\;(\text{mod}\;4)$. In fact, our result concerning the second conjecture holds in a considerably more general setting, which also answers affirmatively a question posed by Bertram [Edward Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory, Ser. A 12 (1972), 368-380] in a particular case. Moreover, we show that any conjugacy class $C$ of derangements in $S_n$ (resp. $A_n$) contains a pair of elements that generate $S_n$ or $A_n$ (resp. $A_n$), unless $C$ is the conjugacy class of fixed-point-free involutions.

The smallest and largest eigenvalues of the Gram matrix induced by the differential neural tangent kernel (DNTK) play a pivotal role in the analysis of over-parameterized PINNs trained by gradient type algorithms. However, a theoretical analysis of the extreme eigenvalues remains completely absent due to the challenge posed by the presence of multiple differential operators. In this work, we provide explicit lower and upper bounds for the extreme eigenvalues of the infinite DNTK matrix for the Poisson equation with the Dirichlet boundary condition for two-layer RePU neural networks without the bias term. The setting is fairly general with respect to the sampling points and input dimension \(d\): \(\delta\)-separated and additionally \(d\geq 3\) when deriving the lower bound of the smallest eigenvalue. These results extend that for the neural tangent kernel, and to the best of our knowledge, represent the first results on the spectrum of the DNTK.

Looking at some monoids and (semi)rings (natural numbers, integers and $p$-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function $f$ on such an algebra to behave like the operations of the algebra. The first way is to preserve congruences or stable preorders. The second way is to demand that, for any (recognizable) set $L$, a suitably chosen lattice (or Boolean algebra) generated by $L$ be closed under inverse images by the function $f$.