Papers

In this paper, we consider Nakao's problem in one space dimension, which couples a semilinear damped wave equation with a semilinear wave equation. For compactly supported energy data satisfying suitable integral sign conditions, we establish sharp lifespan estimates for all powers $p,q>1$. The estimates are classified according to whether the total integral of the initial speed in the wave equation is positive or zero.

This paper investigates the distributed time-varying optimization of stochastic multi-agent systems (SMASs) using only zero-order information. Unlike existing methods that directly couple gradient estimation and optimization updates on a single time scale, this paper constructs a novel stochastic singular perturbation framework by introducing auxiliary fast systems. The proposed scheme naturally forms a slow-fast coupling structure: by introducing auxiliary variables and constructing fast subsystems to generate smooth gradient estimates, while the agent's state evolution, as the slow subsystem, performs distributed optimization and consensus. The convergence of the proposed scheme is analyzed using stochastic singular perturbation techniques and stochastic Lyapunov theory. The results show that the fast subsystem converges rapidly to the instantaneous stochastic gradient estimates, while the slow subsystem achieves practically fixed-time consensus (Pfxc) in probability and asymptotically bounded tracks the time-varying optimal trajectory. Furthermore, this paper establishes explicit bounds to characterize the effects of parameters, stochastic disturbances, and the properties of the objective function on tracking performance. Finally, the theoretical results are validated through numerical simulations.

Hales-Jewett theorem is well known. We prove a primitive recursive bound, in fact an $\clE^5$ one, for the density version. In preparation we deal with what was the main induction step in the old proof of Hales-Jewett as a result in its own right with better bounds. Also in improving our bounds.

Local SGD, also known as Federated Averaging, is a widely used distributed optimization algorithm. Although Local SGD often outperforms alternatives such as Mini-batch SGD in practice, theory still only partially explains when and why local updates help under realistic data heterogeneity. Recent work by [Patel et al., 2025] shows that a bounded second-order heterogeneity assumption captures the efficiency of Local SGD for strongly convex objectives, and conjectures that the same principle extends to the general convex setting. In this paper, we prove this conjecture by establishing an improved convergence guarantee for Local SGD on general convex objectives under bounded second-order heterogeneity. We also improve the best-known lower bounds for Local SGD in this setting, showing that our upper bounds are nearly tight. Together, these results provide a sharper, more fine-grained convergence theory for Local SGD. As a further application of our techniques, we provide a lower bound for serial SGD with replacement, showing how second-order heterogeneity captures the impact of rare high-curvature clients.

We show anchored versions of the Nash inequality for discrete non-local divergence-form operators with degenerate weights. They allow to control the $L^{2}$-norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide on-diagonal heat kernel upper bounds for a class of random conductance models with degenerate jump rates allowing long-range jumps. The results are established on a class of graphs including the integer lattice and possibly correlated supercritical percolation clusters.

This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian curvature equations. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $k$-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations, and by establishing a constant rank theorem, we prove that the resulting $k$-convex hypersurfaces are strictly convex.

We consider a planar ODE system from an important class of models for gene regulatory dynamics. The system depends singularly on the steepness parameters $0<\varepsilon_1,\varepsilon_2 \ll 1$ and converges to a piecewise-smooth system as these parameters tend to zero. Unlike previous studies, we do not assume that $\varepsilon_1 = \varepsilon_2$. As a consequence, the dynamics when $(\varepsilon_1, \varepsilon_2) \to (0,0)$ depends upon how the limit is taken. Using a preliminary blow-up in parameter space, we identify three distinct singular limits. We perform a two-parameter bifurcation analysis in each case, and apply multiple geometric blow-ups in variable and parameter space to determine the bifurcation structure and the associated global dynamics. Bogdanov-Takens bifurcations are revealed in two of three cases, and in one case in particular, a regularised visible-invisible two-fold singularity is shown to organise the unfolding of singular bifurcations in the vicinity of canards. Our results show that the qualitative dynamics and overall sensitivity of the system to parameter variation depends on the relative size of the steepness parameters. More generally, the analytical framework developed herein provides a systematic approach to singular perturbation problems with multiple independent small parameters that should apply well beyond gene regulatory network models.

We classify all 141 finite-volume hyperbolic Coxeter five-dimensional polytopes with eight facets, of which 125 are noncompact. Using maximal-cusp density and a noncompact analog of Bogachev-Douba-Raimbault's argument, we construct infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4, 5, 6, 7, and 9, with the number of commensurability classes growing at least exponentially in volume.

The acyclic dichromatic number of an oriented graph is the minimum size of a vertex-partition such that the digraphs induced by any single part are acyclic, and the oriented bipartite graphs between any two parts are acyclic too. We characterize the subtournaments that must appear in every tournament with sufficiently large acyclic dichromatic number, thereby confirming a conjecture of Bang-Jensen, Picasarri-Arrieta, and Yeo and prove that acyclic dichromatic number satisfies a local to global property.

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce an aperiodic tile set whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is the “Sturmian lattices”, an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction. A classification of Sturmian lattices and complete proofs are given in the full version.

We study twisted Yangians associated with the split symmetric pairs of types BI, CI and DI. We introduce a new presentation of these algebras, which we call the transposed presentation, governed by a twisted reflection equation that interacts naturally with the Gaussian decomposition of the generating matrix. Working entirely within the $R$-matrix presentation, we derive Drinfeld-type current presentations of the special twisted Yangian $SY^{\mathrm{tw}}(\mathfrak{g}_N)$ and of the extended twisted Yangian $X^{\mathrm{tw}}(\mathfrak{g}_N)$, in which the Serre relations are stated in a closed current form. Extracting coefficients recovers the Drinfeld presentation of Lu, Wang and Zhang, in the normalisation of Lu. As a consequence, we establish an explicit isomorphism between the $R$-matrix and Drinfeld presentations conjectured by Lu, Wang and Zhang. As a byproduct, we obtain a tensor product decomposition of $X^{\mathrm{tw}}(\mathfrak{g}_N)$ into the twisted Yangian in the Drinfeld presentation and a polynomial ring in countably many central variables. We also obtain Poincaré-Birkhoff-Witt bases in the Drinfeld generators and the coideal coproduct on the low Drinfeld modes at all but the last node.

We propose and analyze a novel numerical framework for the non-Markovian stochastic Schrödinger equation (NMSSE) based on a low-rank approximation of the bath correlation functions. By decomposing the memory kernel into a finite-dimensional representation, we derive a truncated system of hierarchical equations that effectively balances computational tractability with physical fidelity. A rigorous convergence analysis is established for the hierarchical framework under mild assumptions. We demonstrate that our formulation serves as a mathematical generalization of the Hierarchy of Pure States (HOPS), encompassing it as a special case while offering a more flexible representation of non-Markovian effects. Numerical experiments across several benchmark models are presented to illustrate the validity and efficacy of the proposed method.

This is the second part of a two-paper sequence establishing the global existence of $3D$ relativistic Vlasov-Maxwell system (RVM) for arbitrarily large smooth localized initial data with cylindrical symmetry. In Part II, independently of the results in Part I, we establish two sets of pointwise estimates for the localized electromagnetic field. This is achieved by exploiting cylindrical symmetry, conservation laws, smoothing effects, and structural observations of the electromagnetic field, including its null structure, double null structure, and the magnetic field dichotomy. The first set provides a uniform upper bound for the localized electromagnetic field. The second set offers an upper bound that depends on the distance from the z-axis, providing better control when the electromagnetic field is located further away from the $z$-axis. These two sets of estimates are fundamental to the iterative smoothing scheme (ISS), which is originated and inspired by the work of Klainerman-Staffilani(2002). These two sets were treated as black boxes in Part I. A defining feature of these estimates is that they are assessed concerning the magnitude of the oscillation phase over characteristic time, effectively quantifying the smoothing effect. It follows from these estimates that cases without any smoothing effect are error terms. Addressing this constitutes the initial challenge in the bootstrap argument.

We propose an operator-split Bayesian learning framework for second-order uniformly elliptic Dirichlet problems with unequal numbers of interior and boundary observations. The data consist of noisy measurements of the source in the domain and noisy measurements of the boundary values. Independent Bayesian neural-network (BNN) priors are assigned to these two quantities, and the resulting product posterior is pushed forward through the elliptic solution operator. We prove that the posterior induced by this construction contracts around the true solution. The contraction radius separates a domain contribution, governed by the second-order elliptic operator, from a boundary contribution, governed by the intrinsic dimension of the boundary. Together with the minimax lower bound of \cite{ZhaoLu2026}, this yields a near-minimax upper bound up to logarithmic factors. Our numerical experiments illustrate the propagation of source and boundary uncertainty and the effects of unequal sampling budgets on the posterior reconstruction.

We study strongly indefinite nonlinear Dirac functionals in the lowest partial-wave channel with radial coefficients in $\mathbb R^3$. The symmetric model has the exact power nonlinearity $F_0(x,\psi)=b(|x|)|\psi|^p/p$, where $2<p<3$ and the radial profile $b$ is bounded, continuous, and uniformly positive. Compactness in this finite-angular-mode spinor channel and radial approximation-number estimates give quadratic growth of the symmetric minimax levels. For a non-even localized perturbation of order $1<\tau<p/2$, we impose a semiconvexity condition whose negative curvature is strictly smaller than the spectral gap. Together with the negative quadratic part and the convex exact-power core, this makes every negative spectral fiber uniformly strongly concave. Maximizing along that fiber reduces the problem to a $C^1$ path on the positive spectral space. The polynomial Chambers–Ghoussoub–Bolle deformation theorem then yields infinitely many high-energy critical points of the restricted partial-wave functional; under the channel-invariance hypothesis these are weak solutions of the full Dirac equation. Convex perturbing primitives are covered without a smallness restriction on their amplitude. The same reduction applies after an even Hermitian quadratic term is absorbed into a renormalized Dirac operator, provided the renormalized operator has a gap at zero and the remaining perturbation satisfies the corresponding semiconvexity bound.

A variable-to-variable (V2V) length code parses a source sequence into phrases of variable length and maps each phrase to a binary codeword of, generally, a different random length. After encoding $n$ phrases, the realized compression ratio $R_n=\Lambda_n/\Sigma_n$ – total codeword length over total source-symbol count – is the finite-sample counterpart of the code's asymptotic rate $\rho$, to which it converges only as $n\to\infty$. This paper first derives exact formulas for all integer moments of $R_n$ for a given discrete memoryless source (DMS). Specifically, we obtain a closed-form formula for every moment $\E\{R_n^k\}$ as a one-dimensional integral involving only single-phrase moment generating functions of the pair $(L,\ell)$ – the phrase length, in source symbols, and codeword length, in bits. From these moments we derive an Edgeworth approximation to the cumulative distribution function (CDF) of $R_n$ that is substantially more accurate than the central limit theorem (CLT) approximation. Using the Laplace method of integration, we also derive explicit closed-form formulas for the bias constant $C=\lim_{n\to\infty}n(\E\{R_n\}-\rho)$ and for the variance constant $\lim_{n\to\infty}n\cdot\Var\{R_n\}$. The analysis extends to Markov sources via state-indexed matrices with a redundancy formula obtained in closed form. On the coding-theoretic side, we cast V2V length codes as finite-state encoders and apply a generalized Kraft inequality for a compression-rate lower bound, and give a structural decomposition of the bias coefficient that separates cleanly across variable-to-fixed (V2F) length codes, fixed-to-variable (F2V) length codes, and V2V length codes. Applied to the Khodak code of Bugeaud, Drmota, and Szpankowski, this decomposition shows that its improved performance is reflected in its smaller bias constant.

In a paper by Novi Inverardi and Tagliani from 2024 it is claimed that the log-normal density has maximal entropy in the class of densities having the same moments as the log-normal density. We disprove the claim by finding a family of densities with larger entropy.

Let $f\colon \mathbb{R}\to\mathbb{R}$ be a continuous function. We define the graph welding associated with $f$ as the homeomorphism \[ \varphi = G^{-1}\circ F\colon \mathbb{R}\to\mathbb{R}. \] Here, $F(x)=x+if(x)$ parametrizes the graph of $f$, and $G$ is a conformal mapping from the upper half-plane $\mathbb{H}$ onto one of the two domains bounded by the graph of $f$, admitting a continuous extension to $\mathbb{R}$. In this paper, we investigate how the regularity of the graph function $f$ influences the analytic properties of the associated graph welding $\varphi$. In particular, under certain assumptions that $f$ within Zygmund, $\mathrm{BMO}$, $\mathrm{VMO}$, and Hardy spaces, we establish results on absolute continuity, quasisymmetry, symmetry, strong quasisymmetry, strong symmetry, and the Weil–Petersson property of the associated graph welding $\varphi$. These results clarify the interplay between the regularity of graph functions, the geometry of graph curves, and the boundary behavior of conformal mappings.

When $X$ is a locally compact Euclidean building, the isometry group of $X$ acts by homeomorphisms on the space of $k$-simplices in the visual boundary of $X$. We consider perturbations of these actions for discrete groups of isometries acting with compact quotient on $X$, showing that all small enough perturbations are semi-conjugate to the original action. This proves in particular that, when $Q$ is any parabolic subgroup in a semisimple $p$-adic Lie group $G$, the induced action of a cocompact lattice in $G$ has a topologically stable action on $G/Q$.

A diamond is a graph obtained from \(K_4\) by removing an edge, and a dart is a graph obtained from a diamond by adding a pendant edge to a vertex of degree 3. We prove that every $\{P_6, \text{dart}, K_4\}$-free graph is 6-colorable. This improves the previous bound of 7 due to Hong and Xu \cite{HongXu2025} and resolves their open question on the optimality of the bound. Our result also extends a theorem of Karthick and Mishra~\cite{KarthickMishra2018}, who proved 6-colorability for the class of \(\{P_6, \text{diamond}, K_4\}\)-free graphs.

Identification (ID) coding, introduced by Ahlswede and Dueck, extends Shannon's classical communication paradigm by replacing message reconstruction with hypothesis testing. Instead of decoding the transmitted message, the receiver only decides whether a particular message was sent. A fundamental result of ID theory is the double-exponential growth in the number of identifiable messages with respect to (w.r.t.) the blocklength. This scaling behavior enables fundamentally new communication architectures for large-scale distributed systems and forms a key building block of post-Shannon communication. While ID cannot replace classical communication in general, it is particularly well-suited for scenarios in which full message reconstruction is unnecessary, such as monitoring, alarming, and control systems. In this survey, we review the theoretical foundations of ID coding and discuss emerging communication architectures and application domains based on this paradigm. Particular emphasis is placed on practical use cases, including monitoring systems, special-purpose data storage, joint identification and sensing (JIDAS), semantic communications, mobile-network control systems and networked consensus testing systems. We further highlight recent system concepts, industrial perspectives, and implementation examples that illustrate how ID-based principles can be realized in practical communication systems.

Physics-informed neural networks (PINNs) offer a flexible framework for solving partial differential equations (PDEs), but training can become computationally expensive when a large number of collocation points are required to accurately enforce the governing equations. To alleviate this cost, we introduce multigrid-based parameter-updated PINNs (MPU-PINNs), a coarse-to-fine training strategy that progressively increases the number of training points throughout the learning process. The proposed approach begins by training a neural network on a coarse set of collocation points and then transfers the learned parameters to successively finer levels. This initialization strategy enables the network to capture the global features of the solution at a relatively low computational cost before refining local details with additional training points. To further improve performance for high-frequency problems, we incorporate a scaling technique that mitigates the effects of spectral bias during training. We evaluate MPU-PINNs on several benchmark PDEs, including two- and three-dimensional Poisson equations, a convection-diffusion-reaction equation, and the Helmholtz equation. Numerical experiments indicate that MPU-PINNs greatly reduce training time while achieving accuracy comparable to that of conventional PINNs and other representative variants such as SA-PINNs and XPINNs. The results further suggest that the proposed coarse-to-fine learning strategy substantially decreases the optimization effort required at finer levels. Overall, MPU-PINNs provide an efficient single-network training framework that enhances the computational efficiency and scalability of PINNs for a broad range of PDE problems.

Bosonic invertible and symmetry-protected topological (SPT) phases are well-known to be described by ordinary cohomology groups in low dimensions, but `beyond-cohomology' phases appear in higher dimensions. We make a systematic study of them, and find that the first major non-triviality is a mod-3 phenomenon, and not a mod-2 phenomenon as in the case of fermionic phases. We also point out that this is a dual manifestation of the classic question of Steenrod, namely the issue of the existence of homology cycles without manifold representatives. Thom developed the theory of cobordisms to answer this question, and we explain how the same analysis leads to Dijkgraaf-Witten phases which are nontrivial on general simplicial complexes but become trivial on manifolds.

Topology optimisation (TO) often requires repeated finite element analysis and sensitivity-based material updates, which can be costly when multiple candidate designs are needed under varying physical and design conditions. Generative TO offers a route to rapid design exploration, but existing models may rely on adversarial training, long reverse-diffusion sampling, or external guidance to maintain structural feasibility and physical consistency. This study develops a flow matching-based topology optimisation (FMTO) framework for conditional topology generation. Linear FMTO is first formulated as an endpoint-based baseline by interpolating between a Gaussian source field and the BESO reference topology. To introduce mechanically meaningful intermediate states, a trajectory-aware FMTO formulation is proposed, where volume-fraction-indexed BESO states are used to construct the probability path and target velocity field. This incorporates physics-guided optimisation history into generative flow learning without adding inference-time optimisation. A path–velocity mismatch analysis explains why moderate trajectory weighting can improve generation stability, whereas excessive guidance may over-constrain the learned transport. Numerical examples show that FMTO generates diverse topology candidates with improved compliance-related performance, volume-fraction satisfaction, topology fidelity, and substantially fewer sampling steps than a diffusion-based baseline. Under limited training data, trajectory-aware FMTO achieves the best overall performance with a moderate trajectory weight. Studies on trajectory-anchor density and three-dimensional topology generation further demonstrate the influence of path design and the applicability of the proposed framework beyond two-dimensional problems.

Let $X$ be a normed linear space. A map $P: X \rightarrow X$ is called idempotent if $P^2 = P$. In this paper, we introduce some new classes of idempotent maps (linearity is not assumed) on $X$ and study their properties. A well-known example of an idempotent map is the metric projection $P_K$ onto a Chebyshev subset $K$ of $X$. We provide some necessary conditions for the map $I- P_K$ to be idempotent, where $I$ denotes the identity operator on $X$.

Let F be a field and V be a 2n-dimensional vector space over F. In a previous article, we have proved that if F has more than 2n-2 elements then the greatest possible dimension for an affine space of symplectic forms on V is n(n-1). Here, under the same cardinality assumption we study the spaces that have the critical dimension n(n-1). In particular, if the characteristic of F is not 2 the classification of these spaces up to congruence is reduced to: (1) the classification of nonisotropic quadratic forms over F, up to equivalence and multiplication with a nonzero scalar; (2) the classification of nonisotropic Hermitian forms over all quadratic extensions of F, up to equivalence and multiplication by $-1$. In particular, for quadratically closed fields it is shown that there is exactly one solution up to congruence.

With generalized central binomial coefficients $ \binom{2x}{x}:=\frac{\Gamma(2x+1)}{[\Gamma(x+1)]^2}$ defined through Euler's gamma function, we represent deformed Apéry-like series \[ \mathscr A_{s,n}:=\sum_{k=1}^\infty\left.\!\frac{\partial^n}{\partial x^n}\frac{1}{x^s\binom{2x}{x}}\right|_{x=k} \] by multiple Clausen values (MCVs), which belong to a special class of cyclotomic multiple zeta values (CMZVs) at level $3$. For example, exploiting provable algebraic relations among MCVs, we show that \[\mathscr A_{1,5}=-\frac{9[495L(\chi_{-3},6)-30\pi^{2}L(\chi_{-3},4)-2\pi^{4}L(\chi_{-3},2)]}{4}\]and\[\mathscr A_{4,4}=\frac{352\zeta_{5,3}}{15}+\frac{752537\pi^{8}}{10206000},\]where $ L(\chi_{-3},s):=\sum_{n=0}^\infty\left[(3n+1)^{-s}-(3n+2)^{-s}\right]$ and $\zeta_{5,3}:=\sum_{m>n>0}m^{-5}n^{-3}$.

We establish a sharp parity dichotomy for the equivariant $\mathbb Q$-sliceness and Klein amphichirality of odd-stranded Turk's head knots. When the twisting parameter $q$ is odd, we construct a commuting pair of ambient involutions by lifting explicit symmetries of the signed, arrowed Gauss diagram through the associated abstract link diagram and supporting sphere. This proves that the knots are Klein amphichiral and hence equivariantly $\mathbb Q$-slice. When $q$ is even, we prove that the knots are not equivariantly $\mathbb Q$-slice with respect to any strong inversion. The obstruction is the equivariant Fox-Milnor square condition of Di Prisa-Şavk: we show that the Alexander polynomial is not a Laurent square. The even-$q$ argument combines a Burau square factorization arising from Garside conjugacy with a mod-$2$ Seifert-matrix computation.

We construct cluster structures on the strata $\mathring{S}_w$ of a stratification of the symmetric space $SL_n/SO_n$ over $\mathbb{C}$. To accomplish this, we study foldings of upper cluster algebras and show that the cluster structures on $SL_n/SO_n$ can be obtained from those on $SL_n$ via folding. As a corollary, we show that these cluster structures are compatible with the De Concini Poisson structures, and we construct cluster structures on the variety of symmetric matrices $\text{Sym}_n$.

Logical Kochen-Specker (KS) contextuality is widely regarded as an intrinsic property of specially constructed measurement configurations. We show instead that it can emerge from KS-colorable vector sets through a constructive procedure we call the Hamilton extension. Defined for four-dimensional vector sets, the Hamilton extension associates each real vector with a measurement context while inducing additional measurement contexts among Hamilton-extended children of distinct parent vectors. These emergent contexts fundamentally alter the compatibility structure, transforming KS-colorable configurations into KS-uncolorable ones and recovering logical contextuality lost under apex-vertex augmentation. We establish a sharp and optimal threshold – the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions. Thus, six vectors constitute the smallest parent set capable of generating KS contradiction through this mechanism. Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.