Papers

We classify freeness for finite central arrangements whose underlying matroids are regular. Let $\mathcal A$ be such an arrangement over an arbitrary field, and put $M=M(\mathcal A)$. Then $\mathcal A$ is free if and only if $M$ is supersolvable; equivalently, $M$ admits a nice partition; equivalently, $M$ is the cycle matroid of a chordal simple graph. Thus, for arrangements with regular underlying matroids, freeness has a complete combinatorial classification independent of the base field. We use Seymour's decomposition theorem for regular matroids to prove that freeness forces supersolvability. We also characterize nice partitions of finite simple binary matroids: a partition is nice if and only if it is independent and no line is contained in a single block. Consequently, a finite loopless binary matroid admits a nice partition if and only if it is simple and supersolvable.

Let $(X,\omega)$ be a compact Kähler manifold and let $V\subset X$ be a closed complex submanifold. Coman-Guedj-Zeriahi proposed the problem: is every $\omega|_V$-plurisubharmonic function on $V$ the restriction of an $\omega$-plurisubharmonic function on $X$? In this paper, we solve this problem affirmatively, even for a compact Hermitian manifold.

We introduce the notions of anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras, which arise as nearly associative Levi-Civita products associated with pseudo-Euclidean Lie and Jordan algebras, respectively. We establish a correspondence between these classes of algebras and the Levi-Civita products of their associated Lie and Jordan structures. For anti-left-invariant pseudo-Euclidean nearly associative algebras, we prove that they are nilpotent of index at most five and characterize them as Jacobi–Jordan-admissible nearly associative algebras. We further show that the associated pseudo-Euclidean Jacobi–Jordan algebras are cyclic. Motivated by the classical double extension of Medina and Revoy, we introduce a double extension procedure for this class of algebras and prove that every anti-left-invariant pseudo-Euclidean nearly associative algebra can be obtained from a trivial pseudo-Euclidean algebra by a finite sequence of such extensions. For left-invariant pseudo-Euclidean nearly associative algebras, we prove that the associated pseudo-Euclidean Lie algebras are two-step solvable and cyclic. We then develop block, planar, and linear double extensions and show that every left-invariant pseudo-Euclidean nearly associative algebra can be recursively constructed from a quadratic commutative associative algebra by means of block double extensions. Moreover, we prove that over the field of real numbers, every such algebra can be recursively constructed from a quadratic commutative associative algebra using planar double extensions. These recursive constructions provide a unified framework for describing and classifying pseudo-Euclidean nearly associative algebras in both the anti-left-invariant and left-invariant settings.

Let $\mathbb F_q$ be a finite field and consider quadratic polynomials $f(X)=X^2+bX+c\in\mathbb F_q[X]$ with primitive constant term $c$. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive $c\in\mathbb F_q$, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients $b$ for which $X^2+bX+c$ is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the $(q+1)$-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable $B$, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over $A=\mathbb F_q[C]/(\Phi_{q-1}(C))$. We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of $A[B]$. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.

Let $D(N)$ denote the largest cardinality of a subset of $\{1,\ldots,N\}$ containing no nonzero square difference. While a construction certifying $D(N)\geq (1-o(1))N^{1/2}$ is almost trivial, Erdős conjectured that this bound is sharp up to polylogarithmic factors. This was disproved by Sárközy and later again by Ruzsa, who found an elegant construction showing that $D(N)\geq c\cdot N^{0.733077\dots}$, with an absolute constant $c>0$. His approach was subsequently refined, leading to the previously best known lower bound with exponent $0.7334117\dots$ due to Beigel-Gasarch and, independently, Lewko. However, in the original paper Ruzsa observed that $3/4$ seems to be the natural barrier of his approach. In this paper we develop a new construction leading to the lower bound \[ \liminf_{N\to\infty}\frac{\log D(N)}{\log N} \geq \alpha_*:= 0.7527964558\ldots; \] thus crossing the natural exponent-$3/4$ barrier of Ruzsa's method. The value $0.7527964558\ldots$ arises from a simple optimisation problem and appears to be the limit of the new approach.

Generalized measurements can be implemented projectively after enlarging the Hilbert space, but this dilation changes the available local dimension. We construct a Bell functional with rational coefficients that separates the two measurement models at local dimension two. An explicit three-outcome qubit positive-operator-valued measure with rational matrix entries attains $2\sqrt2+1/100$. On the other hand, all qubit-projective strategies are bounded by $2\sqrt2+\sqrt5/250+\sqrt2/32400$, giving a fully analytic certified gap greater than $1/1000$. To our knowledge, this is the first fully analytic Bell-functional separation between qubit POVMs and qubit projective measurements over arbitrary shared two-qubit states. Lean certificate for the separation theorem is provided for completeness. Separately, an exact level-3 noncommutative sum-of-squares certificate proves that the explicit qubit strategy attains the unrestricted finite-dimensional tensor-product quantum optimum.

We study honest adaptive confidence sets for the regression function in random-design binary regression under $L^2(dx)$ loss. Assuming only known bounds $0<c\leq g\leq C<\infty$ on the unknown design density, we construct asymptotically honest, rate-adaptive confidence sets without requiring $g$ to be smooth. Full adaptation is possible when the range of regression-function smoothness spans at most a factor of two. Over wider smoothness ranges, adaptation is achieved on the usual separated classes at the corresponding testing rates $n^{-2s/(4s+d)}$. A lower bound under the uniform design shows that these separation rates are rate-optimal. This answers a question raised by Mukherjee and Sen (2018).

In this paper, we propose high-order weighted essentially non-oscillatory (WENO)-based semi-implicit Newton-type Gauss-Seidel Lax-Friedrichs fast sweeping methods for solving the generalized Eikonal equation arising in wave propagation through a moving fluid. Building upon the Newton-type framework of Li and Qian (2020), which updates the solution line-wise using Newton's method with a tridiagonal and strictly diagonally dominant Jacobian, we extend the local solver to fifth-, seventh-, and ninth-order accuracy by incorporating high-order WENO approximations of the spatial derivatives into the numerical Hamiltonian. Three alternating sweeping strategies, namely column-wise, row-wise, and column-row-wise, are considered within the Gauss-Seidel iteration framework. Numerical examples in both two and three spatial dimensions demonstrate the efficiency and accuracy of the proposed schemes.

Cover originally proposed a game in which Bob wrote two numbers on pieces of paper and allowed Alice to pick one and look at it. Cover then showed, as Blackwell is thought to have shown previously, that Alice has a strategy involving the selection of a random number which enables her to guess correctly with a probability greater than 0.5 which of the two numbers was larger. Central to Covers reasoning was that Alice could have picked either number to look at with equal probability. In this paper, we investigate the situation in which Bob obtains the two numbers as the result of an experiment based on a probability distribution, such as the spin of a roulette wheel. We then modify this to envision Alice and Bob conducting separate experiments based on probability distributions. Alice conducts one trial of her experiment and uses Covers strategy to guess whether the result from her experiment will be larger or smaller than the result from Bobs experiment whenever he chooses to conduct a single trial of it. The following two results are among those in the paper. (1) If the two numbers are observations of the same experiment, Alice merely has to observe one trial and apply the Cover strategy to have a probability greater than 0.5 of guessing the larger value, even if the second trial has not yet been conducted (2) There are many nontrivial examples where Alice and Bob conduct different experiments, and Alice can observe one trial of her experiment and apply the Cover strategy to have a probability greater than 0.5 of guessing the larger value, even if many of the specifics of the other experiment have not yet been determined and a trial of it has not yet been conducted.

The deformation algorithm based on conservation laws can lift (1+1)-dimensional integrable systems to higher-dimensional counterparts while preserving Lax integrability, yet its generalization to (2+1)-dimensional models has remained an open problem. This paper establishes a unified deformation framework for two (2+1)-dimensional integrable equations: the anisotropic Kadomtsev-Petviashvili (KP) equation and the isotropic Nizhnik-Novikov-Veselov (NNV) equation. By introducing a set of mutually commuting field-dependent deformation operators, we systematically construct infinite families of ($m+3$)-dimensional integrable KP and NNV hierarchies, derive their closed-form Lax pairs, and rigorously verify integrability via the vanishing commutator condition of Lax operators. For each high-dimensional hierarchy, concrete finite-dimensional master systems are obtained by truncating auxiliary spatial variables: a (3+1)-dimensional KP system and a (4+1)-dimensional generalized NNV system. Further symmetry reductions recover the original (2+1)-dimensional KP and NNV equations, and more importantly produce two distinct Harry-Dym (HD)-type reciprocal integrable systems. The KP reduction yields an anisotropic (2+1)-dimensional HD model, while the NNV reduction generates the first spatially isotropic two-space-dimensional HD system reported so far, filling a notable gap in existing literature. Parallel comparison of the KP and NNV branches reveals that the spatial symmetry of the original two-dimensional parent equation directly governs the symmetry properties of its high-dimensional deformations and HD dual subsystems. Our work not only extends the conservation-law deformation conjecture beyond (1+1)-dimensions to accommodate both strong Lax and weak Lax structures, but also provides a universal route to construct reciprocal links for multi-dimensional integrable systems.

We study the orbit decomposition on $\C^n$ generated by diagonal holomorphic $\C^k$-actions in the higher-rank setting of the classical Poincaré–Siegel dichotomy for linear vector fields. The coordinate stratification determines the dimensions and isotropy groups of the leaves and, under a maximal-rank condition, gives a precise description of the orbit structure on every coordinate stratum. For configurations in the Poincaré domain, a separating real direction provides a global conical model of the punctured orbit foliation by its traces on Euclidean spheres. We construct the corresponding radially reparametrized action on a sphere, describe its leaves as homogeneous spaces, and prove that orbit closures are constrained by coordinate supports. In particular, a limit point cannot acquire a new nonzero coordinate, although nonclosed trace leaves may also accumulate within a fixed support stratum. We show that the geometry of the weight configuration determines the complex dimensions of the leaves, whereas the arithmetic of their effective isotropy groups determines their diffeomorphism types. This gives rise to a threshold phenomenon across the coordinate stratification: the diffeomorphism type of the leaves is rigid in the low- and high-dimensional regimes, but becomes arithmetically unstable in the intermediate range $k<|I|<2k$. Finally, we show that these singular foliations admit canonical local transverse holomorphic structures in the spirit of Haefliger's transverse geometry for regular foliations. These structures determine intrinsic transverse pseudogroups, yielding a well-defined local transverse holomorphic geometry for the orbit foliation.

We consider stochastic heat equation (SHE) defined on 1-d torus $\mathbb{T}$ of the form $$\partial_t u=\Delta u+g(u)\dot{W},$$where $\dot{W}$ is a space-time white noise and $g$ is a real-valued function which is uniformly elliptic (i.e., $|g|$ is uniformly bounded away from 0), and is globally $\beta$-Holder continuous for some $\beta\in(0,1)$. We prove that weak uniqueness holds as long as $\beta>\frac{2}{3}$. The same uniqueness holds for vector-valued solutions where the coefficient $G$ has the same dimension as the white noise. Previously, uniqueness of solutions to the SHE with Holder diffusion coefficient was only established for $\beta>\frac{3}{4}$ via a Yamada Watanabe argument by Mytnik and Perkins (arxiv:0809.0248) without assuming $g$ is nonzero. And when $\beta<\frac{3}{4}$, Mueller, Mytnik and Perkins (arXiv:1201.2767) constructed a non-unique SPDE example satisfying $g(0)=0$. A later generalized coupling argument for nondegenerate $g$ also stopped at the same threshold $\frac{3}{4}$. Our result shows that uniform ellipticity of $g$ restores uniqueness to SHEs in the Holder regime where the same SHE with non-elliptic $g$ and the same Holder regularity are often non-unique in law. This constitutes the first general class of SHE weak uniqueness results in the $\beta\in(\frac{2}{3},\frac{3}{4}]$ regime.

How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously published general upper bound improves on the exponential-in-the-square estimate obtained from crossing-number enumeration; we replace it with a factorial-scale upper bound, which is optimal at the level of growth order. The proof turns polygonal self-intersection into a sparse real-algebraic chamber problem in only linearly many dimensions, while a complementary braid construction supplies factorially many distinct knots. The result creates a direct bridge between knot topology, real algebraic geometry, fewnomial structure, and permutation combinatorics.

We construct multiple solutions of the generalized self-dual Abelian Chern-Simons-Higgs equation in a two dimensional flat torus by the topological degree method.

In this paper, we study the distribution of simplices in both discrete and continuous settings. Let $q$ be an odd prime power, let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^d$, and let $2\leq k\leq d-1$. We prove that every set $E\subset\mathbb F_q^d$ with \[ |E|\geq C_{d,k}q^{\beta_{d,k}}, \qquad \beta_{d,k}= \begin{cases} \displaystyle \frac{d+k}{2}-\frac{k-1}{k+1}, & d-k\ \text{even},\\[2mm] \displaystyle \frac{d+k-1}{2}, & d-k\ \text{odd}, \end{cases} \] determines a positive proportion of all ordered nondegenerate $k$-simplex congruence classes. This improves the previous exponent due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2017), and is sharp when $d-k$ is odd. In the Euclidean setting, we prove that if $E\subset\mathbb R^d$ is compact and $\dim_{\mathrm H}(E)>d-1$, then there exists a Frostman probability measure $\mu$, supported on $E$, and a set of pins of full $\mu$-measure such that the pinned distance configuration measure for labeled $(d-1)$-simplices is absolutely continuous at every such pin. We also show that the same conclusion holds when $E\subset\mathbb R^d$ is a compact Salem set with $\dim_{\mathrm H}(E)>k$.

This paper proposes a novel block-regularized splitting (BRS) framework for the efficient solution of indefinite least-squares (ILS) problems. Based on a block-wise regularization strategy incorporated into a matrix splitting scheme, we develop a BRS iterative method together with an effective BRS preconditioner. The convergence of the proposed iterative method is rigorously analyzed, and spectral bounds for the BRS-preconditioned matrix are established. To further enhance computational performance, we introduce a relaxed BRS (RBRS) preconditioner, which provides improved spectral properties and significantly accelerates the convergence of Krylov subspace methods. Extensive numerical experiments on both dense and sparse test problems demonstrate that the proposed BRS and RBRS preconditioners consistently outperform existing approaches in terms of iteration count, computational time, and overall efficiency. These results highlight the effectiveness and robustness of the proposed block-regularized splitting framework for solving large-scale ILS problems.

Unlimited sensing and its extension via modulo hysteresis provide an efficient encoding scheme for high dynamic range signals by folding the signal's amplitude into the range of the analog-to-digital converter prior to sampling. Motivated by shortcomings of the modulo hysteresis model, in this work we introduce generalized modulo encoders that flexibly design the folding by incorporating a general transition function and allowing the folding times to depend on the nonideality of the folding transition. We rigorously study mathematical properties of the respective operators and derive conditions under which a bandlimited signal is uniquely determined by its generalized modulo samples. Finally, we propose a recovery algorithm backed by mathematical guarantees and illustrate our theoretical results through numerical experiments.

We show that Ekeland-type variational principles can be derived from steepest descent curves in metric spaces. From this, we relate the existence of the Calabi flow to the validity of the Ekeland variational principle for the $K$-energy on the space of Kähler potentials.

Motivated by the gap phenomenon for proper holomorphic maps between complex unit balls, this paper investigates smooth CR maps $f$ from an open piece $M$ of the Shilov boundary of the unit ball $B^s$ into the Shilov boundary $S_{r',s'}$ of a higher-rank Type I bounded symmetric domain $\Omega_{r',s'}$. To the best of our knowledge, this is the first work to systematically establish a general gap phenomenon in the higher-rank setting. Our main result demonstrates that when the signature difference $s'-r'$ falls into the interval $k(s-1) \le s'-r' < (k+1)(s-1)$ for some integer $k$, the non-trivial component of $f$ is constrained to a much smaller Type I boundary. Up to automorphisms of the domain and target spaces, $f$ decomposes into the block-diagonal form $$f(z) = \begin{pmatrix} I_{r'-k} & 0 \\ 0 & \phi(z) \end{pmatrix},$$ where the essential component $\phi: M \to S_{k, s'-r'+k}$ is a smooth CR map into the corresponding Shilov boundary. We provide explicit constructions showing that these dimensional bounds are sharp. As an immediate corollary, in the initial gap regime $s-1 \le s' - r' < 2s-2$, the map $f$ reduces to the standard linear embedding.

Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.

The pseudo-quantum representation re-encodes a finite, irreducible, reversible continuous-time Markov chain with $M$ states as a complex-orthogonal flow on a doubled space of dimension $2M$. After uniformisation, a square-root gauge, a doubling of the state space and a diagonal unitary twist, the chain generates an entire one-parameter group $W_z = e^{zK}$ with $z$ complex, and this single group is the object of the paper. Its real slice, $z$ real, reproduces the stochastic semigroup exactly through an explicit decoding whose probabilistic meaning is the parity of the number of ticks of the uniformised chain. Its imaginary slice, $z = i\theta$, is a finite-dimensional unitary quantum system. The chain and the quantum system are therefore not two analogous models but two restrictions of one entire representation. The setup also produces, with no further input, a pseudo-Schroedinger equation, a bilinear von Neumann equation for a complex-symmetric pseudo-density, and a pseudo-Bloch vector equation on a non-compact quadric. We develop the construction from first principles and work out one model completely, the usual symmetric Ehrenfest urn. Its gauged generator equals $(2/n) J_x$, the spin-$n/2$ operator, its relaxation modes are Lorentz boosts, and its imaginary slice is a depth-one quantum circuit whose Born distribution is the classical urn law under the clock $\theta(t) = \arccos(e^{-2t})$. The same clock identifies the Fisher-Rao lift of the urn trajectory with a rigid spin-coherent-state orbit. This is a preliminary simplified version of a longer paper. It treats only finite state spaces and only the symmetric Ehrenfest urn, and it previews without proofs the asymmetric urn, two queues, the symmetric simple exclusion process and the stochastic Ising model. Appendix primers make the quantum, geometric and information-theoretic language self-contained.

We revisit Decomposition-Invariant Conditional Gradient methods, originally introduced by Garber and Meshi in 2016, for minimizing a convex and $\beta$-smooth function over a polytope in $\reals^n$, under an $\alpha$-quadratic growth condition. For 2-level polytopes we design a simple and parameter-free dyadic step-size rule that yields a linear convergence rate which scales with the dimension of the optimal face and not with the ambient dimension as in standard away-step-based conditional gradient methods for polytopes. For general polytopes, under a slightly stronger condition of $\alpha_{\mathrm{F}}$-facial quadratic growth, we introduce a method whose number of iterations to reach an $\epsilon$-approximate solution is of the order $n+\frac{\beta{}D^2}{\alpha{}r^{*2}} + \frac{(d^*+1)\beta{}D^2}{\alpha_{\mathrm{F}}}\log(1/\epsilon)$, where $d^*$ is the dimension of the optimal face, $r^*$ is a separation parameter between the optimal set and faces that do not contain an optimal solution, and $D$ is the diameter of the polytope. This method is also parameter-free and only relies on standard line-search computations. The second result improves upon previous conditional gradient methods, whose number of iterations to $\epsilon$-approximation scales with $\frac{\beta{}D^2n}{\alpha}\log(1/\epsilon)$, in a meaningful regime $\max\{\frac{\alpha}{\alpha_{\rm F}}(d^*+1), \frac{1}{r^{*2}}\} \ll n$

We study optimal data-driven solutions for the stationary diffusive population growth model $-\Delta u = r u$ in a bounded domain $\Omega\subset\mathbb R^N$ with Neumann boundary conditions. Instead of prescribing a functional relation between the position $x$, the net per-capita growth rate $r$ and the population size $u$, we look for a pair $(u,r)\in H^1(\Omega)\times L^\infty(\Omega)$ that fits a given data set in an optimal way measured by a cost functional $I$ and an additional penalty term. We characterize the relaxed cost functional sc$^- I$ by showing that its density is given as the partial lower convex envelope with respect to the variable $r$, and prove the existence of optimal data-driven solutions. Furthermore, we establish a consistency result comparing conventional solutions of $-\Delta u = \varrho(x,u)u$ with optimal data-driven solutions where the data set stems from the functional relation $(x,u)\mapsto \varrho(x,u)$. Finally, as data sets evolve, we prove the convergence of optimal solutions via the $\Gamma$-convergence of the associated cost functionals.

We show that a Borel graph generated by finitely many bounded-to-1 commuting functions is hyperfinite. This recovers a theorem in the forthcoming paper by Naryshkin, Shinko, Weilacher and Yu.

It is well-known the continuity properties of solutions to the wave equation depend strongly on both the initial data and the boundary conditions, particularly across the characteristic lines, where information propagates. Moreover, the continuity properties of solutions to the wave equation depend strongly on the type of boundary condition imposed and on the interaction of the characteristics with the boundary. Across the characteristic lines, singularities and discontinuities propagate according to the geometry of the problem, while the boundary conditions determine how these singularities are re?ected or transmitted. In this study, we will examine the behavior of solutions of the wave equation, corresponding to different boundary conditions, on its characteristic line, using the formulas obtained as a result of d'Alembert's formula, for all the three, Dirichlet, Neumann and Robin, types of boundary conditions.

Distributed target tracking in multi-agent networks plays a critical role in cooperative sensing and autonomous navigation. However, it faces significant challenges in highly dynamic and adversarial setups. This study aims to enhance the resilience of decentralized target tracking algorithms against measurement faults and cyber-physical threats, especially false data injection attacks. We propose a consensus-based estimation algorithm that integrates a nearly-constant-velocity model with saturation-based filtering to suppress impulsive measurement variations and promote robust, distributed state estimation. To counteract adversarial conditions, we incorporate a dynamic false data injection detection and isolation mechanism that uses innovation thresholds to identify and disregard suspicious measurements before they can degrade the global estimate. The effectiveness of the proposed algorithms is demonstrated through a series of simulation-based case studies under both benign and adversarial conditions. The results show that increased network connectivity and higher consensus iteration rates improve estimation accuracy and convergence speed, while properly tuned saturation filters achieve a practical balance between fault suppression and accurate estimation. Furthermore, under localized, coordinated, and transient false data injection attacks, the detection mechanism successfully identifies compromised agents and prevents their data from corrupting the distributed global estimate. Overall, this study illustrates that the proposed algorithm provides a simplified fault-tolerant solution that significantly enhances the accuracy and resilience of distributed target tracking without imposing excessive communication or computational burdens.

In recent years, the prevalence of large-scale data-sets and the demand for sophisti-cated learning models have necessitated the development of efficient distributed ma-chine learning (ML) solutions. Convergence speed is a critical factor influencing the practicality and effectiveness of these distributed frameworks. Recently, non-Lipschitz continuous optimization algorithms have been proposed to improve the slow conver-gence rate of the existing linear solutions. The use of signum-based functions is previ-ously considered in consensus and control literature to reach fast convergence in the prescribed time and also to provide robust algorithms to noisy/outlier data. However, as shown in this work, these algorithms lead to an optimality gap and steady-state re-sidual of the objective function in discrete-time setup. This motivates us to investigate the distributed optimization and ML algorithms in terms of trade-off between conver-gence rate and optimality gap. In this direction, we specifically consider the distributed regression problem and check its convergence rate by applying both linear and non-Lipschitz signum-based functions. We check our distributed regression approach by extensive simulations. Our results show that although adopting signum-based func-tions may give faster convergence, it results in large optimality gaps. The findings pre-sented in this paper may contribute to and advance the ongoing discourse of similar distributed algorithms, e.g., for distributed constrained optimization and distributed estimation.

Curl forces are position-dependent, non-conservative, and non-dissipative forces that, in general, cannot be derived from an ordinary potential energy. Consequently, their equations of motion do not, in general, follow from a standard variational principle. In this paper, we present a dual variational formulation for particle dynamics under curl forces. By introducing variables dual to position and velocity and an auxiliary function, we construct a pre-dual action in which the equations of motion act as constraints. Stationarity with respect to the primal variables defines a dual-to-primal mapping, whose substitution into the pre-dual action gives an action expressed entirely in terms of the dual variables. The Euler–Lagrange equations of the dual action recover both the original equations of motion and their prescribed initial conditions. We also introduce an auxiliary dual Hamiltonian that is conserved along stationary dual trajectories, although it does not represent the physical energy. The formulation is illustrated using two nonlinear curl force fields in two and three dimensions and the classical Ziegler column. These examples demonstrate that non-conservative curl-force dynamics can admit variational descriptions even in the absence of an ordinary potential energy or a conventional Lagrangian.

We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1 < p < \infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available, the global $L^p$ setting has neither. We control the non-local tail through Calderón–Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncation, obtaining order-$(-\epsilon)$ smoothing of the remainder into the Besov scale. Pairing weakly convergent sequences with their canonical Nemyckij duals, we then use the lemma to derive $L^p$ transport equations. For first-order scalar equations we establish the phase-space bicharacteristic (Vlasov) flow of the associated H-distribution when $p \ge 2$; for the quasilinear $p$-wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core $p = 2$ and isolating the structural obstruction to its closure when $p \neq 2$. These results provide functional-analytic tools for tracking the propagation of singularities in degenerate nonlinear and non-local partial differential equations.

Accurate prediction of the nonlinear radiation thermal transfer in composite structures with temperature-dependent properties is significant in high-temperature applications of the materials. This study establishes a high-accuracy multi-scale computational model incorporating novel high-order correction terms for the high-fidelity simulation of nonlinear thermal radiation in composite structures, enabling local balance preserving of heat quantity. Moreover, an explicit convergence rate is also derived for the resulting high-order multi-scale solutions. Furthermore, an efficient multi-scale algorithm consisting of off-line and on-line computation stages is developed for high-accuracy simulation of nonlinear thermal radiation behavior in composite structures, and corresponding convergence analysis is also obtained. Two- and three-dimensional numerical examples are presented to validate the competitive advantages of the proposed multi-scale approach, not only exceptional numerical accuracy, but also reduced computational cost in both storage requirements and computational time.