Papers

In this paper, we give a complete classification, up to isomorphism, of 3-dimensional complex Novikov–Poisson algebras. As an application of the classification, we further prove that every 3-dimensional complex transposed Poisson algebra can be obtained from a Novikov–Poisson algebra except the Lie algebra $\mathfrak{sl}_2(\mathbb C)$. Consequently, Sartayev's conjecture holds in dimension 3, that is, every 3-dimensional complex transposed Poisson algebra is special when regarded as a GD algebra.

This paper introduces a framework that integrates the dynamic stiffness matrix (DSM) with physics-informed neural networks (PINN). The DSM-PINN embeds physical constraints within the model and demonstrates robustness, particularly when addressing limited datasets across diverse investigations. In this approach, deep neural network outputs approximate the displacement fields of element nodes. Unlike the finite element method (FEM), the element shape functions are homogeneous solutions to the governing partial differential equation, forming the basis of the exact dynamic stiffness matrix, thereby avoiding high-order derivative terms. This matrix also serves as a frequency-domain spectral element, resulting in a strong-form PINN. The loss function is produced by connecting neural networks with dynamic stiffness matrices. We focus on utilising PINNs to resolve eigenvalue problems by employing the Wittrick-Williams algorithm, which overcomes the challenge of neural networks failing to converge to higher-order eigenvalues. Additionally, the frequency domain-PINN method is used to analyse structural dynamic responses under moving and impulsive loads, addressing the limitation of neural networks in handling complex numbers. Theoretical convergence stability of the suggested approach is also analysed even DSM is an indefinite matrix after implementing the boundary condition. The numerical results validate the practicality and efficacy of the recommended approach.

We correct Lemma 3.6 and Theorem 3.7 of T.~Bilgin, O.~Kusmus, and R.~M.~Low, \emph{A Characterization of the Unit Group in $\mathbb Z[T\times C_2]$}, Bull.\ Korean Math.\ Soc.\ 53 (2016), no. 4, 1105-1112. In particular, the corrected statement of Theorem 3.7 is that $U_1\bigl(\Z[T\times C_2]\bigr) \cong [F_{33}\rtimes F_5]\rtimes[T\times C_2]. $

This paper presents a macroscopic, or continuum, theory aimed at describing when, where, and why cracks nucleate and propagate in nominally elastic brittle materials under monotonic, quasi-static, but otherwise arbitrary mechanical loads. Motivated by recent insights, the proposed sharp theory posits that: cracks nucleate and propagate exclusively in regions where the strength surface of the material is exceeded, with their evolution dictated by the minimization of the sum of the potential — the elastic energy minus the work done by the externally applied forces — and surface energies. While the theory applies to materials with any elasticity (linear or nonlinear) and any material symmetry (isotropic or anisotropic), attention is restricted here to the most basic case of isotropic elastic brittle materials. For demonstration purposes, the theory is confronted with a set of nine tests that span the entire range of well-settled experimental knowledge on fracture nucleation and propagation — the so-called “Nine Circles of Elastic Brittle Fracture” — on both a hard material (a silicate glass) and a soft material (a synthetic rubber).

Point-mass perturbations and finite-difference operations alter the finite-degree structure of discrete orthogonal polynomials, but their combined effect at the Mehler–Heine scale is not immediate. We study the monic Charlier and Meixner families under a pure Uvarov modification by a fixed mass $A\delta_{0}$ at the endpoint of the support and determine the asymptotic behaviour of their forward and backward differences of arbitrary fixed order. Using a rank-one connection formula, we show that the perturbation coefficient decays factorially in the Charlier case and exponentially, with an algebraic prefactor, in the Meixner case. In both families this decay is faster than every algebraic power of $n^{-1}$ and therefore suppresses the growth produced by any fixed number of finite differences. Consequently, for every fixed $A\geq0$ and $k\in\Nzero$, the mass-modified and classical families have the same locally uniform Mehler–Heine limits in $\C$. Forward differences preserve the reciprocal-Gamma profile up to the factor $(-1)^k$, whereas backward differences translate the limiting argument by $k$, shifting the limiting zero lattice from $\Nzero$ to $k+\Nzero$. We also derive a first-order shift equation for the common forward limit and characterize its entire solution space. Complex-plane portraits and real-axis computations illustrate the predicted limiting profiles, the two zero lattices, and the asymptotic disappearance of the fixed endpoint mass. These results identify a scale-separation mechanism governing the fixed-order asymptotic stability of the Charlier and Meixner families under rank-one endpoint perturbations.

We exhibit a binary linear $[50,40]_2$ code of covering radius $2$, so $\ell_2(10,2)\le 50$, one column below the Kaikkonen–Rosendahl length $51$ that has stood since 2003 and that still seeds the $R=2$ family of Davydov–Marcugini–Pambianco (arXiv:2511.02542). The new matrix admits a $(2,0)$-partition into ten blocks, so Construction $\mathrm{QM}_2^2$ propagates it to exhaustively verified codes of lengths $815$ and $1631$ at $r=18$ and $r=20$, and to the family $n=51\cdot 2^{r/2-5}-1$ of asymptotic density $2601/2048$. The matrices, verifiers, and source are at https://github.com/wustep/maths, pin problems/covering/share/2026-08-24/ at commit 736a38f.

For any given integer $p$, we show that primality of $p$ is equivalent to $\lceil \frac{1}{2}\log_{2}p \rceil$ congruences of binomial coefficients modulo $p$. We also give extensions of this result to several other congruences that are similar in nature.

Let $P$ and $Q$ be independent points chosen uniformly from the interior of a nondegenerate triangle $ABC$, and let $I$ be its incenter. We study the probability that the closed disk with diameter $PQ$ contains $I$. In the language of multivariate statistics, this is the spherical depth of $I$ with respect to the uniform distribution on the triangle. We first give an elementary planar form of the normalized cone-measure construction. If $O$ is an interior point of a convex polygon, then the direction from $O$ to a uniformly distributed interior point has the same law as the direction from $O$ to a boundary point whose density on each side is proportional to the distance from $O$ to that side. Consequently, this boundary point is uniform in arclength if and only if the polygon is tangential with incircle center $O$; for a triangle, this characterizes the incenter. Using this transfer principle, we obtain the closed formula \[\mathrm{SphD}(I) =\left(\frac r s\right)^2 \left[ \frac{8R}{r}-1 -\Gamma(\cos A)-\Gamma(\cos B)-\Gamma(\cos C) \right], \] where $r,R,s$ are the inradius, circumradius, and semiperimeter, and \[ \Gamma(t)=\frac1t-\frac{1-t^2}{t^2}\mathrm{arctanh}\ t \] with continuous values $\Gamma(0)=0$ and $\Gamma(\pm1)=\pm1$. Finally we prove the sharp inequality \[\mathrm{SphD}(I)\le \frac13+\frac{\log 3}{6}, \] with equality if and only if $ABC$ is equilateral.

Let $R(K)$ and $s(K)$ denote the Ramsey number and the stick number of a knot $K$. Johnson proved that $R-s$ is unbounded on $(p-1,p)$ torus knots and asked whether other such families exist. We give an affirmative answer. We construct an explicit eight-stick alternating knot $J$ with $c(J)=6$ and $\operatorname{br}(J)=2$. For $K_n=\#^nJ$, the connected-sum formulas yield $R(K_n)\ge 7n+3$ and $s(K_n)\le 5n+3$.

In this paper, we introduce the RLC-fractal derivative along with the fractal Laplace and fractal Sumudu transforms of it. Then we apply these transformations to solve economic models involving the RLC-fractal derivative.

We study the exceptional component of the space $\mathbb{F}(2,\mathbb{P}^3)$, of codimension-one foliations of degree two on $\mathbb{P}^3$. We describe the geometry of its boundary and prove that it has four irreducible components, all of dimension $12$. Three of these components contain a dense subset given by the orbit of a logarithmic foliation of type $(1,1,2)$, while the fourth contains a family of pull-back type foliations from $\mathbb{P}^2$ whose orbits have dimension $11$.

A group $H$ is said to be a $B$-group if whenever a primitive permutation group $G$ contains a regular subgroup isomorphic to $H$, then $G$ is doubly transitive. Going as far back as William Burnside, several authors have investigated whether certain families of groups are $B$-groups. Using the O'Nan-Scott Theorem as our starting point, we classify the $B$-groups of order $p^4$ where $p$ is a prime.

In this paper, we establish a gluing formula for the pseudo-determinant of the Laplacian on a simple finite graph. We achieve this by using the gluing formula for the determinant of massive Laplacian and the perturbation theory technique. In addition, we apply this gluing relation to derive a gluing formula for the number of spanning trees and rooted spanning forests on simple finite graphs.

We investigate structural and enumerative properties of binary single parity-check product codes. For each $n\geq 2$, $\operatorname{SPC}(n)$ denotes the binary single parity-check code of length $n$, consisting of all binary vectors of length $n$ having even Hamming weight. We determine the generalized Hamming weight hierarchy of the product code $\mathcal{C}_{m,n}=\operatorname{SPC}(m)\otimes\operatorname{SPC}(n)$, whose codewords can be represented as $m\times n$ binary matrices in which every row and every column has even Hamming weight. For the square product $\mathcal{C}_n =\operatorname{SPC}(n)\otimes\operatorname{SPC}(n)$, we also determine the maximum codeword weight and prove that its homogeneous weight enumerator is symmetric if and only if $n$ is even. After characterizing the dual code, we apply the MacWilliams identity in its Walsh–Hadamard formulation to derive an exact closed-form expression for the weight enumerator. By grouping the auxiliary binary vectors according to their Hamming weights, we obtain an explicit formula for each coefficient in terms of binomial coefficients and alternating convolutions. Finally, using Krawtchouk polynomials, we present an exact procedure for computing the full weight distribution without exhaustively enumerating all codewords. Numerical examples illustrate the formulas and verify the resulting computations.

We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over $X\times \mathbb{Z}$. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an $\Omega$-spectrum of invertible phases as conjectured by Kitaev. We then show that the $c=\frac{1}{2}$ chiral Majorana fermion net and the $(E_{8})_{1}$ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on $\mathbb{Z}^{2}$. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.

We prove a quantitative neighboring-denominator variant of the Erdős-Mahler conjecture. Let $p_n/q_n$ be the convergents of an irrational real number $\xi$. If $p_nq_nq_{n+1}$ is $S$-smooth for infinitely many $n$, where $S$ is a fixed finite set of primes, then there exists an effectively computable constant $c=c(S)>0$ such that \[ \log q_{n+1}\gg_{\xi,S} q_n^c \] along those indices. Consequently, $\xi$ is a Liouville number. The proof uses the determinant identity for consecutive convergents and a fixed-base consequence of Yu's theorem on $p$-adic logarithmic forms.

We provide an easy proof that there are no P-points in the models obtained by adding many Silver reals with countable support to a model of CH.

We prove a binomial identity relating the $k$-Lucas and $k$-Fibonacci sequences: for real $k>0$ and integers $m\ge1$, $n\ge1$, a binomial-weighted sum of $k$-Lucas numbers reduces to a multiple of $L_{k,mn}$ when $n$ is even, but to a multiple of $F_{k,mn}$ when $n$ is odd. This parity dependence traces back to the $k$-Lucas Binet formula lacking the normalising factor $1/(\alpha-\beta)$ present for $k$-Fibonacci numbers. The identity supplies, for each parity of $n$ and each $m$, an explicit family of positive weights summing to one; combined with the general annulus principle of Dalal and Govil, this yields a $k$-Lucas annulus containing all the zeros of a complex polynomial of degree $n$, complementing the $k$-Fibonacci-based bounds of Díaz-Barrero, Bidkham-Shashahani, and Kaur.

Lyapunov's direct method is a cornerstone of stability and control, but it hinges on finding a Lyapunov function, a task demanding ingenuity or computation. A key difficulty is that every sub-level set of the function must be forward invariant, coupling its geometry to the system's trajectories. We relax this by replacing invariance with recurrence: a set is recurrent if every trajectory starting in it returns within a finite time. This yields the notion of a Recurrent Lyapunov Function (RLF), whose sub-level sets need only be recurrent. We show that, under mild conditions, RLFs guarantee stability, and we introduce stronger notions yielding asymptotic and exponential stability. We also give norm-based converse theorems: under the corresponding stability conditions, any norm is an RLF for their practical versions. We then develop GPU-based algorithms that certify (practical) stability from trajectory data alone, without a Lyapunov function. Certifying stability up to an $\varepsilon$-neighborhood needs only $O(\log(1/\varepsilon))$ trajectory evaluations, with constants growing as the certified decay rate nears the true one, exposing an intrinsic performance-cost trade-off.

The integration of fixed-route public transit and on-demand mobility services presents both a modeling challenge and a computational opportunity for large-scale network design. We propose a flow-based mixed-integer programming formulation that jointly optimizes transit line planning and service frequencies while explicitly capturing first- and last-mile connectivity via on-demand services, under a fixed operating budget. To achieve tractability at urban scale, we develop a novel column generation heuristic scheme with tailored pricing subproblems. Applied to networks and demand in Boston and Chicago, the framework yields operationally feasible designs that substantially increase demand served. Relative to transit-only and on-demand-only baselines, ridership increases by up to 20.99% and 93.58% in Boston, and by up to 10.63% and 149.84% in Chicago. Compared to a multi-modal benchmark, our approach improves ridership by 5.42% and 5.80% in Boston and Chicago, respectively. These results demonstrate that (i) joint co-design of transit routes, frequencies, and on-demand legs within a unified optimization framework yields substantially greater ridership than single-mode or decoupled approaches under equivalent budget constraints; and (ii) the proposed formulation and column generation pricing scheme admit tractable, operationally feasible, high-performing solutions relative to tested baselines.

In the present work we consider the problem of the homogenization of the Cauchy problem for the equations of the elasticity. We assume the composite material is constructed as a periodic repetition along the chosen axis of the unit cell which models to different materials. The homogenization procedure is based on the operator separation of variables, which gives the new incites for the homogenization of the elasticity equations and provides new way of the homogenization and constructing the homogenized equation.

We investigate the topology generated by modular convergence on vector spaces endowed with a convex modular. Although modular convergence has long played a central role in the theory of modular function spaces, the topology it induces has, to the authors' best knowledge, not previously been studied as an independent mathematical object. Our principal result establishes that the modular topology is compatible with the vector space structure if and only if the underlying modular satisfies the $\Delta_2$-condition. Consequently, the $\Delta_2$-condition admits a purely topological characterization. We also show that the Luxemburg norm topology is the weakest first-countable topology containing all scaled modular topologies. We apply the theory to variable exponent sequence and Lebesgue spaces beyond the classical $\Delta_2$-theory and present an application of our results to the minimization of the Dirichlet integral in $W^{1,p(x)}$ with unbounded exponent $p$, which allows us to solve an open boundary value problem.

In this paper, motivated by discrete-time McKean-Vlasov optimal control problems with law-delay, a class of forward-backward stochastic difference equations is investigated. The main difficulties come from the absence of Ito's formula in the discrete-time setting, as well as the fully coupled structure in which the coefficients depend on both law-delayed and law-anticipated terms. Under suitable monotonicity conditions, the existence and uniqueness of solutions are established. Furthermore, a new monotonicity condition is introduced to address the associated optimal control problem. As an application, the unique optimal control for linear quadratic systems with law-delay is derived.

Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+\lambda/n$ in dimensions $n$ and $n-k$. Uniformly over deletion ranks retaining a fixed positive fraction of observations, the first nonzero difference converges to an explicit phase surface $H_\delta(\lambda)$; its zero curve unifies fixed, sublinear and fixed-fraction deletions, with tangent crossing $12/35$. Through the Khintchine scale-mixture representation, this comparison yields a single compactly supported $C^\infty$ symmetric unimodal parent, independent of $n$, whose Student tail exceeds the equal-scale uniform tail for all sufficiently large $n$ along nominal levels tending to $2\{1-\Phi(\sqrt3)\}$ from below. In contrast, a quantile-ratio order shows that the uniform parent maximizes every even moment and every convergent even power series with nonnegative coefficients. We also derive the fixed-confidence dimension expansion and an exact reversal at $n=6$.

We consider an aggregation-diffusion energy on the sphere and investigate the stability of its equilibria. The energy consists of a porous-medium type nonlinear entropy $\frac{1}{m-1}\int \rho(x)^m\mathrm{d}S(x)$ with $m>1$, together with an interaction energy modeled by a quadratic interaction potential. The energy generalizes the Onsager free energy with dipolar potential, which models polymer orientation. Our study complements the authors' previous work [Nonlinearity 39 (2026), 055011], where the ground states of the energy functional were investigated. In the current paper we extend the previous results on existence of equilibria for $m>2$ from $d=2$ to arbitrary dimension $d \geq 2$. This allows us to present the bifurcation structure (with respect to the interaction strength) of all equilibria in any dimension $d \geq 2$, for all $m>1$. Furthermore, we provide a complete classification of the stability of all equilibria of the energy, by deriving a criterion for stability that can be checked explicitly. In particular, for $m>2$ we identify a saddle-node bifurcation of strictly supported equilibria, and a subcritical pitchfork bifurcation where the uniform distribution loses stability.

We consider the motion of a charged test particle confined to the round unit sphere in a uniform ambient magnetic field. We use an extension of Noether's theorem for systems with magnetic forces to reduce the problem to a quadrature. The rotation number of a trajectory, the azimuth it gains over one oscillation in latitude, is a complete elliptic integral of the third kind in Legendre form. A trajectory closes if and only if that advance is a rational multiple of a full turn. The criterion holds on every level set of the two first integrals. Then, we differentiate the rotation number with respect to the half-cyclotron frequency and sign that derivative on each of the two branches into which a level set divides. The rotation number sees the charge, the mass, the field and the speed through a single dimensionless ratio, that of the half-cyclotron frequency to the speed. A closed trajectory therefore fixes a value of that ratio. We say how many values carry a given rational winding. The poles are attainable on a single level set, where the motion reduces to a pendulum. The field has no flux through the sphere, so it is globally exact. We compute the Mañé strict critical value of the resulting exact magnetic flow. Above that value the closed trajectories are closed Reeb orbits of an explicit contact form.

Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $\mathcal{O}^{\mathfrak{r}\,G}_X$ on $X$ by \begin{align*} \mathcal{O}^{\mathfrak{r}\,G}_X(V):=\{u\in \mathcal{C}(V,G)|\ &u \hbox{ has regularity class $\mathfrak{r}$ on $V$}, \\ &u|_{V\cap\mathrm{int}(X)} \hbox{ is holomorphic}\}. \end{align*} Let $X_0\Subset Z_0\Subset \mathcal{X}$ be a strictly pseudoconvex extension in $\mathcal{X}$, and let $X:= \bar X_0$, $Z:= \bar Z_0$ be the corresponding compact manifolds with boundary in $\mathcal{X}$. Let $K\subset X_0$ be a compact set and $\mathcal{P}$ a holomorphic principal $G$-bundle of class $\mathfrak{r}$ (i.e. an $\mathcal{O}^{\mathfrak{r}\,G}_X$-torsor) on $X\setminus K$. Assuming that (H1) the given strictly pseudoconvex extension $X_0\Subset Z_0$ is non-critical, or that (H2) the underlying topological bundle of $\mathcal{P}$ extends to $Z\setminus K$, we prove that $\mathcal{P}$ admits an extension to $Z\setminus K$. This gives a new proof and a new generalisation of the extension problem stated in the article S. Donaldson, Boundary value problems for Yang-Mills fields, Journal of Geometry and Physics 8, (1992) and studied with different methods in a previous article of the author.

In this paper we prove the following result. Let $\Omega\subset\mathbb R^n, n\geq 3,$ be a bounded, strictly convex, smooth domain and $\varphi: \partial\Omega\rightarrow\mathbb R$ be a smooth function. Then for any $z\in\Omega,$ there exists $c_1=c_1(\Omega, \{z\}, n, \varphi)>0,$ such that if $c\geq c_1$ then the problem: $\sigma_{n-1}(D^2 u)=0$ in $\bar\Omega\setminus\{z\},$ $u|_{\partial\Omega}=\varphi,$ $\lim\limits_{r\rightarrow 0}\sup\limits_{B_r(z)}\frac{u(x)-u(z)}{|x-z|^{(n-2)/(n-1)}}=c,$ admits a smooth solution in $\bar\Omega\setminus\{z\}.$ Moreover, we obtain the optimal a priori estimates for this solution. In particular, we show for any integer $m\geq 0$ there exists $C_m=C_m(m, \Omega, \{z\}, n, \varphi, c)>0$ such that $|D^m u(x)|<C_m|x-z|^{\frac{n-2}{n-1}-m}.$ This work provides the first result demonstrating the existence of smooth solutions to the Dirichlet problem for fully nonlinear elliptic equations with prescribed density in Euclidean space. Previously, only continuous solutions were obtained.

We study optimization problems for separable functionals of the Euclidean or spherical lengths of dual edge pairs in finite extremal unit-diameter configurations in three dimensions. For a fixed diameter graph, these problems lead to nonconvex constrained optimization of the vertex coordinates. We first prove that a convergent sequence of extremal configurations retains an extremal geometric core after coincident points are merged and vertices incident to at most one diameter are removed. A spherical Crofton argument then gives a sharp lower bound for additive concave functionals, attained by the regular tetrahedron. We also analyze the effect of inserting or deleting dangling vertices. For the sum of products of spherical dual-edge lengths, we obtain an exact supremal reformulation of the three-dimensional Blaschke–Lebesgue area problem. The numerical study uses all 10,644 available extremal configurations with at most 16 vertices. We combine direct evaluation with gradient-based local optimization on each fixed graph and reconstruct the intrinsic diameter graph and its dual pairs after vertex collisions. For every supplied graph, the selected verified endpoint contains a regular tetrahedron. These computations do not certify the global maximum for any fixed graph, but they motivate structural conjectures connecting tetrahedral containment with the Blaschke–Lebesgue problem.

We give a product-system description of generalized higher-rank graphs with the weak factorization property. For such a graph with degree-zero groupoid G, we prove that weak factorization is equivalent to multiplication inducing coherent bijections between balanced products of its homogeneous G-G-bisets. Consequently, generalized higher-rank graphs with fixed degree-zero groupoid are equivalent to normalized product systems of groupoid bisets over N^k. For countable left-cancellative graphs, these bisets linearize canonically to product systems of C*(G)-correspondences. Finite alignment implies compact alignment, and the resulting Nica-Toeplitz algebra agrees canonically with Spielberg's full category algebra. Under row-finiteness modulo G and the no-sources condition, the corresponding Cuntz-Pimsner quotient is the boundary groupoid algebra; with injective left actions, the same conclusion holds for the Cuntz-Nica-Pimsner algebra. We also characterize the R-condition by the existence of a strict multiplicative splitting and relate such splittings to higher-rank graph/groupoid Zappa-Szep products. A cancellative rank-two example shows that strict splittings need not exist.