Papers

Cohesive Zone Models (CZMs) are widely used to simulate interface fracture, delamination, adhesive failure, and fiber–matrix debonding in aerospace composite structures. In implicit quasi-static finite element analyses, cohesive softening may introduce negative interface tangents, solution jumps, and Newton-basin mismatch, so the previous converged state can become a poor initial guess for the next increment. This may lead to stagnation, wrong-branch convergence, or repeated step cuts. Existing remedies, including viscous regularization, path following, dynamic relaxation, and manual Newton–Raphson (NR) modification, either alter the effective response, increase cost, or rely on hand-crafted interface rules. This work proposes an Interface-Aware Neural Newton Preconditioner (IA-NNP) for difficult CZM increments. IA-NNP recasts manual NR modification as rule-based interface lifting and generalizes it into a learned, state-dependent interface correction. The method acts only on active interface variables and preserves the original traction–separation law, residual assembly, tangent evaluation, history update, and dissipation checks. Two realizations are developed: IA-NNP-Init for learned initial-guess lifting and IA-NNP-NL for iteration-level nonlinear right preconditioning. Interface graph features encode opening, traction, tangent, damage/history variables, mode mixity, residuals, and neighboring states. The correction is bounded, confidence-gated, and accepted only through the original CZM Newton solve. A root-equivalence property shows that IA-NNP changes the path to convergence but not the discrete CZM solution set. Tests on horizontal, circular, two-interface, and active-front benchmarks show improved difficult-increment convergence, better branch recovery, and fewer failures than standard NR and manual NR modification, while preserving the force–displacement response.

To maintain the antenna apertures offered by 5G massive MIMO systems operating at the sub-6GHz band, known as FR1, 6G base stations (BSs) using the upper-mid band, FR3, should increase the number of antennas by a factor 4-8, giving rise to gigantic MIMO. This poses challenges in terms of processing complexity and interconnection bandwidth. The WAX framework, previously introduced for exploring trade-offs in decentralized architectures, may offer the flexibility needed to tackle these challenges. However, no results have been established on the applicability of this framework in the presence of hard-boundary constraints. The current work explores gigantic MIMO implementations based on a novel adaptation of the WAX framework, where the decentralized processing is performed by non-cooperating hardware modules. These modules may be implemented through state-of-the-art massive MIMO baseband units (BBUs). The results show the potential of the proposed framework towards exploiting trade-offs between complexity and performance in practical gigantic MIMO implementations.

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors. One might wonder: is there a color wheel in which all six of the sections have this property, without all the sections being the same color? We show that the answer is no, not just for the 6-cycle but for any finite connected graph; indeed, for any finite, strongly connected, edge-weighted digraph. The result generalizes the “harmonic lemma" for graphs, replacing the well-behaved averaging function by paint blending, about which almost nothing is assumed. Our proof makes use of a Markov chain stopping rule.

We present an orbit–theoretic reformulation of Galois theory based on the natural action of automorphism groups on fields. Given a field $\mathbf{E}$ and a subgroup $H$ of the automorphism group $\mathrm{Aut}(\mathbf{E})$, we show that algebraic properties of the extension $\mathbf{E}/\mathbf{E}^H$, where $\mathbf{E}^H$ denotes the fixed field of $H$, are encoded in the $H$-orbits arising from the action of $H$ on $\mathbf{E}$. An element $\alpha \in \mathbf{E}$ is algebraic over $\mathbf{E}^H$ if and only if its $H$–orbit is finite. In that case, its minimal polynomial can be explicitly constructed as the product of linear factors over its orbit –a construction that also ensures separability. At the level of field extensions, we prove that $\mathbf{E}/\mathbf{E}^H$ is Galois if and only if all $H$–orbits have finite length, and that $\mathbf{E}/\mathbf{E}^H$ is a finite Galois extension if and only if the lengths of the $H$–orbits are bounded above. This provides a unified orbit–theoretic characterization of algebraicity, separability, normality, and degree. Artin's Lemma is recovered as a direct consequence of this framework. Finally, we show that for simple extensions, the fixed field under a subgroup $H$ of $\mathrm{Aut}(\mathbf{F}(\alpha)/\mathbf{F})$ can be described explicitly by evaluating elementary symmetric polynomials on the $H$–orbit of $\alpha$, provided this orbit is finite. This leads to an effective method for computing fixed fields directly from orbit data. A classical example is included to illustrate the approach.

In this paper, we focus on a new type of non-linear kinetic Fokker-Planck equation where the non-linearity comes from a non-linear diffusion in the velocity variable. The existence of solutions in suitable Lebesgue spaces is proved, together with important entropy estimates on these solutions. We then study the diffusive limit of such equation.

Let $(X,\Theta)$ be a very general principally polarized abelian variety of dimension $g$, and consider the minimal cohomology class $\theta_k=[\Theta]^k/k!$ for $k<g$. We show that the minimal positive multiple of $\theta_k$ which is algebraic is divisible by all primes $p\leq (k+1)/2$. In particular, these minimal multiples grow exponentially with $k$. Our main result follows from [EGFS25] together with a new combinatorial result about $\mathbb F_p$-solutions of certain graphic matroids in their own Albanese graphs.

In nonstandard analysis the halo of a point in a topological space is the intersection of the nonstandard extensions of all its open neighbourhoods. We define a parametric family of modal operators from the halo by varying which elements of the nonstandard extension are admitted as witnesses, and identify four canonical instances. Two recover well-known modalities: the topological closure and the Cantor derivative. A third reduces to Kripke semantics over the specialisation preorder. The fourth, purely nonstandard instance admits only nonstandard witnesses. The Transfer Principle forces it to coincide with the $\omega$-accumulation point operator, a classical topological notion not previously studied in modal logic. Unlike the Cantor derivative, the $\omega$-accumulation operator maps arbitrary sets to closed sets without any separation axiom, yielding an $\omega$-Cantor-Bendixson decomposition on all topological spaces. Axiom 4 holds universally, again without separation conditions. We prove that K4 is the complete logic over infinite spaces, and GL over infinite $\omega$-scattered spaces.

We develop translations from relevant logics into normal modal logics, and use them to clarify structural connections between relevant and modal logic, obtain a few corollary results, and raise questions for future work.

In this article, we investigate the pointed modal logic of reals. We first establish its relational (Kripke) semantics with bounded valuations in the Abelian l-group of real numbers with the distinguished negative constant -1. To study this logic algebraically, we introduce the variety of negatively pointed modal Abelian l-groups, in particular we focus on the strongly pointed members thereof. Constructing complex algebras and canonical frames, we establish a Truth Lemma connecting the relational and algebraic frameworks. Since finitary axiomatizations cannot fully capture the Kripke validities of the reals we introduce further algebraic constraints, in particular including an infinitary Archimedean-style rule. Finally, we prove a corresponding `infinitary algebraic completeness' result for pointed modal Abelian logic with respect to the variety of pointed modal Abelian l-groups.

We define the notion of IK-bisimulation between the relational semantics for the intuitionistic modal logic IK, and prove that IK arises as the IK-bisimulation-invariant fragment of intuitionistic first-order logic. En route, we provide an intrinsic characterisation result of this logic by way of a Hennessy-Milner-style theorem and develop some intuitionistic first-order model theory, including intuitionistic analogues of Los's Theorem, elementary embeddings and countable saturation.

We study interpretations of modal logics in one another where the Boolean connectives are interpreted identically and the modal operator diamond is interpreted by an arbitrary formula A(p). Clearly, such a formula A(p) defines an interpretation of a normal modal logic whenever A(p) is additive (that is, preserves disjunction) and normal (that is, preserves bottom) in the target logic. In the present paper, we provide a complete description of all additive and normal formulas in five prominent modal logics: K, GL, Grz, S4, and S5. For K, GL, and S5, we also describe all additive and normal formulas with parameters.

We study the recently introduced intuitionistic monotone modal logic IM. We first provide a semantic characterisation for a family of natural extensions of IM in terms of constructive neighbourhood models. We then present a calculus for IM and its extensions, obtained by adapting a structured calculus for the classical monotone modal logic M. Based on the calculus, we prove some preliminary results for IM, including its decidability. Our calculus also reveals an interesting analogy between constructive and intuitionistic variants of M and the corresponding variants of K, thereby further justifying IM as a faithful intuitionistic variant of M.

We compute the Clarke subdifferential of the $k$th eigenvalue functional on the space of self-adjoint operators, obtaining a first-variation formula that remains valid even when the eigenvalue lies at the edge of the essential spectrum. This formula provides an effective tool for describing the structure of critical points in eigenvalue optimization problems and can also yield simple proofs of the existence of optimizers. We illustrate these advantages through applications to the optimization of weighted Laplace and Steklov eigenvalues. In particular, we characterize all optimal weights, thereby answering some open questions posed by Kokarev, and give a short proof that such weights exist.

A logics' property is decidable in a class of logics if there exists an algorithm that decides whether a finitely axiomatizable logic in the class has the property. Many properties are undecidable for bimodal logics but decidable for linear tense logics, which leads to a general question on how the interactions of modalities affect the decidability of properties. In this paper, we study the decidability of properties for transitive tense logics and show that most properties are undecidable in the lattice NExt(K4t) of transitive tense logics, including Kripke completeness, the finite model property, and decidability. Our proof method adapts Chagrov's approach of constructing a reduction from an undecidable problem of Minsky machines to the decision problem for logics' properties, yielding a general scheme of proving the undecidability of these properties.

We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory. We define a modal formalism in this language, which we call modal measurable logics. We also introduce a Kripke-like semantics for these logics in measurable spaces taking a designated modal sigma-ideal into consideration. Using a restriction of Jonsson-Tarski duality and a modal extension of the Loomis-Sikorski theorem, we prove completeness of modal measurable logics with respect to this new semantics.

We study layered models, including feedforward networks, ResNets, and transformers, by limiting each layer to a width of $d = 3$, i.e., $\mathbb{R}^3$ as representation space. This allows us to track how a neural network changes low-dimensional topological invariants through its layers. Just about any topological structure may be simplified or even trivialized by simply increasing dimension; e.g., any knot is equivalent to an unknot in $\mathbb{R}^4$. By restricting to $\mathbb{R}^3$, we not only isolate the effects of activation and depth from that of width, we work in a space that lends itself to easy visualization. We focus on linking number here, deferring other invariants like link groups, Milnor's $\bar{\mu}$-invariants, knot types, ambient cobordisms, to a sequel. We provide full proofs and empirical experiments to justify the following insights: When measured by their power to effect changes in linking numbers, the layer-skipping feature in ResNets is as powerful as the attention mechanism in transformers; both ResNets and transformers are strictly more powerful than feedforward neural networks with monotonic activations, which are in turn more powerful than invertible and flow-based models; but replacing monotonic activation with a nonmonotonic one elevates a feedforward network into the same expressivity class as ResNets and transformers. These results suggest that low-dimensional topology can be a useful tool to guide designs of AI architectures. We also generalize our results from $d = 3$ to arbitrary $d > 3$.

A coequivalence relation over a modal logic L is a formula in two tuples of propositional variables of the same length such that the logic L proves it to be an equivalence relation. They were introduced by Ghilardi and Zawadowski in the context of the categorical study of non-classical logics. A coequivalence relation is said to separate variables or to be separating if it corresponds to a collection of formulas, which serve as explicit definitions of quotients. A logic L where all coequivalence relations are separating is said to have the coequivalence separation property (CoSP). Ghilardi and Zawadowski showed that CoSP fails for IPC. In previous work, the second author showed that such a phenomenon happens already in presumably simpler systems like S5. Ghilardi and Zawadowski therefore raised the question whether a weaker property, formulated in categorical terms related to descent theory, was still true. In this paper, we identify the logical meaning of such a property in relation to CoSP. We introduce the notion of local coequivalence relations, which have the additional structure of a local transition term, intuitively capturing the structure of elements lying in the same fiber. We introduce the local coequivalence separation property (LCoSP), and prove it to be equivalent, in good cases, to the almost Barr-exactness of the category dual to finitely presented algebras. We conclude by showing that S5 has the LCoSP.

We investigate logics that generalize both intuitionistic logic and quantum logic. In earlier work, we introduced Ex-logic, an extension of Holliday's fundamental logic that coincides with the intersection of orthologic and the implication-free fragment of intuitionistic logic. In this paper, we add an implication connective to Ex-logic and axiomatize iEx-logic, the intersection of full intuitionistic logic and orthomodular logic with the implication connective interpreted as the Sasaki hook. As a consequence, we obtain a characterization of the lattice of logics extending iEx-logic as the product of the lattice of intermediate logics and the lattice of orthomodular logics. We also explore the robustness of our algebraic approach by briefly discussing extensions of iEx-logic with modal operators.

Optimal spline subspaces are an elegant and efficient tool to remove spurious outliers in isogeometric Galerkin discretizations for the approximation of the spectrum of the Laplace operator. For practical purposes, it is valuable to have a basis construction for such spaces with good computational and spectral properties. We provide a characterization of the bases that enjoy a B-spline-like support structure and whose mass and stiffness matrices are simultaneously diagonalizable. It turns out that these mass and stiffness matrices admit explicitly known closed-form expressions for their eigenvalues, implying that the considered bases are outlier-free. A numerical procedure for constructing such bases is also presented.

We present a numerical study of eigenvector deflation as a means of accelerating the WaveHoltz method for solving the Helmholtz equation. For energy-conserving (Dirichlet or Neumann) boundary conditions the WaveHoltz fixed-point iteration converges slowly at high frequency, requiring approximately $\mathcal{O}(\omega^{2d})$ iterations in $d$ dimensions. We show that deflating the eigenvectors whose eigenvalues lie nearest the driving frequency substantially reduces iteration counts, and we examine two ways of incorporating the eigenvectors: direct eigenvector deflation (DEVD), in which the forcing and iterate are projected against the deflation set, and augmented-Krylov eigenvector deflation (AUKED) using deflated conjugate gradient (DCG), augmented GMRES (AGMRES), and augmented (recycled) BICGSTAB (ABICGSTAB). The required eigenpairs can be computed efficiently with the EigenWave approach, and we demonstrate, in two dimensions, that when the number of deflation vectors grows quadratically with $\omega$ the asymptotic convergence rate remains essentially constant. Because the eigenvectors on structured grids are naturally represented as matrices, we further apply SVD-based compression to reduce their storage. Numerical experiments on single curvilinear grids discretized with summation-by-parts operators, and on overset grids illustrate the robustness and efficiency of the approach, with the deflated solver breaking even against the undeflated solver after as few as two right-hand sides, when accounting for the cost of precomputing the eigenvectors.

Complementary families of polynomials are introduced to generate $C^m$ finite element basis functions of order $p \geq 2m+2$ for arbitrary $m \ge 0$. One family consists of the Hermite splines that serve as the nodal basis functions by ensuring $C^m$ continuity across element boundaries. Explicit formulas for these splines for any $m \ge 0$ are presented on the canonical interval $[0,1]$. The second family is derived on the interval $[-1,1]$ from derivatives of order $m+1$ of the Legendre polynomials of degree $p-m-1$ multiplied by binomial powers of degree $m+1$ at -1 and 1, respectively. These polynomials, related to the ultraspherical polynomials, serve as the interior or bubble basis functions. A relationship between the two families of polynomials is demonstrated. For a particular $m$ and $p$, an interpolant is constructed using these basis pairs together with the roots of the related ultraspherical polynomial and the interval endpoints. A formula for the interpolation error that extends the results for $m=0$ and $m=1$ is given. To prove the formula extensions of the Lagrange interpolants are introduced. A superconvergence result along with the related asymptotic equivalence of the interpolant and finite element solution is proved in the linear case in $H^{m+1}$. Computational results demonstrate the theory for a model problem.

We propose Andrews-Gordon type series for certain level 2 standard modules of type $A^{(2)}_{\textrm{odd}}$, and prove the corresponding sum-product identities except for $A^{(2)}_{6n+3}$. These identities generalize the identities of Göllnitz-Gordon (mod 8), Rogers-Ramanujan (mod 5) and (partially) Nandi (mod 14).

We consider the INDEPENDENT SET RECONFIGURATION problem under the Token Sliding rule. Let $I$ be an independent set of a simple undirected graph $G$. Suppose that each vertex of $I$ has a token placed on it. The tokens are allowed to be moved, one at a time, by sliding along the edges of $G$, so that after each move, the vertices having tokens always form an independent set of $G$. The problem we deal is to decide if we can transform $I$ into $I'$ through a sequence of steps, each of which involves substituting a vertex in the current independent set with one of its neighbours to obtain another independent set. This problem of determining if one independent set of a graph "is reachable" from another independent set of it is known to be PSPACE-hard even for split graphs, planar graphs, and graphs of bounded treewidth. Polynomial time algorithms have been obtained for certain graph classes like trees, interval graphs, claw-free graphs, bipartite permutation graphs, block graphs, and cographs. We present a polynomial time algorithm for the problem on $P_4$-tidy graphs and $(q,q-4)$-graphs, both families of graphs generalizing cographs.

In this paper, we define quantum Stokes matrices of meromorphic linear systems of ordinary differential equations with noncommutative coefficients and a pole of order $p+1$. We prove that these quantum Stokes matrices satisfy natural quantum exchange relations. These relations allow us to interpret the quantum Stokes matrices as an associative algebra homomorphism, which may be viewed as a quantization of the Riemann-Hilbert-Birkhoff map, regarded as a Poisson map, for meromorphic connections.

Many bipartite social networks exhibit pronounced asymmetries in selectivity and matching opportunities: members of one side can afford to be highly selective, while members of the opposite side are forced to accept less desirable matches. While it is natural to try to explain this asymmetry in terms of the intrinsic characteristics of the two sides or other exogenous factors, here we show that such asymmetries can also emerge endogenously through a feedback process generated by the matching process itself: as one side becomes more selective, the other side is pushed to be less selective due to reduced matching opportunities, and vice versa. We develop a model in which individuals repeatedly form one-to-one matches across two groups and adapt their selectivity to achieve a target matching rate. Using both analytic and numerical methods, we show that when encounters are sufficiently frequent, the unique equilibrium is for one group to be highly selective and the other non-selective. This qualitative outcome holds even for heterogeneous groups with overlapping, almost indistinguishable distributions of target matching rates. The model makes several testable predictions, and it provides a mechanism for behavioral differentiation in repeated matching environments, with applications ranging from online dating to hiring and housing markets.

We mainly consider a Liouville-type problem for the three dimensional stationary fractional Navier-Stokes equations with arbitrary asymptotic state $u_\infty$ at infinity. When $u_\infty\neq 0$ and $\frac{1}{2}\leq s<1$, we prove a complete Liouville theorem by establishing some refined $L^p$ estimates for the velocity without relying on perturbation arguments. These new estimates are stronger than the $L^3$ estimates obtained by the classical perturbation framework, we thus can take $u$ as a test function and give a direct and simple proof of Liouville theorem while avoiding some technical fractional calculus. When $u_\infty\neq 0, s=\frac{1}{2}$ or $u_\infty=0,\frac{1}{2}\leq s\leq\frac{5}{6}$, we also prove a complete Liouville theorem by using frequency localization to overcome the obstacles coming from the non-local effects of $(-\Delta)^s$. We wish to emphasize that our method dealing with the case of $u_\infty=0$ is also applicable to dimension $n$ with $n\geq 2$ and $\frac{1}{2}\leq s\leq \frac{n+2}{6}$.

We study a family of Hopf algebras arising as liftings of the Jordan plane over the infinite cyclic group. We determine their centres, prime and primitive spectra, and automorphism groups. We show that every prime ideal is completely prime and that every nonzero ideal intersects the centre nontrivially. We construct explicit simple modules corresponding to all primitive ideals and classify the finite-dimensional simple modules. Finally, we prove that these Hopf algebras satisfy the Dixmier–Moeglin equivalence.

In this paper, we introduce the concept of block sub-additive potentials and study pressures on spaces of average pseudo-orbits for $\mathbb{Z}_+$-actions generated by continuous surjections on compact metric spaces. Relative to each finite open cover, we define topological and measure-theoretic pressures and establish a local variational principle. We prove that the local topological pressure coincides with the local pressure of the induced sub-additive potential. For every invariant measure, the local measure-theoretic pressure is bounded above by the measure-theoretic pressure of the induced potential, and equality holds when the measure is ergodic. Moreover, the global topological and measure-theoretic pressures associated with block sub-additive potentials are shown to be the suprema of their local counterparts over finite open covers. Consequently, the global topological pressure coincides with the pressure of the induced sub-additive potential, yielding a global variational principle.

We develop a macroscopic thermodynamic theory of nematic liquid crystals starting from a kinetic theory of ordered fluids with a collision operator of Bhatnagar-Gross-Krook (BGK) type. The kinetic description incorporates mean-field alignment interactions through a Vlasov potential and relies on a separation of time scales, with orientational relaxation occurring on a faster time scale than translational momentum relaxation. At the continuum level, we establish the balance equations for mass, linear and angular momentum, energy, and entropy. Using the zeroth and first order Chapman-Enskog expansions, we derive a constitutive equation for the Helmholtz free energy and identify the associated structural form of the entropy production rate. We then exploit additional information from the kinetic description to determine a constitutive relation for the entropy production rate itself. Finally, by applying the constrained maximisation procedure of Rajagopal and Srinivasa, we obtain constitutive equations for the Cauchy stress and couple-stress tensors, as well as for the energy and entropy fluxes. In this way we generalise the recent inviscid kinetic theory of Farrell, Russo, and Zerbinati to account for viscous, thermal, and spin-diffusive effects, using the simplest BGK-type approximation of the collision operator. Both compressible and incompressible variants of the theory are presented.

Katz conjectured in a 2018 lecture that the family of curves $y^2=x^d-dx+t$ over the $t$-line is generically ordinary for all sufficiently large primes $p$. We prove that, for every $g\ge 2$ and every nonzero algebraic integer $\alpha$, the genus-$g$ families $C_\alpha: y^2=x^d+\alpha x+t$ where $d\in\{2g+1, 2g+2\}$ are generically ordinary at every prime $p>P^+(d)$, provided that $\alpha$ is nonzero modulo every prime above $p$. The bound $P^+(d)=d^2-4d+2$ if $d$ is odd, and $P^+(d)=(d^2-3d+2)/2$ if $d$ is even. Finally, we show that for every algebraic integer $\alpha$ and every prime $p\equiv 1 \pmod d$, the family $C_\alpha$ is generically ordinary at every prime above $p$.