Papers

We show that the type A Springer representation is realized geometrically in the homology of the complete flag variety by Specht polynomials. For any partition, we identify the classical Specht polynomial generators of the Specht module with the classes of a family of disjoint Levi–Richardson varieties, and this family degenerates to the corresponding Springer fiber. This factors Springer's Schubert positivity problem for Springer fiber components through a chain of positive expansions, from Specht polynomials through the Joseph polynomials to the Schubert cycles. For two-row partitions we make each of these expansions combinatorially explicit, giving manifestly nonnegative Schubert cycle expansions of both the Levi-Richardson cycles and the Springer fiber components. This resolves Springer's question for two-row fibers and proves two conjectures of Precup and Sabando-Alvarez, and identifies the Springer basis with the web basis for two-row Specht modules. As an application, we deduce the Schubert cycle expansions of the components of the Poisson degeneracy locus of the flag variety.

Blind recognition of polar codes from noisy observations is a key problem in non-cooperative signal processing. Although existing blind successive cancellation list (BSCL) recognition exploits channel soft information, it performs two-hypothesis path expansion at every source-bit position, resulting in high complexity. In this paper, we first analyze the first recognition-error positions in the blind successive cancellation (BSC) recognition and observe that they are closely related to the corresponding contribution terms in the existing Bhattacharyya-parameter-based upper bounds. Based on this observation, a critical-set-aided simplified blind successive cancellation list (SBSCL) recognition method is proposed. SBSCL performs two-hypothesis path expansion only at the selected critical-set positions and keeps BSC recognition at the remaining positions, thereby reducing complexity. To improve the reliability of critical-set selection and refine the performance analysis, density-evolution (DE)-based bounds are further developed. Under the ideal SC-consistent condition, the synthetic log-likelihood-ratio (LLR) distributions obtained from density evolution are used to compute the optimized Chernoff coefficient for the upper bound and the overlap coefficient for the lower bound. Simulation results show that the DE-based bounds are tighter than the Bhattacharyya-parameter-based bounds. In the considered settings, the gap between the DE upper and lower bounds is within $1$ dB around a recognition-error probability of $10^{-2}$. Furthermore, SBSCL achieves nearly the same recognition success rate as BSCL, and the size of critical set decreases rapidly as the signal-to-noise ratio (SNR) increases.

The Kourovka Notebook is a long-running collection of open problems in group theory. In this paper we present solutions to eight of its problems. We construct a group with exactly two maximal locally soluble normal subgroups and show that, for every $1 \le k\le n!$, there is a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values. We also give examples showing that group order together with the statistic $\sum_g\varphi(\lvert g\rvert)$ does not determine simplicity, and we construct a surjective non-injective Rota-Baxter operator on a non-abelian group. Further, we determine the group generated by the class transpositions of moduli at most $k$, prove that every power graph of a finite group that is a cograph is chordal, show that the right-relatively convex subgroups of a right-orderable group need not form a sublattice of its subgroup lattice, and disprove a proposed rank inequality for certain $p$-group extensions. All of these solutions were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.

Let $f$ be any arithmetic function and define $S_f(x):=\sum_{n\leq x}f([x/n])$. If the function $f$ is small, namely, $f(n)\ll n^\varepsilon,$ then the error term $E_f(x)$ in the asymptotic formula of $S_f(x)$ has the form $O(x^{1/2+\varepsilon}).$ In this paper, we shall study the mean square of $E_f(x)$ and establish some new results of $E_f(x)$ for some special functions.

We study coalgebra measurings between multiplicative sequences of algebras and the maps induced by them on Hochschild homology. The Hochschild theory of multiplicative sequences is introduced as a functor taking values in graded algebras in the symmetric monoidal category of chain complexes, constructed with the help of the shuffle product. We develop the universal measuring coalgebra, or Sweedler Hom for multiplicative sequences, as well as study several other Sweedler operations in this context. In particular, we obtain an enrichment of multiplicative sequences over cocommutative coalgebras. Using an appropriate theory of bimodules over multiplicative sequences, we study maps induced by comodule measurings on the Hochschild theory with coefficients, as well as the corresponding enriched categories. Finally, we consider measurings and generalized Sweedler operations between multiplicative sequences induced by comultiplicative sequences of coalgebras, and also the maps in Hochschild theory obtained from them.

The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $\nu$ with $\nu[w]\le DP[w]^s$ for every finite prefix $w$ of the tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth. This tree max-flow/min-cut identity makes worst-case probabilities exact and optimizable. Three applications follow. In scheduling, $n$ tasks on one machine are each retried until the next random symbol lies in its success set; every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-\Theta(n)]$, and a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment satisfying a majority of each clause, we find the sharp source threshold $s_*$: below it, on some instances one admissible source forces nontermination for every clause selection; at it, with fixed $D$, the expected number of flips is at most cubic in the initial distance, against linear under independence, with matching instances at distances up to a constant fraction of the number of variables; a fixed margin above it restores linear dependence. For noisy predecessor search with prefix-bounded comparison errors, verified backtracking attains any prescribed error probability at or above a source threshold whose sharpness we leave open, with the same cubic-to-linear query bounds. Restarting on the shared tape restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.

This article is the first paper in the series. In this series of articles, we will examine the influence of the friction factor $\alpha$ at the solid–fluid boundary on the stability of Couette flow. Specifically, we consider the stability of Couette flow in a bounded periodic channel $\mathbb{T} \times [-1,1]$ under Robin-type boundary conditions ($u^2|_{y=\pm 1} = 0$, $[\alpha \partial_n u^1 + u^1]|_{y=\pm1} = f $), where $\alpha$ is the friction factor and $n$ is the unit outer normal vector. In this article, we prove that for a given friction factor $\alpha$, as long as the fluid viscosity coefficient $\nu\ll \alpha$ is sufficiently small, the system is asymptotically stable if the initial perturbation satisfies $\|\omega_{\rm in}\| \leq \epsilon \nu^{1/3}$. Moreover, inviscid damping and enhanced dissipation hold.

We study the higher rank deformed Hermitian-Yang-Mills (dHYM) equations for $SU(2)$-equivariant holomorphic vector bundles over $X\times \mathbb{P}^1$ for $X$ a compact Riemann surface. For a class of vector bundles of vortex type, we establish the equivalence between existence of solutions to higher rank dHYM equations and an appropriate notion of algebro-geometric stability, called $Z$-stability. We show that $Z$-stability is implied by, and conjecturally equivalent to, Bridgeland stability in $D^{b}{\rm Coh}^{SU(2)}(X\times \mathbb{P}^1)$.

We develop a categorical framework for closed dynamical systems in which the abstract pattern of admissible evolutions is separated from its concrete realisation. A closed dynamical system is formulated as a functor $X\colon S\to C$ from a small category $S$, viewed as an abstract evolution shape, to a coefficient category $C$. By varying $S$ and $C$, this single definition encompasses many important examples including autonomous, non-autonomous, switched, hybrid, and stochastic systems. Within this framework, we introduce invariant subsystems, equilibria, and orbits in functorial terms. We then formulate convergence by combining a cosieve-based intrinsic notion of eventuality on the evolution shape with neighbourhood filters of invariant subsystems. Finally, we establish a categorical Lyapunov principle based on categorical sublevel neighbourhoods. This yields abstract stability and convergence criteria that recover the classical Lyapunov method in standard examples.

As generative artificial intelligence enters scientific and professional work, its uncertainty must be defined on the states that matter for inference and decision-making. Language models assign probabilities to words, whereas applications require uncertainty over meaningful states such as diagnoses, hypotheses or operational conditions. We introduce a semantic map: a prespecified, testable bridge from probabilities over verbal responses to a posterior over declared finite states. The language distribution remains unrestricted; held-out calibration connects it to a reference posterior. We derive posterior-error bounds and conditions for existence, conditional uniqueness, presentation stability and stable inverse recovery. This distinction matters because language probabilities depend on prompt wording, while the target posterior should not change under information-equivalent rewording. Experiments use professional market text compiled from Federal Reserve economic and financial series, together with controlled simulations having exact posteriors. Across two fitted language models, language-derived probabilities outperform printed numerical confidence, recover held-out posteriors with valid uncertainty coverage, remain largely stable under paraphrase and respond appropriately to altered evidence. Prompt engineering optimises a wording-dependent response; robust scientific use requires validated stability of application-relevant meaning. The proposed map turns semantic uncertainty in generative systems into an identifiable and testable statistical measurement problem and, when its acceptance conditions hold, yields an auditable posterior estimate.

We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-$d$ free group for $d\ge 2$. We also give a new lower bound for the smallest growth rate of the genus-$g$ surface group for $g\ge 2$, as well as a lower bound for the growth rate associated with a one-relator presentation.

The generalized Fisher transformation maps a non-singular correlation matrix to an unconstrained real vector through the off-diagonal elements of its matrix logarithm. Evaluating its inverse is a computational bottleneck in dynamic correlation and multivariate volatility models. We develop a fast inversion algorithm by characterizing the unknown diagonal as the minimizer of a smooth, strictly convex, and coercive objective. An explicit Hessian and global spectral bounds identify the standard fixed-point iteration as a quasi-Newton method and explain why it can converge slowly near singularity. Every fixed-point step decreases the objective, and the iteration converges from every starting point. These results motivate GFT-FP+N, a hybrid of fixed-point and matrix-free Newton steps that never forms the Jacobian. In benchmarks with up to 1,000 replications per design and dimensions up to 800, GFT-FP+N reduces computation time by up to a factor of forty-five relative to the fixed-point iteration and converged in every replication, including on designs where Broyden's method almost always fails. Julia and R packages are provided.

Energy stability is fundamental in the numerical approximation of phase-field models, but it does not by itself guarantee faithful reproduction of local dynamics. Recent ODE-level studies have shown that energy dissipation alone does not ensure dynamical fidelity for spatially homogeneous phase-field models (Xu and Xu, 2023; Li and Wang, 2026). However, these ODE arguments do not extend directly to spatially inhomogeneous Allen–Cahn solutions: diffusion enters the full residual $\mathcal R_\varepsilon(u):=\Delta u+\varepsilon^{-2}(u-u^3)$ and may change its pointwise sign even when $0<u\leq1$. Motivated by this distinction, we introduce sign-dependent admissible classes determined by the full Allen–Cahn residual and prove that the exact PDE flow preserves the corresponding pointwise monotone-growth or monotone-decay direction. The fully implicit Euler method inherits this structure in its standard unique-solvability regime. We then examine several widely used energy-stable schemes. First-order convex splitting and sufficiently strong first-order stabilization can preserve the correct direction, but on a reduced effective time scale, causing artificial delay or damping. By contrast, second-order convex splitting, stabilized CN/AB, IEQ, and SAV schemes admit monotone data for which large time steps generate wrong-signed pointwise increments despite dissipation of the original or a modified energy. For all four schemes, these strict reversals persist under sufficiently small smooth spatially nonhomogeneous admissible perturbations, so the counterexamples remain valid with genuinely active diffusion. Numerical experiments confirm this classification. Thus, pointwise monotonicity is a local dynamical criterion complementary to energy stability and maximum-bound preservation.

We establish pathwise large-deviation bounds for the empirical-measure flow of a cluster coagulation model with gelling kernels on a locally compact Polish state space. The cluster coagulation process is a measure-valued Markov jump process in which pairs of clusters merge according to a time-inhomogeneous kernel $K(t,x,y,d z)$. Kernels with sufficiently rapid growth may produce gelation: a phase transition in which mass escapes from the finite-cluster population in finite time. We first prove tightness of the empirical flows and identify every subsequential limit as a solution of the time-inhomogeneous multi-type Flory equation, which accounts for post-gelation dynamics through a conserved quantity. We then establish a global large-deviation upper bound on path space with a variational rate function. We also prove a local lower bound near paths that are unique solutions of suitable tilted Flory equations. Finally, we give a convex-dual representation of the upper rate function, including the possible singular part of the representing measure. Our results apply to locally compact Polish state spaces with continuous mass functions, encompassing both the gelling and non-gelling regimes within a unified framework.

A commutative cancellative monoid has the MCD property if every nonempty finite subset has a maximal common divisor, and it has the MCD-finite property if every such subset has only finitely many maximal common divisors up to associates. We study these properties in monoid algebras. We first identify atomic and antimatter families of rank-$1$ torsion-free MCD monoids. We then prove that, for a field $F$ and a torsion-free monoid $M$, the MCD property ascends from $M$ to $F[M]$ whenever the group algebra $F[\mathcal{G}(M)]$ satisfies the ascending chain condition on principal ideals ($\mathcal{G}(M)$ denotes the Grothendieck group of $M$). We also record the pre-Schreier case as a consequence of classical GCD ascent. In contrast, for every prime $p$, we construct a reduced rank-$3$ MCD monoid $M$ such that $\mathbb{F}_p[M]$ is not an MCD domain, and an atomic rank-$1$ MCD monoid whose algebra over $\mathbb{F}_p$ is not atomic. We further prove that the prime reciprocal monoids considered here are MCD-finite and that MCD-finiteness ascends to polynomial extensions. Finally, for every prime $p$, we construct a rank-$1$ torsion-free q-GCD monoid whose algebra over $\mathbb{F}_p$ is not q-GCD, where the q-GCD property requires at most one maximal common divisor up to associates for every nonempty finite subset.

The Diminished Sombor index $(DSO)$ of a simple graph $G$ is a recently introduced degree-based topological index, defined as \[ DSO(G)= \sum_{uv\in E(G)} \frac{\sqrt{d_u^2+d_v^2}} {d_u+d_v} \] In this paper, we establish a collection of new sharp bounds for the $DSO$ index in terms of several fundamental graph parameters and well-known topological indices. In all cases, the proposed bounds rigorously specify the equality conditions, thereby providing a complete characterization of the extremal graphs. These findings deepen the theoretical understanding of the $DSO$ index and provide powerful analytical tools for interdisciplinary applications in mathematical chemistry.

In this paper, we study the factorization of sums of two squares X2 + Y2. We show the existence of linear and quadratic progressions on X that generate both factorizations and alternative decompositions into sums of two squares. We are therefore naturally interested in the number of decompositions into sums of two squares of an integer n and relate it to its number of divisors d(n). Finally, we present an algorithm to obtain these decompositions using the sieve introduced in (1).

Inference-behaviour semantics (I-bS) is a new approach to proof-theoretic semantics, grounded in two inferentialist principles: (1) the use of an expression in reasoning determines its meaning, and (2) a connective is defined by its operational rules. I-bS operationalises these ideas by measuring the syntactic use of a connective in the proof of its definability, against a substructurally minimal derivability relation. I-bS thereby gives the meaning of a connective in terms of its semantic clause, i.e. minimal substructural rule pair. This paper validates and verifies I-bS by applying it to all $10,816$ connective rule pairs that can be formulated in two-dimensional sequent calculi using at most two premiss sequents and at most two active formulae. As a result, we find semantic clauses for exactly $21$ meaningful connectives, namely bottom, top, two negations (intuitionistic and dual-intuitionistic), group and lattice conjunction, disjunction and implication, as well as their converses and inverses. We use these results to precisely map the semantic interrelations among the connectives, across linear, classical, intuitionistic, dual-intuitionistic, minimal, and lattice logic. Most notably, we find that intuitionistic negation, disjunction and implication each capture half of the meaning of their classical counterparts.

Unlike an ordinary fraction with an infinite numerator and denominator, an infinite continued fraction must be irrational. Euler was the first to show that the base of the natural logarithm $e$ is irrational, by numerically estimating its continued fraction, and showing the infinite sequence of convergents $p_n/q_n$ so obtained converged to $e$ using the Ricatti equation. Hermite showed that the recurrence relations for these convergents correspond to the recurrence relations between certain improper integrals, so proving the continued fraction tends to $e$ in the limit of infinite $n$. Here we provide a motivation for the integrals involved and obtain closed form integral representations for $p_n$ and $q_n$ for all $n$. An interesting feature, is that the above results provide a short cut to the standard proof of the irrationality of $\pi$.

In this work, we provide a complete analytical characterization of amplitude modes in a nonlinear Van der Pol network. The phase condition presented in the paper fully classifies all possible configurations, leading to a finite set of discrete solutions. This result provides a complete description of the limit cycles, including all stable oscillatory states for arbitrary network size, without relying on approximate or iterative numerical schemes. In addition, we provide a constructive theorem that yields an iterative scheme for approximating solutions of the corresponding boundary-value problem.

For the circulant graph $C_n(1,2)$ with $n\ge10$ even, the $\F_2$ system requiring every quadrilateral to be unbalanced is consistent and its solutions form exactly four switching classes. We show that the class containing the signing which is $+1$ on step-$1$ edges and $(-1)^i$ on step-$2$ edges has spectrum $\{\pm2\sqrt{\cos^2\theta_k+\cos^2 2\theta_k}\}$ and spectral radius exactly $2\sqrt2$, well below the Kesten bound $2\sqrt3$; that the quadrilateral system is equivalent to alternating triangle fluxes, so that the four classes are coordinatized by $(\tau_0,\alpha)$ and the spectral radius depends only on the Hamilton-cycle holonomy $\alpha$; and that the two twisted classes attain $\rho_-(n)=2\sqrt{\cos^2(\pi/n)+\cos^2(2\pi/n)}<2\sqrt2$. Exhaustive enumeration of all $2^{n+1}$ switching classes for $n\in\{8,10,12,14,16,18\}$ shows that $\rho_-(n)$ is the global minimum in every case, and we conjecture this for all even $n$; the lower bound is a flux-minimization statement in the sense of Lieb's flux-phase theorem. For odd $n$ the quadrilateral system is inconsistent.

The definitions of projective objects and Gorenstein projective objects in the category of heaps of $T$-modules are posed, where $T$ is a truss. It is shown that a heap of $T$-modules $P$ is projective if and only if $\mathcal{G}_{e_p}(P)$ is a projective $R(T)$-module for all $e_p\in P$ and a heap of $T$-modules $M$ is BP Gorenstein projective if and only if $\mathcal{G}_{e_m}(M)$ is a Gorenstein projective $R(T)$-module for all $e_m\in M$. Moreover, we give a functorial description of the BP Gorenstein projective dimension. Finally, it is also proven that a unital truss $T$ is a Gorenstein truss if and only if $R(T)$ is an Iwanaga-Gorenstein ring.

Consider a reductive complex algebraic group $\mathbb{G}$ equipped with an action by a linearly reductive affine group scheme $\mathbb{K}$. The extension of $\mathfrak{p}^*$ by $\mathfrak{p}$ induced by an $(\mathbb{K},\mathfrak{g})$-invariant symmetric non-degenerate bilinear form on $\mathfrak{g}:=Lie(\mathbb{G})$, for a $\mathbb{K}$-invariant parabolic $\mathbb{P}\le \mathbb{G}$, is $\mathbb{K}$-equivariantly isomorphic to the extension obtained via the standard bialgebra structure attached to a $\mathbb{K}$-invariant Cartan/Borel pair $\mathbb{H}\le \mathbb{B}\le \mathbb{P}\le \mathbb{G}$ and the same bilinear form. Associating bundle extensions on an elliptic curve $E$ to said $\mathfrak{p}$-module extensions, this identifies Poisson structures on the smooth locus of the principal-$\mathbb{P}$-bundle moduli space over $E$ respectively defined by Balduzzi (using the former extension) and Feigin-Odesskii (via the standard bialgebra structure). This in particular verifies Feigin-Odesskii's identification of the bialgebra-induced symplectic leaves with the loci of bundles mutually isomorphic after forgetting structure along $\mathbb{P}\le \mathbb{G}$.

This manuscript investigates the Cauchy problem for a class of nonlinear integro-differential equations governing anomalous super-diffusive transport in $\mathbb{R}^N$. The linear dynamics are driven by a dual-scale memory kernel whose Laplace transform is sectorial and exhibits distinct power-law asymptotics at high and low frequencies. This super-diffusive structure precludes the infinite regularizing capacity characteristic of classical parabolic theory; consequently, the associated resolvent operator possesses a heavy algebraic tail in Fourier space, acting as a pseudo-differential operator in the Hörmander class $S^{-2}_{1,0}$ and restricting spatial smoothing. By establishing rigorous $L^q-L^p$ multiplier estimates, the critical Lebesgue threshold $q_c$ for local well-posedness is determined. To demonstrate the sharpness of this threshold, instantaneous norm inflation – and consequent ill-posedness – is proven in the supercritical regime $1 < q < q_c$. Furthermore, tracking the structural crossover to the long-time relaxation parameter resolves the global asymptotic dynamics. The nonlocal Fujita-type critical exponent $\rho_F$ is identified, and global-in-time existence along with algebraic decay is established for small initial data in intersection spaces, provided the nonlinearity remains supercritical and overcomes the structural algebraic barrier connecting the dual scales. This general framework applies directly to canonical physical models, including Cole-Cole fractional retardation and multi-scale Prabhakar memory.

Given a finite dimensional representation $\pi$ of a finitely generated group $G=\langle g_1, \ldots, g_n\rangle$, the associated characteristic polynomial is defined as $Q_\pi(z):=\det(z_0I+z_1\pi(g_1)+\cdots +z_n\pi(g_n))$, and it is known to contain a good amount of structural information about $G$ and $\pi$. This paper is a part of an ongoing project to investigate the number-theoretic properties of the algebraic varieties (called eigensurfaces) $\{z\in \mathbb{C}^{n+1}: Q_\pi(z)=0\}$. Its focus is the distribution of prime triples in the eigensurface $S:=\{z\in \mathbb{C}^3: (z_0+z_1+z_2)^2+z_0z_1=0\}$ associated with the braid group ${\bf B}_3$ and its reduced Burau representation. We prove that such triples occur with higher frequency on $S$ than in the ambient lattice, revealing an unexpected connection between group representation theory and analytic number theory.

The strong chromatic index $\chi'_s(G)$ is the smallest number of colours needed to colour the edges of a graph $G$ so that any two edges at distance at most $2$ receive different colours. Using the local flag algebra framework introduced in a companion paper, we prove $\chi'_s(G) \leq 1.73\,\Delta(G)^2$ for every graph $G$ of maximum degree $\Delta(G)$, $\chi'_s(G) \leq 1.6255\,\Delta(G)^2$ for every bipartite $G$, and $\chi'_s(G) \leq 1.6633\,\Delta_A(G)\,\Delta_B(G)$ for every bipartite $G$ of side maximum degrees $\Delta_A(G), \Delta_B(G)$ with rational $\Delta_B(G)/\Delta_A(G) \in (0, 1]$, provided $\Delta(G)$, $\Delta_A(G)$, $\Delta_B(G)$ are sufficiently large. These three bounds make progress towards three established conjectures: those of Erdős-Nešetřil (1985) for general graphs, Faudree-Gyárfás-Schelp-Tuza (1989) for bipartite graphs, and Brualdi-Quinn Massey (1993) in the asymmetric bipartite setting. Additionally, for the random bipartite graph $G \sim G(n_A, n_B, p)$ at constant $p \in (0,1)$ and bounded aspect ratio $\max(n_A, n_B) = O(\min(n_A, n_B))$, we prove the Brualdi-Quinn Massey bound $\chi'_s(G) \leq \Delta_A(G)\,\Delta_B(G)$ asymptotically almost surely.

Linear attention promises constant-time recurrent inference but degrades sharply on associative recall. We formulate attention recall as a spherical-packing problem and introduce Kernelized Linear Attention Activations (KATA), a framework whose feature maps are derived from first principles by certifying nonnegative attention weights through a self-dual homogeneous cone. Building on this observation, we show that rank-one positive semi-definite (PSD) features offer a favorable capacity–interference tradeoff. KATA recovers a parameter-free convex output gate and characterizes associative capacity through the Welch interference floor. For tolerances above this floor, KATA enlarges the state without adding parameters and admits spherical codes with exponentially many keys in the projection dimension. We implement KATA as fused Triton kernels at two operating points: a flash-attention-style forward up to ${\sim}1.6\times$ FlashAttention-2 throughput, and an exact $O(T)$ chunked-state form that reaches ${\sim}11\times$ FlashAttention-2 forward throughput at $131$k tokens. An associative scan of the first-order feature lowers the inter-chunk recurrence depth to $O(\log(T/C))$ for chunk size $C$ and averages ${\sim}2.4\times$ the throughput of a matched sequential linear-attention baseline. On long-range MQAR and repeated-key overwrite, several KATA variants outperform Gated DeltaNet, with parameter counts and state sizes reported alongside accuracy. Induction preserves near-perfect recall, while kernel benchmarks show that the maps can be implemented efficiently. KATA retains $0.985$ MQAR at a $16\times$ out-of-distribution length, approaching the softmax with roughly one quarter of the KV-cache entries. Experiments on 340M-parameter LLMs reveal a feature-dependent fluency trade-off and clarify how positional embeddings, delta rules, and decay gates interact with feature geometry.

We generalize the results of the first part of the paper by Kiritchenko, Tsfasman, Vlăduţ and Zakharevich on quadratic residue patterns to the case of arbitrary words over the alphabet $\{R, N\}$ of length $l = 4$ and $l = 5$. First, we obtain explicit formulas for the number of occurrences of a given pattern $S$ in the sequence of quadratic residues and nonresidues modulo a prime $p$. These formulas are expressed in terms of Frobenius traces on a finite collection of low-genus hyperelliptic curves. Second, using a generalized Sato-Tate approach, we describe the limiting distributions of the normalized error term for $l = 4$.

We derive residual-based a posteriori error estimates for parabolic surface PDEs on closed evolving surfaces. The main contribution is to prove efficiency and reliability for the proposed error indicator, which bounds the error quantities globally from above and globally in space and locally in time from below. We extend methods for adaptivity for parabolic PDEs on stationary surfaces to allow for non-trivial coarsening on evolving surfaces. Multiple numerical experiments are given, which illustrate the asymptotic behaviour of the error and effectiveness of the refinement and coarsening.

We study phase-field and sharp-interface models for contact line dynamics of a liquid droplet on a solid substrate within a unified variational framework. The motion of the contact line, where liquid, gas, and solid phases meet, poses a fundamental difficulty in continuum modeling due to the classical stress singularity of no-slip hydrodynamics. Phase-field models regularize this singularity by introducing a thin transition layer of thickness and encoding interfacial effects through a Ginzburg-Landau free energy augmented by a wall energy on the substrate. Starting from the total free energy $E = E_b + E_w$, we analyze two phase-field models: the Allen-Cahn equation and the Cahn-Hilliard equation. Using matched asymptotic expansions as $\delta \to 0$, we recover their corresponding sharp interface limits. In the Allen-Cahn case, the limit yields motion by mean curvature with a contact line law driven by deviations of the dynamic contact angle from Young's angle. In the Cahn-Hilliard case, the limit leads to a Mullins-Sekerka problem with the same form of contact line dynamics. A central result of this work is the identification of consistent gradient-flow structures across both models. The Allen-Cahn dynamics correspond to an $L^2$-gradient flow, while the Cahn-Hilliard dynamics correspond to an $H^{-1}$-gradient flow, and both converge to sharp-interface evolutions that preserve the same energy-dissipation structure. This provides a unified interpretation of contact line motion as a consequence of a single variational principle. Finally, we develop energy-stable numerical schemes based on the minimizing movement principle and establish discrete energy dissipation and well-posedness of the fully discrete problem. Numerical examples confirm that both schemes relax toward the same stationary sharp interface solution while their dynamics reflect the different dissipation mechanisms.