Papers

Recent work has shown that classifying large language models (LLMs)' responses can be distinguished by modeling token embeddings as trajectories of a black-box dynamical system (DS) and comparing prediction residuals of two DSs. Despite the empirical success of this dynamical approach, a theoretical understanding of why it works, how well it scales as a function of the token sequence, and when it transfers across embedding models remains lacking. We address these questions by formalizing the classification task as a binary hypothesis test between two stochastic linear DSs. We show that the total variation distance between the stationary marginal distributions of the two DSs can be arbitrarily small even when the dynamics differ substantially, which provides a fundamental accuracy floor for any classifier that ignores token dynamics. We then show that the misclassification probability of DS-based classification decays exponentially in the sequence length $L$, with the decay governed by a dynamical discriminability quantity $\delta^2$ that captures the spectral distance between the two DSs. We also characterize cross-embedding generalization by introducing an approximate intertwining condition between embedding models and establishing a lower bound on the transferable discriminability in terms of the intertwining map's smallest singular value. Together, these results explain the empirical performance of DS-based classification and motivate further investigation into using DS theory to analyze AI systems, in contrast to the more common approach of using AI to model dynamical systems.

There is growing interest in stochastic dynamics in the applied mathematics, interdis-ciplinary science, and artificial intelligence communities. This book is written primarilyfor applied mathematicians and scientists who may not have the necessary backgroundto go directly to advanced reference books or research literature in geometric methodsfor stochastic dynamics. Our goal is to present a few selected geometric methods for understanding solutionsof stochastic differential equations. Some necessary background is reviewed, while moreadvanced materials are referred to relevant literature. It is our hope that this book willprovide a guide to the reader in accessing advanced monographs and research literaturein stochastic dynamics. We have tried to strike a balance between mathematical precisionand accessibility for the readers of this book. For example, some proofs are presented,whereas some are outlined and others are directed to references. Some definitions arepresented in separate paragraphs starting with Definition, but others are introduced lessformally as they occur in the body of the text. This book may be used as a textbook or a reference for advanced doctoral students, postdoctoral fellows, researchers and practitioners in applied mathematics, machinelearning, artificial intelligence, engineering and applied science.

We introduce an extremal invariant associated with Chowla-type order conditions in finite groups. A nonempty subset $S$ of a finite group $G$ is called a Chowla set if every element of $S$ has order greater than $|S|$, and we write $C(G)$ for the maximum cardinality of such a set. We first show that $C(G)$ is determined by the distribution of element orders in $G$. For cyclic groups, we derive an exact divisor formula and characterize the integers $n$ for which $C(\mathbb{Z}/n\mathbb{Z})=\varphi(n)$. We prove that $\liminf_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=1$, whereas $\limsup_{n\to\infty}C(\mathbb{Z}/n\mathbb{Z})/\varphi(n)=\infty$, and we determine the corresponding lower and upper limits under normalization by $n$. For finite abelian groups, we obtain an explicit formula in terms of the invariant-factor decomposition, together with a closed formula for finite abelian $p$-groups. We then develop a linear analogue for finite field extensions. A nonzero $K$-subspace $A$ of an extension $L/K$ is called a Chowla subspace if $[K(a):K]>\dim_K A$ for every nonzero $a\in A$. Since this condition depends on $\dim_K A$, it does not generally require every nonzero element of $A$ to generate $L$ over $K$. Nevertheless, when $L/K$ is finite and separable, we prove the exact formula $C(L/K)=[L:K]-d_{\max}(L/K)$, where $d_{\max}(L/K)$ is the largest degree over $K$ of a proper intermediate field. For finite fields, we give a direct proof in every degree using a normal-basis construction. This work was developed through an expert-guided human-AI collaboration. A reasoning-focused configuration of Co-Scientist was used to explore examples and potential proof strategies. The authors formulated the problem, independently verified and completed all arguments, and wrote the final proofs.

The axiomatic convergence theory for multigrid methods applied to cell-centred finite-difference and finite-volume discretisations rests on two hypotheses: an imbalanced Galerkin condition (G3), which states that $R_{\ell-1}A_\ell P_{\ell-1}=2A_{\ell-1}$ with $R_{\ell-1}=\frac{1}{2}P_{\ell-1}^T$, and a weak approximation property $(A2)_\alpha$ of Bramble type. Under these hypotheses, together with Richardson smoothing, the symmetric W-cycle and the variable V-cycle are known to be uniformly convergent, while the uniform convergence of the standard symmetric V-cycle has remained open. We answer this in the negative by two constructions. First, for every smoothing count $m$ we exhibit hierarchies of every depth satisfying (G3), Richardson admissibility with $C_R=1$, and $(A2)_\alpha$ for every $\alpha\in(0,1]$ with the sharp level-independent constant $C_{A2}^2=4m$, whose symmetric $V(m,m)$-cycle error operator has spectral radius $\theta_m(1+2\theta_m)>1$ already on three levels, where $\theta_m=(1-\frac{1}{4m})^{2m}$, and growing geometrically with the depth; the family shows that any smoothing-count threshold $m_0$ that could restore uniform V-cycle convergence must grow at least quadratically in $C_{A2}$. Second, we prove that the same failure occurs in a completely standard discretisation: the cell-centred finite-volume hierarchy for a one-dimensional diffusion equation with a mesh-aligned coefficient jump $1:\kappa$ and harmonic (Samarskii) interface averaging satisfies (G3) exactly and $(A2)_{1/2}$ with a level-independent constant $C_{A2}=O(\kappa)$, yet for every $\kappa\ge3$ its symmetric $V(1,1)$-cycle with any admissible Richardson parameter, including the optimal one, diverges geometrically in the number of levels. In both constructions the W-cycle remains uniformly contractive, so the hypotheses separate the two cycles. All claims are verified numerically.

When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map? We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit. The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library. Pure schedules approach the relaxed class at rate $O(D^{-1})$ under bounded-variation time dependence and $O(D^{-\vartheta}+D^{-1})$ under Holder dependence of exponent $\vartheta$. Execution arithmetic can reverse this conclusion: full-state write-back introduces a $D\rho_z$ penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law. A fixed-teacher converse makes this rate sharp: for coherent depth-$L$ first-order high-precision comparators, accuracy matching requires $D=\Theta(L)$. Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions. Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training. Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core. Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.

We study one-dimensional particle swarm optimization during stagnation, with two fixed distinct attractors and equal independent uniform acceleration ranges. The position then satisfies a second-order random affine recurrence. For inertia $w$ and acceleration range $c$, we prove that throughout the open mean-square stability region \[ -1<w<1,\qquad c>0,\qquad 12(1-w^2)-c(7-5w)>0, \] no invariant position marginal, and hence no limiting position marginal, can be Gaussian. This solves the open Problem 18 in \cite{ParticleSwarmProblems}. The proof compares the stationary moment equations with the Gaussian moment identities through order eight. A Hermite-polynomial formulation gives explicit fourth- and sixth-order compatibility conditions whose common solutions lie on a degree-$107$ polynomial branch. Exact eighth-order equations exclude every point on that branch. The final certificate is verified using arithmetic modulo $23$ and independently modulo $1{,}000{,}003$. A separate raw-moment implementation produces exact polynomials $Q_4,Q_6,Q_8$ in $(w,c)$ and verifies the same obstruction over the rational numbers. The fourth- and sixth-order curves have a genuine admissible intersection, but the eighth-order condition removes it, showing why low-order Gaussian diagnostics are insufficient. All code, exact polynomials, logs, and plot-validation data are supplied as online resources.

Let $H_\Gamma$ be the Bestvina-Brady group associated to a finite connected graph $\Gamma$. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism $\mathrm{IAut}(A_\Gamma)\cong \mathrm{IAut}(H_\Gamma)$ compatible with the Andreadakis-Johnson filtrations. Second, the quadratic and cubic lower-central relation spaces, together with the separator arrangement detected by the Bieri-Neumann-Strebel invariant, determine a rational associative algebra $\mathscr{C}_\Gamma$. Every integral rank-one square-zero element of this algebra is realized by an automorphism of $H_\Gamma$, and the subgroup generated by these roots has finite index both in the cohomological image of $\mathrm{Aut}(H_\Gamma)$ and in the unit group of an integral order in $\mathscr{C}_\Gamma$. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of $H_\Gamma$. Relative free-product automorphism theory then implies that $\mathrm{Aut}(H_\Gamma)$ and $\mathrm{Out}(H_\Gamma)$ are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that $\mathrm{Aut}(H_\Gamma)$ is finitely presented if and only if $\mathrm{Out}(H_\Gamma)$ is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for $\Gamma_m=C_m\vee K_3$ with $m\geq 5$, $\mathrm{Out}(H_{\Gamma_m})$ is of type $F_\infty$, whereas $\mathrm{Aut}(H_{\Gamma_m})$ is of type $F_3$ but not $F_4$. We also construct a type-$F_\infty$ Bestvina-Brady group whose automorphism and outer automorphism groups are finitely generated but not finitely presented, and show that $H_{C_n}$ is not finitely presented for $n\geq 5$, whereas $\mathrm{Out}(H_{C_n})$ is virtually infinite cyclic.

We introduce the notions of the cohomological dimension and the homological dimension of Lie algebra homomorphisms, extending the corresponding invariants for group homomorphisms. We establish their basic properties, including their behavior under monomorphisms, epimorphisms, pullbacks, composition, and restriction to the image. We also obtain a chain homotopy characterization of the cohomological dimension and compute these invariants explicitly for homomorphisms between free and abelian Lie algebras.

We prove a refinement of Jackson's theorem on the approximation of Lipschitz functions by trigonometric polynomials. Our result precisely characterizes the leading error term associated with Jackson's construction. We do the same for a related construction commonly used in the kernel polynomial method for spectral density estimation, which is slightly better than Jackson's construction in this respect.

We develop a scalable approach to computing the magnitude homology Euler characteristic for directed acyclic graphs based on decategorification. Along with motivating mathematical results and some simple controlled examples, we deploy the Euler characteristic in a proof of concept application to the dynamic analysis of multilayer perceptrons, recovering class-discriminative structure while holding simpler subgraph properties fixed.

Applications of metric magnitude often rely on numerically exact results in order to exploit a connection with information theory. We examine various approaches for scaling the dense linear algebra involved and identify hierarchical low-rank solvers as a preferred approach, with a clear path to scales of $10^5$ points on a single powerful workstation, and larger scales using our containerized CUDA-enabled C++/MPI pipeline.

We study Julia sets arising from non-autonomous iteration of monomials on the Riemann sphere. We give a precise description of the Julia set associated to an arbitrary sequence of monomials in terms of their coefficients and degrees. Combined with the basic invariance relation for non-autonomous Julia sets, this yields a plethora of striking examples in polynomial non-autonomous dynamics: finite Julia sets of arbitrary cardinality, non-trivial Julia sets with non-empty interior, Julia sets that are perfect but not uniformly perfect, and Julia sets with empty interior but positive area. We also describe a connection between autonomous and non-autonomous Julia sets of monomial sequences through Kuratowski limits.

The paper continues the investigation of conditions for the existence of minimal networks in metric spaces. We introduce the notion of a compactly equipped pseudometric space, in which every closed ball is equipped with a compact topology with respect to which the original pseudometric is lower semicontinuous. It is proved that these conditions are sufficient for the existence of minimal parametric networks of any type. This generalizes the corresponding theorems of Ivanov, Tropin and Tuzhilin on the existence of such networks in proper metric spaces. Moreover, it allows us to give a more general formulation of Bednov's theorem on sufficient conditions for the existence of minimal Steiner trees in Banach spaces. We also show which hyperspaces and under what conditions inherit this compactly equipped property.

We present a new algorithm to determine whether a compact orientable surface properly embedded in a compact orientable 3-manifold is incompressible. As a special case, this provides a new algorithm to detect the unknot. The central technique is the use of hierarchies. Unlike previous algorithms, no challenging invariants need to be computed and no lengthy search procedures are required. This algorithm leads to a new proof that determining whether a compact orientable 3-manifold has incompressible boundary is in NP.

We define a coupling product and a draw coproduct on permutations and show that they define a graded, connected, cocommutative, free Hopf algebra $\mathbb{KS}$. In characteristic zero, this implies that $\mathbb{KS}$ is isomorphic to a certain cocommutative Hopf algebra associated with the dual of the coradical filtration of the Malvenuto–Reutenauer Hopf algebra on permutations. The coupling product and draw coproduct are permutation analogues of the product and coproduct on the monomial basis of symmetric functions; therefore, we can say that our presentation is a monomial basis for $\mathbb{KS}$.

We study the size of the largest monochromatic connected component that must appear in any edge-coloring of a random graph. Let $G\sim G(n,p)$ with $p\gg 1/n$ and $p=o(1)$, and write $np=he^h$. We show that, with high probability, every $2$-edge-coloring of $G$ contains a monochromatic connected component of order at least $n-\Theta(ne^{-h})$. Moreover, we construct colorings showing that this bound is best possible up to constant factors. We extend this result to three colors: for $p\gg 1/n$ and $p=o(1)$, with high probability every $3$-edge-coloring of $G$ contains a monochromatic connected component of size at least $\frac{n}{2}-\Theta(1/p)$, and this estimate is again tight up to constant factors. In the bipartite setting $G\sim G(n,n,p)$, under the same assumptions on $p$, we prove an analogous statement: with high probability, every $2$-edge-coloring contains two monochromatic components whose union covers all but $\Theta(ne^{-h})$ vertices, and this bound is asymptotically sharp. Our approach is elementary and is based on analyzing large connected structures across suitably balanced vertex partitions.

We investigated how a magnetic topological defect affects the vacuum polarization of a charged massive scalar field in a flat $(3+1)$-dimensional space-time. The defect was modeled as an impenetrable to matter field finite-thickness tube with magnetic flux inside. We implemented the most general form of the Robin boundary condition on the surface of the magnetic tube, which enables a fully general analysis of the problem. We have found that in flat spacetime, the total vacuum energy generated by a magnetic topological defect depends on the curvature $\xi$, except for special cases corresponding to the Dirichlet and Neumann boundary conditions. By contrast, when Robin's general boundary conditions are imposed, the induced vacuum energy acquires an explicit dependence on the curvature coupling $\xi$, which is significant even in flat space-time. A detailed study of the dependence of the effect on the boundary condition parameter has been carried out. The obtained results highlight the nontrivial role played by boundary conditions in vacuum polarization phenomena.

Topological semantics for affine modal propositional logic is introduced. The interior operator on subsets is replaced with the lower semi-continuous envelope operator on functions. Completeness and affine compactness theorems are proved for this logic.

For a given real matrix $A\in\R^{n\times n}$ of order $n\geq 1$, a Pareto eigenvalue is a scalar $\lambda\in\R$ for which there exists a nonzero vector $x\in\R^n_+$ such that $Ax-\lambda x\in\R^n_+$ and $\langle x,Ax-\lambda x\rangle=0$. It is knwon that a real matrix of order $n=3$ can have at most $9$ distinct Pareto eigenvalues. We study the matrices for which this maximal number is attained and classify the way in which the $9$ Pareto eigenvalues are produced by their supports. A Pareto eigenvalue may be produced by more than one support. We therefore choose one producing support for each distinct values. The main contribution of this paper is to prove that, for every such choice, the numbers of values assigned to supports of sizes $1$, $2$, and $3$ are necessarily $$ (1,5,3),\qquad (2,4,3),\qquad\text{or}\qquad (2,5,2). $$ The proof uses bounds from the order $n=2$ problem and several restrictions on singleton, pair, and full supports. In particular, the graph formed by the pair supports producing $2$ values contains no triangle. These arguments also gives anohter proof that the maximum Pareto capacity in order $3$ is equal to $9$. For each of the $3$ profiles, we give an explicit full-capacity matrix with $9$ regular Pareto eigenvalues. We also prove that each profile occurs on a nonempty Euclidean-open set where every Pareto eigenvalue has a unique producing support.

The game of coding framework was introduced to extend coding-theoretic recovery beyond its traditional limit, under which the number of honest reports must exceed the number of adversarial or corrupted reports. It does so by exploiting the rational behavior of adversarial participants and their incentive to keep the system live. Existing game-of-coding formulations, however, assume that the adversarial-noise distribution is independent of the realized ground-truth computation. This assumption may be restrictive when an informed adversary can adapt its reports to the value being computed. In this paper, we study the game of coding with input-dependent adversarial noise. We introduce a unified multi-node, multidimensional formulation. For every family of conditional adversarial-noise distributions, we construct an input-independent joint noise distribution, and prove that this reduction exactly preserves the probability of acceptance and the accepted mean-squared estimation error. Consequently, the input-dependent and input-independent models have identical achievable performance regions, and the same equilibrium utilities.

We develop a rigorous variational framework for uniaxial nematic liquid crystals interacting with an external electric field in the one-constant Oseen–Frank approximation. Equilibrium configurations are governed by a nonlocal, nonlinear energy with a challenging min-max saddle-point structure. Our first main result reformulates this problem as a pure double-minimization problem. Using convex duality and the Hodge decomposition, we replace the scalar electrostatic potential with a vector potential, yielding a closed-form dual functional with a unique minimizer. This direct energy-minimization principle is advantageous for both theoretical analysis and numerical simulation. Our second result rigorously quantifies the decoupling of the electrostatic back-reaction in the limit of small dielectric anisotropy. We establish a uniform quadratic energy bound between the exact nonlocal energy and its standard local approximation, formally justifying the widespread physics heuristic of neglecting induced depolarization fields. Finally, under a strict coercivity condition, we combine $\Gamma$-convergence, uniform Sobolev regularity, and a perturbative coercivity transfer to prove that physical minimizers converge to limiting harmonic maps at an optimal, quantitative linear rate.

Let $T$ be a countable complete theory with a distinguished unary predicate $P$, and let $T^{P}$ be the theory of the $P$-parts of models of $T$ with the induced structure. $T$ is said to be relatively categorical or categorical over $P$ if any isomorphism between the $P$-parts of two models of $T$ lifts to an isomorphism of the models in question. We study the special case of relative categoricity where $T$ is internal to $T^{P}$ (that is, every model $M$ of $T$ is in the definable closure of $P(M)$ together with additional parameters from $M$). We first give a structure theory for such $T$: after passing to $T^{eq}$ and naming a parameter, $T$ is the same thing as a "pure torsor cover" of $T^{P}$, namely simply adjoining to $T^{P}$ a new sort for a torsor $S$ for a $\emptyset$-definable group $G$ in $T^{P}$, with no additional structure. We discuss relative stability, or stability over $P$, and give a characterization of relative stability, superstability, and $\omega$-stability of $T$ in terms of $H$ having the stable chain condition, superstable chain condition, and $\omega$-stable chain condition, respectively.

We present a proof that determines the optimal value of the universal constant in the thin-shell theorem for log-concave distributions in high dimensions. We prove that for any log-concave random vector $X = (X_1,\ldots,X_n)$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ {\rm Var}( |X|^2 ) \leq 8 n. $$ The constant $8$ is optimal: equality is attained when $X_1,\ldots,X_n$ are independent, identically distributed, standard, centered exponential random variables. Moreover, among isotropic random vectors distributed uniformly on convex bodies in $\mathbb{R}^n$, the quantity ${\rm Var}(|X|^2)$ is maximized by the uniform distribution on a regular simplex. We also provide a corresponding sharp bound on the Hilbert-Schmidt norm of the tensor of $3^{rd}$-moments of isotropic, log-concave distributions. The argument relies on the analysis of a weighted Riemannian manifold associated with log-concave moment measures and the Monge-Ampère equation. This manifold was studied in this context in \cite{lc_moment}. The main improvement over \cite{lc_moment} comes from a concise yet effective analysis of the $3^{rd}$-derivatives tensor of the potential. The proof was found by GPT-5.6 Pro in response to prompts supplied by the first-named author, following general discussions between the two authors concerning log-concave moment measures. The prompts referred to the paper “Logarithmically-concave moment measures I” and suggested bootstrapping a bound on the second trace moment.

Let $\g=\fb\oplus V\oplus\fb^{*}$ be a Medina-Revoy rotational double extension in which an abelian Lie algebra $\fb$ acts by scalar rotations on an orthogonal sum of oriented Euclidean two-planes. We analyse the restriction of the Schouten square $r\mapsto[r,r]$ to the block-rank-one locus $\Rone\subset\fb\wedge V$ and to its nondegenerate open part $\Rplus$. On every marked slice $S_{H}\subset\Rplus$ the Schouten square splits orthogonally into a radial and an angular component. The radial component factors through the standard toric moment map: its image is a simplicial cone, it satisfies a sharp quadratic coercive estimate with an intrinsic optimal constant, and the restricted Schouten map is proper with compact semialgebraic affine fibres. The angular component becomes Laurent monomial in joint eigencoordinates; a marker-cancellation criterion determines its signed exponent configuration, whose integral lattice controls the differential rank, the compact isotropy, its component group, and the Laurent binomial ideal defining the Zariski closure of the complexified image. Smith normal form makes each invariant algorithmic. All statements hold for arbitrary $\dim\fb$ and impose no nonresonance assumption. A corollary of the underlying disjoint-support decomposition classifies the classical Yang-Baxter equation on $\Rone$: for any bivector whose $\fb\wedge V$ component lies in $\Rone$, triangularity forces every block detected by its rotation weight to vanish, regardless of the $\Lambda^{2}\fb$, $\Lambda^{2}V$, or central-wedge terms.

We consider a random walk $\{S_n\}$ with a finite positive drift that is stopped at a random time $\tau$ having an intermediate regularly varying distribution. We assume that the jump distribution is lighter-tailed than the distribution of $\tau$. Under these conditions, we show that the tails of the distributions of $S_{\tau}$ and $M_{\tau} = \max_{k\le \tau} S_k$ are asymptotically equivalent and are determined by the tail of $\tau$, while the random walk $\{S_n\}$ contributes only through the law of large numbers.

Let $\mathcal{F}_n$ be the Farey sequence of order $n$, written in increasing order. Call two fractions $\frac{a}{b} < \frac{c}{d}$ badly ordered if $a < c$ and $b > d$. Let $f(n)$ be the minimum number of Farey fractions strictly between two badly ordered fractions in $\mathcal{F}_n$. We prove $f(n)=\left(\frac{1}{4}+o(1)\right)n$. In the equivalent indexing convention of Erdős Problem 1005, this determines the requested asymptotic constant as $c=1/4$. The upper bound $f(n)\le n/4+O(1)$ was first obtained by Wouter van Doorn; the main result here is the matching lower bound.

Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali–Yan theorem.

We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence—or even, counterintuitively, divergence—in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.

In this paper, we investigate the almost $\eta$-Ricci-Yamabe soliton as a fundamental geometric regulator for a static, spherically symmetric black hole coupled to an imperfect fluid. We have shown that the scaling parameter $\omega(r)$ is governed by thermodynamic friction along the radial vector field, and the geometric coupling with the Hawking temperature: $\alpha(r_H) S_{tt} = 2\pi T_H$ at the horizon. We also derive the Poisson equation along the gradient vector field of the soliton and prove that the flow's kinematic expansion is explicitly dependent on the fluid's equation of state $\rho = \gamma \sigma$. Diverging from traditional methodologies that assume a geometric shape function apriori, we analytically proved the geometric flow endogenously transitions the black hole geometry into a traversable wormhole throat by regularizing of temporal coordinate and satisfying spatial flare-out condition. This transition occurs when fluid enters the dark energy era at $\gamma = -1$ and violates the Null Energy Condition $\rho + \sigma < 0$, with the soliton strictly dominating the local curvature gradient $\omega^{\prime}(r_H) > f^{\prime\prime}(r_H)$, to keep the throat open. Moreover, by smoothly attenuating at spatial infinity, the soliton preserves the exact cosmological spacetime. Finally, through tensorial perturbation analysis, we demonstrate that the geometric flow introduces a localized dissipative mechanism, that the perturbation evolution reduces to damped wave equation, imposing geometric drag on the manifold.

This paper develops and analyzes a least-squares weak Galerkin (LS-WG) finite element method for linear elasticity. By employing weak differential operators, specifically the weak gradient, weak strain tensor, and weak divergence, defined on weak finite element spaces, the proposed framework facilitates the treatment of complex boundary conditions and internal interfaces while avoiding the restrictive discrete inf-sup condition. The resulting formulation is symmetric and positive definite and exhibits robust numerical performance in the nearly incompressible regime. In addition, the proposed method offers exceptional geometric flexibility, allowing implementation on general polytopal (polygonal and polyhedral) meshes. We establish the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a tailored discrete energy norm. Extensive numerical experiments confirm the theoretical convergence rates and demonstrate the method's stability, efficiency, and locking-free performance for nearly incompressible materials.