In this paper we completely classify semifields of order $2^8=256$ containing a nucleus of order $2^4=16$. We introduce new invariants for semifields, and apply new computational techniques for calculating old invariants. Together these make the computational classification significantly quicker.

The geometrisation theorem of 3-manifolds was conjectured by Thurston the 1980s and proved by Perelman in the 2000s. This is an overview on the subject. We explain the content of the theorem and describe its effects in various situations.

The moment-sum-of-squares hierarchy provides a powerful framework for solving non-convex optimal control problems by constructing a sequence of convex semidefinite relaxations. However, when extending these methods to nonlinear partial differential equations (PDEs), a fundamental challenge is the potential existence of a relaxation gap, where the solution to the linear measure formulation using occupation measures fails to correspond to a classical physical solution of the original PDE. In this paper, we prove the absence of a relaxation gap for scalar semilinear parabolic PDEs of the reaction-diffusion type. We do so by showing that each solution to the linear measure equation gives rise to an energy measure-valued (emv) solution in the space of Young measures satisfying suitable energy identities. We then prove that any such emv solution concentrates on the solution to the nonlinear PDE, provided the latter exists and is unique. To the best of our knowledge, this is the first concentration result of this kind for measure-valued solutions of reaction-diffusion PDEs.

Existing scaling laws for Large Language Models (LLMs), predominantly monotonic power laws, fail to explain emerging non-monotonic phenomena such as catastrophic overtraining and quantization-induced degradation, where performance deteriorates despite increased compute. We propose the Shannon Scaling Law, a unified theoretical framework that models LLM training as information transmission over a noisy channel, grounded in the Shannon-Hartley theorem. By mapping model parameters to channel bandwidth and training tokens to signal power, our formulation explicitly captures the interaction between learning signal and intrinsic noise. This perspective reveals a fundamental Shannon capacity for LLMs: scaling model size or data without preserving a sufficient signal-to-noise ratio (SNR) inevitably amplifies noise, inducing a transition from monotonic improvement to U-shaped performance degradation. We validate our theory through experiments on Pythia and OLMo2 under perturbations, including Gaussian noise, quantization and supervised fine-tuning on math, QA and code tasks. The Shannon Scaling Law consistently outperforms classical scaling laws and recent perturbation-aware laws, achieving strong $R^2$ scores and accurately capturing loss basins missed by prior approaches. It also extrapolates: fitted on $\leq$6.9B Pythia models with $\leq$180B tokens, it predicts the unseen 12B model up to 307B tokens at pooled $R^2{=}0.847$, while monotonic baselines collapse.

This paper develops a two-branch multiplicative-coset construction for regular Calderbank-Shor-Steane (CSS) quantum low-density parity-check base matrices. For a target column weight \(J\) and an even row weight \(L\), the method reduces regularity, CSS orthogonality, and same-type 4-cycle exclusion to explicit quotient-coset conditions over a finite field. A normalized exhaustive search for these conditions produces base matrices for several \((J,L)\) pairs, so the construction is not tied to a single degree distribution. The construction separates the finite-length design into two stages: the base matrix fixes the degree distribution and the first girth constraints, and a cyclic lift randomizes edge connections subject to exact algebraic checks. As a detailed example, we carry one \((3,10)\)-regular base through the lift and decoding stages. For this example, the selected 64-fold lift gives a code whose same-type Tanner graphs have girth at least eight, and it also excludes a specified weight-16 nondegenerate logical-support orbit. The resulting instance is a \([[10240,4108,\,10\le d\le32]]\) CSS code. For decoding, we use joint log-domain belief propagation together with low-complexity deterministic post-processing rules for small residual syndromes, including repairs for residual patterns with two unsatisfied checks. The frame error rate (FER) measurements provide finite-length decoding data for this detailed example; at depolarizing probability \(p=0.058\), the post-processing FER is \(1.0\times10^{-7}\).

Meyer defined crystalline measures as tempered distributions $μ$ such that both $μ$ and its Fourier transform $\widehatμ$ are pure-point Radon measures of locally finite support. He conjectured that every crystalline measure is almost periodic as a tempered distribution. Favorov constructed a counterexample and asked whether crystalline measures are at least almost periodic as general distributions. To resolve Favorov's question, we first show that the almost periodicity of a crystalline measure is characterised in terms of its translation boundedness, in any class of Radon measures, tempered distributions, or general distributions. We then construct a crystalline Fourier eigenmeasure that fails to be translation bounded even as a distribution. We finally construct a crystalline measure that fails to be a~Fourier quasicrystal (in particular, it fails to be slowly increasing), but it is an almost periodic tempered distribution whose Fourier transform is even a norm almost periodic measure. Our examples fully resolve the questions of Meyer and Favorov and sharply delineate the class boundary of translation boundedness. They also demonstrate the unusual behaviour of crystalline measures beyond the class of Fourier quasicrystals.

Gradient-flow sampling interprets a Gibbs distribution as the minimizer of an energy functional over probability measures and generates dynamics converging to this target. Under spherical Hellinger-Kantorovich (SHK) geometry, the flow couples transport and reaction and coincides with birth-death Langevin dynamics. In this work, we develop a perturbation theory for SHK gradient flows. For two potentials $V$ and $V^{\prime}$, we compare the associated flows from a common initialization and quantify how potential discrepancies propagate over time. A uniform perturbation bound yields dimension-free, pointwise control of the log-likelihood ratio and Rényi divergence, while additional structure allows us to derive bounds for the KL divergence as well. We apply these results to approximate sampling for the exponential mechanism in differential privacy. The likelihood-ratio control provides explicit time-dependent Pure-DP guarantees for SHK-based samplers, while the KL bound yields Approximate-DP certificates via hockey-stick divergence. We also derive a utility bound separating intrinsic exponential-mechanism suboptimality from finite-time sampling error.

We introduce training-free looped transformers, in which a lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning, continued training, or architectural changes. Unlike prior looped transformer methods that train with the looped structure end-to-end, we retrofit recurrence onto pretrained models at test time. We show that naive block reapplication usually degrades performance, highlighting the importance of the loop application strategy. Motivated by viewing a pre-norm transformer block as a forward Euler step on an ODE, we instead treat looping as a refinement of the same approximation, replacing one large update with smaller damped sub-steps. Across seven dense, sparse MoE, and MLA+MoE model families, our method improves Qwen3-4B-Instruct by +2.64 pp on MMLU-Pro, Qwen3-30B-A3B-Instruct by +1.14 pp on CommonsenseQA, and Moonlight-16B-A3B-Instruct by +1.20 pp on OpenBookQA.

We develop a gradient flow on the space of probability measures defined on matrix-valued parameters induced by regularized Muon, an analytically smoothed version of the idealized Muon optimizer. The key observation is that the regularized orthogonalization map is the gradient of a smooth Fenchel-dual smoothing of the nuclear norm. This identifies the (regularized) Muon update as a mirror/prox step in the update variable, with momentum acting as the dual coordinate. We use this structure to lift Muon from a single matrix parameter to finite-particle probability objectives of the form $J(ρ)=R\left(\int F d ρ\right)$, a setting motivated by mean-field descriptions of neural-network training, and derive the inertial continuous-time limit. Using this structure, we derive the finite-particle continuous-time limit under the inertial scaling of step size and momentum, and then pass to a phase-space mean-field equation over probability laws on parameter-momentum pairs. The resulting flow can be shown to be a damped Hamiltonian probability dynamics whose kinetic energy is induced by the regularized Muon mirror potential. We prove an exact Hamiltonian dissipation identity, showing that the Hamiltonian energy decreases monotonically. While the target objective itself need not be monotone along the inertial Muon dynamics, under additional gradient-dominance, bounded-momentum, and curvature/alignment assumptions, we obtain continuous and discrete-time exponential convergence rates for the objective gap. We also study the well-posedness of the mean-field limit equation and establish propagation of chaos guarantees for the interacting particle system. Finally, we extend the formulation to Hilbert-valued feature maps on product matrix spaces, yielding a blockwise Muon probability flow applicable to smooth transformer mixture-of-experts models.

A zonotope is a linear image of the cube $[-1,1]^m$ for some $m \in \mathbb{N}$. We show that there is a universal constant $C$ such that, for every zonotope $Z\subset \mathbb{R}^d$ and vectors $v_1,\dots,v_n\in Z$, there are signs $x_1,\dots,x_n\in\{-1,1\}$ with \[ \sum_{i=1}^n x_i v_i \in C\sqrt d\, Z. \] This resolves a 2002 question of Schechtman and generalizes Spencer's six standard deviations theorem, which corresponds to the case $Z=[-1,1]^d$.

Let $\mathbb{F}$ be a field and let $M_2(\mathbb{F})$ be the algebra of $2\times 2$ matrices endowed with an involution of the first kind. We study the image of multilinear $*$-polynomials evaluated on $M_2(\mathbb{F})$. For the transpose involution over $\mathbb{R}$, we show that the image is either a proper vector subspace or contains a basis of $M_2(\mathbb{R})$. For the symplectic involution over quadratically closed fields or over $\mathbb{R}$, we prove that the image is always a vector space, namely one of $\{0\}$, $\mathbb{F}$, $sl_2(\mathbb{F})$ or $M_2(\mathbb{F})$. As a byproduct, we complete a theorem of Brešar and Klep describing the linear span of the image of a $*$-polynomial on finite dimensional central simple algebras with involution of the first kind. Their result excluded algebras of dimensions 4 and 16; we settle both cases, extending the description to all dimensions greater than 1 (over $\mathbb{R}$ for the transpose involution, and over quadratically closed fields or $\mathbb{R}$ for the symplectic involution). We also classify all Lie skew-ideals of $M_4(\mathbb{F})$ over fields of characteristic zero.

In industrial scenarios involving multi-agent collective decision-making, centralized decision-making may not be admissible due to restrictive access to individual local information, while the conflicts between participants' self-interest and global performance may also impede collaborative distributed decision-making. This paper proposes a mechanism-driven distributed decision-making method, wherein incentives are employed and designed to motivate participants to collaborate in a distributed fashion even though each participant's decision is driven primarily by self-interest. Focusing on optimization problems with coupled objective functions and coupled constraints, we design a distributed optimization algorithm tailored for this class of problems and provide guarantees for its convergence. Furthermore, we design two incentive mechanisms, the shadow pricing mechanism and the Vickrey-Clarke-Groves mechanism, and demonstrate that participants are willing to engage in distributed collaboration under these mechanisms. The mechanism drives the execution of the distributed algorithm, and the optimal result of distributed computation guides the determination of incentives in the mechanism, both of which are interrelated to form a closed loop. Finally, numerical experiments illustrate the effectiveness of the proposed algorithm and mechanisms.

Bradley-Terry-Luce (BTL) model estimation is a well-established strategy to rank a collection of items given a dataset of pairwise comparisons. Although the theoretical performance of BTL estimation methods, such as spectral and maximum likelihood estimation, is well studied in the regime of uniformly sampled graphs, generalizing such results to a wider class of random graphs has proved challenging. In this work, we investigate the entry-wise error of spectral algorithms against a semi-random adversary that can arbitrarily boost the sampling probabilities of certain edges. We find that the performance of the unweighted spectral method is heavily dependent on the spectral properties of the generated graph. Furthermore, we show that asymptotic performance approaching that of uniformly sampled graphs can be recovered by appropriately reweighting the observed edges to counteract the adversary and restore the spectral gap. Finally, we provide numerical simulations that support our theoretical findings.

The tight-binding (TB) model is a widely adopted approximation scheme for describing light propagation in waveguide arrays. Despite its success, its validity in $\mathcal{PT}$-symmetric systems characterized by strong longitudinal modulation has not been rigorously benchmarked against exact analytical solutions. In this work, we address this gap by performing a comparative analysis between exact continuous solutions derived from $z$-dependent supersymmetric (SUSY) transformations and their corresponding discrete TB approximations. To achieve this, we develop a theoretical model for two PT-symmetric coupled waveguides subject to longitudinal modulation. We then evaluate the performance of the TB framework against the exact SUSY benchmark. Our results delineate the specific validity range of the TB approximation, demonstrating its proficiency in reproducing spatial intensity distributions. However, we also identify its limitations in accurately capturing the complex oscillatory phase dynamics inherent to this non-Hermitian evolution.

We investigate the emergence of non-Markovian dynamics in finite-dimensional open quantum systems governed by Weyl dynamical maps and their convex combinations. Using the Hermite normal form, we provide a complete classification of the subgroups of the discrete phase space $\mathbb{Z}_d \times \mathbb{Z}_d$, establishing the algebraic framework underlying the Weyl maps. We characterize isotropic Weyl dynamical maps that generate Markovian semigroups and show that anisotropic Weyl maps with nonuniform weight distributions cannot possess the semigroup property. Furthermore, we analyze the role of convexity in the generation and suppression of memory effects. Remarkably, we prove that convex combinations of eternally non-Markovian Weyl dephasing maps can generate Markovian semigroups, demonstrating that non-Markovianity is not additive under mixing. Conversely, we establish a general condition under which convex mixtures of $N$ distinct Weyl semigroups exhibit eternal non-Markovianity. In contrast to the qubit Pauli setting, we further identify the existence of irreducible eternally non-Markovian Weyl dephasing maps, namely, individual dynamical maps that display eternal memory effects without requiring any mixing mechanism. Finally, explicit qutrit examples illustrate the transition among Markovian, non-Markovian and eternally non-Markovian regimes. Our results uncover a fundamental connection among finite phase-space algebra, convex structures, and quantum memory effects, thereby extending the theory of non-Markovian dynamics beyond the Pauli framework.

We study the ideal of the algebraic relations among 3-point functions from a combinatorial and topological perspective. We place this problem in the broader setting of incidence toric ideals associated with incidence matrices of t-subsets contained in k-subsets of n elements. Generators of these ideals admit combinatorial interpretations as null t-designs and topological interpretations as balanced orientable normal d-pseudomanifolds without boundary. Generators arising from octahedra play a fundamental role in the structure of these ideals.

Applying the techniques resulting the existence of almost invariant half-spaces, similarity models $\wh T$ can be given for upper triangular operator-matrices $T= \left[\begin{matrix}A&C\\ 0&B\end{matrix}\right]$. The model $\wh T$ is also an operator-matrix, containing two diagonal operators in the general case \cite{ker25}, and the unilateral shift $S$ together with a diagonal operator in the particular case when $A$ is similar to $S$ \cite{ker26}. Well-chosen compressions of operators in the commutant $\{\wh T\}'$ form a linear manifold $\whŁ$ satisfying the condition that every $\wh X\in\wh Ł$ is transformed into $\wh Y$ with $\rank\wh Y\le 2$ by a canonical mapping. Furthermore, a cyclcity property of $\{T\}'$ yields transitivity of $\whŁ$. In \cite{ker25} and \cite{ker26} the 3-dimensional cross-sections of $\whŁ$ have been investigated characterizing the canonical bases occurring in the corresponding subspaces of the matrix-algebra $M_3[\C]$. In this paper new conditions are provided by studying matching of adjacent cross-sections.

In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $Ω$ of an $n$-dimensional Finsler manifold ( $\mathcal{M}, F, \vartheta$ ). We work with Lipschitz functions $ρ_1$ and $ρ_2$ satisfying distance-type properties; in particular, $F(\cdot, \boldsymbol{\nabla} ρ_1) \leq 1$ and $F(\cdot, \boldsymbol{\nabla} ρ_2) \leq 1$ a.e. in $\mathcal{M}$. The singular set is defined by $Γ=ρ_1^{-1}(\{0\})$. The model problem is $-Δ_{p(x)} u+|u|^{q-1} u=0$ in domains of $\mathbb{R}^n \cong \mathbb{R}^d \times \mathbb{R}^{n-d} \cong ρ_1^{-1}(\{0\}) \times ρ_2^{-1}(\{0\})$, where $ρ_1(x)=|(x_{d+1}, \ldots, x_n)|$ and $ρ_2(x)=|(x_1, \ldots, x_d)|$. The main tool in our analysis is the estimate $$ |u(x)| \leq \mathbf{C} ρ_1(x)^{-τ} $$ near $Γ$ for weak solutions $u \in W_{loc}^{1, p(x)}(\barΩ \backslash(Γ\cup Σ) ; \vartheta) \cap L_{loc}^{\infty}(\barΩ \backslash(Γ\cup Σ))$, where the constants $\mathbf{C}>0$ and $τ>0$ converge to positive values as $p^{+} \rightarrow 1$. This estimate is a key ingredient in proving that the singularity at $Γ$ is removable. Moreover, in a bounded domain $Ω$, using this estimate and assuming that, for every variable exponent satisfying $1<p^{-} \leq p^{+}<\min \{2, q+1\}$, there exists a weak solution $u_p \in W_{loc}^{1, p(x)}(Ω; \vartheta) \cap L_{loc}^{\infty}(Ω)$ of $$ -\operatorname{div}\left(|\boldsymbol{\nabla} u_p|_F^{p-2} \boldsymbol{\nabla} u_p\right)+|u_p|^{q-1} u_p=0 \quad \text { in } Ω, $$ we prove that, for every $U \Subset Ω$, there exists a subsequence $\{u_{p_m}\}$, with $p_m^{+} \rightarrow 1$, that converges to a solution $u \in B V(U ; \vartheta) \cap L^{q+1}(U ; \vartheta)$ of $$ -Δ_1 u+|u|^{q-1} u=0 \quad \text { in } U . $$

We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a finite-index subgroup is a central extension of a product of orbifold mapping class groups, and the extension has bounded Euler class. Our theorem is sharp: we exhibit a surface amalgam whose fundamental group has full outer automorphism group which is not a HHG. To prove this, the main technical tool is the fact that a top-dimensional Abelian subgroup of a HHG is a standard flat.

Motivated by majority vertex-colorings of graphs and digraphs and majority edge-colorings of graphs, we introduce two concepts of strong majority colorings. A strong majority vertex-coloring of a graph $G=(V,E)$ is a mapping $c:V\rightarrow C$ such that for every vertex $v\in V$ and every color $α\in C$, at most half of the neighbors of $v$ have color $α$. The strong majority number of $G$, denoted Maj$(G)$, is the least number of colors in such a coloring. We show that Maj$(G)$ can be arbitrarily large and prove a tight upper bound Maj$(G)\le 2Δ(G)+1$ for every graph $G$ without pendant vertices. A strong majority edge-coloring of a graph $G$ is a mapping $c:E\rightarrow C$ such that for every edge $e\in E$ and every color $α\in C$, at most half of the edges adjacent to $e$ have color $α$. The strong majority index of $G$, denoted Maj'$(G)$, is the least number of colors in such a coloring. It is shown that there is an upper constant bound for Maj'$(G)$ of all admissible graphs $G$. We conjecture that this constant is as small as 4 and confirm this conjecture for numerous graph classes.

Given a closed normal subgroup $H$ of a topological group $G$, we address the question of whether the left coarse structure on the quotient group $G/H$ equals the quotient of the left coarse structure on $G$. We provide a counterexample among Polish groups, namely, the mapping class group of the Loch Ness monster surface seen as a quotient of the mapping class group of the punctured Loch Ness monster surface, and establish both equivalent and sufficient conditions for when this holds in special settings. The latter are formulated in terms of liftings of bounded sets, existence of transversals and metrisability of the left coarse structure of $G$ restricted to $H$.

In Poisson Boolean models with deterministic ball grains, the directional visible range from an uncovered point is known to be exponentially distributed in Euclidean and real hyperbolic space. We show that the same phenomenon holds on every simply connected non-compact homogeneous harmonic manifold. The geometric mechanism behind this fact is the affine-linear growth of tube volumes around geodesic segments. As a consequence, we identify the finiteness regime for the expected volume of the visible region, including a geometric interpretation of the critical threshold in the positive-entropy case. We also construct explicit complete non-homogeneous Riemannian manifolds showing that exact exponentiality is tied to exact tube linearity: superlinear tube growth leads to Weibull-type tails, while asymptotic tube linearity still yields an exponential decay rate.

Modern optimal control theory involves adjoining the already known equations of motion of a dynamic system to the objective function using dynamic costates; this is done in order to constrain the optimal control solutions to satisfy the equations of motion. The use of costates increases the number of variables and hence increases the complexity of the problem. On the other hand, variational methods of analytical mechanics finds the equations of motion by minimizing an action functional of the dynamic system, realizing control forces as external input to the system. In this paper a new disruptive approach for computing the optimal control is presented. This approach adopts the variational methods of analytical mechanics to derive equations for the control, in addition to the equations of motion. This is achieved by recognizing the control actuator as part of the dynamic system. In addition to the kinetic energy and potential energy, the action functional in this new approach includes additional energy terms that represent the control energy of the system. Two different methods are presented to write the modified action functional. The proposed approach is a significant departure from the modern optimal control theory, and it eliminates the need for costates when solving for the control. In this paper, a case study is presented to demonstrate the new approach.

In this paper, we investigate the nonlocal problem \begin{equation*}\left\lbrace \begin{aligned} &A_{s} u=(|x|^{-(n-2s)}\ast u^{2_{s}^{\sharp}-1-ε})u^{2_{s}^{\sharp}-2-ε} \quad\quad\hspace{3.5mm} \mbox{in}\hspace{2mm}Ω,\\ &u>0\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{in}\hspace{2mm}Ω,\\ &u=0\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminusΩ, \end{aligned} \right.\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^n$, $0<s<1$, $n\in(2s,\min\{6s,n+2s\})$, $ε>0$ small, $2_{s}^{\sharp}-1=(n+2s)/(n-2s)$ and $A_{s}$ stands for the fractional Laplace operator $(-Δ)^{s}$ in $Ω$ with outside zero Dirichlet boundary condition. The above problem is reduced to the subcritical fractional system $$ A_{s}u=u^{2_{s}^{\sharp}-2-ε}v,\hspace{2mm}A_{s}v=u^{2_{s}^{\sharp}-1-ε},\hspace{2mm}u,v>0\hspace{2mm}\mbox{in}\hspace{2mm}Ω\hspace{2mm}\mbox{and}\hspace{2mm}u=(-Δ)^sv=0\hspace{2mm}\mbox{on}\hspace{2mm}\mathbb{R}^n\setminusΩ.$$ For a general domain $Ω$ or domains with convexity, we first prove a uniform $L^1$ bound away from the boundary and a uniform $L^{\infty}$ bound near the boundary for positive solutions to the general fractional Hartree-type PDEs by applying the moving planes method and integral estimates for the convolution term.Among these results, we study the asymptotic behavior of solutions as $ε\rightarrow0$.These solutions are shown to blow-up at exactly one point $x_0$ and location of this point is characterized. In addition, the shape and exact rates for blowing-up are studied.Finally,we also establish the corresponding main results for solutions of the fractional Brezis-Nirenberg problem involving critical Hartree-type nonlinearity.

This paper presents a framework for assigning intrinsic geometric structures to topological groups using only the data provided by their topological and algebraic structure. The geometrisation spits into small-scale and large-scale components, formalised respectively through local Lipschitz and quasimetric categories that, in turn, are definable from the canonical left uniform and left coarse structures of the group. For Polish groups, the paper characterises metrisability of the left coarse structure in terms of local boundedness, countable coverings by bounded sets, and the existence of compatible coarsely proper left-invariant metrics. It then introduces minimal metrics, which determine local Lipschitz structure, and maximal metrics, which determine quasimetric structure, and provides intrinsic characterisations of both. When both structures exist, they combine into a single canonical Lipschitz structure. Our framework is subsequently applied to specific examples such as homeomorphism groups, non-Archimedean Polish groups and automorphism groups of Fraïssé limits.

We study the Brownian dynamics and linear response of a particle with inertia moving in a 2-dimensional helical landscape imprinted on a cylindrical surface. In the harmonic well approximation, the deterministic motion separates into free propagation along the screw direction and harmonic motion in the transverse screw-normal direction. We show that for isotropic damping this simplification survives in the Langevin description, whereas anisotropic damping along the axial and angular directions couples the stochastic dynamics and destroys separability. The resulting anisotropic model is formulated as a linear Ornstein-Uhlenbeck process in phase space with a zero mode associated with diffusion along the screw coordinate, so that in an infinite system the full phase-space dynamics does not relax to a stationary distribution. To treat transport in this setting, we construct the stationary dynamics in the stable subspace obtained after projecting out the zero mode. This leads to a linear response theory for this system and yields closed analytical expressions for stationary time-correlation functions and the dynamical mobility tensor in both the time and frequency domains. The off-diagonal elements of the mobility tensor describe cross-response between axial forcing and angular motion, and between applied torque and axial transport. Consistent with time reversal symmetry, these cross mobilities are equal and provide a direct dynamical signature of the helical geometry. In addition, a simultaneous application of driving in both the axial and angular direction reveals asymmetry in energy dissipation rate due the helical landscape.

Let $\mathbb{P}$ be an algebraic number field. We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense groups $H\leq \mathrm{SL}(n,\mathbb{P})$, $n$ prime. That is, we present algorithms to find the set of congruence quotients of $H$ modulo all maximal ideals of a finitely generated subring $R$ of $\mathbb{P}$ such that $H\leq \mathrm{SL}(n,R)$. The algorithms have been implemented in GAP. Potential applications are illustrated by a range of experiments in degree $2$, with a special focus on Bianchi groups.

We investigate phaseless inverse scattering problem for the Schrödinger equation and develop reconstruction methods based on the inverse Born series (IBS). We consider three types of phaseless data: the far-field total field, the total field and the far-field scattered field. For phaseless total-field data, we extend the IBS framework and analyze its radius of convergence. In the far-field setting, we propose a Fourier-based reconstruction method that exploits the scattering reciprocity between incident and observation directions to recover Fourier coefficients of the scattering potential. For phaseless scattered-field data, we employ a polarization-based approach to recover phase information and enable IBS reconstruction. Numerical experiments are conducted to validate the proposed methods.

We prove that, among rectangular grid graphs with a fixed number of vertices, the number of spanning trees increases when the side lengths are made more balanced. In particular, among all rectangular grid graphs with $n^2$ vertices, the square $n\times n$ grid has the largest number of spanning trees. The proof starts with the Laplacian product formula, passes to hyperbolic coordinates, and compares logarithms by separating a discrete-concavity term from a positive decreasing residual term.

We study the solution of symmetric positive-definite linear systems by way of families of full- and limited-memory methods. Our contributions are threefold. We first derive new relationships between the conjugate-gradient method (CG) and quasi-Newton methods of the Broyden class that refine existing results, and clarify when those methods generate the same iterates and enjoy quadratic termination. We extend this perspective to the limited-memory BFGS (LBFGS) method. Next, we examine how DIOM, a limited-memory variant of the full orthogonalization Krylov method (FOM), is akin to LBFGS in that it provides a memory lever that is critical in practical performance. Finally, we generalize LBFGS and DIOM to the computation of trust-region steps for unconstrained, potentially nonconvex, optimization. We report numerical experience on positive-definite linear systems and unconstrained optimization problems. The results show that memory is a key algorithmic lever: LBFGS and DIOM are consistently more robust than CG and often achieve comparable accuracy with fewer Hessian-vector products. They emerge as viable alternatives to CG when high accuracy is desirable or when operations with the Hessian are at a premium. The limited-memory SR1 (LSR1) method can be competitive in full-memory form, but its limited-memory variant suffers from discarded curvature information.