Papers

Structure-preserving neural networks are essential for the long-term prediction of Hamiltonian systems from data. Many important Hamiltonian systems in mechanics and control admit symmetry reduction to Lie–Poisson systems, including rigid bodies, underwater vehicles, fluids, plasmas, and optimal control problems. A fundamental challenge in learning such systems is that their dynamics evolve in momentum variables that are typically unobservable, while available data consist only of observable quantities such as configurations and velocities. In optimal control applications, the situation is further complicated because the latent variables contain unobservable co-states and the Hamiltonian may be degenerate, preventing the existence of a corresponding Lagrangian and rendering the encoder-decoder approaches inapplicable. We introduce Latent Lie–Poisson Neural Networks (LLPNNs), a structure-preserving framework for learning Lie–Poisson dynamics directly from observable data. The proposed approach exploits three geometric ingredients: (i) learning either a Hamiltonian decoder or a pseudo-Lagrangian encoder on the active variables, (ii) constructing latent trajectories through a universal Noether invariant arising from Lie–Poisson symmetry reduction, and (iii) reconstructing observable and latent dynamics through Lie–Poisson flows combined with Magnus-based Lie-group updates. The resulting method preserves the geometric structure and is applicable to both regular and degenerate Hamiltonian systems. We demonstrate the method on three examples: a generalized rigid body on SO(3), Kirchhoff's underwater vehicle on SE(3), and an optimal-control problem for interacting vehicles on $SE(2)^N$. Numerical experiments show excellent long-term predictive accuracy, strong robustness to noise, and competitive performance using only modest datasets and lightweight neural-network architectures.

A vertex operator algebra $V_L$ associated with a positive definite even lattice $L$ has a standard integral form, which we denote it by $V_{L,\mathbb{Z}} $. If $F$ is a field of characteristic $p>0$, it is known that $V_{L,F}:= F\otimes_\mathbb{Z} V_{L,\mathbb{Z}}$, a vertex algebra over $F$, is simple if and only if $(p, \det(L))=1$. In this article, we study $V_{L,F}$ when the characteristic of $F$ divides $\det(L)$. We determine the radical $\mathrm{Rad}$ of the invariant bilinear form on $V_{L,F}$, show that it is the unique maximal ideal and study the quotient vertex algebra $V_{L,F}/\mathrm{Rad}$.

We investigate the deterministic hierarchy of prime distribution biases in arithmetic progressions modulo $N$ using a regularized spectral approach. Classical studies on Chebyshev's bias attribute prime races primarily to the accumulation of prime squares $p^2 \equiv 1 \pmod N$, which creates a systematic deficit in quadratic residue classes. However, this classical mechanism fails to explain or distinguish any bias among residue classes sharing identical quadratic residue status (e.g., $3, 5, 7 \pmod 8$). To overcome the long-standing analytical obstacles of jump discontinuities and non-convergent boundary fluctuations inherent in classical Perron-type step-function truncations, we introduce a smooth $C^\infty$ Gaussian mollifier into Weil's explicit formula for Dirichlet $L$-functions. By defining the spectrally normalized individual mollified sums $\widetilde S_T(x, a)$ and adopting the virtual character $\chi_{1,a}(x) := \mathbf{1}_{\{x \equiv 1 \pmod N\}} - \mathbf{1}_{\{x \equiv a \pmod N\}}$, the principal character component $\chi_0$ cancels identically since $1 - \overline{\chi_0}(a) = 0$. This automatic algebraic elimination erases both the universal logarithmic growth $\log x$ and the background noise $\log L(1, \chi^2)$. Under the Deep Riemann Hypothesis (DRH), we uncover a hitherto undetected fine-structure bias (or secondary bias) strictly governed by the special values $\log L(1, \chi)$. We prove that $\widetilde S_T(x, \chi_{1,a}) := \widetilde S_T(x, 1) - \widetilde S_T(x, a) = C_N \cdot \log L(1, \chi_{1,a}) + \mathcal{O}((\log x)/\sqrt x)$ as $x \to \infty$, where $C_N > 0$ depends solely on $N$. Consequently, we establish a deterministic multi-way ranking (such as $7 > 3 > 5 > 1 \pmod 8$) that completely transcends the classical quadratic residue framework.

Let $s>1$, let $M$ be a connected, non-compact, oriented real-analytic manifold, and let $\omega_1,\ldots,\omega_m$ be real-valued closed $1$-forms of Gevrey order $s$ on $M$. We study the differential complex naturally associated with this family on $M\times\mathbb{T}^m$. We prove that its top-degree operator is globally solvable in Roumieu Gevrey ultradistributions, or equivalently that the corresponding top-degree cohomology vanishes. No global hypoellipticity assumption and no arithmetic condition on the periods of the defining forms are required. The proof is carried out in the physical variables and combines fiber translations, a local normal form, and a transport formula along paths in the base manifold. These tools yield propagation of Gevrey regularity, non-confinement of Gevrey singularities, and the support control needed to apply an abstract solvability criterion. The result highlights a sharp contrast with the compact setting, where compatibility conditions are unavoidable and solvability for compatible data may depend on exponential small-denominator conditions.

We use highly oscillatory geometric optics solutions to solve the inverse problem for the system $$ \square \begin{bmatrix} u^{(1)}\\ u^{(2)}\\ \vdots\\ u^{(n)}\end{bmatrix} = \sum_{k\geq l=1}^n \left(Q_0(u^{(k)},u^{(l)}) \begin{bmatrix} q_{1kl}\\q_{2kl}\\\vdots\\q_{nkl} \end{bmatrix} + \sum_{i>j=1}^d Q_{ij}(u^{(k)},u^{(l)})\begin{bmatrix} p_{1ijkl} \\ p_{2ijkl} \\ \vdots \\ p_{nijkl} \end{bmatrix}\right), $$ where $Q_0$ and $Q_{ij}$ are the symmetric and anti-symmetric bilinear null forms. We present solutions in both the linear and weakly nonlinear regimes. In the linear regime, we show that the coefficients of order $h^3$ determine an injective light-ray transform of a vector field which depends on the coefficients $q_{rkl}, p_{rijkl}$. In the weakly nonlinear regime, we see that the coefficients of order $h$ determine the non-abelian light ray transform for matrices associated with the coefficients. While we do not have an injectivity result for this case, we do have one if we assume the coefficients do not depend on the time variable $x_0$, as our coefficients instead determine an injective non-abelian X-ray transform.

The existence of a communal common ground is vital for collective action and coordination in a population, but the micro-level cognitive and social processes that lead to the formation and evolution of common ground at the macro-level are undertheorised and have not been rigorously explored. In this work, we adopt a formal approach and develop an agent-based model that describes repeated grounding attempts between agents interacting on a network, with an explicit distinction between a sender agent and a receiver agent during an interaction involving sharing information. Several key novel features enable us to capture a range of different interaction contexts: we allow for the interaction to result in either acceptance or rejection, the receiver's response may be lost to the sender, and the sender can interpret this lack of response as either acceptance or rejection (or even something in between). A campaign of Monte Carlo simulations reveals how different interaction contexts, as well as the available information for sharing, result in different emergent phenomena, such as a global communal common ground, fragmentation into multiple clusters of differing common ground, and even the total loss of any shared common ground. This work highlights the potential for using mathematical models to study micro-macro links in cultural dynamics, including identifying ways to facilitate interactions to foster the emergence of a global communal common ground.

Given a (not necessarily unital) complete operator system $S$ with a generating cone, there are some studies in literature on the operator system dual $S^\mathrm{d}$ of $S$, i.e., the dual matrix-ordered space $S^*$ equipped with a matrix norm that turns it into a dual operator system satisfying certain universal property. In order to do further study on $S^\mathrm{d}$, one needs to consider the operator system predual construction. More precisely, given a dual operator system $V$ with a generating cone, there is a matrix norm on the predual space $V_*$, that turns it into a complete operator system $V_\#$ satisfying certain universal property. In this article, we show that $V\cong T^\mathrm{d}$ for a complete operator system $T$ if and only if $V$ satisfies a natural property called tightness; in this case, $V\cong (V_\#)^\mathrm{d}$. Furthermore, we establish that if $S$ is tight, then $S\cong W_\#$ for a dual operator system $W$; in fact $S\cong (S^\mathrm{d})_\#$. Since all unital complete operator systems and all $C^*$-algebra are tight, one sees that every unital complete operator system and every $C^*$-algebra is of the form $W_\#$, for a dual operator system $W$.

Let $L:=-\Delta+V$ be a Schrödinger operator on the Euclidean space $\mathbb{R}^n$ with potential $V$ in the reverse Hölder class $RH^{n/2}$ satisfying some mild assumptions that $V$ neither decays too rapidly nor oscillates violently at infinity. In this paper, the authors construct a new system of dyadic cubes $\mathcal{D}^V$ that reflects the intrinsic geometry perturbed by $V$. Then using the quantitative geometric information of $\mathcal{D}^V$, the authors characterize the $L^p$ operator norm of the Riesz potential $L^{-\alpha/2}$ for all $\alpha\in (0,2]$ and $p\in (1,\infty)$. As applications, some quantitative spectral estimates for $L$ are given.

Deep joint source–channel coding (Deep JSCC) can suffer from domain shift between training and deployment data. We study single-source domain adaptation for image classification over wireless channels. A Gaussian example shows that improving channel quality does not necessarily improve target-domain accuracy under a fixed source-trained classifier. We further proposed an information objective to reduce domain information while preserving class information. Motivated by the analysis, we develop a domain-adaptive Deep JSCC method that combines class-wise adversarial alignment with pseudo-label supervised contrastive learning. The former uses all target samples to reduce class-conditional domain differences, while the latter selects high-confidence target samples to obtain class-discriminative received representations. We relate these losses to the information objective, and analyze the effects of pseudo-label errors and confidence-based selection. Experiments on SVHN–MNIST and PACS over AWGN and Rayleigh channels demonstrate improved target-domain classification without additional inference-time networks. The proposed method achieves 98.15% accuracy on SVHN $\rightarrow$ MNIST at 10~dB CSNR.

Let $\mathrm{A}(x_0,y)$ be an algorithm with two inputs: an initial point $x_0$ and an integer parameter $y$, which specifies that $\mathrm{A}(.,.)$ executes at most $y$ iterations or steps. Given an integer $p\ge 1$, $p$ parallel processors search an appropriate value of $T$ for for $A(.)$. Each processor executes an infinite sequence of stages indexed by $i=0,1,2,\ldots$. At stage $i$, processor $j$ is assigned $T_{j,i}=h(j,i),$ where $h:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{R}^{+}$ is a prescribed function. Processor $j$ $(j=0,1,\ldots,p-1)$ then executes $\mathrm{A}(x_0,T_{j,i})$. The efficiency of the parallel framework is characterized by its $(p,\alpha_p)$-approximation guarantee. Specifically, for every integer $T\ge T_0$, there exist a processor $j$ and a stage $i$ such that $T\le T_{j,i}\le T_{j,i}^*<\alpha_p T,$ where $T_{j,i}^*=\sum_{t=0}^{i}T_{j,t}$ denotes the cumulative number of iterations executed by processor $j$ from the beginning to stage $i$. We prove that this framework achieves a $(p,\alpha_p)$-approximation, and a tight lower bound for $\alpha_p$ for all large $p$. We develop arithmetically simple stochastic gradient methods in which every division is of the form $x/2^t$ for some integer $t$, and integrate them into the proposed parallel framework.

This paper proposes a unified modeling framework centered on "state-cost-extremal evolution," referred to as the SMFCE paradigm. The paradigm represents the effective state of a system as a point in a state space, characterizes state cost by a single-valued scalar field, and describes system evolution through local cost descent or local extremum selection. Its mathematical core employs classical tools from differential geometry, gradient systems, and dynamical systems. The contribution of this paper is not to reintroduce these foundational results, but to organize them into two operational modeling principles. On finite-dimensional smooth Riemannian manifolds under appropriate regularity conditions, the path independence of cost differences and the cost monotonicity of negative gradient flows follow directly from classical results, while the descent property of discrete local-minimum transitions follows directly from the definition of a minimum. For same-level iterative emergence and present-centeredness, the paper provides conditional mathematical formulations using, respectively, parameterized gradient systems and bifurcation theory, and Markov processes and receding-horizon optimization. The paper further introduces three layered metrics: cost quantity, directed coupling degree, and commitment fulfillment rate, at the state, transition, path, and system levels. Finally, the paper discusses semantic modeling of large language model training and inference within this framework, explicitly distinguishes classical mathematical results, conditional propositions, and empirical adaptation models, and provides reproducible empirical testing procedures for the two modeling principles.

Let $\mathcal{K}_n$ be the simplex of nonnegative $n \times n$ matrices whose entries sum to $n$. For $A$ in $\mathcal{K}_n$, with row sums $r_1,\ldots,r_n$ and column sums $c_1,\ldots,c_n$, define $\Phi(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\mathrm{per}(A)$. Dittert conjectured that $\Phi$ is uniquely maximized by the uniform matrix $U_n=J_n/n$, with maximum $2-n!/n^n$. We prove the conjecture for every $n \geq 2$. Hwang's theorem already determines every positive maximizer, so the essential problem is to exclude maximizers on the boundary of $\mathcal{K}_n$. A reduction of Cheon and Wanless supplies two zero entries in distinct rows and columns. In order four, we classify all admissible support strata and exclude them by exact Karush-Kuhn-Tucker identities together with a Bernstein-form positivity certificate. For all $n \geq 5$, an attained-stratum averaging argument compresses a hypothetical boundary maximizer to a seven-parameter matrix with one multiplicity $m=n-2$. A symbolic odd-variable expansion, a cofactor identity at the two missing entries, and uniform coefficient inequalities then contradict the one-sided multiplier conditions. This argument is uniform in $m$ and uses no finite enumeration. We also give independent spectral proofs in dimensions 7 through 13 and prescribed-zero proofs in dimensions 14 and 15. Every dimension-dependent calculation is certified over the rationals and recorded explicitly in the appendices.

The $S$-matrix conjecture was formulated by Sloane and Harwit in 1976, motivated by an $A$-optimal design problem arising in spectroscopy. It states that if $A$ is a nonsingular real $n\times n$ matrix whose entries lie in $[0,1]$, then $$\left\|A^{-1}\right\|_F\ge\frac{2n}{n+1}.$$ Moreover, equality holds if and only if $A$ is an $S$-matrix. In this paper, we prove the conjecture by combining a variational trace inequality in odd dimensions with a sharp range-constrained estimate for the Moore–Penrose inverse of a centered matrix in even dimensions.

Karp and Kuznetsov introduced the Meijer–Barnes $K$-function by replacing the Euler gamma factors in the Mellin–Barnes kernel of the Meijer $G$-function with Barnes double gamma factors, but excluded the regime $N=2(m+n)-p-q<0$. We complete this missing case. The kernel decays super-exponentially in two horizontal sectors, and the defining integral converges absolutely for every $z$ on the Riemann surface of the logarithm and every $\alpha\in\mathbb{C}$. There are exactly four admissible contour classes, denoted $K^\rightarrow,K^\leftarrow,K^\uparrow,K^\downarrow$, satisfying the universal relation $K^\uparrow+K^\downarrow=K^\rightarrow+K^\leftarrow$ and hence spanning a space of dimension at most three. Under an explicit closing-arc growth hypothesis, we derive residue expansions and strengthened vanishing theorems; the three basic transformation laws hold unconditionally. We also prove that no vertical contour is admissible for $N<0$, replace the standard vertical-contour Mellin-transform method by a compatible two-index recurrence for residue coefficients, and develop a kernel-level insertion calculus together with analytic transform identities where the required interchange is justified, including integer and Riemann–Liouville fractional derivatives. Finally, we construct a numerical evaluator for the Barnes double gamma function at general $\tau>0$ and validate representative identities and contour formulas to high precision.

We show that, for a given dimension, the set of Spearman's rank correlation matrices and that of linear correlation matrices coincide if and only if the dimension is no larger than nine. For this, we construct an extreme rank-four counterexample in dimension ten and prove its incompatibility using moment identities and Cauchy-Schwarz. Appending unit directions produces counterexamples in every higher dimension. This, together with existing results, completes the dimensional classification and settles a long-standing open question in quantitative risk management.

For every $n\in\mathbb N$, we construct a finite metric subspace $M_n$ of $\ell_\infty^n$ such that the Lipschitz-free space $\mathcal F(M_n)$ contains a linear isometric copy of $\ell_\infty^n$. This answers a question posed by Khan, Mim, and Ostrovskii, who obtained examples in dimensions three and four. As a consequence, the finite-dimensional real Banach spaces that admit a linear isometric embedding into a Lipschitz-free space over a finite metric space are precisely the polyhedral spaces.

Let a finite group $A$ act coprimely on a finite group $G$, and let $C=C_G(A)$. Navarro asked whether an $A$-invariant irreducible character has linear Glauberman–Isaacs correspondent if and only if it is nonvanishing on $C$. We give counterexamples to this equivalence. We also prove that when $G$ is nilpotent, nonvanishing on $C$ implies that the Glauberman–Isaacs correspondent is linear.

Let $N=2^k-1$ and let $\operatorname{wt}(n)$ denote the binary Hamming weight. The Tu-Deng conjecture asserts that, for every $1\le t\le N-1$, at most $2^{k-1}$ pairs $(a,b)\in\{0,\ldots,N-1\}^2$ satisfy $a+b\equiv t\pmod N$ and $\operatorname{wt}(a)+\operatorname{wt}(b)<k$. Partial results are known. We give a complete proof of this conjecture. We first show that the Tu-Deng counts equals the number of cyclic carry solutions for which $\operatorname{wt}(B)-\operatorname{wt}(A)<0$ and $A+t\equiv B\pmod N$. The enumerator of the cyclic carry solutions factors as $$C_v = 1+(X+Y-1)J_v+X^{\operatorname{z}(v)+1}Y^{\operatorname{o}(v)+1},$$ where $t=10v$ is the binary expansion of $t$(least significant bits first) and $J_v$ enumerates the language $$\operatorname{Sub}(v)\mathbin{\dot\cup}\{u\in\partial_1\operatorname{Sub}(v):u<_{\rm lex}v\}.$$ Estimating the strict negative half-plane mass of $C_v$ gives the desired bound.

Let P be the Hantzsche-Wendt (Promislow) group and B(4) the radius-four ball in its standard word metric. Dietrich, Lee, Nies and Vinyals determined the two-sided count: exactly 36 nontrivial units u of F_2[P] with both supp(u) and supp(u^{-1}) in B(4). We determine the one-sided count: exactly 52 nontrivial units with supp(u) in B(4) and no constraint on the inverse. The 16 new units have inverses supported at radius exactly 5; they form two orbits of size 8 under the symmetry group fixing the generating set, and all 52 units have support size 21 on both sides. Completeness is a single propositional unsatisfiability, certified by a DRAT proof checked with drat-trim. As an arithmetic consequence we prove: no unit of Z[P] with support in B(4) has nontrivial reduction modulo 2 – with no bound on the coefficients or on the support of the inverse. Since F_2[P] has 52 nontrivial units on that ball, this separates, in the untwisted setting, the integral group ring from its characteristic-two quotient at the first radius where the unit conjecture fails over a field.

This thesis brings us closer to applying computer-assisted, internal, type-theoretic reasoning to a category, with examples in the category of cubical sets, the category of groupoids, and the category of categories. The steps we make towards this general goal are both in furthering the type theoretic analysis of these examples, as well as implementing computer-assisted syntax-semantic reasoning as part of the HoTTLean project. One key component of this work is the consideration of new exponentiability conditions with respect to a class of maps in a category, and the development of polynomial functors for these maps, making the methods of "algebraic type theory" possible in this very general setting.

In this paper we study a functional differential equation subject to two-point boundary value conditions, where the functional dependence is introduced through an operator of the form \begin{equation*} \sum_{k=1}^{l}\gamma_{k}(t)\mathcal{C}_{k}(u), \end{equation*} where $\mathcal{C}_{k}:C(I) \rightarrow \mathbb{R}$, $k=1, \ldots l$, ($I:=[a,b]$) are linear continuous operators and $\gamma_{k} \in \mathcal{L}^{1}(I)$ for all $k=1, \ldots, l$. This formulation encompasses, among others, equations with piecewise constant arguments and those with integral-type dependence. We analyze this class of equations by deducing and characterizing their Green's function, as well as by computing the related adjoint problem. This approach enables us to establish connections among different types of functional equations and to relate equations with piecewise constant arguments to impulsive differential equations and non local boundary value problems. Next, we develop a series of comparison principles and results that allow us to characterize the regions where the Green's function of the original problem and of its adjoint maintain a constant sign. Finally, we illustrate the theoretical finding with representative examples.

The classification of the maximal subgroups of the Monster group has been completed recently, and explicit generators for each such subgroup (up to conjugacy) have been made available for the software mmgroup, with the exception of the maximal subgroup 59:29, see Dietrich et al. (Adv. Math., 2025; J Algebra, 2026). We provide explicit generators for this last maximal subgroup and comment on the extensive search that led to finding them. Our method is similar to, but significantly more involved than Bray et al.'s (London Math. Soc. J. Comput. Math., 2016) approach for constructing 47:23 in the Baby Monster. Our result allows us to provide a new short proof that the Monster does not have a subgroup PSL2(59), correcting a result of Holmes and Wilson (J. London Math. Soc., 2004).

Given a tessellation of Euclidean or hyperbolic space, the piecewise isometry group is the group whose elements are given by cutting space into finitely many tessellated convex subsets and gluing them back together. Groups of piecewise isometries of tessellations generalize Houghton's groups and Thompson's group $V$, and for cubical tessellations were studied by Bieri and Sach. We prove structure results about groups of piecewise isometries of sufficiently nice tessellations of Euclidean space, such as tessellations associated to crystallographic root systems, in particular proving that they are elementary amenable. Future work in progress will prove finite generation and higher finiteness properties.

Previous works of the author and Sogge [BS17], [BS18] showed the significance of microlocal Kakeya-Nikodym averages in improving $L^p$ bounds on (approximate) eigenfunctions of the Laplacian in the high frequency limit. These averages are formed by taking the $L^2$ norm of an eigenfunction when localized in phase space to a small, frequency-dependent tube about a geodesic segment via a pseudodifferential operator. The former work showed that for values of $p$ beneath the Stein-Tomas exponent, $L^p$ norms are controlled by a supremum over these averages. The latter work then showed that when $(M,g)$ has nonpositive sectional curvatures, there is a logarithmic gain in the averages. In combination, these two works improved the $L^p$ theory for eigenfunctions over the universal bounds of Sogge in this geometric setting. In the present work, we develop sufficient conditions for improving these averages which are more general than nonpositive curvature. Instead our sufficient conditions are rooted in the dynamics of the geodesic flow on the tangent bundle, considering cases where the flow expands and contracts tangent vectors in at least some directions, e.g. partially hyperbolic flows. We make use of Gaussian wave packet (phase space) transforms on the manifold in order to fully appreciate the gain these hypotheses impart on the microlocal averages. In the process, we further develop Gaussian beam approximations to the wave equation in a coordinate invariant manner.

We propose three new particle methods that improve computational efficiency by introducing a generalized Vertical Coordinate Transformation (VCT) for free surface flow problems with complex bottom boundaries. The first method is a bottom boundary-fitted particle method (BF-SPH). The BF-SPH is simply an arrangement of the body-fitted-coordinate system in the finite difference method to the particle method. The BF-SPH can accurately impose the bottom boundary conditions, while a simple procedure is performed by transforming the complex bottom into a flat one. The second method is the bottom boundary-fitted ellipsoidal particle method (BFE-SPH), which combines the BF-SPH with the ellipsoidal particle model proposed by Shibata et al. The BFE-SPH can speed up the particle simulation by choosing a reasonable aspect ratio of ellipsoidal particles. The last method is the $\sigma$-SPH method, which automatically selects the aspect ratios of ellipsoidal particles concerning water depth using the $\sigma$-coordinate system. The $\sigma$-coordinate is often employed in numerical simulations of oceanographic fields, such as in the Princeton Ocean Model. However, this is the first attempt to apply the $\sigma$-coordinate to a particle method. Vertical resolution is required from offshore to the coastal region in oceanographic problems such as tsunamis, especially when conducting detailed analysis using a 3-D particle method. Using the $\sigma$-coordinate allows for a stepwise transition to a naturally efficient coordinate system by referencing water depth. In this paper, we have shown that the above three methods can be generalized as Vertical Coordinate Transformations (VCTs), and the VCTs are successfully achieved by employing SPH(2) with the second-order accuracy of the second-order derivatives, including cross derivatives.

A population of oscillators typically synchronizes because coupling pulls their phases together. Here we consider malthusian oscillators whose coupling is demographic: oscillators are born and die at rates determined by their phases, generating an effective coupling without any phase velocity interaction. We find this coupling can synchronize a population, select a collective frequency, and produce a nongeneric fourth-root onset of coherence. These collective dynamics admit an exact reduction: a population with $m$ frequency classes reduces to $3m$ ordinary differential equations. This is the malthusian analogue of the Ott–Antonsen reduction for Kuramoto oscillators.

We study first-passage percolation on the $\ell$-spread-out line graph, where each vertex $i\in\{0,1,\dots,n\}$ is connected to all others at distance at most $\ell$. Here, we focus on the microscopic regime, with $\ell$ fixed as $n\to\infty$. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time $T_n$ from $0$ to $n$. If the weight distribution has finite variance or a heavy tail with exponent above $2/\ell_c$ where $\ell_c=\ell(\ell+1)/2$, then $T_n$ satisfies a Gaussian CLT with $\sqrt{n}$ scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that $T_n$, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of $T_n$.

We show that the fundamental group of the double of a complete negatively curved locally symmetric manifold along a closed geodesic is linear. More generally, we establish linearity of doubles of torsion-free transverse subgroups (also known in the literature as regular antipodal subgroups) of semisimple Lie groups along biproximal maximal cyclic subgroups.

Several applications for systems of conservation laws of the form $U_t + (\Phi (U) U)_x =0$, $U: R_t\times R_x\rightarrow R^n$ , $n\geq 2$, with $\Phi (U) = \phi (r, \Theta): R^n\rightarrow R$, $r = |U|$, and $\Theta = U/|U|\in S^{n-1}$, are obtained by imposing structural conditions to provide a classification framework for solutions dependent on the form of $\phi(U)$. By prescribing the evolution of a particular eigenvalue, we can categorize classes of such functions, $\phi$, for which this evolution is met, into depending on either a scalar field $z=rK(\Theta)$, where $K: S^{n-1}\rightarrow R$, or on the vector field $\Theta$, and find for which the amplitude of solutions may blow up in finite time. As a consequence, solutions to the corresponding Riemann problems can be divided into those with are classical and those which involve delta-shocks and/or vacuum states.

Let $\Gamma$ and $\Delta$ be finitely generated multiplicative groups of algebraic numbers such that $\Gamma\cap\Delta$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\in \Gamma$ and $y_1,y_2\in \Delta$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \ \ldots, \ x_m+y_m$ with $x_1, \ldots, x_m \in \Gamma$ and $y_1, \ldots, y_m \in \Delta$. For $m=3$ we classify such sums, up to finitely many exceptions, assuming the $abc$-conjecture.