Papers

We show that higher extension spaces between finite-dimensional nilpotent $\mathbb{F}_1$-representations maybe infinite-dimensional, thereby clarifying a misconception in the literature. Our examples arise from cyclic quivers. In particular, for a cyclic quiver $\Delta_n$, we show that $\operatorname{Ext}^3(-,-)$ vanishes for any pair of finite-dimensional nilpotent $\mathbb{F}_1$-representations of $\Delta_n$, while $\operatorname{Ext}^2(-,-)$ is infinite-dimensional for any pair of simple representations.

We rigorously derive the effective large-population, mean-field dynamics of particle-based reactive Langevin dynamics (PBRLD) models. These models extend particle-based stochastic reaction-diffusion (PBSRD) descriptions by incorporating velocities, inertial effects, and underdamped motion. In Isaacson, Liu, Spiliopoulos, and Yao, SIAP 2026, PBRLD models were formulated and shown to recover Doi volume reactivity PBSRD model in the overdamped limit. In this work we prove convergence of the associated measure-valued stochastic processes, representing species concentration fields on position-velocity phase space, to a deterministic mean-field limit. The limiting equations form a novel system of nonlocal kinetic reaction-diffusion partial integro-differential equations, coupling hypoelliptic transport with reaction terms that retain the spatial and velocity structure of the underlying particle interactions.

We study optimal constants and extremizers of smoothing estimates for quantum harmonic oscillators. In particular, we establish harmonic oscillator analogues of free particle results due to Simon (1992), Bez and Sugimoto (2014), and Bez et al. (2015).

In this paper, we establish two loop decomposition theorems for sphere bundles of real vector bundles under appropriate connectivity and characteristic class assumptions. Specifically, we treat two families of sphere bundles: those arising from rank-$(n-1)$ vector bundles and those from rank-$(n-3)$ vector bundles over closed smooth $n$-manifolds.

In this paper, we study the initial-boundary value problem for the nonlinear Schrödinger equation in $\mathbb{R}^{n}_{+}$ \begin{equation*} i\partial_{t}u+\Delta u+\lambda |u|^pu=0, \qquad (x, t) \in \mathbb{R}_{+}^{n} \times \mathbb{R}_{+},\ \ p\in\mathbb{R}_{+} \end{equation*} with nonhomogeneous Dirichlet boundary conditions. For the corresponding linear problem, endpoint Strichartz estimates are derived. For the nonlinear problem, we prove local well-posedness in $H^{s}(\mathbb{R}^{n}_{+})$ with $s\in[0,\frac{5}{2})$ and $p<\frac{4}{n-2s}$. Moreover, global well-posedness is established in the same regularity range. For $s\in[1,\frac{5}{2})$, the one-dimensional global theory of \cite{figment} in $H^{s}(\mathbb{R}_{+})$ is extended to $H^{s}(\mathbb{R}^{n}_{+})$. Additionally, we obtain global solutions in the lower regularity setting $s\in[0,1)$ for the first time. It is noteworthy that for $s=0$, we overcome the lack of mass conservation resulting from the nonzero boundary data and derive the pivotal $L^{2}(\mathbb{R}^{n}_{+})$ a priori estimates.

In this paper, we study the $\mathbb J$-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter $\mu$. This stratification was introduced by Chen-Viehmann and has been expected to serve as an interesting tool for studying basic loci in Shimura varieties. We parametrize the $\mathbb J$-strata whose stabilizers in the Frobenius-twisted centralizer group are parahoric by constructing a natural bijection to combinatorial invariants called small cocharacters. We further prove that the cardinality of these sets is equal to that of a certain subset of the Weyl group orbit of $\mu$. A relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong is also discussed.

We study the geometry of feasible value functions in infinite-horizon partially observable Markov decision processes (POMDPs) under memoryless stochastic policies. Our main contribution is a characterization of the feasible set of value functions as a semi-algebraic set, defined by explicit polynomial inequalities determined by the transition dynamics, observation kernel, and reward structure of the POMDP. This result extends prior work for fully observable Markov decision processes, where the feasible set is known to be a polytope, to the substantially more intricate partially observable setting. In contrast to the polyhedral structure arising in MDPs, partial observability induces fundamentally nonlinear constraints, leading to a richer and more complex geometric structure. Our geometric characterization provides new insight into the landscape of policy optimization in both MDPs and POMDPs, and reveals qualitative phenomena unique to partial observability, including the emergence of isolated local maximizers of the long-term reward and their dependence on the initial state distribution.

In this article, we study the (full or vertical) linearisation problem under a finite Galois étale cover. As an application, we give sufficient conditions for full linearisation near Hopf manifolds and for vertical or full linearisation near hyperelliptic manifolds.

Rotatable antenna (RA) technology has emerged as a promising solution to enhance spectrum efficiency by exploiting additional spatial degrees of freedom (DoFs) in multiple access networks. However, the relative performance superiority among different multiple access schemes remains largely unclear due to the unique capability of RA in reconfiguring the directional gain pattern. In this letter, we conduct a theoretical comparison between non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes in RA-assisted communication systems in terms of transmit power minimization, subject to constraints on antenna rotational range and users' target rates. To address the associated non-convex optimization problem, a particle swarm optimization (PSO) algorithm is employed to optimize the rotational angle. Simulation results demonstrate that RA-assisted schemes significantly reduce transmit power compared to fixed-antenna benchmarks. Furthermore, RA-assisted NOMA may perform worse than time-division multiple access (TDMA) for symmetric user deployments, while it exhibits superior robustness and energy efficiency in asymmetric scenarios.

We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the $\mathsf E_\gamma$-divergence with $\gamma\ge1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the $\mathsf E_\gamma$-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, $\chi^2$, and Rényi divergences. For independent Metropolis–Hastings with target $\pi$, proposal $q$, and unbounded importance weight $w=d\pi/dq$, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core $C_R=\{w\le R\}$ and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile $H_R=\pi(w>R)$, recovering sharp existing moment-based convergence rates when $\mathbb E_q[w^p]<\infty$ for some $p>1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p>1$ exists.

Modeling interactions among multimodal, high-dimensional data is intrinsically challenging due to ultra-high dimensionality and complex dependence structure with high level noise. Screening methods are effective for reducing dimensionality, but most existing approaches shrink only the predictor space while retaining all outcomes. In cross-modal analyses, different outcomes often select different predictor subsets, so the union remains large and the response dimension is unchanged, limiting the practical benefit of screening. This gives rise to heavy computational burdens and poor interpretability. To address these limitations, we propose a new screening framework, Graph Independence Dual Screening (GIDS), which simultaneously reduces the dimensionality of response variables and predictors. We design computationally efficient algorithms that facilitate downstream selection procedures, improving accuracy and scalability, and establish supporting theoretical results. Extensive simulation studies demonstrate that GIDS outperforms existing methods that screen only predictors. To illustrate its utility, we applied GIDS to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset, analyzing interactions between genome-wide 865,353 DNA methylation and 49,386 transcriptomic variables. GIDS reduced the feature space to approximately 9,000 CpGs and 2,000 transcripts, uncovering blockwise interaction structures: clusters of CpG sites and gene transcripts with strong associations. These findings not only improve computational tractability but also yield interpretable biological insights, highlighting coordinated regulatory mechanisms underlying Alzheimer's disease.

We investigate the energetics, interactions, and ordering tendencies of bimerons (cholesteric fingers of the second type, CF–2) and hopfions in thin films of chiral magnets and chiral liquid crystals hosting a conical background state. Although isolated bimerons possess positive eigen-energy with respect to the conical phase, they develop an attractive interaction mediated by the restructuring and partial overlap of their positive-energy shells, i.e., intermediate regions formed relative to the conical state. This attraction promotes the formation of bound pairs and extended bimeron chains, even in parameter regimes where a periodic bimeron lattice is no longer thermodynamically stable. Extending the analysis to three dimensions, we show that circularization of bimerons into hopfions renders their energy finite and gives rise to a well-defined metastability window closely linked to the stability range of cholesteric fingers. Isolated hopfions likewise exhibit an attractive interaction within the conical phase, leading to the formation of hexagonally ordered clusters. The attraction originates from the competition between favorable and unfavorable twist regions and from the energetic cost of the shell structures imposed by the conical background. Despite the presence of attractive pair potentials and cluster formation, we demonstrate that hexagonal hopfion lattices do not exhibit an equilibrium lattice period. Instead, the system evolves toward states in which the conical spiral or the CF–1 phase (cholesteric fingers of the first type) progressively invade the inter-soliton regions, thereby preventing crystallization. Our results reveal a regime of attraction without stable long-range order and clarify the interplay between topology, confinement, and conical-phase frustration in chiral magnet and liquid-crystal thin films.

We study single-error detection and correction for analog codes over $\mathbb{R}$. The key performance measures are the parameters $\Gamma_1(\mathcal{C})$ and $\Gamma_2(\mathcal{C})$, which quantify, respectively, the minimum separation required between large outlying errors that must be detected or located and the magnitude of tolerable perturbations. First, we prove that every real linear $[n,k]$ code $\mathcal{C}$ satisfies \[ \Gamma_1(\mathcal{C})\ge 2\left\lceil\frac{n}{n-k}\right\rceil. \] Moreover, when $k=n-2$, we prove that every real linear $[n,n-2]$ code $\mathcal{C}$ satisfies \[ \Gamma_2(\mathcal{C})\ge \frac{1}{\sin^2(\pi/2n)}. \] Together, these two lower bounds settle all four open problems of Roth concerning the optimality of single-error-detecting and single-error-correcting analog codes. The proof of the first bound is based on a double-induction argument, while the proof of the second combines a zonotope-based geometric characterization of $\Gamma_2(\mathcal{C})$ with a cyclic sine-product inequality. In addition, we construct analog codes with higher fixed redundancy and show that, for every fixed $r\ge 2$, there exists a class of linear $[n,\ge n-r]$ codes over $\mathbb{R}$ such that \[ \Gamma_2(\mathcal{C})\le O\left(n^{1+\frac{1}{r-1}}\right). \] This gives a new upper bound in the fixed-redundancy regime, which was not covered by previously known constructions.

We prove a conjecture of Glasby, Praeger, and Unger concerning the symmetric group $S_{n}$. Let $\pi_{n}$ denote the proportion of elements of $S_{n}$ that are pre-$p$-cycles for some prime $p\in[2, n-3]$. We prove that $\pi_{n} > 1/3$ for all $n\geq 8$.

We study the regularity of spectral measures of self-adjoint operators on infinite weighted graphs in the unimodular setting. This framework encompasses operators in the group algebra of a finitely generated group, random operators whose distribution is quasi-invariant under a group action, and Benjamini–Schramm limits of operators on finite graphs. Under a natural geometric condition on the underlying graph, we prove that the expected spectral measure satisfies a logarithmic Hölder regularity estimate. The proof relies on a strengthened version of the monotone labelling method previously introduced with Sen and Virág to control the pure point part of the spectal measure. Applications include operators in group algebras of indicable groups, Anderson-type models with arbitrary compactly supported potentials on Cayley graphs, anisotropic percolation operators, and operators on quasi-transitive graphs. In particular, our results extend the classical Craig–Simon theorem beyond the Euclidean lattice.

We study a kinematical consequence of the Hopf-algebraic momentum composition law in $\kappa$-Minkowski spacetime. The same curved momentum space can be described in different coordinates. In the bicrossproduct basis the ordered-plane-wave labels are the translation-generator eigenvalues, so the relevant map is one-to-one. In the classical basis, instead, the translation eigenvalues $P_\mu$ are nonlinearly related to the ordered-plane-wave labels $p_\mu$. This relation can fail to be globally one-to-one in a high-momentum region. When a given classical-basis four-momentum admits more than one real auxiliary preimage, the branch-sensitive quantity $P_+\equiv P_0+P_4=\kappa e^{p_0/\kappa}$ enters the coproduct and resolves the branches in two-particle states. Imposing the vanishing total-momentum constraint therefore gives branch-dependent $\kappa$-deformed back-to-back momentum correlations. In a single-branch regime this is just a deformed correlated product, while in a multibranch regime a state specified only by $P_\mu$ can be expanded into distinct auxiliary branches. If $P_\mu$ are taken as the directly meaningful momenta, the physical content is the resulting deformed correlation pattern. If the auxiliary variables $p_\mu$ are assigned operational meaning, the same constrained state can be interpreted as a superposition over different auxiliary branches. We also compare this structure with standard regular self-adjoint nonrelativistic minimal-length models and find no analogous smooth local two-real-branch inversion on their physical domains.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit $t\to+\infty$ with $x=O(t^{1/2})$. We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data $u_0$ in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$ that exhibit generic behavior of the reflection coefficient at the origin.

Motivated by recent developments at the interface of operator algebras and random matrix theory, we propose new conjectures concerning the asymptotic structure of random matrix models of the countable free groups. The first conjecture predicts a random matrix analogue of the Akemann-Ostrand property for free groups, and reveals a succinct approach to recover the Peterson-Thom property for $L(\mathbb{F}_2)$. The second stronger conjecture is motivated by continuous model theory. It predicts that the random embedding of the free group factor into a matrix ultraproduct is existential. We discuss the interesting relationship between these conjectures.

For a finitely generated group $G$, the Bieri–Neumann–Strebel–Renz (BNSR) invariants are subsets of the character sphere of $G$ that govern the finiteness properties of normal subgroups containing the commutator subgroup. We investigate the BNSR invariants of $2$-knot groups and link groups. As an application, we prove Yasuda's conjecture that every non-trivial $2$-plat $2$-knot is non-fibered.

Multiplication table of a ring with identity 1 is said to have the diagonal condition if 1s occur only on the main diagonal. In this paper, we study the diagonal condition in the ring of Gaussian integers $\mathbb{Z}[i]$. Furthermore, we also find the Gaussian integers $\alpha$ so that the rings of Gaussian integers modulo $\alpha$ satisfy the diagonal condition.

While the uncrowding algorithm on set-valued tableaux has long been instrumental in proving the Schur positivity of stable symmetric Grothendieck polynomials, lattice models have emerged as a modern framework for investigating symmetric functions, in particular symmetric Grothendieck polynomials. In this work, we synthesize these combinatorial and lattice-theoretic approaches by defining both the Robinson–Schensted–Knuth (RSK) correspondence and the uncrowding operation directly on a 5-vertex model of Motegi and Sakai and its subsequent reinterpretation by Buciumas, Scrimshaw, and Weber. Our lattice-based RSK formulation yields a powerful new result: the direct construction of the associated crystal structure on the states of the 5-vertex model.

This paper examines three approaches for modeling the dynamics of a flexible-link 2-DoF robotic arm to address unmodeled dynamics not captured by rigid-body models. Two physics informed models combine rigid-body dynamics (RBD) formulations with a Gaussian Mixture Model (GMM) to capture residual model errors and linkage flexibility. A kinematics-based regression model serves as a purely data-driven baseline. Using an open-source dataset, torque predictions are first estimated using Ridge regression on kinematic features, while the physicsbased baseline is constructed from published specifications, and ordinary least-squares regression is subsequently used to estimate the same parameter set directly from data. Results show that the physics-based parameters yield the poorest accuracy, while regularized and least-squares estimators align more closely with measured torques. Residual analysis and error metrics highlight the limitations of purely parametric models for flexible-link systems and underscore the value of regularization and data-driven identification, supporting developments of semi-parametric residual learning methods.

A basic open question motivated by the study of ribbon knot diagrams asks whether every ribbon knot can be presented as a symmetric union. In this article, we give a negative answer to this question by exhibiting a ribbon Montesinos knot which does not admit a symmetric union presentation.

The Distance Geometry Problem (DGP) asks for a geometric realization of a weighted graph with \(n\) vertices in \(\mathbb{R}^K\) such that Euclidean distances between vertices match the given edge weights. When a vertex order where every non-seed vertex has \(K\) predecessors inducing a clique is part of the input, the search space can be discretized via \(K\)-lateration, branching into a binary tree. In this subclass, known as the Combinatorial Discretizable Distance Geometry Problem (Combinatorial DDGP), the goal is to determine the number of realizations satisfying all distance constraints. While the number of realizations is almost always \(2^{n-K}\) when no additional distance constraints are present, a topological solution count in the presence of additional constraints has remained elusive. We develop an algebraic rank-count theory for the feasible binary branch codes, proving that under mirror-separated parameters they form an affine space over \(\mathbb{F}_2\) whenever a viable reference solution exists.

We develop a spectral theory of scale-invariant operators on the multiplicative half-line $(\mathbb{R}_+, dx/x)$. A symmetric kernel $M(x, y)$ satisfying $M(kx, ky) = k^{-a}M(x, y)$ necessarily factorizes as $(xy)^{-a/2}F(x/y)$, where the shape function $F$ depends only on the ratio of its arguments. The Mellin transform diagonalizes such operators: the generalized eigenfunctions are $\psi_\omega(x) = x^{-a/2+i\omega}$, and the eigenvalues are the Mellin multiplier $\tilde{F}(\omega)$. This structure reveals a fundamental decoupling of two exponents. The geometric exponent $a$, carried by the power-law envelope $(xy)^{-a/2}$, governs the matrix scaling under dilation. The spectral exponent $b$, measured from the eigenvalue decay of the finite-dimensional truncation, is an effective quantity determined by the shape of $\tilde{F}(\omega)$. For the explicit kernel $F(t) = c \rho^{|\ln t|}$, the Mellin multiplier is a Lorentzian of width $\sigma = -\ln \rho$, not a power law – so $b$ is generically distinct from $a$. This decoupling provides a precise mathematical characterization of multicriticality: the equality $a = b$ corresponds to a simple critical fixed point of the Renormalization Group, while $a \neq b$ signals the presence of multiple independent scaling dimensions. We prove that the discrete self-similarity condition forces eigenvector collapse on the lattice, motivating the continuum formulation. Finite-size corrections from lattice sampling are quantified numerically.

For $1 \leq a \leq 2$, we say Proposition $(1+a)$ holds if every sufficiently large even integer $N$ can be written as $$N = p + rq, \quad r \leq q^{a-1},$$ where $r$ is either $1$ or prime, and $p,q$ are primes. Thus Proposition $(1+1)$ is essentially the binary Goldbach Conjecture, and Proposition $(1+2)$ is Chen's theorem. We prove unconditionally that Proposition $(1+1.9)$ is true. Assuming the Elliott–Halberstam Conjecture, the exponent $1.9$ can be improved to $1.4$. Analogously, Proposition $(1-a)$ is formulated for the Twin Prime Conjecture. Unconditionally, we prove Proposition $(1-1.75)$, and under the Elliott–Halberstam Conjecture, Proposition $(1-1.4)$. For six decades, a substantial theoretical divide has persisted between Propositions $(1+2)$ and $(1+1)$, and likewise between Propositions $(1-2)$ and $(1-1)$. By constructing new weighted sieves and adopting new analytic tools, this paper establishes a connecting pathway between them and achieves breakthroughs in this line of research.

In this note we provide corrections to Theorem 5.4 of the paper “Isomorphism classes of Drinfeld modules over finite fields”, arXiv:2209.15033. The main theorems of this paper, Theorem A and B in its introduction, are valid as stated; in the proof of Theorem B the argument needs to be modified by replacing the erroneous Theorem 5.4 by the theorem of this note.

Leader-follower gradient iterations, including bilevel and two-time-scale methods, can exhibit large finite-time excursions despite asymptotic stability. For cascaded iterations with block matrix J = [[A,0],[C,D]], symmetric diagonal blocks of norm at most gamma < 1, and coupling norm at most c, we prove an exact, dimension-free minimax formula: at every horizon t, the worst-case amplification equals the spectral norm of [[gamma^t,0],[c t gamma^(t-1),gamma^t]], attained by an aligned two-dimensional quadratic. Consequently, the worst transient is on the order of 1 + c/(1-gamma), and eigenvalues alone cannot predict finite-time gain. Under martingale noise, the aligned system has stationary follower variance sigma^2 c^2 (1+gamma^2)/(1-gamma^2)^3, establishing cubic inverse-gap scaling. For quadratic followers, the coupling is linked to follower sensitivity R by C = (I-D)R. When the leader is isotropic and the follower map is positive semidefinite, amplification is at most 1 plus the operator norm of R, independently of the spectral gaps. A look-ahead Newton update eliminates follower memory and is minimax-optimal among linear updates that track the leader exactly. Experiments validate the bounds and show that accelerated or inexact follower solvers can exceed the sensitivity bound when their effective maps have negative eigenvalues. Code and experiment scripts: https://github.com/Axym-Labs/transient-amplification

This paper presents an efficient reduced order modeling (ROM) framework for simulating spatio-temporal pattern formation in three-species diffusive Lotka-Volterra systems. To alleviate the high computational cost associated with long-time simulations of the high-dimensional full order model (FOM), we apply proper orthogonal decomposition (POD) to project the solution onto a low-dimensional subspace. Further efficiency is achieved through tensorial POD (TPOD), which preserves the quadratic nonlinear structure and enables offline-online decomposition. Numerical experiments demonstrate that both POD and TPOD accurately replicate the key features of spatial segregation patterns while substantially reducing computation time, whereas the TPOD is faster. Additionally, we demonstrate accurate long-time pattern prediction using limited training data.