Papers

Let $p$ be a prime, $G$ be a finite group, $H$ a proper subgroup of $G$ and $V$ a finite dimensional $GF(p)G$-module. The triple $(G,H,V)$ is immutable if and only if $G$ and $H$ have the same orbits on the vectors of $V$. We determine the immutable triples for $G$ a quasisimple group, $H$ a subgroup of $G$ and $V$ a $GF(2)G$-module.

We utilize Stein's method to establish quantitative bounds on the total variation distance between the invariant measure of a drifted nonlocal Markov operator and that of its local counterpart under minimal assumptions on the drifts. The main ingredient is a reduction via Stein's method that transforms the original problem into analyzing growth estimates for solutions to a nonlocal Poisson equation and decay estimates for the invariant measure of the local operator.

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^{*}$. We introduce a hybrid projection method for approximating a common element of four classes of constraints: the set of fixed points of a countable family of generalized nonexpansive-type maps, the solution sets of countably many equilibrium problems, the solution sets of countably many variational inequality problems, and the zero sets of countably many maximal monotone operators. The algorithm combines equilibrium resolvents, variational inequality resolvents, generalized resolvents of maximal monotone operators and a shrinking projection step. Under precise monotonicity, continuity and closedness assumptions, we prove that the generated sequence converges strongly to the generalized projection of the initial point onto the common solution set. We also establish residual convergence, derive convex minimization consequences, present a finite-truncation result, and give an illustrative Hilbert-space specialization showing why the countable setting cannot, in general, be reduced to a finite-family theorem.

In this paper, we study local radial rigidity phenomena for a broad class of general-order elliptic systems on Riemannian manifolds via a reduction to singular ordinary differential equations of Euler type. These rigidity properties have natural applications to several geometric problems, including prescribed curvature problems of Yamabe and Paneitz type in conformal geometry, higher-order analogues arising from harmonic maps, and Calabi-type problems on Kähler manifolds. The method applies also to higher-order nonlinear Poisson systems and, more generally, to weighted divergence-form elliptic systems, yielding local uniqueness and existence results for solutions with prescribed initial jets at the singularity.

Training stability is a key bottleneck in low-precision language model training: efficient low-cost paths can still produce short-lived numerical risks at a small set of operators. We formulate this as runtime stability control and present Gradient Norm-to-Mean Ratio (GNMR), a lightweight controller that compares each recoverable unit's current gradient norm with its historical mean. Together with $\Delta$-GNMR for abrupt short-window increases, GNMR maps local risk signals to bounded recovery actions under a hard $\mathrm{maxO}$ budget and a short lock interval, without changing the numerical format, kernel, or backend recipe. Across activation-quantization stress, DeepSeek-style recipe-level training, and LLaMA-2 13B fine-tuning, GNMR preserves high-fidelity quality with sparse, budgeted recovery. These results support GNMR as a backend-agnostic controller to improve low-precision training stability while preserving low-cost execution.

We prove a flatness theorem for classical stable homogeneous solutions of the $\gamma$-Alt–Phillips free boundary problem in three dimensions in the range $0<\gamma\le 2/3$, where $\gamma$ is the exponent in the energy density $|\nabla u|^2+u^\gamma$. In particular, this implies full regularity of the free boundary for minimizers of the corresponding Alt–Phillips energy in dimension $3$.

Solving Maxwell's equations to obtain explicit analytical representations of electromagnetic fields in open-space settings with general initial conditions and source terms remains a fundamental challenge. We address this problem through an operator-based construction, developing an infinite-order curl-operator expansion that yields analytical solutions as function-to-function mappings from initial data and source terms to electromagnetic fields. The framework is applied to representative cases, including Gaussian initial conditions, Gaussian sources, harmonic sources, and Ricker wavelet sources. In these applications, a family of deformed sine and cosine functions emerges naturally, enabling explicit analytical representations of the solutions. The analysis further shows that these deformed trigonometric functions are intrinsic to the general solution formulas, with their structure determined by the initial conditions and source terms. The results are compared with numerical simulations based on the FDTD method and further applied to a systematic study of mesh refinement effects. This study shows that, for the problems considered, the error measures approach limiting values as the mesh is refined, indicating that further step size reduction alone may produce little additional improvement once the mesh is sufficiently fine. The results demonstrate that the analytical solutions provide theoretical insight into electromagnetic field behavior while serving as practical benchmark solutions for computational electromagnetics.

A cut-edge of a connected graph is an edge whose deletion increases the number of components. In this paper, we first state some conditions for a planar graph to have a few cut-edges. After that, we use the main results to study the conditions for a colored planar graph to have a bounded conflict-free connection number.

Let $\E$ be a uniformly smooth and uniformly convex real Banach space. We study an inertial hybrid retraction method for a countable sequence of mappings satisfying the NST-condition and an approximate $\phi$-Fejér inequality with vanishing errors. We prove that the generated sequence converges strongly to the sunny generalized nonexpansive retraction of the initial point onto the common fixed-point set. The theorem admits vanishing error sequences that need not be summable and therefore contains the summable-error setting as a special case. We also establish a Bregman-projection analogue and provide illustrative examples.

We study perturbation and stability properties of solution mappings associated with convex regularized least-squares problems. We first establish an implicit function theorem for generalized equations governed by maximally monotone operators and smooth perturbations, yielding conditions for local Lipschitz continuity, directional differentiability, and semismoothness of solution mappings. We then characterize the kernel of generalized Hessians of convex functions through the subspace parallel to the subdifferential under $C^2$-cone reducibility assumptions on the conjugate of the regularizer, thereby replacing difficult second-order objects by tractable first-order conditions. As applications, we derive stability results for broad classes of regularized least-squares problems, including weighted polyhedral support-function regularizers and piecewise linear-quadratic penalties. Our framework unifies and extends several recent results on Lipschitz stability for LASSO-type models and monotone generalized equations.

We introduce a family of constants \[ C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac1k\right) - \log n \right), \] which may be regarded as polylogarithmic analogues of Euler's constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants \(C_m\) to special values of certain Dirichlet series involving the Riemann zeta function.

Many stochastic gradient methods are believed not to converge when the noise in stochastic gradients has only a finite $p$-th moment for $p\in\left(1,2\right)$, a setting known as the heavy-tailed noise assumption. However, some recent studies have found that Stochastic Gradient Descent ($\textsf{SGD}$), without any modification to its update rule, can surprisingly converge in expectation for convex problems with bounded domains, highlighting the potential of classical stochastic gradient methods. Inspired by this recent progress, we provide a comprehensive study of stochastic optimization under heavy-tailed noise and establish new in-expectation convergence results for Stochastic Mirror Descent ($\textsf{SMD}$) and Accelerated Stochastic Mirror Descent ($\textsf{ASMD}$) in convex optimization, and for $\textsf{SGD}$ and Stochastic Gradient Descent with Momentum ($\textsf{SGDM}$) in nonconvex optimization. Notably, our results not only hold without algorithmic changes but also avoid restrictive assumptions, such as bounded domains, imposed in prior work. More importantly, our analysis provides a new, elegant, and powerful framework for studying heavy-tailed stochastic optimization, opening a new route to understanding first-order stochastic gradient methods.

This paper studies the regularization of ill-posed inverse problems by deep neural networks (DNNs). We extend architecture-based regularization from shallow networks to deep models by developing a deterministic framework in which the admissible network class is enlarged adaptively and the resulting architecture complexity acts as the regularization mechanism. We propose two discrepancy-principle-driven expanding DNN algorithms to treat the cases where an explicit parameter-radius bound is available and unavailable, respectively. For both algorithms, we prove the finite termination of the adaptive expansion procedure and the convergence of the regularized solutions as the noise level vanishes. In addition, we derive explicit asymptotic bounds on the terminal network architecture, thereby quantifying how the required network complexity scales with the noise level. Numerical experiments on several representative linear and non-linear inverse problems support the theoretical findings and illustrate the practical usefulness of the proposed framework.

In many modern machine learning pipelines, abundant pretrained representations serve as noisy proxy covariates, while task-specific labels remain scarce. We study semi-supervised regression in this setting, and propose a simple two stage estimator that learns kernel eigenfeatures from all proxy covariates and fits a ridge predictor on labeled data. We derive finite sample bounds showing that fast labeled sample rates are recovered when proxy perturbation is controlled and unlabeled proxy covariates are sufficiently abundant. We also show that distribution regression is a direct special case, with analogous guarantees when the finite bag size is large enough. Experiments show consistent gains over supervised and semi-supervised baselines, especially in low label regimes.

Exceptional longest-run behaviour in continued fraction expansions is studied through the interaction between fixed-symbol runs and the overall longest run. For every prescribed partial quotient value and every admissible growth scale, a full Hausdorff dimensional set of irrational numbers is constructed on which the longest run of the prescribed value has exactly the prescribed asymptotic growth and, for every initial length, uniquely realises the overall maximum. It follows that the symbol responsible for the overall longest run can be fixed in advance without any loss of Hausdorff dimension. Thus, the known full-dimensional exceptional-set results for fixed-symbol longest-run growth and for overall longest-run growth are simultaneously strengthened, while the maximising symbol in the overall problem is shown to be fully prescribable.

We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting. In particular, let $\varepsilon$ be a sufficiently small constant, and suppose that $X \in \mathbb R^{n \times n}$ is a Gaussian matrix with a planted rank-$1$ spike, and $E \in \mathbb R^{n \times n}$ is an adversarially chosen matrix supported on an $\varepsilon n \times \varepsilon n$ principal minor. Let $v_{\mathrm{AMP}}(X)$ be the output of an AMP iteration on the uncorrupted matrix $X$. We give a procedure that, given access only to the corrupted matrix $Y = X + E$, computes a vector $v_{\mathrm{ALG}}(Y)$ which is $\tilde{O}(\sqrt{\varepsilon})$-close to $v_{\mathrm{AMP}}(X)$, for any of a class of AMP iterations which includes sparse Principal Component Analysis (PCA), non-negative PCA, and $\mathbb Z_2$ synchronization. Our algorithm consists of a spectral pre-processing step combined with a robust spectral initialization procedure; given these inputs, we prove that (perhaps surprisingly) AMP is robust out-of-the-box.

We characterize dominating sets for weighted Fock spaces $F^p_{\alpha,w}$ induced by Muckenhoupt $A_{\infty}$ weights, which leads to a sufficient condition for reverse Carleson measures for the spaces $F^p_{\alpha,w}$ with restricted $A_{\infty}$ weights. Applications to invertibility of Toeplitz operators, closed range property of Volterra operators, and atomic decomposition of $F^p_{\alpha,w}$ are given.

Let $k$ be a real quadratic field. Let $k_n$ be the $n$-th layer of the $\mathbb{Z}_2$-extension of $k$, and $2^{a_n}$ be the $2$-part of the class number of $k_n$. In this paper, we derive many laws concerning the sequence $(a_0,a_1,\dotsc)$, by iteratively extending the group of cyclotomic units.

This research presents a novel stochastic framework for proactive cybersecurity defense timing under a single attack scenario. The approach models the defense process as a continuous observation mechanism in which the defense instant and the subsequent observation slot follow independent exponential distributions. Laplace-Carson transforms combined with first-excess theory yield the joint detection function that brackets the attack moment. Marginalization under Markovian Poisson arrivals then produces the probability density of the defense moment and conditional expectations of pre-attack and post-attack observation times. These closed-form results enable quantitative assessment of defense timing sensitivity to threat intensity and support precise calibration of observation parameters for low-latency proactive measures. Major contributions include the explicit derivation of marginal distributions and expected values, visualization of defense moment density, and the bridging of stochastic duel methodology with practical cybersecurity applications.

Continuous data assimilation seeks to estimate the state of a dynamical system from partial observations. In many applications, however, the state dynamics are unknown or prohibitively expensive to simulate at the required resolution, leading to model error. Motivated by this challenge and the increasing adoption of machine learning surrogates in data assimilation, this paper develops a unified finite-dimensional analysis of nudging algorithms that employ learned surrogate models of the dynamics. We first establish general conditions on the dynamics and observations that guarantee accurate tracking for nudging with the true dynamics model, both in the noise-free and noisy settings. We then show that nudging algorithms that employ surrogate models retain exponential convergence up to an explicit error floor that quantifies the effects of surrogate approximation error and observation noise. Finally, we analyze surrogate models obtained by learning either the vector field or the short-time solution map of the system, and quantify the amount of training data needed to ensure accurate nudging in the noise-free setting. Numerical experiments support the theory.

Online high-dimensional regression has gained increasing attention in recent years, yet existing methods typically assume that all candidate features, including important ones, are observed from the outset of data collection. This assumption is often violated in real-world scenarios, where new variables become available gradually as data accumulate. To address this gap, we introduce a novel framework, Recurrent Adaptive Variable Selection (RAVAS), for online regression with expanding observability. RAVAS employs a recurrent procedure that dynamically updates feature selection as both the sample size and the observable feature set grow. The algorithm is designed to be computationally efficient and memory-light, relying only on low-dimensional sufficient statistics that are updated online. A key advantage of the method lies in its ability to detect and incorporate important variables that emerge later, thereby mitigating the effect of early-stage missingness. We establish theoretical guarantees on model selection, estimation error, and feature coverage, and develop an adaptive online tuning strategy. Extensive simulations and real-world experiments verify the effectiveness of RAVAS for high-dimensional streaming data.

This paper introduces interval $B_{\pi}^{R^I}$-tensors as a natural extension of $B_{\pi}^{R}$-tensors to the interval setting. We provide two practical verifiable criteria for an interval tensor to be an interval $B_{\pi}^{R^I}$-tensor, one based on endpoint inequalities and another constructing an explicit vector $\pi$. Connections with interval $P$-tensors, positive definite interval tensors, and interval $Z$-tensors are established. Applications in polynomial optimization and interval tensor complementarity problems are briefly discussed.

We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SA-NODEs). We constructively prove that SA-NODEs are capable of exact interpolation of admissible finite datasets, and even satisfy a stronger property that we call simultaneous cell controllability (SCC): their flows can map prescribed disjoint cells into arbitrarily small target balls. This property is the mechanism that upgrades interpolation into quantitative generalization, by allowing SA-NODEs to emulate piecewise-constant nonparametric estimators. Consequently, our risk bounds recover the rates of histogram and nearest-neighbor estimators, provided the network width satisfies a conservative scaling with the sample size. Numerical experiments show that trained SA-NODEs achieve competitive – often lower – test errors than these baselines. Finally, we show that the explicit time dependence is essential. Although two-layer autonomous NODEs can interpolate geometrically nondegenerate datasets, structural obstructions prevent them from achieving SCC. These limitations, further confirmed numerically, support the view that SA-NODEs provide a minimal effective architecture for learning.

We present a lifting-based interface reduction framework for nonlinear transmission and eigenvalue problems. The method represents the solution as a sum of a bulk component and a lifting component that carries the interface jump, thereby reducing the original problem to a nonlinear system posed on the interface. A low-dimensional approximation is obtained by restricting the interface unknown to a finite-dimensional subspace. The corresponding lifting modes are precomputed and reused, leading to a formulation in which the bulk operator remains fixed and the essential behavior is governed by a small number of interface degrees of freedom. For eigenvalue problems, the same framework yields a reduced system in which eigenvalues are identified through the near-singularity of a parameter-dependent interface matrix. The associated eigenvectors reveal dominant interface modes, providing a direct interpretation of the spectral structure. Numerical experiments show that both approximation accuracy and spectral behavior are determined primarily by the interface representation rather than by the bulk discretization. In particular, enriching the interface space rapidly improves accuracy and reveals additional eigenmodes, while mesh refinement alone has limited effect. These results indicate that transmission and eigenvalue problems are effectively governed by a small number of interface modes, offering a simple and computationally efficient perspective on model reduction.

We use sieving arguments to estimate the frequency of $s$-tuples of rational points $$(P_1,\dots,P_s)\in E_1(\mathbb{Q})\times\cdots\times E_s(\mathbb{Q}),$$ where $E_1,\dots,E_s$ are (not necessarily distinct) elliptic curves over $\mathbb{Q}$, for which the product of their denominators is a perfect $\ell$th power for a fixed prime $\ell$. We consider two settings: one in which the points are of the form $n_iP_i+Q_i$ with $n_i$ ranging over an interval, and another in which we take arbitrary points of bounded canonical height. In the special case where all $Q_i$ are the points at infinity, we also obtain better estimates by using a version of the elliptic sieve with elliptic divisibility sequences. Consequently, we derive analogues of these results for various rational functions, providing elliptic analogues of R. de la Bretèche, P. Kurlberg and I. E. Shparlinski (2021).

The Black Scholes equation provides a fundamental model for the no arbitrage pricing of financial derivatives. After finite difference discretisation, the pricing problem can be formulated as a finite dimensional linear algebra problem involving the inverse of a non Hermitian time step matrix. Recent advances in quantum linear algebra algorithms, particularly the generalised quantum signal processing (GQSP)algorithm, enable matrix functions to be implemented through polynomial transformations of a suitable unitary or Hermitian form. In this paper, we develop a Hermitian block embedding method that enables GQSP to be applied to the two dimensional Black Scholes equation. Numerical simulations for two asset European call options are performed to evaluate the proposed approach. GQSP based solutions are benchmarked against the classical polynomial approximation with backward Euler finite difference method, showing close agreement. This indicates that the Hermitian block embedding construction accurately captures the dynamics of the original non Hermitian operator. These results demonstrate the feasibility of combining Hermitian block embeddings with GQSP for multidimensional Black Scholes problems and provide a proof of principle for applying modern quantum linear algebra techniques to option pricing.

Many machine learning techniques rely on approximating a loss function's curvature, but this is notoriously hard to do at the scale of modern deep networks. Surprisingly, no previous work has exploited the curvature constraints that arise from well known weight-space symmetries in loss landscapes. By analytically averaging over group actions that leave the loss invariant, we construct structured Hessian approximations from single gradients that can be tractably estimated, stored, and inverted. The choice of user-specified symmetry group directly governs the trade-off between approximation accuracy and computational cost. Moreover, our framework provides a unifying theoretical lens for viewing existing methods; in particular, a specific choice of symmetry group recovers Shampoo/Muon-like curvature estimates. We validate our method on a range of network architectures, and deploy it to second-order optimization benchmarks, including a small language model. Our curvature estimation framework might find applications in other machine learning problems such as uncertainty estimation, continual learning, compression/pruning, training data attribution, and more.

Let $D$ be a division ring, let $n\geq 2$, and let $f,g:D\to D$ be additive maps satisfying $f(x)x^{-1}+x^n g(x^{-1})=0$ for every nonzero $x\in D$. We establish general vanishing criteria, give an explicit classification over finite fields, and determine the additive-polynomial solutions over infinite fields of positive characteristic. If $\mathbb{F}_q\subseteq Z(D)$, every additive map $D\to D$ admits a canonical decomposition into $\mathbb{F}_q^\times$-weight components, and the identity pairs precisely the weights $r,s$ satisfying $r+s\equiv n+1\pmod{q-1}$. Consequently, for $q=p^m$, the dimension over $\mathbb{F}_q$ of the solution space is the number of ordered pairs $(i,j)$ with $0\leq i,j<m$ satisfying $p^i+p^j\equiv n+1\pmod{q-1}$. Over an infinite field of characteristic $p>0$, the additive-polynomial solutions are exactly the sums of paired Frobenius terms with $p^i+p^j=n+1$. This classification concerns additive-polynomial maps and does not assert a classification of arbitrary additive maps. Prime-field dilation gives complete vanishing in characteristic zero and, in characteristic $p>0$, whenever $p-1$ does not divide $n-1$. In characteristic two, we prove complete vanishing for $n=2$ on every noncommutative division ring. More generally, when $[D:Z(D)]=\infty$, every solution vanishes if the center is infinite. If the center is the finite field $\mathbb{F}_q$, the same conclusion holds for $2\leq n\leq q-1$. The remaining cases are stated explicitly.

Exact distance computation for quantum LDPC (QLDPC) codes plays a central role in validating candidate fault-tolerant quantum-code constructions, yet the computational structure of this problem remains poorly understood. Despite substantial recent progress in QLDPC design, it remains unclear which algorithmic principles govern the practical scalability of exact distance computation and which classes of exact solvers are best suited to this task. To address these questions, we conduct a systematic study of SAT- and MaxSAT-based formulations for exact QLDPC distance computation across representative codes. We further compare these formulations against several established exact-distance approaches in order to better understand the algorithmic landscape of exact QLDPC distance computation. Our study challenges and refines several prevailing intuitions about exact QLDPC distance computation. First, despite the XOR-rich structure of QLDPC parity checks, practical scalability appears to be governed more by the handling of cardinality constraints and optimization bounds than by parity reasoning alone. Accordingly, XOR-aware reasoning does not provide a systematic advantage across our benchmark suite. Second, Brouwer-Zimmermann-style search, long regarded as the benchmark paradigm for exact distance computation in sparse classical codes, no longer maintains its traditional scalability advantage in the QLDPC setting. This finding challenges the expectation that techniques successful for sparse classical codes remain dominant for QLDPC codes. Third, substantial qualitative differences arise even among MaxSAT solvers themselves. Branch-and-bound MaxSAT significantly outperforms unsat-core-based MaxSAT on challenging benchmarks, demonstrating that solver architecture and optimization strategy play a decisive role in practical scalability.

We study Bermudan option pricing under the Gatheral Double Mean-Reverting (GDMR) stochastic volatility model. The model features a variance process together with a stochastic long-run mean variance process and allows Constant Elasticity of Variance (CEV)-type exponents in the diffusion coefficients. This model is attractive since it provides a flexible specification for volatility dynamics. However, the pricing of early-exercise derivatives under the GDMR model remains largely unexplored in the literature. To address this challenge, we adapt a Hybrid Least-Squares Monte Carlo-Partial Differential Equation (LSMC-PDE) framework to the GDMR model and provide a detailed model-specific implementation. Conditioning on simulated variance paths, the pricing problem reduces to a one-dimensional problem in the asset price, which is solved by a Fourier-based approach, while the remaining dependence on the variance variables is approximated by least-squares regression. Our numerical experiments demonstrate that the Hybrid LSMC-PDE approach yields accurate pricing estimates and often lower pricing errors than plain LSMC, particularly for low and moderate numbers of simulation paths, showing the benefit of using the model structure in early-exercise option pricing.