Papers

For a variational problem in a Banach space with quadratic kinetic term and a lower semicontinuous potential, existence of a minimizer is restored by replacing the norm of the space with an equivalent one, arbitrarily close to the original. The form of the problem is preserved; only the norm is perturbed. This extends a result of Ivanov and Zlateva from the convex to the lower semicontinuous case.

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.

A classical universal stabilization formula offers the practitioner no design freedom: it is a single, parameter-free object. We introduce a cost-parametrized family of stabilizing feedback laws, where (1) the user chooses a function that serves as the running cost on control in an inverse-optimal cost functional, and (2) obtains, through a formula, a nonlinear "expander" of a pre-existing universal controller, which solves an infinite-horizon optimal control problem with a meaningful cost on the state. The cost-to-expander formula is a three-step construction, involving, inter alia, cost differentiation and function inversion-overall, a nonlinear infinite-dimensional operator. The cost-to-expander operator is proven Lipschitz, which enables uniform neural operator approximation of the entire family and supports both offline performance exploration and online adaptation. Semiglobal practical asymptotic stability and second-order suboptimality bounds are established under the approximation. The operator learning and its use in semiglobal stabilization are illustrated numerically. We call the result 'half-direct-optimal' because the paper's design is less than a general 'direct optimal' (HJB-inducing) control, but more than the fully inverse optimal, since the user performs minimization for an arbitrary given cost on control. The dual to the half-direct problem we solve is the problem in which the cost on the state is arbitrary and given. This dual problem is easier and outside of the scope of the paper.

A function $f \colon X^k \to X$ on a finite set embeds into a polynomial of total degree $d$ over a commutative ring $R$ if there is an injection $j \colon X \to R$ and a polynomial $g$ of total degree at most $d$ with $j \circ f = g \circ j^k$, where $j^k$ applies $j$ in each coordinate. These are the transition functions of $k$-neighbour cellular automata, and the injection $j$ is an enlargement of the alphabet that preserves the transitions. We prove three results, all verified in Lean~4 with Mathlib~\cite{bacik2026finbin}. Every unary function $f \colon X \to X$ embeds into a polynomial of total degree $1$. Every binary Kronecker delta embeds into a polynomial of total degree $4$. For every $d$ there is a binary function that does not embed into any polynomial of total degree $d$.

Perturbation theory for asymptotically stable systems has received sustained attention, leading to many criteria that preserve stability under uncertainty. A limitation of classical stability analysis, however, is that it is inherently equilibrium-dependent. Contractive systems, by contrast, are defined independently of any equilibrium and admit a rich set of robustness bounds. In this letter, we develop new time-varying perturbation conditions for contractive dynamics that preserve incremental exponential stability of the perturbed system and, in specific regimes, guarantee asymptotic convergence to solutions of the nominal dynamics. As an application, we propose a safety filter for steering a nominally contractive system to a desired equilibrium while avoiding an unsafe set, and we present an example illustrating advantages of the proposed method over existing approaches.

Let $X$ be a compact complex surface. We prove that the skew product associated to a Borel probability measure $\mu$ on $\operatorname{Aut}(X)$ admits a unique invariant measure of maximal fiber entropy, assuming that $\mu$ satisfies a logarithmic integrability condition and that $\operatorname{supp}(\mu)$ generates a non-elementary subgroup of $\operatorname{Aut}(X)$. We describe this measure canonically in terms of the random limit currents constructed by Cantat and Dujardin, and show that its fiber entropy is equal to the Furstenberg exponent of the associated random action on cohomology. Under an exponential moment assumption, we prove that it is mixing.

This paper studies heterogeneous mean-field models in which agent parameters are sampled from a population distribution. We establish an $O(1/M)$ bound on the steady-state mean-square error between the occupancy measure of the $M$-agent system and the corresponding annealed mean-field equilibrium. The analysis extends Stein's method for homogeneous mean-field models and reveals a fundamental difference between homogeneous and heterogeneous systems. While stability of the mean-field dynamics is sufficient in the homogeneous setting, heterogeneous systems further require uniform robustness of the occupancy dynamics with respect to perturbations of the initial condition. The results are illustrated through a heterogeneous SIS epidemic model.

We study high-dimensional linear regression under a general symmetric convex constraint. Rather than imposing a specific sparsity-inducing penalty, we start from an arbitrary sign-symmetric and permutation-invariant convex body $K\subseteq \mathbb R^p$ and construct the sparse convexification hierarchy \[ K^{(s)} = \operatorname{conv}\{v\in K:\|v\|_0\le s\}. \] We propose a penalized least-squares estimator that searches over this hierarchy and adapts to the best sparse convex approximation of the target. Under standard sub-Gaussian assumptions on the random design and noise, we prove an oracle inequality showing that the estimator adapts to the best sparse convex approximation of the target. For an $s$-sparse target, the result yields a squared-error rate governed by the noise level $\sigma$, and the Gaussian width of the sparse convexification $K^{(s)}$. The method applies broadly to symmetric norm balls and can be implemented using oracle access to the Minkowski functional of $K$. As a special case, the framework yields a consistency result for the constrained Lasso.

In their seminal works Herglotz (1905) and Wiechert and Zoeppritz (1907) have solved the so-called Travel Time Tomography Problem (TTTP) in the 1-D case. However, the question about stability estimates and uniqueness theorems for an n-D n>= 2 TTTP with formally determined incomplete input data still mostly stands open after more than one hundred years period. \textquotedblleft Formally determined input data" means that the number p of free variables in the input data equals the number $n$ of free variables in the unknown right hand side of the governing nonliniear eikonal PDE, p=n. Some previous publications demonstrate that it is possible to develop well performed numerical methods for the TTTP with formally determined input data, which indicates the importance of such data for practical applications. This is the first publication in which the above question is addressed. More precisely, we consider a semi-discrete case, in which a PDE generated by the eikonal equation is written in finite differences with respect to n-1 variables. In addition, it is assumed that the solution of that semi-discrete PDE is represented via a truncated Fourier-like series with respect to a special orthonormal basis of functions, which depend only on the position of the point source. Under these conditions, Lipschitz stability estimate is proven, and this estimate implies uniqueness. An important tool of this paper is a new Carleman estimate. Carleman Weighted Spaces are introduced. Carleman estimates were not applied previously to address questions about stability estimates and uniqueness theorems for the TTTP.

In this paper, we consider the Ricci flow with prescribed curvature on infinite graphs, which reads as \begin{equation*}\label{flow-equation3} \frac{d}{dt}\omega(t)=-(\kappa(t)-\kappa^*)\omega(t),~~ t>0, \end{equation*} where $\omega$ is the edge weight, $\kappa$ and $\kappa^*$ are Lin-Lu-Yau Ricci curvature and the prescribed curvature on the set of edges, respectively. First, we establish the existence and uniqueness of the solution to the Ricci flow. Furthermore, we prove the convergence of the Ricci flow for graphs with girth at least 6 under two different conditions. Our convergence result aligns with the conclusion of Rodin and Sullivan (J Differ Geom, 26(2) 1987) that a circle packing in the plane with the hexagonal pattern is the regular hexagonal packing.

This paper investigates the finite generation of the strict closure of rings in arbitrary dimension. For a Noetherian local ring $(R, \mathfrak{m})$, we provide a sufficient condition under which the strict closure $R^*$ is finitely generated as an $R$-module. Using this result, we characterize the finite generation of the strict closure over excellent rings.

This paper investigates the degenerate generalized Stirling operators of the first kind bridging a gap in the operational calculus of the generalized Heisenberg algebra GHA unified with degenerate calculus. As they are the inverse of the degenerate generalized Stirling operators of the second kind, these operators express the monomial operator products in terms of the degenerate factorial operators. We derive key structural and combinatorial properties for these operators, including an explicit product factorization, a fundamental recurrence relation, and an operational shifting identity. Furthermore, we establish the orthogonality relations between the degenerate generalized Stirling operators of the first and second kinds, providing a complete combinatorial framework for functional quantum algebras.

This paper examines wireless-powered Internet of Things (IoT) networks involving multiple unmanned aerial vehicles (UAVs) equipped with backscatter and caching technologies to relay and transmit signals. For data communication and energy harvesting (EH), the source transmits information and power to UAVs using the dynamic time splitting (DTS) method. UAVs use harvested energy for passive communication (backscatter) and for active communication (transmitting information) to the destination. The primary objective is to maximize the total throughput by jointly optimizing the DTS ratio, trajectory, and transmission power, leveraging the UAVs' caching capability. This optimization problem is challenging due to its non-convexity. Therefore, an efficient alternating algorithm using the block coordinate descent (BCD) method is proposed to optimize each variable given the fixed values of the other parameters. By applying the Karush-Kuhn-Tucker (KKT) conditions, we derive a closed-form expression for the optimal DTS ratio, significantly reducing computation time. The optimal values for the other two parameters are determined using the BCD. In order to thoroughly assess the effectiveness of various solutions for the original problem, this paper introduces an approach leveraging a genetic algorithm (GA). The GA in this context employs a one-point crossover method, value mutation, and rank-based selection based on fitness values. Numerical results show that the BCD and GA achieve at least 31% throughput improvement over the benchmarks, with reduced computational time. These findings demonstrate the performance gain and practical feasibility of our solutions in caching-enabled UAV-aided IoT networks.

Post-training quantization (PTQ) converts a trained full-precision model into low-bit weights without task-level retraining, while quantization-aware training (QAT) incorporates quantization into the training loop. Although PTQ is efficient and often accurate at moderate bitwidths, it can fail sharply at aggressive bitwidths; QAT is more expensive but can often recover the lost accuracy. We propose a unified geometric framework that explains both PTQ failure and QAT recovery. We model full-precision training as following a low-loss river inside a wider valley: a normal neighborhood of the river forms a nearly flat basin, while leaving this basin incurs a sharp loss increase. When the quantization grid is comparable to the basin width, local PTQ objectives, including rounding and Hessian-based second-order reconstruction, can select a high-loss deployed quantized point outside the basin even when nearby low-loss quantized points exist. In this regime, straight-through-estimator-based QAT has a useful bias: it evaluates gradients at the deployed quantized weights while updating latent full-precision weights, causing the gradient to sense the valley wall and acquire an inward component that steers subsequent quantized iterates back into the basin. We formalize this mechanism through a local landscape model, construct a geometric PTQ failure mode, and prove finite-time QAT recovery under local quantizer-compatibility assumptions. Experiments across vision and language models under multiple neural-network quantization schemes corroborate the predicted basin-crossing failure of PTQ and the corresponding recovery mechanism of QAT.

The construction of Arf rings and strictly closed rings has been studied widely; however, there has been no clear description of the structure of the strict closure R^* when the integral closure of R is not a finitely generated R-module. In this paper, we investigate the construction and finite generation of the strict closure of rings. We determine its structure when R is a Cohen-Macaulay semi-local ring of dimension one, with dim R_M=1 for every Maximal ideal M in R. Using this, a characterization of the finite generation of the strict closure is given.

This paper investigates a class of partially dissipative viscoelastic Timoshenko systems with memory, where dissipation is induced by a Volterra-type memory term acting only on the shear variable. The well-posedness of weak and strong solutions is established on finite time intervals, including existence, uniqueness, stability, and higher-order regularity under compatibility conditions consistent with mixed boundary conditions. For the numerical approximation, a Fourier spectral fully discrete scheme is constructed: sine and cosine basis expansions are used in space for unknowns satisfying Dirichlet and Neumann boundary conditions, respectively; in time, a central difference scheme is applied to the second-order derivatives, and the composite trapezoidal rule is used to approximate the memory convolution term. Based on a discrete energy method, the positivity of the constructed discrete energy is proved, and the error estimate for the fully discrete scheme with second-order convergence in time and \(q\)-th order in space is established for any q \in \mathbb{N}. Numerical experiments are given to verify the theoretical convergence rates and to compare the dynamic responses of the local and nonlocal models, demonstrating that the memory term effectively captures energy dissipation and vibration attenuation behavior in viscoelastic materials.

We consider the interaction between centralized trading and decentralized Proof of Stake (PoS) blockchain ecosystems. Motivated by the increasing dominance of centralized exchanges and the institutionalization of crypto markets, we study how trading activities on centralized exchanges affect staking behavior, token allocation, and decentralization within a PoS blockchain. We formulate a continuous-time mean field model, where the miners simultaneously act as validators in the PoS protocol and traders in a centralized market with price impact. Under suitable assumptions, we establish the local well-posedness of the mean field system, and derive a semi-explicit characterization of the equilibrium trading strategy. Numerical results suggest that centralized trading activities may enhance staking participation, and promote decentralization of the staking distribution through market incentives – this partially justifies BitMine's buy-and-stake strategy for ETH. We also study the effects of transaction costs and token supply mechanisms on the equilibrium staking ratio and concentration profile. These results illustrate how market microstructure and centralized liquidity provision can exert significant influence on decentralized blockchain protocols.

We study a stochastic multi-armed bandit problem in which the set of available arms expands over time. This setting arises in sequential experimentation when new actions or treatments become available during an ongoing study, making regret against a single best arm in hindsight inappropriate. We instead evaluate performance relative to the best arm currently available, leading to a dynamic-regret criterion for arriving-arm environments. To address the resulting challenges of arrival information discrepancy (AID) and a drifting benchmark (DB), we propose UCB for Arriving Arms (UCB-AA), an elimination-based procedure with an aiding preliminary screening step for newly arrived arms before full competition with incumbent arms. We show that UCB-AA attains regret bounds that depend explicitly on the arrival process, achieves sublinear dynamic regret under regularity conditions on gap evolution, and admits an online extension for unknown horizons. Simulation results show that UCB-AA reduces wasted pulls and maintains a smaller active arm set while preserving competitive regret performance.

We propose LEAF, a learning-enabled ADMM framework for accelerated convex optimization. The key idea is to approximate the Moreau envelope of the objective function using an Input Convex Neural Network (ICNN), resulting in a learned model that preserves convexity and smoothness. This leads to the proposed Moreau Envelope Learning ADMM (MEL-ADMM) and its splitting variant sMEL-ADMM. Unlike existing approaches that learn high-dimensional operators directly, LEAF learns a scalar-valued Moreau envelope, significantly reducing model complexity and improving data efficiency. The framework accommodates a broad class of convex problems with smooth and non-smooth objectives. By embedding convexity explicitly through the ICNN architecture, the proposed approach maintains high approximation accuracy while preserving key structural properties of the optimization problem. Both MEL-ADMM and sMEL-ADMM are developed with theoretical guarantees of convergence and feasibility under the learned model. Rigorous analysis shows that the proposed methods achieve convergence rates comparable to classical ADMM while reducing per-iteration computational cost. Numerical experiments demonstrate up to an order-of-magnitude speedup over state-of-the-art solvers while maintaining low optimality gaps

We classify and count binary necklaces [w] for which the representative-independent minimum nontrivial rotational Hamming distance minDist(w) equals 2, the first positive layer after the periodic case. With a fixed priority convention, the set of nonzero shifts attaining distance 2 determines a classification into half-turn (HT), single-pair (SP), and subgroup-pattern (SUB) classes, while the residual multi-pair (MP) class is empty. We give closed formulas for the HT, SP, and SUB counts and an explicit Mobius-totient divisor-sum formula for |D_2(n)|. We also describe the larger subgroup-shaped branch, which contains the SUB cases together with boundary cases assigned by priority to HT and SP, by a quotient-background normal form and its boundary counts. For odd prime lengths p, every distance-2 necklace is rotation-equivalent to an arithmetic interval on the prime cycle, and |D_2(p)| = (p^2 - 4p + 7)/2.

We present a conjecture for associated varieties of simple affine vertex algebras $L_k(\mathfrak{g})$ attached to a simple Lie algebra $\mathfrak{g}$ of simply-laced type and any rational level $k$ greater than the critical level. The key new ingredient compared to the integral case is the covering duality map introduced by Gao-Liu-Lo-Shahidi. We provide evidence for the conjecture.

Let \(G\) be a finite group and let \(\chi\) be an ordinary irreducible character. We give an elementary algorithm which constructs explicit matrices affording \(\chi\). The regular representation provides a canonical ambient representation, and the usual central idempotent projects onto the \(\chi\)-isotypic component. The main step is then to maximize the squared norm of a diagonal matrix coefficient on the unit sphere of this component. The maximum is \(1/\chi(1)\), and it is attained precisely by vectors whose cyclic span is an irreducible subrepresentation affording \(\chi\). Thus the construction reduces the passage from characters to matrices to a concrete optimization problem. The same extraction method applies inside any smaller ambient representation containing \(\chi\), such as an induced representation from a subgroup. We complement the theory with a discussion on dimension reduction, robust numerical implementation, and an explicit \(S_4\) example.

Warm starts are widely used to accelerate AC optimal power flow (ACOPF) solves, but the impact of different initialization strategies has received limited systematic study, particularly for the primal-dual interior-point methods that dominate large-scale ACOPF algorithms. This paper benchmarks initialization strategies for ACOPF solved with the interior-point solver IPOPT on 19 PGLib-OPF instances (5 to 30,000 buses), testing all 15 non-empty subsets of the primal blocks $\{P_g, Q_g, V_m, V_a\}$ under oracle conditions and three DC-seeded combinations in a practical setting. The experiments show that most partial primal-plus-dual restarts increase solve time or reduce convergence reliability. Among the oracle primal-plus-dual (O-PD) configurations, only the complete restart reliably converges on every baseline-convergent case, reaching a $47.6\%$ median solve-time speedup. Twelve of the 14 partial O-PD combinations have negative median speedups, and several fail repeatedly on larger networks. Decomposing the dual into constraint and bound multipliers shows that coverage, not the presence of duals per se, governs robustness: the full bound-multiplier vector reaches 90.7% convergence and a $+26.8$% median speedup, whereas block-matched coverage (oracle multipliers on some bounds, defaults on the rest) drops to 70.4% and $-31.1$%. Practical DC seeding sometimes helps the AC solve, but the benefit is no longer statistically significant once the DCOPF presolve cost is included in the end-to-end comparison ($p = 0.4171$). For learned warm-start methods, the results support the following target ordering: predict the full primal vector first; if only partial coverage is possible, prioritize voltage variables; and avoid partial or inconsistent dual predictions unless the primal estimate is nearly complete.

The purpose of this paper is to investigate the self-orthogonal embedding problem for linear codes over Z4. We propose several tight bounds on the length of the shortest self-orthogonal embedding over Z4, and determine the exact shortest self-orthogonal embedding length under specific conditions. As an example satisfying these conditions, we establish the exact length of the shortest self-orthogonal embedding for the quaternary Preparata codes. Furthermore, to establish these results, we completely classify the exact length of the shortest doubly even self-orthogonal embedding for binary linear codes in every possible case. Finally, when the shortest self-orthogonal embedding length of a given free code over Z4 is equal to the shortest doubly even self-orthogonal embedding length of its residue code, we present an algorithm to construct all possible shortest self-orthogonal embeddings. With our algorithm, we found twelve linear codes over Z4 whose minimum Lee distances are higher than those of the Z4-linear codes in Aydins database.

Talagrand's correlation inequality provides a quantitative strengthening of the Harris–Kleitman inequality for increasing Boolean functions. Motivated by a Fourier-analytic conjecture of Friedgut, Kahn, Kalai, and Keller, we prove that $$ \mathrm{Cov}(f,g)\ge 2\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2 $$ holds for all increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. The proof combines the reverse Bonami–Beckner inequality with Young's convolution inequality. We also establish a sharp pointwise inequality: for every $n\ge1$, every $0\le\rho\le1$, and every $f,g:\{0,1\}^n\to[0,1]$, the optimal constant $c_{\rho,n}$ for which $$ \left\langle f,T_\rho g \right\rangle\ge c_{\rho,n}\|f*g\|_2^2 $$ holds for all such $f,g$ is $1$ for $0\le\rho\le1/2$, $(2(1-\rho))^n$ for $1/2<\rho<1$, and $0$ for $\rho=1$. Integrating this pointwise inequality yields, for $n\ge1$, the slightly improved bound $$ \mathrm{Cov}(f,g)\ge 4\cdot\frac{n+1}{2n}\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2. $$

Modern generative models often define an entire probability path from a simple prior to the data law, rather than only an endpoint map. Diffusion models follow stochastic denoising paths, flow matching learns transport fields, consistency and distillation methods compress paths into one or a few steps, adversarial models match terminal distributions, and VAEs generate through latent kernels. Existing unifying views mainly describe how such paths are constructed. We study a complementary question: when is a generated probability path self-consistent? We define a self-consistent generative path as a random fixed point of admissible local variational transport corrections. In this framework, a local correction is specified by a random variational transport operator combining a divergence or geometry term, an energy term, and a structural constraint. The framework contains random regularized optimal-transport proximal steps as a structured instance, while also allowing non-OT divergences, latent kernels, adversarial constraints, causal discrete kernels, and terminal one-step maps. The theory yields a random fixed-point path residual (R-FPR), which measures the gap between the actual generated path and an admissible local correction. We prove well-posedness, random fixed-point existence and attraction, non-contractive existence, residual-to-generation error bounds, empirical residual concentration, proxy perturbation bounds, continuous-time limits, and operator-level generalization with model-specific corollaries. The resulting theory turns endpoint matching into path self-consistency testing and provides a residual-control principle for diagnosing failures, regularizing training, and guiding adaptive sampling across diffusion, flow, one-step, VAE, GAN/WGAN, and autoregressive generators.

Classical zero-localization theorems give deterministic certificates that all zeros of a polynomial lie in a prescribed disk, annulus, or related region. When the coefficients are random, each such deterministic certificate becomes a random variable on coefficient space. This paper develops a finite-degree certificate method for random polynomial localization and extends the author's earlier joint work with Mir \cite{SheikhMir2024}. The main result concerns Gaussian polynomials with random leading coefficient: the Cauchy ratios are marginally standard Cauchy, yet they are dependent through their common denominator. We derive the exact dependence-aware certificate integral and prove that its inverse confidence radius has order \(\sqrt{\log n}\), while a fictitious independent-Cauchy model has order \(n\). We also obtain monic coefficient-law certificates, sub-Weibull confidence radii, annular certificates via reversal, Rouché–Chernoff certificates, and an optimized scaled Cauchy envelope. A reproducible Monte Carlo study for monic Gaussian polynomials compares the classical Cauchy radius, the optimized Cauchy envelope, the annular certificate, and the Rouché radius. Across \(5000\) samples for each of \(n=20,50,100\), the Rouché certificate gives the sharpest outer radii, the optimized Cauchy envelope substantially improves the classical Cauchy radius, and all tested certificates satisfy their deterministic containment inequalities numerically.

We introduce a new family of weighted Gaussian $L^2$-Poincaré-type inequalities with explicit sharp constants, optimizers, and corresponding sharp $L^2$-gradient stability estimates. This family substantially extends the classical Gaussian Poincaré inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincaré inequalities do not apply. To overcome this difficulty, we develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, we completely characterize the stability of the $L^2$-Caffarelli–Kohn–Nirenberg (CKN) inequalities by establishing sharp stability estimates, together with the stability of the stability inequality results, throughout the entire parameter range. Previous results were available only in a few special cases. We further establish weighted $L^p$-Poincaré inequalities for all $p>1$, and derive stability estimates for the $L^p$-CKN inequalities for $p\geq 2$ throughout the full parameter regime in which sharp constants and optimizers are known. In contrast, earlier $L^p$ results were restricted to highly limited parameter ranges.

We introduce a quantum deformation $\mathbf{U}(\mathfrak{gl}_n[t])$ of the universal enveloping algebra of the current algebra $\mathfrak{gl}_n[t]$, realized as a Hopf parabolic subalgebra of quantum affine $\mathfrak{gl}_n$. Unlike the Yangian—the standard quantization of the current algebra—our algebra admits a canonical basis. We give a BLM-type realization of \(\mathbf{U}(\mathfrak{gl}_n[t])\) via certain subalgebras of affine quantum Schur algebras, and then construct canonical bases for the modified quantum current algebra $\dot{\mathbf{U}}(\mathfrak{gl}_n[t])$ and for its finite dimensional irreducible graded modules. Moreover, we prove a rigidity theorem: every finite dimensional polynomial irreducible module for quantum affine $\mathfrak{gl}_n$ remains irreducible when restricted to ${\mathbf U}_{\mathbf v}(\frak{gl}_n[t])$ (the specialization of ${\mathbf U}(\frak{gl}_n[t])$ at a non-root-of-unity complex number $\mathbf{v}$); conversely, every finite dimensional affine polynomial irreducible ${\mathbf U}_{\mathbf v}(\frak{gl}_n[t])$-module extends uniquely to a polynomial irreducible module for quantum affine $\mathfrak{gl}_n$. More generally, we classify all finite dimensional polynomial irreducible \({\mathbf U}_{\mathbf v}(\frak{gl}_n[t])\)-modules by \(\mathcal Q(n)\times\Lambda^+(n)\). As an application, we establish a bijection between the isomorphism classes of finite dimensional polynomial irreducible modules of \({\bf U}_{\mathbf v}(\frak{gl}_n[t])\) and those of the Yangian \(Y(\mathfrak{gl}_n)\). Thus \({\bf U}(\mathfrak{gl}_n[t])\) admits canonical bases for its modified form and irreducible graded modules, and its finite dimensional polynomial irreducible modules are in correspondence with those of the Yangian.

The matrix of Lévy's areas of $d$-dimensional Brownian motion is a fundamental object in stochastic analysis. In this article, we study the singular values of this $d \times d$ skew-symmetric random matrix. First, we derive an explicit formula for the density of the singular values and, en passant, present a new short proof of the characteristic function of Lévy's area when $d \ge 3$. This also allows us to extend the well-known formula for the density of Lévy's area to $d \ge 3$. Next, we use these results to characterise the singular spectrum as a determinantal point process with its kernel in explicit form. Finally, we study the asymptotics as $d \to \infty$: the empirical measure of singular values converges to an absolute Cauchy distribution, the largest singular values are of order $d$ with Gaussian fluctuations, the smallest singular values are of order $1/d$, and the local bulk spacings are of order $1/d$, with sine-kernel statistics after rescaling.