Papers

This paper studies the worst-case impact of constrained control inputs: an input seeks to maximize the average cost of some outputs, measured in the $L_2$ or $L_1$ norm, while remaining bounded in terms of other outputs. This problem template subsumes classical metrics such as the $H_\infty$ norm and the output-to-output gain, and arises in adversarial control, security assessment, and robust control. For linear time-invariant systems, we provide an exact semi-definite program (SDP) when there is a single constraint, and SDPs computing upper bounds when there are multiple constraints. We derive sufficient conditions, in terms of system zeros and relative degrees, under which the worst-case cost is unbounded, together with a constructive closed-loop modification that removes the unboundedness. From a security standpoint, unbounded values reveal structural limitations in detecting certain attack inputs. For positive systems, we provide scalable formulations whose complexity grows linearly in the state dimension: a scalable SDP for quadratic costs, and an exact linear program for linear costs. The results extend to nonlinear polynomial systems via a sum-of-squares program. We illustrate the results with numerical examples.

In 1987, Woronowicz asked whether the linear span of irreducible characters is norm dense in the cocommutative part of the ambient C-star algebra of a compact quantum group. After thirty-nine calendar years, we give a negative answer to the universal form of this question, even for compact matrix quantum groups of Kac type. Starting from an infinite finitely generated simple group with property T and a finite bicharacter twist, we construct a compact quantum group whose universal C-star algebra is a full group C-star algebra. A Kazhdan projection is shown to remain cocommutative under the twisted coproduct, while a vector state arising from an induced representation separates this projection from every algebraic cocommutative element. Quantitatively, the distance from the Kazhdan projection to the closed linear span of irreducible characters is at least one half. Moreover, the projection lies in the kernel of the reducing morphism. The obstruction is therefore genuinely universal and disappears after passage to the reduced compact quantum group, where the known character-density theorem remains valid. The construction identifies a sharp boundary between universal and reduced character theory and shows that Kac symmetry alone does not control cocommutative elements in the universal completion.

This study introduces a coupled multi-domain framework to simulate direct current (DC) arcs, integrating thermal, fluid dynamic, and electromagnetic phenomena. We derive a two-dimensional (2D) axisymmetric model by simplifying the magneto-hydrodynamics (MHD) equations and implement it in MATLAB/Simulink. The model captures the spatiotemporal evolution of temperature, fluid velocity, and magnetic fields within a wall-stabilized arc column. Simulations across a 10 A to 1500 A current range show a pronounced radial temperature gradient, with central axis peaks and rapid peripheral decay. The approach's accuracy is validated by aligning the simulated velocity and magnetic fields with established physical paradigms. Additionally, the model accurately reproduces the inverse current-voltage characteristics observed in classical models and empirical studies, while transient analyses confirm rapid millisecond-scale voltage stabilization. This accessible computational model provides a robust foundation for investigating DC arc dynamics and facilitates system-level simulations for advanced circuit breakers, welding technologies, and plasma applications.

We study the problem of finding the norm of the point evaluation operator in the Paley–Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley–Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

Let $H$ be a weak Hopf algebra with bijective antipode. This paper is devoted to the AS-Gorenstein property of restricted faithfully flat weak $H$-Galois extensions $A/B$. We first of all study the cohomologies of weak $H$-Galois extensions and then establish the spectral sequence connecting the Hochschild cohomologies of $A$ and of $B$. Finally we prove that when $A$ is a noetherian affine PI algebra and $B$ is AS-Gorenstein, $A$ shares the AS-Gorenstein property.

In this paper, we present an $hp$-version time-stepping spectral Monte Carlo method for solving semi-linear parabolic equations. The key innovation lies in constructing an exponentially accurate stochastic algorithm that integrates a residual iteration scheme on Gauss-type nodes in both temporal and spatial directions with a reconstruction strategy rooted in spectral methods. To address the long-time simulations and initial singularities that are often challenging for traditional stochastic algorithms (e.g., walk-on-spheres method), we further develop an $hp$-version time-stepping framework that employs multiple time steps and, respectively, geometric time partitions with linearly increasing polynomial degrees to handle these difficulties. Notably, the proposed algorithm bypasses the need to solve linear systems required by traditional spectral methods and remarkably supports parallel computation at both temporal and spatial grid points. We rigorously establish exponential convergence rates for the multistep method within a finite number of iterations. Extensive numerical experiments are conducted to demonstrate the spectral accuracy and computational efficiency of the proposed method in long-time simulations, problems with initial singularities, and a five-dimensional problem, thereby validating the theoretical results.

A finite set $A$ in an additive abelian group is a More Sums Than Differences (MSTD) set if $|A+A| > |A-A|$. We prove that no MSTD set of size $5$ exists in an additive abelian group. Using a computer program inspired by Hegarty, we then obtain classifications of MSTD sets of sizes $6$, $7$, $8$, and $9$ in arbitrary fields. We also investigate the minimal cardinality of MSTD sets that are multiplicative subgroups of $\Z/p\Z$.

Determining when the chromatic symmetric function of a graph is Schur positive is a central problem in algebraic combinatorics. Elementary expansions can help answer this question, but negative coefficients obstruct a direct positivity argument. We develop methods for proving Schur positivity from signed elementary expansions and apply them to chromatic symmetric functions of clique-spiders and three-legged spiders. We first give the Schur alpha-omega lemma, which bounds Schur support. We then give four equivalent sufficient criteria for Schur positivity from a fixed composition-indexed elementary expansion. The matrix, order-ideal, negative Hall, and positive Hall formulations treat positive coefficients as capacities and negative coefficients as demands, with admissible assignments governed by dominance order. These criteria provide certificates of Schur positivity especially when elementary coefficients are negative. For the four basic families of $3$-clique-spiders with total leg length at most one, we obtain complete classifications of both $e$-positivity and Schur positivity. These classifications show how the position of a single leg edge changes positivity. Using selected clique-spider cases and a path-clique bootstrap, we prove that all spiders of the form $S(a,b,2)$ are Schur positive.

In this short note, we characterize the equality case of Bray's conjecture recently proved by Jiang-Li-Wang \cite{JLW}.

This paper is concerned with the asymptotic behavior of the principal eigenvalue $\lambda(D)$ of the elliptic eigenvalue problem \[ -D\Delta_{M}u - a\langle \nabla_M f, \nabla_M u\rangle_g + c u = \lambda(D)u, \] posed on a closed orientable Riemannian manifold $(M,g)$, in the small-diffusion limit $D \to 0^+$. Under the assumption that $f$ is a Morse function on $M$, we establish that the limiting value $\lim_{D\to 0}\lambda(D)$ is completely characterized by the critical points of $f$ and the associated Riemannian Hessian, specifically through the values of $c$ and the Riemannian Hessian eigenvalues at those points.

We prove an interior $C^{1,\alpha}$ estimate for solutions of the homogeneous linearized Monge–Ampère equation in dimension two under the assumption \[ 0<\lambda\leq \det D^2\varphi\leq\Lambda<+\infty. \] No continuity assumption on the Monge–Ampère density is required. Our result is an affine-invariant analogue of the classical Morrey–Nirenberg $C^{1,\alpha}$ estimate in two dimensions. The core of the proof is the partial Legendre transform. After the transform, the first derivatives of the solution are quotients of adjoint solutions for a uniformly elliptic non-divergence form equation. Bauman's Harnack inequality gives the Hölder control of the quotient, while the Jacobian identity of the partial Legendre transform and a Caccioppoli estimate give its local boundedness. As an application, we prove a Liouville theorem for entire solutions with at most linear growth.

In this short paper, we prove that, for any two complex elliptic curves \(E\) and \(F\), the product \(E\times F\) has degree of irrationality \(3\). The upper bound is obtained from compatible cyclic plane-cubic models of \(E\) and \(F\): three diagonal bihomogeneous sections define a dominant rational map \(E\times F\dashrightarrow\mathbb P^2\) of degree \(3\). The lower bound follows by excluding dominant rational maps of degrees \(1\) and \(2\) from an abelian surface to \(\mathbb P^2\).

We establish quantitative versions of the law of rare events and binomial approximations in non-commutative probability settings, including the free, Boolean, and monotone convolution frameworks. Our main results provide explicit error bounds in the non-commutative Wasserstein distance for approximations of convolutions of rare countings by non-commutative Poisson and binomial distributions. These bounds extend classical results from the tensor setting to the non-commutative regime. Our approach relies on a discrete Lindeberg-type interpolation scheme combined with algebraic properties of cumulants adapted to each independence notion. The results presented here fill a gap in the literature concerning explicit rates of convergence in non-commutative limit theorems.

We study when a compactly supported isotropic positive definite function on a Euclidean space remains positive definite after Euclidean distance is replaced by spherical geodesic distance. The corresponding assertion is known in odd dimensions. We prove that it fails in every even dimension, no matter how small a positive upper bound is imposed on the support. An explicit counterexample is also given in dimension two.

We study the geometric quantization of $b^k$-symplectic manifolds using the integrability of Lie algebroids. Using a groupoid index, we define a quantization for $b^k$-symplectic manifolds whose singular locus is a normal crossing divisor and which carry a Hamiltonian action of a compact connected Lie group, generalizing Guillemin–Miranda–Weitsman in a few directions. Firstly, we show this quantization is the index of a $\spinc$-Dirac operator, answering a question of theirs. In particular, it is a finite-dimensional virtual representation for every $k$, whereas their formal quantization is infinite-dimensional when the modular degrees are even. Secondly, our symplectic form can have singularities along hypersurfaces which can have normal crossings. Finally, we prove that quantization commutes with reduction for the Hamiltonian action of a possibly non-abelian compact connected Lie group, when the modular degrees are odd. In the case when the modular degrees are not odd, we give an example when $[Q,R]=0$ fails.

The construction problem of Leibniz bialgebras from Lie bialgebras and perm bialgebras is considered in this paper. We show that there is a Leibniz algebra structure on the tensor product of a Lie algebra and a perm algebra, and elevate this conclusion to the level of bialgebra. We prove that the tensor product of a quadratic Lie algebra and a perm bialgebra has a Leibniz bialgebra structure, and this Leibniz bialgebra structure is coboundary (resp. quasi-triangular, triangular, factorizable) if the original perm bialgebra is coboundary (resp. quasi-triangular, triangular, factorizable). Moreover, we constructed an infinite-dimensional Leibniz bialgebra using the tensor product of a finite-dimensional Lie bialgebra and a quadratic $\bz$-graded perm algebra. Quasi-triangular and triangular infinite-dimensional Leibniz bialgebras are considered.

This manuscript develops a non-parametric and robust framework for estimating the scale of additive noise in weakly sparse systems. The method does not require independence, prescribed dependence, or temporal regularity of the noise sequence. We introduce a class of order-statistic estimators based on comparing the sorted observations with deterministic or random proxies generated from a reference noise distribution. This purely spatial approach avoids preliminary filtering or temporal decorrelation, and therefore preserves the sparsity structure of the latent signal. We establish non-asymptotic concentration inequalities for weighted loss functions, with bounds that separate the contribution of the signal from the discrepancy between the ordered noise and the proxy. We then control this proxy discrepancy in independent and correlated regimes, including heavy-tailed reference laws. Finally, we apply the method to high-frequency observations of continuous-time stochastic processes, obtaining scale estimators for fractional Brownian motion and stable Lévy noise in the presence of lower-variation additive perturbations.

In this paper, we derive optimal order a posteriori error estimates for the variable step-size explicit two-step exponential Adams (E-Adams2) method for parabolic problems. We begin by introducing an E-Adams2 approximation, deffned by the piecewise linear approximate solutions, which leads to suboptimal error estimates. To recover optimal order error estimates, we introducean appropriate reconstruction of the approximation with the second order residual for the explicit E-Adams2 method, which plays key roles in deriving optimal order a posteriori error estimates for the proposed explicit method for linear and semilinear parabolic equations. Various numerical experiments are carried out to verify the correct convergence rates of the a posteriori quantities, and the high efffciency of the adaptive algorithm.

We prove that for every fixed inverse temperature $\beta < 1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_{\beta}\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $\beta$. The main ingredient is a deterministic criterion for optimal-order Poincaré inequalities in general Ising models, established via the integrated Bakry-Émery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincaré inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincaré inequality were generated by GPT-5.6 Sol Ultra.

We prove that the extremal Lyapunov exponents of finite-state Markov $\mathrm{GL}(2,\mathbb{R})$-cocycles are pointwise log-Hölder continuous, jointly in the cocycle matrices and the transition kernel, at every parameter $(A,P)$ satisfying $\lambda_+(A,P)=\lambda_-(A,P)$. Perturbations are taken within a fixed transition graph. The main new ingredient is a Markov perpetuity estimate for the nonsplit triangular case, obtained through a martingale–coboundary decomposition. In the conformal case, the exponent $1/2$ in the logarithmic modulus can be replaced by $1$.

Stacked intelligent metasurfaces (SIMs) have recently emerged as a promising architecture for large-scale beamforming systems, including cell-free massive MIMO (CF-mMIMO), due to their cost-effective wave-domain signal processing capabilities. However, existing algorithms for the joint optimization of digital and SIM-enabled wave-domain beamforming typically incur prohibitive computational complexity. In this work, we propose an efficient alternating optimization (AO) algorithm for weighted sum-rate maximization in SIM-assisted CF-mMIMO systems employing hybrid digital-wave beamforming. Unlike prior approaches that rely on general-purpose optimization solvers or per-element gradient ascent methods, the proposed algorithm updates the digital and wave-domain beamforming variables on a per-access point (AP) or per-SIM-layer basis, enabling closed-form updates at each step. Numerical results demonstrate that the proposed algorithm reduces the computational complexity by more than 99% compared to existing algorithms while achieving nearly identical sum-rate performance.

We attach sequences of matrix spectra to sequences of rational numbers and study them with numerical experiments. The initial segments of the spectrum sequences associated to each of several rational sequences appear to exhibit geometric regularities. Identities originating in the theory of symmetric functions express members of arithmetic sequences $\{h_n\}_{n \ge 0} \thinspace (h_0 = 1)$ as $h_n = |J_{h,n}|/n !$ for particular matrices $J_{h,n}$, where $|\cdot|$ is the determinant. In this setting we studied numerically sequences $\{a(n)\}_{n \ge 1}$ obtained from the Fourier expansions of cusp forms. Within the range of our observations, the sequences of spectra of certain "treated" matrices $\{J^{(c)}_{a,n}\}_{n=1,2,3,...}$ exhibit coherent behavior, meaning that the plots of the minimum moduli among the eigenvalues of the $J^{(c)}_{a,n}$ appear to oscillate or to form approximately straight lines of non-negative slope once $n$ is large enough.

Given a word over a finite alphabet, consider the sequence that counts, for each length, the number of distinct subsequences of that length. In 1976, Chase proved that this sequence is log-concave. His proof uses a triangular array indexed by the prefixes of the word together with a meticulous analysis of ratios of several sums. We instead decompose according to the first letter, which reduces the proof to a weighted average.

Across implicit-neural-representation (INR) architectures and analytic benchmarks we observe that a thoroughly tuned Adam (especially its learning rate (lr), e.g. in a hyperparameter sweep from $lr = 0.05$ to $10^{-8}$) can potentially reach a very low loss even on ill-conditioned loss landscape or converge at a plateau far above the loss attained by second-order methods. This report defines the measured metrics that help determine if Adam can mitigate the ill-conditioning on a given loss landscape. We provide the indicators by which each outcome is determined, that are: the condition number of the Hessian and of the Adam-preconditioned Hessian $D^{-1/2}HD^{-1/2}$ (with the derivation from Adam's update rule), the diagonal mass $\rho$ that distinguishes axis-aligned from cross-coupled ill-conditioning, the negative spectral mass estimated by stochastic Lanczos quadrature, and the gradient energy fractions over curvature bands, including the flat fraction that indicates the Adam stall. A worked out $2\times 2$ example and an illustration show the reasons why a diagonal preconditioning by Adam can remove axis-aligned ill-conditioning by rescaling and why it cannot do the same if the ill-conditioning is cross coupled. In addition, we present a case study of FINER image fitting architecture that goes over the whole loss landscape analysis framework: the fitting architecture description, reasons due to which its landscape stalls Adam at saddles, the measured PSNR values through our tuned baselines to the $120$–$134$\,dB results of the blockwise second order methods, the error maps behind those numbers, and description of the benefits such image fitting accuracy gives in practice.

The Seven Circles Theorem states that if six circles form a closed chain and are tangent to a common circle, then the three lines joining opposite points of tangency on the common circle are concurrent. We extend this result to an arbitrary closed six-circle chain. Given such a chain, we consider two circles, each tangent to one of the two alternating triples of circles in the chain. For each circle in the chain, we then construct a circle tangent to it and to both of these circles. We prove that the three lines joining the centers of the newly constructed circles corresponding to opposite members of the chain are concurrent. In the classical configuration, the two circles associated with the alternating triples coincide with the common circle, while the six newly constructed circles degenerate to the six points of tangency, viewed as circles of radius zero. Thus, the classical Seven Circles Theorem is recovered as a degenerate case of our generalization.

For $n\geq 2$ and $1<p<n$, Ludwig-Xiao-Zhang \cite{LXYZ} asked whether the $L_p$-integral affine surface area of every convex body containing the origin in its interior is bounded above by the ball-normalized multiple of \(L_p\)-variational capacity, and whether the analogous analytic inequality holds for $L_p$-convexified level sets. We answer both questions negatively.

In recent years, there has been a growing interest in comparing the reliability of systems with dependent components. Most of the recent works consider Archimedean copulas to define the dependency structure of systems and examine specific familiy of distributions. In this work, we provide distribution-free results to compare, in the usual stochastic order under some majorization conditions, coherent systems with heterogeneous and dependent components where the dependency structure can be defined by any copula. Applications of our main results for specific copulas and different families of marginal distributions are also given.

We provide a detailed study of equivalent reformulations of the classical condition derivation closedness for weight sequences. This property is crucial in many applications since it ensures that the corresponding weighted spaces satisfy the desired property of stability under taking derivatives. Moreover, it yields technical estimates which are required in various directions and proofs and so the knowledge that different conditions are equivalent to this crucial requirement is important and useful when dealing with weighted spaces. Finally, we construct an explicit (counter-)example which shows that without the standard condition log-convexity the full characterization fails in general and comment on the mixed setting as well.

Starting from the Riemann-Siegel decomposition of $\zeta(s)$ given in Siegel's 1932 paper, we introduce a change of variable, $t=I(T)$, that replaces Siegel's coupled pair "imaginary part $t$ / summation index $m$" with one real index $T$. We then split Siegel's single remainder integral $R$ into two exact pieces, $R_{1ps}$ and $R_{2ps}$, and prove that $R=R_{1ps}+R_{2ps}$, so that $\zeta=\Sigma_1+R_{1ps}+\Sigma_2+R_{2ps}$. Our central observation is that each of these remainders is nothing more than one additional, fractional partial summand appended to its Dirichlet sum: $\zeta(s)=\sum_{n=1}^{m}n^{-s}+\hat{d}_1(m+1)^{-s}+\chi(s)\sum_{n=1}^{m}n^{s-1}+\hat{d}_2\chi(s)(m+1)^{s-1}$, with $\hat{d}_1,\hat{d}_2$ real numbers (always positive on the critical line), the fractions of those two summands that are used. As a corollary, when $\sigma=\frac{1}{2}$ one has $d_1=d_2$ (equivalently $\hat{d}_1=\hat{d}_2$); this fact is formally verified in Lean. Also, with this rescaling the remainder terms are nearly periodic in $T$ with period one, converging to a fixed waveform in the fractional part of $T$. This decomposition of Siegel's $R$ was discovered through experimental mathematics using a spiral visualization of the partial sums, described later in the paper. We also discuss a number of other observations, including what we call the yin yang curves, the zero counting function, and ovals of equal length leg loci.

Over an algebraically closed field of characteristic $p>0$, we prove that smooth Fano threefolds form 105 irreducible deformation families.