Showing 1–50 of 256 results for author: Huang, L

Robust Low-Tubal-Rank Tensor Completion under Cross-Concentrated Sampling

Authors: HanQin Cai, Longxiu Huang, Jing Qin, Chengyue Wu

Abstract: Tensor cross-concentrated sampling (t-CCS) bridges entrywise sampling and t-CUR slice-wise sampling by observing entries only within selected horizontal and lateral slices. Existing t-CCS completion methods, however, assume that the observations are free of gross corruption. In this work, we study robust recovery of a third-order low-tubal-rank tensor from partial t-CCS observations contaminated b… ▽ More Tensor cross-concentrated sampling (t-CCS) bridges entrywise sampling and t-CUR slice-wise sampling by observing entries only within selected horizontal and lateral slices. Existing t-CCS completion methods, however, assume that the observations are free of gross corruption. In this work, we study robust recovery of a third-order low-tubal-rank tensor from partial t-CCS observations contaminated by sparse, arbitrarily large outliers. We propose Robust Iterative t-CUR (R-ItCUR), a tensor-native algorithm that partitions the sampled tensor cross into two exterior blocks and an intersection block, applies adaptive blockwise Welsch correction for outlier suppression, and updates the low-rank component through projected blockwise gradient descent. By operating directly on the sampled cross, R-ItCUR avoids reconstructing the full tensor throughout the iterations, resulting in substantial memory and computational savings. Experiments on synthetic tensors, cardiac MRI data, and three-dimensional seismic data demonstrate accurate recovery and strong robustness to sparse gross corruptions. The results further highlight the importance of explicitly exploiting the cross-concentrated sampling structure in robust tensor completion. △ Less

Submitted 4 August, 2026; v1 submitted 4 August, 2026; originally announced August 2026.

The Existence of Diagonal Quantum Latin Squares with Maximum Cardinality

Authors: Lin Huang, Yang Li

Abstract: A quantum Latin square of order \(n\), denoted by \(\operatorname{QLS}(n)\), is an \(n \times n\) square whose entries are unit column vectors in the \(n\)-dimensional Hilbert space \(\mathcal{H}_n\), such that each row and each column forms an orthonormal basis of \(\mathcal{H}_n\). The cardinality of a QLS($n$) is the number of distinct vectors up to a global phase in the array. A \(\mathrm{QLS}… ▽ More A quantum Latin square of order \(n\), denoted by \(\operatorname{QLS}(n)\), is an \(n \times n\) square whose entries are unit column vectors in the \(n\)-dimensional Hilbert space \(\mathcal{H}_n\), such that each row and each column forms an orthonormal basis of \(\mathcal{H}_n\). The cardinality of a QLS($n$) is the number of distinct vectors up to a global phase in the array. A \(\mathrm{QLS}(n)\) whose main diagonal and anti-diagonal each forms an orthonormal basis of \(\mathcal{H}_n\) is called a diagonal quantum Latin square (\(\mathrm{DQLS}(n)\)). In this paper, we focus on the existence of the \(\mathrm{DQLS}(n)\) with maximum cardinality ($\mathrm{MCDQLS}(n)$). By employing direct constructions based on row-quantum Latin rectangle and special complete mapping, together with the recursive techniques such as the singular direct product construction, We have almost completely determined the existence of \(\mathrm{MCDQLS}(n)\), except for a few exceptional cases. This result is based on the study of the existence of idempotent \(\mathrm{QLS}(n)\) with maximum cardinality (\(\mathrm{MCQLS}(n)\)), and implies an existence result for pandiagonal quantum Latin squares with maximum cardinality (\(\mathrm{MCPQLS}(n)\)). △ Less

Submitted 26 June, 2026; originally announced June 2026.

Second-order $H^1$-norm error analysis for time-fractional advection-dispersion equations based on the fast averaged L1 method

Authors: Liangcai Huang, Lin Li, Pengcheng Xie

Abstract: In this paper, based on the fast averaged L1 method, we present an error analysis for time-fractional advection-dispersion equations with a weak singularity at the initial time. An integrating-factor transformation is introduced to convert the tempered fractional derivative into the standard Caputo derivative, which is more suitable for discretization using the fast averaged L1 method. A sum-of-ex… ▽ More In this paper, based on the fast averaged L1 method, we present an error analysis for time-fractional advection-dispersion equations with a weak singularity at the initial time. An integrating-factor transformation is introduced to convert the tempered fractional derivative into the standard Caputo derivative, which is more suitable for discretization using the fast averaged L1 method. A sum-of-exponentials approximation is then incorporated into the averaged L1 method to reduce computational cost and storage while preserving the desired accuracy. By deriving error estimates for the discrete coefficients and the accumulated truncation errors, we establish the stability and $H^1$-norm convergence analysis, with a convergence order higher than those in the published literature. Numerical examples are tested to validate our theoretical results. The effects of the fractional parameters $α$ and $λ$ on the solution are discussed. The memory effect and long-time tail phenomenon, which are known to exist in real systems yet cannot be captured by classical integer-order equations, are again found in the current fractional case. △ Less

Submitted 20 June, 2026; originally announced June 2026.

Comments: 25 pages, 12 figures

MSC Class: 65M06; 65M12; 65M15; 65M70

Multiple positive solutions of a quasilinear Schrödinger-Poisson system with concave and convex nonlinearities

Authors: Lanxin Huang, Xinqi Zhou

Abstract: In this paper, we consider the quasilinear Schrödinger-Poisson system with concave and convex nonlinearities \begin{align*} \begin{cases} -Δ_{p} u+λV(x)|u|^{p-2}u + μφ|u|^{p-2}u= a(x)|u|^{m-2}u + b(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3}, -Δφ=|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*} where $λ>0, ~μ>0$, $\frac{3}{2}<p<3$, $1< q<p < m < 2p$ and… ▽ More In this paper, we consider the quasilinear Schrödinger-Poisson system with concave and convex nonlinearities \begin{align*} \begin{cases} -Δ_{p} u+λV(x)|u|^{p-2}u + μφ|u|^{p-2}u= a(x)|u|^{m-2}u + b(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3}, -Δφ=|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*} where $λ>0, ~μ>0$, $\frac{3}{2}<p<3$, $1< q<p < m < 2p$ and $Δ_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. We assume that $V(x) \in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well, while $a(x)$ and $b(x)$ are allowed to be sign-changing and satisfy some suitable assumptions in $\mathbb{R}^3$. By using the Ekeland's variational principle and combining the constraint approach, we prove that the system admits two positive solutions. △ Less

Submitted 17 June, 2026; originally announced June 2026.

Silting subcategories and (co)torsion pairs associated to extended hearts

Authors: Liangwei Huang, Haicheng Zhang

Abstract: We establish the poset isomorphisms between $(d+1)$-term silting subcategories, functorially finite $s$-torsion pairs in the $d$-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. As an application, we also give dg algebra versions of these bijections, which establish the poset isomorphisms between $τ$-tilting pairs, $(d+1)$-term silting complexes, and functorially… ▽ More We establish the poset isomorphisms between $(d+1)$-term silting subcategories, functorially finite $s$-torsion pairs in the $d$-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. As an application, we also give dg algebra versions of these bijections, which establish the poset isomorphisms between $τ$-tilting pairs, $(d+1)$-term silting complexes, and functorially finite $s$-torsion pairs. △ Less

Submitted 14 June, 2026; v1 submitted 11 June, 2026; originally announced June 2026.

Comments: 19 pages

The distance from functions in BMO to BLO

Authors: Hua Huang, Long Huang, Ciqiang Zhuo

Abstract: Let BMO and BLO denote the spaces of all locally integrable real-valued functions on $\mathbb{R}^n$ with bounded mean oscillation and bounded lower oscillation, respectively. It is well known that $$L^\infty(\mathbb{R}^n)\subsetneqq {\rm BLO}\subsetneqq {\rm BMO}.$$ In 1978, Garnett and Jones gave distance formulas of $f\in {\rm BMO}$ to $L^\infty(\mathbb{R}^n)$ and recently, Angrisani studied the… ▽ More Let BMO and BLO denote the spaces of all locally integrable real-valued functions on $\mathbb{R}^n$ with bounded mean oscillation and bounded lower oscillation, respectively. It is well known that $$L^\infty(\mathbb{R}^n)\subsetneqq {\rm BLO}\subsetneqq {\rm BMO}.$$ In 1978, Garnett and Jones gave distance formulas of $f\in {\rm BMO}$ to $L^\infty(\mathbb{R}^n)$ and recently, Angrisani studied the distance of $f\in {\rm BLO}$ to $L^\infty(\mathbb{R}^n)$. In this paper, we characterize the distance from any given function $f \in {\rm BMO}$ to BLO via the Muckenhoupt weight class $A_p$ as follows \begin{center} dist\,($f$,\ BLO)\,$\sim\inf\left\{ξ\in(0,\infty):\ e^{-\frac fξ}\in A_p\ \mathrm{for\ some\ }p\in(1,\infty)\right\}$. \end{center} Two equivalent representations of this distance are also established in terms of exponential form and the infimum of the constant in a variant of John--Nirenberg inequality, respectively. △ Less

Submitted 15 June, 2026; v1 submitted 8 June, 2026; originally announced June 2026.

Revisiting CUR Perturbation Analysis: A Local Tangent-Space Expansion

Authors: Longxiu Huang

Abstract: CUR decompositions approximate a matrix using selected columns, rows, and their intersection. Classical CUR theory provides exactness results for low-rank matrices and perturbation bounds controlled by the size of the noise. In this work we develop a local perturbation expansion for a fixed-index rank-truncated CUR map near an admissible rank-\(r\) matrix. We show that the Fréchet derivative of th… ▽ More CUR decompositions approximate a matrix using selected columns, rows, and their intersection. Classical CUR theory provides exactness results for low-rank matrices and perturbation bounds controlled by the size of the noise. In this work we develop a local perturbation expansion for a fixed-index rank-truncated CUR map near an admissible rank-\(r\) matrix. We show that the Fréchet derivative of the rank-truncated CUR map is a sampling-induced oblique tangent-space projector determined by the selected rows and columns. Consequently, the local recovery error for an underlying low-rank matrix is governed not by the full perturbation norm alone, but by the image of the perturbation under this sampling-induced tangent projector. In particular, perturbations that are invisible to the selected rows and columns are removed to first order. We compare this behavior with the classical local expansion of the rank-\(r\) SVD truncation. SVD removes orthogonal-normal perturbations to first order, whereas rank-truncated CUR removes perturbations in the kernel of the sampling-induced oblique tangent projector. Numerical experiments illustrate these regimes and confirm the predicted first- and second-order local rates. △ Less

Submitted 13 May, 2026; originally announced May 2026.

MSC Class: 15A23; 15A18; 65F30; 65F35

Principal-agent problems with adverse selection: A stochastic target problem formulation

Authors: Guillermo Alonso Alvarez, Ibrahim Ekren, Liwei Huang

Abstract: We study a principal-agent problem with adverse selection, where the principal does not know the agent's true cost but must design a contract to optimize a specific criterion. Unlike standard screening frameworks that allow for self-selection, we assume the principal can only offer a unique contract. We show that the agent's optimization problem can be reformulated as a stochastic target problem.… ▽ More We study a principal-agent problem with adverse selection, where the principal does not know the agent's true cost but must design a contract to optimize a specific criterion. Unlike standard screening frameworks that allow for self-selection, we assume the principal can only offer a unique contract. We show that the agent's optimization problem can be reformulated as a stochastic target problem. After characterizing the credible domain of this target problem, we show that the principal's objective can be solved as a stochastic optimal control problem with partial information and state constraints. The description of the credible domain also allows us to obtain the value of screening contracts. △ Less

Submitted 16 May, 2026; v1 submitted 1 May, 2026; originally announced May 2026.

Robust Spectral Recovery for Dynamical Sampling

Authors: HanQin Cai, Longxiu Huang, Tianming Wang, Juntao You

Abstract: We study the spectral recovery problem for dynamical sampling on a finite cyclic grid. Given time snapshots obtained from a fixed uniform spatial subsampling of the orbit $x_{\ell}=A^{\ell}f$, we aim to recover the spectrum of the unknown circular convolution operator $A$. However, in the presence of outliers, even in only a few snapshots, existing approaches often struggle to recover the spectrum… ▽ More We study the spectral recovery problem for dynamical sampling on a finite cyclic grid. Given time snapshots obtained from a fixed uniform spatial subsampling of the orbit $x_{\ell}=A^{\ell}f$, we aim to recover the spectrum of the unknown circular convolution operator $A$. However, in the presence of outliers, even in only a few snapshots, existing approaches often struggle to recover the spectrum. We address this challenge by proposing a novel robust spectral recovery model in the presence of time-sparse corruptions. We propose a robust pipeline that lifts the problem to a sequence of robust low-rank Hankel recovery and completion tasks, followed by Prony-type spectral estimation. Numerical experiments confirm the accurate spectral recovery of the proposed approach and exhibit its superior robustness against state-of-the-art under various settings. △ Less

Submitted 10 April, 2026; originally announced April 2026.

Comments: 2026 IEEE International Symposium on Information Theory

Electrostatic skeletons and condition of strict descent

Authors: Linhang Huang

Abstract: Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this pap… ▽ More Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons. △ Less

Submitted 4 April, 2026; originally announced April 2026.

Comments: 29 pages, 22 figures

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

Authors: Ziang Chen, Liqiang Huang, Mengxuan Yang, Shengxuan Zhou

Abstract: We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural ne… ▽ More We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension. △ Less

Submitted 22 February, 2026; originally announced February 2026.

Quantile Randomized Kaczmarz Algorithm with Whitelist Trust Mechanism

Authors: Sofiia Shvaiko, Longxiu Huang, Elizaveta Rebrova

Abstract: Randomized Kaczmarz (RK) is a simple and fast solver for consistent overdetermined systems, but it is known to be fragile under noise. We study overdetermined $m\times n$ linear systems with a sparse set of corrupted equations, $ {\bf A}{\bf x}^\star = {\bf b}, $where only $\tilde{\bf b} = {\bf b} + \boldsymbol{\varepsilon}$ is observed with $\|\boldsymbol{\varepsilon}\|_0 \le βm$. The recently in… ▽ More Randomized Kaczmarz (RK) is a simple and fast solver for consistent overdetermined systems, but it is known to be fragile under noise. We study overdetermined $m\times n$ linear systems with a sparse set of corrupted equations, $ {\bf A}{\bf x}^\star = {\bf b}, $where only $\tilde{\bf b} = {\bf b} + \boldsymbol{\varepsilon}$ is observed with $\|\boldsymbol{\varepsilon}\|_0 \le βm$. The recently introduced QuantileRK (QRK) algorithm addresses this issue by testing residuals against a quantile threshold, but computing a per-iteration quantile across many rows is costly. In this work we (i) reanalyze QRK and show that its convergence rate improves monotonically as the corruption fraction $β$ decreases; (ii) propose a simple online detector that flags and removes unreliable rows, which reduces the effective $β$ and speeds up convergence; and (iii) make the method practical by estimating quantiles from a small random subsample of rows, preserving robustness while lowering the per-iteration cost. Simulations on imaging and synthetic data demonstrate the efficiency of the proposed method. △ Less

Submitted 12 February, 2026; originally announced February 2026.

Comments: Accepted by ICASSP 2026

A numerical study for tempered time-fractional advection-dispersion equation on graded meshes

Authors: Liangcai Huang, Lin Li, Shujuan Lü

Abstract: In this paper, we develop a second-order accurate time-stepping scheme for the tempered time-fractional advection-dispersion equation based on a sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional derivative. To effectively resolve the weak initial-time singularity at t=0, graded temporal meshes are employed. A fully discrete scheme is constructed by coupli… ▽ More In this paper, we develop a second-order accurate time-stepping scheme for the tempered time-fractional advection-dispersion equation based on a sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional derivative. To effectively resolve the weak initial-time singularity at t=0, graded temporal meshes are employed. A fully discrete scheme is constructed by coupling the proposed half-time-level temporal discretization with a finite difference method in space. Compared with the classical L1 scheme, the proposed SOE-based method achieves the same global convergence order while reducing both storage requirements and computational cost. Specifically, the storage demand is reduced from O(MN) to O(MN_exp), and the computational complexity is lowered from O(MN^2) to O(MN N_exp), where M and N denote the numbers of spatial and temporal grid points, respectively, and N_exp is the number of exponential terms used in the SOE approximation. The unique solvability, stability and accuracy of the resulting scheme are rigorously analyzed. Several numerical results are presented to confirm the sharpness of the error analysis and to demonstrate the efficiency of the proposed method. △ Less

Submitted 9 February, 2026; originally announced February 2026.

MSC Class: 65M06; 65M12; 35R11; 35S10

Continuized Nesterov Momentum Achieves the $O(\varepsilon^{-7/4})$ Complexity in Smooth Nonconvex Optimization

Authors: Julien Hermant, Jean-François Aujol, Charles Dossal, Lorick Huang, Aude Rondepierre, Irène Waldspurger

Abstract: For first-order optimization of non-convex functions with Lipschitz-continuous gradient and Hessian, the best-known complexity for reaching an $\varepsilon$-approximation of a stationary point is $\mathcal{O}(\varepsilon^{-7/4})$. The existing algorithms achieving this bound are based on momentum, but are always complemented with safeguard mechanisms that erase the accumulated momentum if a certai… ▽ More For first-order optimization of non-convex functions with Lipschitz-continuous gradient and Hessian, the best-known complexity for reaching an $\varepsilon$-approximation of a stationary point is $\mathcal{O}(\varepsilon^{-7/4})$. The existing algorithms achieving this bound are based on momentum, but are always complemented with safeguard mechanisms that erase the accumulated momentum if a certain condition is violated. Whether such momentum-control mechanisms are fundamentally necessary has remained an open question. We show that randomizing the parameters enables one to achieve this complexity in expectation when using momentum without any of such mechanisms, and we improve the numerical constant factor of the bound in the case of a large enough number of iterations. From an analysis perspective, we do so by leveraging the continuized method, which interprets the algorithm as a realization of a continuous-time stochastic differential equation (SDE) involving a Poisson process. We show that this SDE converges in probability to the Heavy Ball ordinary differential equation when the stepsize goes to zero, paralleling the behavior of more classical instances of Nesterov momentum. △ Less

Submitted 28 July, 2026; v1 submitted 5 February, 2026; originally announced February 2026.

A finite-termination algorithm for testing copositivity over the positive semidefinite cone

Authors: Lei Huang, Lingling Xie

Abstract: This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous polynomial is copositive over the positive semidefinite cone, the… ▽ More This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous polynomial is copositive over the positive semidefinite cone, the algorithm provides a certificate; otherwise, it returns a vector that refutes copositivity. Building on a similar idea, we further propose an algorithm to test copositivity over the direct product of the positive semidefinite cone and the nonnegative orthant. Preliminary numerical experiments demonstrate the effectiveness of the proposed methods. △ Less

Submitted 10 January, 2026; originally announced January 2026.

Liouville-type Theorems for Stable Solutions of the Hénon-Lane-Emden System

Authors: Long-Han Huang, Wenming Zou

Abstract: We investigate the Hénon-Lane-Emden system defined by $- Δu=|x|^a |v|^{p-1}v$ and $- Δv=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the Hénon-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that $0 < \min\,\{p, q\} < 1$, or… ▽ More We investigate the Hénon-Lane-Emden system defined by $- Δu=|x|^a |v|^{p-1}v$ and $- Δv=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical case. Then we prove that the Hénon-Lane-Emden conjecture is valid for solutions stable outside a compact set, provided that $0 < \min\,\{p, q\} < 1$, or $0 \leq a - b \leq (N-2)(p - q)$, or $N \leq \frac{2(p+q+2)}{pq-1} + 10$. Additional Liouville-type theorems for the subcritical case are also obtained. Furthermore, we address the supercritical case. To our knowledge, these results constitute the first Liouville-type theorems for this class of solutions in the Hénon-Lane-Emden system. As a by-product, several existing results in the literature are refined. △ Less

Submitted 18 December, 2025; originally announced December 2025.

Comments: 40 pages. To appear in J. London Math. Soc

MSC Class: 35B09; 35B40; 35B33

Continuized Nesterov Acceleration for Non-Convex Optimization

Authors: Julien Hermant, Jean-François Aujol, Charles Dossal, Lorick Huang, Aude Rondepierre

Abstract: In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding str… ▽ More In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers. △ Less

Submitted 6 January, 2026; v1 submitted 18 December, 2025; originally announced December 2025.

A New Fast Finite Difference Scheme for Tempered Time Fractional Advection-Dispersion Equation with a Weak Singularity at Initial Time

Authors: Liangcai Huang, Shujuan Lü

Abstract: In this paper, we propose a new second-order fast finite difference scheme in time for solving the Tempered Time Fractional Advection-Dispersion Equation. Under the assumption that the solution is nonsmooth at the initial time, we investigate the uniqueness, stability, and convergence of the scheme. Furthermore, we prove that the scheme achieves second-order convergence in both time and space. Fin… ▽ More In this paper, we propose a new second-order fast finite difference scheme in time for solving the Tempered Time Fractional Advection-Dispersion Equation. Under the assumption that the solution is nonsmooth at the initial time, we investigate the uniqueness, stability, and convergence of the scheme. Furthermore, we prove that the scheme achieves second-order convergence in both time and space. Finally, corresponding numerical examples are provided. △ Less

Submitted 19 December, 2025; v1 submitted 17 December, 2025; originally announced December 2025.

MSC Class: 26A33 and 35R11 and 65M12

Contracting with discretionary bonuses

Authors: Guillermo Alonso Alvarez, Ibrahim Ekren, Liwei Huang

Abstract: We study a continuous time contracting model in which a principal hires a risk averse agent to manage a project over a finite horizon and provides sequential payments whose timing is endogenously determined. The resulting nonzero-sum interaction between the principal and the agent is reformulated as a mixed control and stopping problem. Using numerical simulations, we investigate how factors such… ▽ More We study a continuous time contracting model in which a principal hires a risk averse agent to manage a project over a finite horizon and provides sequential payments whose timing is endogenously determined. The resulting nonzero-sum interaction between the principal and the agent is reformulated as a mixed control and stopping problem. Using numerical simulations, we investigate how factors such as the relative impatience of the parties and the number of bonus payments influence the principal's value and the structure of the optimal bonus payment scheme. A notable finding is that, in some contractual environments, the principal optimally offers a sign-on bonus to front-load incentives. △ Less

Submitted 28 November, 2025; originally announced November 2025.

Low-Discrepancy Set Post-Processing via Gradient Descent

Authors: François Clément, Linhang Huang, Woorim Lee, Cole Smidt, Braeden Sodt, Xuan Zhang

Abstract: The construction of low-discrepancy sets, used for uniform sampling and numerical integration, has recently seen great improvements based on optimization and machine learning techniques. However, these methods are computationally expensive, often requiring days of computation or access to GPU clusters. We show that simple gradient descent-based techniques allow for comparable results when starting… ▽ More The construction of low-discrepancy sets, used for uniform sampling and numerical integration, has recently seen great improvements based on optimization and machine learning techniques. However, these methods are computationally expensive, often requiring days of computation or access to GPU clusters. We show that simple gradient descent-based techniques allow for comparable results when starting with a reasonably uniform point set. Not only is this method much more efficient and accessible, but it can be applied as post-processing to any low-discrepancy set generation method for a variety of standard discrepancy measures. △ Less

Submitted 13 November, 2025; originally announced November 2025.

Capacitary Muckenhoupt Weight, BMO and BLO Spaces with Hausdorff Content, Factorization Theorems and Applications

Authors: Long Huang, Yangzhi Zhang, Ciqiang Zhuo

Abstract: Let $δ\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ spa… ▽ More Let $δ\in(0,n]$, $p\in[1,\infty)$, $\mathcal H_{\infty}^δ$ denote the Hausdorff content on $\mathbb R^n$, and $\mathcal A_{p,δ}$ be the capacitary Muckenhoupt weight class. We are interested in understanding the relationship between the capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$ and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ spaces for all dimension $δ\in(0,n]$, and further to comprehend the structure of these two spaces. Our main result shows that $\mathcal A_{p,δ}$ for $p\in(1,\infty)$ is equivalent to the BMO spaces, while $\mathcal A_{1,δ}$ is equivalent to the BLO spaces, and consequently yields the factorization theorems for these BMO and BLO spaces via capacitary Hardy--Littlewood maximal operators, which essentially extend main results of Coifman and Rochberg in 1980 beyond measure theory. As applications, by establishing some capacitary weighted John--Nirenberg inequalities, we obtain the equivalence between capacitary weighted BMO or BLO spaces and ${\rm{BMO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ or ${\rm{BLO}}(\mathbb R^n, \mathcal H_{\infty}^δ)$ respectively. These results reveal deep connections between $\mathcal A_{p,δ}$ and BMO or BLO spaces with Hausdorff content, beyond the classical measure-theoretic settings. We develop some approaches in the proofs and using a new observation, that is, the additivity of measures and linearity of integrals are superfluous for the corresponding classical theory. △ Less

Submitted 1 February, 2026; v1 submitted 2 November, 2025; originally announced November 2025.

Comments: 39 pages, 2 figures; comments are welcome

Monotonicity of Causal Killing Vectors and Geometry of ADM Mass Minimizers

Authors: Sven Hirsch, Lan-Hsuan Huang

Abstract: We address two problems concerning the ADM mass-minimizing initial data sets. First, we show that the equality case of the positive mass theorem embeds into a pp-wave spacetime. Second, we show that positive Bartnik mass minimizers embed into strongly stationary vacuum spacetimes, thereby confirming the Bartnik stationary vacuum conjecture. A key ingredient is a new monotonicity formula for the Lo… ▽ More We address two problems concerning the ADM mass-minimizing initial data sets. First, we show that the equality case of the positive mass theorem embeds into a pp-wave spacetime. Second, we show that positive Bartnik mass minimizers embed into strongly stationary vacuum spacetimes, thereby confirming the Bartnik stationary vacuum conjecture. A key ingredient is a new monotonicity formula for the Lorentzian length of a causal Killing vector field, which, among other applications, yields a strong maximum principle for the length. △ Less

Submitted 11 October, 2025; originally announced October 2025.

Capacitary Muckenhoupt Weights and Weighted Norm Inequalities for Hardy-Littlewood Maximal Operators

Authors: Long Huang, Yangzhi Zhang, Ciqiang Zhuo

Abstract: Let $\mathcal H_{\infty}^δ$ denote the Hausdorff content of dimension $δ\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$-norm inequality with $p\in(1,\infty)$ and the weighted weak $L^1$-norm inequality on Hardy-Littlewood maximal operators associated with Hausdorff contents… ▽ More Let $\mathcal H_{\infty}^δ$ denote the Hausdorff content of dimension $δ\in(0,n]$ defined on subsets of $\mathbb R^n$. The principal problem, considered in this paper, is to characterize the non-negative function $w$ for which the weighted $L^p$-norm inequality with $p\in(1,\infty)$ and the weighted weak $L^1$-norm inequality on Hardy-Littlewood maximal operators associated with Hausdorff contents hold true. To achieve this, we introduce a class of capacitary Muckenhoupt weights depending on the dimension $δ$, denoted as $\mathcal A_{p,δ}$, which enjoys the strict monotonicity on the dimension index $δ$. Then we show that, for any $p\in(1,\infty)$ and $δ\in(0,n]$, the weighted $L^p$-norm inequality holds true if and only if $w\in\mathcal A_{p,δ}$, and the weighted weak $L^1$-norm inequality holds true if and only if $w\in\mathcal A_{1,δ}$ by a new approach developed in this paper. As the second objective, applying this new approach, the seminal properties of classical Muckenhoupt $A_p$ weights, such as the reverse Hölder inequality [R. R. Coifman and C. Fefferman, Studia Math. 51 (1974), 241-250], the self-improving property [B. Muckenhoupt, Trans. Amer. Math. Soc. 165 (1972), 207-226], and Jones' factorization theorem [P. W. Jones, Ann. of Math. (2) 111 (1980), 511-530], are all established within the framework of capacitary Muckenhoupt weight class $\mathcal A_{p,δ}$. Finally, we also show that the maximal operator is bounded on the weak weighted Choquet-Lebesgue space $L_w^{p,\infty}(\mathbb R^n,{\mathcal H}_\infty^δ)$ if and only if $w\in\mathcal A_{p,δ}$ with $p\in(1,\infty)$ and $δ\in(0,n]$. △ Less

Submitted 18 December, 2025; v1 submitted 28 September, 2025; originally announced September 2025.

Comments: 50 pages; quasicontinuity is removed in Theorem 1.7, 1.9 and 1.12

Randomized Space-Time Sampling for Affine Graph Dynamical Systems

Authors: Le Gong, Longxiu Huang

Abstract: This paper investigates the problem of dynamical sampling for graph signals influenced by a constant source term. We consider signals evolving over time according to a linear dynamical system on a graph, where both the initial state and the source term are bandlimited. We introduce two random space-time sampling regimes and analyze the conditions under which stable recovery is achievable. While ou… ▽ More This paper investigates the problem of dynamical sampling for graph signals influenced by a constant source term. We consider signals evolving over time according to a linear dynamical system on a graph, where both the initial state and the source term are bandlimited. We introduce two random space-time sampling regimes and analyze the conditions under which stable recovery is achievable. While our framework extends recent work on homogeneous dynamics, it addresses a fundamentally different setting where the evolution includes a constant source term. This results in a non-orthogonal-diagonalizable system matrix, rendering classical spectral techniques inapplicable and introducing new challenges in sampling design, stability analysis, and joint recovery of both the initial state and the forcing term. A key component of our analysis is the spectral graph weighted coherence, which characterizes the interplay between the sampling distribution and the graph structure. We establish sampling complexity bounds ensuring stable recovery via the Restricted Isometry Property (RIP), and develop a robust recovery algorithm with provable error guarantees. The effectiveness of our method is validated through extensive experiments on both synthetic and real-world datasets. △ Less

Submitted 20 September, 2025; originally announced September 2025.

Many lemniscates with large diameter

Authors: Linhang Huang

Abstract: We prove that for every $0 < c < 4$ and every $N \in \mathbb{N}$ there exists a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ such that the set $\{z \in \mathbb{C} : |p(z)| \leq 1\}$ has at least $N$ connected components with diameter at least $c$. This answers a question of Erdős. We prove that for every $0 < c < 4$ and every $N \in \mathbb{N}$ there exists a monic polynomial $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0$ such that the set $\{z \in \mathbb{C} : |p(z)| \leq 1\}$ has at least $N$ connected components with diameter at least $c$. This answers a question of Erdős. △ Less

Submitted 16 September, 2025; v1 submitted 15 September, 2025; originally announced September 2025.

Comments: 4 pages, 3 figures

Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

Authors: Ziang Chen, Liqiang Huang

Abstract: We study the complexity of approximating high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube using Barron spaces. Under suitable Barron assumptions on the coefficients and forcing term, we prove that the solutions can be approximated to any prescribed accuracy \(\varepsilon>0\) by two-layer neural networks whose widths and relevant parameters are… ▽ More We study the complexity of approximating high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube using Barron spaces. Under suitable Barron assumptions on the coefficients and forcing term, we prove that the solutions can be approximated to any prescribed accuracy \(\varepsilon>0\) by two-layer neural networks whose widths and relevant parameters are bounded by \(\mathcal{O}\bigl(d^{C|\log\varepsilon|}\bigr)\). Consequently, we identify a class of elliptic PDEs that can be approximated by shallow neural networks without suffering from the curse of dimensionality. △ Less

Submitted 1 August, 2026; v1 submitted 10 August, 2025; originally announced August 2025.

MSC Class: 68T07; 65N99; 35J25; 35C20

Free semigroupoid algebras and the first cohomology groups

Authors: Linzhe Huang, Minghui Ma

Abstract: This paper investigates derivations of the free semigroupoid algebra $\mathfrak{L}_G$ of a countable or uncountable directed graph $G$ and its norm-closed version, the tensor algebra $\mathcal{A}_G$. We first prove a weak Dixmier approximation theorem for $\mathfrak{L}_G$ when $G$ is strongly connected. Using the theorem, we show that if every connected component of $G$ is strongly connected, then… ▽ More This paper investigates derivations of the free semigroupoid algebra $\mathfrak{L}_G$ of a countable or uncountable directed graph $G$ and its norm-closed version, the tensor algebra $\mathcal{A}_G$. We first prove a weak Dixmier approximation theorem for $\mathfrak{L}_G$ when $G$ is strongly connected. Using the theorem, we show that if every connected component of $G$ is strongly connected, then every bounded derivation $δ$ from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is of the form $δ=δ_T$ for some $T\in\mathfrak{L}_G$ with $\|T\|\leqslant\|δ\|$. For any finite directed graph $G$, we also show that the first cohomology group $H^1(\mathcal{A}_G,\mathfrak{L}_G)$ vanishes if and only if every connected component of $G$ is either strongly connected or a fruit tree. To handle infinite directed graphs, we introduce the alternating number and propose \Cref{conj intro-in-tree}. Suppose every connected component of $G$ is not strongly connected. We show that if every bounded derivation from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is inner, then every connected component of $G$ is a generalized fruit tree and the alternating number $A(G)$ of $G$ is finite. The converse is also true if the conjecture holds. Finally, we provide some examples of free semigroupoid algebras together with their nontrivial first cohomology groups. △ Less

Submitted 30 July, 2025; originally announced July 2025.

Comments: 38 pages

MSC Class: 46L10; 46K50; 47B47

A remark for fully non-linear elliptic equations on compact almost Hermitian manifolds

Authors: Liding Huang

Abstract: In this paper, we generalize the definition of sub-slope, introduced by Guo-Song, to almost Hermitian manifolds and prove the existence of solutions for a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian quotient equation and the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting. In this paper, we generalize the definition of sub-slope, introduced by Guo-Song, to almost Hermitian manifolds and prove the existence of solutions for a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian quotient equation and the deformed Hermitian-Yang-Mills equation in the almost Hermitian setting. △ Less

Submitted 19 June, 2025; originally announced June 2025.

Lagrange multiplier expressions for matrix polynomial optimization and tight relaxations

Authors: Lei Huang, Jiawang Nie, Jiajia Wang, Lingling Xie

Abstract: This paper studies matrix constrained polynomial optimization. We investigate how to get explicit expressions for Lagrange multiplier matrices from the first order optimality conditions. The existence of these expressions can be shown under the nondegeneracy condition. Using Lagrange multiplier matrix expressions, we propose a strengthened Moment-SOS hierarchy for solving matrix polynomial optimiz… ▽ More This paper studies matrix constrained polynomial optimization. We investigate how to get explicit expressions for Lagrange multiplier matrices from the first order optimality conditions. The existence of these expressions can be shown under the nondegeneracy condition. Using Lagrange multiplier matrix expressions, we propose a strengthened Moment-SOS hierarchy for solving matrix polynomial optimization. Under some general assumptions, we show that this strengthened hierarchy is tight, or equivalently, it has finite convergence. We also study how to detect tightness and how to extract optimizers. Numerical experiments are provided to show the efficiency of the strengthened hierarchy. △ Less

Submitted 10 January, 2026; v1 submitted 14 June, 2025; originally announced June 2025.

Uniqueness and dimension for the geodesic of the critical long-range percolation metric

Authors: Jian Ding, Zherui Fan, Lu-Jing Huang

Abstract: By recent works of Bäumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points, for which an important ingredient of independent interest is the continuity of the metric distribution. In addition, we establish the Hausdorff dimension of the geo… ▽ More By recent works of Bäumler [2] and of the authors of this paper [5], the (limiting) random metric for the critical long-range percolation was constructed. In this paper, we prove the uniqueness of the geodesic between two fixed points, for which an important ingredient of independent interest is the continuity of the metric distribution. In addition, we establish the Hausdorff dimension of the geodesics. △ Less

Submitted 12 June, 2025; originally announced June 2025.

Comments: 18 pages, 5 figures

MSC Class: 60K35; 82B27; 82B43

Quantitative Hardy--Littlewood maximal inequalities and Wiener--Stein theorem on p.c.f. fractals

Authors: Long Huang, Jinjun Li, Xiaofeng Wang

Abstract: Let $K\subset \mathbb{R}^d$ be a post-critically finite (p.c.f.) self-similar set with Hausdorff dimension $s$, and $μ$ be a self-similar probability measure supported on $K$. Let $H^α_μ$, $0<α\le s$, be the Hausdorff content on $K$, and $M_{\mathcal{D}}^μ$ be the Hardy--Littlewood maximal operator defined on $K$ associated with its basic cubes $\mathcal{D}$. In this paper, we establish quantitati… ▽ More Let $K\subset \mathbb{R}^d$ be a post-critically finite (p.c.f.) self-similar set with Hausdorff dimension $s$, and $μ$ be a self-similar probability measure supported on $K$. Let $H^α_μ$, $0<α\le s$, be the Hausdorff content on $K$, and $M_{\mathcal{D}}^μ$ be the Hardy--Littlewood maximal operator defined on $K$ associated with its basic cubes $\mathcal{D}$. In this paper, we establish quantitative strong type and weak type Hardy--Littlewood maximal inequalities on fractal set $K$ with respect to $H^α_μ$ for all range $0<α\le s$. As applications, the Lebesgue differentiation theorem on $K$ is proved. Moreover, via the Hardy--Littlewood maximal operator $M_{\mathcal{D}}^μ$, we characterize the Lebesgue--Choquet space $L^p(K,H^α_μ)$ and the Zygmund space $L\log L(K,μ)$. To be exact, given $α/s< p\le \infty$, we discover that \[ \text{$f\in L^p(K,H^α_μ)$ if and only if $M_{\mathcal{D}}^μf\in L^p(K,H^α_μ)$}\] and, for $f\in L^1(K,μ)$ with $K$ satisfying the strong separation condition, \[\text{$M_{\mathcal{D}}^μf\in L^1(K,μ)$ if and only if $f\in L\log L(K,μ)$}.\] That is, Wiener's $L\log L$ inequality and its converse inequality due to Stein in 1969 are extended to fractal set $K$ with respect to $μ$. △ Less

Submitted 12 January, 2026; v1 submitted 8 June, 2025; originally announced June 2025.

Comments: 28 pages, 5 figures; the main results are improved by weakening SSC to p.c.f

MSC Class: 28A80; 47G10; 42B35

Spectral dimensions for one-dimensional critical long-range percolation

Authors: Zherui Fan, Lu-Jing Huang

Abstract: Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+δ)$, where $δ\in (0,1)$ is the e… ▽ More Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+δ)$, where $δ\in (0,1)$ is the exponent of the effective resistance in the LRP model, as derived in [10, Theorem 1.1]. Our work addresses an open question from [7, Section 5]. △ Less

Submitted 6 June, 2025; v1 submitted 20 May, 2025; originally announced May 2025.

Comments: 14 pages

MSC Class: 60K35; 82B27; 82B43

Collet-Eckmann type conditions and conformal welding of unicritical quadratic laminations

Authors: Linhang Huang

Abstract: In this paper, we introduce a Collet-Eckmann type condition for the unicritical laminations on the unit circle. We prove that this condition implies the lamination admits a Hölder continuous conformal welding which produces a Julia set for some unicritical polynomial. In consequence, we present a new proof that almost all angles on the unit circle produce quadratic polynomials with Hölder Fatou co… ▽ More In this paper, we introduce a Collet-Eckmann type condition for the unicritical laminations on the unit circle. We prove that this condition implies the lamination admits a Hölder continuous conformal welding which produces a Julia set for some unicritical polynomial. In consequence, we present a new proof that almost all angles on the unit circle produce quadratic polynomials with Hölder Fatou components, without the use of Beurling's theorem. △ Less

Submitted 5 May, 2025; originally announced May 2025.

Comments: 31 pages, 14 figures

The polynomial growth of effective resistances in one-dimensional critical long-range percolation

Authors: Jian Ding, Zherui Fan, Lu-Jing Huang

Abstract: We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to… ▽ More We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 0 to $[-n,n]^c$ and from the interval $[-n,n]$ to $[-2n,2n]^c$ (conditioned on no edge joining $[-n,n]$ and $[-2n,2n]^c$) both grow like $n^{δ(β)}$ for some $δ(β)\in (0,1)$. △ Less

Submitted 6 June, 2025; v1 submitted 30 April, 2025; originally announced April 2025.

Comments: 68 pages, 11 figures

MSC Class: 60K35; 82B27; 82B43

On convex domains maximizing the gradient of the torsion function

Authors: Linhang Huang

Abstract: We consider the solution of $-Δu = 1$ on convex domains $Ω\subset \mathbb{R}^2$ subject to Dirichlet boundary conditions $u =0$ on $\partial Ω$. Our main concern is the behavior of $\|\nabla u\|_{L^{\infty}}$, also known as the maximum shear stress in Elasticity Theory and first investigated by Saint Venant in 1856. We consider the two shape optimization problems… ▽ More We consider the solution of $-Δu = 1$ on convex domains $Ω\subset \mathbb{R}^2$ subject to Dirichlet boundary conditions $u =0$ on $\partial Ω$. Our main concern is the behavior of $\|\nabla u\|_{L^{\infty}}$, also known as the maximum shear stress in Elasticity Theory and first investigated by Saint Venant in 1856. We consider the two shape optimization problems $\| \nabla u\|_{L^{\infty}}/ |Ω|^{1/2}$ and $\| \nabla u\|_{L^{\infty}}/ H^1( \partial Ω)$. Numerically, the extremal domain for each functional looks a bit like the rounded letter `D'. We prove that (1) either the extremal domain does not have a $C^{2 + \varepsilon}$ boundary or (2) there exists an infinite set of points on $\partial Ω$ where the curvature vanishes. Either scenario seems curious and is rarely encountered for such problems. The techniques are based on finding a representation of the functional using only conformal geometry and classic perturbation arguments. △ Less

Submitted 6 May, 2025; v1 submitted 9 April, 2025; originally announced April 2025.

Comments: 31 pages, 8 figures

Optimization over the weakly Pareto set and multi-task learning

Authors: Lei Huang, Jiawang Nie, Jiajia Wang

Abstract: We study the optimization problem over the weakly Pareto set of a convex multiobjective optimization problem given by polynomial functions. Using Lagrange multiplier expressions and the weight vector, we give three types of representations for the weakly Pareto set. Using these representations, we reformulate the optimization problem over the weakly Pareto set as a polynomial optimization problem.… ▽ More We study the optimization problem over the weakly Pareto set of a convex multiobjective optimization problem given by polynomial functions. Using Lagrange multiplier expressions and the weight vector, we give three types of representations for the weakly Pareto set. Using these representations, we reformulate the optimization problem over the weakly Pareto set as a polynomial optimization problem. We then apply the Moment--SOS hierarchy to solve it and analyze its convergence properties under certain conditions. Numerical experiments are provided to demonstrate the effectiveness of our methods. Applications in multi-task learning are also presented. △ Less

Submitted 31 March, 2025; originally announced April 2025.

Matrix Pencil-Based Analysis of Multirate Simulation Schemes

Authors: Liya Huang, Georgios Tzounas

Abstract: This paper focuses on multirate time-domain simulations of power system models. It proposes a matrix pencil-based approach to evaluate the spurious numerical deformation introduced into power system dynamics by a given multirate integration scheme. Moreover, it considers the problem of multirate partitioning and discusses a strategy for allocating state and algebraic variables to fast and slow sub… ▽ More This paper focuses on multirate time-domain simulations of power system models. It proposes a matrix pencil-based approach to evaluate the spurious numerical deformation introduced into power system dynamics by a given multirate integration scheme. Moreover, it considers the problem of multirate partitioning and discusses a strategy for allocating state and algebraic variables to fast and slow subsystems based on modal participation factors (PFs). The suitability and features of the proposed approach are illustrated through numerical simulations that assess the accuracy effects of interfacing, as well as of various prediction and solution methods. △ Less

Submitted 17 June, 2025; v1 submitted 24 March, 2025; originally announced March 2025.

Quantum inequalities and their applications

Authors: Linzhe Huang

Abstract: In recent years, various quantum inequalities have been established on quantum symmetries in the framework of quantum Fourier analysis. We provide a detailed introduction to quantum inequalities including Hausdorff-Young inequality, Young's inequality, uncertainty principles, entropic convolution inequalities etc on subfactors, an important type of quantum symmetries. We cite several applications… ▽ More In recent years, various quantum inequalities have been established on quantum symmetries in the framework of quantum Fourier analysis. We provide a detailed introduction to quantum inequalities including Hausdorff-Young inequality, Young's inequality, uncertainty principles, entropic convolution inequalities etc on subfactors, an important type of quantum symmetries. We cite several applications of the complete positivity of the comultiplication in category theory and subfactor theory, which indicate the fundamental differences between quantum inequalities and non-commutative inequalities. We also review the Perron-Frobenius theorem together with the algebraic structures of eigenvector spaces. △ Less

Submitted 6 May, 2025; v1 submitted 17 February, 2025; originally announced February 2025.

Comments: Need major modification

MSC Class: 46L37; 43A30

Weak uniqueness for the PDE governing the joint law of a diffusion and its running supremum

Authors: Laure Coutin, Lorick Huang, Monique Pontier

Abstract: In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system's behavior and further validating… ▽ More In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system's behavior and further validating the approach introduced in [8]. △ Less

Submitted 17 January, 2025; originally announced January 2025.

Topology automaton and Hölder equivalence of Barański carpets

Authors: Yunjie Zhu, Liang-yi Huang, chunbo Cheng

Abstract: The study of Lipschitz equivalence of fractals is a very active topic in recent years. In 2023, Huang \emph{et al.} (\textit{Topology automaton of self-similar sets and its applications to metrical classifications}, Nonlinearity \textbf{36} (2023), 2541-2566.) studied the Hölder and Lipschitz equivalence of a class of p.c.f. self-similar sets which are not totally disconnected. The main tool they… ▽ More The study of Lipschitz equivalence of fractals is a very active topic in recent years. In 2023, Huang \emph{et al.} (\textit{Topology automaton of self-similar sets and its applications to metrical classifications}, Nonlinearity \textbf{36} (2023), 2541-2566.) studied the Hölder and Lipschitz equivalence of a class of p.c.f. self-similar sets which are not totally disconnected. The main tool they used is the so called topology automaton. In this paper, we define topology automaton for Barański carpets, and we show that the method used in Huang \emph{et al.} still works for the self-affine and non-p.c.f. settings. As an application, we obtain a very general sufficient condition for Barański carpets to be Hölder (or Lipschitz) equivalent. △ Less

Submitted 14 January, 2025; v1 submitted 1 December, 2024; originally announced December 2024.

Sequential optimal contracting in continuous time

Authors: Guillermo Alonso Alvarez, Erhan Bayraktar, Ibrahim Ekren, Liwei Huang

Abstract: In this paper we study a principal-agent problem in continuous time with multiple lump-sum payments (contracts) paid at different deterministic times. We reduce the non-zero sum Stackelberg game between the principal and agent to a standard stochastic optimal control problem. We apply our result to a benchmark model for which we investigate how different inputs (payment frequencies, payments' dist… ▽ More In this paper we study a principal-agent problem in continuous time with multiple lump-sum payments (contracts) paid at different deterministic times. We reduce the non-zero sum Stackelberg game between the principal and agent to a standard stochastic optimal control problem. We apply our result to a benchmark model for which we investigate how different inputs (payment frequencies, payments' distribution, discounting factors, agent's reservation utility) affect the principal's value and agent's optimal compensations. △ Less

Submitted 6 November, 2024; originally announced November 2024.

Direct Adaptive Control of Grid-Connected Power Converters via Output-Feedback Data-Enabled Policy Optimization

Authors: Feiran Zhao, Ruohan Leng, Linbin Huang, Huanhai Xin, Keyou You, Florian Dörfler

Abstract: Power electronic converters are becoming the main components of modern power systems due to the increasing integration of renewable energy sources. However, power converters may become unstable when interacting with the complex and time-varying power grid. In this paper, we propose an adaptive data-driven control method to stabilize power converters by using only online input-output data. Our cont… ▽ More Power electronic converters are becoming the main components of modern power systems due to the increasing integration of renewable energy sources. However, power converters may become unstable when interacting with the complex and time-varying power grid. In this paper, we propose an adaptive data-driven control method to stabilize power converters by using only online input-output data. Our contributions are threefold. First, we reformulate the output-feedback control problem as a state-feedback linear quadratic regulator (LQR) problem with a controllable non-minimal state, which can be constructed from past input-output signals. Second, we propose a data-enabled policy optimization (DeePO) method for this non-minimal realization to achieve efficient output-feedback adaptive control. Third, we use high-fidelity simulations to verify that the output-feedback DeePO can effectively stabilize grid-connected power converters and quickly adapt to the changes in the power grid. △ Less

Submitted 8 April, 2025; v1 submitted 6 November, 2024; originally announced November 2024.

Existence of static vacuum extensions for Bartnik boundary data near Schwarzschild spheres

Authors: Spyros Alexakis, Zhongshan An, Ahmed Ellithy, Lan-Hsuan Huang

Abstract: We obtain existence and local uniqueness of asymptotically flat, static vacuum extensions for Bartnik data on a sphere near the data of a sphere of symmetry in a Schwarzschild manifold. We obtain existence and local uniqueness of asymptotically flat, static vacuum extensions for Bartnik data on a sphere near the data of a sphere of symmetry in a Schwarzschild manifold. △ Less

Submitted 5 November, 2024; v1 submitted 4 November, 2024; originally announced November 2024.

Comments: 17 pages

Phase Group Categories of Bimodule Quantum Channels

Authors: Linzhe Huang, Chunlan Jiang, Zhengwei Liu, Jinsong Wu

Abstract: In this paper, we study the quantum channel on a von Neuamnn algebra $\mathcal{M}$ preserving a von Neumann subalgebra $\mathcal{N}$, namely an $\mathcal{N}$-$\mathcal{N}$-bimodule unital completely positive map. By introducing the relative irreducibility of a bimodule quantum channel, we show that its eigenvalues with modulus 1 form a finite cyclic group, called its phase group. Moreover, the cor… ▽ More In this paper, we study the quantum channel on a von Neuamnn algebra $\mathcal{M}$ preserving a von Neumann subalgebra $\mathcal{N}$, namely an $\mathcal{N}$-$\mathcal{N}$-bimodule unital completely positive map. By introducing the relative irreducibility of a bimodule quantum channel, we show that its eigenvalues with modulus 1 form a finite cyclic group, called its phase group. Moreover, the corresponding eigenspaces are invertible $\mathcal{N}$-$\mathcal{N}$-bimodules, which encode a categorification of the phase group. When $\mathcal{N}\subset \mathcal{M}$ is a finite-index irreducible subfactor of type II$_1$, we prove that any bimodule quantum channel is relatively irreducible for the intermediate subfactor of its fixed points. In addition, we can reformulate and prove these results intrinsically in subfactor planar algebras without referring to the subfactor using the methods of quantum Fourier analysis. △ Less

Submitted 27 January, 2026; v1 submitted 4 November, 2024; originally announced November 2024.

Comments: 27pages, close to the published version, typos are fixed

MSC Class: 46L37; 43A30

Journal ref: Sci. China Math. (2025)

Random space-time sampling and reconstruction of sparse bandlimited graph diffusion field

Authors: Longxiu Huang, Dongyang Li, Sui Tang, Qing Yao

Abstract: In this work, we investigate the sampling and reconstruction of spectrally $s$-sparse bandlimited graph signals governed by heat diffusion processes. We propose a random space-time sampling regime, referred to as {randomized} dynamical sampling, where a small subset of space-time nodes is randomly selected at each time step based on a probability distribution. To analyze the recovery problem, we e… ▽ More In this work, we investigate the sampling and reconstruction of spectrally $s$-sparse bandlimited graph signals governed by heat diffusion processes. We propose a random space-time sampling regime, referred to as {randomized} dynamical sampling, where a small subset of space-time nodes is randomly selected at each time step based on a probability distribution. To analyze the recovery problem, we establish a rigorous mathematical framework by introducing the parameter \textit{the dynamic spectral graph weighted coherence}. This key parameter governs the number of space-time samples needed for stable recovery and extends the idea of variable density sampling to the context of dynamical systems. By optimizing the sampling probability distribution, we show that as few as $\mathcal{O}(s \log(k))$ space-time samples are sufficient for accurate reconstruction in optimal scenarios, where $k$ denotes the bandwidth of the signal. Our framework encompasses both static and dynamic cases, demonstrating a reduction in the number of spatial samples needed at each time step by exploiting temporal correlations. Furthermore, we provide a computationally efficient and robust algorithm for signal reconstruction. Numerical experiments validate our theoretical results and illustrate the practical efficacy of our proposed methods. △ Less

Submitted 23 October, 2024; originally announced October 2024.

Comments: 23 pages

MSC Class: 94A20; 94A12

Adversarial Network Optimization under Bandit Feedback: Maximizing Utility in Non-Stationary Multi-Hop Networks

Authors: Yan Dai, Longbo Huang

Abstract: Stochastic Network Optimization (SNO) concerns scheduling in stochastic queueing systems. It has been widely studied in network theory. Classical SNO algorithms require network conditions to be stationary with time, which fails to capture the non-stationary components in many real-world scenarios. Many existing algorithms also assume knowledge of network conditions before decision, which rules out… ▽ More Stochastic Network Optimization (SNO) concerns scheduling in stochastic queueing systems. It has been widely studied in network theory. Classical SNO algorithms require network conditions to be stationary with time, which fails to capture the non-stationary components in many real-world scenarios. Many existing algorithms also assume knowledge of network conditions before decision, which rules out applications where unpredictability presents. Motivated by these issues, we consider Adversarial Network Optimization (ANO) under bandit feedback. Specifically, we consider the task of *i)* maximizing some unknown and time-varying utility function associated to scheduler's actions, where *ii)* the underlying network is a non-stationary multi-hop one whose conditions change arbitrarily with time, and *iii)* only bandit feedback (effect of actually deployed actions) is revealed after decisions. Our proposed `UMO2` algorithm ensures network stability and also matches the utility maximization performance of any "mildly varying" reference policy up to a polynomially decaying gap. To our knowledge, no previous ANO algorithm handled multi-hop networks or achieved utility guarantees under bandit feedback, whereas ours can do both. Technically, our method builds upon a novel integration of online learning into Lyapunov analyses: To handle complex inter-dependencies among queues in multi-hop networks, we propose meticulous techniques to balance online learning and Lyapunov arguments. To tackle the learning obstacles due to potentially unbounded queue sizes, we design a new online linear optimization algorithm that automatically adapts to loss magnitudes. To maximize utility, we propose a bandit convex optimization algorithm with novel queue-dependent learning rate scheduling that suites drastically varying queue lengths. Our new insights in online learning can be of independent interest. △ Less

Submitted 28 August, 2024; originally announced August 2024.

Positive solutions with prescribed mass for a planar Choquard equation with critical growth

Authors: Ling Huang, Giulio Romani

Abstract: We study normalised solutions for a Choquard equation in the plane with polynomial Riesz kernel and exponential nonlinearities, which are critical in the sense of Trudinger-Moser. For all prescribed values of the mass, we prove existence of a positive radial solution by a variational argument, which exploits a delicate analysis on the mountain pass level. Under an additional monotonicity assumptio… ▽ More We study normalised solutions for a Choquard equation in the plane with polynomial Riesz kernel and exponential nonlinearities, which are critical in the sense of Trudinger-Moser. For all prescribed values of the mass, we prove existence of a positive radial solution by a variational argument, which exploits a delicate analysis on the mountain pass level. Under an additional monotonicity assumption on the nonlinearity, such a solution turns out to be also a ground state in $H^1(\mathbb R^2)$. Our work extends the results by Dou, Huang, and Zhong (J Geom Anal 34(10):317, 2024) to the Choquard setting, improving in several directions those by Deng and Yu in (Z Angew Math Phys 74(3):103, 2023). △ Less

Submitted 28 April, 2025; v1 submitted 30 July, 2024; originally announced July 2024.

Comments: Some improvements in the proof of the main result, the statement of which remained unchanged

MSC Class: 35J20; 35J91; 35Q55; 35R09; 35B33

Journal ref: Z. Angew. Math. Phys. 76, 96 (2025)

Hochschild cohomology for free semigroup algebras

Authors: Linzhe Huang, Minghui Ma, Xiaomin Wei

Abstract: This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_α\}_{α\inΛ}$, with the left regular free semigroup algebra $\mathfrak{L}_Λ$ and the non-commutative disc algebra $\mathfrak{A}_Λ$ serving as two typical examples. We establish that all derivations of these algebras are automatically continuous. By introducing a novel computational… ▽ More This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set $\{z_α\}_{α\inΛ}$, with the left regular free semigroup algebra $\mathfrak{L}_Λ$ and the non-commutative disc algebra $\mathfrak{A}_Λ$ serving as two typical examples. We establish that all derivations of these algebras are automatically continuous. By introducing a novel computational approach, we demonstrate that the first Hochschild cohomology group of $\mathfrak{A}_Λ$ with coefficients in $\mathfrak{L}_Λ$ is zero. Utilizing the Cesàro operators and conditional expectations, we show that the first normal cohomology group of $\mathfrak{L}_Λ$ is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when $|Λ|<\infty$. These methods extend to compute the cohomology groups of a specific class of operator algebras generated by the left regular representations of cancellative semigroups, which notably include Thompson's semigroup. △ Less

Submitted 19 July, 2024; originally announced July 2024.

Comments: 19 pages

MSC Class: 47L10; 46K50; 47B47

The mass-mixed case for normalized solutions to NLS equations in dimension two

Authors: Daniele Cassani, Ling Huang, Cristina Tarsi, Xuexiu Zhong

Abstract: \noindent We are concerned with positive normalized solutions $(u,λ)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -Δu+λu=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying the mass constraint $$\int_{\mathbb{R}^2}|u|^2\, dx=c^2\ .$$ We are interested in the so-called mass mixed case in which $f$ has $L^2$-subcritical growth at zero and critical growth a… ▽ More \noindent We are concerned with positive normalized solutions $(u,λ)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -Δu+λu=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying the mass constraint $$\int_{\mathbb{R}^2}|u|^2\, dx=c^2\ .$$ We are interested in the so-called mass mixed case in which $f$ has $L^2$-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: one is a local minimizer and the second one is of mountain pass type. We also investigate the asymptotic behavior of solutions approaching the zero mass case, namely when $c\to 0^+$. △ Less

Submitted 14 July, 2024; originally announced July 2024.

Locally measure preserving property of bi-Lipschitz maps between Moran sets

Authors: Liang-yi Huang, Shishuang Liu

Abstract: In literature it is shown that bi-Lipschitz maps between self-similar sets or self-affine sets enjoy a locally measure preserving property, namely, if $f:(E,μ)\to (F,ν)$ is a bi-Lipschitz map, then the Radon-Nykodym derivative $df^*ν/dμ$ is a constant function on a subset $E'\subset E$ with $μ(E')>0$, where $f^*ν(\cdot)=ν(f(\cdot))$. Indeed, this measure preserving property plays an important role… ▽ More In literature it is shown that bi-Lipschitz maps between self-similar sets or self-affine sets enjoy a locally measure preserving property, namely, if $f:(E,μ)\to (F,ν)$ is a bi-Lipschitz map, then the Radon-Nykodym derivative $df^*ν/dμ$ is a constant function on a subset $E'\subset E$ with $μ(E')>0$, where $f^*ν(\cdot)=ν(f(\cdot))$. Indeed, this measure preserving property plays an important role in Lipschitz classification of fractal sets. In this paper, we show that such measure preserving property also holds for bi-Lipschitz maps between two Moran sets in a certain class. △ Less

Submitted 28 July, 2024; v1 submitted 14 July, 2024; originally announced July 2024.