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class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span class="pagination-ellipsis">…</span></li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.11415">arXiv:2502.11415</a> <span> [<a href="https://arxiv.org/pdf/2502.11415">pdf</a>, <a href="https://arxiv.org/ps/2502.11415">ps</a>, <a href="https://arxiv.org/format/2502.11415">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Weak Closed-loop Solvability for Discrete-time Linear-Quadratic Optimal Control </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Sun%2C+Y">Yue Sun</a>, <a href="/search/math?searchtype=author&query=Wu%2C+X">Xianping Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.11415v1-abstract-short" style="display: inline;"> In this paper, the open-loop, closed-loop, and weak closed-loop solvability for discrete-time linear-quadratic (LQ) control problem is considered due to the fact that it is always open-loop optimal solvable if the LQ control problem is closed-loop optimal solvable but not vice versa. The contributions are two-fold. On the one hand, the equivalent relationship between the closed-loop optimal solvab… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.11415v1-abstract-full').style.display = 'inline'; document.getElementById('2502.11415v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.11415v1-abstract-full" style="display: none;"> In this paper, the open-loop, closed-loop, and weak closed-loop solvability for discrete-time linear-quadratic (LQ) control problem is considered due to the fact that it is always open-loop optimal solvable if the LQ control problem is closed-loop optimal solvable but not vice versa. The contributions are two-fold. On the one hand, the equivalent relationship between the closed-loop optimal solvability and the solution of the generalized Riccati equation is given. On the other hand, when the system is merely open-loop solvable, we have found the equivalent existence form of the optimal solution by perturbation method, which is said to be a weak closed-loop solution. Moreover, it obtains that there is an open-loop optimal control with a linear feedback form of the state. The essential technique is to solve the forward and backward difference equations by iteration. An example sheds light on the theoretical results established. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.11415v1-abstract-full').style.display = 'none'; document.getElementById('2502.11415v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> org </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.11280">arXiv:2502.11280</a> <span> [<a href="https://arxiv.org/pdf/2502.11280">pdf</a>, <a href="https://arxiv.org/format/2502.11280">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Instrumentation and Methods for Astrophysics">astro-ph.IM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Single-Impulse Reachable Set in Arbitrary Dynamics Using Polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhou%2C+X">Xingyu Zhou</a>, <a href="/search/math?searchtype=author&query=Armellin%2C+R">Roberto Armellin</a>, <a href="/search/math?searchtype=author&query=Qiao%2C+D">Dong Qiao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiangyu Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.11280v1-abstract-short" style="display: inline;"> This paper presents a method to determine the reachable set (RS) of spacecraft after a single velocity impulse with an arbitrary direction, which is appropriate for the RS in both the state and observation spaces under arbitrary dynamics, extending the applications of current RS methods from two-body to arbitrary dynamics. First, the single-impulse RS model is generalized as a family of two-variab… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.11280v1-abstract-full').style.display = 'inline'; document.getElementById('2502.11280v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.11280v1-abstract-full" style="display: none;"> This paper presents a method to determine the reachable set (RS) of spacecraft after a single velocity impulse with an arbitrary direction, which is appropriate for the RS in both the state and observation spaces under arbitrary dynamics, extending the applications of current RS methods from two-body to arbitrary dynamics. First, the single-impulse RS model is generalized as a family of two-variable parameterized polynomials in the differential algebra scheme. Then, using the envelope theory, the boundary of RS is identified by solving the envelope equation. A framework is proposed to reduce the complexity of solving the envelope equation by converting it to the problem of searching the root of a one-variable polynomial. Moreover, a high-order local polynomial approximation for the RS envelope is derived to improve computational efficiency. The method successfully determines the RSs of two near-rectilinear halo orbits in the cislunar space. Simulation results show that the RSs in both state and observation spaces can be accurately approximated under the three-body dynamics, with relative errors of less than 0.0658%. In addition, using the local polynomial approximation, the computational time for solving the envelope equation is reduced by more than 84%. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.11280v1-abstract-full').style.display = 'none'; document.getElementById('2502.11280v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.11038">arXiv:2502.11038</a> <span> [<a href="https://arxiv.org/pdf/2502.11038">pdf</a>, <a href="https://arxiv.org/format/2502.11038">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> A robust and p-hacking-proof significance test under variance uncertainty </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xifeng Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+S">Shuzhen Yang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+J">Jianfeng Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.11038v1-abstract-short" style="display: inline;"> P-hacking poses challenges to traditional hypothesis testing. In this paper, we propose a robust method for the one-sample significance test that can protect against p-hacking from sample manipulation. Precisely, assuming a sequential arrival of the data whose variance can be time-varying and for which only lower and upper bounds are assumed to exist with possibly unknown values, we use the modern… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.11038v1-abstract-full').style.display = 'inline'; document.getElementById('2502.11038v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.11038v1-abstract-full" style="display: none;"> P-hacking poses challenges to traditional hypothesis testing. In this paper, we propose a robust method for the one-sample significance test that can protect against p-hacking from sample manipulation. Precisely, assuming a sequential arrival of the data whose variance can be time-varying and for which only lower and upper bounds are assumed to exist with possibly unknown values, we use the modern theory of sublinear expectation to build a testing procedure which is robust under such variance uncertainty, and can protect the significance level against potential data manipulation by an experimenter. It is shown that our new method can effectively control the type I error while preserving a satisfactory power, yet a traditional rejection criterion performs poorly under such variance uncertainty. Our theoretical results are well confirmed by a detailed simulation study. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.11038v1-abstract-full').style.display = 'none'; document.getElementById('2502.11038v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.10782">arXiv:2502.10782</a> <span> [<a href="https://arxiv.org/pdf/2502.10782">pdf</a>, <a href="https://arxiv.org/format/2502.10782">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Propagation of chaos and Razumikhin theorem for the nonlinear McKean-Vlasov SFDEs with common noise </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+X">Xing Chen</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoyue Li</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+C">Chenggui Yuan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.10782v1-abstract-short" style="display: inline;"> As the limit equations of mean-field particle systems perturbed by common environmental noise, the McKean-Vlasov stochastic differential equations with common noise have received a lot of attention. Moreover, past dependence is an unavoidable natural phenomenon for dynamic systems in life sciences, economics, finance, automatic control, and other fields. Combining the two aspects above, this paper… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.10782v1-abstract-full').style.display = 'inline'; document.getElementById('2502.10782v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.10782v1-abstract-full" style="display: none;"> As the limit equations of mean-field particle systems perturbed by common environmental noise, the McKean-Vlasov stochastic differential equations with common noise have received a lot of attention. Moreover, past dependence is an unavoidable natural phenomenon for dynamic systems in life sciences, economics, finance, automatic control, and other fields. Combining the two aspects above, this paper delves into a class of nonlinear McKean-Vlasov stochastic functional differential equations (MV-SFDEs) with common noise. The well-posedness of the nonlinear MV-SFDEs with common noise is first demonstrated through the application of the Banach fixed-point theorem. Secondly, the relationship between the MV-SFDEs with common noise and the corresponding functional particle systems is investigated. More precisely, the conditional propagation of chaos with an explicit convergence rate and the stability equivalence are studied. Furthermore, the exponential stability, an important long-time behavior of the nonlinear MV-SFDEs with common noise, is derived. To this end, the It么 formula involved with state and measure is developed for the MV-SFDEs with common noise. Using this formula, the Razumikhin theorem is proved, providing an easy-to-implement criterion for the exponential stability. Lastly, an example is provided to illustrate the result of the stability. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.10782v1-abstract-full').style.display = 'none'; document.getElementById('2502.10782v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.08799">arXiv:2502.08799</a> <span> [<a href="https://arxiv.org/pdf/2502.08799">pdf</a>, <a href="https://arxiv.org/ps/2502.08799">ps</a>, <a href="https://arxiv.org/format/2502.08799">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Strong completeness of SDEs and non-explosion for RDEs whose coefficients have unbounded derivatives </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xue-Mei Li</a>, <a href="/search/math?searchtype=author&query=Ying%2C+K">Kexing Ying</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.08799v1-abstract-short" style="display: inline;"> We establish a non-explosion result for rough differential equations (RDEs) in which both the noise and drift coefficients together with their derivatives are allowed to grow at infinity. Additionally, we prove the existence of a bi-continuous solution flow for stochastic differential equations (SDEs). In the case of RDEs with additive noise, we show that our result is optimal by providing a count… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.08799v1-abstract-full').style.display = 'inline'; document.getElementById('2502.08799v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.08799v1-abstract-full" style="display: none;"> We establish a non-explosion result for rough differential equations (RDEs) in which both the noise and drift coefficients together with their derivatives are allowed to grow at infinity. Additionally, we prove the existence of a bi-continuous solution flow for stochastic differential equations (SDEs). In the case of RDEs with additive noise, we show that our result is optimal by providing a counterexample. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.08799v1-abstract-full').style.display = 'none'; document.getElementById('2502.08799v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.04276">arXiv:2502.04276</a> <span> [<a href="https://arxiv.org/pdf/2502.04276">pdf</a>, <a href="https://arxiv.org/format/2502.04276">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> Gaussian Process Regression for Inverse Problems in Linear PDEs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a>, <a href="/search/math?searchtype=author&query=Lange-Hegermann%2C+M">Markus Lange-Hegermann</a>, <a href="/search/math?searchtype=author&query=Rai%C5%A3%C4%83%2C+B">Bogdan Rai牛膬</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.04276v1-abstract-short" style="display: inline;"> This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer alg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.04276v1-abstract-full').style.display = 'inline'; document.getElementById('2502.04276v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.04276v1-abstract-full" style="display: none;"> This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer algebra software. An example application includes identifying the wave speed from noisy data for classical wave equations, which are widely used in physics. The method achieves high accuracy while enhancing computational efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.04276v1-abstract-full').style.display = 'none'; document.getElementById('2502.04276v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.04155">arXiv:2502.04155</a> <span> [<a href="https://arxiv.org/pdf/2502.04155">pdf</a>, <a href="https://arxiv.org/format/2502.04155">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computers and Society">cs.CY</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> User-Friendly Game-Theoretic Modeling and Analysis of Multi-Modal Transportation Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zambrano%2C+M">Margarita Zambrano</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinling Li</a>, <a href="/search/math?searchtype=author&query=Fiorista%2C+R">Riccardo Fiorista</a>, <a href="/search/math?searchtype=author&query=Zardini%2C+G">Gioele Zardini</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.04155v1-abstract-short" style="display: inline;"> The evolution of existing transportation systems, mainly driven by urbanization and increased availability of mobility options, such as private, profit-maximizing ride-hailing companies, calls for tools to reason about their design and regulation. To study this complex socio-technical problem, one needs to account for the strategic interactions of the stakeholders involved in the mobility ecosyste… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.04155v1-abstract-full').style.display = 'inline'; document.getElementById('2502.04155v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.04155v1-abstract-full" style="display: none;"> The evolution of existing transportation systems, mainly driven by urbanization and increased availability of mobility options, such as private, profit-maximizing ride-hailing companies, calls for tools to reason about their design and regulation. To study this complex socio-technical problem, one needs to account for the strategic interactions of the stakeholders involved in the mobility ecosystem. In this paper, we present a game-theoretic framework to model multi-modal mobility systems, focusing on municipalities, service providers, and travelers. Through a user-friendly, Graphical User Interface, one can visualize system dynamics and compute equilibria for various scenarios. The framework enables stakeholders to assess the impact of local decisions (e.g., fleet size for services or taxes for private companies) on the full mobility system. Furthermore, this project aims to foster STEM interest among high school students (e.g., in the context of prior activities in Switzerland, and planned activities with the MIT museum). This initiative combines theoretical advancements, practical applications, and educational outreach to improve mobility system design. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.04155v1-abstract-full').style.display = 'none'; document.getElementById('2502.04155v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.00788">arXiv:2502.00788</a> <span> [<a href="https://arxiv.org/pdf/2502.00788">pdf</a>, <a href="https://arxiv.org/ps/2502.00788">ps</a>, <a href="https://arxiv.org/format/2502.00788">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Explicit positivity preserving numerical method for linear stochastic volatility models driven by $伪$-stable process </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaotong Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+W">Wei Liu</a>, <a href="/search/math?searchtype=author&query=Mao%2C+X">Xuerong Mao</a>, <a href="/search/math?searchtype=author&query=Tian%2C+H">Hongjiong Tian</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yue Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.00788v1-abstract-short" style="display: inline;"> In this paper, we introduce a linear stochastic volatility model driven by $伪$-stable processes, which admits a unique positive solution. To preserve positivity, we modify the classical forward Euler-Maruyama scheme and analyze its numerical properties. The scheme achieves a strong convergence order of $1/伪$. Numerical simulations are presented at the end to verify theoretical results. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.00788v1-abstract-full" style="display: none;"> In this paper, we introduce a linear stochastic volatility model driven by $伪$-stable processes, which admits a unique positive solution. To preserve positivity, we modify the classical forward Euler-Maruyama scheme and analyze its numerical properties. The scheme achieves a strong convergence order of $1/伪$. Numerical simulations are presented at the end to verify theoretical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.00788v1-abstract-full').style.display = 'none'; document.getElementById('2502.00788v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H35; 65C30; 60J76 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.16096">arXiv:2501.16096</a> <span> [<a href="https://arxiv.org/pdf/2501.16096">pdf</a>, <a href="https://arxiv.org/format/2501.16096">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A New Approach for Fourier Extension Based on Weighted Generalized Inverse </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhao%2C+Z">Zhenyu Zhao</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yanfei Wang</a>, <a href="/search/math?searchtype=author&query=Yagola%2C+A+G">Anatoly G. Yagola</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xusheng Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.16096v2-abstract-short" style="display: inline;"> This paper examines the Fourier extension from a new perspective of solving the compact operator equation with perturbed data. By converting the approximation target from the best approximate solution to the weighted best approximate solution, the oscillation in the extended region has been overcome. The error estimation of the solution is theoretically established. Furthermore, we point out the d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.16096v2-abstract-full').style.display = 'inline'; document.getElementById('2501.16096v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.16096v2-abstract-full" style="display: none;"> This paper examines the Fourier extension from a new perspective of solving the compact operator equation with perturbed data. By converting the approximation target from the best approximate solution to the weighted best approximate solution, the oscillation in the extended region has been overcome. The error estimation of the solution is theoretically established. Furthermore, we point out the difficulties faced by the original weighted operator in calculation due to the limitation of machine precision and propose an effective correction operator. The relevant parameters involved in the method are further tested, and finally the effectiveness of the method is verified through numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.16096v2-abstract-full').style.display = 'none'; document.getElementById('2501.16096v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 42A10; 65T40; 65T50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.12864">arXiv:2501.12864</a> <span> [<a href="https://arxiv.org/pdf/2501.12864">pdf</a>, <a href="https://arxiv.org/ps/2501.12864">ps</a>, <a href="https://arxiv.org/format/2501.12864">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> The $r$-chain minimal and maximal excludant sizes of an overpartition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+T+Y">Thomas Y. He</a>, <a href="/search/math?searchtype=author&query=Huang%2C+C+S">C. S. Huang</a>, <a href="/search/math?searchtype=author&query=Li%2C+H+X">H. X. Li</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+X">X. Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.12864v1-abstract-short" style="display: inline;"> Recently, Andrews and Newman studied the minimal excludant of a partition, which is the smallest positive integer that is not a part of a partition. Chern introduced the maximal excludant of a partition, which is the largest nonnegative integer smaller than the largest part that is not a part of a partition. Bhoria, Eyyunnib, Maji and Li investigated the $r$-chain minimal and maximal excludants of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.12864v1-abstract-full').style.display = 'inline'; document.getElementById('2501.12864v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.12864v1-abstract-full" style="display: none;"> Recently, Andrews and Newman studied the minimal excludant of a partition, which is the smallest positive integer that is not a part of a partition. Chern introduced the maximal excludant of a partition, which is the largest nonnegative integer smaller than the largest part that is not a part of a partition. Bhoria, Eyyunnib, Maji and Li investigated the $r$-chain minimal and maximal excludants of a partition. In this article, we consider the $r$-chain minimal and maximal excludant sizes of an overpartition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.12864v1-abstract-full').style.display = 'none'; document.getElementById('2501.12864v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.10670">arXiv:2501.10670</a> <span> [<a href="https://arxiv.org/pdf/2501.10670">pdf</a>, <a href="https://arxiv.org/format/2501.10670">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Computing Capacity-Cost Functions for Continuous Channels in Wasserstein Space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyang Li</a>, <a href="/search/math?searchtype=author&query=Andrei%2C+V+C">Vlad C. Andrei</a>, <a href="/search/math?searchtype=author&query=M%C3%B6nich%2C+U+J">Ullrich J. M枚nich</a>, <a href="/search/math?searchtype=author&query=Liu%2C+F">Fan Liu</a>, <a href="/search/math?searchtype=author&query=Boche%2C+H">Holger Boche</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.10670v2-abstract-short" style="display: inline;"> This paper investigates the problem of computing capacity-cost (C-C) functions for continuous channels. Motivated by the Kullback-Leibler divergence (KLD) proximal reformulation of the classical Blahut-Arimoto (BA) algorithm, the Wasserstein distance is introduced to the proximal term for the continuous case, resulting in an iterative algorithm related to the Wasserstein gradient descent. Practica… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.10670v2-abstract-full').style.display = 'inline'; document.getElementById('2501.10670v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.10670v2-abstract-full" style="display: none;"> This paper investigates the problem of computing capacity-cost (C-C) functions for continuous channels. Motivated by the Kullback-Leibler divergence (KLD) proximal reformulation of the classical Blahut-Arimoto (BA) algorithm, the Wasserstein distance is introduced to the proximal term for the continuous case, resulting in an iterative algorithm related to the Wasserstein gradient descent. Practical implementation involves moving particles along the negative gradient direction of the objective function's first variation in the Wasserstein space and approximating integrals by the importance sampling (IS) technique. Such formulation is also applied to the rate-distortion (R-D) function for continuous source spaces and thus provides a unified computation framework for both problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.10670v2-abstract-full').style.display = 'none'; document.getElementById('2501.10670v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted to IEEE International Conference on Communications 2025</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.06147">arXiv:2501.06147</a> <span> [<a href="https://arxiv.org/pdf/2501.06147">pdf</a>, <a href="https://arxiv.org/ps/2501.06147">ps</a>, <a href="https://arxiv.org/format/2501.06147">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Uniform local well-posedness and liviscid limit for the KdV-Burgers equation on $\mathbb{T}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xintong Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.06147v1-abstract-short" style="display: inline;"> This article investigates the uniform well-posedness and inviscid limit problem for the Korteweg-de Vries-Burgers equation on a torus, we consider the KdV-Burgers equation $$\partial_t u(t,x)+\partial_x^3u(t,x)-\varepsilon\partial_x^2u(t,x)=\dfrac{1}{2}\partial_x(u(t,x))^2, \quad u(0)=蠁, $$ where $\varepsilon\in(0, 1]$ represents the diffusion coefficient, and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.06147v1-abstract-full').style.display = 'inline'; document.getElementById('2501.06147v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.06147v1-abstract-full" style="display: none;"> This article investigates the uniform well-posedness and inviscid limit problem for the Korteweg-de Vries-Burgers equation on a torus, we consider the KdV-Burgers equation $$\partial_t u(t,x)+\partial_x^3u(t,x)-\varepsilon\partial_x^2u(t,x)=\dfrac{1}{2}\partial_x(u(t,x))^2, \quad u(0)=蠁, $$ where $\varepsilon\in(0, 1]$ represents the diffusion coefficient, and $u(t,x):\mathbb{R}^{+}\times\mathbb{T}\rightarrow \mathbb{R}$ is a real-valued function, we show that it is uniformly local well-posed in $H^s$ with $s\geq 0$ for all $\varepsilon\in[0,1]$. Moreover, we prove that there exists some $T>0$ such that for any $s\geq 0$, its solution converges in $C([0,T];H^s)$ to that of the KdV equation if $\varepsilon$ tends to $0$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.06147v1-abstract-full').style.display = 'none'; document.getElementById('2501.06147v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.05145">arXiv:2501.05145</a> <span> [<a href="https://arxiv.org/pdf/2501.05145">pdf</a>, <a href="https://arxiv.org/ps/2501.05145">ps</a>, <a href="https://arxiv.org/format/2501.05145">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On Maximum Induced Forests of the Balanced Bipartite Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ghalavand%2C+A">Ali Ghalavand</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xueliang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.05145v1-abstract-short" style="display: inline;"> We are examining a specific type of graph called a balanced bipartite graph. The balanced bipartite graph, such as $\mathcal{B}$, has two parts, $V_1$ and $V_2$, each containing $n$ vertices, for a total of $2n$ vertices. The degree of a vertex $v$ in $V_1\cup\,V_2$ is denoted by $d_\mathcal{B}(v)$. The minimum degree of any vertex in the graph $\mathcal{B}$ is represented by $未(\mathcal{B})$. If… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.05145v1-abstract-full').style.display = 'inline'; document.getElementById('2501.05145v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.05145v1-abstract-full" style="display: none;"> We are examining a specific type of graph called a balanced bipartite graph. The balanced bipartite graph, such as $\mathcal{B}$, has two parts, $V_1$ and $V_2$, each containing $n$ vertices, for a total of $2n$ vertices. The degree of a vertex $v$ in $V_1\cup\,V_2$ is denoted by $d_\mathcal{B}(v)$. The minimum degree of any vertex in the graph $\mathcal{B}$ is represented by $未(\mathcal{B})$. If $S$ is a subset of $V_1\cup\,V_2$, then the subgraph of $\mathcal{B}$ induced by $S$ is the graph that has $S$ as its vertex set and contains all the edges of $\mathcal{B}$ that have both endpoints in $S$. This subgraph is denoted by $\mathcal{B}[S]$. The forest number of a graph $\mathcal{B}$ is the size of the largest subset of vertices of $\mathcal{B}$ that form an induced forest. We use $f(\mathcal{B})$ to represent the forest number of graph $\mathcal{B}$. A decycling set or a feedback vertex set of a graph is a set of vertices whose removal results in a forest. The smallest possible size of a decycling set of $\mathcal{B}$ is represented by $\nabla(\mathcal{B})$. Finding the decycling number of $\mathcal{B}$ is equivalent to determining the largest order of an induced forest, i.e., $f(\mathcal{B})+\nabla(\mathcal{B})=2n$. In this essay, we study the structure and cardinality of the largest subsets of vertices of graph $\mathcal{B}$ that form induced forests. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.05145v1-abstract-full').style.display = 'none'; document.getElementById('2501.05145v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.04603">arXiv:2501.04603</a> <span> [<a href="https://arxiv.org/pdf/2501.04603">pdf</a>, <a href="https://arxiv.org/ps/2501.04603">ps</a>, <a href="https://arxiv.org/format/2501.04603">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Infinite Horizon Fully Coupled Nonlinear Forward-Backward Stochastic Difference Equations and their Application to LQ Optimal Control Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+X">Xinyu Ma</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+Q">Qingxin Meng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.04603v1-abstract-short" style="display: inline;"> This paper focuses on the study of infinite horizon fully coupled nonlinear forward-backward stochastic difference equations (FBS$\bigtriangleup$Es). Firstly, we establish a pair of priori estimates for the solutions to forward stochastic difference equations (FS$\bigtriangleup$Es) and backward stochastic difference equations (BS$\bigtriangleup$Es) respectively. Then, to achieve broader applicabil… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04603v1-abstract-full').style.display = 'inline'; document.getElementById('2501.04603v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.04603v1-abstract-full" style="display: none;"> This paper focuses on the study of infinite horizon fully coupled nonlinear forward-backward stochastic difference equations (FBS$\bigtriangleup$Es). Firstly, we establish a pair of priori estimates for the solutions to forward stochastic difference equations (FS$\bigtriangleup$Es) and backward stochastic difference equations (BS$\bigtriangleup$Es) respectively. Then, to achieve broader applicability, we utilize a set of domination-monotonicity conditions which are more lenient than general ones. Using these conditions, we apply continuation methods to prove the unique solvability of infinite horizon fully coupled FBS$\bigtriangleup$Es and derive a set of solution estimates. Furthermore, our results have considerable implications for a variety of related linear quadratic (LQ) problems, especially when the stochastic Hamiltonian system is consistent with FBS$\bigtriangleup$Es satisfying these introduced domination-monotonicity conditions. Thus, by solving the associated stochastic Hamiltonian system, we can derive an explicit expression for the unique optimal control. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04603v1-abstract-full').style.display = 'none'; document.getElementById('2501.04603v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:2410.01749</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.04245">arXiv:2501.04245</a> <span> [<a href="https://arxiv.org/pdf/2501.04245">pdf</a>, <a href="https://arxiv.org/ps/2501.04245">ps</a>, <a href="https://arxiv.org/format/2501.04245">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> A symmetric function approach to log-concavity of independence polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+E+Y+H">Ethan Y. H. Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+G+M+X">Grace M. X. Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+A+L+B">Arthur L. B. Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zhong-Xue Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.04245v1-abstract-short" style="display: inline;"> As introduced by Gutman and Harary, the independence polynomial of a graph serves as the generating polynomial of its independent sets. In 1987, Alavi, Malde, Schwenk and Erd艖s conjectured that the independence polynomials of all trees are unimodal. In this paper we come up with a new way for proving log-concavity of independence polynomials of graphs by means of their chromatic symmetric function… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04245v1-abstract-full').style.display = 'inline'; document.getElementById('2501.04245v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.04245v1-abstract-full" style="display: none;"> As introduced by Gutman and Harary, the independence polynomial of a graph serves as the generating polynomial of its independent sets. In 1987, Alavi, Malde, Schwenk and Erd艖s conjectured that the independence polynomials of all trees are unimodal. In this paper we come up with a new way for proving log-concavity of independence polynomials of graphs by means of their chromatic symmetric functions, which is inspired by a result of Stanley connecting properties of polynomials to positivity of symmetric functions. This method turns out to be more suitable for treating trees with irregular structures, and as a simple application we show that all spiders have log-concave independence polynomials, which provides more evidence for the above conjecture. Moreover, we present two symmetric function analogues of a basic recurrence formula for independence polynomials, and show that all pineapple graphs also have log-concave independence polynomials. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04245v1-abstract-full').style.display = 'none'; document.getElementById('2501.04245v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C69; 05E05; 05C05; 05C15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.00726">arXiv:2501.00726</a> <span> [<a href="https://arxiv.org/pdf/2501.00726">pdf</a>, <a href="https://arxiv.org/format/2501.00726">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Enhancing Unsupervised Feature Selection via Double Sparsity Constrained Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xiu%2C+X">Xianchao Xiu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+A">Anning Yang</a>, <a href="/search/math?searchtype=author&query=Huang%2C+C">Chenyi Huang</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinrong Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+W">Wanquan Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.00726v1-abstract-short" style="display: inline;"> Unsupervised feature selection (UFS) is widely applied in machine learning and pattern recognition. However, most of the existing methods only consider a single sparsity, which makes it difficult to select valuable and discriminative feature subsets from the original high-dimensional feature set. In this paper, we propose a new UFS method called DSCOFS via embedding double sparsity constrained opt… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.00726v1-abstract-full').style.display = 'inline'; document.getElementById('2501.00726v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.00726v1-abstract-full" style="display: none;"> Unsupervised feature selection (UFS) is widely applied in machine learning and pattern recognition. However, most of the existing methods only consider a single sparsity, which makes it difficult to select valuable and discriminative feature subsets from the original high-dimensional feature set. In this paper, we propose a new UFS method called DSCOFS via embedding double sparsity constrained optimization into the classical principal component analysis (PCA) framework. Double sparsity refers to using $\ell_{2,0}$-norm and $\ell_0$-norm to simultaneously constrain variables, by adding the sparsity of different types, to achieve the purpose of improving the accuracy of identifying differential features. The core is that $\ell_{2,0}$-norm can remove irrelevant and redundant features, while $\ell_0$-norm can filter out irregular noisy features, thereby complementing $\ell_{2,0}$-norm to improve discrimination. An effective proximal alternating minimization method is proposed to solve the resulting nonconvex nonsmooth model. Theoretically, we rigorously prove that the sequence generated by our method globally converges to a stationary point. Numerical experiments on three synthetic datasets and eight real-world datasets demonstrate the effectiveness, stability, and convergence of the proposed method. In particular, the average clustering accuracy (ACC) and normalized mutual information (NMI) are improved by at least 3.34% and 3.02%, respectively, compared with the state-of-the-art methods. More importantly, two common statistical tests and a new feature similarity metric verify the advantages of double sparsity. All results suggest that our proposed DSCOFS provides a new perspective for feature selection. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.00726v1-abstract-full').style.display = 'none'; document.getElementById('2501.00726v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.00471">arXiv:2501.00471</a> <span> [<a href="https://arxiv.org/pdf/2501.00471">pdf</a>, <a href="https://arxiv.org/format/2501.00471">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Alternating minimization for square root principal component pursuit </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Deng%2C+S">Shengxiang Deng</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xudong Li</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yangjing Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.00471v1-abstract-short" style="display: inline;"> Recently, the square root principal component pursuit (SRPCP) model has garnered significant research interest. It is shown in the literature that the SRPCP model guarantees robust matrix recovery with a universal, constant penalty parameter. While its statistical advantages are well-documented, the computational aspects from an optimization perspective remain largely unexplored. In this paper, we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.00471v1-abstract-full').style.display = 'inline'; document.getElementById('2501.00471v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.00471v1-abstract-full" style="display: none;"> Recently, the square root principal component pursuit (SRPCP) model has garnered significant research interest. It is shown in the literature that the SRPCP model guarantees robust matrix recovery with a universal, constant penalty parameter. While its statistical advantages are well-documented, the computational aspects from an optimization perspective remain largely unexplored. In this paper, we focus on developing efficient optimization algorithms for solving the SRPCP problem. Specifically, we propose a tuning-free alternating minimization (AltMin) algorithm, where each iteration involves subproblems enjoying closed-form optimal solutions. Additionally, we introduce techniques based on the variational formulation of the nuclear norm and Burer-Monteiro decomposition to further accelerate the AltMin method. Extensive numerical experiments confirm the efficiency and robustness of our algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.00471v1-abstract-full').style.display = 'none'; document.getElementById('2501.00471v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">32 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C06; 90C25; 90C26; 90C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.20703">arXiv:2412.20703</a> <span> [<a href="https://arxiv.org/pdf/2412.20703">pdf</a>, <a href="https://arxiv.org/format/2412.20703">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> The Restricted Inverse Optimal Value Problem under Weighted Bottle-neck Hamming distance on trees </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+Q">Qiao Zhang</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiao Li</a>, <a href="/search/math?searchtype=author&query=Guan%2C+X">Xiucui Guan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.20703v2-abstract-short" style="display: inline;"> We consider the Restricted Inverse Optimal Value Problem (RIOVSP) on trees under weighted bottleneck Hamming distance, denoted as (RIOVSPT$_{BH}$). The problem aims to minimize the total cost under weighted bottle-neck Hamming distance such that the length of the shortest root-leaf path of the tree is lower-bounded by a given value by adjusting the length of some edges. Additionally, the specified… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.20703v2-abstract-full').style.display = 'inline'; document.getElementById('2412.20703v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.20703v2-abstract-full" style="display: none;"> We consider the Restricted Inverse Optimal Value Problem (RIOVSP) on trees under weighted bottleneck Hamming distance, denoted as (RIOVSPT$_{BH}$). The problem aims to minimize the total cost under weighted bottle-neck Hamming distance such that the length of the shortest root-leaf path of the tree is lower-bounded by a given value by adjusting the length of some edges. Additionally, the specified lower bound must correspond to the length of a particular root-leaf path. Through careful analysis of the problem's structural properties, we develop an algorithm with $O(n\log n)$ time complexity to solve (RIOVSPT$_{BH}$). Furthermore, by removing the path-length constraint, we derive the Minimum Cost Shortest Path Interdiction Problem on Trees (MCSPIT), for which we present an $O(n\log n)$ time algorithm that operates under weighted bottleneck Hamming distance. Extensive computational experiments demonstrate the efficiency and effectiveness of both algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.20703v2-abstract-full').style.display = 'none'; document.getElementById('2412.20703v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.19066">arXiv:2412.19066</a> <span> [<a href="https://arxiv.org/pdf/2412.19066">pdf</a>, <a href="https://arxiv.org/format/2412.19066">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> FFCG: Effective and Fast Family Column Generation for Solving Large-Scale Linear Program </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yi-Xiang Hu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+F">Feng Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+S">Shaoang Li</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yifang Zhao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang-Yang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.19066v1-abstract-short" style="display: inline;"> Column Generation (CG) is an effective and iterative algorithm to solve large-scale linear programs (LP). During each CG iteration, new columns are added to improve the solution of the LP. Typically, CG greedily selects one column with the most negative reduced cost, which can be improved by adding more columns at once. However, selecting all columns with negative reduced costs would lead to the a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.19066v1-abstract-full').style.display = 'inline'; document.getElementById('2412.19066v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.19066v1-abstract-full" style="display: none;"> Column Generation (CG) is an effective and iterative algorithm to solve large-scale linear programs (LP). During each CG iteration, new columns are added to improve the solution of the LP. Typically, CG greedily selects one column with the most negative reduced cost, which can be improved by adding more columns at once. However, selecting all columns with negative reduced costs would lead to the addition of redundant columns that do not improve the objective value. Therefore, selecting the appropriate columns to add is still an open problem and previous machine-learning-based approaches for CG only add a constant quantity of columns per iteration due to the state-space explosion problem. To address this, we propose Fast Family Column Generation (FFCG) -- a novel reinforcement-learning-based CG that selects a variable number of columns as needed in an iteration. Specifically, we formulate the column selection problem in CG as an MDP and design a reward metric that balances both the convergence speed and the number of redundant columns. In our experiments, FFCG converges faster on the common benchmarks and reduces the number of CG iterations by 77.1% for Cutting Stock Problem (CSP) and 84.8% for Vehicle Routing Problem with Time Windows (VRPTW), and a 71.4% reduction in computing time for CSP and 84.0% for VRPTW on average compared to several state-of-the-art baselines. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.19066v1-abstract-full').style.display = 'none'; document.getElementById('2412.19066v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> I.2.6 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.19008">arXiv:2412.19008</a> <span> [<a href="https://arxiv.org/pdf/2412.19008">pdf</a>, <a href="https://arxiv.org/ps/2412.19008">ps</a>, <a href="https://arxiv.org/format/2412.19008">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> On the minimal parabolic induction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyu Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.19008v1-abstract-short" style="display: inline;"> Motivated by Beilinson-Bernstein's proof of the Jantzen conjectures, we define the minimal parabolic induction functor for Kac-Moody algebras, and establish some basic properties. As applications of the formal theory, we examine first extension groups between simple highest weight modules in the category of weight modules, and analyze the annihilators of some simple highest weight modules. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.19008v1-abstract-full" style="display: none;"> Motivated by Beilinson-Bernstein's proof of the Jantzen conjectures, we define the minimal parabolic induction functor for Kac-Moody algebras, and establish some basic properties. As applications of the formal theory, we examine first extension groups between simple highest weight modules in the category of weight modules, and analyze the annihilators of some simple highest weight modules. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.19008v1-abstract-full').style.display = 'none'; document.getElementById('2412.19008v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.17107">arXiv:2412.17107</a> <span> [<a href="https://arxiv.org/pdf/2412.17107">pdf</a>, <a href="https://arxiv.org/format/2412.17107">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Grams: Gradient Descent with Adaptive Momentum Scaling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cao%2C+Y">Yang Cao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoyu Li</a>, <a href="/search/math?searchtype=author&query=Song%2C+Z">Zhao Song</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.17107v1-abstract-short" style="display: inline;"> We introduce \textbf{Gr}adient Descent with \textbf{A}daptive \textbf{M}omentum \textbf{S}caling (\textbf{Grams}), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is u… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.17107v1-abstract-full').style.display = 'inline'; document.getElementById('2412.17107v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.17107v1-abstract-full" style="display: none;"> We introduce \textbf{Gr}adient Descent with \textbf{A}daptive \textbf{M}omentum \textbf{S}caling (\textbf{Grams}), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.17107v1-abstract-full').style.display = 'none'; document.getElementById('2412.17107v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.17074">arXiv:2412.17074</a> <span> [<a href="https://arxiv.org/pdf/2412.17074">pdf</a>, <a href="https://arxiv.org/ps/2412.17074">ps</a>, <a href="https://arxiv.org/format/2412.17074">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Interplay between the local metric dimension and the clique number of a graph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ghalavand%2C+A">Ali Ghalavand</a>, <a href="/search/math?searchtype=author&query=Klav%C5%BEar%2C+S">Sandi Klav啪ar</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xueliang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.17074v1-abstract-short" style="display: inline;"> The local metric dimension ${\rm dim}_l$ in relation to the clique number $蠅$ is investigated. It is proved that if $蠅(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the graphs $G$ with ${\rm dim}_l(G) = n(G)-3$ are listed (with no condition on the clique number). It is proved that if $蠅(G)=n(G)-2$, then… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.17074v1-abstract-full').style.display = 'inline'; document.getElementById('2412.17074v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.17074v1-abstract-full" style="display: none;"> The local metric dimension ${\rm dim}_l$ in relation to the clique number $蠅$ is investigated. It is proved that if $蠅(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the graphs $G$ with ${\rm dim}_l(G) = n(G)-3$ are listed (with no condition on the clique number). It is proved that if $蠅(G)=n(G)-2$, then $n(G)-4 \leq {\rm dim}_l(G)\leq n(G)-3$, and all graphs are divided into two groups depending on which of the options applies. The conjecture asserting that for any graph $G$ we have ${\rm dim}_l(G) \leq \left[(蠅(G)-2)/(蠅(G)-1)\right] \cdot n(G)$ is proved for all graphs with $蠅(G)\in\{n(G)-1,n(G)-2,n(G)-3\}$. A negative answer is given for the problem whether every planar graph fulfills the inequality ${\rm dim}_l(G) \leq \lceil (n(G)+1)/2 \rceil$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.17074v1-abstract-full').style.display = 'none'; document.getElementById('2412.17074v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.17070">arXiv:2412.17070</a> <span> [<a href="https://arxiv.org/pdf/2412.17070">pdf</a>, <a href="https://arxiv.org/ps/2412.17070">ps</a>, <a href="https://arxiv.org/format/2412.17070">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Decoupled Functional Central Limit Theorems for Two-Time-Scale Stochastic Approximation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+Y">Yuze Han</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang Li</a>, <a href="/search/math?searchtype=author&query=Liang%2C+J">Jiadong Liang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zhihua Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.17070v3-abstract-short" style="display: inline;"> In two-time-scale stochastic approximation (SA), two iterates are updated at different rates, governed by distinct step sizes, with each update influencing the other. Previous studies have demonstrated that the convergence rates of the error terms for these updates depend solely on their respective step sizes, a property known as decoupled convergence. However, a functional version of this decoupl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.17070v3-abstract-full').style.display = 'inline'; document.getElementById('2412.17070v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.17070v3-abstract-full" style="display: none;"> In two-time-scale stochastic approximation (SA), two iterates are updated at different rates, governed by distinct step sizes, with each update influencing the other. Previous studies have demonstrated that the convergence rates of the error terms for these updates depend solely on their respective step sizes, a property known as decoupled convergence. However, a functional version of this decoupled convergence has not been explored. Our work fills this gap by establishing decoupled functional central limit theorems for two-time-scale SA, offering a more precise characterization of its asymptotic behavior. To achieve these results, we leverage the martingale problem approach and establish tightness as a crucial intermediate step. Furthermore, to address the interdependence between different time scales, we introduce an innovative auxiliary sequence to eliminate the primary influence of the fast-time-scale update on the slow-time-scale update. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.17070v3-abstract-full').style.display = 'none'; document.getElementById('2412.17070v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.16263">arXiv:2412.16263</a> <span> [<a href="https://arxiv.org/pdf/2412.16263">pdf</a>, <a href="https://arxiv.org/ps/2412.16263">ps</a>, <a href="https://arxiv.org/format/2412.16263">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Low-rank matrix recovery via nonconvex optimization methods with application to errors-in-variables matrix regression </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a>, <a href="/search/math?searchtype=author&query=Wu%2C+D">Dongya Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.16263v1-abstract-short" style="display: inline;"> We consider the nonconvex regularized method for low-rank matrix recovery. Under the assumption on the singular values of the parameter matrix, we provide the recovery bound for any stationary point of the nonconvex method by virtue of regularity conditions on the nonconvex loss function and the regularizer. This recovery bound can be much tighter than that of the convex nuclear norm regularized m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.16263v1-abstract-full').style.display = 'inline'; document.getElementById('2412.16263v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.16263v1-abstract-full" style="display: none;"> We consider the nonconvex regularized method for low-rank matrix recovery. Under the assumption on the singular values of the parameter matrix, we provide the recovery bound for any stationary point of the nonconvex method by virtue of regularity conditions on the nonconvex loss function and the regularizer. This recovery bound can be much tighter than that of the convex nuclear norm regularized method when some of the singular values are larger than a threshold defined by the nonconvex regularizer. In addition, we consider the errors-in-variables matrix regression as an application of the nonconvex optimization method. Probabilistic consequences and the advantage of the nonoconvex method are demonstrated through verifying the regularity conditions for specific models with additive noise and missing data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.16263v1-abstract-full').style.display = 'none'; document.getElementById('2412.16263v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.15880">arXiv:2412.15880</a> <span> [<a href="https://arxiv.org/pdf/2412.15880">pdf</a>, <a href="https://arxiv.org/ps/2412.15880">ps</a>, <a href="https://arxiv.org/format/2412.15880">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang Li</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yong Luo</a>, <a href="/search/math?searchtype=author&query=Sun%2C+J">Jun Sun</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.15880v1-abstract-short" style="display: inline;"> In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.15880v1-abstract-full" style="display: none;"> In this paper, we will prove some rigidity theorems for blow up limits to Type II singularities of Lagrangian mean curvature flow with zero Maslov class or almost calibrated Lagrangian mean curvature flows, especially for Lagrangian translating solitons in any dimension. These theorems generalized previous corresponding results from two dimensional case to arbitrarily dimensional case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15880v1-abstract-full').style.display = 'none'; document.getElementById('2412.15880v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">All comments are welcome! 16 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.15613">arXiv:2412.15613</a> <span> [<a href="https://arxiv.org/pdf/2412.15613">pdf</a>, <a href="https://arxiv.org/ps/2412.15613">ps</a>, <a href="https://arxiv.org/format/2412.15613">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Representation of finite order solutions to linear differential equations with exponential sum coefficients </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xing-Yu Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jun Wang</a>, <a href="/search/math?searchtype=author&query=Wen%2C+Z">Zhi-Tao Wen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.15613v1-abstract-short" style="display: inline;"> We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solut… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15613v1-abstract-full').style.display = 'inline'; document.getElementById('2412.15613v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.15613v1-abstract-full" style="display: none;"> We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the condition that there exists a finite order solution of (+) with exponential sum coefficients having rational frequencies and constant coefficients, we give the precise form of all finite order solutions, which are exponential sums. It is a partial answer to Gol'dberg-Ostrovski菒 Problem and Problem 5 in \cite{HITW2022} since exponential sums are of completely regular growth. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15613v1-abstract-full').style.display = 'none'; document.getElementById('2412.15613v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 34M05; Secondary 30D35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.13792">arXiv:2412.13792</a> <span> [<a href="https://arxiv.org/pdf/2412.13792">pdf</a>, <a href="https://arxiv.org/ps/2412.13792">ps</a>, <a href="https://arxiv.org/format/2412.13792">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Spectral radius of graphs of given size with forbidden a fan graph $F_6$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+J">Jing Gao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xueliang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.13792v1-abstract-short" style="display: inline;"> Let $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an $F_{2k+1}$-free or $F_{2k+2}$-free graph, then $位(G)\leq \frac{k-1+\sqrt{4m-k^2+1}}{2}$ and the equality holds if and only if… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.13792v1-abstract-full').style.display = 'inline'; document.getElementById('2412.13792v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.13792v1-abstract-full" style="display: none;"> Let $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an $F_{2k+1}$-free or $F_{2k+2}$-free graph, then $位(G)\leq \frac{k-1+\sqrt{4m-k^2+1}}{2}$ and the equality holds if and only if $G\cong K_k\vee\left(\frac{m}{k}-\frac{k-1}{2}\right)K_1$. Recently, Li et al. in [arXiv:2409.15918] showed that the above conjecture holds for $k\geq 3$. The only left case is for $k=2$, which corresponds to $F_5$ or $F_6$. Since the case of $F_5$ was solved by Yu et al. in [arXiv:2404.03423] and Zhang and Wang in [On the spectral radius of graphs without a gem, Discrete Math. 347 (2024) 114171]. So, one needs only to deal with the case of $F_6$. In this paper, we solve the only left case by determining the maximum spectral radius of $F_6$-free graphs with size $m\geq 88$, and the corresponding extremal graph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.13792v1-abstract-full').style.display = 'none'; document.getElementById('2412.13792v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C35; 05C50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.11341">arXiv:2412.11341</a> <span> [<a href="https://arxiv.org/pdf/2412.11341">pdf</a>, <a href="https://arxiv.org/format/2412.11341">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Coupling-based Convergence Diagnostic and Stepsize Scheme for Stochastic Gradient Descent </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang Li</a>, <a href="/search/math?searchtype=author&query=Xie%2C+Q">Qiaomin Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.11341v1-abstract-short" style="display: inline;"> The convergence behavior of Stochastic Gradient Descent (SGD) crucially depends on the stepsize configuration. When using a constant stepsize, the SGD iterates form a Markov chain, enjoying fast convergence during the initial transient phase. However, when reaching stationarity, the iterates oscillate around the optimum without making further progress. In this paper, we study the convergence diagn… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.11341v1-abstract-full').style.display = 'inline'; document.getElementById('2412.11341v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.11341v1-abstract-full" style="display: none;"> The convergence behavior of Stochastic Gradient Descent (SGD) crucially depends on the stepsize configuration. When using a constant stepsize, the SGD iterates form a Markov chain, enjoying fast convergence during the initial transient phase. However, when reaching stationarity, the iterates oscillate around the optimum without making further progress. In this paper, we study the convergence diagnostics for SGD with constant stepsize, aiming to develop an effective dynamic stepsize scheme. We propose a novel coupling-based convergence diagnostic procedure, which monitors the distance of two coupled SGD iterates for stationarity detection. Our diagnostic statistic is simple and is shown to track the transition from transience stationarity theoretically. We conduct extensive numerical experiments and compare our method against various existing approaches. Our proposed coupling-based stepsize scheme is observed to achieve superior performance across a diverse set of convex and non-convex problems. Moreover, our results demonstrate the robustness of our approach to a wide range of hyperparameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.11341v1-abstract-full').style.display = 'none'; document.getElementById('2412.11341v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 30 figures, to be published in AAAI 2025</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.10045">arXiv:2412.10045</a> <span> [<a href="https://arxiv.org/pdf/2412.10045">pdf</a>, <a href="https://arxiv.org/format/2412.10045">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Numerical Analysis of Multi-patch Discontinuous Galerkin Isogeometric Method for Full-potential Electronic Structure Calculations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoxu Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+X">Xucheng Meng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.10045v1-abstract-short" style="display: inline;"> In this paper, we study the multi-patch discontinuous Galerkin isogeometric (DG-IGA) approximations for full-potential electronic structure calculations. We decompose the physical domain into several subdomains, represent each part of the wavefunction separately using B-spline basis functions, possibly with different degrees, on varying mesh sizes, and then combine them by DG methods. We also prov… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.10045v1-abstract-full').style.display = 'inline'; document.getElementById('2412.10045v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.10045v1-abstract-full" style="display: none;"> In this paper, we study the multi-patch discontinuous Galerkin isogeometric (DG-IGA) approximations for full-potential electronic structure calculations. We decompose the physical domain into several subdomains, represent each part of the wavefunction separately using B-spline basis functions, possibly with different degrees, on varying mesh sizes, and then combine them by DG methods. We also provide a rigorous {\em a priori} error analysis of the DG-IGA approximations for linear eigenvalue problems. Furthermore, this work offers a unified analysis framework for the DG-IGA method applied to a class of elliptic eigenvalue problems. Finally, we present several numerical experiments to verify our theoretical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.10045v1-abstract-full').style.display = 'none'; document.getElementById('2412.10045v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.09253">arXiv:2412.09253</a> <span> [<a href="https://arxiv.org/pdf/2412.09253">pdf</a>, <a href="https://arxiv.org/ps/2412.09253">ps</a>, <a href="https://arxiv.org/format/2412.09253">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Traveling waves to a logarithmic chemotaxis model with fast diffusion and singularities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaowen Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+D">Dongfang Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jingyu Li</a>, <a href="/search/math?searchtype=author&query=Mei%2C+M">Ming Mei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.09253v2-abstract-short" style="display: inline;"> This paper is concerned with a chemotaxis model with logarithmic sensitivity and fast diffusion, which possesses strong singularities for the sensitivity at zero-concentration of chemical signal, and for the diffusion at zero-population of cells, respectively. The main purpose is to show the existence of traveling waves connecting the singular zero-end-state, and particularly, to show the asymptot… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09253v2-abstract-full').style.display = 'inline'; document.getElementById('2412.09253v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.09253v2-abstract-full" style="display: none;"> This paper is concerned with a chemotaxis model with logarithmic sensitivity and fast diffusion, which possesses strong singularities for the sensitivity at zero-concentration of chemical signal, and for the diffusion at zero-population of cells, respectively. The main purpose is to show the existence of traveling waves connecting the singular zero-end-state, and particularly, to show the asymptotic stability of these traveling waves. The challenge of the problem is the interaction of two kinds of singularities involved in the model: one is the logarithmic singularity of the sensitivity; and the other is the power-law singularity of the diffusivity. To overcome the singularities for the wave stability, some new techniques of weighted energy method are introduced artfully. Numerical simulations are also carried out, which further confirm our theoretical stability results, in particular, the numerical results indicate that the effect of fast diffusion to the structure of traveling waves is essential, which causes the traveling waves much steeper like shock waves. This new phenomenon is a first observation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09253v2-abstract-full').style.display = 'none'; document.getElementById('2412.09253v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 14 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35C07; 35B35; 35Q92; 92C17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.09242">arXiv:2412.09242</a> <span> [<a href="https://arxiv.org/pdf/2412.09242">pdf</a>, <a href="https://arxiv.org/ps/2412.09242">ps</a>, <a href="https://arxiv.org/format/2412.09242">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4208/jms.v57n3.24.06">10.4208/jms.v57n3.24.06 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Transition layers to chemotaxis-consumption models with volume-filling effect </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaowen Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jingyu Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.09242v2-abstract-short" style="display: inline;"> We are interested in the dynamical behaviors of solutions to a parabolic-parabolic chemotaxis-consumption model with a volume-filling effect on a bounded interval, where the physical no-flux boundary condition for the bacteria and mixed Dirichlet-Neumann boundary condition for the oxygen are prescribed. By taking a continuity argument, we first show that the model admits a unique nonconstant stead… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09242v2-abstract-full').style.display = 'inline'; document.getElementById('2412.09242v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.09242v2-abstract-full" style="display: none;"> We are interested in the dynamical behaviors of solutions to a parabolic-parabolic chemotaxis-consumption model with a volume-filling effect on a bounded interval, where the physical no-flux boundary condition for the bacteria and mixed Dirichlet-Neumann boundary condition for the oxygen are prescribed. By taking a continuity argument, we first show that the model admits a unique nonconstant steady state. Then we use Helly's compactness theorem to show that the asymptotic profile of steady state is a transition layer as the chemotactic coefficient goes to infinity. Finally, based on the energy method along with a cancellation structure of the model, we show that the steady state is nonlinearly stable under appropriate perturbations. Moreover, we do not need any assumption on the parameters in showing the stability of steady state. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09242v2-abstract-full').style.display = 'none'; document.getElementById('2412.09242v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q92; 35B35; 92C17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Mathematical Study,57 (2024), pp. 331-357 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.08027">arXiv:2412.08027</a> <span> [<a href="https://arxiv.org/pdf/2412.08027">pdf</a>, <a href="https://arxiv.org/format/2412.08027">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Error estimate for the first order energy stable scheme of Q-tensor nematic model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+J">Jin Huang</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiao Li</a>, <a href="/search/math?searchtype=author&query=Ji%2C+G">Guanghua Ji</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.08027v1-abstract-short" style="display: inline;"> We present rigorous error estimates towards a first-order unconditionally energy stable scheme designed for 3D hydrodynamic Q-tensor model of nematic liquid crystals. This scheme combines the scalar auxiliary variable (SAV), stabilization and projection method together. The unique solvability and energy dissipation of the scheme are proved. We further derive the boundness of numerical solution in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08027v1-abstract-full').style.display = 'inline'; document.getElementById('2412.08027v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.08027v1-abstract-full" style="display: none;"> We present rigorous error estimates towards a first-order unconditionally energy stable scheme designed for 3D hydrodynamic Q-tensor model of nematic liquid crystals. This scheme combines the scalar auxiliary variable (SAV), stabilization and projection method together. The unique solvability and energy dissipation of the scheme are proved. We further derive the boundness of numerical solution in L^{\infty} norm with mathematical deduction. Then, we can give the rigorous error estimate of order O(未t) in the sense of L2 norm, where 未t is the time step.Finally, we give some numerical simulations to demonstrate the theoretical analysis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08027v1-abstract-full').style.display = 'none'; document.getElementById('2412.08027v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.04538">arXiv:2412.04538</a> <span> [<a href="https://arxiv.org/pdf/2412.04538">pdf</a>, <a href="https://arxiv.org/format/2412.04538">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Communication Compression for Distributed Learning without Control Variates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ortega%2C+T">Tomas Ortega</a>, <a href="/search/math?searchtype=author&query=Huang%2C+C">Chun-Yin Huang</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoxiao Li</a>, <a href="/search/math?searchtype=author&query=Jafarkhani%2C+H">Hamid Jafarkhani</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.04538v1-abstract-short" style="display: inline;"> Distributed learning algorithms, such as the ones employed in Federated Learning (FL), require communication compression to reduce the cost of client uploads. The compression methods used in practice are often biased, which require error feedback to achieve convergence when the compression is aggressive. In turn, error feedback requires client-specific control variates, which directly contradicts… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04538v1-abstract-full').style.display = 'inline'; document.getElementById('2412.04538v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.04538v1-abstract-full" style="display: none;"> Distributed learning algorithms, such as the ones employed in Federated Learning (FL), require communication compression to reduce the cost of client uploads. The compression methods used in practice are often biased, which require error feedback to achieve convergence when the compression is aggressive. In turn, error feedback requires client-specific control variates, which directly contradicts privacy-preserving principles and requires stateful clients. In this paper, we propose Compressed Aggregate Feedback (CAFe), a novel distributed learning framework that allows highly compressible client updates by exploiting past aggregated updates, and does not require control variates. We consider Distributed Gradient Descent (DGD) as a representative algorithm and provide a theoretical proof of CAFe's superiority to Distributed Compressed Gradient Descent (DCGD) with biased compression in the non-smooth regime with bounded gradient dissimilarity. Experimental results confirm that CAFe consistently outperforms distributed learning with direct compression and highlight the compressibility of the client updates with CAFe. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04538v1-abstract-full').style.display = 'none'; document.getElementById('2412.04538v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 68W10; 68W15; 68W40; 90C06; 90C35; 90C26 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.1.6; F.2.1; E.4 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.04480">arXiv:2412.04480</a> <span> [<a href="https://arxiv.org/pdf/2412.04480">pdf</a>, <a href="https://arxiv.org/ps/2412.04480">ps</a>, <a href="https://arxiv.org/format/2412.04480">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Physics">physics.comp-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Learning Generalized Diffusions using an Energetic Variational Approach </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lu%2C+Y">Yubin Lu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaofan Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Chun Liu</a>, <a href="/search/math?searchtype=author&query=Tang%2C+Q">Qi Tang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yiwei Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.04480v1-abstract-short" style="display: inline;"> Extracting governing physical laws from computational or experimental data is crucial across various fields such as fluid dynamics and plasma physics. Many of those physical laws are dissipative due to fluid viscosity or plasma collisions. For such a dissipative physical system, we propose two distinct methods to learn the corresponding laws of the systems based on their energy-dissipation laws, a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04480v1-abstract-full').style.display = 'inline'; document.getElementById('2412.04480v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.04480v1-abstract-full" style="display: none;"> Extracting governing physical laws from computational or experimental data is crucial across various fields such as fluid dynamics and plasma physics. Many of those physical laws are dissipative due to fluid viscosity or plasma collisions. For such a dissipative physical system, we propose two distinct methods to learn the corresponding laws of the systems based on their energy-dissipation laws, assuming either continuous data (probability density) or discrete data (particles) are available. Our methods offer several key advantages, including their robustness to corrupted observations, their easy extension to more complex physical systems, and the potential to address higher-dimensional systems. We validate our approach through representative numerical examples and carefully investigate the impacts of data quantity and data property on the model discovery. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04480v1-abstract-full').style.display = 'none'; document.getElementById('2412.04480v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.00527">arXiv:2412.00527</a> <span> [<a href="https://arxiv.org/pdf/2412.00527">pdf</a>, <a href="https://arxiv.org/format/2412.00527">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cen%2C+S">Siyu Cen</a>, <a href="/search/math?searchtype=author&query=Jin%2C+B">Bangti Jin</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiyao Li</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+Z">Zhi Zhou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.00527v1-abstract-short" style="display: inline;"> In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. W… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.00527v1-abstract-full').style.display = 'inline'; document.getElementById('2412.00527v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.00527v1-abstract-full" style="display: none;"> In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.00527v1-abstract-full').style.display = 'none'; document.getElementById('2412.00527v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages, 18 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.17544">arXiv:2411.17544</a> <span> [<a href="https://arxiv.org/pdf/2411.17544">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A Network Flow Approach to Optimal Scheduling in Supply Chain Logistics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yichen Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Huanbo Zhang</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+C">Chunhong Yuan</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiangyu Li</a>, <a href="/search/math?searchtype=author&query=Jiang%2C+Z">Zuowen Jiang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.17544v2-abstract-short" style="display: inline;"> In the evolving digital landscape, network flow models have transcended traditional applications to become integral in diverse sectors, including supply chain management. This research develops a robust network flow model for semiconductor wafer supply chains, optimizing resource allocation and addressing maximum flow challenges in production and logistics. The model incorporates the stochastic na… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17544v2-abstract-full').style.display = 'inline'; document.getElementById('2411.17544v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.17544v2-abstract-full" style="display: none;"> In the evolving digital landscape, network flow models have transcended traditional applications to become integral in diverse sectors, including supply chain management. This research develops a robust network flow model for semiconductor wafer supply chains, optimizing resource allocation and addressing maximum flow challenges in production and logistics. The model incorporates the stochastic nature of wafer batch transfers and employs a dual-layer optimization framework to reduce variability and exceedance probabilities in finished goods. Empirical comparisons reveal significant enhancements in cost efficiency, productivity, and resource utilization, with a 20% reduction in time and production costs, and a 10% increase in transportation and storage capacities. The model's efficacy is underscored by a 15% decrease in transportation time and a 6700 kg increase in total capacity, demonstrating its capability to resolve logistical bottlenecks in semiconductor manufacturing. This study concludes that network flow models are a potent tool for optimizing supply chain logistics, offering a 23% improvement in resource utilization and a 13% boost in accuracy. The findings provide valuable insights for supply chain logistics optimization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17544v2-abstract-full').style.display = 'none'; document.getElementById('2411.17544v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.16398">arXiv:2411.16398</a> <span> [<a href="https://arxiv.org/pdf/2411.16398">pdf</a>, <a href="https://arxiv.org/format/2411.16398">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyi Li</a>, <a href="/search/math?searchtype=author&query=Shi%2C+J">Jialu Shi</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Q">Qiheng Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.16398v1-abstract-short" style="display: inline;"> We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $纬\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, El… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16398v1-abstract-full').style.display = 'inline'; document.getElementById('2411.16398v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.16398v1-abstract-full" style="display: none;"> We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $纬\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for $纬\in (\frac{d+2}{2d},1)$. The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Pr茅vost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16398v1-abstract-full').style.display = 'none'; document.getElementById('2411.16398v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C81; 60F10; 60G70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14921">arXiv:2411.14921</a> <span> [<a href="https://arxiv.org/pdf/2411.14921">pdf</a>, <a href="https://arxiv.org/format/2411.14921">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> A boundary Harnack principle and its application to analyticity of 3D Brownian intersection exponents </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+Y">Yifan Gao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyi Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yifan Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+R">Runsheng Liu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+X">Xiangyi Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14921v1-abstract-short" style="display: inline;"> We show that a domain in $\mathbb{R}^3$ with the trace of a 3D Brownian motion removed almost surely satisfies the boundary Harnack principle (BHP). Then, we use it to prove that the intersection exponents for 3D Brownian motion are analytic. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14921v1-abstract-full" style="display: none;"> We show that a domain in $\mathbb{R}^3$ with the trace of a 3D Brownian motion removed almost surely satisfies the boundary Harnack principle (BHP). Then, we use it to prove that the intersection exponents for 3D Brownian motion are analytic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14921v1-abstract-full').style.display = 'none'; document.getElementById('2411.14921v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">49 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60J65 (Primary) 31B25; 31B05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14828">arXiv:2411.14828</a> <span> [<a href="https://arxiv.org/pdf/2411.14828">pdf</a>, <a href="https://arxiv.org/ps/2411.14828">ps</a>, <a href="https://arxiv.org/format/2411.14828">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+J">Juan Liu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+N">Nan-Jing Huang</a>, <a href="/search/math?searchtype=author&query=Long%2C+X">Xian-Jun Long</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xue-song Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14828v1-abstract-short" style="display: inline;"> This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14828v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14828v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14828v1-abstract-full" style="display: none;"> This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving the unconstrained optimization problem. Under certain conditions, we demonstrate that this continuous-time dynamical system exponentially converges to the optimal solution of the inequality constrained convex optimization problem. Moreover, we derive several discrete-time algorithms from this continuous-time framework and obtain their optimal convergence rates. Finally, we present numerical experiments to validate the effectiveness of the proposed algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14828v1-abstract-full').style.display = 'none'; document.getElementById('2411.14828v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.13868">arXiv:2411.13868</a> <span> [<a href="https://arxiv.org/pdf/2411.13868">pdf</a>, <a href="https://arxiv.org/format/2411.13868">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computation and Language">cs.CL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Robust Detection of Watermarks for Large Language Models Under Human Edits </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang Li</a>, <a href="/search/math?searchtype=author&query=Ruan%2C+F">Feng Ruan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+H">Huiyuan Wang</a>, <a href="/search/math?searchtype=author&query=Long%2C+Q">Qi Long</a>, <a href="/search/math?searchtype=author&query=Su%2C+W+J">Weijie J. Su</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.13868v1-abstract-short" style="display: inline;"> Watermarking has offered an effective approach to distinguishing text generated by large language models (LLMs) from human-written text. However, the pervasive presence of human edits on LLM-generated text dilutes watermark signals, thereby significantly degrading detection performance of existing methods. In this paper, by modeling human edits through mixture model detection, we introduce a new m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13868v1-abstract-full').style.display = 'inline'; document.getElementById('2411.13868v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13868v1-abstract-full" style="display: none;"> Watermarking has offered an effective approach to distinguishing text generated by large language models (LLMs) from human-written text. However, the pervasive presence of human edits on LLM-generated text dilutes watermark signals, thereby significantly degrading detection performance of existing methods. In this paper, by modeling human edits through mixture model detection, we introduce a new method in the form of a truncated goodness-of-fit test for detecting watermarked text under human edits, which we refer to as Tr-GoF. We prove that the Tr-GoF test achieves optimality in robust detection of the Gumbel-max watermark in a certain asymptotic regime of substantial text modifications and vanishing watermark signals. Importantly, Tr-GoF achieves this optimality \textit{adaptively} as it does not require precise knowledge of human edit levels or probabilistic specifications of the LLMs, in contrast to the optimal but impractical (Neyman--Pearson) likelihood ratio test. Moreover, we establish that the Tr-GoF test attains the highest detection efficiency rate in a certain regime of moderate text modifications. In stark contrast, we show that sum-based detection rules, as employed by existing methods, fail to achieve optimal robustness in both regimes because the additive nature of their statistics is less resilient to edit-induced noise. Finally, we demonstrate the competitive and sometimes superior empirical performance of the Tr-GoF test on both synthetic data and open-source LLMs in the OPT and LLaMA families. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13868v1-abstract-full').style.display = 'none'; document.getElementById('2411.13868v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.11737">arXiv:2411.11737</a> <span> [<a href="https://arxiv.org/pdf/2411.11737">pdf</a>, <a href="https://arxiv.org/ps/2411.11737">ps</a>, <a href="https://arxiv.org/format/2411.11737">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Randomization-based Z-estimation for evaluating average and individual treatment effects </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Qu%2C+T">Tianyi Qu</a>, <a href="/search/math?searchtype=author&query=Du%2C+J">Jiangchuan Du</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinran Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.11737v1-abstract-short" style="display: inline;"> Randomized experiments have been the gold standard for drawing causal inference. The conventional model-based approach has been one of the most popular ways for analyzing treatment effects from randomized experiments, which is often carried through inference for certain model parameters. In this paper, we provide a systematic investigation of model-based analyses for treatment effects under the ra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11737v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11737v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11737v1-abstract-full" style="display: none;"> Randomized experiments have been the gold standard for drawing causal inference. The conventional model-based approach has been one of the most popular ways for analyzing treatment effects from randomized experiments, which is often carried through inference for certain model parameters. In this paper, we provide a systematic investigation of model-based analyses for treatment effects under the randomization-based inference framework. This framework does not impose any distributional assumptions on the outcomes, covariates and their dependence, and utilizes only randomization as the "reasoned basis". We first derive the asymptotic theory for Z-estimation in completely randomized experiments, and propose sandwich-type conservative covariance estimation. We then apply the developed theory to analyze both average and individual treatment effects in randomized experiments. For the average treatment effect, we consider three estimation strategies: model-based, model-imputed, and model-assisted, where the first two can be sensitive to model misspecification or require specific ways for parameter estimation. The model-assisted approach is robust to arbitrary model misspecification and always provides consistent average treatment effect estimation. We propose optimal ways to conduct model-assisted estimation using generally nonlinear least squares for parameter estimation. For the individual treatment effects, we propose to directly model the relationship between individual effects and covariates, and discuss the model's identifiability, inference and interpretation allowing model misspecification. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11737v1-abstract-full').style.display = 'none'; document.getElementById('2411.11737v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.11215">arXiv:2411.11215</a> <span> [<a href="https://arxiv.org/pdf/2411.11215">pdf</a>, <a href="https://arxiv.org/ps/2411.11215">ps</a>, <a href="https://arxiv.org/format/2411.11215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Hypergeometric $\ell$-adic sheaves for reductive groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xuanyou Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.11215v2-abstract-short" style="display: inline;"> We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11215v2-abstract-full').style.display = 'inline'; document.getElementById('2411.11215v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11215v2-abstract-full" style="display: none;"> We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we are able to study the hypergeometric $\ell$-adic sheaf via the hypergeometric $\mathcal D$-module. We apply our results to the estimation of the hypergeometric exponential sum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11215v2-abstract-full').style.display = 'none'; document.getElementById('2411.11215v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Revised version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Secondary 14M27; 11L07. Primary 14F10; 14F20; Secondary 14M27; 11L07. Primary 14F10; 14F20; Secondary 14M27; 11L07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.11115">arXiv:2411.11115</a> <span> [<a href="https://arxiv.org/pdf/2411.11115">pdf</a>, <a href="https://arxiv.org/ps/2411.11115">ps</a>, <a href="https://arxiv.org/format/2411.11115">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Numerical integrations of stochastic contact Hamiltonian systems via stochastic contact Hamilton-Jacobi equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhan%2C+Q">Qingyi Zhan</a>, <a href="/search/math?searchtype=author&query=Duan%2C+J">Jinqiao Duan</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaofan Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Lijin Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.11115v1-abstract-short" style="display: inline;"> Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving schemes are constructed by the stochastic contact… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11115v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11115v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11115v1-abstract-full" style="display: none;"> Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving schemes are constructed by the stochastic contact Hamilton-Jacobi equation. A general numerical approximation method of the stochastic contact Hamilton-Jacobi equation is devised, and the convergent order theorem is provided, too. Numerical tests are shown to confirm the theoretical results and the usability of proposed approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11115v1-abstract-full').style.display = 'none'; document.getElementById('2411.11115v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 11 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C50; 65C30; 65P20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.08311">arXiv:2411.08311</a> <span> [<a href="https://arxiv.org/pdf/2411.08311">pdf</a>, <a href="https://arxiv.org/ps/2411.08311">ps</a>, <a href="https://arxiv.org/format/2411.08311">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> A generalization of the martingale property of entropy production in stochastic systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiangting Li</a>, <a href="/search/math?searchtype=author&query=Chou%2C+T">Tom Chou</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.08311v1-abstract-short" style="display: inline;"> By decoupling forward and backward stochastic trajectories, we develop a family of martingales and work theorems for the same stochastic process. We achieve this by introducing an alternative work theorem derivation that uses tools from stochastic calculus instead of path integrals. Our derivation applies to both overdamped and underdamped Langevin dynamics and generalizes work theorems so that th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08311v1-abstract-full').style.display = 'inline'; document.getElementById('2411.08311v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.08311v1-abstract-full" style="display: none;"> By decoupling forward and backward stochastic trajectories, we develop a family of martingales and work theorems for the same stochastic process. We achieve this by introducing an alternative work theorem derivation that uses tools from stochastic calculus instead of path integrals. Our derivation applies to both overdamped and underdamped Langevin dynamics and generalizes work theorems so that they connect new quantities in stochastic processes, potentially revealing new applications in dissipative systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08311v1-abstract-full').style.display = 'none'; document.getElementById('2411.08311v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.07271">arXiv:2411.07271</a> <span> [<a href="https://arxiv.org/pdf/2411.07271">pdf</a>, <a href="https://arxiv.org/format/2411.07271">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Multi-hop Upstream Anticipatory Traffic Signal Control with Deep Reinforcement Learning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaocan Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+X">Xiaoyu Wang</a>, <a href="/search/math?searchtype=author&query=Smirnov%2C+I">Ilia Smirnov</a>, <a href="/search/math?searchtype=author&query=Sanner%2C+S">Scott Sanner</a>, <a href="/search/math?searchtype=author&query=Abdulhai%2C+B">Baher Abdulhai</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.07271v2-abstract-short" style="display: inline;"> Coordination in traffic signal control is crucial for managing congestion in urban networks. Existing pressure-based control methods focus only on immediate upstream links, leading to suboptimal green time allocation and increased network delays. However, effective signal control inherently requires coordination across a broader spatial scope, as the effect of upstream traffic should influence sig… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.07271v2-abstract-full').style.display = 'inline'; document.getElementById('2411.07271v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.07271v2-abstract-full" style="display: none;"> Coordination in traffic signal control is crucial for managing congestion in urban networks. Existing pressure-based control methods focus only on immediate upstream links, leading to suboptimal green time allocation and increased network delays. However, effective signal control inherently requires coordination across a broader spatial scope, as the effect of upstream traffic should influence signal control decisions at downstream intersections, impacting a large area in the traffic network. Although agent communication using neural network-based feature extraction can implicitly enhance spatial awareness, it significantly increases the learning complexity, adding an additional layer of difficulty to the challenging task of control in deep reinforcement learning. To address the issue of learning complexity and myopic traffic pressure definition, our work introduces a novel concept based on Markov chain theory, namely \textit{multi-hop upstream pressure}, which generalizes the conventional pressure to account for traffic conditions beyond the immediate upstream links. This farsighted and compact metric informs the deep reinforcement learning agent to preemptively clear the multi-hop upstream queues, guiding the agent to optimize signal timings with a broader spatial awareness. Simulations on synthetic and realistic (Toronto) scenarios demonstrate controllers utilizing multi-hop upstream pressure significantly reduce overall network delay by prioritizing traffic movements based on a broader understanding of upstream congestion. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.07271v2-abstract-full').style.display = 'none'; document.getElementById('2411.07271v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 tables, 11 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06656">arXiv:2411.06656</a> <span> [<a href="https://arxiv.org/pdf/2411.06656">pdf</a>, <a href="https://arxiv.org/ps/2411.06656">ps</a>, <a href="https://arxiv.org/format/2411.06656">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On moments of the error term of the multivariable k-th divisor functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+Z">Zhen Guo</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.06656v1-abstract-short" style="display: inline;"> Suppose $k\geqslant3$ is an integer. Let $蟿_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}蟿_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+伪_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<伪_k<1$ ar… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06656v1-abstract-full').style.display = 'inline'; document.getElementById('2411.06656v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06656v1-abstract-full" style="display: none;"> Suppose $k\geqslant3$ is an integer. Let $蟿_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}蟿_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+伪_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<伪_k<1$ are computable constants. In this paper we study the mean square of $螖_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $螖_{r,3}(x)$. We also get an upper bound for the third power moment of $螖_{r,3}(x)$. Moreover, we study the first power moment of $螖_{r,3}(x)$ and then give a result for the sign changes of it. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06656v1-abstract-full').style.display = 'none'; document.getElementById('2411.06656v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06494">arXiv:2411.06494</a> <span> [<a href="https://arxiv.org/pdf/2411.06494">pdf</a>, <a href="https://arxiv.org/ps/2411.06494">ps</a>, <a href="https://arxiv.org/format/2411.06494">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> On hydrostatic limit of Beris-Edwards system in a thin strip </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=De+Anna%2C+F">Francesco De Anna</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xingyu Li</a>, <a href="/search/math?searchtype=author&query=Paicu%2C+M">Marius Paicu</a>, <a href="/search/math?searchtype=author&query=Zarnescu%2C+A">Arghir Zarnescu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.06494v2-abstract-short" style="display: inline;"> In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06494v2-abstract-full').style.display = 'inline'; document.getElementById('2411.06494v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06494v2-abstract-full" style="display: none;"> In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains in the limit a partly decoupled system. For the fluid part we obtain the Prandtl system while for the $Q$-tensors we obtain a non-standard system, involving fluids components and a non-standard combination of partly dissipative equations and algebraic constraints. We prove the convergence of the rescaled system and the well-posedness of the limit in Sobolev spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06494v2-abstract-full').style.display = 'none'; document.getElementById('2411.06494v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.03972">arXiv:2411.03972</a> <span> [<a href="https://arxiv.org/pdf/2411.03972">pdf</a>, <a href="https://arxiv.org/format/2411.03972">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Toward end-to-end quantum simulation for protein dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zhenning Liu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiantao Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+C">Chunhao Wang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jin-Peng Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.03972v1-abstract-short" style="display: inline;"> Modeling and simulating the protein folding process overall remains a grand challenge in computational biology. We systematically investigate end-to-end quantum algorithms for simulating various protein dynamics with effects, such as mechanical forces or stochastic noises. We offer efficient quantum simulation algorithms to produce quantum encoding of the final states, history states, or density m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03972v1-abstract-full').style.display = 'inline'; document.getElementById('2411.03972v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03972v1-abstract-full" style="display: none;"> Modeling and simulating the protein folding process overall remains a grand challenge in computational biology. We systematically investigate end-to-end quantum algorithms for simulating various protein dynamics with effects, such as mechanical forces or stochastic noises. We offer efficient quantum simulation algorithms to produce quantum encoding of the final states, history states, or density matrices of inhomogeneous or stochastic harmonic oscillator models. For the read-in setting, we design (i) efficient quantum algorithms for initial state preparation, utilizing counter-based random number generator and rejection sampling, and (ii) depth-efficient approaches for molecular structure loading. Both are particularly important in handling large protein molecules. For the read-out setting, our algorithms estimate various classical observables, such as energy, low vibration modes, density of states, correlation of displacement, and optimal control of molecular dynamics. We also show classical numerical experiments focused on estimating the density of states and applying the optimal control to facilitate conformation changes to verify our arguments on potential quantum speedups. Overall, our study demonstrates that the quantum simulation of protein dynamics can be a solid end-to-end application in the era of early or fully fault-tolerant quantum computing. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03972v1-abstract-full').style.display = 'none'; document.getElementById('2411.03972v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">61 pages, 10 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.03599">arXiv:2411.03599</a> <span> [<a href="https://arxiv.org/pdf/2411.03599">pdf</a>, <a href="https://arxiv.org/format/2411.03599">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Structure-preserving quantum algorithms for linear and nonlinear Hamiltonian systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+H">Hsuan-Cheng Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiantao Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.03599v2-abstract-short" style="display: inline;"> Hamiltonian systems of ordinary and partial differential equations are fundamental across modern science and engineering, appearing in models that span virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the symplectic structure, which preserves geometric properties like phase-space volume over time and energy conservation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03599v2-abstract-full').style.display = 'inline'; document.getElementById('2411.03599v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03599v2-abstract-full" style="display: none;"> Hamiltonian systems of ordinary and partial differential equations are fundamental across modern science and engineering, appearing in models that span virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the symplectic structure, which preserves geometric properties like phase-space volume over time and energy conservation over an extended period. In this paper, we present quantum algorithms that incorporate symplectic integrators, ensuring the preservation of this key structure. We demonstrate how these algorithms maintain the symplectic properties for both linear and nonlinear Hamiltonian systems. Additionally, we provide a comprehensive theoretical analysis of the computational complexity, showing that our approach offers both accuracy and improved efficiency over classical algorithms. These results highlight the potential application of quantum algorithms for solving large-scale Hamiltonian systems while preserving essential physical properties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03599v2-abstract-full').style.display = 'none'; document.getElementById('2411.03599v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.01911">arXiv:2411.01911</a> <span> [<a href="https://arxiv.org/pdf/2411.01911">pdf</a>, <a href="https://arxiv.org/ps/2411.01911">ps</a>, <a href="https://arxiv.org/format/2411.01911">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Contraction property on complex hyperbolic ball </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoshan Li</a>, <a href="/search/math?searchtype=author&query=Su%2C+G">Guicong Su</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.01911v2-abstract-short" style="display: inline;"> We prove an isoperimetric inequalitie on the complex hyperbolic ball with Assumption \ref{assumption}}. As an application, we prove a contraction property for the holomorphic functions in Hardy and weighted Bergman spaces on the complex hyperbolic ball with this assumption. The results can be seen as partial generalization of Kulikov's result on the complex hyperbolic plane. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.01911v2-abstract-full" style="display: none;"> We prove an isoperimetric inequalitie on the complex hyperbolic ball with Assumption \ref{assumption}}. As an application, we prove a contraction property for the holomorphic functions in Hardy and weighted Bergman spaces on the complex hyperbolic ball with this assumption. The results can be seen as partial generalization of Kulikov's result on the complex hyperbolic plane. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01911v2-abstract-full').style.display = 'none'; document.getElementById('2411.01911v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Li%2C+X&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a 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