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class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li> <a href="/search/?searchtype=author&query=Luo%2C+Y&start=250" class="pagination-link " aria-label="Page 6" aria-current="page">6 </a> </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/2503.14913">arXiv:2503.14913</a> <span> [<a href="https://arxiv.org/pdf/2503.14913">pdf</a>, <a href="https://arxiv.org/format/2503.14913">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 PINN-enriched finite element method for linear elliptic problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+X">Xiao Chen</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yixin Luo</a>, <a href="/search/math?searchtype=author&query=Chen%2C+J">Jingrun Chen</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="2503.14913v1-abstract-short" style="display: inline;"> In this paper, we propose a hybrid method that combines finite element method (FEM) and physics-informed neural network (PINN) for solving linear elliptic problems. This method contains three steps: (1) train a PINN and obtain an approximate solution $u_胃$; (2) enrich the finite element space with $u_胃$; (3) obtain the final solution by FEM in the enriched space. In the second step, the enriched s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.14913v1-abstract-full').style.display = 'inline'; document.getElementById('2503.14913v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.14913v1-abstract-full" style="display: none;"> In this paper, we propose a hybrid method that combines finite element method (FEM) and physics-informed neural network (PINN) for solving linear elliptic problems. This method contains three steps: (1) train a PINN and obtain an approximate solution $u_胃$; (2) enrich the finite element space with $u_胃$; (3) obtain the final solution by FEM in the enriched space. In the second step, the enriched space is constructed by addition $v + u_胃$ or multiplication $v \cdot u_胃$, where $v$ belongs to the standard finite element space. We conduct the convergence analysis for the proposed method. Compared to the standard FEM, the same convergence order is obtained and higher accuracy can be achieved when solution derivatives are well approximated in PINN. Numerical examples from one dimension to three dimensions verify these theoretical results. For some examples, the accuracy of the proposed method can be reduced by a couple of orders of magnitude compared to the standard FEM. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.14913v1-abstract-full').style.display = 'none'; document.getElementById('2503.14913v1-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.09087">arXiv:2503.09087</a> <span> [<a href="https://arxiv.org/pdf/2503.09087">pdf</a>, <a href="https://arxiv.org/ps/2503.09087">ps</a>, <a href="https://arxiv.org/format/2503.09087">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"> Shortest Circuits in Homology Classes of Graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Ye Luo</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="2503.09087v1-abstract-short" style="display: inline;"> Recently, the study of circuits and cycles within the homology classes of graphs has attracted considerable research interest. However, the detection and counting of shorter circuits in homology classes, especially the shortest ones, remain underexplored. This paper aims to fill this gap by solving the problem of detecting and counting the shortest cycles in homology classes, leveraging the concep… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.09087v1-abstract-full').style.display = 'inline'; document.getElementById('2503.09087v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.09087v1-abstract-full" style="display: none;"> Recently, the study of circuits and cycles within the homology classes of graphs has attracted considerable research interest. However, the detection and counting of shorter circuits in homology classes, especially the shortest ones, remain underexplored. This paper aims to fill this gap by solving the problem of detecting and counting the shortest cycles in homology classes, leveraging the concept of direction-consistent circuits and extending classical results on Eulerian circuits such as Hierholzer's algorithm and the BEST theorem. As an application, we propose the one-carrier transportation routing problem and relate it to a circuit detection problem in graph homology. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.09087v1-abstract-full').style.display = 'none'; document.getElementById('2503.09087v1-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">14 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C38; 05C45; 05C50; 90C27 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.05442">arXiv:2503.05442</a> <span> [<a href="https://arxiv.org/pdf/2503.05442">pdf</a>, <a href="https://arxiv.org/format/2503.05442">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"> 3-path-connectivity of bubble-sort star graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yi-Lu Luo</a>, <a href="/search/math?searchtype=author&query=Deng%2C+Y">Yun-Ping Deng</a>, <a href="/search/math?searchtype=author&query=Sun%2C+Y">Yuan 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="2503.05442v1-abstract-short" style="display: inline;"> Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $T$ be a subset of $ V(G)$ with cardinality $|T|\geq2$. A path connecting all vertices of $T$ is called a $T$-path of $G$. Two $T$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=T$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote by $蟺_G(T)$ the maximum number of internally disjoint $T$- pa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.05442v1-abstract-full').style.display = 'inline'; document.getElementById('2503.05442v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.05442v1-abstract-full" style="display: none;"> Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $T$ be a subset of $ V(G)$ with cardinality $|T|\geq2$. A path connecting all vertices of $T$ is called a $T$-path of $G$. Two $T$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=T$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote by $蟺_G(T)$ the maximum number of internally disjoint $T$- paths in G. Then for an integer $\ell$ with $\ell\geq2$, the $\ell$-path-connectivity $蟺_\ell(G)$ of $G$ is formulated as $\min\{蟺_G(T)\,|\,T\subseteq V(G)$ and $|T|=\ell\}$. In this paper, we study the $3$-path-connectivity of $n$-dimensional bubble-sort star graph $BS_n$. By deeply analyzing the structure of $BS_n$, we show that $蟺_3(BS_n)=\lfloor\frac{3n}2\rfloor-3$, for any $n\geq3$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.05442v1-abstract-full').style.display = 'none'; document.getElementById('2503.05442v1-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.05440">arXiv:2503.05440</a> <span> [<a href="https://arxiv.org/pdf/2503.05440">pdf</a>, <a href="https://arxiv.org/ps/2503.05440">ps</a>, <a href="https://arxiv.org/format/2503.05440">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> A path description for $\varepsilon$-characters of representations of type $A$ restricted quantum loop algebras at roots of unity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=An%2C+X">Xiao-Juan An</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jian-Rong Li</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yan-Feng Luo</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+W">Wen-Ting 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="2503.05440v1-abstract-short" style="display: inline;"> Fix $\varepsilon^{2\ell}=1$ with $\ell \geq 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\varepsilon$-characters of any simple module with highest $l$-weight of degr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.05440v1-abstract-full').style.display = 'inline'; document.getElementById('2503.05440v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.05440v1-abstract-full" style="display: none;"> Fix $\varepsilon^{2\ell}=1$ with $\ell \geq 2$. In this paper, we show that all finite-dimensional simple modules of any restricted quantum loop algebra $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ in a certain category can be transformed into snake modules. We obtain an effective and concrete path description for $\varepsilon$-characters of any simple module with highest $l$-weight of degree two and any Kirillov-Reshetikhin module of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$. As an application of our path description, we obtain a necessary and sufficient condition for the tensor product of two fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. Additionally, we obtain a necessary condition for the tensor product of two or more fundamental representations of $U_{\varepsilon}^{\rm res}({L\mathfrak{sl}_{n+1}})$ to be irreducible. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.05440v1-abstract-full').style.display = 'none'; document.getElementById('2503.05440v1-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.02982">arXiv:2503.02982</a> <span> [<a href="https://arxiv.org/pdf/2503.02982">pdf</a>, <a href="https://arxiv.org/ps/2503.02982">ps</a>, <a href="https://arxiv.org/format/2503.02982">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Performance">cs.PF</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"> Heavy-traffic Optimality of Skip-the-Longest-Queues in Heterogeneous Parallel Service Systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yishun Luo</a>, <a href="/search/math?searchtype=author&query=Zubeldia%2C+M">Martin Zubeldia</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="2503.02982v1-abstract-short" style="display: inline;"> We consider a discrete-time parallel service system consisting of $n$ heterogeneous single server queues with infinite capacity. Jobs arrive to the system as an i.i.d. process with rate proportional to $n$, and must be immediately dispatched in the time slot that they arrive. The dispatcher is assumed to be able to exchange messages with the servers to obtain their queue lengths and make dispatchi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.02982v1-abstract-full').style.display = 'inline'; document.getElementById('2503.02982v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.02982v1-abstract-full" style="display: none;"> We consider a discrete-time parallel service system consisting of $n$ heterogeneous single server queues with infinite capacity. Jobs arrive to the system as an i.i.d. process with rate proportional to $n$, and must be immediately dispatched in the time slot that they arrive. The dispatcher is assumed to be able to exchange messages with the servers to obtain their queue lengths and make dispatching decisions, introducing an undesirable communication overhead. In this setting, we propose a ultra-low communication overhead load balancing policy dubbed $k$-Skip-the-$d$-Longest-Queues ($k$-SLQ-$d$), where queue lengths are only observed every $k(n-d)$ time slots and, between observations, incoming jobs are sent to a queue that is not one of the $d$ longest ones at the time that the queues were last observed. For this policy, we establish conditions on $d$ for it to be throughput optimal and we show that, under that condition, it is asymptotically delay-optimal in heavy-traffic for arbitrarily low communication overheads (i.e., for arbitrarily large $k$). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.02982v1-abstract-full').style.display = 'none'; document.getElementById('2503.02982v1-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> 4 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.18131">arXiv:2502.18131</a> <span> [<a href="https://arxiv.org/pdf/2502.18131">pdf</a>, <a href="https://arxiv.org/ps/2502.18131">ps</a>, <a href="https://arxiv.org/format/2502.18131">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="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> An approximate solution of a case of perturbed Fokker-Planck equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yan Luo</a>, <a href="/search/math?searchtype=author&query=Sheng%2C+K">Kaicheng Sheng</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="2502.18131v1-abstract-short" style="display: inline;"> This paper focuses on finding an approximate solution of a kind of Fokker-Planck equation with time-dependent perturbations. A formulation of the approximate solution of the equation is constructed, and then the existence of the formulation is proved. The related Hamiltonian dynamical system explains the estimations. Our work provides a more comprehensive understanding of the behaviour of systems… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.18131v1-abstract-full').style.display = 'inline'; document.getElementById('2502.18131v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.18131v1-abstract-full" style="display: none;"> This paper focuses on finding an approximate solution of a kind of Fokker-Planck equation with time-dependent perturbations. A formulation of the approximate solution of the equation is constructed, and then the existence of the formulation is proved. The related Hamiltonian dynamical system explains the estimations. Our work provides a more comprehensive understanding of the behaviour of systems described by this Fokker-Planck equation and the corresponding stochastic differential equation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.18131v1-abstract-full').style.display = 'none'; document.getElementById('2502.18131v1-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 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/2502.16956">arXiv:2502.16956</a> <span> [<a href="https://arxiv.org/pdf/2502.16956">pdf</a>, <a href="https://arxiv.org/ps/2502.16956">ps</a>, <a href="https://arxiv.org/format/2502.16956">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Jordan property for automorphism groups of compact varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yujie Luo</a>, <a href="/search/math?searchtype=author&query=Meng%2C+S">Sheng Meng</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+D">De-Qi 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="2502.16956v1-abstract-short" style="display: inline;"> In this note, we report some recent progress on the Jordan property for (birational) automorphism groups of projective varieties and compact complex varieties. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.16956v1-abstract-full" style="display: none;"> In this note, we report some recent progress on the Jordan property for (birational) automorphism groups of projective varieties and compact complex varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.16956v1-abstract-full').style.display = 'none'; document.getElementById('2502.16956v1-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> 24 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">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14J50; 32M05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.06765">arXiv:2502.06765</a> <span> [<a href="https://arxiv.org/pdf/2502.06765">pdf</a>, <a href="https://arxiv.org/ps/2502.06765">ps</a>, <a href="https://arxiv.org/format/2502.06765">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> <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="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Are all models wrong? Fundamental limits in distribution-free empirical model falsification </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=M%C3%BCller%2C+M+M">Manuel M. M眉ller</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetian Luo</a>, <a href="/search/math?searchtype=author&query=Barber%2C+R+F">Rina Foygel Barber</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="2502.06765v1-abstract-short" style="display: inline;"> In statistics and machine learning, when we train a fitted model on available data, we typically want to ensure that we are searching within a model class that contains at least one accurate model -- that is, we would like to ensure an upper bound on the model class risk (the lowest possible risk that can be attained by any model in the class). However, it is also of interest to establish lower bo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.06765v1-abstract-full').style.display = 'inline'; document.getElementById('2502.06765v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.06765v1-abstract-full" style="display: none;"> In statistics and machine learning, when we train a fitted model on available data, we typically want to ensure that we are searching within a model class that contains at least one accurate model -- that is, we would like to ensure an upper bound on the model class risk (the lowest possible risk that can be attained by any model in the class). However, it is also of interest to establish lower bounds on the model class risk, for instance so that we can determine whether our fitted model is at least approximately optimal within the class, or, so that we can decide whether the model class is unsuitable for the particular task at hand. Particularly in the setting of interpolation learning where machine learning models are trained to reach zero error on the training data, we might ask if, at the very least, a positive lower bound on the model class risk is possible -- or are we unable to detect that "all models are wrong"? In this work, we answer these questions in a distribution-free setting by establishing a model-agnostic, fundamental hardness result for the problem of constructing a lower bound on the best test error achievable over a model class, and examine its implications on specific model classes such as tree-based methods and linear regression. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.06765v1-abstract-full').style.display = 'none'; document.getElementById('2502.06765v1-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 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/2502.06218">arXiv:2502.06218</a> <span> [<a href="https://arxiv.org/pdf/2502.06218">pdf</a>, <a href="https://arxiv.org/ps/2502.06218">ps</a>, <a href="https://arxiv.org/format/2502.06218">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> The basic locus of ramified unitary Rapoport-Zink space at maximal vertex level </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+Q">Qiao He</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yu Luo</a>, <a href="/search/math?searchtype=author&query=Shi%2C+Y">Yousheng Shi</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="2502.06218v1-abstract-short" style="display: inline;"> We construct the Bruhat-Tits stratification of the ramified unitary Rapoport-Zink space, with the level being the stabilizer of a vertex lattice. We develop the local model theory for Bruhat-Tits strata, proving their normality and Cohen-Macaulayness, and provide precise dimension formulas. Additionally, we establish an explicit isomorphism between Bruhat-Tits strata and Deligne-Lusztig varieties,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.06218v1-abstract-full').style.display = 'inline'; document.getElementById('2502.06218v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.06218v1-abstract-full" style="display: none;"> We construct the Bruhat-Tits stratification of the ramified unitary Rapoport-Zink space, with the level being the stabilizer of a vertex lattice. We develop the local model theory for Bruhat-Tits strata, proving their normality and Cohen-Macaulayness, and provide precise dimension formulas. Additionally, we establish an explicit isomorphism between Bruhat-Tits strata and Deligne-Lusztig varieties, revealing new phenomena beyond the previously studied Coxeter-type cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.06218v1-abstract-full').style.display = 'none'; document.getElementById('2502.06218v1-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 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/2502.02915">arXiv:2502.02915</a> <span> [<a href="https://arxiv.org/pdf/2502.02915">pdf</a>, <a href="https://arxiv.org/ps/2502.02915">ps</a>, <a href="https://arxiv.org/format/2502.02915">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 a trace formula of counting Eulerian cycles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Ye Luo</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="2502.02915v1-abstract-short" style="display: inline;"> We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the trace sum of certain twisted vertex and edge adjacency matrices of the graph. Moreover, we show that reduction of computation can be achieved by taking into acc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.02915v1-abstract-full').style.display = 'inline'; document.getElementById('2502.02915v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.02915v1-abstract-full" style="display: none;"> We make connections of a counting problem of Eulerian cycles for undirected graphs to homological spectral graph theory, and formulate explicitly a trace formula that identifies the number of Eulerian circuits on an Eulerian graph with the trace sum of certain twisted vertex and edge adjacency matrices of the graph. Moreover, we show that reduction of computation can be achieved by taking into account symmetries related to twisted adjacency matrices induced by spectral antisymmetry and graph automorphisms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.02915v1-abstract-full').style.display = 'none'; document.getElementById('2502.02915v1-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 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">25 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C50; 05C38 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.18021">arXiv:2501.18021</a> <span> [<a href="https://arxiv.org/pdf/2501.18021">pdf</a>, <a href="https://arxiv.org/ps/2501.18021">ps</a>, <a href="https://arxiv.org/format/2501.18021">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> </div> </div> <p class="title is-5 mathjax"> Codimension 1 transfer maps of K theoretic indexes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetong Luo</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="2501.18021v1-abstract-short" style="display: inline;"> Let $M$ be a closed spin manifold and $N$ be a codimension 1 submanifold of it. Given certain homotopy conditions, Zeidler shows that the Rosenberg index of $N$ is an obstruction to the existence of positive scalar curvature on $M$. He further gives a transfer map between the K groups of the group $C^*$ algebras of the foundemental group. The transfer map maps the Rosenberg index of $M$ to the one… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.18021v1-abstract-full').style.display = 'inline'; document.getElementById('2501.18021v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.18021v1-abstract-full" style="display: none;"> Let $M$ be a closed spin manifold and $N$ be a codimension 1 submanifold of it. Given certain homotopy conditions, Zeidler shows that the Rosenberg index of $N$ is an obstruction to the existence of positive scalar curvature on $M$. He further gives a transfer map between the K groups of the group $C^*$ algebras of the foundemental group. The transfer map maps the Rosenberg index of $M$ to the one of $N$. In this note, we present an alternative formulation of the transfer map using maps between $C^*$ algebras, and give an analogus result for the codimension 1 transfer of higher K theoretic signatures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.18021v1-abstract-full').style.display = 'none'; document.getElementById('2501.18021v1-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> 29 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">Comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 19K56 (Primary); 46L80 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.08175">arXiv:2501.08175</a> <span> [<a href="https://arxiv.org/pdf/2501.08175">pdf</a>, <a href="https://arxiv.org/ps/2501.08175">ps</a>, <a href="https://arxiv.org/format/2501.08175">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> Maximal green sequences for $\mathcal{Q}^N$ quivers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+J">Jingmin Guo</a>, <a href="/search/math?searchtype=author&query=Duan%2C+B">Bing Duan</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yanfeng Luo</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="2501.08175v1-abstract-short" style="display: inline;"> We introduce $\mathcal{Q}^N$ quivers and construct maximal green sequences for these quivers. We prove that any finite connected full subquiver of the quivers defined by Hernandez and Leclerc, arising in monoidal categorifications of cluster algebras, is a special case of $\mathcal{Q}^N$ quivers. Moreover, we prove that the trees of oriented cycles introduced by Garver and Musiker are special case… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.08175v1-abstract-full').style.display = 'inline'; document.getElementById('2501.08175v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.08175v1-abstract-full" style="display: none;"> We introduce $\mathcal{Q}^N$ quivers and construct maximal green sequences for these quivers. We prove that any finite connected full subquiver of the quivers defined by Hernandez and Leclerc, arising in monoidal categorifications of cluster algebras, is a special case of $\mathcal{Q}^N$ quivers. Moreover, we prove that the trees of oriented cycles introduced by Garver and Musiker are special cases of $\mathcal{Q}^N$ quivers. This result resolves an open problem proposed by Garver and Musiker, providing a construction of maximal green sequences for quivers that are trees of oriented cycles. Furthermore, we prove that quivers that are mutation equivalent to an orientation of a type AD Dynkin diagram can also be recognized as special cases of $\mathcal{Q}^N$ quivers. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.08175v1-abstract-full').style.display = 'none'; document.getElementById('2501.08175v1-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">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> 13F60; 16G20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.04035">arXiv:2501.04035</a> <span> [<a href="https://arxiv.org/pdf/2501.04035">pdf</a>, <a href="https://arxiv.org/ps/2501.04035">ps</a>, <a href="https://arxiv.org/format/2501.04035">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"> Knudsen boundary layer equations with incoming boundary condition: full range of cutoff collision kernels and Mach numbers of the far field </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+N">Ning Jiang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yi-Long Luo</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yulong Wu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+T">Tong Yang</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="2501.04035v1-abstract-short" style="display: inline;"> This paper establishes tahe existence and uniqueness of the nonlinear Knudsen layer equation with incoming boundary conditions. It is well-known that the solvability conditions of the problem vary with the Mach number of the far Maxwellian $\mathcal{M}^\infty$. We consider full ranges of cutoff collision kernels (i.e., $- 3 < 纬\leq 1$) and all the Mach numbers of the far field in the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04035v1-abstract-full').style.display = 'inline'; document.getElementById('2501.04035v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.04035v1-abstract-full" style="display: none;"> This paper establishes tahe existence and uniqueness of the nonlinear Knudsen layer equation with incoming boundary conditions. It is well-known that the solvability conditions of the problem vary with the Mach number of the far Maxwellian $\mathcal{M}^\infty$. We consider full ranges of cutoff collision kernels (i.e., $- 3 < 纬\leq 1$) and all the Mach numbers of the far field in the $L^\infty_{x,v}$ framework. Additionally, the solution exhibits exponential decay $\exp \{- c x^\frac{2}{3 - 纬} - c |v|^2 \}$ for some $c > 0$. To address the general angular cutoff collision kernel, we introduce a $(x,v)$-mixed weight $蟽$. The proof is essentially bsed on adding an artificial damping term. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.04035v1-abstract-full').style.display = 'none'; document.getElementById('2501.04035v1-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 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: substantial text overlap with arXiv:2407.02852</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q20; 76P05; 35F30; 35B45; 35A01; 35A02 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.01751">arXiv:2501.01751</a> <span> [<a href="https://arxiv.org/pdf/2501.01751">pdf</a>, <a href="https://arxiv.org/ps/2501.01751">ps</a>, <a href="https://arxiv.org/format/2501.01751">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Gap theorems for complete submanifolds in the hyperbolic space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+J">Jianling Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yong Luo</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="2501.01751v1-abstract-short" style="display: inline;"> Based on the seminal Simons' formula, Shen \cite{Shen} and Lin-Xia \cite{LX} obtained gap theorems for compact minimal submanifolds in the unit sphere in the late 1980's. Then due to the effect of Xu \cite{Xu}, Ni \cite{Ni}, Yun \cite{Yun} and Xu-Gu \cite{XuG}, we achieved a comprehensive understanding of gap phenomena of complete submanifolds with parallel mean curvature vector field in the spher… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.01751v1-abstract-full').style.display = 'inline'; document.getElementById('2501.01751v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.01751v1-abstract-full" style="display: none;"> Based on the seminal Simons' formula, Shen \cite{Shen} and Lin-Xia \cite{LX} obtained gap theorems for compact minimal submanifolds in the unit sphere in the late 1980's. Then due to the effect of Xu \cite{Xu}, Ni \cite{Ni}, Yun \cite{Yun} and Xu-Gu \cite{XuG}, we achieved a comprehensive understanding of gap phenomena of complete submanifolds with parallel mean curvature vector field in the sphere or in the Euclidean space. But such kind of results in case of the hyperbolic space were obtained by Wang-Xia \cite{XiaW}, Lin-Wang \cite{LW} and Xu-Xu \cite{XX} until relatively recently and are not quite complete so far. In this paper first we continue to study gap theorems for complete submanifolds with parallel mean curvature vector field in the hyperbolic space, which generalize or extend several results in the literature. Second we prove a gap theorem for complete hypersurfaces with constant scalar curvature $n(1-n)$ in the hyperbolic space, which extends related results due to Bai-Luo \cite{BL2} in cases of the Euclidean space and the unit sphere. Such kind of results in case of the hyperbolic space are more complicated, due to some extra bad terms in the Simons' formula, and one of main ingredients of our proofs is an estimate for the first eigenvalue of complete submanifolds in the hyperbolic space obtained by Lin \cite{Lin}. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.01751v1-abstract-full').style.display = 'none'; document.getElementById('2501.01751v1-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">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">All comments are welcome! 17 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/2412.16639">arXiv:2412.16639</a> <span> [<a href="https://arxiv.org/pdf/2412.16639">pdf</a>, <a href="https://arxiv.org/format/2412.16639">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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"> Stability analysis of the nonlinear pendulums under stochastic perturbations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yan Luo</a>, <a href="/search/math?searchtype=author&query=Sheng%2C+K">Kaicheng Sheng</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="2412.16639v1-abstract-short" style="display: inline;"> We consider a nonlinear pendulum whose suspension point undergoes stochastic vibrations in its plane of motion. Stochastic vibrations are constructed by stochastic differential equations with random periodic solutions. Averaging over these stochastic vibrations can be simplified with ergodicity. We give a complete description of the bifurcations of phase portraits of the averaged Hamiltonian syste… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.16639v1-abstract-full').style.display = 'inline'; document.getElementById('2412.16639v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.16639v1-abstract-full" style="display: none;"> We consider a nonlinear pendulum whose suspension point undergoes stochastic vibrations in its plane of motion. Stochastic vibrations are constructed by stochastic differential equations with random periodic solutions. Averaging over these stochastic vibrations can be simplified with ergodicity. We give a complete description of the bifurcations of phase portraits of the averaged Hamiltonian system. The bifurcation curves of the stochastic perturbed Hamiltonian system are shown numerically. Estimations between the averaged system and the exact system are calculated. The correspondence of the averaged system to the exact system is explained through the Poincar茅 return map. Studying the averaged Hamiltonian system provided important information for the exact stochastic perturbed Hamiltonian system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.16639v1-abstract-full').style.display = 'none'; document.getElementById('2412.16639v1-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 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/2412.15643">arXiv:2412.15643</a> <span> [<a href="https://arxiv.org/pdf/2412.15643">pdf</a>, <a href="https://arxiv.org/ps/2412.15643">ps</a>, <a href="https://arxiv.org/format/2412.15643">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"> Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yong Luo</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+X">Xianjing Zheng</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="2412.15643v1-abstract-short" style="display: inline;"> In this paper we study eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds. In particular we establish some universal inequality for eigenvalues of the Dirichlet Laplacian on the hyperbolic space $\mathbb{H}^n(-1)$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.15643v1-abstract-full" style="display: none;"> In this paper we study eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds. In particular we establish some universal inequality for eigenvalues of the Dirichlet Laplacian on the hyperbolic space $\mathbb{H}^n(-1)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15643v1-abstract-full').style.display = 'none'; document.getElementById('2412.15643v1-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">Submitted, 21 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/2412.15636">arXiv:2412.15636</a> <span> [<a href="https://arxiv.org/pdf/2412.15636">pdf</a>, <a href="https://arxiv.org/ps/2412.15636">ps</a>, <a href="https://arxiv.org/format/2412.15636">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"> Inequalities for eigenvalues of Laplacian and biharmonic operators on submanifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yong Luo</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+X">Xianjing Zheng</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="2412.15636v1-abstract-short" style="display: inline;"> In this paper we study eigenvalues of Laplacian and biharmonic operators on compact domains in complete manifolds. We establish several new inequalities for eigenvalues of Laplacian and biharmonic operators respectively by using Sobolev type inequalities. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.15636v1-abstract-full" style="display: none;"> In this paper we study eigenvalues of Laplacian and biharmonic operators on compact domains in complete manifolds. We establish several new inequalities for eigenvalues of Laplacian and biharmonic operators respectively by using Sobolev type inequalities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.15636v1-abstract-full').style.display = 'none'; document.getElementById('2412.15636v1-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">Accepted by Jounral of Mathematical Study, 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/2412.07287">arXiv:2412.07287</a> <span> [<a href="https://arxiv.org/pdf/2412.07287">pdf</a>, <a href="https://arxiv.org/ps/2412.07287">ps</a>, <a href="https://arxiv.org/format/2412.07287">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"> Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in $H^{-\f12}$ space with polynomial tail </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+L">Ling-Bing He</a>, <a href="/search/math?searchtype=author&query=Ji%2C+J">Jie Ji</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yue Luo</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="2412.07287v3-abstract-short" style="display: inline;"> We demonstrate that the spatially homogeneous Landau-Coulomb equation exhibits global existence and uniqueness around the space $H^{-\f12}_3\cap L^1_{7}\cap L\log L$. Additionally, we furnish several quantitative assessments regarding the smoothing estimates in weighted Sobolev spaces. As a result, we confirm that the solution exhibits a $ C^\infty $ but not $ H^\infty $ smoothing effect in th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.07287v3-abstract-full').style.display = 'inline'; document.getElementById('2412.07287v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.07287v3-abstract-full" style="display: none;"> We demonstrate that the spatially homogeneous Landau-Coulomb equation exhibits global existence and uniqueness around the space $H^{-\f12}_3\cap L^1_{7}\cap L\log L$. Additionally, we furnish several quantitative assessments regarding the smoothing estimates in weighted Sobolev spaces. As a result, we confirm that the solution exhibits a $ C^\infty $ but not $ H^\infty $ smoothing effect in the velocity variable for any positive time, when the initial data possesses a polynomial tail. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.07287v3-abstract-full').style.display = 'none'; document.getElementById('2412.07287v3-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 10 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">54 pages, 0 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 82B40; 35B65; 35H20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.17227">arXiv:2411.17227</a> <span> [<a href="https://arxiv.org/pdf/2411.17227">pdf</a>, <a href="https://arxiv.org/format/2411.17227">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Uniformization of gasket Julia sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yusheng Luo</a>, <a href="/search/math?searchtype=author&query=Ntalampekos%2C+D">Dimitrios Ntalampekos</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="2411.17227v1-abstract-short" style="display: inline;"> The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set can be quasiconformally uniformized by a round gasket if and only if it is a fat gasket, i.e., boundaries of Fatou components intersect tangentially. We also p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17227v1-abstract-full').style.display = 'inline'; document.getElementById('2411.17227v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.17227v1-abstract-full" style="display: none;"> The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set can be quasiconformally uniformized by a round gasket if and only if it is a fat gasket, i.e., boundaries of Fatou components intersect tangentially. We also prove that a Julia set can be uniformized by a round gasket with a David homeomorphism if and only if every Fatou component is a quasidisk; equivalently, there are no parabolic cycles of multiplicity 2. Our theorem applies to show that gasket Julia sets and limit sets of Kleinian groups can be locally quasiconformally homeomorphic, although globally this is conjectured to be false. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17227v1-abstract-full').style.display = 'none'; document.getElementById('2411.17227v1-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 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, 9 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 37F10; 37F31; Secondary 30C62; 37F30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.17093">arXiv:2411.17093</a> <span> [<a href="https://arxiv.org/pdf/2411.17093">pdf</a>, <a href="https://arxiv.org/ps/2411.17093">ps</a>, <a href="https://arxiv.org/format/2411.17093">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> </div> </div> <p class="title is-5 mathjax"> The Schur-Weyl duality and Invariants for classical Lie superalgebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yang Luo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yongjie 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="2411.17093v1-abstract-short" style="display: inline;"> In this article, we provide a comprehensive characterization of invariants of classical Lie superalgebras from the super-analog of the Schur-Weyl duality in a unified way. We establish $\mathfrak{g}$-invariants of the tensor algebra $T(\mathfrak{g})$, the supersymmetric algebra $S(\mathfrak{g})$, and the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ of a classical Lie superalgebra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17093v1-abstract-full').style.display = 'inline'; document.getElementById('2411.17093v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.17093v1-abstract-full" style="display: none;"> In this article, we provide a comprehensive characterization of invariants of classical Lie superalgebras from the super-analog of the Schur-Weyl duality in a unified way. We establish $\mathfrak{g}$-invariants of the tensor algebra $T(\mathfrak{g})$, the supersymmetric algebra $S(\mathfrak{g})$, and the universal enveloping algebra $\mathrm{U}(\mathfrak{g})$ of a classical Lie superalgebra $\mathfrak{g}$ corresponding to every element in centralizer algebras and their relationship under supersymmetrization. As a byproduct, we prove that the restriction on $T(\mathfrak{g})^{\mathfrak{g}}$ of the projection from $T(\mathfrak{g})$ to $\mathrm{U}(\mathfrak{g})$ is surjective, which enables us to determine the generators of the center $\mathcal{Z}(\mathfrak{g})$ except for $\mathfrak{g}=\mathfrak{osp}_{2m|2n}$. Additionally, we present an alternative algebraic proof of the triviality of $\mathcal{Z}(\mathfrak{p}_n)$. The key ingredient involves a technique lemma related to the symmetric group and Brauer diagrams. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.17093v1-abstract-full').style.display = 'none'; document.getElementById('2411.17093v1-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">31 pages, 18 figures, comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> (2020): 17B35; 17B10; 20C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14203">arXiv:2411.14203</a> <span> [<a href="https://arxiv.org/pdf/2411.14203">pdf</a>, <a href="https://arxiv.org/format/2411.14203">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Piecewise quasiconformal dynamical systems of the unit circle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yusheng Luo</a>, <a href="/search/math?searchtype=author&query=Ntalampekos%2C+D">Dimitrios Ntalampekos</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="2411.14203v1-abstract-short" style="display: inline;"> We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result generalizes the main result of arXiv:2010.11256, which deals with piecewise analytic maps. As applications, we provide a classification of piecewise quasiconformal maps o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14203v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14203v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14203v1-abstract-full" style="display: none;"> We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result generalizes the main result of arXiv:2010.11256, which deals with piecewise analytic maps. As applications, we provide a classification of piecewise quasiconformal maps of the circle up to quasisymmetric conjugacy, we prove a general conformal mating theorem for Blaschke products, and we study the quasiconformal geometry of parabolic basins. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14203v1-abstract-full').style.display = 'none'; document.getElementById('2411.14203v1-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> <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">38 pages, 9 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 30J10; 37F10; 37F31; Secondary 30C62; 30C65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.10748">arXiv:2411.10748</a> <span> [<a href="https://arxiv.org/pdf/2411.10748">pdf</a>, <a href="https://arxiv.org/format/2411.10748">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"> Classification and Nondegeneracy of Cubic Nonlinear Schr枚dinger System in $\mathbb{R}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+Y">Yujin Guo</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yong Luo</a>, <a href="/search/math?searchtype=author&query=Wei%2C+J">Juncheng Wei</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="2411.10748v1-abstract-short" style="display: inline;"> We study the following one-dimensional cubic nonlinear Schr枚dinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-渭_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $渭_1\leq渭_2\leq\cdots\leq渭_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10748v1-abstract-full').style.display = 'inline'; document.getElementById('2411.10748v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.10748v1-abstract-full" style="display: none;"> We study the following one-dimensional cubic nonlinear Schr枚dinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-渭_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $渭_1\leq渭_2\leq\cdots\leq渭_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on the explicit values of $渭_1\leq渭_2\leq渭_3<0$, there exist two different classes of normalized solutions $u=(u_1, u_2, u_3)$ satisfying $\int _{R}u_i^2dx=1$ for all $i=1, 2, 3$, which are completely different from the case $N=2$; (iii). The linearized operator at any nontrivial solution of the system is non-degenerate. The conjectures on the explicit classification and nondegeneracy of solutions for the system are also given for the case $N>3$. These address the questions of [R. Frank, D. Gontier and M. Lewin, CMP, 2021], where the complete classification and uniqueness results for the system were already proved for the case $N=2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10748v1-abstract-full').style.display = 'none'; document.getElementById('2411.10748v1-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 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">40 pages, 1 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.02232">arXiv:2411.02232</a> <span> [<a href="https://arxiv.org/pdf/2411.02232">pdf</a>, <a href="https://arxiv.org/ps/2411.02232">ps</a>, <a href="https://arxiv.org/format/2411.02232">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="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"> Two-loop Loewner potentials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yan Luo</a>, <a href="/search/math?searchtype=author&query=Maibach%2C+S">Sid Maibach</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="2411.02232v1-abstract-short" style="display: inline;"> We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively expressing it in terms of normalized Brownian loop measure, zeta-regularized determinants of the Laplacian, an integral formula generalizing universal Liouville action,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.02232v1-abstract-full').style.display = 'inline'; document.getElementById('2411.02232v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.02232v1-abstract-full" style="display: none;"> We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively expressing it in terms of normalized Brownian loop measure, zeta-regularized determinants of the Laplacian, an integral formula generalizing universal Liouville action, and Loewner-Kufarev energy of a foliation. Moreover, we prove that the potential is finite if and only if both loops are Weil-Petersson quasicircles, that it is an Onsager-Machlup functional for the two-loop SLE, and a variational formula involving Schwarzian derivatives. Addressing the question of minimization of the two-loop Loewner potential, we find that any such minimizers must be pairs of circles. However, the potential is not bounded, diverging to negative infinity as the circles move away from each other and to positive infinity as the circles merge, thus preventing a definition of two-loop Loewner energy for the prospective large deviations theory for the two-loop SLE. To remedy the divergence, we study a way of generalizing the two-loop Loewner potential by taking into account how conformal field theory (CFT) partition functions depend on the modulus of the annulus between the loops. This generalization is motivated by the correspondence between SLE and CFT, and it also emerges from the geometry of the real determinant line bundle as introduced by Kontsevich and Suhov. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.02232v1-abstract-full').style.display = 'none'; document.getElementById('2411.02232v1-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> 4 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">36 pages, 5 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/2410.22647">arXiv:2410.22647</a> <span> [<a href="https://arxiv.org/pdf/2410.22647">pdf</a>, <a href="https://arxiv.org/ps/2410.22647">ps</a>, <a href="https://arxiv.org/format/2410.22647">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> </div> </div> <p class="title is-5 mathjax"> Adaptive Robust Confidence Intervals </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetian Luo</a>, <a href="/search/math?searchtype=author&query=Gao%2C+C">Chao Gao</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="2410.22647v1-abstract-short" style="display: inline;"> This paper studies the construction of adaptive confidence intervals under Huber's contamination model when the contamination proportion is unknown. For the robust confidence interval of a Gaussian mean, we show that the optimal length of an adaptive interval must be exponentially wider than that of a non-adaptive one. An optimal construction is achieved through simultaneous uncertainty quantifica… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22647v1-abstract-full').style.display = 'inline'; document.getElementById('2410.22647v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.22647v1-abstract-full" style="display: none;"> This paper studies the construction of adaptive confidence intervals under Huber's contamination model when the contamination proportion is unknown. For the robust confidence interval of a Gaussian mean, we show that the optimal length of an adaptive interval must be exponentially wider than that of a non-adaptive one. An optimal construction is achieved through simultaneous uncertainty quantification of quantiles at all levels. The results are further extended beyond the Gaussian location model by addressing a general family of robust hypothesis testing. In contrast to adaptive robust estimation, our findings reveal that the optimal length of an adaptive robust confidence interval critically depends on the distribution's shape. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22647v1-abstract-full').style.display = 'none'; document.getElementById('2410.22647v1-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> 29 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.10882">arXiv:2410.10882</a> <span> [<a href="https://arxiv.org/pdf/2410.10882">pdf</a>, <a href="https://arxiv.org/ps/2410.10882">ps</a>, <a href="https://arxiv.org/format/2410.10882">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"> Type number for orders of level (N_1,N_2) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yifan Luo</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+H">Haigang 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="2410.10882v1-abstract-short" style="display: inline;"> Let $N_1=p_1^{2u_1+1}...p_w^{2u_w+1}$, where the $p_i$ are distinct primes, $u_1,...,u_w$ are nonnegative integers and $w$ is an odd integer, and $N_2$ be a positive integer such that $\gcd(N_1,N_2)=1$. In this paper, we give an explicit formula for the type number, i.e. the number of isomorphism classes, of orders of level $(N_1, N_2)$. The method of proof involves the Siegel-Weil formula for ter… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.10882v1-abstract-full').style.display = 'inline'; document.getElementById('2410.10882v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.10882v1-abstract-full" style="display: none;"> Let $N_1=p_1^{2u_1+1}...p_w^{2u_w+1}$, where the $p_i$ are distinct primes, $u_1,...,u_w$ are nonnegative integers and $w$ is an odd integer, and $N_2$ be a positive integer such that $\gcd(N_1,N_2)=1$. In this paper, we give an explicit formula for the type number, i.e. the number of isomorphism classes, of orders of level $(N_1, N_2)$. The method of proof involves the Siegel-Weil formula for ternary quadratic forms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.10882v1-abstract-full').style.display = 'none'; document.getElementById('2410.10882v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">arXiv admin note: text overlap with arXiv:2402.17443</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11E20; 11R52; 11F37; 11E41 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.06182">arXiv:2410.06182</a> <span> [<a href="https://arxiv.org/pdf/2410.06182">pdf</a>, <a href="https://arxiv.org/format/2410.06182">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> On the lattice of the weak factorization systems on a finite lattice </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yongle Luo</a>, <a href="/search/math?searchtype=author&query=Rognerud%2C+B">Baptiste Rognerud</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="2410.06182v1-abstract-short" style="display: inline;"> We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer systems. As an application we find a lower bound for the number of transfer systems on a boolean lattice. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.06182v1-abstract-full" style="display: none;"> We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer systems. As an application we find a lower bound for the number of transfer systems on a boolean lattice. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06182v1-abstract-full').style.display = 'none'; document.getElementById('2410.06182v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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, comments are welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.04500">arXiv:2410.04500</a> <span> [<a href="https://arxiv.org/pdf/2410.04500">pdf</a>, <a href="https://arxiv.org/ps/2410.04500">ps</a>, <a href="https://arxiv.org/format/2410.04500">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Regular models of ramified unitary Shimura varieties at maximal parahoric level </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+Q">Qiao He</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yu Luo</a>, <a href="/search/math?searchtype=author&query=Shi%2C+Y">Yousheng Shi</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="2410.04500v1-abstract-short" style="display: inline;"> We use the idea of splitting models to define and study a semi-stable model for unitary Shimura varieties of signature $(n-1,1)$ with maximal parahoric level structure at ramified primes. In this case, the ``naive'' splitting model defined by Pappas and Rapoport fails to be flat in a crucial way. We prove that the genuine splitting model in this case is flat with semi-stable reduction. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.04500v1-abstract-full" style="display: none;"> We use the idea of splitting models to define and study a semi-stable model for unitary Shimura varieties of signature $(n-1,1)$ with maximal parahoric level structure at ramified primes. In this case, the ``naive'' splitting model defined by Pappas and Rapoport fails to be flat in a crucial way. We prove that the genuine splitting model in this case is flat with semi-stable reduction. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04500v1-abstract-full').style.display = 'none'; document.getElementById('2410.04500v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.15860">arXiv:2409.15860</a> <span> [<a href="https://arxiv.org/pdf/2409.15860">pdf</a>, <a href="https://arxiv.org/ps/2409.15860">ps</a>, <a href="https://arxiv.org/format/2409.15860">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"> Solitons, scattering and blow-up for the nonlinear Schr枚dinger equation with combined power-type nonlinearities on $\mathbb{R}^d\times\mathbb{T}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Forcella%2C+L">Luigi Forcella</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yongming Luo</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Z">Zehua Zhao</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="2409.15860v1-abstract-short" style="display: inline;"> We investigate the long time dynamics of the nonlinear Schr枚dinger equation (NLS) with combined powers on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$. Different from the previously studied NLS-models with single power on the waveguide manifolds, where the non-scale-invariance is mainly due to the mixed nature of the underlying domain, the non-scale-invariance of the present model is both… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.15860v1-abstract-full').style.display = 'inline'; document.getElementById('2409.15860v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.15860v1-abstract-full" style="display: none;"> We investigate the long time dynamics of the nonlinear Schr枚dinger equation (NLS) with combined powers on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$. Different from the previously studied NLS-models with single power on the waveguide manifolds, where the non-scale-invariance is mainly due to the mixed nature of the underlying domain, the non-scale-invariance of the present model is both geometrical and structural. By considering different combinations of the nonlinearities, we establish both qualitative and quantitative properties of the soliton, scattering and blow-up solutions. As one of the main novelties of the paper compared to the previous results for the NLS with single power, we particularly construct two different rescaled families of variational problems, which leads to an NLS with single power in different limiting profiles respectively, to establish the periodic dependence results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.15860v1-abstract-full').style.display = 'none'; document.getElementById('2409.15860v1-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> 24 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">arXiv admin note: text overlap with arXiv:2205.04969</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.05299">arXiv:2409.05299</a> <span> [<a href="https://arxiv.org/pdf/2409.05299">pdf</a>, <a href="https://arxiv.org/format/2409.05299">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Earth and Planetary Astrophysics">astro-ph.EP</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"> Stability analysis of spatial perturbed elliptic restricted 3-body problem with double-averaging </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yan Luo</a>, <a href="/search/math?searchtype=author&query=Sheng%2C+K">Kaicheng Sheng</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="2409.05299v1-abstract-short" style="display: inline;"> This paper investigates the secular motion of a massless asteroid within the framework of the double-averaged elliptic restricted three-body problem. By employing Poincar茅 variables, we analyze the stability properties of asteroid orbits in the presence of planetary perturbations. Our study reveals that periodic orbits identified in the planar configuration maintain stability in the spatial pertur… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.05299v1-abstract-full').style.display = 'inline'; document.getElementById('2409.05299v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.05299v1-abstract-full" style="display: none;"> This paper investigates the secular motion of a massless asteroid within the framework of the double-averaged elliptic restricted three-body problem. By employing Poincar茅 variables, we analyze the stability properties of asteroid orbits in the presence of planetary perturbations. Our study reveals that periodic orbits identified in the planar configuration maintain stability in the spatial perturbed problem across a wide range of parameter values. These findings, supported by numerical simulations, contribute to a deeper understanding of asteroid dynamics and have implications for studying exoplanetary systems with highly eccentric host stars. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.05299v1-abstract-full').style.display = 'none'; document.getElementById('2409.05299v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.10875">arXiv:2408.10875</a> <span> [<a href="https://arxiv.org/pdf/2408.10875">pdf</a>, <a href="https://arxiv.org/format/2408.10875">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Half grid diagrams and Thompson links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yangxiao Luo</a>, <a href="/search/math?searchtype=author&query=Wan%2C+S">Shunyu Wan</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="2408.10875v1-abstract-short" style="display: inline;"> We define half grid diagrams and prove every link is half grid presentable by constructing a canonical half grid pair (which gives rise to a grid diagram of some special type) associated with an element in the oriented Thompson group. We show that this half grid construction is equivalent to Jones' construction of oriented Thompson links. Using this equivalence, we relate the (oriented) Thompson i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.10875v1-abstract-full').style.display = 'inline'; document.getElementById('2408.10875v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.10875v1-abstract-full" style="display: none;"> We define half grid diagrams and prove every link is half grid presentable by constructing a canonical half grid pair (which gives rise to a grid diagram of some special type) associated with an element in the oriented Thompson group. We show that this half grid construction is equivalent to Jones' construction of oriented Thompson links. Using this equivalence, we relate the (oriented) Thompson index to several classical topological link invariants, and give both the lower and upper bounds of the maximal Thurston-Bennequin number of a knot in terms of the oriented Thompson index. Moreover, we give a one-to-one correspondence between half grid diagrams and elements in symmetric groups and give a new description of link group using two elements in a symmetric group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.10875v1-abstract-full').style.display = 'none'; document.getElementById('2408.10875v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">43 pages, 25 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/2408.10344">arXiv:2408.10344</a> <span> [<a href="https://arxiv.org/pdf/2408.10344">pdf</a>, <a href="https://arxiv.org/format/2408.10344">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</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"> Disk patterns, quasi-duality and the uniform bounded diameter conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yusheng Luo</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yongquan 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="2408.10344v1-abstract-short" style="display: inline;"> We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a conjecture of Minsky in the reflection group setting. This result can be interpreted as a uniform rigidity theorem for disk patterns. Our method also establishes a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.10344v1-abstract-full').style.display = 'inline'; document.getElementById('2408.10344v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.10344v1-abstract-full" style="display: none;"> We show that the diameter of the image of the skinning map on the deformation space of an acylindrical reflection group is bounded by a constant depending only on the topological complexity of the components of its boundary, answering a conjecture of Minsky in the reflection group setting. This result can be interpreted as a uniform rigidity theorem for disk patterns. Our method also establishes a connection between the diameter of the skinning image and certain discrete extremal width on the Coxeter graph of the reflection group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.10344v1-abstract-full').style.display = 'none'; document.getElementById('2408.10344v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">50 pages, 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/2408.00204">arXiv:2408.00204</a> <span> [<a href="https://arxiv.org/pdf/2408.00204">pdf</a>, <a href="https://arxiv.org/format/2408.00204">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yusheng Luo</a>, <a href="/search/math?searchtype=author&query=Lyubich%2C+M">Mikhail Lyubich</a>, <a href="/search/math?searchtype=author&query=Mukherjee%2C+S">Sabyasachi Mukherjee</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="2408.00204v1-abstract-short" style="display: inline;"> In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. Th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.00204v1-abstract-full').style.display = 'inline'; document.getElementById('2408.00204v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.00204v1-abstract-full" style="display: none;"> In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. These matings emerge naturally as degenerate (anti-)polynomial-like maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps. Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou's vision of a unified theory of groups and maps. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.00204v1-abstract-full').style.display = 'none'; document.getElementById('2408.00204v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30C10; 30F35; 37F05; 37F10; 37F31; 37F32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.11272">arXiv:2407.11272</a> <span> [<a href="https://arxiv.org/pdf/2407.11272">pdf</a>, <a href="https://arxiv.org/format/2407.11272">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Vision and Pattern Recognition">cs.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Differentiable Voxelization and Mesh Morphing </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yihao Luo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yikai Wang</a>, <a href="/search/math?searchtype=author&query=Xiang%2C+Z">Zhengrui Xiang</a>, <a href="/search/math?searchtype=author&query=Xiu%2C+Y">Yuliang Xiu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+G">Guang Yang</a>, <a href="/search/math?searchtype=author&query=Yap%2C+C">ChoonHwai Yap</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="2407.11272v2-abstract-short" style="display: inline;"> In this paper, we propose the differentiable voxelization of 3D meshes via the winding number and solid angles. The proposed approach achieves fast, flexible, and accurate voxelization of 3D meshes, admitting the computation of gradients with respect to the input mesh and GPU acceleration. We further demonstrate the application of the proposed voxelization in mesh morphing, where the voxelized mes… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11272v2-abstract-full').style.display = 'inline'; document.getElementById('2407.11272v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.11272v2-abstract-full" style="display: none;"> In this paper, we propose the differentiable voxelization of 3D meshes via the winding number and solid angles. The proposed approach achieves fast, flexible, and accurate voxelization of 3D meshes, admitting the computation of gradients with respect to the input mesh and GPU acceleration. We further demonstrate the application of the proposed voxelization in mesh morphing, where the voxelized mesh is deformed by a neural network. The proposed method is evaluated on the ShapeNet dataset and achieves state-of-the-art performance in terms of both accuracy and efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11272v2-abstract-full').style.display = 'none'; document.getElementById('2407.11272v2-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.07366">arXiv:2407.07366</a> <span> [<a href="https://arxiv.org/pdf/2407.07366">pdf</a>, <a href="https://arxiv.org/ps/2407.07366">ps</a>, <a href="https://arxiv.org/format/2407.07366">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"> Counting Permutations in $S_{2n}$ and $S_{2n+1}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuewen Luo</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="2407.07366v1-abstract-short" style="display: inline;"> Let $伪(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $伪(2n+1) = (2n+1) 伪(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents a combinatorial proof for this conjecture. At the same time, we demonstrate that all permutations with an even number of even cycles in both $S_{2n}$ and $S_{2n+1}$ can be ca… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.07366v1-abstract-full').style.display = 'inline'; document.getElementById('2407.07366v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.07366v1-abstract-full" style="display: none;"> Let $伪(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $伪(2n+1) = (2n+1) 伪(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents a combinatorial proof for this conjecture. At the same time, we demonstrate that all permutations with an even number of even cycles in both $S_{2n}$ and $S_{2n+1}$ can be categorized into three distinct types that correspond to each other. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.07366v1-abstract-full').style.display = 'none'; document.getElementById('2407.07366v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.04987">arXiv:2407.04987</a> <span> [<a href="https://arxiv.org/pdf/2407.04987">pdf</a>, <a href="https://arxiv.org/ps/2407.04987">ps</a>, <a href="https://arxiv.org/format/2407.04987">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"> Anisotropic Finsler $N$-Laplacian Liouville equation in convex cones </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dai%2C+W">Wei Dai</a>, <a href="/search/math?searchtype=author&query=Gui%2C+C">Changfeng Gui</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">YunPeng Luo</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="2407.04987v2-abstract-short" style="display: inline;"> We consider the anisotropic Finsler $N$-Laplacian Liouville equation \[-螖^{H}_{N}u=e^u \qquad {\rm{in}}\,\, \mathcal{C},\] where $N\geq2$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone including $\mathbb{R}^{N}$, the half space $\mathbb{R}^{N}_{+}$ and $\frac{1}{2^{m}}$-space $\mathbb{R}^{N}_{2^{-m}}:=\{x\in\mathbb{R}^{N}\mid x_{1},\cdots,x_{m}>0\}$ ($m=1,\cdots,N$), and the anisotro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.04987v2-abstract-full').style.display = 'inline'; document.getElementById('2407.04987v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.04987v2-abstract-full" style="display: none;"> We consider the anisotropic Finsler $N$-Laplacian Liouville equation \[-螖^{H}_{N}u=e^u \qquad {\rm{in}}\,\, \mathcal{C},\] where $N\geq2$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone including $\mathbb{R}^{N}$, the half space $\mathbb{R}^{N}_{+}$ and $\frac{1}{2^{m}}$-space $\mathbb{R}^{N}_{2^{-m}}:=\{x\in\mathbb{R}^{N}\mid x_{1},\cdots,x_{m}>0\}$ ($m=1,\cdots,N$), and the anisotropic Finsler $N$-Laplacian $螖^{H}_{N}$ is induced by a positively homogeneous function $H(x)$ of degree $1$. All solutions to the Finsler $N$-Laplacian Liouville equation with finite mass are completely classified. In particular, if $H(尉)=|尉|$, then the Finsler $N$-Laplacian $螖^{H}_{N}$ reduces to the regular $N$-Laplacian $螖_N$. Our result is a counterpart in the limiting case $p=N$ of the classification results in \cite{CFR} for the critical anisotropic $p$-Laplacian equations with $1<p<N$ in convex cones, and also extends the classification results in \cite{CK,CL,CW,CL2,E} for Liouville equation in the whole space $\mathbb{R}^{N}$ to general convex cones. In our proof, besides exploiting the anisotropic isoperimetric inequality inside convex cones, we have also proved and applied the radial Poincar茅 type inequality (Lemma \ref{A1}), which are key ingredients in the proof and of their own importance and interests. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.04987v2-abstract-full').style.display = 'none'; document.getElementById('2407.04987v2-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 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">38 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J92; 35B06; 35B40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.02852">arXiv:2407.02852</a> <span> [<a href="https://arxiv.org/pdf/2407.02852">pdf</a>, <a href="https://arxiv.org/ps/2407.02852">ps</a>, <a href="https://arxiv.org/format/2407.02852">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"> Knudsen boundary layer equations for full ranges of cutoff collision kernels: Maxwell reflection boundary with all accommodation coefficients in [0,1] </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+N">Ning Jiang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yi-Long Luo</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yulong 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="2407.02852v2-abstract-short" style="display: inline;"> In this paper, we prove the existence and uniqueness of the Knudsen layer equation imposed on Maxwell reflection boundary condition with full ranges of cutoff collision kernels and accommodation coefficients (i.e., $- 3 < 纬\leq 1$ and $0 \leq 伪_* \leq 1$, respectively) in the $L^\infty_{x,v}$ framework. Moreover, the solution enjoys the exponential decay… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.02852v2-abstract-full').style.display = 'inline'; document.getElementById('2407.02852v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.02852v2-abstract-full" style="display: none;"> In this paper, we prove the existence and uniqueness of the Knudsen layer equation imposed on Maxwell reflection boundary condition with full ranges of cutoff collision kernels and accommodation coefficients (i.e., $- 3 < 纬\leq 1$ and $0 \leq 伪_* \leq 1$, respectively) in the $L^\infty_{x,v}$ framework. Moreover, the solution enjoys the exponential decay $\exp \{- c x^\frac{2}{3 - 纬} - c |v|^2 \}$ for some $c > 0$. In order to study the general angular cutoff collision kernel $-3 < 纬\leq 1$, we should introduce a $(x,v)$-mixed weight $蟽$. The biggest difficulty in this paper is the nondissipative boundary condition, hence, the boundary temperature and velocity $(T_w, u_w)$ on $\{ x = 0 \}$ and $(T, \mathfrak{u})$ on $\{ x = + \infty \}$ do not guarantee the nonnegativity of the $L^2$ boundary energy. We also do not assume that $(T_w, u_w)$ and $(T, \mathfrak{u})$ are very closed to each other. We first derive the Nondissipative boundary lemma to pull the boundary energy to the interior weighted $L^2$ norms with higher power of $x$-polynomial weights. Then a so-called spatial-velocity indices iteration approach is developed to shift the higher power $x$-polynomial weights to $|v|$-polynomial weights. Finally, we construct an interleaved iteration process such that the boundary energy is successfully dominated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.02852v2-abstract-full').style.display = 'none'; document.getElementById('2407.02852v2-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 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">88 pages, two figures, all comments wellcome! Yulong Wu made important comtributions on the revised version. So his name is added as co-author</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.15107">arXiv:2405.15107</a> <span> [<a href="https://arxiv.org/pdf/2405.15107">pdf</a>, <a href="https://arxiv.org/ps/2405.15107">ps</a>, <a href="https://arxiv.org/format/2405.15107">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="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Is Algorithmic Stability Testable? A Unified Framework under Computational Constraints </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetian Luo</a>, <a href="/search/math?searchtype=author&query=Barber%2C+R+F">Rina Foygel Barber</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="2405.15107v1-abstract-short" style="display: inline;"> Algorithmic stability is a central notion in learning theory that quantifies the sensitivity of an algorithm to small changes in the training data. If a learning algorithm satisfies certain stability properties, this leads to many important downstream implications, such as generalization, robustness, and reliable predictive inference. Verifying that stability holds for a particular algorithm is th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15107v1-abstract-full').style.display = 'inline'; document.getElementById('2405.15107v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.15107v1-abstract-full" style="display: none;"> Algorithmic stability is a central notion in learning theory that quantifies the sensitivity of an algorithm to small changes in the training data. If a learning algorithm satisfies certain stability properties, this leads to many important downstream implications, such as generalization, robustness, and reliable predictive inference. Verifying that stability holds for a particular algorithm is therefore an important and practical question. However, recent results establish that testing the stability of a black-box algorithm is impossible, given limited data from an unknown distribution, in settings where the data lies in an uncountably infinite space (such as real-valued data). In this work, we extend this question to examine a far broader range of settings, where the data may lie in any space -- for example, categorical data. We develop a unified framework for quantifying the hardness of testing algorithmic stability, which establishes that across all settings, if the available data is limited then exhaustive search is essentially the only universally valid mechanism for certifying algorithmic stability. Since in practice, any test of stability would naturally be subject to computational constraints, exhaustive search is impossible and so this implies fundamental limits on our ability to test the stability property for a black-box algorithm. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15107v1-abstract-full').style.display = 'none'; document.getElementById('2405.15107v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.18967">arXiv:2404.18967</a> <span> [<a href="https://arxiv.org/pdf/2404.18967">pdf</a>, <a href="https://arxiv.org/ps/2404.18967">ps</a>, <a href="https://arxiv.org/format/2404.18967">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On the Boston's Unramified Fontaine-Mazur Conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yufan Luo</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="2404.18967v1-abstract-short" style="display: inline;"> This paper studies the Unramified Fontaine-Mazur Conjecture for $ p $-adic Galois representations and its generalizations. We prove some basic cases of the conjecture and provide some useful criterions for verifying it. In addition, we propose several different strategies to attack the conjecture and reduce it to some special cases. We also prove many new results of the conjecture in the two-dimen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.18967v1-abstract-full').style.display = 'inline'; document.getElementById('2404.18967v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.18967v1-abstract-full" style="display: none;"> This paper studies the Unramified Fontaine-Mazur Conjecture for $ p $-adic Galois representations and its generalizations. We prove some basic cases of the conjecture and provide some useful criterions for verifying it. In addition, we propose several different strategies to attack the conjecture and reduce it to some special cases. We also prove many new results of the conjecture in the two-dimensional case. Furthermore, we also study the unramified Galois deformation rings. Assuming the Unramified Fontaine-Mazur conjecture, we prove that the generic fiber of the unramified deformation ring is a finite direct product of fields. In particular, the unramified deformation ring has only finitely many $\overline{\mathbb{Q}}_{p}$-valued points. We also give some counterexamples to the so-called dimension conjecture for Galois deformation rings assuming the conjecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.18967v1-abstract-full').style.display = 'none'; document.getElementById('2404.18967v1-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> 29 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">44 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11F80; 11R32; 20E18; 22E35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.10723">arXiv:2404.10723</a> <span> [<a href="https://arxiv.org/pdf/2404.10723">pdf</a>, <a href="https://arxiv.org/ps/2404.10723">ps</a>, <a href="https://arxiv.org/format/2404.10723">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On the moduli description of ramified unitary Local models of signature (n-1,1) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yu Luo</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="2404.10723v1-abstract-short" style="display: inline;"> We give a moduli description for the ramified unitary local model of signature (n-1,1) with any parahoric level structure when the residue field has characteristic not equals 2, confirming a conjecture of Smithling. As applications, we can present moduli descriptions for: (1) ramified unitary Pappas-Zhu local models with any parahoric level; (2) the irreducible components of their special fiber wi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.10723v1-abstract-full').style.display = 'inline'; document.getElementById('2404.10723v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.10723v1-abstract-full" style="display: none;"> We give a moduli description for the ramified unitary local model of signature (n-1,1) with any parahoric level structure when the residue field has characteristic not equals 2, confirming a conjecture of Smithling. As applications, we can present moduli descriptions for: (1) ramified unitary Pappas-Zhu local models with any parahoric level; (2) the irreducible components of their special fiber with maximal parahoric level; (3) integral model of ramified unitary Shimura varieties for any (quasi-)parahoric level. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.10723v1-abstract-full').style.display = 'none'; document.getElementById('2404.10723v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">arXiv admin note: text overlap with arXiv:1405.1079 by other authors</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.06061">arXiv:2404.06061</a> <span> [<a href="https://arxiv.org/pdf/2404.06061">pdf</a>, <a href="https://arxiv.org/ps/2404.06061">ps</a>, <a href="https://arxiv.org/format/2404.06061">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 preconditioned iteration method for solving saddle point problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+J">Juan Zhang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yiyi Luo</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="2404.06061v1-abstract-short" style="display: inline;"> This paper introduces a preconditioned method designed to comprehensively address the saddle point system with the aim of improving convergence efficiency. In the preprocessor construction phase, a technical approach for solving the approximate inverse matrix of sparse matrices is presented. The effectiveness of the proposed method is demonstrated through numerical examples, emphasizing its effica… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06061v1-abstract-full').style.display = 'inline'; document.getElementById('2404.06061v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.06061v1-abstract-full" style="display: none;"> This paper introduces a preconditioned method designed to comprehensively address the saddle point system with the aim of improving convergence efficiency. In the preprocessor construction phase, a technical approach for solving the approximate inverse matrix of sparse matrices is presented. The effectiveness of the proposed method is demonstrated through numerical examples, emphasizing its efficacy in approximating the inverse matrix. Furthermore, the preprocessing technology includes a low-rank processing step, effectively reducing algorithmic complexity. Numerical experiments validate the effectiveness and feasibility of PSLR-GMRES in solving the saddle point system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06061v1-abstract-full').style.display = 'none'; document.getElementById('2404.06061v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.06018">arXiv:2404.06018</a> <span> [<a href="https://arxiv.org/pdf/2404.06018">pdf</a>, <a href="https://arxiv.org/ps/2404.06018">ps</a>, <a href="https://arxiv.org/format/2404.06018">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"> Preprocessed GMRES for fast solution of linear equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+J">Juan Zhang</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yiyi Luo</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="2404.06018v1-abstract-short" style="display: inline;"> The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an outer iteration to solve the linear equation system. These three algorithms can indirectly generate preprocessing matrices, which are used for solving equation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06018v1-abstract-full').style.display = 'inline'; document.getElementById('2404.06018v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.06018v1-abstract-full" style="display: none;"> The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an outer iteration to solve the linear equation system. These three algorithms can indirectly generate preprocessing matrices, which are used for solving equation systems. In addition, we provide corresponding convergence analysis and numerical examples. Through numerical examples, we demonstrate the effectiveness and feasibility of these preprocessing methods. Furthermore, in the Kaczmarz algorithm, we introduce both constant step size and adaptive step size, and extend the parameter range of the Kaczmarz algorithm to $伪\in(0,\infty)$. We also study the solution rate of linear equation systems using different step sizes. Numerical examples show that both constant step size and adaptive step size have higher solution efficiency than the solving algorithm without preprocessing. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06018v1-abstract-full').style.display = 'none'; document.getElementById('2404.06018v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.16838">arXiv:2403.16838</a> <span> [<a href="https://arxiv.org/pdf/2403.16838">pdf</a>, <a href="https://arxiv.org/format/2403.16838">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</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"> Thompson's group $F$, tangles, and link homology </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Krushkal%2C+V">Vyacheslav Krushkal</a>, <a href="/search/math?searchtype=author&query=Liles%2C+L">Louisa Liles</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yangxiao Luo</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="2403.16838v1-abstract-short" style="display: inline;"> We extend a construction of Jones to associate $(n, n)$-tangles with elements of Thompson's group $F$ and prove that it is asymptotically faithful as $n \to\infty$. Using this construction we show that the oriented Thompson group $\vec F$ admits a lax group action on a category of Khovanov's chain complexes. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.16838v1-abstract-full" style="display: none;"> We extend a construction of Jones to associate $(n, n)$-tangles with elements of Thompson's group $F$ and prove that it is asymptotically faithful as $n \to\infty$. Using this construction we show that the oriented Thompson group $\vec F$ admits a lax group action on a category of Khovanov's chain complexes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.16838v1-abstract-full').style.display = 'none'; document.getElementById('2403.16838v1-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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, many 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/2403.11062">arXiv:2403.11062</a> <span> [<a href="https://arxiv.org/pdf/2403.11062">pdf</a>, <a href="https://arxiv.org/format/2403.11062">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"> A Simple Mixture Policy Parameterization for Improving Sample Efficiency of CVaR Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yudong Luo</a>, <a href="/search/math?searchtype=author&query=Pan%2C+Y">Yangchen Pan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+H">Han Wang</a>, <a href="/search/math?searchtype=author&query=Torr%2C+P">Philip Torr</a>, <a href="/search/math?searchtype=author&query=Poupart%2C+P">Pascal Poupart</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="2403.11062v3-abstract-short" style="display: inline;"> Reinforcement learning algorithms utilizing policy gradients (PG) to optimize Conditional Value at Risk (CVaR) face significant challenges with sample inefficiency, hindering their practical applications. This inefficiency stems from two main facts: a focus on tail-end performance that overlooks many sampled trajectories, and the potential of gradient vanishing when the lower tail of the return di… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.11062v3-abstract-full').style.display = 'inline'; document.getElementById('2403.11062v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.11062v3-abstract-full" style="display: none;"> Reinforcement learning algorithms utilizing policy gradients (PG) to optimize Conditional Value at Risk (CVaR) face significant challenges with sample inefficiency, hindering their practical applications. This inefficiency stems from two main facts: a focus on tail-end performance that overlooks many sampled trajectories, and the potential of gradient vanishing when the lower tail of the return distribution is overly flat. To address these challenges, we propose a simple mixture policy parameterization. This method integrates a risk-neutral policy with an adjustable policy to form a risk-averse policy. By employing this strategy, all collected trajectories can be utilized for policy updating, and the issue of vanishing gradients is counteracted by stimulating higher returns through the risk-neutral component, thus lifting the tail and preventing flatness. Our empirical study reveals that this mixture parameterization is uniquely effective across a variety of benchmark domains. Specifically, it excels in identifying risk-averse CVaR policies in some Mujoco environments where the traditional CVaR-PG fails to learn a reasonable policy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.11062v3-abstract-full').style.display = 'none'; document.getElementById('2403.11062v3-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">RLC 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.01550">arXiv:2403.01550</a> <span> [<a href="https://arxiv.org/pdf/2403.01550">pdf</a>, <a href="https://arxiv.org/format/2403.01550">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Discrete Mathematics">cs.DM</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Spectral Antisymmetry of Twisted Graph Adjacency </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Ye Luo</a>, <a href="/search/math?searchtype=author&query=Roy%2C+A">Arindam Roy</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="2403.01550v2-abstract-short" style="display: inline;"> We address a prime counting problem across the homology classes of a graph, presenting a graph-theoretical Dirichlet-type analogue of the prime number theorem. The main machinery we have developed and employed is a spectral antisymmetry theorem, revealing that the spectra of the twisted graph adjacency matrices have an antisymmetric distribution over the character group of the graph with a special… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.01550v2-abstract-full').style.display = 'inline'; document.getElementById('2403.01550v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.01550v2-abstract-full" style="display: none;"> We address a prime counting problem across the homology classes of a graph, presenting a graph-theoretical Dirichlet-type analogue of the prime number theorem. The main machinery we have developed and employed is a spectral antisymmetry theorem, revealing that the spectra of the twisted graph adjacency matrices have an antisymmetric distribution over the character group of the graph with a special character called the canonical character being an extremum. Additionally, we derive some trace formulas based on the twisted adjacency matrices as part of our analysis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.01550v2-abstract-full').style.display = 'none'; document.getElementById('2403.01550v2-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">51 pages. The setting has been changed from simple graphs to graphs. Some terminologies have also been changed to more commonly used terms, such as "paths" changed to "walks"</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C50; 05C38; 11M41 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.18903">arXiv:2402.18903</a> <span> [<a href="https://arxiv.org/pdf/2402.18903">pdf</a>, <a href="https://arxiv.org/format/2402.18903">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"> An Adaptive Hybrid Genetic and Large Neighborhood Search Approach for Multi-Attribute Vehicle Routing Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+W">Weiting Liu</a>, <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yunqi Luo</a>, <a href="/search/math?searchtype=author&query=Yu%2C+Y">Yugang Yu</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="2402.18903v1-abstract-short" style="display: inline;"> Known for its dynamic utilization of destroy and repair operators, the Adaptive Large Neighborhood Search (ALNS) seeks to unearth high-quality solutions and has thus gained widespread acceptance as a meta-heuristic tool for tackling complex Combinatorial Optimization Problems (COPs). However, challenges arise when applying uniform parameters and acceptance criteria to diverse instances of the same… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.18903v1-abstract-full').style.display = 'inline'; document.getElementById('2402.18903v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.18903v1-abstract-full" style="display: none;"> Known for its dynamic utilization of destroy and repair operators, the Adaptive Large Neighborhood Search (ALNS) seeks to unearth high-quality solutions and has thus gained widespread acceptance as a meta-heuristic tool for tackling complex Combinatorial Optimization Problems (COPs). However, challenges arise when applying uniform parameters and acceptance criteria to diverse instances of the same COP, resulting in inconsistent performance outcomes. To address this inherent limitation, we propose the Adaptive Hybrid Genetic Search and Large Neighborhood Search (AHGSLNS), a novel approach designed to adapt ALNS parameters and acceptance criteria to the specific nuances of distinct COP instances. Our evaluation focuses on the Multi-Attribute Vehicle Routing Problem, a classical COP prevalent in real-world semi-automated storage and retrieval robotics systems. Empirical findings showcase that AHGSLNS not only competes effectively with ALNS under varying parameters but also exhibits superior performance in terms of convergence and stability. In alignment with our dedication to research transparency, the implementation of the proposed approach will be made publicly available. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.18903v1-abstract-full').style.display = 'none'; document.getElementById('2402.18903v1-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> 29 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.17443">arXiv:2402.17443</a> <span> [<a href="https://arxiv.org/pdf/2402.17443">pdf</a>, <a href="https://arxiv.org/ps/2402.17443">ps</a>, <a href="https://arxiv.org/format/2402.17443">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"> The classification and representations of positive definite ternary quadratic forms of level 4N </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yifan Luo</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+H">Haigang 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="2402.17443v1-abstract-short" style="display: inline;"> Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive definite ternary quadratic forms of level $4N$ explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.17443v1-abstract-full').style.display = 'inline'; document.getElementById('2402.17443v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.17443v1-abstract-full" style="display: none;"> Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive definite ternary quadratic forms of level $4N$ explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level $4N$, which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level $4N$. As applications, we derive an explicit base of Eisenstein series space of modular forms of weight $3/2$ and level $4N$, and give new proofs of some interesting identities involving representation number of ternary quadratic forms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.17443v1-abstract-full').style.display = 'none'; document.getElementById('2402.17443v1-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> 27 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11E20; 11R52; 11R29; 11F50; 11F37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.12709">arXiv:2402.12709</a> <span> [<a href="https://arxiv.org/pdf/2402.12709">pdf</a>, <a href="https://arxiv.org/format/2402.12709">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On quasiconformal non-equivalence of gasket Julia sets and limit sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yusheng Luo</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yongquan 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="2402.12709v1-abstract-short" style="display: inline;"> This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.12709v1-abstract-full" style="display: none;"> This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12709v1-abstract-full').style.display = 'none'; document.getElementById('2402.12709v1-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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 without references, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> MPIM-Bonn-2024; Stony Brook IMS #2024/02 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.07388">arXiv:2402.07388</a> <span> [<a href="https://arxiv.org/pdf/2402.07388">pdf</a>, <a href="https://arxiv.org/ps/2402.07388">ps</a>, <a href="https://arxiv.org/format/2402.07388">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> <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="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> The Limits of Assumption-free Tests for Algorithm Performance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yuetian Luo</a>, <a href="/search/math?searchtype=author&query=Barber%2C+R+F">Rina Foygel Barber</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="2402.07388v3-abstract-short" style="display: inline;"> Algorithm evaluation and comparison are fundamental questions in machine learning and statistics -- how well does an algorithm perform at a given modeling task, and which algorithm performs best? Many methods have been developed to assess algorithm performance, often based around cross-validation type strategies, retraining the algorithm of interest on different subsets of the data and assessing i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.07388v3-abstract-full').style.display = 'inline'; document.getElementById('2402.07388v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.07388v3-abstract-full" style="display: none;"> Algorithm evaluation and comparison are fundamental questions in machine learning and statistics -- how well does an algorithm perform at a given modeling task, and which algorithm performs best? Many methods have been developed to assess algorithm performance, often based around cross-validation type strategies, retraining the algorithm of interest on different subsets of the data and assessing its performance on the held-out data points. Despite the broad use of such procedures, the theoretical properties of these methods are not yet fully understood. In this work, we explore some fundamental limits for answering these questions with limited amounts of data. In particular, we make a distinction between two questions: how good is an algorithm $A$ at the problem of learning from a training set of size $n$, versus, how good is a particular fitted model produced by running $A$ on a particular training data set of size $n$? Our main results prove that, for any test that treats the algorithm $A$ as a ``black box'' (i.e., we can only study the behavior of $A$ empirically), there is a fundamental limit on our ability to carry out inference on the performance of $A$, unless the number of available data points $N$ is many times larger than the sample size $n$ of interest. (On the other hand, evaluating the performance of a particular fitted model is easy as long as a holdout data set is available -- that is, as long as $N-n$ is not too small.) We also ask whether an assumption of algorithmic stability might be sufficient to circumvent this hardness result. Surprisingly, we find that this is not the case: the same hardness result still holds for the problem of evaluating the performance of $A$, aside from a high-stability regime where fitted models are essentially nonrandom. Finally, we also establish similar hardness results for the problem of comparing multiple algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.07388v3-abstract-full').style.display = 'none'; document.getElementById('2402.07388v3-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.06197">arXiv:2402.06197</a> <span> [<a href="https://arxiv.org/pdf/2402.06197">pdf</a>, <a href="https://arxiv.org/ps/2402.06197">ps</a>, <a href="https://arxiv.org/format/2402.06197">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Recurrence relations of Exceptional Laurent biorthogonal polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+Y">Yu Luo</a>, <a href="/search/math?searchtype=author&query=Tsujimoto%2C+S">Satoshi Tsujimoto</a>, <a href="/search/math?searchtype=author&query=Yang%2C+H">Hao Yang</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="2402.06197v1-abstract-short" style="display: inline;"> Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue problems. In this paper, we discuss the recurrence relations satisfied by these exceptional Laurent biorthogonal polynomials and provide a type of recurrence re… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.06197v1-abstract-full').style.display = 'inline'; document.getElementById('2402.06197v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.06197v1-abstract-full" style="display: none;"> Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue problems. In this paper, we discuss the recurrence relations satisfied by these exceptional Laurent biorthogonal polynomials and provide a type of recurrence relations with $3l_0+4$ terms explicitly, where the parameter $l_0$ corresponds to the degree of the polynomial part in the seed function used in the Darboux transformation. In the proof of these recurrence relations, the backward operator which maps an exceptional polynomial into a classical one plays a significant role. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.06197v1-abstract-full').style.display = 'none'; document.getElementById('2402.06197v1-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">32 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 33C45; 33C47; 42C05 </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" 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