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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Lin%2C+H&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Lin%2C+H&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Lin%2C+H&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Lin%2C+H&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Lin%2C+H&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Lin%2C+H&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li> <a href="/search/?searchtype=author&query=Lin%2C+H&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/2503.10984">arXiv:2503.10984</a> <span> [<a href="https://arxiv.org/pdf/2503.10984">pdf</a>, <a href="https://arxiv.org/format/2503.10984">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Other Statistics">stat.OT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> The Problem of the Priors, or Posteriors? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hanti Lin</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="2503.10984v1-abstract-short" style="display: inline;"> The problem of the priors is well known: it concerns the challenge of identifying norms that govern one's prior credences. I argue that a key to addressing this problem lies in considering what I call the problem of the posteriors -- the challenge of identifying norms that directly govern one's posterior credences, which then induce constraints on the priors via the diachronic requirement of condi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.10984v1-abstract-full').style.display = 'inline'; document.getElementById('2503.10984v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.10984v1-abstract-full" style="display: none;"> The problem of the priors is well known: it concerns the challenge of identifying norms that govern one's prior credences. I argue that a key to addressing this problem lies in considering what I call the problem of the posteriors -- the challenge of identifying norms that directly govern one's posterior credences, which then induce constraints on the priors via the diachronic requirement of conditionalization. This forward-looking approach can be summarized as: Think ahead, work backward. Although this idea can be traced to Freedman (1963), Carnap (1963), and Shimony (1970), it has received little attention in philosophy. In this paper, I initiate a systematic defense of forward-looking Bayesianism, addressing potential objections from more traditional views (both subjectivist and objectivist) and arguing for its advantages. In particular, I develop a specific approach to forward-looking Bayesianism -- one that treats the convergence of posterior credences to the truth as a fundamental rather than derived normative requirement. This approach, called convergentist Bayesianism, is argued to be crucial for a Bayesian foundation of Ockham's razor and related inference methods in statistics and machine learning. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.10984v1-abstract-full').style.display = 'none'; document.getElementById('2503.10984v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 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/2502.17802">arXiv:2502.17802</a> <span> [<a href="https://arxiv.org/pdf/2502.17802">pdf</a>, <a href="https://arxiv.org/ps/2502.17802">ps</a>, <a href="https://arxiv.org/format/2502.17802">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Almost Representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huaxin Lin</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="2502.17802v1-abstract-short" style="display: inline;"> Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a countable discrete amenable group. We prove the following: For any $蔚>0,$ any finite subset ${\cal F}\subset G,$ and $0<蟽\le 1,$ there exists $未>0,$ finite subsets… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.17802v1-abstract-full').style.display = 'inline'; document.getElementById('2502.17802v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.17802v1-abstract-full" style="display: none;"> Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a countable discrete amenable group. We prove the following: For any $蔚>0,$ any finite subset ${\cal F}\subset G,$ and $0<蟽\le 1,$ there exists $未>0,$ finite subsets ${\cal G}\subset G$ and ${\cal S}\subset {\bf C}[G]$ satisfying the following property: For any map $蠁: G\to U(B(H))$ such that $$ \|蠁(fg)-蠁(f)蠁(g)\|<未\,\,\,for\,\, all\,\, f,g\in {\cal G}\,\,\, and \,\,\, \|蟺\circ \tilde 蠁(x)\|\ge 蟽\|x\|\,\,\, for\,\, all\,\, x\in {\cal S}, $$ there is a group homomorphism $h: G\to U(B(H))$ such that $$ \|蠁(f)-h(f)\|<蔚\,\,\, for\,\,\, all\,\,\, f\in {\cal F}, $$ where $\tilde 蠁$ is the linear extension of $蠁$ on the group ring ${\bf C}[G]$ and $蟺: B(H)\to B(H)/{\cal K}$ is the quotient map. A counterexample is given that the fullness condition above cannot be removed. We actually prove a more general result for separable amenable $C^*$-algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.17802v1-abstract-full').style.display = 'none'; document.getElementById('2502.17802v1-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">MSC Class:</span> 46L35; 43A07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.18656">arXiv:2501.18656</a> <span> [<a href="https://arxiv.org/pdf/2501.18656">pdf</a>, <a href="https://arxiv.org/ps/2501.18656">ps</a>, <a href="https://arxiv.org/format/2501.18656">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"> Extremal distance spectral radius of graphs with fixed size </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongying Lin</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+B">Bo 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="2501.18656v1-abstract-short" style="display: inline;"> Let $m$ be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size $m$. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eig… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.18656v1-abstract-full').style.display = 'inline'; document.getElementById('2501.18656v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.18656v1-abstract-full" style="display: none;"> Let $m$ be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size $m$. After partial results due to Friedland and Stanley, Rowlinson completely confirmed the conjecture. The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We investigate the problem to determine the connected graphs with minimum distance spectral radius in the class of graphs with size $m$. Given $m$, there is exactly one positive integer $n$ such that ${n-1\choose 2} <m\leq {n\choose 2}$. We establish some structural properties of the extremal graphs for all $m$ and solve the problem for ${n-1\choose 2}+\max\{\frac{n-6}{2},1\}\le m\leq {n\choose 2}$. We give a conjecture for the remaining case. To prove the main results, we also determine the the complements of forests of fixed order with large and small distance spectral radius. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.18656v1-abstract-full').style.display = 'none'; document.getElementById('2501.18656v1-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">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.13305">arXiv:2501.13305</a> <span> [<a href="https://arxiv.org/pdf/2501.13305">pdf</a>, <a href="https://arxiv.org/ps/2501.13305">ps</a>, <a href="https://arxiv.org/format/2501.13305">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> <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="Category Theory">math.CT</span> </div> </div> <p class="title is-5 mathjax"> RTT presentation of coideal subalgebra of quantized enveloping algebra of type CI </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yingwen Zhang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongda Lin</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.13305v1-abstract-short" style="display: inline;"> The pair consisting of a quantum group and its corresponding coideal subalgebra, known as a quantum symmetric pair, was developed independently by M. Noumi and G. Letzter through different approaches. The purpose of this paper is threefold. First, for symmetric pairs $(\mathfrak{sp}_{2n},\mathfrak{gl}_n)$, we construct a coideal subalgebra $U_q^{tw}(\mathfrak{gl}_n)$ of the quantized enveloping al… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.13305v1-abstract-full').style.display = 'inline'; document.getElementById('2501.13305v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.13305v1-abstract-full" style="display: none;"> The pair consisting of a quantum group and its corresponding coideal subalgebra, known as a quantum symmetric pair, was developed independently by M. Noumi and G. Letzter through different approaches. The purpose of this paper is threefold. First, for symmetric pairs $(\mathfrak{sp}_{2n},\mathfrak{gl}_n)$, we construct a coideal subalgebra $U_q^{tw}(\mathfrak{gl}_n)$ of the quantized enveloping algebra of type CI using the $R$-matrix presentation, based on the work of Noumi. Second, we derive a Poincar茅-Birkhoff-Witt(PBW) basis for $U_q^{tw}(\mathfrak{gl}_n)$ by the $\mathbb{A}$-form approach. As a consequence of the isomorphism btween $U_q^{tw}(\mathfrak{gl}_n)$ and the $\imath$quantum group $\mathcal{U}^{\imath}$, our method also yields the PBW basis for the $\imath$quantum group of type CI. Finally, as an application of the $R$-matrix presentation, we construct a Poisson algebra $\mathcal{P}_n$ associated with $U_q^{tw}(\mathfrak{gl}_n)$, and explicitly describe the action of the braid group $\mathcal{B}_n$ on the elements of $\mathcal{P}_n$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.13305v1-abstract-full').style.display = 'none'; document.getElementById('2501.13305v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <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">33 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/2501.05800">arXiv:2501.05800</a> <span> [<a href="https://arxiv.org/pdf/2501.05800">pdf</a>, <a href="https://arxiv.org/ps/2501.05800">ps</a>, <a href="https://arxiv.org/format/2501.05800">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"> Quantum Berezinian for the Twisted Super Yangian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongda Lin</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yongjie Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.05800v1-abstract-short" style="display: inline;"> Motivated by an open problem proposed in Molev's book \cite[Section 2.16, Example 16]{Mo07}, we investigate the quantum Berezinian $\mathfrak{B}^{tw}(u)$ associated with the twisted super Yangian, which is a coideal sub-superalgebra of the super Yangian of the general linear Lie superalgebra. We provide an explicit formulation of $\mathfrak{B}^{tw}(u)$, and we also construct the center of the twis… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.05800v1-abstract-full').style.display = 'inline'; document.getElementById('2501.05800v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.05800v1-abstract-full" style="display: none;"> Motivated by an open problem proposed in Molev's book \cite[Section 2.16, Example 16]{Mo07}, we investigate the quantum Berezinian $\mathfrak{B}^{tw}(u)$ associated with the twisted super Yangian, which is a coideal sub-superalgebra of the super Yangian of the general linear Lie superalgebra. We provide an explicit formulation of $\mathfrak{B}^{tw}(u)$, and we also construct the center of the twisted super Yangian. This construction enables us to define the special twisted super Yangian, which is isomorphic to the quotient of the twisted super Yangian by its center. Moreover, we demonstrate the quantum Sylvester theorem for both the generator matrix and the quantum Berezinian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.05800v1-abstract-full').style.display = 'none'; document.getElementById('2501.05800v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <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">37 pages, comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B37; 17D10; 20G42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.12787">arXiv:2412.12787</a> <span> [<a href="https://arxiv.org/pdf/2412.12787">pdf</a>, <a href="https://arxiv.org/ps/2412.12787">ps</a>, <a href="https://arxiv.org/format/2412.12787">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"> Maximize the Steklov eigenvalue of trees </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+D">Da 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="2412.12787v1-abstract-short" style="display: inline;"> We study the maximal Steklov eigenvalues of trees with given number of boundary vertices and total number of vertices. Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply-connected Riemannian manifolds of non-positive sectional curvature. Let $蟽_{k,\text{max}}(b, n)$ be the maximal of $k$-th Steklov eigenvalue of trees with $b$ leaves and $n$ vertices. We determine that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.12787v1-abstract-full').style.display = 'inline'; document.getElementById('2412.12787v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.12787v1-abstract-full" style="display: none;"> We study the maximal Steklov eigenvalues of trees with given number of boundary vertices and total number of vertices. Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply-connected Riemannian manifolds of non-positive sectional curvature. Let $蟽_{k,\text{max}}(b, n)$ be the maximal of $k$-th Steklov eigenvalue of trees with $b$ leaves and $n$ vertices. We determine that $$ 蟽_{2, \text{max}} (b, n) = \begin{cases} \frac{2}{n-1}, & b=2, n\geq 3, \frac{1}{r}, & b \geq 3, br + k, 3 - b \leq k \leq 2, r \in \mathbb{Z}_+, \frac{1}{r+1-\frac{1}{b}}, & b \geq 3, n = br + 2, r \in \mathbb{Z}_+, \end{cases} $$ and we characterize the trees attaining this bound. For $k \geq 3$, we show that $蟽_{k, \text{max}} (b, n) = 1$. We also give a lower bound on the maximal Steklov eigenvalues of trees with given diameter and total number of vertices. Our work can be regarded as a completion of the work by He--Hua [Upper bounds for the Steklov eigenvalues on trees, Calc. Var. Partial Differential Equations (2022)] and Yu--Yu [Monotonicity of Steklov eigenvalues on graphs and applications, Calc. Var. Partial Differential Equations (2024)]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.12787v1-abstract-full').style.display = 'none'; document.getElementById('2412.12787v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C05; 47A75; 49J40; 49R05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.12446">arXiv:2411.12446</a> <span> [<a href="https://arxiv.org/pdf/2411.12446">pdf</a>, <a href="https://arxiv.org/ps/2411.12446">ps</a>, <a href="https://arxiv.org/format/2411.12446">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Fiber products under toric flops and flips </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+T">Tsung-Chen Chen</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hui-Wen Lin</a>, <a href="/search/math?searchtype=author&query=Wang%2C+S">Sz-Sheng 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="2411.12446v2-abstract-short" style="display: inline;"> Let $危$ and $危'$ be two refinements of a fan $危_0$ and $f \colon X_危 \dashrightarrow X_{危'}$ be the birational map induced by $X_危 \rightarrow X_{危_0} \leftarrow X_{危'}$. We show that the graph closure $\overline螕_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality. In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theore… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12446v2-abstract-full').style.display = 'inline'; document.getElementById('2411.12446v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.12446v2-abstract-full" style="display: none;"> Let $危$ and $危'$ be two refinements of a fan $危_0$ and $f \colon X_危 \dashrightarrow X_{危'}$ be the birational map induced by $X_危 \rightarrow X_{危_0} \leftarrow X_{危'}$. We show that the graph closure $\overline螕_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality. In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_危\mathop{\times}\limits_{X_{危_0}}X_{危'}$ is in general not toric, though it is still irreducible and $X_{\rm red} = \overline螕_f$. A complete numerical criterion to ensure $X = X_{\rm red}$ is given for 3-folds, which is fulfilled when $X_危$ has at most terminal singularities. In this case, we further conclude that $X$ is normal. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12446v2-abstract-full').style.display = 'none'; document.getElementById('2411.12446v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.22632">arXiv:2410.22632</a> <span> [<a href="https://arxiv.org/pdf/2410.22632">pdf</a>, <a href="https://arxiv.org/ps/2410.22632">ps</a>, <a href="https://arxiv.org/format/2410.22632">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"> Upper bounds of Steklov eigenvalues on graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=Liu%2C+L">Lianping Liu</a>, <a href="/search/math?searchtype=author&query=You%2C+Z">Zhe You</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+D">Da 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="2410.22632v1-abstract-short" style="display: inline;"> Let $螖$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $蟽_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $蟽_2 = \mathcal{O}\left(\frac{螖(g+1)^3}{|B|}\right)$ for graphs of orientable genus $g$ if… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22632v1-abstract-full').style.display = 'inline'; document.getElementById('2410.22632v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.22632v1-abstract-full" style="display: none;"> Let $螖$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $蟽_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $蟽_2 = \mathcal{O}\left(\frac{螖(g+1)^3}{|B|}\right)$ for graphs of orientable genus $g$ if $|B| \geq \max\{3 \sqrt{g},|V|^{\frac{1}{4} + 蔚}, 9\}$ for some $蔚> 0$. This can be seen as a discrete analogue of Karpukhin's bound. Secondly, we prove that $蟽_2 \leq \frac{8螖+4X}{|B|}$ based on planar crossing number $X$. Thirdly, we show that $蟽_2 \leq \frac{|B|}{|B|-1} \cdot 未_B$, where $未_B$ denotes the minimum degree for boundary vertices in $B$. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22632v1-abstract-full').style.display = 'none'; document.getElementById('2410.22632v1-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> <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, 2 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/2410.21887">arXiv:2410.21887</a> <span> [<a href="https://arxiv.org/pdf/2410.21887">pdf</a>, <a href="https://arxiv.org/format/2410.21887">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Graphs with positive Lin-Lu-Yau curvature without Quadrilateral </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=You%2C+Z">Zhe You</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="2410.21887v1-abstract-short" style="display: inline;"> The definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using graph Laplacian was given in M眉nch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtain… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21887v1-abstract-full').style.display = 'inline'; document.getElementById('2410.21887v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.21887v1-abstract-full" style="display: none;"> The definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using graph Laplacian was given in M眉nch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtained from a triangle and $K_{1,3}$ by adding a matching between every leave of $K_{1,3}$ and a vertex of the triangle. In this paper, we classify all of connected $C_4$-free graphs with positive LLY curvature for minimum degree at least 2: the cycles $C_3,C_5$, the friendship graphs $F_2,F_3$ and $T$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21887v1-abstract-full').style.display = 'none'; document.getElementById('2410.21887v1-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> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C99; 05C81; 51F99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.21755">arXiv:2410.21755</a> <span> [<a href="https://arxiv.org/pdf/2410.21755">pdf</a>, <a href="https://arxiv.org/ps/2410.21755">ps</a>, <a href="https://arxiv.org/format/2410.21755">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"> Braid group action and quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+X">Xianghua Wu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongda Lin</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian 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="2410.21755v2-abstract-short" style="display: inline;"> In this paper, we investigate the structure of the quantum affine superalgebra associated with the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m\geqslant 1$. The Drinfeld-Jimbo presentation for this algebra, denoted as $U_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$, was originally introduced by H. Yamane. We provide the definition of the Drinfeld presentation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21755v2-abstract-full').style.display = 'inline'; document.getElementById('2410.21755v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.21755v2-abstract-full" style="display: none;"> In this paper, we investigate the structure of the quantum affine superalgebra associated with the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m\geqslant 1$. The Drinfeld-Jimbo presentation for this algebra, denoted as $U_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$, was originally introduced by H. Yamane. We provide the definition of the Drinfeld presentation $\mathcal{U}_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$. To establish the isomorphism between the Drinfeld-Jimbo presentation and the Drinfeld presentation of the quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$, we introduce a braid group action to define quantum root vectors of quantum superalgebras. Specifically, we present an efficient method for verifying the isomorphism between two presentations of the quantum affine superalgebra associated with the type $\mathfrak{osp}(2m+1|2n)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21755v2-abstract-full').style.display = 'none'; document.getElementById('2410.21755v2-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 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">32 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/2410.21116">arXiv:2410.21116</a> <span> [<a href="https://arxiv.org/pdf/2410.21116">pdf</a>, <a href="https://arxiv.org/ps/2410.21116">ps</a>, <a href="https://arxiv.org/format/2410.21116">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> The existence of biregular spanning subgraphs in bipartite graphs via spectral radius </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fan%2C+D">Dandan Fan</a>, <a href="/search/math?searchtype=author&query=Gu%2C+X">Xiaofeng Gu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2410.21116v1-abstract-short" style="display: inline;"> Biregular bipartite graphs have been proven to have similar edge distributions to random bipartite graphs and thus have nice pseudorandomness and expansion properties. Thus it is quite desirable to find a biregular bipartite spanning subgraph in a given bipartite graph. In fact, a theorem of Ore implies a structural characterization of such subgraphs in bipartite graphs. In this paper, we demonstr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21116v1-abstract-full').style.display = 'inline'; document.getElementById('2410.21116v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.21116v1-abstract-full" style="display: none;"> Biregular bipartite graphs have been proven to have similar edge distributions to random bipartite graphs and thus have nice pseudorandomness and expansion properties. Thus it is quite desirable to find a biregular bipartite spanning subgraph in a given bipartite graph. In fact, a theorem of Ore implies a structural characterization of such subgraphs in bipartite graphs. In this paper, we demonstrate the existence of biregular bipartite spanning subgraphs in bipartite graphs by employing spectral radius. We also study the existence of spanning trees with restricted degrees and edge-disjoint spanning trees in bipartite graphs via spectral radius. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21116v1-abstract-full').style.display = 'none'; document.getElementById('2410.21116v1-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 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/2410.00648">arXiv:2410.00648</a> <span> [<a href="https://arxiv.org/pdf/2410.00648">pdf</a>, <a href="https://arxiv.org/ps/2410.00648">ps</a>, <a href="https://arxiv.org/format/2410.00648">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> A strengthening on consecutive odd cycles in graphs of given minimum degree </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hao Lin</a>, <a href="/search/math?searchtype=author&query=Wang%2C+G">Guanghui Wang</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+W">Wenling 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="2410.00648v1-abstract-short" style="display: inline;"> Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when $k$ is even. In this paper, we confirm this conjecture for any $k\in \mathbb N$. Moreover, we also improve some previous results about cycl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00648v1-abstract-full').style.display = 'inline'; document.getElementById('2410.00648v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.00648v1-abstract-full" style="display: none;"> Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when $k$ is even. In this paper, we confirm this conjecture for any $k\in \mathbb N$. Moreover, we also improve some previous results about cycles of consecutive lengths. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00648v1-abstract-full').style.display = 'none'; document.getElementById('2410.00648v1-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 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">10 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/2409.12436">arXiv:2409.12436</a> <span> [<a href="https://arxiv.org/pdf/2409.12436">pdf</a>, <a href="https://arxiv.org/format/2409.12436">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"> Approximate Resolution of Stochastic Choice-based Discrete Planning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jiajie Zhang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y+H">Yun Hui Lin</a>, <a href="/search/math?searchtype=author&query=Berbeglia%2C+G">Gerardo Berbeglia</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="2409.12436v1-abstract-short" style="display: inline;"> Stochastic choice-based discrete planning is a broad class of decision-making problems characterized by a sequential decision-making process involving a planner and a group of customers. The firm or planner first decides a subset of options to offer to the customers, who, in turn, make selections based on their utilities of those options. This problem has extensive applications in many areas, incl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.12436v1-abstract-full').style.display = 'inline'; document.getElementById('2409.12436v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.12436v1-abstract-full" style="display: none;"> Stochastic choice-based discrete planning is a broad class of decision-making problems characterized by a sequential decision-making process involving a planner and a group of customers. The firm or planner first decides a subset of options to offer to the customers, who, in turn, make selections based on their utilities of those options. This problem has extensive applications in many areas, including assortment planning, product line design, and facility location. A key feature of these problems is that the firm cannot fully observe the customers' utilities or preferences, which results in intrinsic and idiosyncratic uncertainties. Most works in the literature have studied a specific type of uncertainty, resulting in customized decision models that are subsequently tackled using ad-hoc algorithms designed to exploit the specific model structure. In this paper we propose a modeling framework capable of solving this family of sequential problems that works for a large variety of uncertainties. We then leverage an approximation scheme and develop an adaptable mixed-integer linear programming method. To speed up the solution process, we further develop an efficient decomposition approach. We show that our solution framework can yield solutions proven to be (near-)optimal for a broad class of problems. We illustrate this by applying our approach to three classical application problems: constrained assortment optimization and two facility location problems. Through extensive computational experiments, we demonstrate the performance of our approach in terms of both solution quality and computational speed, and provide computational insights. In particular, when we use our method to solve the constrained assortment optimization problem under the Exponomial choice model, it improves the state-of-the-art. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.12436v1-abstract-full').style.display = 'none'; document.getElementById('2409.12436v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 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/2408.14102">arXiv:2408.14102</a> <span>  </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Spectral Tur谩n problem for $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongying Lin</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="2408.14102v3-abstract-short" style="display: inline;"> We determine the maximum index and the signed graphs with the maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 3$, as well as the second maximum index and the signed graphs with the second maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 4$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.14102v3-abstract-full" style="display: none;"> We determine the maximum index and the signed graphs with the maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 3$, as well as the second maximum index and the signed graphs with the second maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 4$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.14102v3-abstract-full').style.display = 'none'; document.getElementById('2408.14102v3-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 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">The conclusion is not right. A new version is under work</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.09823">arXiv:2408.09823</a> <span> [<a href="https://arxiv.org/pdf/2408.09823">pdf</a>, <a href="https://arxiv.org/format/2408.09823">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="Metric Geometry">math.MG</span> </div> </div> <p class="title is-5 mathjax"> Graphs with nonnegative Bakry-脡mery curvature without Quadrilateral </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=You%2C+Z">Zhe You</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="2408.09823v1-abstract-short" style="display: inline;"> The definition of Ricci curvature on graphs in Bakry-脡mery's sense based on curvature dimension condition was introduced by Lin and Yau [\emph{Math. Res. Lett.}, 2010]. Hua and Lin [\emph{Comm. Anal. Geom.}, 2019] classified unweighted graphs satisfying the curvature dimension condition $CD(0,\infty)$ whose girth are at least five. In this paper, we classify all of connected unweighted normalized… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09823v1-abstract-full').style.display = 'inline'; document.getElementById('2408.09823v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.09823v1-abstract-full" style="display: none;"> The definition of Ricci curvature on graphs in Bakry-脡mery's sense based on curvature dimension condition was introduced by Lin and Yau [\emph{Math. Res. Lett.}, 2010]. Hua and Lin [\emph{Comm. Anal. Geom.}, 2019] classified unweighted graphs satisfying the curvature dimension condition $CD(0,\infty)$ whose girth are at least five. In this paper, we classify all of connected unweighted normalized $C_4$-free graphs satisfying curvature dimension condition $CD(0,\infty)$ for minimum degree at least 2 and the case with non-normalized Laplacian without degree condition.. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09823v1-abstract-full').style.display = 'none'; document.getElementById('2408.09823v1-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">MSC Class:</span> 05C99; 51F99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.10398">arXiv:2407.10398</a> <span> [<a href="https://arxiv.org/pdf/2407.10398">pdf</a>, <a href="https://arxiv.org/ps/2407.10398">ps</a>, <a href="https://arxiv.org/format/2407.10398">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"> Proof of Lew's conjecture on the spectral gaps of simplicial complexes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhan%2C+X">Xiongfeng Zhan</a>, <a href="/search/math?searchtype=author&query=Huang%2C+X">Xueyi Huang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2407.10398v2-abstract-short" style="display: inline;"> As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let $X$ be a simplicial complex on vertex set $V$ of size $n$, and let $X(k)$ denote the set of all $k$-dimensional simplices of $X$. The $k$-th spectral gap $渭_k(X)$ is the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. For an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.10398v2-abstract-full').style.display = 'inline'; document.getElementById('2407.10398v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.10398v2-abstract-full" style="display: none;"> As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let $X$ be a simplicial complex on vertex set $V$ of size $n$, and let $X(k)$ denote the set of all $k$-dimensional simplices of $X$. The $k$-th spectral gap $渭_k(X)$ is the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. For any $k\geq -1$, Lew [J. Combin. Theory Ser. A 169 (2020) 105127] established a lower bound for $渭_k(X)$: $$渭_k(X)\geq (d+1)\left(\min_{蟽\in X(k)}掳_X(蟽)+k+1\right)-dn\geq (d+1)(k+1)-dn,$$ where $掳_X(蟽)$ and $d$ denote the degree of $蟽$ in $X$ and the maximal dimension of a missing face of $X$, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the $k$-th spectral gap, $(d+1)(k+1)-dn$, for some $k$, thereby confirming a conjecture proposed by Lew. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.10398v2-abstract-full').style.display = 'none'; document.getElementById('2407.10398v2-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 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">15 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.08301">arXiv:2407.08301</a> <span> [<a href="https://arxiv.org/pdf/2407.08301">pdf</a>, <a href="https://arxiv.org/ps/2407.08301">ps</a>, <a href="https://arxiv.org/format/2407.08301">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> The first Steklov eigenvalue of planar graphs and beyond </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+D">Da 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="2407.08301v3-abstract-short" style="display: inline;"> The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let $D$ and $未惟$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $位_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, 未惟)$. In this paper, using the circle packing theorem and conformal mapping, we first… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08301v3-abstract-full').style.display = 'inline'; document.getElementById('2407.08301v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.08301v3-abstract-full" style="display: none;"> The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let $D$ and $未惟$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $位_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, 未惟)$. In this paper, using the circle packing theorem and conformal mapping, we first show that $位_2 \leq 8D / |未惟|$ for planar graphs. This can be seen as a discrete analogue of Kokarev's bound [Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math. (2014)], that is, $位_2 < 8蟺/ |\partial 惟|$ for compact surfaces with boundary of genus $0$. Let $B$ and $L$ be the maximum block size and the diameter of a block graph $\mathcal{G}$ repsectively. Secondly, we prove that $位_2 \leq B^2 (D-1)/ |未惟|$ and $位_2 \leq (2L + (L-2)(B-2))/L^2$ for block graphs, which extend the results on trees by He and Hua [Upper bounds for the Steklov eigenvalues on trees, Calc. Var. Partial Differential Equations (2022)]. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08301v3-abstract-full').style.display = 'none'; document.getElementById('2407.08301v3-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 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">4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47A75; 49J40; 49R05; 05C10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.09800">arXiv:2406.09800</a> <span> [<a href="https://arxiv.org/pdf/2406.09800">pdf</a>, <a href="https://arxiv.org/format/2406.09800">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 class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3842/SIGMA.2024.105">10.3842/SIGMA.2024.105 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> $R$-Matrix Presentation of Quantum Affine Superalgebra for Type $\mathfrak{osp}(2m+1|2n)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+X">Xianghua Wu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongda Lin</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian 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="2406.09800v3-abstract-short" style="display: inline;"> In our preceding research, we introduced the Drinfeld presentation of the quantum affine superalgebra associated to the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m>0$. We provided the isomorphism between its Drinfeld-Jimbo presentation and Drinfeld presentation using braid group actions as a fundamental method. Based on this work, our current study delves into its $R$-matrix… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09800v3-abstract-full').style.display = 'inline'; document.getElementById('2406.09800v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.09800v3-abstract-full" style="display: none;"> In our preceding research, we introduced the Drinfeld presentation of the quantum affine superalgebra associated to the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m>0$. We provided the isomorphism between its Drinfeld-Jimbo presentation and Drinfeld presentation using braid group actions as a fundamental method. Based on this work, our current study delves into its $R$-matrix presentation, wherein we establish a clear isomorphism between the $R$-matrix presentation and the Drinfeld presentation. In particular, our contribution extends the investigations of Jing, Liu and Molev concerning quantum affine algebra in type BCD to the realm of supersymmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09800v3-abstract-full').style.display = 'none'; document.getElementById('2406.09800v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIGMA 20 (2024), 105, 38 pages </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.07307">arXiv:2406.07307</a> <span> [<a href="https://arxiv.org/pdf/2406.07307">pdf</a>, <a href="https://arxiv.org/ps/2406.07307">ps</a>, <a href="https://arxiv.org/format/2406.07307">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The effective cone conjecture for Calabi--Yau pairs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gachet%2C+C">C茅cile Gachet</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hsueh-Yung Lin</a>, <a href="/search/math?searchtype=author&query=Stenger%2C+I">Isabel Stenger</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Long 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="2406.07307v1-abstract-short" style="display: inline;"> We formulate an {\it effective cone conjecture} for klt Calabi--Yau pairs $(X,螖)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,螖)$. Assuming the existence of good minimal models in dimension $\dim(X)$, known to hold in dimension up to $3$, we prove that the effective cone conjecture for… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.07307v1-abstract-full').style.display = 'inline'; document.getElementById('2406.07307v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.07307v1-abstract-full" style="display: none;"> We formulate an {\it effective cone conjecture} for klt Calabi--Yau pairs $(X,螖)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms $\mathrm{PsAut}(X,螖)$. Assuming the existence of good minimal models in dimension $\dim(X)$, known to hold in dimension up to $3$, we prove that the effective cone conjecture for $(X,螖)$ is equivalent to the Kawamata--Morrison--Totaro movable cone conjecture for $(X,螖)$. As an application, we show that the movable cone conjecture unconditionally holds for the smooth Calabi--Yau threefolds introduced by Schoen and studied by Namikawa, Grassi and Morrison. We also show that for such a Calabi--Yau threefold $X$, all of its minimal models, apart from $X$ itself, have rational polyhedral nef cones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.07307v1-abstract-full').style.display = 'none'; document.getElementById('2406.07307v1-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> 11 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.05135">arXiv:2406.05135</a> <span> [<a href="https://arxiv.org/pdf/2406.05135">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Robotics">cs.RO</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"> Smart Navigation System for Parking Assignment at Large Events: Incorporating Heterogeneous Driver Characteristics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cheng%2C+X">Xi Cheng</a>, <a href="/search/math?searchtype=author&query=Su%2C+G">Gaofeng Su</a>, <a href="/search/math?searchtype=author&query=Feng%2C+S">Siyuan Feng</a>, <a href="/search/math?searchtype=author&query=Liu%2C+K">Ke Liu</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+C">Chen Zhu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hui Lin</a>, <a href="/search/math?searchtype=author&query=Song%2C+J">Jilin Song</a>, <a href="/search/math?searchtype=author&query=Chen%2C+J">Jianan 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="2406.05135v1-abstract-short" style="display: inline;"> Parking challenges escalate significantly during large events such as concerts or sports games, yet few studies address dynamic parking lot assignments for such occasions. This paper introduces a smart navigation system designed to optimize parking assignments swiftly during large events, utilizing a mixed search algorithm that accounts for the heterogeneous characteristics of drivers. We conducte… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.05135v1-abstract-full').style.display = 'inline'; document.getElementById('2406.05135v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.05135v1-abstract-full" style="display: none;"> Parking challenges escalate significantly during large events such as concerts or sports games, yet few studies address dynamic parking lot assignments for such occasions. This paper introduces a smart navigation system designed to optimize parking assignments swiftly during large events, utilizing a mixed search algorithm that accounts for the heterogeneous characteristics of drivers. We conducted simulations in the Berkeley city area during the "Big Game" to validate our system and demonstrate the benefits of our innovative parking assignment approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.05135v1-abstract-full').style.display = 'none'; document.getElementById('2406.05135v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.20766">arXiv:2405.20766</a> <span> [<a href="https://arxiv.org/pdf/2405.20766">pdf</a>, <a href="https://arxiv.org/ps/2405.20766">ps</a>, <a href="https://arxiv.org/format/2405.20766">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"> Long cycles and spectral radii in planar graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xu%2C+P">Ping Xu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=Fang%2C+L">Longfei Fang</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="2405.20766v1-abstract-short" style="display: inline;"> There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu (1994) and Sanders (1996) respectively proved that every 4-connected planar graph contains a cycle of length $n-1, n-2$ and $n-3$. Chen, Fan and Yu (2004) further… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20766v1-abstract-full').style.display = 'inline'; document.getElementById('2405.20766v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.20766v1-abstract-full" style="display: none;"> There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu (1994) and Sanders (1996) respectively proved that every 4-connected planar graph contains a cycle of length $n-1, n-2$ and $n-3$. Chen, Fan and Yu (2004) further conjectured that every 4-connected planar graph contains a cycle of length $\ell$ for $\ell\in\{n,n-1,\ldots,n-25\}$ and they verified that $\ell\in \{n-4, n-5, n-6\}$. When we remove the ``4-connected" condition, how to guarantee the existence of a long cycle in a planar graph? A natural question asks by adding a spectral radius condition: What is the smallest constant $C$ such that for sufficiently large $n$, every graph $G$ of order $n$ with spectral radius greater than $C$ contains a long cycle in a planar graph? In this paper, we give a stronger answer to the above question. Let $G$ be a planar graph with order $n\geq 1.8\times 10^{17}$ and $k\leq \lfloor\log_2(n-3)\rfloor-8$ be a non-negative integer, we show that if $蟻(G)\geq 蟻(K_2\vee(P_{n-2k-4}\cup 2P_{k+1}))$ then $G$ contains a cycle of length $\ell$ for every $\ell\in \{n-k, n-k-1, \ldots, 3\}$ unless $G\cong K_2\vee(P_{n-2k-4}\cup 2P_{k+1})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20766v1-abstract-full').style.display = 'none'; document.getElementById('2405.20766v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C50; 05C35; 05C45 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.20583">arXiv:2405.20583</a> <span> [<a href="https://arxiv.org/pdf/2405.20583">pdf</a>, <a href="https://arxiv.org/format/2405.20583">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Geometry">cs.CG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> The Gestalt Computational Model by Persistent Homology </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yu Chen</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongwei Lin</a>, <a href="/search/math?searchtype=author&query=Yan%2C+J">Jiacong Yan</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="2405.20583v2-abstract-short" style="display: inline;"> Widely employed in cognitive psychology, Gestalt theory elucidates basic principles in visual perception. However, the Gestalt principles are validated mainly by psychological experiments, lacking quantitative research supports and theoretical coherence. In this paper, we utilize persistent homology, a mathematical tool in computational topology, to develop a unified computational model for Gestal… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20583v2-abstract-full').style.display = 'inline'; document.getElementById('2405.20583v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.20583v2-abstract-full" style="display: none;"> Widely employed in cognitive psychology, Gestalt theory elucidates basic principles in visual perception. However, the Gestalt principles are validated mainly by psychological experiments, lacking quantitative research supports and theoretical coherence. In this paper, we utilize persistent homology, a mathematical tool in computational topology, to develop a unified computational model for Gestalt principles, addressing the challenges of quantification and computation. On the one hand, the Gestalt computational model presents quantitative supports for Gestalt theory. On the other hand, it shows that the Gestalt principles can be uniformly calculated using persistent homology, thus developing a coherent theory for Gestalt principles in computation. Moreover, it is anticipated that the Gestalt computational model can serve as a significant computational model in the field of computational psychology, and help the understanding of human being visual perception. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20583v2-abstract-full').style.display = 'none'; document.getElementById('2405.20583v2-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 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/2405.20056">arXiv:2405.20056</a> <span> [<a href="https://arxiv.org/pdf/2405.20056">pdf</a>, <a href="https://arxiv.org/ps/2405.20056">ps</a>, <a href="https://arxiv.org/format/2405.20056">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> A unified approach to the spectral radius, connectivity and edge-connectivity of graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yu Wang</a>, <a href="/search/math?searchtype=author&query=Li%2C+D">Dan Li</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2405.20056v2-abstract-short" style="display: inline;"> For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $魏^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$ connected components in $G\!-\!S$ and each component has at least $h+1$ vertices. Denote by $\mathcal{G}_{n,未}^{魏_r^h}$ the set of graphs with $h$-extra $r$-component c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20056v2-abstract-full').style.display = 'inline'; document.getElementById('2405.20056v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.20056v2-abstract-full" style="display: none;"> For two integers $r\geq 2$ and $h\geq 0$, the \emph{$h$-extra $r$-component connectivity} $魏^h_r(G)$ of a graph $G$ is defined to be the minimum size of a subset of vertices whose removal disconnects $G$, and there are at least $r$ connected components in $G\!-\!S$ and each component has at least $h+1$ vertices. Denote by $\mathcal{G}_{n,未}^{魏_r^h}$ the set of graphs with $h$-extra $r$-component connectivity $魏^h_r(G)$ and minimum degree $未$. The following problem concerning spectral radius was proposed by Brualdi and Solheid [On the spectral radius of complementary acyclic matrices of zeros and one, SIAM J. Algebra Discrete Methods 7 (1986) 265-272]: Given a set of graphs $\mathscr{S}$, find an upper bound for the spectral radius of graphs in $\mathscr{S}$ and characterize the graphs in which the maximal spectral radius is attained. We study this question for $\mathscr{S}=\mathcal{G}_{n,未}^{魏_r^h}$ where $r\geq 2$ and $h\geq 0$. Fan, Gu and Lin [$l$-connectivity, $l$-edge-connectivity and spectral radius of graphs, \emph{arXiv}:2309.05247] give the answer to $r\geq 2$ and $h=0$. In this paper, we solve this problem completely for $r\geq 2$ and $h\geq1$. Moreover, we also investigate analogous problems for the edge version. Our results can break the restriction of the extremum structure of the conditional connectivity. This implies some previous results in connectivity and edge-connectivity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20056v2-abstract-full').style.display = 'none'; document.getElementById('2405.20056v2-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 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/2405.08306">arXiv:2405.08306</a> <span> [<a href="https://arxiv.org/pdf/2405.08306">pdf</a>, <a href="https://arxiv.org/format/2405.08306">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> </div> </div> <p class="title is-5 mathjax"> Flight Path Optimization with Optimal Control Method </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Su%2C+G">Gaofeng Su</a>, <a href="/search/math?searchtype=author&query=Cheng%2C+X">Xi Cheng</a>, <a href="/search/math?searchtype=author&query=Feng%2C+S">Siyuan Feng</a>, <a href="/search/math?searchtype=author&query=Liu%2C+K">Ke Liu</a>, <a href="/search/math?searchtype=author&query=Song%2C+J">Jilin Song</a>, <a href="/search/math?searchtype=author&query=Chen%2C+J">Jianan Chen</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+C">Chen Zhu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hui Lin</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="2405.08306v2-abstract-short" style="display: inline;"> This paper is based on a crucial issue in the aviation world: how to optimize the trajectory and controls given to the aircraft in order to optimize flight time and fuel consumption. This study aims to provide elements of a response to this problem and to define, under certain simplifying assumptions, an optimal response, using Constrained Finite Time Optimal Control(CFTOC). The first step is to d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08306v2-abstract-full').style.display = 'inline'; document.getElementById('2405.08306v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.08306v2-abstract-full" style="display: none;"> This paper is based on a crucial issue in the aviation world: how to optimize the trajectory and controls given to the aircraft in order to optimize flight time and fuel consumption. This study aims to provide elements of a response to this problem and to define, under certain simplifying assumptions, an optimal response, using Constrained Finite Time Optimal Control(CFTOC). The first step is to define the dynamic model of the aircraft in accordance with the controllable inputs and wind disturbances. Then we will identify a precise objective in terms of optimization and implement an optimization program to solve it under the circumstances of simulated real flight situation. Finally, the optimization result is validated and discussed by different scenarios. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.08306v2-abstract-full').style.display = 'none'; document.getElementById('2405.08306v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 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/2404.13389">arXiv:2404.13389</a> <span> [<a href="https://arxiv.org/pdf/2404.13389">pdf</a>, <a href="https://arxiv.org/ps/2404.13389">ps</a>, <a href="https://arxiv.org/format/2404.13389">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"> Eigenvalues and graph minors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhai%2C+M">Mingqing Zhai</a>, <a href="/search/math?searchtype=author&query=Fang%2C+L">Longfei Fang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2404.13389v2-abstract-short" style="display: inline;"> Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$, $\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified phenomena on general minors. Every… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.13389v2-abstract-full').style.display = 'inline'; document.getElementById('2404.13389v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.13389v2-abstract-full" style="display: none;"> Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$, $\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified phenomena on general minors. Every graph $G$ on $n$ vertices with spectral radius $蟻\geq spex(n,H_{minor})$ contains either an $H$ minor or a spanning book $K_{纬_H}\nabla(n-纬_H)K_1$, where $纬_H=|H|-伪(H)-1$. Furthermore, assume that $G$ is $H$-minor free and $螕^*_s(H)$ is the family of $s$-vertex irreducible induced subgraphs of $H$, then $G$ minus its $纬_H$ dominating vertices is $螕^*_{伪(H)+1}(H)$-minor saturate, and it is further edge-maximal if $螕^*_{伪(H)+1}(H)$ is a connected family. As applications, we obtain some known results on minors mentioned above. We also determine the extremal values for some other minors, such as flowers, wheels, generalized books and complete multi-partite graphs. Our results extend some conjectures on planar graphs, outer-planar graphs and $K_{s,t}$-minor free graphs. To obtain the results, we combine stability method, spectral techniques and structural analyses. Especially, we give an exploration of using absorbing method in spectral extremal problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.13389v2-abstract-full').style.display = 'none'; document.getElementById('2404.13389v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 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/2404.06685">arXiv:2404.06685</a> <span> [<a href="https://arxiv.org/pdf/2404.06685">pdf</a>, <a href="https://arxiv.org/ps/2404.06685">ps</a>, <a href="https://arxiv.org/format/2404.06685">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Spectral expansion properties of pseudorandom bipartite graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fan%2C+D">Dandan Fan</a>, <a href="/search/math?searchtype=author&query=Gu%2C+X">Xiaofeng Gu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2404.06685v1-abstract-short" style="display: inline;"> An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have nice pseudorandom and expansion properties when the second largest adjacency eigenvalue is not large. In this paper, we prove several explicit properties of bire… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06685v1-abstract-full').style.display = 'inline'; document.getElementById('2404.06685v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.06685v1-abstract-full" style="display: none;"> An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have nice pseudorandom and expansion properties when the second largest adjacency eigenvalue is not large. In this paper, we prove several explicit properties of biregular bipartite graphs from spectral perspectives. In particular, we show that for any $(a,b)$-biregular bipartite graph $G$, if the spectral gap is greater than $\frac{2(k-1)}{\sqrt{(a+1)(b+1)}}$, then $G$ is $k$-edge-connected; and if the spectral gap is at least $\frac{2k}{\sqrt{(a+1)(b+1)}}$, then $G$ has at least $k$ edge-disjoint spanning trees. We also prove that if the spectral gap is at least $\frac{(k-1)\max\{a,b\}}{2\sqrt{ab - (k-1)\max\{a,b\}}}$, then $G$ is $k$-connected for $k\ge 2$; and if the spectral gap is at least $\frac{6k+2\max\{a,b\}}{\sqrt{(a-1)(b-1)}}$, then $G$ has at least $k$ edge-disjoint spanning 2-connected subgraphs. We have stronger results in the paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.06685v1-abstract-full').style.display = 'none'; document.getElementById('2404.06685v1-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/2404.01007">arXiv:2404.01007</a> <span> [<a href="https://arxiv.org/pdf/2404.01007">pdf</a>, <a href="https://arxiv.org/format/2404.01007">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Geometry">cs.CG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> Extraction of Singular Patterns from a Vector Field via Persistent Path Homology </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yu Chen</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongwei Lin</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="2404.01007v2-abstract-short" style="display: inline;"> The extraction of singular patterns is a fundamental problem in theoretical and practical domains due to the ability of such patterns to detect the intrinsic characteristics of vector fields. In this study, we propose an approach for extracting singular patterns from discrete planar vector fields. Our method involves converting the planar discrete vector field into a specialized digraph and comput… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.01007v2-abstract-full').style.display = 'inline'; document.getElementById('2404.01007v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.01007v2-abstract-full" style="display: none;"> The extraction of singular patterns is a fundamental problem in theoretical and practical domains due to the ability of such patterns to detect the intrinsic characteristics of vector fields. In this study, we propose an approach for extracting singular patterns from discrete planar vector fields. Our method involves converting the planar discrete vector field into a specialized digraph and computing its one-dimensional persistent path homology. By analyzing the persistence diagram, we can determine the location of singularity and segment a region of the singular pattern, which is referred to as a singular polygon. Moreover, the variations of singular patterns can also be analyzed. The experimental results demonstrate the effectiveness of our method in analyzing the centers and impact areas of tropical cyclones, positioning the dip poles from geomagnetic fields, and measuring variations of singular patterns between vector fields. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.01007v2-abstract-full').style.display = 'none'; document.getElementById('2404.01007v2-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 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/2403.16932">arXiv:2403.16932</a> <span> [<a href="https://arxiv.org/pdf/2403.16932">pdf</a>, <a href="https://arxiv.org/ps/2403.16932">ps</a>, <a href="https://arxiv.org/format/2403.16932">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> On the Maximum Theta Series over Unimodular Lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bollauf%2C+M+F">Maiara F. Bollauf</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hsuan-Yin Lin</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="2403.16932v2-abstract-short" style="display: inline;"> The theta series of a lattice has been extensively studied in the literature and is closely related to a critical quantity widely used in the fields of physical layer security and cryptography, known as the flatness factor, or equivalently, the smoothing parameter of a lattice. Both fields raise the fundamental question of determining the (globally) maximum theta series over a particular set of vo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.16932v2-abstract-full').style.display = 'inline'; document.getElementById('2403.16932v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.16932v2-abstract-full" style="display: none;"> The theta series of a lattice has been extensively studied in the literature and is closely related to a critical quantity widely used in the fields of physical layer security and cryptography, known as the flatness factor, or equivalently, the smoothing parameter of a lattice. Both fields raise the fundamental question of determining the (globally) maximum theta series over a particular set of volume-one lattices, namely, the stable lattices. In this work, we present a property of unimodular lattices, a subfamily of stable lattices, to verify that the integer lattice $\mathbb{Z}^{n}$ achieves the largest possible value of theta series over the set of unimodular lattices. Such a result moves towards proving a conjecture recently stated by Regev and Stephens-Davidowitz: any unimodular lattice, except for those lattices isomorphic to $\mathbb{Z}^{n}$, has a strictly smaller theta series than that of $\mathbb{Z}^{n}$. Our techniques are mainly based on studying the ratio of the theta series of a unimodular lattice to the theta series of $\mathbb{Z}^n$, called the secrecy ratio. We relate the Regev and Stephens-Davidowitz conjecture with another conjecture for unimodular lattices, known in the literature as the Belfiore-Sol茅 conjecture. The latter assumes that the secrecy ratio of any unimodular lattice has a symmetry point, which is exactly where the global minimum of the secrecy ratio is achieved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.16932v2-abstract-full').style.display = 'none'; document.getElementById('2403.16932v2-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.17253">arXiv:2402.17253</a> <span> [<a href="https://arxiv.org/pdf/2402.17253">pdf</a>, <a href="https://arxiv.org/ps/2402.17253">ps</a>, <a href="https://arxiv.org/format/2402.17253">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"> Hardy type inequalities on manifolds with nonnegative Ricci curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dong%2C+Y">Yuxin Dong</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hezi Lin</a>, <a href="/search/math?searchtype=author&query=Lu%2C+L">Lingen Lu</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="2402.17253v2-abstract-short" style="display: inline;"> We prove the Heisenberg-Pauli-Weyl inequality, Hardy-Sobolev inequality, and Caffarelli-Kohn-Nirenberg (CKN) inequality on manifolds with nonnegative Ricci curvature and Euclidean volume growth, of dimension n>=3. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.17253v2-abstract-full" style="display: none;"> We prove the Heisenberg-Pauli-Weyl inequality, Hardy-Sobolev inequality, and Caffarelli-Kohn-Nirenberg (CKN) inequality on manifolds with nonnegative Ricci curvature and Euclidean volume growth, of dimension n>=3. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.17253v2-abstract-full').style.display = 'none'; document.getElementById('2402.17253v2-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 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">Comments:</span> <span class="has-text-grey-dark mathjax">All 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/2402.16419">arXiv:2402.16419</a> <span> [<a href="https://arxiv.org/pdf/2402.16419">pdf</a>, <a href="https://arxiv.org/ps/2402.16419">ps</a>, <a href="https://arxiv.org/format/2402.16419">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 the spectral extremal problem of planar graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+X">Xiaolong Wang</a>, <a href="/search/math?searchtype=author&query=Huang%2C+X">Xueyi Huang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2402.16419v1-abstract-short" style="display: inline;"> The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3) (1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257] independently) conjectured that $K_2 + P_{n-2}$ is the unique graph attaining the maximum spectral radius among all planar graphs on $n$ vertices, where $K_2 + P_{n-2}$ i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.16419v1-abstract-full').style.display = 'inline'; document.getElementById('2402.16419v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.16419v1-abstract-full" style="display: none;"> The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3) (1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257] independently) conjectured that $K_2 + P_{n-2}$ is the unique graph attaining the maximum spectral radius among all planar graphs on $n$ vertices, where $K_2 + P_{n-2}$ is the graph obtained from $K_2\cup P_{n-2}$ by adding all possible edges between $K_2$ and $P_{n-2}$. In 2017, Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] confirmed this conjecture for all sufficiently large $n$. In this paper, we consider the spectral extremal problem for planar graphs without specified subgraphs. For a fixed graph $F$, let $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ denote the set of graphs attaining the maximum spectral radius among all $F$-free planar graphs on $n$ vertices. We describe a rough sturcture for the connected extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ when $F$ is a planar graph not contained in $K_{2,n-2}$. As applications, we determine the extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,W_k)$, $\mathrm{SPEX}_{\mathcal{P}}(n,F_k)$ and $\mathrm{SPEX}_{\mathcal{P}}(n,(k+1)K_2)$ for all sufficiently large $n$, where $W_k$, $F_k$ and $(k+1)K_2$ are the wheel graph of order $k$, the friendship graph of order $2k+1$ and the disjoint union of $k+1$ copies of $K_2$, respectively. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.16419v1-abstract-full').style.display = 'none'; document.getElementById('2402.16419v1-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 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">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.12609">arXiv:2402.12609</a> <span> [<a href="https://arxiv.org/pdf/2402.12609">pdf</a>, <a href="https://arxiv.org/ps/2402.12609">ps</a>, <a href="https://arxiv.org/format/2402.12609">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Existence of Approximately Macroscopically Unique States </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huaxin Lin</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="2402.12609v1-abstract-short" style="display: inline;"> Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the C*-algebra of bounded operators on $H.$ Suppose that $T_1,T_2,..., T_n$ are self-adjoint operators in $B(H).$ We show that, if commutators $[T_i, T_j]$ are sufficiently small in norm, then ``Approximately Macroscopically Unique" states always exist for any values in a synthetic spectrum of the $n$-tuple of self-adjoint opera… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12609v1-abstract-full').style.display = 'inline'; document.getElementById('2402.12609v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.12609v1-abstract-full" style="display: none;"> Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the C*-algebra of bounded operators on $H.$ Suppose that $T_1,T_2,..., T_n$ are self-adjoint operators in $B(H).$ We show that, if commutators $[T_i, T_j]$ are sufficiently small in norm, then ``Approximately Macroscopically Unique" states always exist for any values in a synthetic spectrum of the $n$-tuple of self-adjoint operators. This is achieved under the circumstance for which the $n$-tuple may not be approximated by commuting ones. This answers a question proposed by David Mumford for measurements in quantum theory. If commutators are not small in norm but small modulo compact operators, then ``Approximate Macroscopic Uniqueness" states also exist. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12609v1-abstract-full').style.display = 'none'; document.getElementById('2402.12609v1-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> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.16879">arXiv:2401.16879</a> <span> [<a href="https://arxiv.org/pdf/2401.16879">pdf</a>, <a href="https://arxiv.org/ps/2401.16879">ps</a>, <a href="https://arxiv.org/format/2401.16879">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</span> </div> </div> <p class="title is-5 mathjax"> Optimal Control of a Stochastic Power System -- Algorithms and Mathematical Analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zhen Wang</a>, <a href="/search/math?searchtype=author&query=Xi%2C+K">Kaihua Xi</a>, <a href="/search/math?searchtype=author&query=Cheng%2C+A">Aijie Cheng</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H+X">Hai Xiang Lin</a>, <a href="/search/math?searchtype=author&query=van+Schuppen%2C+J+H">Jan H. van Schuppen</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="2401.16879v1-abstract-short" style="display: inline;"> The considered optimal control problem of a stochastic power system, is to select the set of power supply vectors which infimizes the probability that the phase-angle differences of any power flow of the network, endangers the transient stability of the power system by leaving a critical subset. The set of control laws is restricted to be a periodically recomputed set of fixed power supply vectors… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.16879v1-abstract-full').style.display = 'inline'; document.getElementById('2401.16879v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.16879v1-abstract-full" style="display: none;"> The considered optimal control problem of a stochastic power system, is to select the set of power supply vectors which infimizes the probability that the phase-angle differences of any power flow of the network, endangers the transient stability of the power system by leaving a critical subset. The set of control laws is restricted to be a periodically recomputed set of fixed power supply vectors based on predictions of power demand for the next short horizon. Neither state feedback nor output feedback is used. The associated control objective function is Lipschitz continuous, nondifferentiable, and nonconvex. The results of the paper include that a minimum exists in the value range of the control objective function. Furthermore, it includes a two-step procedure to compute an approximate minimizer based on two key methods: (1) a projected generalized subgradient method for computing an initial vector, and (2) a steepest descent method for approximating a local minimizer. Finally, it includes two convergence theorems that an approximation sequence converges to a local minimum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.16879v1-abstract-full').style.display = 'none'; document.getElementById('2401.16879v1-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">24 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 93E20; 90C30; and 90C26 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.05786">arXiv:2401.05786</a> <span> [<a href="https://arxiv.org/pdf/2401.05786">pdf</a>, <a href="https://arxiv.org/ps/2401.05786">ps</a>, <a href="https://arxiv.org/format/2401.05786">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Spectral extremal results on trees </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fang%2C+L">Longfei Fang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=Shu%2C+J">Jinlong Shu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zhiyuan 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="2401.05786v2-abstract-short" style="display: inline;"> Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its ind… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.05786v2-abstract-full').style.display = 'inline'; document.getElementById('2401.05786v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.05786v2-abstract-full" style="display: none;"> Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where `$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioab膬, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228--2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $蟻(G)\geq 蟻(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $蟻(K_{q,n-q})\leq {\rm spex}(n,F)\leq 蟻(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.05786v2-abstract-full').style.display = 'none'; document.getElementById('2401.05786v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.04018">arXiv:2401.04018</a> <span> [<a href="https://arxiv.org/pdf/2401.04018">pdf</a>, <a href="https://arxiv.org/ps/2401.04018">ps</a>, <a href="https://arxiv.org/format/2401.04018">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Almost commuting self-adjoint operators and measurements </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huaxin Lin</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="2401.04018v3-abstract-short" style="display: inline;"> We study the problem when an almost commuting $n$-tuple self-adjoint operators in an infinite dimensional separable Hilbert space $H$ is close to an $n$-tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.04018v3-abstract-full').style.display = 'inline'; document.getElementById('2401.04018v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.04018v3-abstract-full" style="display: none;"> We study the problem when an almost commuting $n$-tuple self-adjoint operators in an infinite dimensional separable Hilbert space $H$ is close to an $n$-tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when $n\ge 3$ is negative even the associated Fredholm index vanishes. In the case that $n=2,$ we show that a pair of almost commuting self-adjoint operators in an infinite dimensional separable Hilbert space is close to a commuting pair of self-adjoint operators if and only if a corresponding Fredholm index vanishes outside of an essential synthetic-spectrum. This is an attempt to solve a problem proposed by David Mumford related to quantum theory and measurements. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.04018v3-abstract-full').style.display = 'none'; document.getElementById('2401.04018v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">v2 is a revision</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46L05; 46L30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.17021">arXiv:2312.17021</a> <span> [<a href="https://arxiv.org/pdf/2312.17021">pdf</a>, <a href="https://arxiv.org/ps/2312.17021">ps</a>, <a href="https://arxiv.org/format/2312.17021">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Predual Spaces of Hardy Spaces Related to Fractional Schr枚dinger Operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Q">Qiumeng Li</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Haibo Lin</a>, <a href="/search/math?searchtype=author&query=Yang%2C+S">Sibei 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="2312.17021v1-abstract-short" style="display: inline;"> Let $n\in\mathbb{N}$ and $伪\in(0,\min\{2,n\})$. For any $a\in[a^\ast,\infty)$, the fractional Schr枚dinger operator $L_伪$ is defined by \begin{equation*} L_伪:=(-螖)^{伪/2}+a{|x|}^{-伪}, \end{equation*} where $a^*:=-{\frac{2^伪螕((d+伪)/4)^2}{螕((d-伪)/4)^2}}$. Let $纬\in[0,\frac伪{n})$. In this paper, we introduce the VMO-type spaces $\mathrm{VMO}_{L_伪}^纬(\mathbb{R}^{n})$ associated with $L_伪$, and character… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17021v1-abstract-full').style.display = 'inline'; document.getElementById('2312.17021v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.17021v1-abstract-full" style="display: none;"> Let $n\in\mathbb{N}$ and $伪\in(0,\min\{2,n\})$. For any $a\in[a^\ast,\infty)$, the fractional Schr枚dinger operator $L_伪$ is defined by \begin{equation*} L_伪:=(-螖)^{伪/2}+a{|x|}^{-伪}, \end{equation*} where $a^*:=-{\frac{2^伪螕((d+伪)/4)^2}{螕((d-伪)/4)^2}}$. Let $纬\in[0,\frac伪{n})$. In this paper, we introduce the VMO-type spaces $\mathrm{VMO}_{L_伪}^纬(\mathbb{R}^{n})$ associated with $L_伪$, and characterize these spaces via some tent spaces. We also prove that, for any given $p\in(\frac{n}{n+伪},1]$, the space $\mathrm{VMO}_{L_伪}^{\frac{1}{p}-1} (\mathbb{R}^{n})$ is the predual space of the Hardy space $H_{L_伪}^p\left(\mathbb{R}^{n}\right)$ related to $L_伪$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.17021v1-abstract-full').style.display = 'none'; document.getElementById('2312.17021v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.15902">arXiv:2312.15902</a> <span> [<a href="https://arxiv.org/pdf/2312.15902">pdf</a>, <a href="https://arxiv.org/ps/2312.15902">ps</a>, <a href="https://arxiv.org/format/2312.15902">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"> Eigenvalues and factors: a survey </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fan%2C+D">Dandan Fan</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</a>, <a href="/search/math?searchtype=author&query=Lu%2C+H">Hongliang Lu</a>, <a href="/search/math?searchtype=author&query=O%2C+S">Suil O</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="2312.15902v1-abstract-short" style="display: inline;"> A factor of a graph is a spanning subgraph satisfying some given conditions. An earlier survey of factors can be traced back to the Akiyama and Kano [J. Graph Theory, 1985, 9: 1-42] in which they described the characterization of factors in (bipartite) graphs and digraphs, respectively. Soon after, Kouider and Vestergaard summarized the findings related to connected factors [Graphs Combin., 2005,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.15902v1-abstract-full').style.display = 'inline'; document.getElementById('2312.15902v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.15902v1-abstract-full" style="display: none;"> A factor of a graph is a spanning subgraph satisfying some given conditions. An earlier survey of factors can be traced back to the Akiyama and Kano [J. Graph Theory, 1985, 9: 1-42] in which they described the characterization of factors in (bipartite) graphs and digraphs, respectively. Soon after, Kouider and Vestergaard summarized the findings related to connected factors [Graphs Combin., 2005, 21(1): 1-26]. Plummer extended the aforementioned research by providing a comprehensive overview of progress made in the study of graph factors and factorization from 1985 to 2003 [Discrete Math., 2007, 7-8: 791-821]. In this paper, we aim to summarize the relevant results regarding factors from the perspective of eigenvalues. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.15902v1-abstract-full').style.display = 'none'; document.getElementById('2312.15902v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </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 are very 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/2311.15462">arXiv:2311.15462</a> <span> [<a href="https://arxiv.org/pdf/2311.15462">pdf</a>, <a href="https://arxiv.org/ps/2311.15462">ps</a>, <a href="https://arxiv.org/format/2311.15462">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Double duals and Hilbert modules </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huaxin Lin</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="2311.15462v1-abstract-short" style="display: inline;"> Let $A$ be a $C^*$-algebra, $H$ be a Hilbert $A$-module and $K(H)$ be the closure of the set of finite rank module maps. We show that the $W^*$-algebra of all bounded $A^{**}$-module maps on the smallest self-dual Hilbert $A^{**}$-module containing $H$ is isomorphic to $K(H)^{**}$ as $W^*$-algebras. We also show that the unit ball of $H$ is closed in $H^\sharp,$ the dual of $H,$ in an $A$-weak top… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.15462v1-abstract-full').style.display = 'inline'; document.getElementById('2311.15462v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.15462v1-abstract-full" style="display: none;"> Let $A$ be a $C^*$-algebra, $H$ be a Hilbert $A$-module and $K(H)$ be the closure of the set of finite rank module maps. We show that the $W^*$-algebra of all bounded $A^{**}$-module maps on the smallest self-dual Hilbert $A^{**}$-module containing $H$ is isomorphic to $K(H)^{**}$ as $W^*$-algebras. We also show that the unit ball of $H$ is closed in $H^\sharp,$ the dual of $H,$ in an $A$-weak topology of $H^\sharp$ as well as dense in the unit ball of $H^\sharp$ in a weak*-topology and some versions of Kaplansky density theorem for Hilbert $C^*$-modules. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.15462v1-abstract-full').style.display = 'none'; document.getElementById('2311.15462v1-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, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46L08; 46L05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.14238">arXiv:2311.14238</a> <span> [<a href="https://arxiv.org/pdf/2311.14238">pdf</a>, <a href="https://arxiv.org/ps/2311.14238">ps</a>, <a href="https://arxiv.org/format/2311.14238">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> A review of the Elliott program of classification of simple amenable C*-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gong%2C+G">Guihua Gong</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huaxin Lin</a>, <a href="/search/math?searchtype=author&query=Niu%2C+Z">Zhuang Niu</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="2311.14238v1-abstract-short" style="display: inline;"> We give a brief survey of the development of the Elliott program of classification of separable simple amenable $C^*$-algebras. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.14238v1-abstract-full" style="display: none;"> We give a brief survey of the development of the Elliott program of classification of separable simple amenable $C^*$-algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.14238v1-abstract-full').style.display = 'none'; document.getElementById('2311.14238v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </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">This is a version of the Frontier Science Award Lecture at the First International Congress of Basic Science held in July 2023</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46L35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.08814">arXiv:2311.08814</a> <span> [<a href="https://arxiv.org/pdf/2311.08814">pdf</a>, <a href="https://arxiv.org/ps/2311.08814">ps</a>, <a href="https://arxiv.org/format/2311.08814">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> The quotient spaces of topological groups with a $q$-point </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xie%2C+L">Li-Hong Xie</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hai-Hua Lin</a>, <a href="/search/math?searchtype=author&query=Li%2C+P">Piyu Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.08814v1-abstract-short" style="display: inline;"> In this paper, we study the uniformities on the double coset spaces in topological groups. As an implication, the quotient spaces of topological groups with a $q$-point are studied. It mainly shows that: (1) Suppose that $G$ is a topological group with a $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and quasi-perfect preimage of a metrizable space; in part… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.08814v1-abstract-full').style.display = 'inline'; document.getElementById('2311.08814v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.08814v1-abstract-full" style="display: none;"> In this paper, we study the uniformities on the double coset spaces in topological groups. As an implication, the quotient spaces of topological groups with a $q$-point are studied. It mainly shows that: (1) Suppose that $G$ is a topological group with a $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and quasi-perfect preimage of a metrizable space; in particular, $G/H$ is an $M$-space. (2) Suppose that $G$ is a topological group with a strict $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and sequentially perfect preimage of a metrizable space. (3) Suppose that $G$ is a topological group with a strong $q$-point and $H$ is a closed subgroup of $G$; then the quotient space $G/H$ is an open and strongly sequentially perfect preimage of a metrizable space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.08814v1-abstract-full').style.display = 'none'; document.getElementById('2311.08814v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </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">17</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 54A20; 54H11; 54B15; 54C10; 54E15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.00345">arXiv:2311.00345</a> <span> [<a href="https://arxiv.org/pdf/2311.00345">pdf</a>, <a href="https://arxiv.org/ps/2311.00345">ps</a>, <a href="https://arxiv.org/format/2311.00345">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Some characterizations of $蠅$-balanced topological groups with a $q$-point </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+D">Deng-Bin Chen</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hai-Hua Lin</a>, <a href="/search/math?searchtype=author&query=Xie%2C+L">Li-Hong Xie</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.00345v1-abstract-short" style="display: inline;"> In this paper, we study some characterizations of $q$-spaces, strict $q$-spaces and strong $q$-spaces under $蠅$-balanced topological groups as follows: (1) A topological group $G$ is $蠅$-balanced and a $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a countably compact invariant subgroup $H$ which is of countable character in $G$, such that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00345v1-abstract-full').style.display = 'inline'; document.getElementById('2311.00345v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.00345v1-abstract-full" style="display: none;"> In this paper, we study some characterizations of $q$-spaces, strict $q$-spaces and strong $q$-spaces under $蠅$-balanced topological groups as follows: (1) A topological group $G$ is $蠅$-balanced and a $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a countably compact invariant subgroup $H$ which is of countable character in $G$, such that $H \subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is quasi-perfect and the quotient group $G/H$ is metrizable. (2) A topological group $G$ is $蠅$-balanced and a strict $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a closed sequentially compact invariant subgroup $H$ which is of countable character in $G$, such that $H \subseteq O$ and the canonical quotient mapping $p:G\rightarrow G/H$ is sequential-perfect and the quotient group $G/H$ is metrizable. (3) A topological group $G$ is $蠅$-balanced and a strong $q$-space if and only if for each open neighborhood $O$ of the identity in $G$, there is a closed sequentially compact invariant subgroup $H$ of countable character $\{V_{n}:n\in 蠅\} $, such that $H \subseteq O$ and $\{V_{n}:n\in蠅\}$ is a strong $q$-sequence at each $ y\in H $, in $G$ such that the canonical quotient mapping $p:G\rightarrow G/H$ is strongly sequential-perfect and the quotient group $G/H$ is metrizable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00345v1-abstract-full').style.display = 'none'; document.getElementById('2311.00345v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </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">11</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.12327">arXiv:2310.12327</a> <span>  </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"> Representation type of cyclotomic quiver Hecke algebras of type $C^{(1)}_{\ell}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Huang Lin</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="2310.12327v3-abstract-short" style="display: inline;"> We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type C. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type C, by using the quiver we construct. We als… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12327v3-abstract-full').style.display = 'inline'; document.getElementById('2310.12327v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.12327v3-abstract-full" style="display: none;"> We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type C. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type C, by using the quiver we construct. We also determine the Morita equivalence classes and graded decomposition matrices of certain representation-finite and tame cyclotomic KLR algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12327v3-abstract-full').style.display = 'none'; document.getElementById('2310.12327v3-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </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">This article has been withdrawn and no more revised version will be updated. There are two reason for this: 1. Section 1-4 are parallel to the well-known facts in affine type A with slight modifications, 2. a gap on decomposition matrices appear in the last section</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.09523">arXiv:2310.09523</a> <span> [<a href="https://arxiv.org/pdf/2310.09523">pdf</a>, <a href="https://arxiv.org/ps/2310.09523">ps</a>, <a href="https://arxiv.org/format/2310.09523">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"> Toughness and spectral radius in graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yuanyuan Chen</a>, <a href="/search/math?searchtype=author&query=Fan%2C+D">Dandan Fan</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2310.09523v1-abstract-short" style="display: inline;"> The Brouwer's toughness conjecture states that every $d$-regular connected graph always has $t(G)>\frac{d}位-1$ where $位$ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness $蟿(G)$ of a graph $G$. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be $蟿$-tough (… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.09523v1-abstract-full').style.display = 'inline'; document.getElementById('2310.09523v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.09523v1-abstract-full" style="display: none;"> The Brouwer's toughness conjecture states that every $d$-regular connected graph always has $t(G)>\frac{d}位-1$ where $位$ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness $蟿(G)$ of a graph $G$. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be $蟿$-tough ($蟿\geq 2$ is an integer) and to be $蟿$-tough ($\frac{1}蟿$ is a positive integer) with minimum degree $未$, respectively. Additionally, we also investigate a analogous problem concerning balanced bipartite graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.09523v1-abstract-full').style.display = 'none'; document.getElementById('2310.09523v1-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.05085">arXiv:2310.05085</a> <span> [<a href="https://arxiv.org/pdf/2310.05085">pdf</a>, <a href="https://arxiv.org/ps/2310.05085">ps</a>, <a href="https://arxiv.org/format/2310.05085">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Spectral extremal results on edge blow-up of graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fang%2C+L">Longfei Fang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2310.05085v3-abstract-short" style="display: inline;"> Let ${\rm ex}(n,F)$ and ${\rm spex}(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. The value ${\rm spex}(n,F)$ is called the spectral extremal value of $F$. Nikiforov [J. Graph Theory 62 (2009) 362--368] gave the spectral Stability Lemma, which implies that for every $\varepsilon>0$, sufficiently large $n$ and a non-bipartite graph $H$ with c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.05085v3-abstract-full').style.display = 'inline'; document.getElementById('2310.05085v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.05085v3-abstract-full" style="display: none;"> Let ${\rm ex}(n,F)$ and ${\rm spex}(n,F)$ be the maximum size and maximum spectral radius of an $F$-free graph of order $n$, respectively. The value ${\rm spex}(n,F)$ is called the spectral extremal value of $F$. Nikiforov [J. Graph Theory 62 (2009) 362--368] gave the spectral Stability Lemma, which implies that for every $\varepsilon>0$, sufficiently large $n$ and a non-bipartite graph $H$ with chromatic number $蠂(H)$, the extremal graph for ${\rm spex}(n,H)$ can be obtained from the Tur谩n graph $T_{蠂(H)-1}(n)$ by adding and deleting at most $\varepsilon n^2$ edges. It is still a challenging problem to determine the exact spectral extremal values of many non-bipartite graphs. Given a graph $F$ and an integer $p\geq 2$, the edge blow-up of $F$, denoted by $F^{p+1}$, is the graph obtained from replacing each edge in $F$ by a $K_{p+1}$ where the new vertices of $K_{p+1}$ are all distinct. In this paper, we determine the exact spectral extremal values of the edge blow-up of all non-bipartite graphs and provide the asymptotic spectral extremal values of the edge blow-up of all bipartite graphs for sufficiently large $n$, which can be seen as a spectral version of the theorem on ${\rm ex}(n,F^{p+1})$ given by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. As applications, on the one hand, we generalize several previous results on ${\rm spex}(n,F^{p+1})$ for $F$ being a matching and a star for $p\geq 3$. On the other hand, we obtain the exact values of ${\rm spex}(n,F^{p+1})$ for $F$ being a path, a cycle and a complete graph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.05085v3-abstract-full').style.display = 'none'; document.getElementById('2310.05085v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.04980">arXiv:2310.04980</a> <span> [<a href="https://arxiv.org/pdf/2310.04980">pdf</a>, <a href="https://arxiv.org/ps/2310.04980">ps</a>, <a href="https://arxiv.org/format/2310.04980">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="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"> On the virtual invariants of zero entropy groups of compact K盲hler manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dinh%2C+T">Tien-Cuong Dinh</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Hsueh-Yung Lin</a>, <a href="/search/math?searchtype=author&query=Oguiso%2C+K">Keiji Oguiso</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="2310.04980v1-abstract-short" style="display: inline;"> Let $X$ be a compact K盲hler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -魏(X)$, where $魏(X)$ is the Kodaira dimension of $X$. Modulo the main conje… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04980v1-abstract-full').style.display = 'inline'; document.getElementById('2310.04980v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.04980v1-abstract-full" style="display: none;"> Let $X$ be a compact K盲hler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -魏(X)$, where $魏(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -魏(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04980v1-abstract-full').style.display = 'none'; document.getElementById('2310.04980v1-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </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">This paper contains results from an earlier version of arXiv:1810.04827. They are completely revised with new proofs. 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/2309.05247">arXiv:2309.05247</a> <span> [<a href="https://arxiv.org/pdf/2309.05247">pdf</a>, <a href="https://arxiv.org/ps/2309.05247">ps</a>, <a href="https://arxiv.org/format/2309.05247">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"> l-connectivity, l-edge-connectivity and spectral radius of graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fan%2C+D">Dandan Fan</a>, <a href="/search/math?searchtype=author&query=Gu%2C+X">Xiaofeng Gu</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2309.05247v1-abstract-short" style="display: inline;"> Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. Discrete Math. 35 (2021) 948--952] proved a tight lower bound on toughness of regular graphs in terms of the second… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.05247v1-abstract-full').style.display = 'inline'; document.getElementById('2309.05247v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.05247v1-abstract-full" style="display: none;"> Let G be a connected graph. The toughness of G is defined as t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of components of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. Discrete Math. 35 (2021) 948--952] proved a tight lower bound on toughness of regular graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] then studied the toughness of simple graphs from the spectral radius perspective. While the toughness is an important concept in graph theory, it is also very interesting to study |S| for which c(G-S)\geq l for a given integer l\geq 2. This leads to the concept of the l-connectivity, which is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Gu [European J. Combin. 92 (2021) 103255] discovered a lower bound on the l-connectivity of regular graphs via the second largest absolute eigenvalue. As a counterpart, we discover the connection between the l-connectivity of simple graphs and the spectral radius. We also study similar problems for digraphs and an edge version. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.05247v1-abstract-full').style.display = 'none'; document.getElementById('2309.05247v1-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> 11 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.04101">arXiv:2309.04101</a> <span> [<a href="https://arxiv.org/pdf/2309.04101">pdf</a>, <a href="https://arxiv.org/ps/2309.04101">ps</a>, <a href="https://arxiv.org/format/2309.04101">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> The largest eigenvalue of $\mathcal{C}_4^{-}$-free signed graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yongang Wang</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huiqiu Lin</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="2309.04101v1-abstract-short" style="display: inline;"> Let $\mathcal{C}_{k}^{-}$ be the set of all negative $C_k$. For odd cycle, Wang, Hou and Li [29] gave a spectral condition for the existence of negative $C_3$ in unbalanced signed graphs. For even cycle, we determine the maximum index among all $\mathcal{C}_4^{-}$-free unbalanced signed graphs and completely characterize the extremal signed graph in this paper. This could be regarded as a signed g… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.04101v1-abstract-full').style.display = 'inline'; document.getElementById('2309.04101v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.04101v1-abstract-full" style="display: none;"> Let $\mathcal{C}_{k}^{-}$ be the set of all negative $C_k$. For odd cycle, Wang, Hou and Li [29] gave a spectral condition for the existence of negative $C_3$ in unbalanced signed graphs. For even cycle, we determine the maximum index among all $\mathcal{C}_4^{-}$-free unbalanced signed graphs and completely characterize the extremal signed graph in this paper. This could be regarded as a signed graph version of the results by Nikiforov [23] and Zhai and Wang [37]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.04101v1-abstract-full').style.display = 'none'; document.getElementById('2309.04101v1-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.16375">arXiv:2307.16375</a> <span> [<a href="https://arxiv.org/pdf/2307.16375">pdf</a>, <a href="https://arxiv.org/format/2307.16375">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="Distributed, Parallel, and Cluster Computing">cs.DC</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"> UniAP: Unifying Inter- and Intra-Layer Automatic Parallelism by Mixed Integer Quadratic Programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hao Lin</a>, <a href="/search/math?searchtype=author&query=Wu%2C+K">Ke Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jie Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jun Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+W">Wu-Jun Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.16375v5-abstract-short" style="display: inline;"> Distributed learning is commonly used for training deep learning models, especially large models. In distributed learning, manual parallelism (MP) methods demand considerable human effort and have limited flexibility. Hence, automatic parallelism (AP) methods have recently been proposed for automating the parallel strategy optimization process. Existing AP methods suffer from sub-optimal solutions… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.16375v5-abstract-full').style.display = 'inline'; document.getElementById('2307.16375v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.16375v5-abstract-full" style="display: none;"> Distributed learning is commonly used for training deep learning models, especially large models. In distributed learning, manual parallelism (MP) methods demand considerable human effort and have limited flexibility. Hence, automatic parallelism (AP) methods have recently been proposed for automating the parallel strategy optimization process. Existing AP methods suffer from sub-optimal solutions because they do not jointly optimize the two categories of parallel strategies (i.e., inter-layer parallelism and intra-layer parallelism). In this paper, we propose a novel AP method called UniAP, which unifies inter- and intra-layer automatic parallelism by mixed integer quadratic programming. To the best of our knowledge, UniAP is the first parallel method that can jointly optimize the two categories of parallel strategies to find an optimal solution. Experimental results show that UniAP outperforms state-of-the-art methods by up to 3.80$\times$ in throughput and reduces strategy optimization time by up to 107$\times$ across five Transformer-based models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.16375v5-abstract-full').style.display = 'none'; document.getElementById('2307.16375v5-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 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </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">16 pages, 10 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.15558">arXiv:2307.15558</a> <span> [<a href="https://arxiv.org/pdf/2307.15558">pdf</a>, <a href="https://arxiv.org/ps/2307.15558">ps</a>, <a href="https://arxiv.org/format/2307.15558">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Extensions of C*-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gabe%2C+J">James Gabe</a>, <a href="/search/math?searchtype=author&query=Lin%2C+H">Huaxin Lin</a>, <a href="/search/math?searchtype=author&query=Ng%2C+P+W">Ping Wong Ng</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="2307.15558v1-abstract-short" style="display: inline;"> Let $A$ be a separable amenable $C^*$-algebra and $B$ a non-unital and $蟽$-unital simple $C^*$-algebra with continuous scale ($B$ need not be stable). We classify, up to unitary equivalence, all essential extensions of the form $0 \rightarrow B \rightarrow D \rightarrow A \rightarrow 0$ using KK theory. There are characterizations of when the relation of weak unitary equivalence is the same as t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.15558v1-abstract-full').style.display = 'inline'; document.getElementById('2307.15558v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.15558v1-abstract-full" style="display: none;"> Let $A$ be a separable amenable $C^*$-algebra and $B$ a non-unital and $蟽$-unital simple $C^*$-algebra with continuous scale ($B$ need not be stable). We classify, up to unitary equivalence, all essential extensions of the form $0 \rightarrow B \rightarrow D \rightarrow A \rightarrow 0$ using KK theory. There are characterizations of when the relation of weak unitary equivalence is the same as the relation of unitary equivalence, and characterizations of when an extension is liftable (a.k.a.~trivial or split). In the case where $B$ is purely infinite, an essential extension $蟻: A \rightarrow M(B)/B$ is liftable if and only if $[蟻]=0$ in $KK(A, M(B)/B)$. When $B$ is stably finite, the extension $蟻$ is often not liftable when $[蟻]=0$ in $KK(A, M(B)/B).$ Finally, when $B$ additionally has tracial rank zero and when $A$ belongs to a sufficiently regular class of unital separable amenable $C^*$-algebras, we have a version of the Voiculescu noncommutative Weyl--von Neumann theorem: Suppose that $桅, 唯: A \rightarrow M(B)$ are unital injective homomorphisms such that $桅(A) \cap B = 唯(A) \cap B = \{ 0 \}$ and $蟿\circ 桅= 蟿\circ 唯$ for all $蟿\in T(B),$ {the tracial state space of $B.$} Then there exists a sequence $\{ u_n \}$ of unitaries in $M(B)$ such that (i) $u_n 桅(a) u_n^* - 唯(a) \in B$ for all $a \in A$ and $n \geq 1$, (ii) $\| u_n 桅(a) u_n^* - 唯(a) \| \rightarrow 0$ as $n \rightarrow \infty$ for all $a \in A$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.15558v1-abstract-full').style.display = 'none'; document.getElementById('2307.15558v1-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 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </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">89 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/2307.08200">arXiv:2307.08200</a> <span> [<a href="https://arxiv.org/pdf/2307.08200">pdf</a>, <a href="https://arxiv.org/format/2307.08200">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Ternary Stochastic Geometry Theory for Performance Analysis of RIS-Assisted UDN </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongchi Lin</a>, <a href="/search/math?searchtype=author&query=yu%2C+Q">Qiyue 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="2307.08200v3-abstract-short" style="display: inline;"> Currently, network topology becomes increasingly complex with the increased number of various network nodes, bringing in the challenge of network design and analysis. Most of the current studies are deduced based on the binary system stochastic geometry, overlooking the coupling and collaboration among nodes. This limitation makes it difficult to accurately analyze network systems, such as reconfi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.08200v3-abstract-full').style.display = 'inline'; document.getElementById('2307.08200v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.08200v3-abstract-full" style="display: none;"> Currently, network topology becomes increasingly complex with the increased number of various network nodes, bringing in the challenge of network design and analysis. Most of the current studies are deduced based on the binary system stochastic geometry, overlooking the coupling and collaboration among nodes. This limitation makes it difficult to accurately analyze network systems, such as reconfigurable intelligent surface (RIS) assisted ultra-dense network (UDN). To address this issue, we propose a dual coordinate system analysis method, by using dual observation points and their established coordinates. The concept of a typical triangle that consists of a base station (BS), a RIS, and a user equipment (UE) is defined as the fundamental unit of analysis for ternary stochastic geometry. This triangle comprises the base station, the RIS, and the user equipment (UE). Furthermore, we extend Campbell's theorem and propose an approximate probability generating function for ternary stochastic geometry. Utilizing the theoretical framework of ternary stochastic geometry, we derive and analyze performance metrics of a RIS-assisted UDN system, such as coverage probability, area spectral efficiency, area energy efficiency, and energy coverage efficiency. Simulation results show that RIS can significantly enhance system performance, particularly for UEs with high signal-to-interference-plus-noise ratios, exhibiting a phenomenon similar to the Matthew effect. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.08200v3-abstract-full').style.display = 'none'; document.getElementById('2307.08200v3-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 10 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2306.14244">arXiv:2306.14244</a> <span> [<a href="https://arxiv.org/pdf/2306.14244">pdf</a>, <a href="https://arxiv.org/ps/2306.14244">ps</a>, <a href="https://arxiv.org/format/2306.14244">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"> Largest and Least H-Eigenvalues of Symmetric Tensors and Hypergraphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lin%2C+H">Hongying Lin</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+L">Lu Zheng</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+B">Bo 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="2306.14244v1-abstract-short" style="display: inline;"> In tensor eigenvalue problems, one is likely to be more interested in H-eigenvalues of tensors. The largest H-eigenvalue of a nonnegative tensor or of a uniform hypergraph is the spectral radius of the tensor or of the uniform hypergraph. We find upper bounds and lower bounds (interlacing inequalities) for the largest H-eigenvalue of a principal subtensor of a symmetric zero diagonal tensor that i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.14244v1-abstract-full').style.display = 'inline'; document.getElementById('2306.14244v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.14244v1-abstract-full" style="display: none;"> In tensor eigenvalue problems, one is likely to be more interested in H-eigenvalues of tensors. The largest H-eigenvalue of a nonnegative tensor or of a uniform hypergraph is the spectral radius of the tensor or of the uniform hypergraph. We find upper bounds and lower bounds (interlacing inequalities) for the largest H-eigenvalue of a principal subtensor of a symmetric zero diagonal tensor that is of even order or nonnegative, as well as lower bounds for the largest H-eigenvalue of a uniform hypergraph with some vertices or edges removed. We also investigate similar problems for the least H-eigenvalues. We give examples to verify the sharpness of the bounds or in some cases for uniform hypergraphs, we characterize the equality. Particularly, for a connected linear $k$-uniform hypergraph $G$ with $v\in V(G)$, we give a sharp lower bound for the spectral radius of $G-v$ in terms of the spectral radius of $G$ and the degree of $v$ and characterize the extremal hypergraphs, and show that the maximum spectral radius of the subhypergraphs with one vertex removed is greater than or equal to the spectral radius of the hypergraph minus one, which is attained if and only if it is a Steiner system $S(2,k,n)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.14244v1-abstract-full').style.display = 'none'; document.getElementById('2306.14244v1-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 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2023. </p> </li> </ol> <nav class="pagination is-small is-centered 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