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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=Chu%2C+C&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Chu%2C+C&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Chu%2C+C&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </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/2502.06399">arXiv:2502.06399</a> <span> [<a href="https://arxiv.org/pdf/2502.06399">pdf</a>, <a href="https://arxiv.org/format/2502.06399">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A Linearly Convergent Algorithm for Computing the Petz-Augustin Information </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chun-Neng Chu</a>, <a href="/search/math?searchtype=author&query=Tseng%2C+W">Wei-Fu Tseng</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yen-Huan Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.06399v1-abstract-short" style="display: inline;"> We propose an iterative algorithm for computing the Petz-Augustin information of order $伪\in(1/2,1)\cup(1,\infty)$. The optimization error is guaranteed to converge at a rate of $O\left(\vert 1-1/伪\vert^T\right)$, where $T$ is the number of iterations. Let $n$ denote the cardinality of the input alphabet of the classical-quantum channel, and $d$ the dimension of the quantum states. The algorithm h… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.06399v1-abstract-full').style.display = 'inline'; document.getElementById('2502.06399v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.06399v1-abstract-full" style="display: none;"> We propose an iterative algorithm for computing the Petz-Augustin information of order $伪\in(1/2,1)\cup(1,\infty)$. The optimization error is guaranteed to converge at a rate of $O\left(\vert 1-1/伪\vert^T\right)$, where $T$ is the number of iterations. Let $n$ denote the cardinality of the input alphabet of the classical-quantum channel, and $d$ the dimension of the quantum states. The algorithm has an initialization time complexity of $O\left(n d^{3}\right)$ and a per-iteration time complexity of $O\left(n d^{2}+d^3\right)$. To the best of our knowledge, this is the first algorithm for computing the Petz-Augustin information with a non-asymptotic convergence guarantee. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.06399v1-abstract-full').style.display = 'none'; document.getElementById('2502.06399v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <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">QIP 2025 poster</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.09315">arXiv:2409.09315</a> <span> [<a href="https://arxiv.org/pdf/2409.09315">pdf</a>, <a href="https://arxiv.org/format/2409.09315">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Existence of 5 minimal tori in 3-spheres of positive Ricci curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+A+C">Adrian Chun-Pong Chu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yangyang 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="2409.09315v1-abstract-short" style="display: inline;"> In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. We confirm this conjecture for 3-spheres of positive Ricci curvature. While our proof uses min-max theory, the underlying heuristics are largely inspired by mean curvature flow. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.09315v1-abstract-full" style="display: none;"> In 1989, B. White conjectured that every Riemannian 3-sphere has at least 5 embedded minimal tori. We confirm this conjecture for 3-spheres of positive Ricci curvature. While our proof uses min-max theory, the underlying heuristics are largely inspired by mean curvature flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.09315v1-abstract-full').style.display = 'none'; document.getElementById('2409.09315v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">104 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/2403.12628">arXiv:2403.12628</a> <span> [<a href="https://arxiv.org/pdf/2403.12628">pdf</a>, <a href="https://arxiv.org/ps/2403.12628">ps</a>, <a href="https://arxiv.org/format/2403.12628">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</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/10.1007/s11425-016-5135-4">10.1007/s11425-016-5135-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A geometric characterisation of real C*-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</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="2403.12628v1-abstract-short" style="display: inline;"> We characterise the positive cone of a real C*-algebra geometrically. Given an open cone $惟$ in a real Banach space $V$, with closure $\overline 惟$, we show that $惟$ is the interior of the positive cone of a unital real C*-algebra if and only if it is a Finsler symmetric cone with an orientable extension, which is equivalent to the condition that $V$ is, in an equivalent norm, the hermitian part o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.12628v1-abstract-full').style.display = 'inline'; document.getElementById('2403.12628v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.12628v1-abstract-full" style="display: none;"> We characterise the positive cone of a real C*-algebra geometrically. Given an open cone $惟$ in a real Banach space $V$, with closure $\overline 惟$, we show that $惟$ is the interior of the positive cone of a unital real C*-algebra if and only if it is a Finsler symmetric cone with an orientable extension, which is equivalent to the condition that $V$ is, in an equivalent norm, the hermitian part of a unital real C*-algebra with positive cone $\overline惟$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.12628v1-abstract-full').style.display = 'none'; document.getElementById('2403.12628v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46L05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Science China Mathematics 2023 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.07741">arXiv:2309.07741</a> <span> [<a href="https://arxiv.org/pdf/2309.07741">pdf</a>, <a href="https://arxiv.org/format/2309.07741">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"> A strong multiplicity one theorem in min-max theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+A+C">Adrian Chun-Pong Chu</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yangyang 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="2309.07741v2-abstract-short" style="display: inline;"> It was asked by Marques-Neves [MN17, Section 9] which min-max p-widths of the unit 3-sphere lie strictly between $2蟺^2$ and $8蟺$. We show that the 10th to the 13th widths do. More generally, we strengthen X. Zhou's multiplicity one theorem: We prove that in any closed manifold of dimension between 3 and 7 with a bumpy metric or a metric of positive Ricci curvature, for any min-max p-width, there… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.07741v2-abstract-full').style.display = 'inline'; document.getElementById('2309.07741v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.07741v2-abstract-full" style="display: none;"> It was asked by Marques-Neves [MN17, Section 9] which min-max p-widths of the unit 3-sphere lie strictly between $2蟺^2$ and $8蟺$. We show that the 10th to the 13th widths do. More generally, we strengthen X. Zhou's multiplicity one theorem: We prove that in any closed manifold of dimension between 3 and 7 with a bumpy metric or a metric of positive Ricci curvature, for any min-max p-width, there exists a minimizing sequence of sweepouts that only detects multiplicity one minimal hypersurfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.07741v2-abstract-full').style.display = 'none'; document.getElementById('2309.07741v2-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 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/2308.05923">arXiv:2308.05923</a> <span> [<a href="https://arxiv.org/pdf/2308.05923">pdf</a>, <a href="https://arxiv.org/format/2308.05923">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Genus one singularities in mean curvature flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+A+C">Adrian Chun-Pong Chu</a>, <a href="/search/math?searchtype=author&query=Sun%2C+A">Ao Sun</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2308.05923v3-abstract-short" style="display: inline;"> We show that for certain one-parameter families of initial conditions in $\mathbb R^3$, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinke… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.05923v3-abstract-full').style.display = 'inline'; document.getElementById('2308.05923v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.05923v3-abstract-full" style="display: none;"> We show that for certain one-parameter families of initial conditions in $\mathbb R^3$, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.05923v3-abstract-full').style.display = 'none'; document.getElementById('2308.05923v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">37 pages. Accepted by Geometry & Topology. v3: We added a remark to address the genus one example by Buzano-Nguyen-Schulz and their numerical simulation</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.06943">arXiv:2209.06943</a> <span> [<a href="https://arxiv.org/pdf/2209.06943">pdf</a>, <a href="https://arxiv.org/ps/2209.06943">ps</a>, <a href="https://arxiv.org/format/2209.06943">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="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</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/10.1007/s10231-023-01419-7">10.1007/s10231-023-01419-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Horofunctions and metric compactification of noncompact Hermitian symmetric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Cueto-Avellaneda%2C+M">Mar铆a Cueto-Avellaneda</a>, <a href="/search/math?searchtype=author&query=Lemmens%2C+B">Bas Lemmens</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="2209.06943v1-abstract-short" style="display: inline;"> Given a Hermitian symmetric space $M$ of noncompact type, we give a complete description of the horofunctions in the metric compactification of $M$ with respect to the Carath茅odory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan structure, called a $\mathrm{JB}^*$-triple. The Carath茅odory distance $蟻$ on $D$ has a Finsler stru… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.06943v1-abstract-full').style.display = 'inline'; document.getElementById('2209.06943v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.06943v1-abstract-full" style="display: none;"> Given a Hermitian symmetric space $M$ of noncompact type, we give a complete description of the horofunctions in the metric compactification of $M$ with respect to the Carath茅odory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan structure, called a $\mathrm{JB}^*$-triple. The Carath茅odory distance $蟻$ on $D$ has a Finsler structure. It is the integrated distance of the Carath茅odory differential metric, and the norm $\|\cdot\|$ in the realisation is the Carath茅odory norm with respect to the origin $0\in D$. We also identify the horofunctions of the metric compactification of $(V,\|\cdot\|)$ and relate its geometry and global topology to the closed dual unit ball (i.e., the polar of $D$). Moreover, we show that the exponential map $\exp_0 \colon V \longrightarrow D$ at $0\in D$ extends to a homeomorphism between the metric compactifications of $(V,\|\cdot\|)$ and $(D,蟻)$, preserving the geometric structure. Consequently, the metric compactification of $M$ admits a concrete realisation as the closed dual unit ball of $(V,\|\cdot\|)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.06943v1-abstract-full').style.display = 'none'; document.getElementById('2209.06943v1-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 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </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">39 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32M15; 17C65; 46L70; 53C60 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ann. Mat. Pura Appl. (4), (203)(4), (2024), 1703-1751 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.06577">arXiv:2208.06577</a> <span> [<a href="https://arxiv.org/pdf/2208.06577">pdf</a>, <a href="https://arxiv.org/format/2208.06577">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> A free boundary minimal surface via a 6-sweepout </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+A+C">Adrian Chun-Pong Chu</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="2208.06577v2-abstract-short" style="display: inline;"> We prove that the Almgren-Pitts 6-width of the unit 3-ball is less than $2蟺$. We also prove that there exists a free boundary minimal surface in the unit 3-ball that has genus at most 1, index at most 5, area less than $2蟺$, and is not the equatorial disk or the critical catenoid. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.06577v2-abstract-full" style="display: none;"> We prove that the Almgren-Pitts 6-width of the unit 3-ball is less than $2蟺$. We also prove that there exists a free boundary minimal surface in the unit 3-ball that has genus at most 1, index at most 5, area less than $2蟺$, and is not the equatorial disk or the critical catenoid. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.06577v2-abstract-full').style.display = 'none'; document.getElementById('2208.06577v2-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 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted by J. Geom. Anal. Abstract updated</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.07636">arXiv:2206.07636</a> <span> [<a href="https://arxiv.org/pdf/2206.07636">pdf</a>, <a href="https://arxiv.org/format/2206.07636">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Graphics">cs.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</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/10.1016/j.cag.2022.07.004">10.1016/j.cag.2022.07.004 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> SHREC 2022: Fitting and recognition of simple geometric primitives on point clouds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Romanengo%2C+C">Chiara Romanengo</a>, <a href="/search/math?searchtype=author&query=Raffo%2C+A">Andrea Raffo</a>, <a href="/search/math?searchtype=author&query=Biasotti%2C+S">Silvia Biasotti</a>, <a href="/search/math?searchtype=author&query=Falcidieno%2C+B">Bianca Falcidieno</a>, <a href="/search/math?searchtype=author&query=Fotis%2C+V">Vlassis Fotis</a>, <a href="/search/math?searchtype=author&query=Romanelis%2C+I">Ioannis Romanelis</a>, <a href="/search/math?searchtype=author&query=Psatha%2C+E">Eleftheria Psatha</a>, <a href="/search/math?searchtype=author&query=Moustakas%2C+K">Konstantinos Moustakas</a>, <a href="/search/math?searchtype=author&query=Sipiran%2C+I">Ivan Sipiran</a>, <a href="/search/math?searchtype=author&query=Nguyen%2C+Q">Quang-Thuc Nguyen</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chi-Bien Chu</a>, <a href="/search/math?searchtype=author&query=Nguyen-Ngoc%2C+K">Khoi-Nguyen Nguyen-Ngoc</a>, <a href="/search/math?searchtype=author&query=Vo%2C+D">Dinh-Khoi Vo</a>, <a href="/search/math?searchtype=author&query=To%2C+T">Tuan-An To</a>, <a href="/search/math?searchtype=author&query=Nguyen%2C+N">Nham-Tan Nguyen</a>, <a href="/search/math?searchtype=author&query=Le-Pham%2C+N">Nhat-Quynh Le-Pham</a>, <a href="/search/math?searchtype=author&query=Nguyen%2C+H">Hai-Dang Nguyen</a>, <a href="/search/math?searchtype=author&query=Tran%2C+M">Minh-Triet Tran</a>, <a href="/search/math?searchtype=author&query=Qie%2C+Y">Yifan Qie</a>, <a href="/search/math?searchtype=author&query=Anwer%2C+N">Nabil Anwer</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="2206.07636v2-abstract-short" style="display: inline;"> This paper presents the methods that have participated in the SHREC 2022 track on the fitting and recognition of simple geometric primitives on point clouds. As simple primitives we mean the classical surface primitives derived from constructive solid geometry, i.e., planes, spheres, cylinders, cones and tori. The aim of the track is to evaluate the quality of automatic algorithms for fitting and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.07636v2-abstract-full').style.display = 'inline'; document.getElementById('2206.07636v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.07636v2-abstract-full" style="display: none;"> This paper presents the methods that have participated in the SHREC 2022 track on the fitting and recognition of simple geometric primitives on point clouds. As simple primitives we mean the classical surface primitives derived from constructive solid geometry, i.e., planes, spheres, cylinders, cones and tori. The aim of the track is to evaluate the quality of automatic algorithms for fitting and recognising geometric primitives on point clouds. Specifically, the goal is to identify, for each point cloud, its primitive type and some geometric descriptors. For this purpose, we created a synthetic dataset, divided into a training set and a test set, containing segments perturbed with different kinds of point cloud artifacts. Among the six participants to this track, two are based on direct methods, while four are either fully based on deep learning or combine direct and neural approaches. The performance of the methods is evaluated using various classification and approximation measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.07636v2-abstract-full').style.display = 'none'; document.getElementById('2206.07636v2-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 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 68U05; 68U07; 65D18; 65D17 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.1.2; I.3.5; I.5.4 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Computers & Graphics 107 (2022) 32-49 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.03360">arXiv:2011.03360</a> <span> [<a href="https://arxiv.org/pdf/2011.03360">pdf</a>, <a href="https://arxiv.org/ps/2011.03360">ps</a>, <a href="https://arxiv.org/format/2011.03360">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> A Gleason-Kahane-呕elazko theorem for reproducing kernel Hilbert spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</a>, <a href="/search/math?searchtype=author&query=Hartz%2C+M">Michael Hartz</a>, <a href="/search/math?searchtype=author&query=Mashreghi%2C+J">Javad Mashreghi</a>, <a href="/search/math?searchtype=author&query=Ransford%2C+T">Thomas Ransford</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="2011.03360v3-abstract-short" style="display: inline;"> We establish the following Hilbert-space analogue of the Gleason-Kahane-呕elazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $螞$ is a linear functional on $\mathcal{H}$ such that $螞(1)=1$ and $螞(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then $螞$ is multiplicative, in the sense that $螞(fg)=螞(f)螞(g)$ for all… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.03360v3-abstract-full').style.display = 'inline'; document.getElementById('2011.03360v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.03360v3-abstract-full" style="display: none;"> We establish the following Hilbert-space analogue of the Gleason-Kahane-呕elazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with a normalized complete Pick kernel, and if $螞$ is a linear functional on $\mathcal{H}$ such that $螞(1)=1$ and $螞(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then $螞$ is multiplicative, in the sense that $螞(fg)=螞(f)螞(g)$ for all $f,g\in\mathcal{H}$ such that $fg\in\mathcal{H}$. Moreover $螞$ is automatically continuous. We give examples to show that the theorem fails if the hypothesis of a complete Pick kernel is omitted. We also discuss conditions under which $螞$ has to be a point evaluation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.03360v3-abstract-full').style.display = 'none'; document.getElementById('2011.03360v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46E22 (primary); 46H40 (secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.06499">arXiv:2006.06499</a> <span> [<a href="https://arxiv.org/pdf/2006.06499">pdf</a>, <a href="https://arxiv.org/ps/2006.06499">ps</a>, <a href="https://arxiv.org/format/2006.06499">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Siegel domains over Finsler symmetric cones </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</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="2006.06499v1-abstract-short" style="display: inline;"> Let $惟$ be a proper open cone in a real Banach space $V$. We show that the tube domain $V \oplus i惟$ over $惟$ is biholomorphic to a bounded symmetric domain if and only if $惟$ is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that $V$ is a unital JB-algebra in an equivalent norm and $惟$ is the interior of $\{v^2: v\in V\}$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.06499v1-abstract-full" style="display: none;"> Let $惟$ be a proper open cone in a real Banach space $V$. We show that the tube domain $V \oplus i惟$ over $惟$ is biholomorphic to a bounded symmetric domain if and only if $惟$ is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that $V$ is a unital JB-algebra in an equivalent norm and $惟$ is the interior of $\{v^2: v\in V\}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.06499v1-abstract-full').style.display = 'none'; document.getElementById('2006.06499v1-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58B20; 32M15; 22E65; 17C65; 46B40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2004.02284">arXiv:2004.02284</a> <span> [<a href="https://arxiv.org/pdf/2004.02284">pdf</a>, <a href="https://arxiv.org/ps/2004.02284">ps</a>, <a href="https://arxiv.org/format/2004.02284">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"> Product of truncated Hankel and truncated Toeplitz operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="2004.02284v1-abstract-short" style="display: inline;"> A truncated Toeplitz operator is the compression of a classical Toeplitz operator on the Hardy space to a model space. A truncated Hankel operator is the compression of a Hankel operator on the Hardy space to the orthogonal complement of a model space. We study the product of a truncated Hankel operator and a truncated Toeplitz operator, and characterize when such a product is zero or compact. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2004.02284v1-abstract-full" style="display: none;"> A truncated Toeplitz operator is the compression of a classical Toeplitz operator on the Hardy space to a model space. A truncated Hankel operator is the compression of a Hankel operator on the Hardy space to the orthogonal complement of a model space. We study the product of a truncated Hankel operator and a truncated Toeplitz operator, and characterize when such a product is zero or compact. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.02284v1-abstract-full').style.display = 'none'; document.getElementById('2004.02284v1-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.05202">arXiv:2001.05202</a> <span> [<a href="https://arxiv.org/pdf/2001.05202">pdf</a>, <a href="https://arxiv.org/ps/2001.05202">ps</a>, <a href="https://arxiv.org/format/2001.05202">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Randomized Bregman Coordinate Descent Methods for Non-Lipschitz Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+T">Tianxiang Gao</a>, <a href="/search/math?searchtype=author&query=Lu%2C+S">Songtao Lu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jia Liu</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chris Chu</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="2001.05202v1-abstract-short" style="display: inline;"> We propose a new \textit{randomized Bregman (block) coordinate descent} (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global Lipschitz-continuous (partial) gradient assumption. Under the notion of relative smoothness based on the Bregman distance, we prove that every limit point of the gene… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.05202v1-abstract-full').style.display = 'inline'; document.getElementById('2001.05202v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.05202v1-abstract-full" style="display: none;"> We propose a new \textit{randomized Bregman (block) coordinate descent} (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global Lipschitz-continuous (partial) gradient assumption. Under the notion of relative smoothness based on the Bregman distance, we prove that every limit point of the generated sequence is a stationary point. Further, we show that the iteration complexity of the proposed method is $O(n\varepsilon^{-2})$ to achieve $蔚$-stationary point, where $n$ is the number of blocks of coordinates. If the objective is assumed to be convex, the iteration complexity is improved to $O(n蔚^{-1} )$. If, in addition, the objective is strongly convex (relative to the reference function), the global linear convergence rate is recovered. We also present the accelerated version of the RBCD method, which attains an $O(n\varepsilon^{-1/纬} )$ iteration complexity for the convex case, where the scalar $纬\in [1,2]$ is determined by the \textit{generalized translation variant} of the Bregman distance. Convergence analysis without assuming the global Lipschitz-continuous (partial) gradient sets our results apart from the existing works in the composite problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.05202v1-abstract-full').style.display = 'none'; document.getElementById('2001.05202v1-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 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </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">First draft</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.07527">arXiv:1912.07527</a> <span> [<a href="https://arxiv.org/pdf/1912.07527">pdf</a>, <a href="https://arxiv.org/ps/1912.07527">ps</a>, <a href="https://arxiv.org/format/1912.07527">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Leveraging Two Reference Functions in Block Bregman Proximal Gradient Descent for Non-convex and Non-Lipschitz Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+T">Tianxiang Gao</a>, <a href="/search/math?searchtype=author&query=Lu%2C+S">Songtao Lu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jia Liu</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chris Chu</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="1912.07527v1-abstract-short" style="display: inline;"> In the applications of signal processing and data analytics, there is a wide class of non-convex problems whose objective function is freed from the common global Lipschitz continuous gradient assumption (e.g., the nonnegative matrix factorization (NMF) problem). Recently, this type of problem with some certain special structures has been solved by Bregman proximal gradient (BPG). This inspires us… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.07527v1-abstract-full').style.display = 'inline'; document.getElementById('1912.07527v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.07527v1-abstract-full" style="display: none;"> In the applications of signal processing and data analytics, there is a wide class of non-convex problems whose objective function is freed from the common global Lipschitz continuous gradient assumption (e.g., the nonnegative matrix factorization (NMF) problem). Recently, this type of problem with some certain special structures has been solved by Bregman proximal gradient (BPG). This inspires us to propose a new Block-wise two-references Bregman proximal gradient (B2B) method, which adopts two reference functions so that a closed-form solution in the Bregman projection is obtained. Based on the relative smoothness, we prove the global convergence of the proposed algorithms for various block selection rules. In particular, we establish the global convergence rate of $O(\frac{\sqrt{s}}{\sqrt{k}})$ for the greedy and randomized block updating rule for B2B, which is $O(\sqrt{s})$ times faster than the cyclic variant, i.e., $O(\frac{s}{\sqrt{k}} )$, where $s$ is the number of blocks, and $k$ is the number of iterations. Multiple numerical results are provided to illustrate the superiority of the proposed B2B compared to the state-of-the-art works in solving NMF problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.07527v1-abstract-full').style.display = 'none'; document.getElementById('1912.07527v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </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">Submit to TSP</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1910.04830">arXiv:1910.04830</a> <span> [<a href="https://arxiv.org/pdf/1910.04830">pdf</a>, <a href="https://arxiv.org/ps/1910.04830">ps</a>, <a href="https://arxiv.org/format/1910.04830">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"> Which de Branges-Rovnyak spaces have complete Nevanlinna-Pick property? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1910.04830v2-abstract-short" style="display: inline;"> We characterize the de Branges-Rovnyak spaces with complete Nevanlinna-Pick property. Our method relies on the general theory of reproducing kernel Hilbert spaces. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.04830v2-abstract-full" style="display: none;"> We characterize the de Branges-Rovnyak spaces with complete Nevanlinna-Pick property. Our method relies on the general theory of reproducing kernel Hilbert spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.04830v2-abstract-full').style.display = 'none'; document.getElementById('1910.04830v2-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 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.01532">arXiv:1903.01532</a> <span> [<a href="https://arxiv.org/pdf/1903.01532">pdf</a>, <a href="https://arxiv.org/format/1903.01532">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"> Hierarchical Distributed Framework for EV Charging Scheduling Using Exchange Problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Khaki%2C+B">Behnam Khaki</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chicheng Chu</a>, <a href="/search/math?searchtype=author&query=Gadh%2C+R">Rajit Gadh</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="1903.01532v1-abstract-short" style="display: inline;"> In this paper, a distributed trilayer multi-agent framework is proposed for optimal electric vehicle charging scheduling (EVCS). The framework reduces the negative effects of electric vehicle charging demand on the electrical grids. To solve the scheduling problem, a novel hierarchical distributed EV charging scheduling (HDEVCS) is developed as the \textit{exchange problem}, where the agents are c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.01532v1-abstract-full').style.display = 'inline'; document.getElementById('1903.01532v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.01532v1-abstract-full" style="display: none;"> In this paper, a distributed trilayer multi-agent framework is proposed for optimal electric vehicle charging scheduling (EVCS). The framework reduces the negative effects of electric vehicle charging demand on the electrical grids. To solve the scheduling problem, a novel hierarchical distributed EV charging scheduling (HDEVCS) is developed as the \textit{exchange problem}, where the agents are clustered based on their coupling constraints. According to the separability of the agents' objectives and the clusters' coupled constraints, HDEVCS is solved efficiently in a distributed manner by the alternating direction method of multipliers. Comparing to the exiting trilayer methods, HDEVCS reduces the convergence time and the iteration numbers since its structure allows the agents to update their primal optimization variable simultaneously. The performance of HDEVCS is evaluated by numerical simulation of two small- and large- scale case studies consisting of $306$ and $9051$ agents, respectively. The results verify the scalability and efficiency of the proposed method, as it reduces the convergence time and iteration numbers by $60\%$ compared to the state-of-the-art methods, flattens the load profile and decreases the charging cost considerably without violating the grid feeders' capacity. The significant outcome of our method is the accommodation of a large EV population without investment in grid expansion. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.01532v1-abstract-full').style.display = 'none'; document.getElementById('1903.01532v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.02847">arXiv:1812.02847</a> <span> [<a href="https://arxiv.org/pdf/1812.02847">pdf</a>, <a href="https://arxiv.org/format/1812.02847">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"> Hierarchical Distributed EV Charging Scheduling in Distribution Grids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Khaki%2C+B">Behnam Khaki</a>, <a href="/search/math?searchtype=author&query=Chung%2C+Y">Yu-Wei Chung</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chicheng Chu</a>, <a href="/search/math?searchtype=author&query=Gadh%2C+R">Rajit Gadh</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="1812.02847v2-abstract-short" style="display: inline;"> In this paper, a hierarchical distributed method consisting of two iterative procedures is proposed for optimal electric vehicle charging scheduling (EVCS) in the distribution grids. In the proposed method, the distribution system operator (DSO) aims at reducing the grid loss while satisfying the power flow constraints. This is achieved by a consensus-based iterative procedure with the EV aggregat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.02847v2-abstract-full').style.display = 'inline'; document.getElementById('1812.02847v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.02847v2-abstract-full" style="display: none;"> In this paper, a hierarchical distributed method consisting of two iterative procedures is proposed for optimal electric vehicle charging scheduling (EVCS) in the distribution grids. In the proposed method, the distribution system operator (DSO) aims at reducing the grid loss while satisfying the power flow constraints. This is achieved by a consensus-based iterative procedure with the EV aggregators (Aggs) located in the grid buses. The goal of aggregators, which are equipped with the battery energy storage (BES), is to reduce their electricity cost by optimal control of BES and EVs. As Aggs' optimization problem increases dimensionally by increasing the number of EVs, they solved their problem through another iterative procedure with their customers. This procedure is implementable by exploiting the mathematical properties of the problem and rewriting Aggs' optimization problem as the \textit{sharing problem}, which is solved efficiently by the alternating direction method of multipliers (ADMM). To validate the performance, the proposed method is applied to IEEE-13 bus system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.02847v2-abstract-full').style.display = 'none'; document.getElementById('1812.02847v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </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 has been accepted to IEEE PES General Meeting 2019</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.08397">arXiv:1810.08397</a> <span> [<a href="https://arxiv.org/pdf/1810.08397">pdf</a>, <a href="https://arxiv.org/ps/1810.08397">ps</a>, <a href="https://arxiv.org/format/1810.08397">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"> Infinitely many solutions of a class of elliptic equations with variable exponent </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chang-Mu Chu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+H">Haidong Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.08397v1-abstract-short" style="display: inline;"> This paper is concerned with the $p(x)$-Laplacian equation of the form \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -螖_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ 惟,\\ u=0, &\mbox{on}\ \partial 惟, \end{array}\right. \end{equation} where $惟\subset\R^N$ is a smooth bounded domain, $1<p^-=\min_{x\in\overline惟}p(x)\leq p(x)\leq\max_{x\in\overline惟}p(x)=p^+<N$,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.08397v1-abstract-full').style.display = 'inline'; document.getElementById('1810.08397v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.08397v1-abstract-full" style="display: none;"> This paper is concerned with the $p(x)$-Laplacian equation of the form \begin{equation}\label{eq0.1} \left\{\begin{array}{ll} -螖_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ 惟,\\ u=0, &\mbox{on}\ \partial 惟, \end{array}\right. \end{equation} where $惟\subset\R^N$ is a smooth bounded domain, $1<p^-=\min_{x\in\overline惟}p(x)\leq p(x)\leq\max_{x\in\overline惟}p(x)=p^+<N$, $1\leq r(x)<p^{*}(x)=\frac{Np(x)}{N-p(x)}$, $r^-=\min_{x\in \overline惟}r(x)<p^-$, $r^+=\max_{x\in\overline惟}r(x)>p^+$ and $Q: \overline惟\to\R$ is a nonnegative continuous function. We prove that \eqref{eq0.1} has infinitely many small solutions and infinitely many large solutions by using the Clark's theorem and the symmetric mountain pass lemma. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.08397v1-abstract-full').style.display = 'none'; document.getElementById('1810.08397v1-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 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.01058">arXiv:1810.01058</a> <span> [<a href="https://arxiv.org/pdf/1810.01058">pdf</a>, <a href="https://arxiv.org/ps/1810.01058">ps</a>, <a href="https://arxiv.org/format/1810.01058">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"> Reducing Subspaces of de Branges-Rovnyak Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1810.01058v1-abstract-short" style="display: inline;"> For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operato… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.01058v1-abstract-full').style.display = 'inline'; document.getElementById('1810.01058v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.01058v1-abstract-full" style="display: none;"> For $b\in H^\infty_1$, the closed unit ball of $H^\infty$, the de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a Hilbert space contractively contained in the Hardy space $H^2$ that is invariant by the backward shift operator $S^*$. We consider the reducing subspaces of the operator $S^{*2}|_{\mathcal{H}(b)}$. When $b$ is an inner function, $S^{*2}|_{\mathcal{H}(b)}$ is a truncated Toepltiz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where $b$ is extreme. We prove that if $b$ is extreme but not inner, then $S^{*2}|_{\mathcal{H}(b)}$ is reducible if and only if $b$ is even or odd, and describe the structure of reducing subspaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.01058v1-abstract-full').style.display = 'none'; document.getElementById('1810.01058v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1808.03416">arXiv:1808.03416</a> <span> [<a href="https://arxiv.org/pdf/1808.03416">pdf</a>, <a href="https://arxiv.org/ps/1808.03416">ps</a>, <a href="https://arxiv.org/format/1808.03416">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Infinite dimensional holomorphic homogeneous regular domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Kim%2C+K">Kang-Tae Kim</a>, <a href="/search/math?searchtype=author&query=Kim%2C+S">Sejun Kim</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="1808.03416v3-abstract-short" style="display: inline;"> We extend the concept of a finite dimensional {\it holomorphic homogeneous regular} (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.03416v3-abstract-full').style.display = 'inline'; document.getElementById('1808.03416v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.03416v3-abstract-full" style="display: none;"> We extend the concept of a finite dimensional {\it holomorphic homogeneous regular} (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing function of all HHR bounded symmetric domains, including the two exceptional domains. We also show that uniformly elliptic domains in Hilbert spaces are HHR. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.03416v3-abstract-full').style.display = 'none'; document.getElementById('1808.03416v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, 0 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 58B12; 32H02; 32M15; 32T05; Secondary 17C65; 32F45 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Geom. Anal. 30 (2020) 223-247 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1807.08366">arXiv:1807.08366</a> <span> [<a href="https://arxiv.org/pdf/1807.08366">pdf</a>, <a href="https://arxiv.org/ps/1807.08366">ps</a>, <a href="https://arxiv.org/format/1807.08366">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"> Hilbert Spaces Contractively Contained in Weighted Bergman Spaces on the Unit Disk </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1807.08366v2-abstract-short" style="display: inline;"> Sub-Bergman Hilbert spaces are analogues of de Branges-Rovnyak spaces in the Bergman space setting. They are reproducing kernel Hilbert spaces contractively contained in the Bergman space of the unit disk. K. Zhu analyzed sub-Bergman Hilbert spaces associated with finite Blaschke products, and proved that they are norm equivalent to the Hardy space. Later S. Sultanic found a different proof of Zhu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.08366v2-abstract-full').style.display = 'inline'; document.getElementById('1807.08366v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1807.08366v2-abstract-full" style="display: none;"> Sub-Bergman Hilbert spaces are analogues of de Branges-Rovnyak spaces in the Bergman space setting. They are reproducing kernel Hilbert spaces contractively contained in the Bergman space of the unit disk. K. Zhu analyzed sub-Bergman Hilbert spaces associated with finite Blaschke products, and proved that they are norm equivalent to the Hardy space. Later S. Sultanic found a different proof of Zhu's result, which works in weighted Bergman space settings as well. In this paper, we give a new approach to this problem and obtain a stronger result. Our method relies on the theory of reproducing kernel Hilbert spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.08366v2-abstract-full').style.display = 'none'; document.getElementById('1807.08366v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1806.11383">arXiv:1806.11383</a> <span> [<a href="https://arxiv.org/pdf/1806.11383">pdf</a>, <a href="https://arxiv.org/ps/1806.11383">ps</a>, <a href="https://arxiv.org/format/1806.11383">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Density of Polynomials in Sub-Bergman Hilbert Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1806.11383v1-abstract-short" style="display: inline;"> The sub-Bergman Hilbert spaces are analogues of de BrangesRovnyak spaces in the Bergman space setting. We prove that the polynomials are dense in sub-Bergman Hilbert spaces. This answers the question posted by Zhu in the affirmative. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.11383v1-abstract-full" style="display: none;"> The sub-Bergman Hilbert spaces are analogues of de BrangesRovnyak spaces in the Bergman space setting. We prove that the polynomials are dense in sub-Bergman Hilbert spaces. This answers the question posted by Zhu in the affirmative. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.11383v1-abstract-full').style.display = 'none'; document.getElementById('1806.11383v1-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 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.08938">arXiv:1802.08938</a> <span> [<a href="https://arxiv.org/pdf/1802.08938">pdf</a>, <a href="https://arxiv.org/ps/1802.08938">ps</a>, <a href="https://arxiv.org/format/1802.08938">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> DID: Distributed Incremental Block Coordinate Descent for Nonnegative Matrix Factorization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+T">Tianxiang Gao</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chris Chu</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="1802.08938v1-abstract-short" style="display: inline;"> Nonnegative matrix factorization (NMF) has attracted much attention in the last decade as a dimension reduction method in many applications. Due to the explosion in the size of data, naturally the samples are collected and stored distributively in local computational nodes. Thus, there is a growing need to develop algorithms in a distributed memory architecture. We propose a novel distributed algo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.08938v1-abstract-full').style.display = 'inline'; document.getElementById('1802.08938v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.08938v1-abstract-full" style="display: none;"> Nonnegative matrix factorization (NMF) has attracted much attention in the last decade as a dimension reduction method in many applications. Due to the explosion in the size of data, naturally the samples are collected and stored distributively in local computational nodes. Thus, there is a growing need to develop algorithms in a distributed memory architecture. We propose a novel distributed algorithm, called \textit{distributed incremental block coordinate descent} (DID), to solve the problem. By adapting the block coordinate descent framework, closed-form update rules are obtained in DID. Moreover, DID performs updates incrementally based on the most recently updated residual matrix. As a result, only one communication step per iteration is required. The correctness, efficiency, and scalability of the proposed algorithm are verified in a series of numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.08938v1-abstract-full').style.display = 'none'; document.getElementById('1802.08938v1-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, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Accepted by AAAI 2018</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1711.03086">arXiv:1711.03086</a> <span> [<a href="https://arxiv.org/pdf/1711.03086">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Real-Time Bi-directional Electric Vehicle Charging Control with Distribution Grid Implementation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xiong%2C+Y">Yingqi Xiong</a>, <a href="/search/math?searchtype=author&query=Khaki%2C+B">Behnam Khaki</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chi-cheng Chu</a>, <a href="/search/math?searchtype=author&query=Gadh%2C+R">Rajit Gadh</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="1711.03086v1-abstract-short" style="display: inline;"> As electric vehicle (EV) adoption is growing year after year, there is no doubt that EVs will occupy a significant portion of transporting vehicle in the near future. Although EVs have benefits for environment, large amount of un-coordinated EV charging will affect the power grid and degrade power quality. To alleviate negative effects of EV charging load and turn them to opportunities, a decentra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1711.03086v1-abstract-full').style.display = 'inline'; document.getElementById('1711.03086v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1711.03086v1-abstract-full" style="display: none;"> As electric vehicle (EV) adoption is growing year after year, there is no doubt that EVs will occupy a significant portion of transporting vehicle in the near future. Although EVs have benefits for environment, large amount of un-coordinated EV charging will affect the power grid and degrade power quality. To alleviate negative effects of EV charging load and turn them to opportunities, a decentralized real-time control algorithm is developed in this paper to provide optimal scheduling of EV bi-directional charging. To evaluate the performance of the proposed algorithm, numerical simulation is performed based on real-world EV user data, and power flow analysis is carried out to show how the proposed algorithm improve power grid steady state operation. . The results show that the implementation of proposed algorithm can effectively coordinate bi-directional charging by 30% peak load shaving, more than 2% of voltage drop reduction, and 40% transmission line current decrease. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1711.03086v1-abstract-full').style.display = 'none'; document.getElementById('1711.03086v1-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 November, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.04010">arXiv:1709.04010</a> <span> [<a href="https://arxiv.org/pdf/1709.04010">pdf</a>, <a href="https://arxiv.org/ps/1709.04010">ps</a>, <a href="https://arxiv.org/format/1709.04010">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Bounded Composition Operators and Multipliers of Some Reproducing Kernel Hilbert Spaces on the Bidisk </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1709.04010v2-abstract-short" style="display: inline;"> We study the boundedness of composition operators on the bidisk using reproducing kernels. We show that a composition operator is bounded on the Hardy space of the bidisk if some associated function is a positive kernel. This positivity condition naturally leads to the study of the sub-Hardy Hilbert spaces of the bidisk, which are analogs of de Branges-Rovnyak spaces on the unit disk. We discuss m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.04010v2-abstract-full').style.display = 'inline'; document.getElementById('1709.04010v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.04010v2-abstract-full" style="display: none;"> We study the boundedness of composition operators on the bidisk using reproducing kernels. We show that a composition operator is bounded on the Hardy space of the bidisk if some associated function is a positive kernel. This positivity condition naturally leads to the study of the sub-Hardy Hilbert spaces of the bidisk, which are analogs of de Branges-Rovnyak spaces on the unit disk. We discuss multipliers of those spaces and obtain some classes of bounded composition operators on the bidisk. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.04010v2-abstract-full').style.display = 'none'; document.getElementById('1709.04010v2-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 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.04008">arXiv:1709.04008</a> <span> [<a href="https://arxiv.org/pdf/1709.04008">pdf</a>, <a href="https://arxiv.org/ps/1709.04008">ps</a>, <a href="https://arxiv.org/format/1709.04008">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"> Normal Truncated Toeplitz Operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1709.04008v2-abstract-short" style="display: inline;"> The characterization of normal truncated Toepltiz operators is first given by Chalendar and Timotin. We give an elementary proof of their result without using the algebraic properties of truncated Toeplitz operators. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.04008v2-abstract-full" style="display: none;"> The characterization of normal truncated Toepltiz operators is first given by Chalendar and Timotin. We give an elementary proof of their result without using the algebraic properties of truncated Toeplitz operators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.04008v2-abstract-full').style.display = 'none'; document.getElementById('1709.04008v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.03610">arXiv:1707.03610</a> <span> [<a href="https://arxiv.org/pdf/1707.03610">pdf</a>, <a href="https://arxiv.org/ps/1707.03610">ps</a>, <a href="https://arxiv.org/format/1707.03610">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Infinite dimensional Jordan algebras and symmetric cones </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</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="1707.03610v1-abstract-short" style="display: inline;"> A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.03610v1-abstract-full" style="display: none;"> A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.03610v1-abstract-full').style.display = 'none'; document.getElementById('1707.03610v1-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 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17C65; 22E65; 46B40; 46H70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1703.04552">arXiv:1703.04552</a> <span> [<a href="https://arxiv.org/pdf/1703.04552">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Distributed, Parallel, and Cluster Computing">cs.DC</span> </div> </div> <p class="title is-5 mathjax"> Distributed Optimal Vehicle Grid Integration Strategy with User Behavior Prediction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xiong%2C+Y">Yingqi Xiong</a>, <a href="/search/math?searchtype=author&query=Wang%2C+B">Bin Wang</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Chi-cheng Chu</a>, <a href="/search/math?searchtype=author&query=Gadh%2C+R">Rajit Gadh</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="1703.04552v1-abstract-short" style="display: inline;"> With the increasing of electric vehicle (EV) adoption in recent years, the impact of EV charging activities to the power grid becomes more and more significant. In this article, an optimal scheduling algorithm which combines smart EV charging and V2G gird service is developed to integrate EVs into power grid as distributed energy resources, with improved system cost performance. Specifically, an o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.04552v1-abstract-full').style.display = 'inline'; document.getElementById('1703.04552v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1703.04552v1-abstract-full" style="display: none;"> With the increasing of electric vehicle (EV) adoption in recent years, the impact of EV charging activities to the power grid becomes more and more significant. In this article, an optimal scheduling algorithm which combines smart EV charging and V2G gird service is developed to integrate EVs into power grid as distributed energy resources, with improved system cost performance. Specifically, an optimization problem is formulated and solved at each EV charging station according to control signal from aggregated control center and user charging behavior prediction by mean estimation and linear regression. The control center collects distributed optimization results and updates the control signal, periodically. The iteration continues until it converges to optimal scheduling. Experimental result shows this algorithm helps fill the valley and shave the peak in electric load profiles within a microgrid, while the energy demand of individual driver can be satisfied. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.04552v1-abstract-full').style.display = 'none'; document.getElementById('1703.04552v1-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, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2017. </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">IEEE PES General Meeting 2017</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.08848">arXiv:1612.08848</a> <span> [<a href="https://arxiv.org/pdf/1612.08848">pdf</a>, <a href="https://arxiv.org/ps/1612.08848">ps</a>, <a href="https://arxiv.org/format/1612.08848">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Horoballs and iteration of holomorphic maps on bounded symmetric domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Rigby%2C+M">Michael Rigby</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="1612.08848v1-abstract-short" style="display: inline;"> Given a fixed-point free compact holomorphic self-map $f$ on a bounded symmetric domain $D$, which may be infinite dimensional, we establish the existence of a family $\{H(尉, 位)\}_{位>0}$ of convex $f$-invariant domains at a point $尉$ in the boundary $\partial D$ of $D$, which generalises completely Wolff's theorem for the open unit disc in $\mathbb{C}$. Further, we construct horoballs at $尉$ and s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.08848v1-abstract-full').style.display = 'inline'; document.getElementById('1612.08848v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.08848v1-abstract-full" style="display: none;"> Given a fixed-point free compact holomorphic self-map $f$ on a bounded symmetric domain $D$, which may be infinite dimensional, we establish the existence of a family $\{H(尉, 位)\}_{位>0}$ of convex $f$-invariant domains at a point $尉$ in the boundary $\partial D$ of $D$, which generalises completely Wolff's theorem for the open unit disc in $\mathbb{C}$. Further, we construct horoballs at $尉$ and show that they are exactly the $f$-invariant domains when $D$ is of finite rank. Consequently, we show in the latter case that the limit functions of the iterates $(f^n)$ with weakly closed range all accumulate in one single boundary component of $\partial D$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.08848v1-abstract-full').style.display = 'none'; document.getElementById('1612.08848v1-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, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32H50; 32M15; 17C65; 58B12; 58C10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1609.09648">arXiv:1609.09648</a> <span> [<a href="https://arxiv.org/pdf/1609.09648">pdf</a>, <a href="https://arxiv.org/ps/1609.09648">ps</a>, <a href="https://arxiv.org/format/1609.09648">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"> Separably injective $C_蟽$-spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Li%2C+L">Lei 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="1609.09648v1-abstract-short" style="display: inline;"> We show that a (complex) $C_蟽$-space is separably injective if and only if it is linearly isometric to the Banach space $C_0(惟)$ of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space $惟$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1609.09648v1-abstract-full" style="display: none;"> We show that a (complex) $C_蟽$-space is separably injective if and only if it is linearly isometric to the Banach space $C_0(惟)$ of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space $惟$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1609.09648v1-abstract-full').style.display = 'none'; document.getElementById('1609.09648v1-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 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1604.00432">arXiv:1604.00432</a> <span> [<a href="https://arxiv.org/pdf/1604.00432">pdf</a>, <a href="https://arxiv.org/ps/1604.00432">ps</a>, <a href="https://arxiv.org/format/1604.00432">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"> Separably injective C*-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Li%2C+L">Lei 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="1604.00432v1-abstract-short" style="display: inline;"> We show that a C*-algebra is a $1$-separably injective Banach space if, and only if, it is linearly isometric to the Banach space $C_0(惟)$ of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space $惟$. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1604.00432v1-abstract-full" style="display: none;"> We show that a C*-algebra is a $1$-separably injective Banach space if, and only if, it is linearly isometric to the Banach space $C_0(惟)$ of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space $惟$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.00432v1-abstract-full').style.display = 'none'; document.getElementById('1604.00432v1-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 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1512.03347">arXiv:1512.03347</a> <span> [<a href="https://arxiv.org/pdf/1512.03347">pdf</a>, <a href="https://arxiv.org/ps/1512.03347">ps</a>, <a href="https://arxiv.org/format/1512.03347">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Cohomology of Jordan triples via Lie algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Russo%2C+B">Bernard Russo</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="1512.03347v1-abstract-short" style="display: inline;"> We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1512.03347v1-abstract-full" style="display: none;"> We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1512.03347v1-abstract-full').style.display = 'none'; document.getElementById('1512.03347v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 December, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, to appear in: Topics in Functional Analysis and Algebra, Proceedings of a special session of the USA-Uzbekistan Conference on Analysis and Mathematical Physics, CSU Fullerton, May 20-23, 2014, Contemporary Mathematics, 2016</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 17C65; 18G60; Secondary 46L70; 16W10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.06731">arXiv:1503.06731</a> <span> [<a href="https://arxiv.org/pdf/1503.06731">pdf</a>, <a href="https://arxiv.org/ps/1503.06731">ps</a>, <a href="https://arxiv.org/format/1503.06731">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> A Note on the Spectral Area of Toeplitz Operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</a>, <a href="/search/math?searchtype=author&query=Khavinson%2C+D">Dmitry Khavinson</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="1503.06731v2-abstract-short" style="display: inline;"> In this note, we show that for hyponormal Toeplitz operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.06731v2-abstract-full" style="display: none;"> In this note, we show that for hyponormal Toeplitz operators, there exists a lower bound for the area of the spectrum. This extends the known estimate for the spectral area of Toeplitz operators with an analytic symbol. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.06731v2-abstract-full').style.display = 'none'; document.getElementById('1503.06731v2-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, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.1517">arXiv:1412.1517</a> <span> [<a href="https://arxiv.org/pdf/1412.1517">pdf</a>, <a href="https://arxiv.org/ps/1412.1517">ps</a>, <a href="https://arxiv.org/format/1412.1517">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Amenability, Reiter's condition and Liouville property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1412.1517v2-abstract-short" style="display: inline;"> We show that the Liouville property and Reiter's condition are equivalent for semigroupoids. This result applies to semigroups as well as semigroup actions. In the special case of measured groupoids and locally compact groupoids, our result proves Kaimanovich's conjecture of the equivalence of amenability and the Liouville property. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.1517v2-abstract-full" style="display: none;"> We show that the Liouville property and Reiter's condition are equivalent for semigroupoids. This result applies to semigroups as well as semigroup actions. In the special case of measured groupoids and locally compact groupoids, our result proves Kaimanovich's conjecture of the equivalence of amenability and the Liouville property. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.1517v2-abstract-full').style.display = 'none'; document.getElementById('1412.1517v2-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 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2014. </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">To appear in Journal of Functional Analysis</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 20L05; 43A05; Secondary 20M30; 22A22; 45E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1403.6513">arXiv:1403.6513</a> <span> [<a href="https://arxiv.org/pdf/1403.6513">pdf</a>, <a href="https://arxiv.org/ps/1403.6513">ps</a>, <a href="https://arxiv.org/format/1403.6513">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Asymptotic Bohr Radius for the Polynomials in One Complex Variable </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1403.6513v2-abstract-short" style="display: inline;"> We consider the Bohr radius $R_n$ for the class of complex polynomials in one variable of degree at most $n$. It was conjectured by R. Fournier in 2008 that $R_n={1\over 3}+{蟺^2\over {3n^2}}+o({1\over n^2})$. We shall prove this conjecture is true in this paper. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1403.6513v2-abstract-full" style="display: none;"> We consider the Bohr radius $R_n$ for the class of complex polynomials in one variable of degree at most $n$. It was conjectured by R. Fournier in 2008 that $R_n={1\over 3}+{蟺^2\over {3n^2}}+o({1\over n^2})$. We shall prove this conjecture is true in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.6513v2-abstract-full').style.display = 'none'; document.getElementById('1403.6513v2-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 April, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 March, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2014. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1403.2338">arXiv:1403.2338</a> <span> [<a href="https://arxiv.org/pdf/1403.2338">pdf</a>, <a href="https://arxiv.org/ps/1403.2338">ps</a>, <a href="https://arxiv.org/format/1403.2338">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"> Compact Product of Hankel and Toeplitz Operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1403.2338v1-abstract-short" style="display: inline;"> In this paper, we study the product of a Hankel operator and a Toeplitz operator on the Hardy space. We give necessary and sufficient conditions of when such a product $H_f T_g$ is compact. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1403.2338v1-abstract-full" style="display: none;"> In this paper, we study the product of a Hankel operator and a Toeplitz operator on the Hardy space. We give necessary and sufficient conditions of when such a product $H_f T_g$ is compact. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.2338v1-abstract-full').style.display = 'none'; document.getElementById('1403.2338v1-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 March, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2014. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1310.2136">arXiv:1310.2136</a> <span> [<a href="https://arxiv.org/pdf/1310.2136">pdf</a>, <a href="https://arxiv.org/ps/1310.2136">ps</a>, <a href="https://arxiv.org/format/1310.2136">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> A Characterization of $\mathcal{M}_{A_1}(\mathbb{R}^n)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cheng Chu</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="1310.2136v1-abstract-short" style="display: inline;"> We characterize the set of all measurable functions on $\RR^n$ possessing an $A_1$ majorant, denoted as $\cM_{A_1}(\RR^n)$, by certain Banach function spaces. We prove that a function has an $A_1$ majorant if and only if it belongs to some Banach function space for which the Hardy-Littlewood maximal operator is bounded. This answers the question posted by G. Knese, J. M$^{c}$Carthy, and K. Moen. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1310.2136v1-abstract-full" style="display: none;"> We characterize the set of all measurable functions on $\RR^n$ possessing an $A_1$ majorant, denoted as $\cM_{A_1}(\RR^n)$, by certain Banach function spaces. We prove that a function has an $A_1$ majorant if and only if it belongs to some Banach function space for which the Hardy-Littlewood maximal operator is bounded. This answers the question posted by G. Knese, J. M$^{c}$Carthy, and K. Moen. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.2136v1-abstract-full').style.display = 'none'; document.getElementById('1310.2136v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2013. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1305.6808">arXiv:1305.6808</a> <span> [<a href="https://arxiv.org/pdf/1305.6808">pdf</a>, <a href="https://arxiv.org/ps/1305.6808">ps</a>, <a href="https://arxiv.org/format/1305.6808">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</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="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1140/epjc/s10052-013-2586-4">10.1140/epjc/s10052-013-2586-4 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Instanton String and M-Wave in Multiple M5-Branes System </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</a>, <a href="/search/math?searchtype=author&query=Isono%2C+H">Hiroshi Isono</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="1305.6808v1-abstract-short" style="display: inline;"> We consider the non-abelian self-dual two-form theory arXiv:1203.4224 and find new exact solutions. Our solutions are supported by Yang-Mills (anti)instantons in 4-dimensions and describe wave moving in null directions. We argue and provide evidence that these instanton string solutions correspond to M-wave (MW) on the worldvolume of multiple M5-branes. When dimensionally reduced on a circle, the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.6808v1-abstract-full').style.display = 'inline'; document.getElementById('1305.6808v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1305.6808v1-abstract-full" style="display: none;"> We consider the non-abelian self-dual two-form theory arXiv:1203.4224 and find new exact solutions. Our solutions are supported by Yang-Mills (anti)instantons in 4-dimensions and describe wave moving in null directions. We argue and provide evidence that these instanton string solutions correspond to M-wave (MW) on the worldvolume of multiple M5-branes. When dimensionally reduced on a circle, the MW/M5 system is reduced to the D0/D4 system with the D0-branes represented by the Yang-Mills instanton of the D4-branes Yang-Mills gauge theory. We show that this picture is precisely reproduced by the dimensional reduction of our instanton string solutions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1305.6808v1-abstract-full').style.display = 'none'; document.getElementById('1305.6808v1-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 May, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2013. </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">12 pages. LaTeX</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1210.1799">arXiv:1210.1799</a> <span> [<a href="https://arxiv.org/pdf/1210.1799">pdf</a>, <a href="https://arxiv.org/ps/1210.1799">ps</a>, <a href="https://arxiv.org/format/1210.1799">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Localization of Rota-Baxter algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chenghao Chu</a>, <a href="/search/math?searchtype=author&query=Guo%2C+L">Li Guo</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="1210.1799v1-abstract-short" style="display: inline;"> A commutative Rota-Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota-Baxter algebra, we extend the central concept of localization for commutative algebras to commutative Rota-Baxter algebras. The existence of such a localization is proved and, under mild conditio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.1799v1-abstract-full').style.display = 'inline'; document.getElementById('1210.1799v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1210.1799v1-abstract-full" style="display: none;"> A commutative Rota-Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota-Baxter algebra, we extend the central concept of localization for commutative algebras to commutative Rota-Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit constructions are obtained. The existence of tensor products of commutative Rota-Baxter algebras is also proved and the compatibility of localization and tensor product of Rota-Baxter algebras is established. We further study Rota-Baxter coverings and show that they form a Gr枚thendieck topology. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.1799v1-abstract-full').style.display = 'none'; document.getElementById('1210.1799v1-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 October, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 13B30; 13F99; 16W99 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Pure Appl. Algebra, 218 (2014), 237-251 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1010.2896">arXiv:1010.2896</a> <span> [<a href="https://arxiv.org/pdf/1010.2896">pdf</a>, <a href="https://arxiv.org/ps/1010.2896">ps</a>, <a href="https://arxiv.org/format/1010.2896">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Sheaves and $K$-theory for $\mathbb{F}_1$-schemes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chenghao Chu</a>, <a href="/search/math?searchtype=author&query=Lorscheid%2C+O">Oliver Lorscheid</a>, <a href="/search/math?searchtype=author&query=Santhanam%2C+R">Rekha Santhanam</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="1010.2896v2-abstract-short" style="display: inline;"> This paper is devoted to the open problem in $\mathbb{F}_1$-geometry of developing $K$-theory for $\mathbb{F}_1$-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for $\mathcal{M}_0$-schemes and $\mathbb{F}_1$-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1010.2896v2-abstract-full').style.display = 'inline'; document.getElementById('1010.2896v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1010.2896v2-abstract-full" style="display: none;"> This paper is devoted to the open problem in $\mathbb{F}_1$-geometry of developing $K$-theory for $\mathbb{F}_1$-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for $\mathcal{M}_0$-schemes and $\mathbb{F}_1$-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over $\mathbb{F}_1$. Special attention is paid to two aspects particular to $\mathbb{F}_1$-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillen's Q-construction to a definition of $G$-theory and $K$-theory for $\mathbb{F}_1$-schemes. A comparison with Waldhausen's $S_{\bullet}$-construction yields the ring structure of $K$-theory. In particular, we generalize Deitmar's $K$-theory of monoids and show that $K_*(\Spec\mathbb{F}_1)$ realizes the stable homotopy of the spheres as a ring spectrum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1010.2896v2-abstract-full').style.display = 'none'; document.getElementById('1010.2896v2-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 January, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 October, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2010. </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 paper got extended by two new section treating the $K$-theory spectrum and the ring structure of the $K$-theory spectrum. This is the final version as in print. 67 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14A20; 18F20; 18F25; 19D06; 19D10; 19E08; 20M14; 20M30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1009.3235">arXiv:1009.3235</a> <span> [<a href="https://arxiv.org/pdf/1009.3235">pdf</a>, <a href="https://arxiv.org/ps/1009.3235">ps</a>, <a href="https://arxiv.org/format/1009.3235">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> </div> </div> <p class="title is-5 mathjax"> On the Algebraic K-theory of Monoids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chenghao Chu</a>, <a href="/search/math?searchtype=author&query=Morava%2C+J">Jack Morava</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="1009.3235v1-abstract-short" style="display: inline;"> Let $A$ be a not necessarily commutative monoid with zero such that projective $A$-acts are free. This paper shows that the algebraic K-groups of $A$ can be defined using the +-construction and the Q-construction. It is shown that these two constructions give the same K-groups. As an immediate application, the homotopy invariance of algebraic K-theory of certain affine $\mathbb{F}_1$-schemes is ob… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.3235v1-abstract-full').style.display = 'inline'; document.getElementById('1009.3235v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1009.3235v1-abstract-full" style="display: none;"> Let $A$ be a not necessarily commutative monoid with zero such that projective $A$-acts are free. This paper shows that the algebraic K-groups of $A$ can be defined using the +-construction and the Q-construction. It is shown that these two constructions give the same K-groups. As an immediate application, the homotopy invariance of algebraic K-theory of certain affine $\mathbb{F}_1$-schemes is obtained. From the computation of $K_2(A),$ where $A$ is the monoid associated to a finitely generated abelian group, the universal central extension of certain groups are constructed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.3235v1-abstract-full').style.display = 'none'; document.getElementById('1009.3235v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 September, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2010. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1004.1513">arXiv:1004.1513</a> <span> [<a href="https://arxiv.org/pdf/1004.1513">pdf</a>, <a href="https://arxiv.org/ps/1004.1513">ps</a>, <a href="https://arxiv.org/format/1004.1513">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</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="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</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/10.1007/JHEP02(2011)037">10.1007/JHEP02(2011)037 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Cartan-Weyl 3-algebras and the BLG Theory II: Strong-Semisimplicity and Generalized Cartan-Weyl 3-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</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="1004.1513v2-abstract-short" style="display: inline;"> One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1004.1513v2-abstract-full').style.display = 'inline'; document.getElementById('1004.1513v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1004.1513v2-abstract-full" style="display: none;"> One of the most important questions in the Bagger-Lambert-Gustavsson (BLG) theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie 3-algebra should be chosen such that the corresponding BLG model is unitary and admits fuzzy 3-sphere as a solution. In this paper we propose another new condition: the Lie 3-algebras of use must be connected to the semisimple Lie algebras describing the gauge symmetry of D-branes via a certain reduction condition. We show that this reduction condition leads to a natural generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397. Similar to a Cartan-Weyl 3-algebra, a generalized Cartan-Weyl 3-algebra processes a set of step generators characterized by non-degenerate roots. However, its Cartan subalgebra is non-abelian in general. We give reasons why having a non-abelian Cartan subalgebra may be just right to allow for fuzzy 3-sphere solution in the corresponding BLG models. We propose that generalized Cartan-Weyl 3-algebras is the right class of metric Lie 3-algebras to be used in the BLG theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1004.1513v2-abstract-full').style.display = 'none'; document.getElementById('1004.1513v2-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 August, 2010; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 April, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2010. </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">28 pages. LaTeX. v2. typos corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> DCPT-10/07 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 1102:037,2011 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1004.1397">arXiv:1004.1397</a> <span> [<a href="https://arxiv.org/pdf/1004.1397">pdf</a>, <a href="https://arxiv.org/ps/1004.1397">ps</a>, <a href="https://arxiv.org/format/1004.1397">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</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="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</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/10.1007/JHEP10(2010)050">10.1007/JHEP10(2010)050 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</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="1004.1397v2-abstract-short" style="display: inline;"> As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^伪that are characterized by a root space of non-deg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1004.1397v2-abstract-full').style.display = 'inline'; document.getElementById('1004.1397v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1004.1397v2-abstract-full" style="display: none;"> As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^伪that are characterized by a root space of non-degenerate one-forms 伪. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1004.1397v2-abstract-full').style.display = 'none'; document.getElementById('1004.1397v2-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 August, 2010; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 April, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2010. </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">LaTeX. 34 pages.v2. deleted some distracting paragraphs in the introduction to bring more out the main results of the paper. typos corrected and references added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> DCPT-10/05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 1010:050,2010 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0903.0865">arXiv:0903.0865</a> <span> [<a href="https://arxiv.org/pdf/0903.0865">pdf</a>, <a href="https://arxiv.org/ps/0903.0865">ps</a>, <a href="https://arxiv.org/format/0903.0865">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</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/10.1112/blms/bdp068">10.1112/blms/bdp068 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Eigenvalue decay of operators on harmonic function spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bandtlow%2C+O+F">Oscar F. Bandtlow</a>, <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</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="0903.0865v1-abstract-short" style="display: inline;"> Let $惟$ be an open set in $\R^d$ $(d > 1)$ and $h(惟)$ the Fr茅chet space of harmonic functions on $惟$. Given a bounded linear operator $L :h(惟)\to h(惟)$, we show that its eigenvalues $位_n$, arranged in decreasing order and counting multiplicities, satisfy $|位_n|\leq K\exp(-cn^{1/(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0903.0865v1-abstract-full" style="display: none;"> Let $惟$ be an open set in $\R^d$ $(d > 1)$ and $h(惟)$ the Fr茅chet space of harmonic functions on $惟$. Given a bounded linear operator $L :h(惟)\to h(惟)$, we show that its eigenvalues $位_n$, arranged in decreasing order and counting multiplicities, satisfy $|位_n|\leq K\exp(-cn^{1/(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0903.0865v1-abstract-full').style.display = 'none'; document.getElementById('0903.0865v1-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, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2009. </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">AMS-LaTeX, 14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 47B38 (Primary) 47B07; 47B06; 46E10; 31B05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0708.0419">arXiv:0708.0419</a> <span> [<a href="https://arxiv.org/pdf/0708.0419">pdf</a>, <a href="https://arxiv.org/ps/0708.0419">ps</a>, <a href="https://arxiv.org/format/0708.0419">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="Differential Geometry">math.DG</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/10.4153/CJM-2011-026-1">10.4153/CJM-2011-026-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the Geometry of the Moduli Space of Real Binary Octics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+K+C+K">Kenneth C. K. Chu</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="0708.0419v1-abstract-short" style="display: inline;"> The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are, up to commensur… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0708.0419v1-abstract-full').style.display = 'inline'; document.getElementById('0708.0419v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0708.0419v1-abstract-full" style="display: none;"> The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are, up to commensurability, discrete hyperbolic reflection groups, and their Vinberg diagrams are computed. We conclude with a simple proof that the moduli space of GIT-stable real binary octics itself cannot be a real hyperbolic orbifold. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0708.0419v1-abstract-full').style.display = 'none'; document.getElementById('0708.0419v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 August, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2007. </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">23 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32G13 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Can. J. Math.-J. Can. Math. 63 (2011) 755-797 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0201188">arXiv:math/0201188</a> <span> [<a href="https://arxiv.org/pdf/math/0201188">pdf</a>, <a href="https://arxiv.org/ps/math/0201188">ps</a>, <a href="https://arxiv.org/format/math/0201188">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Normal contractive projections preserve type </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Cho-Ho Chu</a>, <a href="/search/math?searchtype=author&query=Neal%2C+M">Matthew Neal</a>, <a href="/search/math?searchtype=author&query=Russo%2C+B">Bernard Russo</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="math/0201188v1-abstract-short" style="display: inline;"> Given a JBW*-triple Z and a normal contractive projection P on Z, we show that the (Murray-von Neumann) type of each summand of P(Z) is dominated by the type of Z. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0201188v1-abstract-full" style="display: none;"> Given a JBW*-triple Z and a normal contractive projection P on Z, we show that the (Murray-von Neumann) type of each summand of P(Z) is dominated by the type of Z. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0201188v1-abstract-full').style.display = 'none'; document.getElementById('math/0201188v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 January, 2002; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2002. </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">26 pages, latex, submitted in July 2001</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46L70; 46L10; 17C65; 32M15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Operator Theory 51 (2004) 281-301 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/q-alg/9612031">arXiv:q-alg/9612031</a> <span> [<a href="https://arxiv.org/pdf/q-alg/9612031">pdf</a>, <a href="https://arxiv.org/ps/q-alg/9612031">ps</a>, <a href="https://arxiv.org/format/q-alg/9612031">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/10.1142/S0217751X97002929">10.1142/S0217751X97002929 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Poisson Algebra of Differential Forms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</a>, <a href="/search/math?searchtype=author&query=Ho%2C+P">Pei-Ming Ho</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="q-alg/9612031v2-abstract-short" style="display: inline;"> We give a natural definition of a Poisson Differential Algebra. Consistence conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on differential calculus in a simple canonical form by a coordinate transformation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9612031v2-abstract-full').style.display = 'inline'; document.getElementById('q-alg/9612031v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="q-alg/9612031v2-abstract-full" style="display: none;"> We give a natural definition of a Poisson Differential Algebra. Consistence conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on differential calculus in a simple canonical form by a coordinate transformation. This is in analogy with the standard Darboux's theorem for symplectic geometry. For certain cases there exists a realization of the exterior derivative through a certain canonical one-form. All the above are carried out similarly for the case of a complex Poisson Differential Algebra. The case of one complex dimension is treated in details and interesting features are noted. A conclusion is made in the last section. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9612031v2-abstract-full').style.display = 'none'; document.getElementById('q-alg/9612031v2-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 January, 1997; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 December, 1996; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 1996. </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">Latex, 14 pages, no figure, typos corrected, references added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int.J.Mod.Phys. 12 (1997) 5573-5587 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/q-alg/9510021">arXiv:q-alg/9510021</a> <span> [<a href="https://arxiv.org/pdf/q-alg/9510021">pdf</a>, <a href="https://arxiv.org/ps/q-alg/9510021">ps</a>, <a href="https://arxiv.org/format/q-alg/9510021">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="High Energy Physics - Theory">hep-th</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/10.1007/s002880050233">10.1007/s002880050233 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Geometry of the Quantum Complex Projective Space $CP_q(N)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</a>, <a href="/search/math?searchtype=author&query=Ho%2C+P">Pei-Ming Ho</a>, <a href="/search/math?searchtype=author&query=Zumino%2C+B">Bruno Zumino</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="q-alg/9510021v1-abstract-short" style="display: inline;"> The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum K盲hler manifold. The braiding of several copies of $CP_q(N)$ is introduced and the anharmonic ratios of four collinear points are shown to be invariant under… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9510021v1-abstract-full').style.display = 'inline'; document.getElementById('q-alg/9510021v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="q-alg/9510021v1-abstract-full" style="display: none;"> The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum K盲hler manifold. The braiding of several copies of $CP_q(N)$ is introduced and the anharmonic ratios of four collinear points are shown to be invariant under quantum projective transformations. They provide the building blocks of all projective invariants. The Poisson limit is also described. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9510021v1-abstract-full').style.display = 'none'; document.getElementById('q-alg/9510021v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 October, 1995; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 1995. </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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> UCB-PTH-95/38 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Z.Phys. C72 (1996) 163-170 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/q-alg/9507013">arXiv:q-alg/9507013</a> <span> [<a href="https://arxiv.org/pdf/q-alg/9507013">pdf</a>, <a href="https://arxiv.org/ps/q-alg/9507013">ps</a>, <a href="https://arxiv.org/format/q-alg/9507013">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="High Energy Physics - Theory">hep-th</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/10.1142/S0217732396000357">10.1142/S0217732396000357 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Braided Quantum 2-Sphere </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</a>, <a href="/search/math?searchtype=author&query=Ho%2C+P">Pei-Ming Ho</a>, <a href="/search/math?searchtype=author&query=Zumino%2C+B">Bruno Zumino</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="q-alg/9507013v1-abstract-short" style="display: inline;"> In a recent paper the quantum 2-sphere $S^2_q$ was described as a quantum complex manifold. Here we consider several copies of $S^2_q$ and derive their braiding commutation relations. The braiding is extended to the differential and to the integral calculus on the spheres. A quantum analogue of the classical anharmonic ratio of four points on the sphere is given, which is invariant under the coa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9507013v1-abstract-full').style.display = 'inline'; document.getElementById('q-alg/9507013v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="q-alg/9507013v1-abstract-full" style="display: none;"> In a recent paper the quantum 2-sphere $S^2_q$ was described as a quantum complex manifold. Here we consider several copies of $S^2_q$ and derive their braiding commutation relations. The braiding is extended to the differential and to the integral calculus on the spheres. A quantum analogue of the classical anharmonic ratio of four points on the sphere is given, which is invariant under the coaction of $SU_q(2)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9507013v1-abstract-full').style.display = 'none'; document.getElementById('q-alg/9507013v1-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 July, 1995; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 1995. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> LBL-37506, UCB-PTH-95/25 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mod.Phys.Lett. A11 (1996) 307-316 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/q-alg/9504003">arXiv:q-alg/9504003</a> <span> [<a href="https://arxiv.org/pdf/q-alg/9504003">pdf</a>, <a href="https://arxiv.org/ps/q-alg/9504003">ps</a>, <a href="https://arxiv.org/format/q-alg/9504003">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"> The quantum 2-sphere as a complex quantum manifold </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</a>, <a href="/search/math?searchtype=author&query=Ho%2C+P">Pei-Ming Ho</a>, <a href="/search/math?searchtype=author&query=Zumino%2C+B">Bruno Zumino</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="q-alg/9504003v1-abstract-short" style="display: inline;"> We describe the quantum sphere of Podle艣 for $c=0$ by means of a stereographic projection which is analogous to that which exhibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with \SU coefficients as well as under the action of \SU vector fields. Going… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9504003v1-abstract-full').style.display = 'inline'; document.getElementById('q-alg/9504003v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="q-alg/9504003v1-abstract-full" style="display: none;"> We describe the quantum sphere of Podle艣 for $c=0$ by means of a stereographic projection which is analogous to that which exhibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with \SU coefficients as well as under the action of \SU vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the complex quantum plane. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9504003v1-abstract-full').style.display = 'none'; document.getElementById('q-alg/9504003v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 April, 1995; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 1995. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> LBL-37055, UCB-PTH-95/10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/q-alg/9502005">arXiv:q-alg/9502005</a> <span> [<a href="https://arxiv.org/pdf/q-alg/9502005">pdf</a>, <a href="https://arxiv.org/ps/q-alg/9502005">ps</a>, <a href="https://arxiv.org/format/q-alg/9502005">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="High Energy Physics - Theory">hep-th</span> </div> </div> <p class="title is-5 mathjax"> Realization of Vector fields for Quantum Groups as Pseudodifferential Operators on Quantum Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chu%2C+C">Chong-Sun Chu</a>, <a href="/search/math?searchtype=author&query=Zumino%2C+B">Bruno Zumino</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="q-alg/9502005v1-abstract-short" style="display: inline;"> The vector fields of the quantum Lie algebra are described for the quantum groups $GL_q(N), SL_q(N)$ and $SO_q(N)$ as pseudodifferential operators on the linear quantum spaces covariant under the corresponding quantum group. Their expressions are simple and compact. It is pointed out that these vector fields satisfy certain characteristic polynomial identities. The real forms $SU_q(N)$ and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9502005v1-abstract-full').style.display = 'inline'; document.getElementById('q-alg/9502005v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="q-alg/9502005v1-abstract-full" style="display: none;"> The vector fields of the quantum Lie algebra are described for the quantum groups $GL_q(N), SL_q(N)$ and $SO_q(N)$ as pseudodifferential operators on the linear quantum spaces covariant under the corresponding quantum group. Their expressions are simple and compact. It is pointed out that these vector fields satisfy certain characteristic polynomial identities. The real forms $SU_q(N)$ and $SO_q(N,R)$ are discussed in detail. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('q-alg/9502005v1-abstract-full').style.display = 'none'; document.getElementById('q-alg/9502005v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 February, 1995; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 1995. </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, Latex, no figures, to appear in the Proceedings of the XX International Conference on Group Theory Methods in Physics, Toyonaka, Japan (1994)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> LBL-36746, UCB-PTH-95/04 </p> </li> </ol> 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