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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <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/2410.18863">arXiv:2410.18863</a> <span> [<a href="https://arxiv.org/pdf/2410.18863">pdf</a>, <a href="https://arxiv.org/format/2410.18863">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"> Exploring a Geometric Conjecture, Some Properties of Blaschke Products, and the Geometry of Curves Formed by Them </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Celik%2C+M">Mehmet Celik</a>, <a href="/search/math?searchtype=author&query=Duguin%2C+M">Mathis Duguin</a>, <a href="/search/math?searchtype=author&query=Guo%2C+J">Jia Guo</a>, <a href="/search/math?searchtype=author&query=Luo%2C+D">Dianlun Luo</a>, <a href="/search/math?searchtype=author&query=Spinelli%2C+K">Kamryn Spinelli</a>, <a href="/search/math?searchtype=author&query=Zeytuncu%2C+Y+E">Yunus E. Zeytuncu</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+Z">Zhuoyu Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.18863v1-abstract-short" style="display: inline;"> In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18863v1-abstract-full').style.display = 'inline'; document.getElementById('2410.18863v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.18863v1-abstract-full" style="display: none;"> In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains constant. In this paper, we provide a proof of Reznik's conjecture and present a formula for calculating the total sum. Additionally, we demonstrate the algebraic structures formed by various sets of products and the geometric properties of polygons and ellipses created by these products. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.18863v1-abstract-full').style.display = 'none'; document.getElementById('2410.18863v1-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages. This article was written as part of the Polymath Jr. program in the summer of 2022</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30J10; 53A04 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.08608">arXiv:2406.08608</a> <span> [<a href="https://arxiv.org/pdf/2406.08608">pdf</a>, <a href="https://arxiv.org/format/2406.08608">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Integral representation and approximation of L-functions associated to Hecke cusp eigenforms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+A">An Huang</a>, <a href="/search/math?searchtype=author&query=Spinelli%2C+K">Kamryn Spinelli</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.08608v1-abstract-short" style="display: inline;"> We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives. Together with the Euler product… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.08608v1-abstract-full').style.display = 'inline'; document.getElementById('2406.08608v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.08608v1-abstract-full" style="display: none;"> We derive a family of approximations for L-functions of Hecke cusp eigenforms, according to a recipe first described by Matiyasevich for the Riemann xi function. We show that these approximations converge to the true L-function, and along the way we demonstrate error formulas which may be used to investigate analytic properties of the L-function and its derivatives. Together with the Euler product expansion of the L-function, the family of approximations also encodes some of the key features of the L-function such as its functional equation. Finally, we derive via Mellin transforms a convolution-type formula which leads to precise error bounds in terms of the incomplete gamma function. This formula can be interpreted as an alternative definition for the approximation and sheds light on Matiyasevich's procedure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.08608v1-abstract-full').style.display = 'none'; document.getElementById('2406.08608v1-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11F11 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.12413">arXiv:2110.12413</a> <span> [<a href="https://arxiv.org/pdf/2110.12413">pdf</a>, <a href="https://arxiv.org/ps/2110.12413">ps</a>, <a href="https://arxiv.org/format/2110.12413">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> CR embeddability of quotients of the Rossi sphere via spectral theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bosch%2C+H">Henry Bosch</a>, <a href="/search/math?searchtype=author&query=Gonzales%2C+T">Tyler Gonzales</a>, <a href="/search/math?searchtype=author&query=Spinelli%2C+K">Kamryn Spinelli</a>, <a href="/search/math?searchtype=author&query=Udell%2C+G">Gabe Udell</a>, <a href="/search/math?searchtype=author&query=Zeytuncu%2C+Y+E">Yunus E. Zeytuncu</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="2110.12413v1-abstract-short" style="display: inline;"> We look at the action of finite subgroups of $\operatorname{SU}(2)$ on $S^3$, viewed as a CR manifold, both with the standard CR structure as the unit sphere in $\mathbb{C}^2$ and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of $\operatorname{SU}(2)$ to the asymptotic distri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.12413v1-abstract-full').style.display = 'inline'; document.getElementById('2110.12413v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.12413v1-abstract-full" style="display: none;"> We look at the action of finite subgroups of $\operatorname{SU}(2)$ on $S^3$, viewed as a CR manifold, both with the standard CR structure as the unit sphere in $\mathbb{C}^2$ and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of $\operatorname{SU}(2)$ to the asymptotic distribution of the Kohn Laplacian's eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.12413v1-abstract-full').style.display = 'none'; document.getElementById('2110.12413v1-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 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.11970">arXiv:2103.11970</a> <span> [<a href="https://arxiv.org/pdf/2103.11970">pdf</a>, <a href="https://arxiv.org/ps/2103.11970">ps</a>, <a href="https://arxiv.org/format/2103.11970">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"> Manifolds with bounded integral curvature and no positive eigenvalue lower bounds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Anderson%2C+C+C">Connor C. Anderson</a>, <a href="/search/math?searchtype=author&query=Oliv%C3%A9%2C+X+R">Xavier Ramos Oliv茅</a>, <a href="/search/math?searchtype=author&query=Spinelli%2C+K">Kamryn Spinelli</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="2103.11970v2-abstract-short" style="display: inline;"> We provide an explicit construction of a sequence of closed surfaces with uniform bounds on the diameter and on $L^p$ norms of the curvature, but without a positive lower bound on the first non-zero eigenvalue of the Laplacian $位_1$. This example shows that the assumption of smallness of the $L^p$ norm of the curvature is a necessary condition to derive Lichnerowicz and Zhong-Yang type estimates u… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.11970v2-abstract-full').style.display = 'inline'; document.getElementById('2103.11970v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.11970v2-abstract-full" style="display: none;"> We provide an explicit construction of a sequence of closed surfaces with uniform bounds on the diameter and on $L^p$ norms of the curvature, but without a positive lower bound on the first non-zero eigenvalue of the Laplacian $位_1$. This example shows that the assumption of smallness of the $L^p$ norm of the curvature is a necessary condition to derive Lichnerowicz and Zhong-Yang type estimates under integral curvature conditions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.11970v2-abstract-full').style.display = 'none'; document.getElementById('2103.11970v2-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 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58J50 (Primary) 53C21; 58J60 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2010.04568">arXiv:2010.04568</a> <span> [<a href="https://arxiv.org/pdf/2010.04568">pdf</a>, <a href="https://arxiv.org/ps/2010.04568">ps</a>, <a href="https://arxiv.org/format/2010.04568">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> A Tauberian Approach to Weyl's Law for the Kohn Laplacian on Spheres </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bosch%2C+H">Henry Bosch</a>, <a href="/search/math?searchtype=author&query=Gonzales%2C+T">Tyler Gonzales</a>, <a href="/search/math?searchtype=author&query=Spinelli%2C+K">Kamryn Spinelli</a>, <a href="/search/math?searchtype=author&query=Udell%2C+G">Gabe Udell</a>, <a href="/search/math?searchtype=author&query=Zeytuncu%2C+Y+E">Yunus E. Zeytuncu</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="2010.04568v1-abstract-short" style="display: inline;"> We compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2010.04568v1-abstract-full" style="display: none;"> We compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.04568v1-abstract-full').style.display = 'none'; document.getElementById('2010.04568v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 32V05; 32V20 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> 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