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</div> <div class="control"> <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/2503.15440">arXiv:2503.15440</a> <span> [<a href="https://arxiv.org/pdf/2503.15440">pdf</a>, <a href="https://arxiv.org/ps/2503.15440">ps</a>, <a href="https://arxiv.org/format/2503.15440">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Counting the number of elements in the nilradical of a parabolic subalgebra of $\mathfrak{gl}_n(\mathbb F_q)$ with a specified Jordan form </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</a>, <a href="/search/math?searchtype=author&query=Salmasian%2C+H">Hadi Salmasian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.15440v1-abstract-short" style="display: inline;"> Let $螞$ be a composition of $n$, and let $\mathfrak u_螞(\mathbb F_q)$ denote the nilradical of the standard parabolic subalgebra of $\mathfrak{gl}_n(\mathbb F_q)$ associated with $螞$. Using a parabolic variation of Borodin's division algorithm we prove that, up to an explicit polynomial factor in $q$, the number of elements in $\mathfrak u_螞(\mathbb F_q)$ of Jordan type $渭$ is equal to the coeffic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15440v1-abstract-full').style.display = 'inline'; document.getElementById('2503.15440v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.15440v1-abstract-full" style="display: none;"> Let $螞$ be a composition of $n$, and let $\mathfrak u_螞(\mathbb F_q)$ denote the nilradical of the standard parabolic subalgebra of $\mathfrak{gl}_n(\mathbb F_q)$ associated with $螞$. Using a parabolic variation of Borodin's division algorithm we prove that, up to an explicit polynomial factor in $q$, the number of elements in $\mathfrak u_螞(\mathbb F_q)$ of Jordan type $渭$ is equal to the coefficient of the monomial $\mathbf x^螞$ in the specialization of the dual Macdonald polynomial $\mathrm Q_{渭'}(\mathbf x;q^{-1},t)$ at $t=0$. Our proof is shorter and more direct than the one given by Karp and Thomas. We give three applications of the above theorem. First, we demonstrate that a recurrence relation of Kirillov for the number of strictly upper triangular matrices of Jordan type $渭$ is a straightforward consequence of the above result combined with the Cauchy identity for Macdonald polynomials. Second, we give an explicit formula for the number of $X\in \mathfrak u_螞(\mathbb F_q)$ that satisfy $X^2=0$. In the special case $螞=(1^n)$, our formula is different from the one conjectured by Kirillov and Melnikov (which was proved by Ekhad and Zeilberger). We obtain a new proof of the Kirillov-Melnikov-Ekhad-Zeilberger formula using our latter result and a formula for two-rowed Macdonald polynomials by Jing and J贸zefiak. Third, we compute the number of double cosets $\mathsf U_螞(\mathbb F_q)\backslash\mathsf{GL}_n(\mathbb F_q)/\mathsf U_螕(\mathbb F_q)$ where $\mathsf U_螞$ and $\mathsf U_螕$ denote unipotent radicals of the standard parabolics of $\mathsf{GL}_n(\mathbb F_q)$ associated to compositions $螞$ and $螕$ of $n$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15440v1-abstract-full').style.display = 'none'; document.getElementById('2503.15440v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05; 15B33; 05A30; 20G40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2204.12412">arXiv:2204.12412</a> <span> [<a href="https://arxiv.org/pdf/2204.12412">pdf</a>, <a href="https://arxiv.org/ps/2204.12412">ps</a>, <a href="https://arxiv.org/format/2204.12412">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</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="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Polynomiality of the faithful dimension of nilpotent groups over finite truncated valuation rings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</a>, <a href="/search/math?searchtype=author&query=Rumiantsau%2C+D">Dzmitry Rumiantsau</a>, <a href="/search/math?searchtype=author&query=Salmasian%2C+H">Hadi Salmasian</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="2204.12412v2-abstract-short" style="display: inline;"> The faithful dimension of a finite group $\mathrm G$ over $\mathbb C$, denoted by $m_\mathrm{faithful}(\mathrm G)$, is the smallest integer $n$ such that $\mathrm G$ can be embedded in $\mathrm{GL}_n(\mathbb C)$. Continuing our previous work (arXiv:1712.02019), we address the problem of determining the faithful dimension of a finite $p$-group of the form $\mathcal G_R:=\exp(\mathfrak g_R)$ associa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.12412v2-abstract-full').style.display = 'inline'; document.getElementById('2204.12412v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2204.12412v2-abstract-full" style="display: none;"> The faithful dimension of a finite group $\mathrm G$ over $\mathbb C$, denoted by $m_\mathrm{faithful}(\mathrm G)$, is the smallest integer $n$ such that $\mathrm G$ can be embedded in $\mathrm{GL}_n(\mathbb C)$. Continuing our previous work (arXiv:1712.02019), we address the problem of determining the faithful dimension of a finite $p$-group of the form $\mathcal G_R:=\exp(\mathfrak g_R)$ associated to $\mathfrak g_R:=\mathfrak g \otimes_\mathbb Z R $ in the Lazard correspondence, where $\mathfrak g$ is a nilpotent $\mathbb Z$-Lie algebra and $R$ ranges over finite truncated valuation rings. Our first main result is that if $R$ is a finite field with $p^f$ elements and $p$ is sufficiently large, then $m_\mathrm{faithful}(\mathcal G_R)=fg(p^f)$ where $g(T)$ belongs to a finite list of polynomials $g_1,\ldots,g_k$, with non-negative integer coefficients. The list of polynomials is uniquely determined by the Lie algebra $\mathfrak g$. Furthermore, for $1\leq i\leq k$ the set of pairs $(p,f)$ for which $g=g_i$ is a finite union of Cartesian products $\mathcal P\times \mathcal F$, where $\mathcal P$ is a Frobenius set of prime numbers and $\mathcal F$ is a subset of $\mathbb N$ that belongs to the Boolean algebra generated by arithmetic progressions. Next we formulate a conjectural polynomiality property for $m_\mathrm{faithful}(\mathcal G_R)$ in the more general setting where $R$ is a finite truncated valuation ring, and prove special cases of this conjecture. In particular, we show that for a vast class of Lie algebras $\mathfrak g $ that are defined by partial orders, $m_\mathrm{faithful}(\mathcal G_R)$ is given by a single polynomial-type formula. Finally, we compute $m_\mathrm{faithful}(\mathcal G_R)$ precisely in the case where $\mathfrak g$ is the free metabelian nilpotent Lie algebra of class $c$ on $n$ generators and $R$ is a finite truncated valuation ring. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.12412v2-abstract-full').style.display = 'none'; document.getElementById('2204.12412v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">27 pages, to appear in Transactions of the American Mathematical Society</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G05; 20C15; 14G05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.02019">arXiv:1712.02019</a> <span> [<a href="https://arxiv.org/pdf/1712.02019">pdf</a>, <a href="https://arxiv.org/format/1712.02019">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</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/S0010437X19007462">10.1112/S0010437X19007462 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</a>, <a href="/search/math?searchtype=author&query=Salmasian%2C+H">Hadi Salmasian</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="1712.02019v3-abstract-short" style="display: inline;"> Given a finite group $\mathrm{G}$ and a field $K$, the faithful dimension of $\mathrm{G}$ over $K$ is defined to be the smallest integer $n$ such that $\mathrm{G}$ embeds into $\mathrm{GL}_n(K)$. In this paper we address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_q:=\exp(\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q)$ associated to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.02019v3-abstract-full').style.display = 'inline'; document.getElementById('1712.02019v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.02019v3-abstract-full" style="display: none;"> Given a finite group $\mathrm{G}$ and a field $K$, the faithful dimension of $\mathrm{G}$ over $K$ is defined to be the smallest integer $n$ such that $\mathrm{G}$ embeds into $\mathrm{GL}_n(K)$. In this paper we address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_q:=\exp(\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q)$ associated to $\mathfrak{g}_q:=\mathfrak{g} \otimes_\mathbb{Z}\mathbb{F}_q$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of $\mathscr{G}_p$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such on each part the faithful dimension of $\mathscr{G}_q$ for $q:=p^f$ is equal to $f g(p^f)$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.02019v3-abstract-full').style.display = 'none'; document.getElementById('1712.02019v3-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 April, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Compositio Math. 155 (2019) 1618-1654 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1701.06024">arXiv:1701.06024</a> <span> [<a href="https://arxiv.org/pdf/1701.06024">pdf</a>, <a href="https://arxiv.org/ps/1701.06024">ps</a>, <a href="https://arxiv.org/format/1701.06024">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Polynomial configurations in sets of positive upper density over local fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</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="1701.06024v4-abstract-short" style="display: inline;"> Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{R}^m$ with respect to the portion of the graph of $F$ de… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.06024v4-abstract-full').style.display = 'inline'; document.getElementById('1701.06024v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1701.06024v4-abstract-full" style="display: none;"> Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{R}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. We conclude that if $I \subseteq \mathbb{R}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ for an unbounded set of values $s \in \mathbb{R}$. It follows that the Borel chromatic number of the Cayley graph of $\mathbb{R}^m$ with respect to the set $\{ \pm F(s): s \in \mathbb{R} \}$ is infinite. Analogous results are also proven when $\mathbb{R}$ is replaced by the field of $p$-adic numbers $\mathbb{Q}_p$. At the end, we will also the existence of real analytic functions $f_1, \dots, f_m$, for which the analogous statements no longer hold. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1701.06024v4-abstract-full').style.display = 'none'; document.getElementById('1701.06024v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">Minor corrections and modifications</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.02271">arXiv:1602.02271</a> <span> [<a href="https://arxiv.org/pdf/1602.02271">pdf</a>, <a href="https://arxiv.org/ps/1602.02271">ps</a>, <a href="https://arxiv.org/format/1602.02271">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On equality of ranks of local components of automorphic representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Salmasian%2C+H">Hadi Salmasian</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="1602.02271v3-abstract-short" style="display: inline;"> We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li, Dvorsky--Sahi, and Kobayashi--Savin. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian uni… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.02271v3-abstract-full').style.display = 'inline'; document.getElementById('1602.02271v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.02271v3-abstract-full" style="display: none;"> We prove that the local components of an automorphic representation of an adelic semisimple group have equal rank in the sense defined earlier by the second author. Our theorem is an analogue of the results previously obtained by Howe, Li, Dvorsky--Sahi, and Kobayashi--Savin. Unlike previous works which are based on explicit matrix realizations and existence of parabolic subgroups with abelian unipotent radicals, our proof works uniformly for all of the (classical as well as exceptional) groups under consideration. Our result is an extension of the statement known for several semisimple groups that if at least one local component of an automorphic representation is a minimal representation, then all of its local components are minimal. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.02271v3-abstract-full').style.display = 'none'; document.getElementById('1602.02271v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">Remark 9.6 is modified. A corrigendum will also appear in International Mathematical Research Notices</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1511.02427">arXiv:1511.02427</a> <span> [<a href="https://arxiv.org/pdf/1511.02427">pdf</a>, <a href="https://arxiv.org/format/1511.02427">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On the chromatic number of structured Cayley graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</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="1511.02427v3-abstract-short" style="display: inline;"> In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.02427v3-abstract-full').style.display = 'inline'; document.getElementById('1511.02427v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1511.02427v3-abstract-full" style="display: none;"> In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower bounds on the chromatic number of these graphs. This provides a lower bound for the chromatic number of Cayley graphs of the regular graphs associated to the ring of $n\times n$ matrices over finite fields. Using Weil's bound for Kloosterman sums we will also prove an analogous result for $\mathrm{SL}_2$ over finite rings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.02427v3-abstract-full').style.display = 'none'; document.getElementById('1511.02427v3-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 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 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">arXiv admin note: text overlap with arXiv:1507.05300</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1507.05300">arXiv:1507.05300</a> <span> [<a href="https://arxiv.org/pdf/1507.05300">pdf</a>, <a href="https://arxiv.org/format/1507.05300">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On a generalization of the Hadwiger-Nelson problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</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="1507.05300v4-abstract-short" style="display: inline;"> For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an edge if and only if $Q(v-w)=1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson proble… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.05300v4-abstract-full').style.display = 'inline'; document.getElementById('1507.05300v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1507.05300v4-abstract-full" style="display: none;"> For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an edge if and only if $Q(v-w)=1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field $F$ of characteristic zero, the Borel chromatic number of $\mathrm{QG}_Q$ is infinite if and only if $Q$ represents zero non-trivially over $F$. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.05300v4-abstract-full').style.display = 'none'; document.getElementById('1507.05300v4-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 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 July, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">This is the final version. Accepted in Israel Journal of Mathematics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.00626">arXiv:1505.00626</a> <span> [<a href="https://arxiv.org/pdf/1505.00626">pdf</a>, <a href="https://arxiv.org/format/1505.00626">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Minimal dimension of faithful representations for $p$-groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</a>, <a href="/search/math?searchtype=author&query=Salmasian%2C+H">Hadi Salmasian</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="1505.00626v3-abstract-short" style="display: inline;"> For a group $G$, we denote by $m_{faithful}(G)$, the smallest dimension of a faithful complex representation of $G$. Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the maximal ideal $\mathfrak{p}$. In this paper, we compute the precise value of $m_{faithful}(G)$ when $G$ is the Heisenberg group over $\mathcal{O}/\mathfrak{p}^n$. We then use the Weil represen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.00626v3-abstract-full').style.display = 'inline'; document.getElementById('1505.00626v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.00626v3-abstract-full" style="display: none;"> For a group $G$, we denote by $m_{faithful}(G)$, the smallest dimension of a faithful complex representation of $G$. Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the maximal ideal $\mathfrak{p}$. In this paper, we compute the precise value of $m_{faithful}(G)$ when $G$ is the Heisenberg group over $\mathcal{O}/\mathfrak{p}^n$. We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over $\mathcal{O}/\mathfrak{p}^n$ and many of its subgroups. By a theorem of Karpenko and Merkurjev, our result yields the precise value of the essential dimension of the latter finite groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.00626v3-abstract-full').style.display = 'none'; document.getElementById('1505.00626v3-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 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">Final version. To appear in "Journal of group theory"</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1403.3722">arXiv:1403.3722</a> <span> [<a href="https://arxiv.org/pdf/1403.3722">pdf</a>, <a href="https://arxiv.org/format/1403.3722">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Karimianpour%2C+C">Camelia Karimianpour</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</a>, <a href="/search/math?searchtype=author&query=Salmasian%2C+H">Hadi Salmasian</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="1403.3722v3-abstract-short" style="display: inline;"> Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the prime ideal $\mathfrak{p}$ and let $G={\bf G}\left(\mathcal{O}/\mathfrak{p}^n\right)$ be the adjoint Chevalley group. Let $m_f(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone-von Neumann theorem, we determine a lower bound for $m_f(G)$ which is asymptotically the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.3722v3-abstract-full').style.display = 'inline'; document.getElementById('1403.3722v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1403.3722v3-abstract-full" style="display: none;"> Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the prime ideal $\mathfrak{p}$ and let $G={\bf G}\left(\mathcal{O}/\mathfrak{p}^n\right)$ be the adjoint Chevalley group. Let $m_f(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone-von Neumann theorem, we determine a lower bound for $m_f(G)$ which is asymptotically the same as the results of Landazuri, Seitz and Zalesskii for split Chevalley groups over $\mathbb{F}_q$. Our result yields a conceptual explanation of the exponents that appear in the aforementioned results <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.3722v3-abstract-full').style.display = 'none'; document.getElementById('1403.3722v3-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 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 March, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">Final version. To appear in "Groups, Geometry, and Dynamics"</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1310.1643">arXiv:1310.1643</a> <span> [<a href="https://arxiv.org/pdf/1310.1643">pdf</a>, <a href="https://arxiv.org/format/1310.1643">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On the Erdos-Ko-Rado property for finite Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</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="1310.1643v5-abstract-short" style="display: inline;"> Let a finite group $G$ act transitively on a finite set $X$. A subset $S\subseteq G$ is said to be {\it intersecting} if for any $s_1,s_2\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\it weak Erd艖s-Ko-Rado} property, if the cardinality of any intersecting set is at most $|G|/|X|$. If, moreover, any maximal intersecting set is a coset of a point stabilizer, th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.1643v5-abstract-full').style.display = 'inline'; document.getElementById('1310.1643v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1310.1643v5-abstract-full" style="display: none;"> Let a finite group $G$ act transitively on a finite set $X$. A subset $S\subseteq G$ is said to be {\it intersecting} if for any $s_1,s_2\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\it weak Erd艖s-Ko-Rado} property, if the cardinality of any intersecting set is at most $|G|/|X|$. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the {\it strong Erd艖s-Ko-Rado} property. In this paper we will investigate the weak and strong Erd艖s-Ko-Rado property and attempt to classify the groups whose all transitive actions have these properties. In particular, we show that a group with the weak Erd艖s-Ko-Rado property is solvable and that a nilpotent group with the strong Erd艖s-Ko-Rado property is product of a $2$-group and an abelian group of odd order. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.1643v5-abstract-full').style.display = 'none'; document.getElementById('1310.1643v5-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 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 October, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">This is the final version. To appear in the Journal of Algebraic Combinatorics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1309.7482">arXiv:1309.7482</a> <span> [<a href="https://arxiv.org/pdf/1309.7482">pdf</a>, <a href="https://arxiv.org/ps/1309.7482">ps</a>, <a href="https://arxiv.org/format/1309.7482">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"> Mertens's theorem for splitting primes and more </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Freiberg%2C+T">Tristan Freiberg</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="1309.7482v1-abstract-short" style="display: inline;"> Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is also quite adaptable, and it is readily combined with well-known results from prime number theory. We demonstrate this by proving a version of the theorem for… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.7482v1-abstract-full').style.display = 'inline'; document.getElementById('1309.7482v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1309.7482v1-abstract-full" style="display: none;"> Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is also quite adaptable, and it is readily combined with well-known results from prime number theory. We demonstrate this by proving a version of the theorem for primes in arithmetic progressions with uniformity in the modulus, as well as a non-abelian analogue of this. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.7482v1-abstract-full').style.display = 'none'; document.getElementById('1309.7482v1-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 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">This is a survey paper</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1202.4194">arXiv:1202.4194</a> <span> [<a href="https://arxiv.org/pdf/1202.4194">pdf</a>, <a href="https://arxiv.org/ps/1202.4194">ps</a>, <a href="https://arxiv.org/format/1202.4194">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Quasi-Random profinite groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</a>, <a href="/search/math?searchtype=author&query=Mallahi-Karai%2C+K">Keivan Mallahi-Karai</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="1202.4194v5-abstract-short" style="display: inline;"> We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\mathrm{SL}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\mathrm{Sp}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1202.4194v5-abstract-full').style.display = 'inline'; document.getElementById('1202.4194v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1202.4194v5-abstract-full" style="display: none;"> We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\mathrm{SL}_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\mathrm{Sp}_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups $\mathrm{SL}_{k}({\mathbb{Z}_p})$ and $\mathrm{Sp}_{2k}(\mathbb{Z}_p)$. We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1202.4194v5-abstract-full').style.display = 'none'; document.getElementById('1202.4194v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 August, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 February, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">This is the final version. To appear in Glasgow Mathematical Journal</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20P05; 20F; 20C33 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1202.2047">arXiv:1202.2047</a> <span> [<a href="https://arxiv.org/pdf/1202.2047">pdf</a>, <a href="https://arxiv.org/format/1202.2047">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> The Density of a family of monogenic number fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bardestani%2C+M">Mohammad Bardestani</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="1202.2047v2-abstract-short" style="display: inline;"> A monogenic polynomial $f$ is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime $q$, using the Chebotarev density theorem, we will show the density of primes $p$, such that $t^q-p$ is monogenic, is bigger or equal than $(q-1)/q$. We will also prove that, when $q=3$, the density of primes $p$, which $\mathbb{Q}(\sqrt[3]{p})$ is non-m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1202.2047v2-abstract-full').style.display = 'inline'; document.getElementById('1202.2047v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1202.2047v2-abstract-full" style="display: none;"> A monogenic polynomial $f$ is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime $q$, using the Chebotarev density theorem, we will show the density of primes $p$, such that $t^q-p$ is monogenic, is bigger or equal than $(q-1)/q$. We will also prove that, when $q=3$, the density of primes $p$, which $\mathbb{Q}(\sqrt[3]{p})$ is non-monogenic, is at least 1/9. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1202.2047v2-abstract-full').style.display = 'none'; document.getElementById('1202.2047v2-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 June, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 February, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">Various corrections. Submitted</span> </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"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

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