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2022-07-01
The moduli spaces of Lorentzian left-invariant metrics on three-dimensional unimodular simply connected Lie groups
Mohamed Boucetta, Abdelmounaim ChakkarAbstract : Let $\mathrm{G}$ be an arbitrary, connected, simply connected and unimodular Lie group of dimension $3$. On the space $\mathfrak{M}(\mathrm{G})$ of left-invariant Lorentzian metrics on $\mathrm{G}$, there exists a natural action of the group ${\rm Aut}(\mathrm{G})$ of automorphisms of $\mathrm{G}$, so it yields an equivalence relation $\backsimeq$ on $\mathfrak{M}(\mathrm{G})$, in the following way: $h_1\backsimeq h_2 \Leftrightarrow h_2=\phi^{*}(h_1) \;\textrm{for some}\; \phi \in {\rm Aut}(\mathrm{G}).$ In this paper a procedure to compute the orbit space ${\rm Aut}(\mathrm{G})/\mathfrak{M}(\mathrm{G})$ (so called moduli space of $\mathfrak{M}(\mathrm{G})$) is given.
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2022-07-01
On $3^k$-regular cubic partitions
Nayandeep Deka Baruah, Hirakjyoti DasAbstract : Recently, Gireesh, Shivashankar, and Naika \cite{GSN20} found some infinite classes of congruences for the 3- and the 9-regular cubic partitions modulo powers of 3. We extend their study to all the $3^k$-regular cubic partitions. We also find new families of congruences.
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2022-07-01
Foliations from left orders
Hyungryul Baik, Sebastian Hensel, Chenxi WuAbstract : We describe a construction which takes as an input a left order of the fundamental group of a manifold, and outputs a (singular) foliation of this manifold which is analogous to a taut foliation. We investigate this construction in detail in dimension $2$, and exhibit connections to various problems in dimension $3$.
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2022-07-01
Uniqueness of quasi-roots in right-angled Artin groups
Eon-Kyung Lee, Sang-Jin LeeAbstract : We introduce the notion of quasi-roots and study their \linebreak uniqueness in right-angled Artin groups.
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2022-07-01
Gorenstein quasi-resolving subcategories
Weiqing Cao, Jiaqun WeiAbstract : In this paper, we introduce the notion of Gorenstein quasi-resolving subcategories (denoted by $\mathcal{GQR}_{\mathcal{X}}(\mathcal{A})$) in term of quasi-resolving subcategory $\mathcal{X}$. We define a resolution dimension relative to the Gorenstein quasi-resolving categories $\mathcal{GQR}_{\mathcal{X}}(\mathcal{A})$. In addition, we study the stability of $\mathcal{GQR}_{\mathcal{X}}(\mathcal{A})$ and apply the obtained properties to special subcategories and in particular to modules categories. Finally, we use the restricted flat dimension of right $B$-module $M$ to characterize the finitistic dimension of the endomorphism algebra $B$ of a $\mathcal{GQ}_{\mathcal{X}}$-projective $A$-module $M$.
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2022-07-01
Sharp Ore-type conditions for the existence of an even $[4,b]$-factor in a graph
Eun-Kyung Cho, Su-Ah Kwon, Suil OAbstract : Let $a$ and $b$ be positive even integers. An even $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $d_H(v)$ is even and $a \le d_H(v) \le b$. Let $\kappa(G)$ be the minimum size of a vertex set $S$ such that $G-S$ is disconnected or one vertex, and let $\sigma_2(G)=\min_{uv \notin E(G)}(d(u)+d(v))$. In 2005, Matsuda proved an Ore-type condition for an $n$-vertex graph satisfying certain properties to guarantee the existence of an even $[2,b]$-factor. In this paper, we prove that for an even positive integer $b$ with $b \ge 6$, if $G$ is an $n$-vertex graph such that $n \ge b+5$, $\kappa(G) \ge 4$, and $\sigma_2(G) \ge \frac{8n}{b+4}$, then $G$ contains an even $[4,b]$-factor; each condition on $n$, $\kappa(G)$, and $\sigma_2(G)$ is sharp.
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2022-07-01
A note on the zeros of Jensen polynomials
Young-One Kim, Jungseob LeeAbstract : Sufficient conditions for the Jensen polynomials of the derivatives of a real entire function to be hyperbolic are obtained. The conditions are given in terms of the growth rate and zero distribution of the function. As a consequence some recent results on Jensen polynomials, relevant to the Riemann hypothesis, are extended and improved.
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2022-07-01
C*-algebraic Schur product theorem, P\'{O}lya-Szeg\H{O}-Rudin question and Novak's conjecture
Krishnanagara Mahesh~KrishnaAbstract : Striking result of Vyb\'{\i}ral \cite{VYBIRAL} says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vyb\'{\i}ral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vyb\'{\i}ral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate P\'{o}lya-Szeg\H{o}-Rudin question for the C*-algebraic Schur product of positive matrices.
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2022-07-01
Finite $p$-groups whose normal closures of non-normal subgroups have two orders
Lifang WangAbstract : We describe the structure of finite $p$-groups in which all normal closures of non-normal subgroups have two orders for $p>2$.
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2022-07-01
Regularity relative to a hereditary torsion theory for modules over a commutative ring
Lei Qiao, Kai ZuoAbstract : In this paper, we introduce and study regular rings relative to the hereditary torsion theory $w$ (a special case of a well-centered torsion theory over a commutative ring), called $w$-regular rings. We focus mainly on the $w$-regularity for $w$-coherent rings and $w$-Noetherian rings. In particular, it is shown that the $w$-coherent $w$-regular domains are exactly the Pr\"ufer $v$-multiplication domains and that an integral domain is $w$-Noetherian and $w$-regular if and only if it is a Krull domain. We also prove the $w$-analogue of the global version of the Serre--Auslander-Buchsbaum Theorem. Among other things, we show that every $w$-Noetherian $w$-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak $w$-projective dimension of a $w$-Noetherian ring is 0, 1, or $\infty$.
July, 2022 | Volume 59, No. 4
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