Abstract : For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.
Abstract : Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}}, \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vertex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.
Abstract : Let $D$ be an integral domain with quotient field $K$, $Pic(D)$ be the ideal class group of $D$, and $X$ be an indeterminate. A polynomial overring of $D$ means a subring of $K[X]$ containing $D[X]$. In this paper, we study almost Dedekind domains which are polynomial overrings of a principal ideal domain $D$, defined by the intersection of $K[X]$ and rank-one discrete valuation rings with quotient field $K(X)$, and their ideal class groups. Next, let $\mathbb{Z}$ be the ring of integers, $\mathbb{Q}$ be the field of rational numbers, and $\mathfrak{G}_f$ be the set of finitely generated abelian groups (up to isomorphism). As an application, among other things, we show that there exists an overring $R$ of $\mathbb{Z}[X]$ such that (i) $R$ is a Bezout domain, (ii) $R \cap \mathbb{Q}[X]$ is an almost Dedekind domain, (iii) $Pic(R \cap \mathbb{Q}[X]) = \bigoplus_{G \in \mathfrak{G}_f}G$, (iv) for each $G \in \mathfrak{G}_f$, there is a multiplicative subset $S$ of $\mathbb{Z}$ such that $R_S \cap \mathbb{Q}[X]$ is a Dedekind domain with $Pic(R_S \cap \mathbb{Q}[X]) = G$, and (v) every invertible integral ideal $I$ of $R \cap \mathbb{Q}[X]$ can be written uniquely as $I = X^nQ_1^{e_1} \cdots Q_k^{e_k}$ for some integer $n \geq 0$, maximal ideals $Q_i$ of $R \cap \mathbb{Q}[X]$, and integers $e_i \neq 0$. We also completely characterize the almost Dedekind polynomial overrings of $\mathbb{Z}$ containing Int$(\mathbb{Z})$.
Abstract : For two dimensional lattices, a Gaussian basis achieves all two successive minima. For dimension larger than two, constructing a pairwise Gaussian basis is useful to compute short vectors of the lattice. For three dimensional lattices, Semaev showed that one can convert a pairwise Gaussian basis to a basis achieving all three successive minima by one simple reduction. A pairwise Gaussian basis can be obtained from a given basis by executing Gauss algorithm for each pair of basis vectors repeatedly until it returns a pairwise Gaussian basis. In this article, we prove a necessary and sufficient condition for a pairwise Gaussian basis to achieve the first $k$ successive minima for three dimensional lattices for each $k\in\{1,2,3\}$ by modifying Semaev's condition. Our condition directly checks whether a pairwise Gaussian basis contains the first $k$ shortest independent vectors for three dimensional lattices. LLL is the most basic lattice basis reduction algorithm and we study how to use LLL to compute a pairwise Gaussian basis. For $\delta\ge 0.9$, we prove that LLL($\delta$) with an additional simple reduction turns any basis for a three dimensional lattice into a pairwise SV-reduced basis. By using this, we convert an LLL reduced basis to a pairwise Gaussian basis in a few simple reductions. Our result suggests that the LLL algorithm is quite effective to compute a basis with all three successive minima for three dimensional lattices.
Abstract : 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.
Abstract : In this paper, we study the boundedness of a class of inhomogeneous Journ\'{e}'s product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journ\'{e}'s product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.
Abstract : In this paper, we introduce the notion of stress-energy tensor $Q$ of the traceless Ricci tensor for Riemannian manifolds $(M^n, g)$, and investigate harmonicity of Riemannian curvature tensor and Weyl curvature tensor when $(M, g)$ satisfies some geometric structure such as critical point equation or vacuum static equation for smooth functions.
Abstract : There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a $C^2$ strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.
Abstract : We introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable compact spaces which are generalizations of expansivity and shadowing, respectively, for metric spaces. The main result is to generalize the Smale's spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces.
Abstract : 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 $\infty$.
Shivani Dubey, Mukund Madhav Mishra, Ashutosh Pandey
J. Korean Math. Soc. 2022; 59(3): 635-648
https://doi.org/10.4134/JKMS.j210462
Xiaolei Zhang
J. Korean Math. Soc. 2023; 60(3): 521-536
https://doi.org/10.4134/JKMS.j220055
Qingfang Shi, Xinli Zhang
J. Korean Math. Soc. 2023; 60(3): 587-618
https://doi.org/10.4134/JKMS.j220285
Shaoting Xie, Jiandong Yin
J. Korean Math. Soc. 2022; 59(6): 1229-1254
https://doi.org/10.4134/JKMS.j220202
Peter Jaehyun Cho , Gyeongwon Oh
J. Korean Math. Soc. 2023; 60(1): 167-193
https://doi.org/10.4134/JKMS.j220242
Jun-ichi Inoguchi, Ji-Eun Lee
J. Korean Math. Soc. 2023; 60(6): 1303-1336
https://doi.org/10.4134/JKMS.j230105
Byung Hee An, Youngjin Cho
J. Korean Math. Soc. 2023; 60(6): 1171-1213
https://doi.org/10.4134/JKMS.j220343
Diego Conti, Federico A. Rossi, Romeo Segnan Dalmasso
J. Korean Math. Soc. 2023; 60(5): 1135-1136
https://doi.org/10.4134/JKMS.j230073
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