Abstract : Let $f(z)$ be a primitive holomorphic cusp form and $g(z)$ be a Maass cusp form. In this paper, we give quantitative results for the sign changes of coefficients of triple product $L$-functions $L(f\times f\times f, s)$ and $L(f\times f\times g, s)$.
Abstract : We consider a vector bundle map $F\colon E_{1}\longrightarrow E_{2}$ between Lie algebroids $E_{1}$ and $E_{2}$ over arbitrary bases $M_{1}$ and $M_{2}$. We associate to it different notions of curvature which we call A-curvature, Q-curvature, P-curvature, and S-curvature using the different characterizations of Lie algebroid structure, namely Lie algebroid, Q-manifold, Poisson and Schouten structures. We will see that these curvatures generalize the ordinary notion of curvature defined for a vector bundle, and we will prove that these curvatures are equivalent, in the sense that $F$ is a morphism of Lie algebroids if and only if one (and hence all) of these curvatures is null. In particular we get as a corollary that $F$ is a morphism of Lie algebroids if and only if the corresponding map is a morphism of Poisson manifolds (resp. Schouten supermanifolds).
Abstract : We use Borcherds products to give a new construction of the weight~$3$paramodular nonlift eigenform~$f_N$ for levels~$N=61,73,79$.We classify the congruences of~$f_N$ to Gritsenko lifts.We provide techniques that compute eigenvalues to supportfuture modularity applications.Our method does not compute Hecke eigenvalues from Fouriercoefficients but instead uses elliptic modular forms, specificallythe restrictions of Gritsenko lifts and their images under the slashoperator to modular curves.
Abstract : Let $Q$ be an integral positive definite quadratic form of level $N$ in $2k(\geq4)$ variables. We assume that $(-1)^kN$ is a fundamental discriminant and the associated character $\chi$ of $Q$ is primitive of conductor $N$. Under our assumption, we find the pairs $(k,N)$ such that the dimension of spaces of cusp forms of weight $k$ and level $N$ with Nebentypus $\chi$ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs $(k,N)$, we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level $N$ in $2k$ variables, which are expressed in terms of divisor functions and their convolution sums.
Abstract : Let $R$ be an associative ring with an identity, $ \sigma $ be an automorphism, and $\delta$ be a $\sigma$-derivation of $R$. In this article, we describe all (nilpotent) associated primes of the skew inverse Laurent series ring $ R((x^{-1}$; $\sigma,\delta)) $ in terms of the (nilpotent) associated primes of $R.$
Abstract : In this study, we give weighted mean and weighted Gaussian curvatures of two types of timelike general rotational surfaces with non-null plane meridian curves in four-dimensional Minkowski space $\mathbb{E}_{1}^{4}$ with density $e^{\lambda_{1}x^{2}+\lambda_{2}y^{2}+\lambda_{3}z^{2}+\lambda_{4}t^{2}},$ where $\lambda_{i}$ ($i=1,2,3,4$) are not all zero. We give some results about weighted minimal and weighted flat timelike general rotational surfaces in $\mathbb{E}_{1}^{4}$ with density. Also, we construct some examples for these surfaces.
Abstract : The objective which drives the writing of this article is to study the behavior of the algebraic transfer for ranks $h\in \{6,\, 7,\, 8\}$ across various internal degrees. More precisely, we prove that the algebraic transfer is an isomorphism in certain bidegrees. A noteworthy aspect of our research is the rectification of the results outlined by M. Moetele and M. F. Mothebe in [East-West J. of Mathematics \textbf{18} (2016), 151--170]. This correction focuses on the $\mathcal A$-generators for the polynomial algebra $\mathbb Z/2[t_1, t_2, \ldots, t_h]$ in degree thirteen and the ranks $h$ mentioned above. As direct consequences, we are able to confirm the Singer conjecture for the algebraic transfer in the cases under consideration. Especially, we affirm that \textit{the decomposable element $h_6Ph_2 \in {\rm Ext}_{\mathcal A}^{6, 80}(\mathbb Z/2, \mathbb Z/2)$ does not reside within the image of the sixth algebraic transfer}. This event carries significance as it enables us to either strengthen or refute the Singer conjecture, which is relevant to the behavior of the algebraic transfer. Additionally, we also show that the indecomposable element $$q\in {\rm Ext}_{\mathcal A}^{6, 38}(\mathbb Z/2, \mathbb Z/2)$$ is not detected by the sixth algebraic transfer. \textit{Prior to this research, no other authors had delved into the Singer conjecture for these cases. The significant and remarkable advancement made in this paper regarding the investigation of Singer's conjecture for ranks $h,\, 6\leq h\leq 8,$ highlights a deeper understanding of the enigmatic nature of ${\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb Z/2, \mathbb Z/2)$}.
Abstract : In this paper, in one spatial dimension, we study the convergence rate for Schr\"{o}dinger operators with complex time $P_{a,\gamma}^{t}$, which is defined by $$P_{a,\gamma}^{t}f(x)=S_{a}^{t+it^{\gamma}}f(x) =\int_{\mathbb{R}} e^{ix\xi}e^{it|\xi|^{a}}e^{-t^{\gamma}|\xi|^{a}} \hat{f}(\xi)d\xi,$$ where $\gamma>0$ and $a>0.$ The convergence rate for Schr\"{o}dinger operators with complex time is different from that of classical Schr\"{o}dinger operators in Cao-Fan-Wang (Illinois J. Math. 62: 365--380, 2018).
Abstract : For a domain $D \subset \mathbb C^n$ and an admissible weight $\mu$ on it, we consider the weighted Bergman kernel $K_{D, \mu}$ and the corresponding weighted Bergman metric on $D$. In particular, motivated by work of Mok, Ng, Chan--Yuan and Chan--Xiao--Yuan among others, we study the nature of holomorphic isometries from the disc $\mathbb D \subset \mathbb C$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb D}^{-d}$ for some integer $d \ge 0$. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied in detail include those in which the isometry takes values in $\mathbb D^n$ and $\mathbb D \times \mathbb B^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented.
Abstract : We show that under a mild thickness condition, every Tits pentagon arises through a folding process from a free module of rank~$5$ over a unitary associative ring of stable range~$1$.
Ya Gao , Yanling Gao, Jing Mao , Zhiqi Xie
J. Korean Math. Soc. 2024; 61(1): 183-205
https://doi.org/10.4134/JKMS.j230283
Hieu Van Ha, Vu Anh Le, Tu Thi Cam Nguyen, Hoa Duong Quang
J. Korean Math. Soc. 2023; 60(4): 835-858
https://doi.org/10.4134/JKMS.j220384
Bokhee Im, Jonathan D. H. Smith
J. Korean Math. Soc. 2023; 60(1): 91-113
https://doi.org/10.4134/JKMS.j220182
Yoosik Kim
J. Korean Math. Soc. 2023; 60(5): 1109-1133
https://doi.org/10.4134/JKMS.j230098
Railane Antonia, Henrique de Lima, Márcio Santos
J. Korean Math. Soc. 2024; 61(1): 41-63
https://doi.org/10.4134/JKMS.j220523
Gyu Whan Chang, Jun Seok Oh
J. Korean Math. Soc. 2023; 60(2): 407-464
https://doi.org/10.4134/JKMS.j220271
Sang-Bum Yoo
J. Korean Math. Soc. 2024; 61(1): 161-181
https://doi.org/10.4134/JKMS.j230278
Salah Gomaa Elgendi, Amr Soleiman
J. Korean Math. Soc. 2024; 61(1): 149-160
https://doi.org/10.4134/JKMS.j230263
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