Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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  • 2023-07-31

    Hypersurfaces with prescribed mean curvature in measure metric space

    Zhengmao Chen

    Abstract : For any given function $f$, we focus on the so-called prescribed mean curvature problem for the measure $e^{-f(|x|^2)}dx$ provided that $e^{-f(|x|^2)}\in L^1(\mathbb{R}^{n+1})$. More precisely, we prove that there exists a smooth hypersurface $M$ whose metric is $ds^2=d\rho^2+\rho^2d\xi^2$ and whose mean curvature function is \begin{equation*} \frac{1}{n}\frac{u^p}{\rho^\beta}e^{f(\rho^2)}\psi(\xi) \end{equation*} for any given real constants $p$, $\beta$ and functions $f$ and $\psi$ where $u$ and $\rho$ are the support function and radial function of $M$, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere $\mathbb{S}^{n}$, \begin{equation*} \sum\limits_{i,j}(\delta_{ij}-\frac{\rho_i\rho_j}{\rho^2+|\nabla\rho|^2})(-\rho_{ji} +\frac{2}{\rho}\rho_j\rho_i +\rho\delta_{ji})=\psi\frac{\rho^{2p+2-n-\beta} e^{f(\rho^2)}}{(\rho^2+|\nabla \rho|^2)^{\frac{p}{2}}} \end{equation*} under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)=\frac{t}{2}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

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  • 2023-07-31

    Some results on 2-strongly Gorenstein projective modules and related rings

    Dong Chen, Kui Hu

    Abstract : In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring $R$ is 2-strongly Gorenstein hereditary if and only if every ideal of $R$ is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.

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  • 2023-07-31

    Toeplitz-type operators on the Fock space $F_{\alpha}^{2}$

    Chunxu Xu, Tao Yu

    Abstract : Let $j$ be a nonnegative integer. We define the Toeplitz-type operators $T_{a}^{(j)}$ with symbol $a\in L^{\infty}(C)$, which are variants of the traditional Toeplitz operators obtained for $j=0$. In this paper, we study the boundedness of these operators and characterize their compactness in terms of its Berezin transform.

  • 2023-07-31

    A model of retirement and consumption-portfolio choice

    Junkee Jeon, Hyeng Keun Koo

    Abstract : In this study we propose a model of optimal retirement, consumption and portfolio choice of an individual agent, which encompasses a large class of the models in the literature, and provide a methodology to solve the model. Different from the traditional approach, we consider the problems before and after retirement simultaneously and identify the difference in the dual value functions as the utility value of lifetime labor. The utility value has an option nature, namely, it is the maximized value of choosing the retirement time optimally and we discover it by solving a variational inequality. Then, we discover the dual value functions by using the utility value. We discover the value function and optimal policies by establishing a duality between the value function and the dual value function. The model and approach offer a significant advantage for computation of optimal policies for a large class of problems.

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  • 2022-03-31

    Positive expansivity, chain transitivity, rigidity, and specification on general topological spaces

    Thiyam Thadoi Devi, Khundrakpam Binod Mangang

    Abstract : We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform $h$-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff $h$-shadowing.

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  • 2023-05-31

    Bach almost solitons in paraSasakian geometry

    Uday Chand De, Gopal Ghosh

    Abstract : If a paraSasakian manifold of dimension $(2n+1)$ represents Bach almost solitons, then the Bach tensor is a scalar multiple of the metric tensor and the manifold is of constant scalar curvature. Additionally it is shown that the Ricci operator of the metric $g$ has a constant norm. Next, we characterize 3-dimensional paraSasakian manifolds admitting Bach almost solitons and it is proven that if a 3-dimensional paraSasakian manifold admits Bach almost solitons, then the manifold is of constant scalar curvature. Moreover, in dimension 3 the Bach almost solitons are steady if $r=-6$; shrinking if $r>-6$; expanding if $r

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  • 2023-01-31

    UN rings and group rings

    Kanchan Jangra, Dinesh Udar

    Abstract : A ring $R$ is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal $I$ of a UN ring $R$. Matrix rings over UN rings are discussed and it is obtained that for a commutative ring $R$, a matrix ring $M_n(R)$ is UN if and only if $R$ is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group $G$ and a field $K$ for the group algebra $KG$ to be UN. Then we extend the results obtained for $KG$ to the group ring $RG$ over a ring $R$ (which may not necessarily be a field).

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  • 2023-07-31

    On semi-regular injective modules and strong dedekind rings

    Renchun Qu

    Abstract : The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring $R$ is called strong Dedekind if every semi-regular ideal is $Q_0$-invertible, and an $R$-module $E$ is called a semi-regular injective module provided ${\rm Ext}^1_R(T,E)=0$ for every $\mathcal{Q}$-torsion module $T$. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of $R$-modules. Moreover, we introduce and study the semi-regular global dimensions $sr$-gl.dim$(R)$ of commutative rings $R$. Finally, we obtain that a ring $R$ is a ${\rm DQ}$-ring if and only if $sr$-gl.dim$(R)=0$, and a ring $R$ is a strong Dedekind ring if and only if $sr$-gl.dim$(R)\leq 1$, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

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  • 2023-07-31

    Residual supersingular Iwasawa theory over quadratic imaginary fields

    Parham Hamidi

    Abstract : Let $ p $ be an odd prime. Let $ E $ be an elliptic curve defined over a quadratic imaginary field, where $ p $ splits completely. Suppose $ E $ has supersingular reduction at primes above $ p $. Under appropriate hypotheses, we extend the results of \cite{SujFil} to $ \mathbb{Z}_p^{2} $-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed $ \mu $-invariants of one elliptic curve implies the vanishing of the signed $ \mu $-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

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  • 2023-01-31

    On Chowla's hypothesis implying that $L(s,\chi)>0$ for $s>0$ for real characters $\chi$

    St\'ephane R. Louboutin

    Abstract : Let $L(s,\chi)$ be the Dirichlet $L$-series associated with an $f$-periodic complex function $\chi$. Let $P(X)\in {\mathbb C}[X]$. We give an expression for $\sum_{n=1}^f \chi (n)P(n)$ as a linear combination of the $L(-n,\chi)$'s for $0\leq n<\deg P(X)$. We deduce some consequences pertaining to the Chowla hypothesis implying that $L(s,\chi )>0$ for $s>0$ for real Dirichlet characters $\chi$. To date no extended numerical computation on this hypothesis is available. In fact by a result of R. C. Baker and H. L. Montgomery we know that it does not hold for almost all fundamental discriminants. Our present numerical computation shows that surprisingly it holds true for at least $65\%$ of the real, even and primitive Dirichlet characters of conductors less than $10^6$. We also show that a generalized Chowla hypothesis holds true for at least $72\%$ of the real, even and primitive Dirichlet characters of conductors less than $10^6$. Since checking this generalized Chowla's hypothesis is easy to program and relies only on exact computation with rational integers, we do think that it should be part of any numerical computation verifying that $L(s,\chi )>0$ for $s>0$ for real Dirichlet characters $\chi$. To date, this verification for real, even and primitive Dirichlet characters has been done only for conductors less than $2\cdot 10^5$.

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March, 2024
Vol.61 No.2

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