Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

• 2022-05-01

Stabilized-penalized collocated finite volume scheme for incompressible biofluid flows

Nasserdine Kechkar, Mohammed Louaar

Abstract : In this paper, a stabilized-penalized collocated finite volume (SPCFV) scheme is developed and studied for the stationary generalized Navier-Stokes equations with mixed Dirichlet-traction boundary conditions modelling an incompressible biological fluid flow. This method is based on the lowest order approximation (piecewise constants) for both velocity and pressure unknowns. The stabilization-penalization is performed by adding discrete pressure terms to the approximate formulation. These simultaneously involve discrete jump pressures through the interior volume-boundaries and discrete pressures of volumes on the domain boundary. Stability, existence and uniqueness of discrete solutions are established. Moreover, a convergence analysis of the nonlinear solver is also provided. Numerical results from model tests are performed to demonstrate the stability, optimal convergence in the usual $L^2$ and discrete $H^1$ norms as well as robustness of the proposed scheme with respect to the choice of the given traction vector.

• 2021-05-01

General iterative algorithms for monotone inclusion, variational inequality and fixed point problems

Jong Soo Jung

Abstract : In this paper, we introduce two general iterative algorithms (one implicit algorithm and one explicit algorithm) for finding a common element of the solution set of the variational inequality problems for a continuous monotone mapping, the zero point set of a set-valued maximal monotone operator, and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed iterative algorithms to a common point of three sets, which is a solution of a certain variational inequality. Further, we find the minimum-norm element in common set of three sets.

• 2021-03-01

Topological stability and shadowing property for group actions on metric spaces

Yinong Yang

Abstract : In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for group actions on metric spaces and give a version of Walters's stability theorem for group actions on locally compact metric spaces. Moreover, we show that if $G$ is a finitely generated virtually nilpotent group and there exists $g\in G$ such that if $T_g$ is expansive and has the shadowing property, then $T$ is topologically stable.

• 2020-09-01

Some finiteness results for co-associated primes of generalized local homology modules and applications

Yen Ngoc Do, Tri Minh Nguyen, Nam Tuan Tran

Abstract : We prove some results about the finiteness of co-associated primes of generalized local homology modules inspired by a conjecture of Grothendieck and a question of Huneke. We also show some equivalent properties of minimax local homology modules. By duality, we get some properties of Herzog's generalized local cohomology modules.

• 2020-11-01

Static and related critical spaces with harmonic curvature and three Ricci eigenvalues

Jongsu Kim

Abstract : In this article we make a local classification of $n$-dimensional Riemannian manifolds $(M,g)$ with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution $f$ to the following equation \begin{align} \label{0002bxu} \nabla df = f(r -\frac{R}{n-1} g) + x \cdot r+ y(R) g, \end{align} where $\nabla$ is the Levi-Civita connection of $g$, $r$ is the Ricci tensor of $g$, $x$ is a constant and $y(R)$ a function of the scalar curvature $R$. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, either {\rm (i)} or {\rm (ii)} below holds; {\rm (i)} $(V, g, f+x)$ is a static space and isometric to a domain in the Riemannian product of an Einstein manifold $N$ and a static space $(W, g_W, f+x)$, where $g_W$ is a warped product metric of an interval and an Einstein manifold. {\rm (ii)} $(V, g)$ is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

• 2022-05-01

Construction of a solution of split equality variational inequality problem for pseudomonotone mappings in Banach spaces

Getahun Bekele Wega

Abstract : The purpose of this paper is to introduce an iterative algorithm for approximating a solution of split equality variational inequality problem for pseudomonotone mappings in the setting of Banach spaces. Under certain conditions, we prove a strong convergence theorem for the iterative scheme produced by the method in real reflexive Banach spaces. The assumption that the mappings are uniformly continuous and sequentially weakly continuous on bounded subsets of Banach spaces are dispensed with. In addition, we present an application of our main results to find solutions of split equality minimum point problems for convex functions in real reflexive Banach spaces. Finally, we provide a numerical example which supports our main result. Our results improve and generalize many of the results in the literature.

• 2021-03-01

A natural topological manifold structure of phase tropical hypersurfaces

Young Rock Kim, Mounir Nisse

Abstract : First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.

• 2020-07-01

Equidistribution of higher dimensional generalized Dedekind sums and exponential sums

Hi-joon Chae, Byungheup Jun, Jungyun Lee

Abstract : We consider generalized Dedekind sums in dimension $n$, defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in $\FR/\FZ$.

• 2022-05-01

On the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $\Phi_{p^s}(x)$ or $\Phi_{p^s q^t}(x)$

Jung Hee Cheon, Dongwoo Kim, Duhyeong Kim, Keewoo Lee