Journal of the
Korean Mathematical Society
JKMS

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

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  • 2021-01-01

    Mathematical analysis of an ``SIR'' epidemic model in a continuous reactor - deterministic and probabilistic approaches

    Miled El Hajji, Sayed Sayari, Abdelhamid Zaghdani

    Abstract : In this paper, a mathematical dynamical system involving both deterministic (with or without delay) and stochastic ``SIR'' epidemic model with nonlinear incidence rate in a continuous reactor is considered. A profound qualitative analysis is given. It is proved that, for both deterministic models, if $\R_d > 1$, then the endemic equilibrium is globally asymptotically stable. However, if $\R_d \leq 1$, then the disease-free equilibrium is globally asymptotically stable. Concerning the stochastic model, the Feller's test combined with the canonical probability method were used in order to conclude on the long-time dynamics of the stochastic model. The results improve and extend the results obtained for the deterministic model in its both forms. It is proved that if $\R_s > 1$, the disease is stochastically permanent with full probability. However, if $\R_s \leq 1$, then the disease dies out with full probability. Finally, some numerical tests are done in order to validate the obtained results.

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  • 2021-07-01

    Weighted estimates for certain rough operators with applications to vector valued inequalities

    Feng Liu, Qingying Xue

    Abstract : Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.

  • 2021-05-01

    Kenmotsu manifolds satisfying the Fischer-Marsden equation

    Sudhakar Kr Chaubey, Uday Chand De, Young Jin Suh

    Abstract : The present paper deals with the study of Fischer-Marsden conjecture on a Kenmotsu manifold. It is proved that if a Kenmotsu metric satisfies $\mathfrak{L}^{*}_{g}(\lambda)=0$ on a $(2n+1)$-dimensional Kenmotsu manifold $M^{2n+1}$, then either $\xi \lambda=- \lambda$ or $M^{2n+1}$ is Einstein. If $n=1$, $M^3$ is locally isometric to the hyperbolic space $H^{3}(-1)$.

  • 2021-05-01

    On $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes and their weight enumerators

    Maheshanand Bhaintwal, Soumak Biswas

    Abstract : In this paper we study the algebraic structure of $\mathbb{Z}_p\mathbb{Z}_p[u]/$ $\langle u^k\rangle$-cyclic codes, where $u^k=0$ and $p$ is a prime. A $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-linear code of length $(r+s)$ is an $R_k$-submodule of $\mathbb{Z}_p^r \times R_k^s$ with respect to a suitable scalar multiplication, where $R_k = \mathbb{Z}_p[u]/\langle u^k\rangle$. Such a code can also be viewed as an $R_k$-submodule of $\mathbb{Z}_p[x]/\langle x^r-1\rangle \times R_k[x]/\langle x^s-1\rangle$. A new Gray map has been defined on $\mathbb{Z}_p[u]/\langle u^k\rangle$. We have considered two cases for studying the algebraic structure of $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered $(r,p)=1$ and $(s,p)\neq 1$, and in the second case we consider $(r,p)=1$ and $(s,p)=1$. We have established the MacWilliams identity for complete weight enumerators of $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-linear codes. Examples have been given to construct $\mathbb{Z}_p\mathbb{Z}_p[u]/\langle u^k\rangle$-cyclic codes, through which we get codes over $\mathbb{Z}_p$ using the Gray map. Some optimal $p$-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.

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  • 2021-03-01

    A van der Corput type lemma for oscillatory integrals with H\"older amplitudes and its applications

    Hussain Al-Qassem, Leslie Cheng, Yibiao Pan

    Abstract : We prove a decay estimate for oscillatory integrals with \linebreak H\"older amplitudes and polynomial phases. The estimate allows us to answer certain questions concerning the uniform boundedness of oscillatory singular integrals on various spaces.

  • 2021-03-01

    Rigidity characterizations of complete Riemannian manifolds with $\alpha$-Bach-flat

    Guangyue Huang, Qianyu Zeng

    Abstract : For complete manifolds with $\alpha$-Bach tensor (which is defined by \eqref{1-Int-2}) flat, we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics have also been characterized by some $L^{\frac{n}{2}}$-integral inequalities. Furthermore, we also give some rigidity characterizations for constant sectional curvature.

  • 2021-03-01

    Parameter Marcinkiewicz integral and its commutator on generalized Orlicz-Morrey spaces

    Guanghui Lu

    Abstract : The aim of this paper is to mainly establish the sufficient and necessary conditions for the boundedness of the commutator $\mathcal{M}^{\rho}_{\Omega, b}$ which is generated by the parameter Marcinkiwicz integral $\mathcal{M}^{\rho}_{\Omega}$ and the Lipschitz function $b$ on generalized Orlicz-Morrey space $L^{\Phi,\varphi}(\mathbb{R}^{d})$ in the sense of the Adams type result (or Spanne type result). Moreover, the necessary conditions for the parameter Marcinkiewizcz integral $\mathcal{M}^{\rho}_{\Omega}$ on the $L^{\Phi,\varphi}(\mathbb{R}^{d})$, and the commutator $[b,\mathcal{M}^{\rho}_{\Omega}]$ generated by the $\mathcal{M}^{\rho}_{\Omega}$ and the space $\mathrm{BMO}$ on the $L^{\Phi,\varphi}(\mathbb{R}^{d})$, are also obtained, respectively.

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  • 2021-09-01

    Eventual shadowing for chain transitive sets of $C^1$ generic dynamical systems

    Manseob Lee

    Abstract : We show that given any chain transitive set of a $C^1$ generic diffeomorphism $f$, if a diffeomorphism $f$ has the eventual shadowing property on the locally maximal chain transitive set, then it is hyperbolic. Moreover, given any chain transitive set of a $C^1$ generic vector field $X$, if a vector field $X$ has the eventual shadowing property on the locally maximal chain transitive set, then the chain transitive set does not contain a singular point and it is hyperbolic. We apply our results to conservative systems (volume-preserving diffeomorphisms and divergence-free vector fields).

  • 2021-03-01

    Symplectic fillings of quotient surface singularities and minimal model program

    Hakho Choi, Heesang Park, Dongsoo Shin

    Abstract : We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic varieties.

  • 2021-01-01

    Every abelian group is the class group of a ring of Krull type

    Gyu Whan Chang

    Abstract : Let $Cl(A)$ denote the class group of an arbitrary integral domain $A$ introduced by Bouvier in 1982. Then $Cl(A)$ is the ideal class (resp., divisor class) group of $A$ if $A$ is a Dedekind or a Pr\"ufer (resp., Krull) domain. Let $G$ be an abelian group. In this paper, we show that there is a ring of Krull type $D$ such that $Cl(D) = G$ but $D$ is not a Krull domain. We then use this ring to construct a Pr\"ufer ring of Krull type $E$ such that $Cl(E) = G$ but $E$ is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.

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May, 2022
Vol.59 No.3

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