Korean Journal of Mathematics

On the relations for limiting case of selection with equilibrium and mutation of diploid model

2024-09-05T12:52:34+09:00

Assume that at a certain locus there are three genotypes and that for every one progeny produced by an $I^A I^A$ homozygote, the heterozygote $I^A I^B$ produces. Choi find the adapted partial equations for the model of selection against heterozygotes and in case that the allele frequency changes after one generation of selection when there is overdominance. Also he find the partial differential equation of general type of selection at diploid model and it also shall apply to actual examples. This is a very meaningful result in that it can be applied in any model ([1], [2]).

In this paper, we start with the limiting case of selection against recessive alleles. For the time being, assume that the trajectories of $p_t$ and $q_t$ at time $t$ can be approximated by paths which are continuous and therefore we have a diffusion process. We shall find the relations for time $t$, $p_t$ and $q_t$ and apply to equilibrium state and mutation.

Generalized hilbert operator on bergman spaces

2024-12-15T17:18:51+09:00

We consider the generalized Hilbert operator $\mathcal{H_\beta}$ for $\beta\geq 0$ and find the condition on the parameter $\beta$ for which the operator $\mathcal{H_\beta}$ is bounded on the Bergman space $ A^p$ for $2<p<\infty$. Also, we estimates the upper bound of the norm on $A^p$. Further shows that $\mathcal{H_\beta}$ is not bounded on $ A^2$ for $0\leq\beta<1.$

A characterization of $S_1$-projective modules

2024-11-30T18:59:04+09:00

Recently, Zhao, Pu, Chen, and Xiao introduced and investigated novel concepts regarding $S$-torsion exact sequences, $S$-torsion commutative diagrams, and $S_i$-projective modules (for $i = 1, 2$) in the context of a commutative ring $R$ and a multiplicative subset $S$ of $R$. Their research included various results, such as proving that an $R$-module is $S_1$-projective if it is $S$-torsion isomorphic to a projective module. In this paper, we further examine properties of $S$-torsion exact sequences and $S$-torsion commutative diagrams, and we establish the equivalence between an $R$-module being $S_1$-projective and its $S$-torsion isomorphism to a projective module.

On some ideals defined by an arithmetic sequence

2024-11-30T18:01:25+09:00

This paper investigates properties of ideals in the affine and homogeneous projective coordinate rings of the plane, defined using arithmetic sequence
\begin{equation*}
\{a_\ell = a+\ell d ~|~ \ell \ge 0 \}
\end{equation*}
for some positive integers $a$ and $d$. Specifically, we study two types of ideals:
$I(a,d)$ is generated by $D(a,d)$ in $K[x,y]$ and $J(a,d)$ is generated by $E(a,d)$ in $K[x,y,z]$ where
\begin{equation*}
D(a,d) =\{f_\ell = x^{a_{\ell}}-y^{a_{\ell+1}} ~|~ \ell\ge 0\}
\end{equation*}
and
\begin{equation*}
E(a,d) =\{F_\ell = x^{a_{\ell}} z^d -y^{a_{\ell+1}} ~|~ \ell\ge 0\}.
\end{equation*}
This paper provides detailed answers to several problems, including finding finite generating sets, describing the zero locus of these ideals, and determining their Hilbert functions. Finally, the Castelnuovo-Mumford regularity and the minimal free resolution of the homogeneous coordinate ring and multi secant line are discussed.

The $n$-generalized composition operators from Zygmund spaces to $Q_K(p,q)$ spaces

2024-07-22T13:54:27+09:00

The boundedness and compactness of the so-called $n$-generalized composition operator $c^{\mathfrak{g},n}_\varphi$ from the class of Zygmund-type spaces into $Q_K(p,q)$ spaces are characterized in this paper.

Hermite-Hadamard type inequalities for preinvex functions with applications

2023-09-26T09:46:49+09:00

In this article, we establish new Hermite-Hadamard Type inequalities for functions whose first derivative in absolute value are preinvex. Further, we give some application of our obtained results to some special means of real numbers. Moreover, we discuss several special cases of the results obtained in this paper.

Rough $\mathcal{I}$-convergence of sequences in probabilistic normed spaces

2023-09-03T17:41:11+09:00

In this paper, we have studied the idea of rough $\mathcal{I}$-convergence in probabilistic normed spaces which is indeed a generalized version as compared to the notion of rough $\mathcal{I}$-convergence in normed linear spaces. On the other way, it is also a generalization of rough statistical convergence in probabilistic normed spaces. Furthermore, we have defined the notion of rough $\mathcal{I}$-cluster points and have proved some important results associated with the set of rough $\mathcal{I}$-limits of a sequence in the same space.