Abstract : Let $k\geqslant 2$ be an integer, $S^k=\{1^k,2^k,3^k,\ldots\}$ and $B=\{b_1,b_2,b_3,\ldots\}$ be an additive complement of $S^k$, which means all sufficiently large integers can be written as the sum of an element of $S^k$ and an element of $B$. In this paper we prove that $$\limsup_{n\rightarrow \infty}\frac{\Gamma\left(2-\frac{1}{k}\right)^{\frac{k}{k-1}}\Gamma\left(1+\frac{1}{k}\right) ^{\frac{k}{k-1}}n^{\frac{k}{k-1}}-b_n}{n} \geqslant \frac{k}{2(k-1)}\frac{\Gamma\left(2-\frac{1}{k}\right)^2}{\Gamma\left(2-\frac{2}{k}\right)},$$ where $\Gamma(\cdot)$ is Euler's Gamma function.
Abstract : We discuss the wellposedness of the Neumann problem on a half-space for the Kohn-Laplacian in the Heisenberg group. We then construct the Neumann function and explicitly represent the solution of the associated inhomogeneous problem.
Abstract : In this paper, the existence and uniqueness for the global solution of neutral stochastic functional differential equation is investigated under the locally Lipschitz condition and the contractive condition. The implicit iterative methodology and the Lyapunov-Razumikhin theorem are used. The stability analysis for such equations is also applied. One numerical example is provided to illustrate the effectiveness of the theoretical results obtained.
Abstract : In this paper, we study the Birkhoff's ergodic theorem on geodesic metric spaces, especially on Hadamard spaces, using the notion of weighted inductive means. Also, we study a deterministic weighted sequence for the weighted Birkhoff's ergodic theorem in Hadamard spaces.
Abstract : Let $\mathcal{M}_{\alpha}$ be a bilinear fractional maximal operator and $BM_{\alpha}$ be a fractional maximal operator associated with the bilinear Hilbert transform. In this paper, the compactness on weighted Lebesgue spaces are considered for commutators of bilinear fractional maximal operators; these commutators include the fractional maximal linear commutators $\mathcal{M}_{\alpha,b}^{j}$ and $BM_{\alpha, b}^{j} $ $(j=1,2)$, the fractional maximal iterated commutator $\mathcal{M}_{\alpha,\vec{b}}$, and $BM_{\alpha, \vec{b}}$, where $b\in{\rm BMO}(\mathbb{R}^{d})$ and $\vec{b}=(b_{1},b_{2})\in{\rm BMO}(\mathbb{R}^{d})\times {\rm BMO}(\mathbb{R}^{d})$. In particular, we improve the well-known results to a larger scale for $1/2
Abstract : Let $G$ be a non-discrete locally compact abelian group, and $\mu$ be a transformable and translation bounded Radon measure on $G$. In this paper, we construct a Segal algebra $S_{\mu}(G)$ in $L^1(G)$ such that the generalized Poisson summation formula for $\mu$ holds for all $f\in S_{\mu}(G)$, for all $x\in G$. For the definitions of transformable and translation bounded Radon measures and the generalized Poisson summation formula, we refer to L. Argabright and J. Gil de Lamadrid's monograph in 1974.
Abstract : We consider the set of points with historic behavior (which is also called the irregular set) for continuous flows and suspension flows. In this paper under the hypothesis that $(X_t)_t$ is a continuous flow on a $d$-dimensional Riemaniann closed manifold $M$ $(d \geq 2)$ with gluing orbit property, we prove that the set of points with historic behavior in a compact and invariant subset $\Delta$ of $M$ is either empty or is a Baire residual subset on $\Delta$. We also prove that the set of points with historic behavior of a suspension flows over a homeomorphism satisfyng the gluing orbit property is either empty or Baire residual and carries full topological entropy.
Abstract : It is known that the maximal invariant set of a continuous iterative dynamical system in a compact Hausdorff space is equal to the intersection of its forward image sets, which we will call the {\it first minimal image set}. In this article, we investigate the corresponding relation for a class of discontinuous self maps that are on the verge of continuity, or {\it topologically almost continuous endomorphisms}. We prove that the iterative dynamics of a topologically almost continuous endomorphisms yields a chain of minimal image sets that attains a unique transfinite {\it length}, which we call the {\it maximal invariance order}, as it stabilizes itself at the maximal invariant set. We prove the converse, too. Given ordinal number $\xi$, there exists a topologically almost continuous endomorphism $f$ on a compact Hausdorff space $X$ with the maximal invariance order $\xi$. We also discuss some further results regarding the maximal invariance order as more layers of topological restrictions are added.
Abstract : Let $D$ be an integral domain with quotient field $K$, $Pic(D)$ be the ideal class group of $D$, and $X$ be an indeterminate. A polynomial overring of $D$ means a subring of $K[X]$ containing $D[X]$. In this paper, we study almost Dedekind domains which are polynomial overrings of a principal ideal domain $D$, defined by the intersection of $K[X]$ and rank-one discrete valuation rings with quotient field $K(X)$, and their ideal class groups. Next, let $\mathbb{Z}$ be the ring of integers, $\mathbb{Q}$ be the field of rational numbers, and $\mathfrak{G}_f$ be the set of finitely generated abelian groups (up to isomorphism). As an application, among other things, we show that there exists an overring $R$ of $\mathbb{Z}[X]$ such that (i) $R$ is a Bezout domain, (ii) $R \cap \mathbb{Q}[X]$ is an almost Dedekind domain, (iii) $Pic(R \cap \mathbb{Q}[X]) = \bigoplus_{G \in \mathfrak{G}_f}G$, (iv) for each $G \in \mathfrak{G}_f$, there is a multiplicative subset $S$ of $\mathbb{Z}$ such that $R_S \cap \mathbb{Q}[X]$ is a Dedekind domain with $Pic(R_S \cap \mathbb{Q}[X]) = G$, and (v) every invertible integral ideal $I$ of $R \cap \mathbb{Q}[X]$ can be written uniquely as $I = X^nQ_1^{e_1} \cdots Q_k^{e_k}$ for some integer $n \geq 0$, maximal ideals $Q_i$ of $R \cap \mathbb{Q}[X]$, and integers $e_i \neq 0$. We also completely characterize the almost Dedekind polynomial overrings of $\mathbb{Z}$ containing Int$(\mathbb{Z})$.
Abstract : Let $\mathcal{A}$ be the collection of analytic functions $f$ defined in $\mathbb{D}:=\left\{\xi\in\mathbb{C}:|\xi|
Nguyen Thi Anh Hang, Michael Hoff, Hoang Le Truong
J. Korean Math. Soc. 2022; 59(1): 87-103
https://doi.org/10.4134/JKMS.j210140
Jung Hee Cheon, Yongha Son, Donggeon Yhee
J. Korean Math. Soc. 2022; 59(1): 35-51
https://doi.org/10.4134/JKMS.j200650
Yuhui Liu
J. Korean Math. Soc. 2022; 59(3): 439-448
https://doi.org/10.4134/JKMS.j190679
Jung Pil Park, Yong-Su Shin
J. Korean Math. Soc. 2022; 59(1): 71-85
https://doi.org/10.4134/JKMS.j200690
Jung Pil Park, Yong-Su Shin
J. Korean Math. Soc. 2022; 59(1): 71-85
https://doi.org/10.4134/JKMS.j200690
Hien Minh Huynh
J. Korean Math. Soc. 2022; 59(1): 151-170
https://doi.org/10.4134/JKMS.j210258
Nguyen Thi Anh Hang, Michael Hoff, Hoang Le Truong
J. Korean Math. Soc. 2022; 59(1): 87-103
https://doi.org/10.4134/JKMS.j210140
Jundong Zhou
J. Korean Math. Soc. 2022; 59(1): 53-69
https://doi.org/10.4134/JKMS.j200665
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