Abstract : Let $\mathbb F_q$ be a finite field with $q$ elements. A function $f: \mathbb F_q^d\times \mathbb F_q^d \to \mathbb F_q$ is called a Mattila--Sj\"{o}lin type function of index $\gamma \in \mathbb R$ if $\gamma$ is the smallest real number such that whenever $|E|\geq Cq^{\gamma}$ for a sufficiently large constant $C$, the set $f(E,E):=\{f(x,y): x, y\in E\}$ is equal to $\mathbb F_q$. In this article, we construct an example of a Mattila--Sj\"{o}lin type function $f$ and provide its index, generalizing the result of Cheong, Koh, Pham and Shen [1].
Abstract : In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of $\mathcal{U}$ admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.
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.
Abstract : In this paper, we introduce and study preresolving subcategories in an extriangulated category~$\mathscr{C}$. Let $\mathcal{Y}$ be a $\mathcal{Z}$-preresolving subcategory of $\mathscr{C}$ admitting a $\mathcal{Z}$-proper $\xi$-generator $\mathcal{X}$. We give the characterization of $\mathcal{Z}\text{-}{\rm proper}~\mathcal{Y}$-resolution dimension of an object in $\mathscr{C}$. Next, for an object $A$ in $\mathscr{C}$, if the $\mathcal{Z}\text{-}{\rm proper}~\mathcal{Y}$-resolution~dimension of $A$ is at most $n$, then all ``$n$-$\mathcal{X}$-syzygies" of $A$ are objects in $\mathcal{Y}$. Finally, we prove that $A$ has a $\mathcal{Z}$-proper $\mathcal{X}$-resolution if and only if $A$ has a $\mathcal{Z}$-proper $\mathcal{Y}$-resolution. As an application, we introduce $(\mathcal{X},\mathcal{Z})$-Gorenstein~subcategory $\mathcal{GX}_{\mathcal{Z}}(\xi)$ of $\mathscr{C}$ and prove that $\mathcal{GX}_{\mathcal{Z}}(\xi)$ is both $\mathcal{Z}$-resolving subcategory and $\mathcal{Z}$-coresolving subcategory of $\mathscr{C}$.
Abstract : The scaled inverse of a nonzero element $a(x)\in \mathbb{Z}[x]/f(x)$, where $f(x)$ is an irreducible polynomial over $\mathbb{Z}$, is the element $b(x)\in \mathbb{Z}[x]/f(x)$ such that $a(x)b(x)=c \pmod{f(x)}$ for the smallest possible positive integer scale $c$. In this paper, we investigate the scaled inverse of $(x^i-x^j)$ modulo cyclotomic polynomial of the form $\Phi_{p^s}(x)$ or $\Phi_{p^s q^t}(x)$, where $p, q$ are primes with $p<q$ and $s, t$ are positive integers. Our main results are that the coefficient size of the scaled inverse of $(x^i-x^j)$ is bounded by $p-1$ with the scale $p$ modulo $\Phi_{p^s}(x)$, and is bounded by $q-1$ with the scale not greater than $q$ modulo $\Phi_{p^s q^t}(x)$. Previously, the analogous result on cyclotomic polynomials of the form $\Phi_{2^n}(x)$ gave rise to many lattice-based cryptosystems, especially, zero-knowledge proofs. Our result provides more flexible choice of cyclotomic polynomials in such cryptosystems. Along the way of proving the theorems, we also prove several properties of $\{x^k\}_{k\in\mathbb{Z}}$ in $\mathbb{Z}[x]/\Phi_{pq}(x)$ which might be of independent interest.
Abstract : In this paper, we introduce the concepts of (quasi-)weakly almost periodic point and minimal center of attraction for $\mathbb{Z}^d_+$-actions, explore the connections of levels of the topological structure the orbits of (quasi-)weakly almost periodic points and discuss the relations between (quasi-)weakly almost periodic point and minimal center of attraction. Especially, we investigate the chaotic dynamics near or inside the minimal center of attraction of a point in the cases of $S$-generic setting and non $S$-generic setting, respectively. Actually, we show that weakly almost periodic points and quasi-weakly almost periodic points have distinct topological structures of the orbits and we prove that if the minimal center of attraction of a point is non $S$-generic, then there exist certain Li-Yorke chaotic properties inside the involved minimal center of attraction and sensitivity near the involved minimal center of attraction; if the minimal center of attraction of a point is $S$-generic, then there exist stronger Li-Yorke chaotic (Auslander-Yorke chaotic) dynamics and sensitivity ($\aleph_0$-sensitivity) in the involved minimal center of attraction.
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 : 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 : We introduce the notion of quasi-roots and study their uniqueness in right-angled Artin groups.
Abstract : In this paper, the weighted $L^{p}$ boundedness of multilinear commutators and multilinear iterated commutators generated by the multilinear singular integral operators with generalized kernels and BMO functions is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels under certain conditions.
Soyoon Bak, Philsu Kim, Sangbeom Park
J. Korean Math. Soc. 2022; 59(5): 891-909
https://doi.org/10.4134/JKMS.j210701
Yoon Kyung Park
J. Korean Math. Soc. 2023; 60(2): 395-406
https://doi.org/10.4134/JKMS.j220180
Jiling Cao, Beidi Peng, Wenjun Zhang
J. Korean Math. Soc. 2022; 59(6): 1153-1170
https://doi.org/10.4134/JKMS.j210728
Allami Benyaiche, Ismail Khlifi
J. Korean Math. Soc. 2022; 59(6): 1139-1151
https://doi.org/10.4134/JKMS.j210669
Peter Jaehyun Cho , Gyeongwon Oh
J. Korean Math. Soc. 2023; 60(1): 167-193
https://doi.org/10.4134/JKMS.j220242
Byung Hee An, Youngjin Cho
J. Korean Math. Soc. 2023; 60(6): 1171-1213
https://doi.org/10.4134/JKMS.j220343
yezhou Li, heqing sun
J. Korean Math. Soc. 2023; 60(4): 859-876
https://doi.org/10.4134/JKMS.j220473
U\u{g}ur Sert
J. Korean Math. Soc. 2023; 60(3): 565-586
https://doi.org/10.4134/JKMS.j220129
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