Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2022-05-01
On weighted compactness of commutators of bilinear fractional maximal operator
Qianjun He, Juan ZhangAbstract : 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
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2022-05-01
The ideal class group of polynomial overrings of the ring of integers
Gyu Whan ChangAbstract : 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$(\mathbb{Z})$.
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2022-05-01
Stabilized-penalized collocated finite volume scheme for incompressible biofluid flows
Nasserdine Kechkar, Mohammed LouaarAbstract : 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.
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2022-03-01
Cohomology of torsion and completion of $N$-complexes
Pengju Ma, Xiaoyan YangAbstract : We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes using the $\check{\mathrm{C}}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the categories of cohomologically $\mathfrak{a}$-torsion $N$-complexes and cohomologically $\mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative Noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $\otimes$ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.
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2022-03-01
Green's additive complement problem for $k$-th powers
Yuchen Ding, Li-Yuan WangAbstract : 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.
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2022-03-01
A deep learning algorithm for optimal investment strategies under Merton's framework
Daeyung Gim, Hyungbin ParkAbstract : This paper treats Merton's classical portfolio optimization problem for a market participant who invests in safe assets and risky assets to maximize the expected utility. When the state process is a $d$-dimensional Markov diffusion, this problem is transformed into a problem of solving a Hamilton--Jacobi--Bellman (HJB) equation. The main purpose of this paper is to solve this HJB equation by a deep learning algorithm: the deep Galerkin method, first suggested by J. Sirignano and K. Spiliopoulos. We then apply the algorithm to get the solution to the HJB equation and compare with the result from the finite difference method.
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2022-05-01
Boundedness of Calder\'{o}n-Zygmund operators on inhomogeneous product Lipschitz spaces
Shaoyong He, Taotao ZhengAbstract : In this paper, we study the boundedness of a class of inhomogeneous Journ\'{e}'s product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journ\'{e}'s product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.
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2022-03-01
Constructions of Segal algebras in $L^1(G)$ of LCA groups $G$ in which a generalized Poisson summation formula holds
Jyunji Inoue, Sin-Ei TakahasiAbstract : 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.
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2022-03-01
Historic behavior for flows with the gluing orbit property
Heides Lima de SantanaAbstract : 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.
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2022-03-01
Moment estimate and existence for the solution of neutral stochastic functional differential equation
Huabin Chen, Qunjia WanAbstract : 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.
Most Read
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Hardy type estimates for Riesz transforms associated with Schr\"{o}dinger operators on the Heisenberg group
Chunfang Gao
J. Korean Math. Soc. 2022; 59(2): 235-254
https://doi.org/10.4134/JKMS.j200484 -
Margin-based generalization for classifications with input noise
Hi Jun Choe, Hayeong Koh, Jimin Lee
J. Korean Math. Soc. 2022; 59(2): 217-233
https://doi.org/10.4134/JKMS.j200406 -
The exceptional set of one prime square and five prime cubes
Yuhui Liu
J. Korean Math. Soc. 2022; 59(3): 439-448
https://doi.org/10.4134/JKMS.j190679 -
Gorenstein sequences of high socle degrees
Jung Pil Park, Yong-Su Shin
J. Korean Math. Soc. 2022; 59(1): 71-85
https://doi.org/10.4134/JKMS.j200690
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On cylindrical smooth rational Fano fourfolds
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 -
Katok-Hasselblatt-kinematic expansive flows
Hien Minh Huynh
J. Korean Math. Soc. 2022; 59(1): 151-170
https://doi.org/10.4134/JKMS.j210258 -
Gorenstein sequences of high socle degrees
Jung Pil Park, Yong-Su Shin
J. Korean Math. Soc. 2022; 59(1): 71-85
https://doi.org/10.4134/JKMS.j200690 -
The exceptional set of one prime square and five prime cubes
Yuhui Liu
J. Korean Math. Soc. 2022; 59(3): 439-448
https://doi.org/10.4134/JKMS.j190679
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