Abstract : A bounded linear operator $T$ on a separable infinite dimensional Banach space $X$ is called strongly hypercyclic if $$X\backslash\{0\}\subseteq \bigcup_{n=0}^{\infty}T^n(U)$$ for all nonempty open sets $U\subseteq X$. We show that if $T$ is strongly hypercyclic, then so are $T^n$ and $cT$ for every $n\ge 2$ and each unimodular complex number $c$. These results are similar to the well known Ansari and Le\'{o}n-M\"{u}ller theorems for hypercyclic operators. We give some results concerning multiplication operators and weighted composition operators. We also present a result about the invariant subset problem.
Abstract : In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels $h$ and sphere kernels $\Omega$ by assuming $h\in{\triangle}_{\gamma}(\mathbb{R}_+)$ and $\Omega\in\mathcal{WG}_\beta({\rm S}^{n-1})$ for some $\gamma>1$ and $\beta>1$. Here $\Omega\in\mathcal{WG}_\beta({\rm S}^{n-1})$ denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.
Abstract : In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of $Lu(\mathbf{x})=a_{ij}(\mathbf{x})D_{ij}u(\mathbf{x})+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ or $Lu(\mathbf{x})=D_{i}(a_{ij}(\mathbf{x})$ $D_{j}u(\mathbf{x}))+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term $f$ of the equation.
Abstract : In this paper, we main study the strong law of large numbers and complete convergence for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables, by using the Marcinkiewicz-Zygmund type moment inequality and Roenthal type moment inequality for AANA random variables. As an application, the complete consistency for the weighted linear estimator of nonparametric regression models based on AANA errors is obtained. Finally, some numerical simulations are carried out to verify the validity of our theoretical result.
Abstract : In \cite{S:3} the author finds a graded minimal free resolution of the $2$-nd order symbolic power of a star configuration in $\P^n$ of any codimension $r$. In this paper, we find that of any $m$-th order symbolic power of a star configuration in $\P^n$ of codimension $2$, which generalizes the result of Galetto, Geramita, Shin, and Van Tuyl in \cite[Theorem 5.3]{GGSV:1}. Furthermore, we extend it to the $m$-th order symbolic power of a star configuration in $\P^n$ of any codimension $r$ for $m=3,4$, which also generalizes the result of Biermann et al. in \cite[Corollaries 4.6 and 5.7]{BDGMNORS}. We also suggest how to find a graded minimal free resolution of the $m$-th order symbolic power of a star configuration in $\P^n$ of any codimension $r$ for $m\ge 5$.
Abstract : The spherical buildings associated with absolutely simple algebraic groups of relative rank~$2$ are all Moufang polygons. Tits polygons are a more general class of geometric structures that includes Moufang polygons as a special case. Dagger-sharp Tits $n$-gons exist only for $n=3$, $4$, $6$ and~$8$. Moufang octagons were classified by Tits. We show here that there are no dagger-sharp Tits octagons that are not Moufang. As part of the proof it is shown that the same conclusion holds for a certain class of dagger-sharp Tits quadrangles.
Abstract : In this paper, we consider the problem of finding the hypersurface $M^{n}$ in the Euclidean $\left( n+1\right)$-space $\mathbb{R}^{n+1}$ that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper halfspace $\mathbb{R}_{+}^{3}\left( \mathbf{u} \right) $ with lowest gravity center, for a fixed unit vector $\mathbf{u}\in \mathbb{R}^{3}$. We first state that a singular minimal cylinder $M^{n}$ in $\mathbb{R}^{n+1}$ is either a hyperplane or a $\alpha $-catenary cylinder. It is also shown that this result remains true when $M^{n}$ is a translation hypersurface and $\mathbf{u}$ is a horizantal vector. As a further application, we prove that a singular minimal translation graph in $\mathbb{R }^{3}$ of the form $z=f(x)+g(y+cx),$ $c\in \mathbb{R-\{}0\},$ with respect to a certain horizantal vector $\mathbf{u}$ is either a plane or a $\alpha $- catenary cylinder.
Abstract : A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. {\bf 48}, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an $(m+2)$-dimensional space and spherical curves in an $(m+1)$-dimensional space.
Abstract : In this article, we study the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model. This model describes the interaction of two complex scalar fields and one gauge field, and generalizes the classical Klein-Gordon equation coupled with the Maxwell electrodynamics. We prove that there exist infinitely many standing wave solutions for $p\in (2,6)$ which are radially symmetric. Here, $p$ comes from the exponent of the potential of scalar fields. We also prove the nonexistence of nontrivial solutions for the critical case $p=6$.
Abstract : Let $C[0,T]$ denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval $[0,T]$. Define $Z_{\vec e,n}:C[0,T]\to\mathbb R^{n+1}$ by \begin{align*} Z_{\vec e,n}(x)=\left(x(0),\int_0^Te_1(t)dx(t),\ldots,\int_0^Te_n(t)dx(t)\right), \end{align*} where $e_1, \ldots,e_n$ are of bounded variations on $[0,T]$. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on $C[0,T]$ with the conditioning function $Z_{\vec e,n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on $C[0,T]$ which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval $[0,T]$.
Gyu Whan Chang
J. Korean Math. Soc. 2022; 59(3): 571-594
https://doi.org/10.4134/JKMS.j210419
Nak Eun Cho, Anbhu Swaminathan, Lateef Ahmad Wani
J. Korean Math. Soc. 2022; 59(2): 353-365
https://doi.org/10.4134/JKMS.j210246
Shaoyong He, Taotao Zheng
J. Korean Math. Soc. 2022; 59(3): 469-494
https://doi.org/10.4134/JKMS.j210115
Neil Epstein, Jay Shapiro
J. Korean Math. Soc. 2021; 58(6): 1311-1325
https://doi.org/10.4134/JKMS.j200475
Shaoyong He, Taotao Zheng
J. Korean Math. Soc. 2022; 59(3): 469-494
https://doi.org/10.4134/JKMS.j210115
Gyu Whan Chang
J. Korean Math. Soc. 2022; 59(3): 571-594
https://doi.org/10.4134/JKMS.j210419
Neil Epstein, Jay Shapiro
J. Korean Math. Soc. 2021; 58(6): 1311-1325
https://doi.org/10.4134/JKMS.j200475
Shivani Dubey, Mukund Madhav Mishra, Ashutosh Pandey
J. Korean Math. Soc. 2022; 59(3): 635-648
https://doi.org/10.4134/JKMS.j210462
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