Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors

Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we further introduce skew centrosymmetric tensors and centrosymmetric Cauchy tensors, and discuss properties of these three classes of structured tensors. Some sufficient and necessary conditions for a tensor to be centrosymmetric or skew centrosymmetric are given. We show that, a general tensor can always be expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric tensor. Some sufficient and necessary conditions for a Cauchy tensor to be centrosymmetric or skew centrosymmetric are also given. Spectral properties on H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and centrosymmetric Cauchy tensors are discussed. Some further questions on these tensors are raised

Electromigration dispersion in Capillary Electrophoresis

In a previous paper (S. Ghosal and Z. Chen Bull. Math. Biol. 2010, vol. 72, pg. 2047) it was shown that the evolution of the solute concentration in capillary electrophoresis is described by a nonlinear wave equation that reduced to Burger's equation if the nonlinearity was weak. It was assumed that only strong electrolytes (fully dissociated) were present. In the present paper it is shown that the same governing equation also describes the situation where the electrolytic buffer consists of a single weak acid (or base). A simple approximate formula is derived for the dimensionless peak variance which is shown to agree well with published experimental data.Comment: 10 pages, 2 figure

Parameterized Synthetic Image Data Set for Fisheye Lens

Based on different projection geometry, a fisheye image can be presented as a parameterized non-rectilinear image. Deep neural networks(DNN) is one of the solutions to extract parameters for fisheye image feature description. However, a large number of images are required for training a reasonable prediction model for DNN. In this paper, we propose to extend the scale of the training dataset using parameterized synthetic images. It effectively boosts the diversity of images and avoids the data scale limitation. To simulate different viewing angles and distances, we adopt controllable parameterized projection processes on transformation. The reliability of the proposed method is proved by testing images captured by our fisheye camera. The synthetic dataset is the first dataset that is able to extend to a big scale labeled fisheye image dataset. It is accessible via: http://www2.leuphana.de/misl/fisheye-data-set/.Comment: 2018 5th International Conference on Information Science and Control Engineerin

Global Heat Kernel Estimates for Fractional Laplacians in Unbounded Open Sets

In this paper, we derive global sharp heat kernel estimates for symmetric alpha-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C^{1,1} open sets in R^d: half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for p_D(t,x,y) for all t>0 and x, y\in D. Our approach is based on the idea that for x and y in $D$ far from the boundary and t sufficiently large, we can compare p_D(t,x,y) to the heat kernel in a well understood open set: either a half-space or R^d; while for the general case we can reduce them to the above case by pushing $x$ and $y$ inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets

On unique extension of time changed reflecting Brownian motions

Let $D$ be an unbounded domain in \RR^d with $d\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on $\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\overline D$. We further show that if there is some $r>0$ so that $D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.Comment: To appear in Ann. Inst. Henri Poincare Probab. Statis

Heat Kernels for Non-symmetric Non-local Operators

We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.Comment: Survey article. To appear as a chapter in "Recent Developments in the Nonlocal Theory" by De Gruyte