On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process ${(X_t)}_{t\ge 0}$,

$\underset{t\to 0}{\lim }?\frac{1}{t}({\mathbb{E}}^{x}f(X_t)?f(x)),x\in {\mathbb{R}}^d,$

for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x\in {\mathbb{R}}^d$ if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of ${(X_t)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.     

New classes of second order difference equations solvable by factorization method

Under some assumptions we find a general solution of the factorization problem for a family of second order difference equations.     

On the normalized Laplacian of Möbius phenylene chain and its applications

Let denote a molecular graph of linear [n] phenylene with n hexagons and n squares, and let the Möbius phenylene chain be the graph obtained from the by identifying the opposite lateral edges in reversed way. Utilizing the decomposition theorem of the normalized Laplacian characteristic polynomial, we study the normalized Laplacian spectrum of , which consists of the eigenvalues of two symmetric matrices ℒ _R and ℒ _Q of order 3n. By investigating the relationship between the roots and coefficients of the characteristic polynomials of the two matrices above, we obtain an explicit closed-form formula of the multiplicative degree-Kirchhoff index as well as the number of spanning trees of . Furthermore, we determine the limited value for the quotient of the multiplicative degree-Kirchhoff index and the Gutman index of .     

Automatic hierarchy classification in venation networks using directional morphological filtering for hierarchical structure traits extraction

The extraction of vein traits from venation networks is of great significance to the development of a variety of research fields, such as evolutionary biology. However, traditional studies normally target to the extraction of reticulate structure traits (ReSTs), which is not sufficient enough to distinguish the difference between vein orders. For hierarchical structure traits (HiSTs), only a few tools have made attempts with human assistance, and obviously are not practical for large-scale traits extraction. Thus, there is a necessity to develop the method of automated vein hierarchy classification, raising a new challenge yet to be addressed. We propose a novel vein hierarchy classification method based on directional morphological filtering to automatically classify vein orders. Different from traditional methods, our method classify vein orders from highly dense venation networks for the extraction of traits with ecological significance. To the best of our knowledge, this is the first attempt to automatically classify vein hierarchy. To evaluate the performance of our method, we prepare a soybean transmission image dataset (STID) composed of 1200 soybean leaf images and the vein orders of these leaves are manually coarsely annotated by experts as ground truth. We apply our method to classify vein orders of each leaf in the dataset. Compared with ground truth, the proposed method achieves great performance, while the average deviation on major vein is less than 5 pixels and the average completeness on second-order veins reaches 54.28%.     

Some extensions of the operator entropy type inequalities

In this paper, we define the generalised relative operator entropy and investigate some of its properties such as subadditivity and homogeneity. As application of our result, we obtain the information inequality. In continuation, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond–Pe?ari? method.     

Optimal recovery of integral operators and its applications

In this paper we present the solution to a problem of recovering a rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE’s.     

基于迈克尔逊干涉原理测量金属热膨胀系数仪器的改进

在利用迈克尔逊干涉原理测量金属热膨胀系数的基础上,运用CCD成像系统取代原有观察屏,更利于观察并记录数据。