Product Type Operators Involving Radial Derivative Operator Acting between Some Analytic Function Spaces

Let N denote the set of all positive integers and N0=N∪{0}. For m∈N, let Bm={z∈Cm:|z|<1} be the open unit ball in the m−dimensional Euclidean space Cm. Let H(Bm) be the space of all analytic functions on Bm. For an analytic self map ξ=(ξ1,ξ2,…,ξm) on Bm and ϕ1,ϕ2,ϕ3∈H(Bm), we have a product type operator Tϕ1,ϕ2,ϕ3,ξ which is basically a combination of three other operators namely composition operator Cξ, multiplication operator Mϕ and radial derivative operator R. We study the boundedness and compactness of this operator mapping from weighted Bergman–Orlicz space AσΨ into weighted type spaces Hω∞ and Hω,0∞.

The second-order composed radial derivatives of perturbation mappings of parametric set-valued optimization problems

In the paper, we study the generalized differentiability in set-valued optimization, namely stydying the second-order composed radial derivative of a given set-valued mapping. Inspired by the adjacent cone and the higher-order radial con in Anh NLH et al. (2011), we introduce the second-order composed radial derivative. Then, its basic properties are investigated and relationships between the second-order compsoed radial derivative of a given set-valued mapping and that of its profile are obtained. Finally, applications of this derivative to sensitivity analysis are studied. In detail, we work on a parametrized set-valued optimization problem concerning Pareto solutions. Based on the above-mentioned results, we find out sensitivity analysis for Pareto solution mapping of the problem. More precisely, we establish the second-order composed radial derivative for the perturbation mapping (here, the perturbation means the Pareto solution mapping concerning some parameter). Some examples are given to illustrate our results. The obtained results are new and improve the existing ones in the literature.

IDENTIFICATION OF BURIED ARCHEOLOGICAL OBJECTS WITH RADIAL DERIVATIVES OF MICRO GRAVITY DATA

The development of recent gravimetric technology allows us to measure gravity anomalies with accuracy of micro Gal. Micro gravity is able to detect very small gravity anomalies such as anomaly due to buried archeological objects below the earth surface. Radial Derivatives of gravity data is used to sharpen anomaly due to lateral changes of density contrast. Horizontal derivatives carried out by previous researchers have some weaknesses, i.e. the loss of derivative values in certain directions and inconsistence values at the source boundary of the same anomaly edge. To solve the horizontal derivative problem, a radial derivative is made. Radial derivative is derivative of gravity anomaly over horizontal distance in the radial direction from a certain point which is considered as the center of anomaly. There are two kind of radial derivative i.e. First Radial Derivative (FRD) and Second Radial Derivative (SRD). Blade Pattern is another way to enrich the ability of SRD to detect boundary of anomaly source. Synthetic gravity data of buried archeological object was made by counting the response of forward modelling. All of programs and calculation of the models in this research is performed based on Matlab® program. The results of the tests on the synthetic data of the model show that the radial derivative is able to detect the boundaries in buried temples due to density contrast. The advantage of radial derivatives which is a horizontal derivative in the direction of radial compared to ordinary horizontal derivatives is the ability to detect vertical boundaries of various anomaly due to horizontal layers and capable of showing density contrast in almost all directions.