Abstract:
Computational modeling offers an opportunity for a better understanding and investigation of thermal transfer mechanisms. It can be used for the optimization of the electron beam melting process and for obtaining new materials with improved characteristics that have many applications in the power industry, medicine, instrument engineering, electronics, etc. A time-dependent 3D axis-symmetrical heat model for simulation of thermal transfer in metal ingots solidified in a water-cooled crucible at electron beam melting and refining (EBMR) is developed. The model predicts the change in the temperature field in the casting ingot during the interaction of the beam with the material. A modified Pismen-Rekford numerical scheme to discretize the analytical model is developed. These equation systems, describing the thermal processes and main characteristics of the developed numerical method, are presented. In order to optimize the technological regimes, different criteria for better refinement and obtaining dendrite crystal structures are proposed. Analytical problems of mathematical optimization are formulated, discretized and heuristically solved by cluster methods. Using important for the practice simulation results, suggestions can be made for EBMR technology optimization. The proposed tool is important and useful for studying, control, optimization of EBMR process parameters and improving of the quality of the newly produced materials.

Abstract:
In the paper we study weak and strong invariance of differential inclusions with fixed time impulses and with state constraints.We also investigate some properties of the solution set of impulsive system without state constraints. When the right-hand side is one sided Lipschitz we prove also the relaxation theorem and study the funnel equation of the reachable set.

Abstract:
In the paper, we study weak invariance of differential inclusions with non-fixed time impulses under compactness type assumptions. When the right-hand side is one sided Lipschitz an extension of the well known relaxation theorem is proved. In this case also necessary and sufficient condition for strong invariance of upper semi continuous systems are obtained. Some properties of the solution set of impulsive system (without constrains) in appropriate topology are investigated.

Abstract:
The notions of relaxed submonotone and relaxedmonotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. Forexample, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.

Abstract:
In the paper we study the continuity properties of the solution set of upper semicontinuous differential inclusions in both regularly and singularly perturbed case. Using a kind of dissipative type of conditions introduced in [1] we obtain lower semicontinuous dependence of the solution sets. Moreover new existence result for lower semicontinuous differential inclusions is proved.

Abstract:
We consider the symmetric Poissonian two-armed bandit problem. For the case of switching arms, only one of which creates reward, we solve explicitly the Bellman equation for a β-discounted reward and prove that a myopic policy is optimal.

Abstract:
The value function in the optimal detection problem for jump-times of a Poisson process satisfies a special system of functional-differential equations. In this paper, we investigate the system and prove the existence and uniqueness of its solution.

Abstract:
We extend some Hardy-type inequalities with general kernels to arbitrary time scales using multivariable convex functions. Some classical and new inequalities are deduced seeking applications. 1. Introduction The significant Hardy inequality is published in [1] (1952) (both in the continuous and discrete settings). More general Hardy integral inequalities have been studied in continuous cases. We notice only [2–6] and the references therein. Recently, this inequality is studied in discrete case, and some variants of it are proved in case of time scales [7–9]. In [10], the authors study Hardy-type inequalities using convex functions of one variable with general kernels to arbitrary time scales. The aim of this paper is to provide Hardy-type inequalities using multivariable convex function with general kernels to arbitrary time scales. Notice that time scales include continuous and discrete time cases under unified approach. Firstly, we recall necessary preliminary facts needed afterward. The main results are given in Section 3. Section 4 is devoted to some inequalities with certain kernels. In the last section, we discuss some particular cases of Hardy-type inequalities. 2. Preliminaries First, we recall the basic concepts used in the paper and refer the interested reader to [11] for the theory of time scales. A time scale is any nonempty closed subset of the real line . On nondensity points we define forward, respectively, backward jump operators as The point is said to be right-scattered if and left-scattered if , respectively. Clearly, is right-dense if and left dense if , respectively. Let ; we define -dimensional time scale by the Cartesian product of given time scales , , as Evidently, is a complete metric space with distance as follows: Now we are going to describe the construction of Lebesgue measure in . We refer to [12–14] for the theory of measure spaces and measurable functions on time scales. Let be the family of all -dimensional time scale intervals in ; that is, with , , and for all . Let be the set function that assigns to each -dimensional time scale interval its volume as follows: Let . If there exists finite or countable system of pairwise disjoint -dimensional time scale intervals with , then the outer measure of is defined by If there is no such covering of , then . A subset of is said to be measurable (or -measurable) if holds for all , where . The family of all -measurable subsets of is a -algebra generated by . The restriction of to , which we denote by , is a -additive measure on . Clearly, and for each . The measure (called the

Microcystins
cause acute hepatotoxicity and chronic liver tumor promotion. This study presents
the results of HPLC DAD analyses and their LC-MS confirmation of samples from five
Bulgarian water bodies (reservoirs Stoudena, Pchelina, Bistritsa and lakes Dourankoulak,
Vaya). The total concentration of microcystins in water samples ranged from 0.1
to 26.5 μg/l. The amount of microcystins
in the biomasses ranged from 11.4 to 49.6 μg/g (d.w.). The high percent of positive
samples in which the most toxic microcystin-LR is recorded, can serve as a strong
alarm for the necessity of a serious study and relevant discussion of the problem
with responsible authorities at national level.

Abstract:
Differential inclusions with compact, upper semi-continuous, not necessarily convex right-hand sides in R^n are studied. Under a weakened monotonicity-type condition the existence of solutions is proved.