### Article

## Modelling of Liquid Nuclei Generation for Field-Emission Silicon Nanocathode

This paper presents the results of mathematical modeling of heat transfer in the field emission process in a conic cathode of small dimensions with its possible melting considered. It is shown that the possibility of melting is determined by the cathode vertex angle. The melting is modeled in the framework of the phase field system using the proposed methods for the formation of the liquid phase zone.

A new mathematical model of heat transfer in silicon field emission pointed cathode of small dimensions is constructed which permits taking its partial melting into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type problems. The approach used by the authors is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the obtained mathematical model including its parallel implementation. The results of numerical simulation conclude the book.

The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

In the present work the results of different scenario of the cliff of Cape Canaille hypothetic collapse (South of France) are presented. Three scenarios were considered: falling of one block, falling of several blocks in one time and debris flow avalanche. The analysis of the entire scenario was done.

New quasi – 3D numerical model for thermal analysis of the BGA packages is presented. The general 3D heat transfer problem is correctly transformed to the set of 2D equations for temperature distributions in different layers of the package. The complexity and CPU time of the thermal analysis are many times reduced. The results of BGA package thermal modeling are presented.

The computational model of the temperature sensors integrated on the IC chip with power transistors is developed. The 2D/3D problem of sensor placement is mathematically described by the classic heat transfer equation coupled with the equation for current density distribution. It is shown that parasitic effects of sensor current displacement and thermo-emf generation resulting from a temperature gradients (Seebeck effect) must be taken into account. For this purpose the special differential equation is introduced. The examples of point- and strip-like temperature sensors modeling for power BJTs and ICs are demonstrated.

Phenomenological model of temperature field evolution and effect of the relaxation processes caused by thermally activated defects and inertia of the medium on the thermal conductivity of solids under the action of intense energy flows is presented

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

A form for an unbiased estimate of the coefficient of determination of a linear regression model is obtained. It is calculated by using a sample from a multivariate normal distribution. This estimate is proposed as an alternative criterion for a choice of regression factors.