Application of mathematical modelling methods for acoustic images reconstruction

The article considers the reconstruction of images by Synthetic Aperture Focusing Technique (SAFT). The work compares additive and multiplicative methods for processing signals received from antenna array. We have proven that the multiplicative method gives a better resolution. The study includes the estimation of beam trajectories for antenna arrays using analytical and numerical methods. We have shown that the analytical estimation method allows decreasing the image reconstruction time in case of linear antenna array implementation

Finite Element Modelling of Rubber-Metal Vibration Isolators with Holes for the Vibration Protection System of Buildings

The article presents a calculation of rubber-metal vibration isolators with five holes of different diameters using a software package that implements the finite element method. A comparative analysis of the Eigen frequencies of rubber-metal vibration isolators with five holes (one in the center, 4 symmetrically at the corners) and without holes is presented. Finite element models of a rubber-metal vibration isolator with and without holes are modeled, and their characteristics are analyzed. The results show that vibration isolators with several symmetrically located holes have several advantages in a number of parameters to vibration isolators without holes, and, therefore, can be used for vibration isolation of buildings, especially in the case of delayed installation of vibration protection

Local Variational Principle

A generalization of the Gibbs-Bogoliubov-Feynman inequality for spinless particles is proven and then illustrated for the simple model of a symmetric double-well quartic potential. The method gives a pointwise lower bound for the finite-temperature density matrix and it can be systematically improved by the Trotter composition rule. It is also shown to produce groundstate energies better than the ones given by the Rayleigh-Ritz principle as applied to the groundstate eigenfunctions of the reference potentials. Based on this observation, it is argued that the Local Variational Principle performs better than the equivalent methods based on the centroid path idea and on the Gibbs-Bogoliubov-Feynman variational principle, especially in the range of low temperatures.Comment: 15 pages, 5 figures, one more section adde

Pseudogap phase formation in the crossover from Bose-Einstein condensation to BCS superconductivity

A phase diagram for a 2D metal with variable carrier density has been derived. It consists of a normal phase, where the order parameter is absent; a so-called ``abnormal normal'' phase where this parameter is also absent but the mean number of composite bosons (bound pairs) exceeds the mean number of free fermions; a pseudogap phase where the absolute value of the order parameter gradually increases but its phase is a random value, and finally a superconducting (here Berezinskii-Kosterlitz-Thouless) phase. The characteristic transition temperatures between these phases are found. The chemical potential and paramagnetic susceptibility behavior as functions of the fermion density and the temperature are also studied. An attempt is made to qualitatively compare the resulting phase diagram with the features of underdoped high-$T_{c}$ superconducting compounds above their critical temperature.Comment: 26 pages, revtex, 5 EMTeX figures; more discussion and references added; to be published in JET

Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts

We show that G\"odel's negative results concerning arithmetic, which date back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites paradox") pose the questions of the use of fuzzy sets and of the effect of a measuring device on the experiment. The consideration of these facts led, in thermodynamics, to a new one-parameter family of ideal gases. In turn, this leads to a new approach to probability theory (including the new notion of independent events). As applied to economics, this gives the correction, based on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are added. arXiv admin note: significant text overlap with arXiv:1111.610

Mathematical Conception of "Phenomenological" Equilibrium Thermodynamics

In the paper, the principal aspects of the mathematical theory of equilibrium thermodynamics are distinguished. It is proved that the points of degeneration of a Bose gas of fractal dimension in the momentum space coincide with critical points or real gases, whereas the jumps of critical indices and the Maxwell rule are related to the tunnel generalization of thermodynamics. Semiclassical methods are considered for the tunnel generalization of thermodynamics and also for the second and ultrasecond quantization (operators of creation and annihilation of pairs). To every pure gas there corresponds a new critical point of the limit negative pressure below which the liquid passes to a dispersed state (a foam). Relations for critical points of a homogeneous mixture of pure gases are given in dependence on the concentration of gases.Comment: 37 pages, 9 figure, more precise explanations, more references. arXiv admin note: substantial text overlap with arXiv:1202.525