Abstract:
The theory of Bogoliubov is generalized for the case of a weakly-interacting Bose-gas in harmonic trap. A set of nonlinear matrix equations is obtained to make the diagonalization of Hamiltonian possible. Its perturbative solution is used for the calculation of the energy and the condensate fraction of the model system to show the applicability of the method.

Abstract:
The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is considered providing the model of harmonically trapped Bose-gas. The results are generalized for the system with power energy spectrum.

Abstract:
A two-parametric fractional statistics is proposed, which can be used to model a weakly-interacting Bose-system. It is shown that the parameters of the introduced weakly nonadditive Polychronakos statistics can be linked to effects of interactions as well as finite-size corrections. The calculations of the specific heat and condensate fraction of the model system corresponding to harmonically trapped Rb-87 atoms are made. The behavior of the specific heat of three-dimensional isotropic harmonic oscillators with respect to the values of the statistics parameters is studied in the temperature domain including the BEC-like phase transition point.

Abstract:
An approach is proposed to analyze an interacting bosonic system using two-time temperature Green's functions on the collective variables. Two systems are studied: liquid helium-4 and the Yukawa Bose-liquid being a model of the nuclear matter. The suggested decoupling in the equations of motion for Green's functions yields a good description of the elementary excitation spectrum in the long-wavelength limit.

Abstract:
A detailed bibliography related to physics at the University of Lviv (Leopolis, Lemberg, Lw\'ow) in 18th-19th centuries is presented. Over ninety works of various types are listed with a large share being illustrated by title or starting pages. Brief biographical accounts of the authors are given to put their works in the context of the University history.

Abstract:
In the paper, two-parametric models of fractional statistics are proposed in order to determine the functional form of the distribution function of free anyons. From the expressions of the second and third virial coefficients, an approximate correspondence is shown to hold for three models, namely, the nonextensive Polychronakos statistics and both the incomplete and the nonextensive modifications of the Haldane--Wu statistics. The difference occurs only in the fourth virial coefficient leading to a small correction in the equation of state. For the two generalizations of the Haldane--Wu statistics, the solutions for the statistics parameters $g,q$ exist in the whole domain of the anyonic parameter $\alpha\in[0;1]$, unlike the nonextensive Polychronakos statistics. It is suggested that the search for the expression of the anyonic distribution function should be made within some modifications of the Haldane--Wu statistics.

Abstract:
In the paper, the quantum-statistical approach is used to estimate the number of restricted plane partitions of an integer $n$ with the number of parts not exceeding some finite $N$. The analogy between this number-theoretical problem and the enumeration of microstates of the ideal two-dimensional Bose-gas is used. The conjectured expression for the number of restricted plane partitions shows a good agreement between calculated and exact values for $n=10\div20$.

Abstract:
A new set of parameters to describe the word frequency behavior of texts is proposed. The analogy between the word frequency distribution and the Bose-distribution is suggested and the notion of "temperature" is introduced for this case. The calculations are made for English, Ukrainian, and the Guinean Maninka languages. The correlation between in-deep language structure (the level of analyticity) and the defined parameters is shown to exist.

Abstract:
In the paper, a complex statistical characteristics of a Ukrainian novel is given for the first time. The distribution of word-forms with respect to their size is studied. The linguistic laws by Zipf-Mandelbrot and Altmann-Menzerath are analyzed.