The Rank-Frequency Analysis for the Functional Style Corpora in the Ukrainian Language

We use the rank-frequency analysis for the estimation of Kernel Vocabulary size within specific corpora of Ukrainian. The extrapolation of high-rank behaviour is utilized for estimation of the total vocabulary size.Comment: 8 page

Phase transition in a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameter

For a system of 1D harmonic oscillators obeying Polychronakos statistics with a complex parameters the emergence of a phase transition is reported and temperature dependences of energy and heat capacity are studied in detail. Estimations towards a possibility to check the obtained jumps in the specific heat are made

The relation between fractional statistics and finite bosonic systems in one-dimensional case

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

Lviv period for Smoluchowski: Science, teaching, and beyond

A major part of Marian Smoluchowski's achievements in science corresponds to the period of his work at the University of Lviv. Since this part is well described in the literature, in the paper the emphasis is made on some less known activities of this outstanding scientist: his teaching, his organizational efforts, and even his hobbies. The list of publications corresponding to the Lviv period is given.iльшiсть наукових досягнень Марiана Смолуховського припадає на час його роботи у Львiвському унiверситетi. Оскiльки ця сторона дiяльностi вченого добре описана в лiтературi, то у статтi зроблено наголос на менш вiдомих напрямках роботи цього видатного вченого: викладання й орґанiзацiйнi заходи i навiть його хобi. Наведено також перелiк публiкацiй львiвського перiоду

Effective Hamiltonian and excitation spectrum of harmonically trapped bosons

An approach is proposed to obtain an effective Hamiltonian of a harmonically trapped Bose-system. Such a Hamiltonian is quadratic in the creation–annihilation operators and certain approximations allow to simplify higher (three and four operator) products to the required form. After the Hamiltonian diagonalization, the expression for the excitation spectrum is obtained containing in particular temperature-dependent corrections. Numerical calculations are made for a one-dimensional system. Some prospects towards the extension of the suggested approach to study binary bosonic mixtures are briefly discussed

Asymptotic formulas for integer partitions within the approach of microcanonical ensemble

The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary (one-dimensional) partitions, the correction to the leading asymptotic is obtained. The estimate for the number of two-dimensional (plane) partitions coincides with known asymptotic results.Розглянуто задачу про розбиття цiлих чисел у межах мiкроканонiчного пiдходу, який ґрунтується на аналогiї мiж цiєю задачею з теорiї чисел i обчисленням кiлькостi мiкростанiв багатобозонної системи. Для звичайних (одновимiрних) розбиттiв отримано поправку до головної асимптотики. Оцiнка кiлькостi двовимiрних (плоских) розбиттiв добре узгоджується з вiдомими асимптотичними результатами

Fractional statistics and finite bosonic system: A one-dimensional case

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.Comment: 10 page