Rank one HCIZ at high temperature: interpolating between classical and free convolutions

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretation in terms of another associated family of distribution indexed by c, called the Markov-Krein transform: the c-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.

A Computational Model of Maxwell’s Distribution for Undergraduates

In this paper we employ a numerical approach to perform simulations of Maxwell distribution for several dimensionalities, based on the Central Limit Theorem. We show that by increasing the number of molecules of the gas N, the simulated distributions tend toward the respective theoretical distributions. Also, we observed that by increasing the model temperature n, the distribution shifted toward higher speeds, in agreement with theoretical results. The numerical simulations provide a physical definition of the concept of temperature. The codes used to perform the simulations are quite easy to construct and implement, while the results strikingly satisfy theoretical expectations. Furthermore, the actual approach makes it possible to skip the mathematical details and explain the distribution by just following the algorithm of simulations. We recommend such approach as a demonstrative tool that can be shown in a lecture class thus enriching the teaching quality and improving students’ understanding.

Calculate central limit theorem for the number of empty cells after allocation of particles

This work belongs to the field of limit theorems for separable statistics. In particular, this paper considers the number of empty cells after placing particles in a finite number of cells, where each particle is placed in a polynomial scheme. The statistics under consideration belong to the class of separable statistics, which were previously considered in (Mirakhmedov: 1985), where necessary statements for the layout of particles in a countable number of cells were proved. The same scheme was considered in (Asimov: 1982), in which the conditions for the asymptotic normality of random variables were established. In this paper, the asymptotic normality of the statistics in question is proved and an estimate of the remainder term in the central limit theorem is obtained. In summary, the demand for separable statistics systems is growing day by day to address large-scale databases or to facilitate user access to data management. Because such systems are not only used for data entry and storage, they also describe their structure: file collection supports logical consistency; provides data processing language; restores data after various interruptions; database management systems allow multiple users.