This is the page of the United Seminar of the Department of Probability Theory of the Faculty of Mechanics and Mathematics of Moscow State University. The permanent website of the seminar is here. The seminar is a continuation of the research seminar of the Department of Probability Theory under the leadership of A.N. Kolmogorov and B.V. Gnedenko.

Upcoming reports

Past reports

December 11, 16:45 msk

Hanchao Wang, Shandong University, China

The Majorizing Measure Theorems for Log-concave-tailed Canonical Process

The Majorizing measure theorem for Gaussian processes is famous. This talk will introduce my work on majorizing measure theorem for log-concave tailed canonical processes. We fully characterize the expected suprema of log-concave-tailed canonical processes. In particular, we obtain a new majorizing measure theorem and its dual version.

The recording: YouTube

November 27, 16:45 msk

Elmira Kalimulina, Lomonosov MSU, Russia

Queueing Networks with Node Dynamics: A Survey of Models, New Results, and Open Problems

This talk addresses queueing networks with a large number of nodes, where node dynamics (e.g., failures and recoveries) lead to changes in network topology. Special attention is devoted to sparse networks, which most accurately represent real-world systems. The presentation will begin with a survey of the most prominent approaches in queueing theory, discussing their advantages and limitations. Subsequently, more general models overcoming the constraints of standard methods will be proposed, accompanied by an analysis of their properties. The talk will conclude with the results of simulation studies, a discussion of open problems, and an outline of key challenges relevant to practical applications.

The recording: YouTube

November 13, 16:45 msk

Stanislav Molchanov, UNC Charlotte, USA; Higher School of Economics, Russia

Spectral theory of the Schrödinger type operators on the Exner’s periodic quantum graphs

Quantum (or metric) graphs are a natural class of physical systems intermediate between lattices Z^d with the lattice Hamiltonians and the continuous models with the phase space R^d and classical Schrödinger operators H = -Δ + σV(x) on L^2(R^d). In the simplest form such a graph Γ^d is a cubic lattice Z^d with 1D edges connecting the neighboring vertices. The edges are equipped by the Euclidean metric, they are connected by the Kirchhoff’s gluing conditions at the vertices. P. Exner introduced this kind of graphs and operators in 60th before the development of the general theory. This topic is popular today mainly because of applications to the optical computers. One of the most important properties of corresponding periodic Hamiltonians is the existence of the infinitely many spectral gaps. It means that such graphs demonstrate (in contrast with the classical theory in R^d, d≥2) the semi-conductor properties for arbitrary high frequencies. The talk will contain the review of recent results on the structure of the Brownian motion on Γ^d and the spectra of the Schrödinger type operators with the decreasing or increasing to ∞ potentials. The most interesting (probably) are the theorems about point spectrum (localization) inside the gaps of the periodic background under random perturbations.

The recording: YouTube

October 30, 16:45 msk

Ciprian Tudor, University of Lille, France

Multidimensional Stein-Malliavin calculus for Gaussian and Gamma distribution

We develop an extension of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the probability distributions of (X,Y) and (Z,Y), where X,Y are arbitrary random vectors and Z ~ N(0, σ^2) (or it is Gamma distributed) is independent of Y. We apply this method to study the asymptotic independence for sequences of random vectors.

The recording: YouTube

October 16, 16:45 msk

Jordan Stoyanov, Bulgarian Academy of Sciences, Sofia, Gwo Dong Lin, Academia Sinica, Taipei

Normal distribution: recent not so well-known results and some open questions 

The main discussion in this talk will be on the following distributions: normal, half-normal, log-normal, inverse Gaussian, skew-normal, log-skew-normal, exp-normal. Of interest to us, and in general, are properties such as characterizations, moment-determinacy, infinite divisibility, comparison. We are going to report a series of well-selected results obtained by ourselves or by other authors over a couple of decades. The results are interesting, some are known, others in the best case are only 'slightly known', but definitely 'not well-known'. All results, statements, corollaries, etc., will be formulated completely and clearly, useful hints will be given, and in two cases, full proofs provided. We hope that such results and the methods used would be useful for university students and teachers. Professional researchers may challenge seven clearly formulated open questions.

The recording: YouTube

October 2, 16:45 msk

Yaroslav Sergeyev, University of Calabria, Rende, Italy and Lobachevsky State University, Nizhni Novgorod, Russia

Computations with numerical infinities and infinitesimals and infinitesimal probabilities

In Kolmogorov’s probability axioms, the axiom of additivity is formulated using notions of countable sets and sigma-algebras. In models having an infinite number of events, e.g., in continuous models, this can lead to some ungracious situations where possible events corresponding to sets of measure zero have probability zero whereas in the discrete finite case the probability zero is assigned to the impossible event only. 

In this lecture, a recent award-winning computational methodology (not related to non-standard analysis) is described (see [1-4]). It allows people to work with infinities and infinitesimals in a unique framework and in all the situations requiring these notions not only symbolically, but also numerically on a computer. The new methodology is based on the Euclid’s Common Notion “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite).  

It is shown that the methodology allows one to create a new viewpoint on probability and to avoid a number of paradoxes (see [3-7]) related to infinity, infinitesimals, and probability (among other paradoxes that can be avoided we find classical paradoxes of Hilbert, Galilei, Torricelli, etc.). One of the strong advantages of this methodology consists of its usefulness in practical applications (see [1, 4, 8, 9]) that can be implemented on a new kind of supercomputer called the Infinity Computer patented in several countries. It works numerically with numbers that can have several infinite and infinitesimal parts using a special floating-point representation. It should be also stressed that the methodology is already successfully taught in colleges in several countries (see, e.g., [7, 10] and references given therein). We conclude by emphasizing that reviews, videos, and more than 70 papers of authors using this methodology in their research areas can be downloaded from https://www.theinfinitycomputer.com

Selected references

1.     Sergeyev Ya.D. Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences, 4(2), 219–320, 2017.

2.     Sergeyev Ya.D., Arithmetic of Infinity, Edizioni Orizzonti Meridionali, CS, 2003 (2nd ed. 2013).

3.     Sergeyev Ya.D. Some paradoxes of infinity revisited, Mediterr. Journal of Mathem., 19, article 143, 2022.

4.     Sergeyev Ya. D. A new look at infinitely large and infinitely small quantities: Methodological foundations and practical calculations with these numbers on a computer. Informatics and Education, 36(8), 5–22, 2021. (In Russian.)

5.     Calude C. S., Dumitrescu M. Infinitesimal probabilities based on Grossone, SN Comput. Sci., 1, 36, 2020.

6.     Rizza D. A Study of Mathematical Determination through Bertrand’s Paradox, Philosophia Mathematica, 26(3), 375–395, 2018.

7.     Nasr L. Students’ resolutions of some paradoxes of infinity in the lens of the Grossone methodology. Informatics and Education, 38(1), 83–91. 2023.

8.     Sergeyev Ya.D., De Leone R. (eds.) Numerical Infinities and Infinitesimals in Optimization, Springer, Cham, 2022. 

9.     Falcone A., Garro A., Mukhametzhanov M.S., Sergeyev Ya.D. Advantages of the usage of the Infinity Computer for reducing the Zeno behavior in hybrid system models, Soft Computing, 27(12), 8189-8208, 2023.

10.  Rizza D., Primi Passi nell’Aritmetica dell’Infinito, Bonomo Editore, Bologna, 2023.

The recording: YouTube