Mathematics, Informatics, and Their Applications in Natural Sciences and Engineering

David Gordeziani was born in Tbilisi on December 9 of 1937. Since 1945 attended Tbilisi Secondary School No 1, which he graduated with the gold medal in 1956. In the same year he became a student of the faculty of Mechanics and Mathematics at Ivane Javakhishvili Tbilisi State University (TSU).

In this paper we develop a method to associate a function curvature to human teeth roots.These functions are calculated from radiographs using mathematical software. We use this procedure to compare the curvatures of the roots with the curvatures of the resin blocks that are used for learning. The comparison is made by calculating the distances of the corresponding functions in the \(L_1\) metric.

The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of charged Fredholm-Stieltjes integral equations, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, including approximation of the relevant eigenfunctions. Starting from the problem of a string charged by a finite number of cursors, a survey including the extensions to the 2D and 3D dimensional problems is presented.

In this paper, we consider a new nonlinear continuous mathematical model of linguistic globalization. Two categories of the world’s population are considered: a category that hinders and a category conducive to the dominant position of the English language.With a positive demographic factor of the population, which prevents globalization and the negative demographic factor of the population contributing to globalization, it is shown that the dynamic system describing this process allows for the existence of two topologically not equivalent phase portraits (a stable node, a limit cycle). Under certain restrictions on the parameters of the model, the theorem on the absence of periodic trajectories of the dynamical system is proved and an asymptotically stable equilibrium position (limit cycle) is found. Thus, it is established that complete linguistic globalization is impossible if the demographic factor of the category of the world population contributing to the dominance of the English language is non-positive. Full linguistic globalization is possible only if the demographic factor is positive for the category of the world’s population, which contributes to the dominance of the English language and a certain restriction on the parameters of the model associated with the coefficient of assimilation.

In many engineering problems one has to deal with two materials which can induce local singularities (infinite stresses) as soon as the interface reaches the boundary. This is the case of water pipes with an inner coating for avoiding rusting. Because the wave velocity is smaller in the coating, it appears Love waves which can be used in order to detect defects because they propagate further as in a wave guide. They can be cracks corresponding to a disconnection at the interface between the two materials. In order to detect them, one can use measurements performed at the extremities of the pipes even if the signal in the numerical model, is very much perturbed by the singularities appearing at the interface between the two media at the extremities of the pipe. The phenomenon is amplified when one considers an artificial truncation of the structure and adding transparent boundary conditions in order to avoid reflection for simulating what happens in the full structure. The goal of this paper is to focus on a numerical method which can be used for the analysis of the influence of the singularities on the signal processing analysis. First of all, we give a mathematical description of the singularity met in our problem. Then, we define the extension to our case of the method introduced for cracks by G. Fix. It consists in adding the singular function to finite element functional space used in a classical numerical simulation. The main point of the paper is then to analyze, in a mathematical framework, the error estimates on the coefficients of the singularities with respect to the mesh size. Few numerical tests illustrate the mathematical results obtained for the problem we are dealing with.

Let \(\varOmega \) be a bounded domain in \(\mathbb {R}^n\) with boundary \(\delta \varOmega \) and consider the eigenvalue problem with Dirichlet boundary conditions, i.e. \(u\left| _{\delta \varOmega }\right. =0\). Its set of eigenvalues, \(0<\lambda _1\le \lambda _2\le \cdots \le \lambda _k\le \cdots \)—each eigenvalue being repeated according to (algebraic) multiplicity—is countable and the eigenvalue counting function may be defined as for a given positive \(\lambda \). The modified Weyl-Berry conjecture for the asymptotics of the eigenvalues of the Laplacian on bounded open subsets of the line (fractal strings) then states that with \(\left| \varOmega \right| _1\) being the one-dimensional Lebesgue measure of \(\varOmega \) and \(d\in [0,1]\) the Minkowski dimension of the boundary. Based upon a matrix representation of the Laplacian, it will be shown how to obtain some of the key results on the one-dimensional modified Weyl-Berry conjecture through elementary methods.

One type of integro-differential systems arising in mathematical modeling of the process of penetration of the magnetic field into a substance is studied. The model is based on the system of Maxwell equations. Uniqueness and large time behavior of solution of the corresponding initial-boundary value problem for the aforementioned model are given. Convergence of the fully discrete scheme is proved. A wide class of nonlinearity is studied.

The present exposition is a contribution to the third conference AMINSE. Our topic is related to research teams from the I. Vekua Institute of Applied Mathematics and Iv. Javakhishvili Tbilisi State University who are working in the tradition of I. Vekua. The authors are happy to give an overview of in memory of Professor David Gordeziani who contributed so much for building up mathematical institutions in Tbilisi to attractive centers of research in many areas, including singular analysis, with members Jaiani, Chinchaldze, Duduchava, Natroshvili and many others.

In this work the unsolvability of certain equations is studied in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain. Winterhof et al. considered the equations for \(\gamma \in {\mathbb Z}\). We first extend this result to \(\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m]\) by using a new method from algebraic number theory. Then we present its applications to \(\gamma \)-Butson-Hadamard matrices, \(\gamma \)-Conference matrices and type \(\gamma \) nearly perfect sequences for \(\gamma \in {\mathbb R}\cap {\mathbb Z}[\zeta _m] \).

The model based investigation of immune mediated disorders is a complex and evolving field. We herein report about further development of the mathematical model of rheumatoid arthritis. We improved previously reported model (Odisharia et al. in Vekua Inst Appl Math 31:107–110, 2017, [1]) by providing treatment component. Tocilizumab is considered as a drug for treatment. The model explores the functional dynamics of cartilage destruction during disease progression, in which a model deciphers the interactions between B and T lymphocytes. Coefficients of model equations allow the adaptation of the model to an individual case of patients that is its advantage. These coefficients can be calculated automatically based on the results of blood clinical analysis. Developed software simulates treatment process which will help in process of choosing optimal treatment scheme.

This writing is a short account of a talk I gave in Tbilisi (Georgia) on December 8, 2017, within the framework of AMINSE2017; it is based on material to be found in a recent paper of mine with the same title (Podio-Guidugli in Discret Contin Dyn Syst Ser S 10(6), 2017, [1]) and includes the statements of insofar unpublished results of mine to be prooved elsewhere.