Vakhtang Kvaratskhelia

We study the unconditional convergence of series in Banach spaces. We consider series of special type (Hadamard series), obtain the condition of their unconditional convergence, and discuss some of their applications. Further, we examine the almost sure unconditional convergence of random series in Banach spaces and, in the case of Gaussian series, we establish the relationship between the almost sure unconditional convergence and the geometry of Banach spaces. We also consider some probabilistic problems related to the convergence of series.

The textbook discusses such fundamental problems of the theory of algorithms as the need to specify the concept of an algorithm, computable and recursive algorithms, Turing and URM machines, the concept of algorithm complexity, algorithmically unsolvable problems, and others.

The textbook is intended for students of informatics, mathematics, physics, engineering and economic fields of higher education.

A sufficient condition is given for the a.s. unconditional convergence of Gaussian series in Banach spaces with unconditional bases not containing $j_\infty^n$l's uniformly. By the a.s. unconditional convergence of random series we understand the convergence of all rearrangements of the series on the same set of total probability.

The a.s. unconditionally convergent random series are investigated. The connection of the a.s. unconditionally convergence with the geometry of spaces is established.

A summable sequence $(a_n)$ in a Banach space $X$  is called $l_p$-canonical, $1 \le p <\infty$, if $a_n  = \alpha_n ve_n, n = 1,2, \ldots$ where $(a_n) \in l_p, v : l_p \to X$ is a  continuous  linear  operator  and  $(e_n)$  is  the  natural  basis of $l_p$. We are showing that a summable sequence $(a_n)$ in $X$  is $l_p$-canonical if and only if the  operator  $u : c_0 \to X$, with $ue_n = a_n, n=1,2,\ldots$ is $p$-summing. It follows that in a given Banach space $X$ any summable sequence is $l_p$-canonical if and only if any continuous linear operator from $c_0$ to $X$  is $p$-summing.  The last assertion implies the following statement obtained previously in [V.V. Kvaratskhelia, On unconditional convergence of random series in Banach spaces. Lect. Notes Math., 828 (1980), 162-166]:  in  a  given  Banach  space  $X$  any  summable sequence is $l_p$-canonical for certain $p, 2 \le p < \infty$ if and only if $X$   does  not  contain  $l_\infty^n$'s uniformly. For the spaces with a given cotype $p$ we are obtaining  the more precise results  showing , in  particular, that in cotype 2 spaces any summable sequence is  $l_2$-canonical, while in $l_p$, with $2 \le p < \infty$,  not  any summable sequence is $l_p$-canonical.t any summable sequence is $l_p$-canonical.

In the paper characterization of the cross-covariance operators in Banach spaces is presented. The notion of the coefficient of correlation in Banach spaces is introduced and its well-known property in one dimensional case is proved for the general case.

In the paper the weak and strong linear approximations of random elements with values    in Banach spaces are investigated

A criterion is proved for unconditional convergence of a specific series constructed by Hadamard matrices in the Banach spaces  $l_p$

In Banach spaces lp for $p \ge 2$ a series of a specific type, Sylvester series, converges only if it converges unconditionally. For $1 \le p < 2$ there exist Sylvester series that converge but fail to converge unconditionally.

The numerical characteristics for Hadamard and Sylvester matrices are introduced and some estimates for the maximum of partial sums related to these matrices are proved.

The numerical characteristics for Hadamard and Sylvester matrices are introduced and some estimates for the maximum of partial sums related to these matrices are proved.

The author defines a parameter $\varrho^{(n)}$ connected with an $n\times n$ matrix $A=(a_{ki}))$ and a normalized basis $(\varphi_k)$ of a Banach space $X$ by

$$\varrho^{(n)}=\max _{1\le m\le 2^n}\Vert\sum_{i=1}^{2^n}\sum_k=1^m a_{ki}\varphi_i\Vert.$$

Throughout it is assumed that $(\varphi_k)$ is subsymmetric with constant 1. The first part deals with the case where $A$ is a Sylvester matrix. The main result gives an upper estimate for the parameter in terms of the expressions $\Vert\sum_k=1^n\varphi_k\Vert$. As corollaries, several other estimates for $\varrho^{(n)}$  and similar parameters are obtained. The value of $\varrho^{(n)}$  for $l_1$ is given. A general lower estimate is also presented. In a next part, the parameter is evaluated for general Hadamard matrices. Finally an isomorphic characterization of $l_1$ in terms of the asymptotics of the defined parameters is given. All results are stated without proofs.

In the paper, we introduce a characteristic of the Sylvester matrix and find its explicit values. We describe some applications of the introduced characteristic

It is proved that in every infinite-dimensional normed space for any positive integer r there is an "unconditional sequence" of r elements. The proof uses the Brunel-Sucheston spreading model

The method of construction of sign invariant pair of elements in normed spaces is presented

The characterization of Banach spaces with finite cotype is given in terms of a.s. unconditionally convergence of weakly sub-Gaussian random series

The elementary proof of a relation between moments of measurable functions of different orders is given. This result can be used in the study of the connection of a.e. unconditional convergence of functional series in a Banach space with the geometry of the space

We give a survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces. Some new results and examples are also given

Characterizations of the class of Banach spaces isomorphing to the space $c_0$ , as well as to the class of Banach spaces not containing $l_\infty^n$ 's uniformly, are obtained in terms of almost surely unconditional convergence of weakly sub-Gaussian random series. Under almost surely unconditional convergence of random series, convergence of all permutations on the same set of full probability is understood. The questions of almost surely unconditional and weak absolute convergence in the spaces isomorphing to $c_0$ are investigated as well.

Characterization of the Banach spaces isomorphic to the Banach space $c_0$ c is obtained in terms of unconditionally converging series

We show that in any two-dimensional normed space there exists a collection of vectors $x_1, x_2, \ldots, x_n, n\ge 1$, such that the greedy algorithm for estimation of ( ) 1 1 $\min_\pi \max_{1\le k\le n} \Vert \sum_{i=1} ^k x_{\pi(i)}\Vert$ fails to be optimal.

The life and work of the prominent Georgian mathematician, Professor Davit Kveselava, is described.

In a Banach lattice, the convergence of a series of absolute values $\sum_{k\ge 1} |x_k|$ implies the unconditional convergence of the series $\sum_{k\ge 1} x_k$. The converse assertion is valid only in Banach lattices order-isomorphic to $M$-spaces. In this paper a new proof of this fact using Sylvester series is given.

It is shown that every infinite-dimensional real Banach space $X$ contains a sequence $(x_n)_{n∈ℕ}$ with the following properties: (a) Some subsequence of $(∑_{k=1}^n x_k)_{n∈ℕ}$ converges in $X$ and sup_{n∈ℕ} \Vert \sum_{k=1}^n x_k \Vert \le 1$; (b) $\sum_{k=1} ^\infty \Vert x_k \Vert^p<\infty$ for every $p\in ]2,\infty[$; (c) for any permutation $\pi:ℕ\to ℕ$ and any sequence $(\theta_n)_{n∈ℕ}$ with $\theta_n\in \{-1,1\}, n=1,2,\ldots$, the series $\sum_{k=1| ^\infty \theta_k x_{\pi(k)}$ diverges in $X$. This result implies, in particular, that the rearrangement theorem and the Dvoretzky–Hanani theorem fail drastically for infinite-dimensional Banach spaces.

We call a Gaussian random element $\eta$ in a Banach space $X$ with a Schauder basis $e=(e_n)$ diagonally canonical (for short, $D$-canonical) with respect to $e$ if the distribution of $η$ coincides with the distribution of a random element having the form $B\xi$, where $\xi$ is a Gaussian random element in $X$, whose $e$-components are stochastically independent and $B:X\to X$ is a continuous linear mapping. In this paper we show that if $X=l_p, 1\le p<\infty$ and $p\neq 2$, or $X=c_0$, then there exists a Gaussian random element $\eta$ in $X$, which is not $D$-canonical with respect to the natural basis of $X$. We derive this result in the case when $X=l_p, 2<p<\infty$, or $X=c_0$ from the following statement, an analogue which was known earlier only for Banach spaces without an unconditional Schauder basis: if $X=lp, 2<p<\infty$, or $X=c_0$, then there exists a Gaussian random element $\eta$ in $X$ such that the distribution of $\eta$ does not coincide with the distribution of the sum of almost surely convergent in $X$ series $\sum_{n=1} ^\infty x_n g_n$, where $(x_n)$ is an unconditionally summable sequence of elements of $X$ and $(g_n)$ is a sequence of stochastically independent standard Gaussian random variables.

We introduce numerical characteristics of Sylvester and Hadamard matrices and give their estimates and some of their applications.

In this paper a sufficient and a necessary condition

for unconditional convergence of a series in a Banach space with

an unconditional basis are analyzed.

The life and work of the prominent Georgian mathematician, academician Niko (Nicholas) Vakhania, is described.

We give a characterization of weakly subgaussian random elements that are $\gamma$-subgaussian in infinite-dimensional Banach and Hilbert spaces.

Maximal inequalities for the signed vector summands are established. Probabilistic estimations for the sets of appropriate signs are given. By use of the “transference technique” appropriate maximal inequalities are derived for the permutations. One application for orthogonal and Hadamard matrices is suggested.

In this work we investigate performance of the bus transit system of city Tbilisi based on the statistical analysis of the passenger flow during the year 2019. In order to detect the changes in the system during the transitional period 2018–2019, some statistics of the mentioned period are compared with that of 2017, investigated by the joint research project of Tbilisi City Hall and an international engineering and consulting group SYSTRA. Passenger flow during 2019 is also analyzed with regard to the working and festive days and by seasonal trends as well.

We prove an analogue of Khinchin’s theorem for weakly correlated random elements with values in the spaces $l_p, 1\le p<\infty$.

The paper is dedicated to the 100th anniversary of the Georgian Technical University and describes the activities of Muskhelishvili Institute of Computational Mathematics.

An elementary proof of a property of convergent series consisting of non-increasing non-negative real numbers is proposed.

The problem of Compact Vector Summation (CVS) consists in finding an upper estimation for $r(x, \pi_{min})$, the minimum of radii of spheres that contain the trajectory of partial sums for a collection of vectors $x=(x_1,\ldots,x_n)$ of a normed space under an optimal permutation of $(x_1,\ldots,x_n)$. Finding explicitly a permutation $\pi$ that ensures an estimation found is another part of the CVS-problem. The CVS-problem found many applications in analysis (sum range of a conditionally convergent series; Kolmogorov Conjecture on rearrangements of orthonormal systems, etc.). CVS-problem also found applications in scheduling theory (problem of

reroute sequence planning in telecommunication networks; volume calendar planning, etc.). We suggest an effective algorithmic method for finding

an optimal permutation in CVS and estimation of $r(x, \pi_{min})$.

Hadamard transform is an important tool for the investigati-

on of some problems of Quantum Computing, Coding Theo-

ry and Cryptology, Statistics, Image Analysis, Signal Proces-

sing, Fault-Tolerant Systems, Analysis of Stock Market Da-

ta, Combinatorial Designs and so on. Here we present one

numerical property of Hadamard matrices

A new numerical characterization for Hadamard matrices is introduced. Its estimations for different norms are established by use of appropriate maximal inequalities for the signed vector summands.

In this communication a non-local modified characteristic problem for a second order quasi-linear equation with real characteristics is investigated.

We give a short survey concerning sub-Gaussian random elements in a Banach space and prove a statement about the induced operator of a bounded random element in a Hilbert space.