Gogi Rauli Pantsulaia

• Doctor of Physics and Mathematics
• Senior Researcher at Tbilisi State University

~For $m \ge 1$, it is considered Ornstein-Uhlenbeck process in ${\bf C}[-l,l[$ defined by the stochastic differential equation $$ d\Psi(t,x,\omega)=\sum_{n=0}^{2m} A_n\frac{\partial^{n}}{\partial x^{n}}\Psi(t,x,\omega)dt +\sigma d W(t,\omega) I_{[-l,l[}(x) ~((t,x,\omega) \in [0,+\infty[\times [-l,l[ \times \Omega) $$ with initial condition $$\Psi(0...

We consider the Wiener process with drift $$ dX_t=\mu dt +\sigma d W_t $$ with initial value problem $X_0=x_0$, where $x_0 \in R$, $ \mu \in R$ and $\sigma > 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, the infinite-sample consistent estimates of each unknown parameter of the Wiene...

A certain version of the Erdos problem is studied. More precisely, it is proved that ¨ there does not exist a finite constant c such that each plane set with an outer Lebesgue measure greater than c contains the vertices of a triangle of area 1. It is shown that a sentence ”each plane set E with Lebesgue outer measure +∞ contains the vertices of a...

The purpose of the present talk is an introduction of a concept of the Dirac delta function defined on the class of all continuous functions over $R^{\infty}$ equipped with Tychonoff topology and a representation of this functional in terms of infinite-dimensional Lebesgue measures in $R^{\infty}$.

It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-\theta t} + \mu (1-e^{-\theta t}) + \sigma \int_0^t e^{-\theta (t-s)} dW_s$, where $x_0 \in R$, $\theta>0$, $ \mu \in R$ and $\sigma > 0$ are parameters. By use values $(z_k)_{k \in N}$ of corresponding trajectories at a fixed positive moment $t$, a consistent estimate of each unknown par...

A representation of the Dirac delta function in $ \mathcal{C}(R^{\infty})$ in terms of infinite-dimensional Lebesgue measures in $R^{\infty}$ is obtained and some it's properties are studied in this paper.

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(\lambda_t)_{t \in [0,1]}$ of invariant extensions of the linear Lebesgue measure $\lambda$ in $[0.1]$, and it is shown that the $\lambda_t^{\infty}$ measure of the set of all $\lambda_t$-uniformly distributed se...

We present the proof of a certain version of Kolmogorov strong law of large numbers which differs from Kolmogorov’s original proof.

Let $\mathbf{Q}$ be a set of all rational numbers of $[0,1]$ and $F \subseteq [0,1]\cap \mathbf{Q}$ be finite. Let $f : [0,1] \to R$ be Lebesgue integrable, continuous almost everywhere and locally bounded on $[0,1] \setminus F$. Assume that for every $\beta \in F$ there is some neighbourhood $U$ of $\beta$ such that $f$ is either bounded or monoto...

By using the method developed in the paper [Georg. Inter. J. Sci. Tech., Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions

In the present paper we consider the following Satisfaction Problem of Consumers Demands (SPCD): The supplier must supply the measurable system of the measure mk to the k-th consumer at the moment tk for 1 � k � n. The measure of the supplied measurable system is changed under action of some dynamical system; What minimal measure of measurable syst...

In the present paper we consider the following Satisfaction Problem of Consumers Demands (SPCD): {\it The supplier must supply the measurable system of the measure $m_k$ to the $k$-th consumer at the moment $t_k$ for $1 \le k \le n$. The measure of the supplied measurable system is changed under action of some dynamical system; What minimal measure...

It is introduced a certain approach for equipment of an arbitrary set of the cardinality of the continuum by structures of Polish groups and two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki's certain question(2012) {\it what are the real $k$-dimensional manifolds with at least two different Lie g...

We give a certain approach which allows us to equip an arbitrary set of the cardinality of the continuum by structures of Polish groups and consider various examples of Haar measure spaces.

We give a certain approach which allows us to equip an arbitrary set of the cardinality of the continuum by structures of Polish groups and consider various examples of Haar measure spaces.

In [1], the concept of increasing families of finite subsets uniformly distributed in infinite-dimensional rectangles has been introduced and a certain infinite generalization of the Weyl's famous result (cf. [2], Theorem 1.1, p. 2) has been obtained. In this talk we introduce Riemann integrability with respect to product measures for real-valued f...

In this talk we study the structure of the set of all uniformly distributed sequences on [􀀀-1/2 ; 1/2 ] from the point of view of shyness.

We describe a certain approach which allows us to equip an arbitrary set of the cardinality of the continuum by structures of Polish groups and consider various examples of Haar measure spaces.

In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2], Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that $\mu$-almost every element of $\mathbf{R}^{\infty}$ is uniformly distributed in $[-\frac{1}{2}, \frac{1}{2}]$, where $\mu$ denotes Yamasaki-Kharazishvili measure in $\mathbf{R}^{\infty}$ for which $\mu([-...

By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in $R^{\infty}$ described in [G.R. Pantsulaia, On uniformly distributed sequences of an increasing family of finite sets in infinite-dimensional rectangles, Real Anal. Exchange. 36 (2) (2010/2011), 325--340 ], a new approach for...

In Solovay model it is shown that the duality principle between the measure and the Baire category holds true with respect to the sentence - "The domain of an arbitrary generalized integral for a vector-function is of first category."

The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive...

t is shown that $ \overline{\lim}\widetilde{T_n} := \inf_n \sup_{m \ge n}\widetilde{T_m}$ and $ \underline{\lim}\widetilde{T_n} := \sup_n \inf_{m \ge n}\widetilde{T_m}$ are objective infinite sample well-founded estimates of a useful signal $\theta$ in one-dimensional linear stochastic model $ \xi_k=\theta+\Delta_k ~(k \in \mathbb{N}), $ where $\#(...

By using the notion of a Haar ambivalent set introduced by Balka, Buczolich and Elekes (2012), essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish non-locally-compact group admitting an invariant metric and relations between them are studied in this paper...

By using properties of Markov homogeneous chains and Banach measure in $\mathrm{N}$, it is proved that a relative frequency of even numbers in the sequence of $n$-th coordinates of all Collatz sequences is equal to the number $\frac{2}{3}+\frac{(-1)^{n+1}}{3\times 2^{n+1}}.$ It is shown also that an analogous numerical characteristic for numbers of...

In the present report, the notion of a Haar ambivalent introduced by Balka, Buczolich and Elekes in 2012 is used in studying the properties of some infinite sample statistics and in explaining why the null hypothesis is sometimes rejected for ”almost every”(in the sense of Christensen) infinite sample by some hypothesis testing of maximal reliabili...

The notion of a Haar null set introduced by Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke, has been used, in the last two decades, in studying exceptional sets in diverse areas, including analysis, dynamic systems, group theory, and descriptive set theory. In the present paper, the not...

It is shown that for the vector space R^ N (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a σ-finite non-trivial Borel measure that is translation invaria...

For a group Γ of all rotations of the plane R^2 about it's origin, by using the technique developed in a paper [Kharazishvili A. B., Small sets in uncountable abelian groups. Acta Univ. Lodz. Folia Math. No. 7 (1995), 31–39] it is proved an existence of a partition of the plane R^2 into absolutely Γ-negligible subsets of R^2 for which an intersecti...

A certain version of the Erdös problem is studied. More precisely, it is proved that there does not exist a finite constant c such that each plane set with an outer Lebesgue measure greater than c contains the vertices of a triangle of area 1. It is shown that a sentence "each plane set E with Lebesgue outer measure +∞ contains the vertices of a tr...

By using an infinite-dimensional "Lebesgue measure" in an infinite-dimensional separable Banach space B with Schauder basis a solution of a heat equation with initial value problem on B is constructed. Properties of uniformly distributed real-valued sequences in an interval of the real axis are used for a construction of a certain algorithm which g...

By using the technique of "Fourier differential operator" in R ∞ and Laplace transforms, a representation in a multiple trigonometric series of the solution of a certain generalized heat equation of many variables is obtained.

Some classes of real-valued functions defined on a metric space V equipped with a nonzero sigma-finite diffused Borel measure µ were introduced and relationships between them (in the sense of inclusion) are studied. In particular, it is shown that when V is a Polish metric space then the properties of µ-massiveness along trajectories of all continu...

It is proved an existence of maximal ”small” plane sets in R^2 which contain only the vertices of a triangle of area less than one. It is shown also that the closing of each maximal ”small” plane set in R^2 contains the vertices of a triangle of area one.

By using the method developed in the paper [G.Pantsulaia, G.Giorgadze, On some applications of infinite-dimensional cellular matrices, {\it Georg. Inter. J. Sci. Tech., Nova Science Publishers,} Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of t...

By using the method developed in the paper [G.Pantsulaia, G.Giorgadze, On some applications of infinite-dimensional cellular matrices, {\it Georg. Inter. J. Sci. Tech., Nova Science Publishers,} Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the particular solution of the linear non-homogeneous ordinary d...

This book explores a number of new applications of invariant quasi-finite diffused Borel measures in Polish groups for a solution of various problems stated by famous mathematicians (for example, Carmichael, Erdos, Fremlin, Darji and so on). By using natural Borel embeddings of an infinite-dimensional function space into the standard topological ve...

It is given an effective construction of the strong objective infinite-sample well-founded estimate of a useful signal in the linear one-dimensional stochastic model.

This article presents main results of investigations of the authors which were obtained during the last five years by the partially support on the Shota Rustaveli National Science Foundation (Grant no. 31–24). These results are Liouville-type theorems and describe the behavior of various phase motions in terms of ordinary and standard “Lebesgue mea...

The separation problem for a family of Borel and Baire G-powers of shift measures on \( \mathbb{R} \) is studied for an arbitrary infinite additive group G by using the technique developed in [L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974)], [ A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980)]...

We study a structure of uniformly distributed sequences on [-1 2,1 2] in terms of Yamasaki measure μ. In particular, we show that μ-almost every element of ℝ ∞ is uniformly distributed on [-1 2,1 2].

By using a structure of the "Fourier differential operator" in R ∞ , we describe a new and essentially different approach for a solution of the old functional problem posed by R. D. Carmichael in 1936. More precisely, under some natural restrictions, we express in an explicit form the general solution of the linear inhomogeneous differen-tial equat...

We construct a quasi-finite non-sigma-finite translation-invariant generator of shy sets in the Euclidean plane R^2 such that a value of that generator on an arbitrary planar circle curve coincides with its length. We introduce a notion of a maximal generator of Borel shy sets in Polish topological vector spaces and show that such a generator no al...

By using results from a paper [G.R. Pantsulaia, On ordinary and standard Lebesgue measures on R ∞ , Bull. Pol. Acad. Sci. Math. 57 (3-4) (2009), 209–222] and an approach based in a paper [T. Gill, A.Kirtadze, G.Pantsulaia , A.Plichko, The existence and uniqueness of translation invariant measures in separable Banach spaces, Functiones et Approximat...

By using Martin Axiom, we prove that σ-ideals of Preiss-Tiser generalized shy sets in a Polish topological vector space, Mankiewicz generalized shy sets and Baker gen-eralized shy sets in the topological vector space of all real-valued sequences equipped with Tychonoff topology are closed under an operation taking a union fewer than c elements of t...

It is proved that the duality principle between the measure and category is valid with respect to the sentence P defined by: For every two Polish groups G_1 and G_2 , and for every Haar null set Y ⊂ G_1 we have (∀X)(X ⊆ G_2 → Y × X is Haar null in G_1 × G_2). By using that approach which differs from the approach of M. P. Cohen (2012), it is shown...

By using a technique developed in [1], we describe a new and essentially different approach for a solution of the old functional problem (a) posed by R. D. Carmichael in [2](cf. p. 199). More precisely, under some general restrictions, we express in an explicit form the general solution of the non-homogeneous ordinary differential equation of the i...

By using a technique of invariant measures developed by Oxtoby [J. C. Oxtoby, Invariant measures in groups which are not locally compact, Trans. Amer. Math. Soc. 60 (1946), 215-237] for a Polish group, which is dense in itself, we prove an existence of a quasi-finite left (or right) invariant generator of left (or right)-shy sets in an entire group...

In this paper we investigate the foundations for analysis in infinitely-many (independent) variables. We give a topological approach to the construction of the regular $\s$-finite Kirtadze-Pantsulaia measure on $\R^\iy$ (the usual completion of the Yamasaki-Kharazishvili measure), which is an infinite dimensional version of the classical method of...

By using a structure of the "Fourier differential operator" in R^∞ , under some general restrictions we describe a new and essentially different approach for a construction of a particular solution of the non-homogeneous differential equation of the higher order with real constant coefficients and give its representation in an explicit form.

We show that Gardner transition kernel (see, [1], Example 1.4, p. 972) is such an example of a modified uniformly orthogonal transition kernel which is not modified completely orthogonal. This answers negatively to the Problem 2 posed in [3].

This book explores a new concept of T-shy sets in Radon metric groups which is an extension of the concepts of null sets introduced by J.R. Christensen, J. Mycielski and B. R. Hunt, T. Sauer and J. A. Yorke for Polish groups. This book considers various interesting applications of the theory of infinite-dimensional cellular matrices for a solution...

We consider a certain generalization of the von Foerster-Lasota differential equation and by using the technique of infinite-dimensional cellular matrices (the so-called Maclaurin differential operators) give its solution in an explicit form. We study the behaviour of corresponding motions in ℝ ∞ in terms of ordinary and standard “Lebesgue measures...

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on ℝ∞ and Rogers-Fremlin measures...

The concepts of uniformly distributed sequences of an increasing family of finite sets and Riemann integrability are considered in terms of the “Lebesgue measure” on infinite-dimensional rectangles in \(R^{\infty}\) and infinite-dimensional versions of famous results of Lebesgue and Weyl are proved.

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on R∞ and Rogers–Fremlin measures...

We construct an example of two non-σ-finite translation invariant measures on a certain σ-algebra of subsets of the real plane R ^2 such that in the theory ZFC, these measures are mutually singular if and only if Continuum Hypothesis is true. Also, in the theory ZF +DC we give two examples of modified uniformly orthogonal transition kernels such th...

By using the technique of infinite-dimensional cellular matrices we obtain solutions of various initial condition problems and study behavior of corresponding mo-tions in R^∞ in the sense of ordinary and standard "Lebesgue measures". words and phrases: Maclaurin's and Fourier's differential operators, Formal solutions, Phase flows, Riemann's conjec...

We prove that "Lebesgue "-null sets in R^∞ are not preserved under Lipschitz isomorphisms. A similar result is obtained in an infinite-dimensional separable Banach space with a Schauder basis.

We prove an existence of a left-invariant quasi-finite Borel measure on the product of an arbitrary family of locally compact Hausdorff groups that are not compact. 2010 Mathematics Subject Classification: Primary 28xx; Secondary 28Bxx, 28Cxx Key words and phrases: Locally compact Group, Haar measure, left-invariant measure Let (X i , B i , µ i) i∈...

We consider a new approach to prove that a number 1 n is a Witsenhausen-Kalai constant for the measure μ n on S n . Under the Witsenhausen conjecture, we prove a certain geometric inequality on the sphere S n . Some computations in Mathcad show that this result does not contradict the Witsenhausen conjecture [P. Frankl and R. M. Wilson, Combinatori...

This book explores a number of new constructions of generators of shy sets (Mankiewicz generator, Preiss-Tiser generator, Baker generator, Kharazishvili generator, ordinary and standard Lebesgue measures, etc), which naturally generate classes of null sets playing an important role in studying the properties of a function space. It includes several...

Let (α n (k) ) n∈N be an infinite sequence of different integer numbers for every k∈ℕ. Then a set of all sequences (x k ) k∈ℕ in ℝ ∞ for which a sequence of increasing sets (Y n ((x k ) k∈ℕ )) n∈ℕ is λ-uniformly distributed on ∏ k∈ℕ [a k ,b k ]), where Y n ((x k ) k∈ℕ )=∏ k=1 n ⋃ j=1 n {〈α j (k) x k 〉(b k -a k )} + a k ×∏ k∈ℕ∖{1,...,n} {a k } and λ...

Let α be an arbitrary infinite set for which the condition card(α ℵ 0 )=card(α) holds, where ℵ 0 denotes a cardinality of the set of all natural numbers. Let (H i ) i∈α be a sequence of locally compact σ-compact Polish topological groups and μ i be a continuous H i -left (right or two-sided)-invariant Haar measure on H i for which μ i (H i )>1 (i∈α...

We prove that the Komjath Axiom KA is equivalent to the negation of Continuum Hypothesis {reversed not sign}CH in the Martin-Solovay model ZFC & MA. Following [11], we deduce that a system of axioms ZFC & MA & KA is consistent in such a way that the Komjath Conjecture [6, Question 1.1, p.2] is valid. Also, we prove that in the theory ZFC an axiom K...

Let (H, ⊙) be an uncountable group for which card(H@0) = card(H). By the generalizing method of independent families of sets [30], we elaborate one method for constructing of various families of left-H-invariant diffused non-elementary nonseparable probability measures on entire group and consider its some realizations. Also, we give some applicati...

The concepts of uniformly distributed sequences of an increasing family of finite sets and Riemann integrability are considered in terms of the “Lebesgue measure” on infinite-dimensional rectangles in R∞ and infinite-dimensional versions of famous results of Lebesgue and Weyl are proved. Mathematical Reviews subject classification: Primary: 28xx, 0...

We show that Gardner transition kernel (see, [Gardner R.J. A note on conditional distributions and orthogonal measures. Ann. Probab., 10, 3 (1982), 877-878], Example 1.4, p. 972) is such an example of a modified uniformly orthogonal transition kernel which is not modified completely orthogonal. This answers negatively to the Problem 2 posed in [Mau...

We show that if S is a planar set of positive two-dimensional Lebesgue measure l , then for arbitrary c, the S contains the vertices of a triangle of area c if and only if it is not bounded by a natural norm. Also, we consider a certain generalized problem of P. Erosh asking whether there exists a positive constant c such that every planar set with...

Specialized in the Solovay model, we prove that any Haar null set in ℝ n (n>2) can be decomposed into two at most 1-dimensional Haar null sets. Moreover, we describe an algorithm which gives a partition {E k :1≤k≤n} of the Lebesgue null set E⊂ℝ n such that e k is transverse to E k for 1≤k≤n. This partition answers H. Shi’s question [Measure-theoret...

For n ε N, let GnP&T be an n-dimensional Preiss-Tišer generator in an infinite- dimensional topological vector space V (cf. [7]). For D ⊇ V , we establish the validity of the following formula dim(D) = inf{m - 1: G(m)P&T (D) = 0}, where D denotes a closed convex hull of the set D in V.

Using N.Lusin's arithmetic example of a non-Borel analytic set [LUZIN, N. N., Lekcii ob analiticheskih mnoestvah i ih priloeniyah. (Russian) [Lectures on analytic sets and their applications.] Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953. 359 pp.], an example of an uncountable family of polygons is constructed in the Euclidean plane R^2 such t...

We consider a concept of the uniformly distribution for increasing sequences of finite subsets in an infinite-dimensional rectangle, and by using the technique of the infinite-dimensional Lebesgue measure, introduce a notion of Reimann integrability for functions defined on the entire rectangle. Further, we prove an infinite-dimensional version of...

Our method of construction of measures is a concept of strict standard and strict ordinary products of an infinite family of (no only σ-finite) measures, which allows us to construct Mankiewicz and Preiss-Tíser generators on R ∞ . We show that if f : R ∞ → R is a Lipschitz function and R^ (N) is a group of all eventually zero sequences, then, in So...

Let G be a complete metric group and T be such a subgroup of its Borel automorphisms group (), G A which contains all left and right shifts of the G. We introduce notions of-T cm and-T shy sets and demonstrate that they constitute a-σ ideal and coincide in Radon metric groups. This result extends main results established in [6], [13], [20]. For a B...

We give an expansion of a real-valued square integrable (with respect to the infinite-dimensional "Lebesgue measure" λ [1]) function on an infinite-dimensional rectangle R (with λ(R) < +∞) in R ∞ into an infinite-dimensional multiple trigono-metric series. Also, we consider an analogous question for real-valued square inte-grable (with respect to t...

We consider a new approach to describe a group of admissible translations for Gaussian Baire measures on R I for an arbitrary parameter set I. As a consequence, we show that Cameron-Martin formula does not determine the group of all admissible translations of the Wiener measure defined on the [0, 2π].

It is proved that the sentence “For a fixed n>1 and for each extension λ of the completion μ ¯ n of the surface measure μ n there is a Witsenhausen-Kalai constant w (n,λ) for λ” is independent from the theory ZF + DC, where ZF denotes Zermelo-Fraenkel system of axioms and DC denotes an axiom of dependent choice. If n=2k, then we show that 1 2μ(S n...

Let α be an infinite parameter set, and let (α i ) i∈I be its partition such that α i is a non-empty finite subset for every i∈I. For j∈α, let μ j be a σ-finite Borel measure defined on a Polish metric space (E j ,ρ j ). We introduce a concept of standard (α i ) i∈I -product of measures (μ j ) j∈α and investigate its properties. As a consequence, w...