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Gogi Pantsulaia

Publications and source records attributed to Gogi Pantsulaia.

11 recordsLinked to original sources

On a consistent estimator of a useful signal in Ornstein-Uhlenbeck model in $\mathbb{C}[-l,l[$

~It is considered a transmittion process of a useful signal in Ornstein-Uhlenbeck model in $\mathbb{C}[-l,l[$ defined by the stochastic differential equation $$ dΨ(t,x,ω)=\sum_{n=0}^{2m} A_n\frac{\partial^{n}}{\partial x^{n}}Ψ(t,x,ω)dt +σd W(t,ω) $$ with initial condition $$Ψ(0,x,ω)=Ψ_0(x) \in FD^{(0)}[-l,l[, $$ where $m \ge 1$, $(A_n)_{0 \le n \le 2m} \in \mathbb{R}^+\times \mathbb{R}^{2m-1}$,$~((t,x,ω) \in [0,+\infty[\times [-l,l[ \times Ω)$, $σ\in \mathbb{R}^+$, $\mathbb{C}[-l,l[$ is Banach space of all real-valued bounded continuous functions on $[-l,l[$, $FD^{(0)}[-l,l[ \subset \mathbb{C}[-l,l[ $ is class of all real-valued bounded continuous functions on $[-l,l[$ whose Fourier series converges to himself everywhere on $[-l,l[$, $(W(t,ω))_{t \ge 0}$ is a Wiener process and $Ψ_0(x)$ is a useful signal. By use a sequence of transformed signals $(Z_k)_{k \in N}=(Ψ(t_0,x,ω_k))_{k \in N}$ at moment $t_0>0$, consistent and infinite-sample consistent estimations of the useful signal $Ψ_0$ is constructed under assumption that parameters $(A_n)_{0 \le n \le 2m}$ and $σ$ are known. Animation and simulation of the Ornstein-Uhlenbeck process in $\mathbb{C}[-l,l[$ and an estimation of a useful signal are also presented.

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Satisfaction Problem of Consumers Demands measured by ordinary "Lebesgue measures" in $R^{\infty}$

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 time $t_k$ for $1 \le k \le n$. The measure of the supplied measurable system is changed under action of some dynamical system, What is a minimal measure of measurable system which must take the supplier at the initial time $t=0$ to satisfy demands of all consumers ?} In this paper we consider Satisfaction Problem of Consumers Demands measured by ordinary "Lebesgue measures" in $R^{\infty}$ for various dynamical systems in $R^{\infty}$. In order to solve this problem we use Liouville type theorems for them which describes the dependence between initial and resulting measures of the entire system.

math.CA↗

Infinite-sample consistent estimations of parameters of the Wiener process with drift

We consider the Wiener process with drift $$ dX_t=μdt +σd W_t $$ with initial value problem $X_0=x_0$, where $x_0 \in R$, $ μ\in R$ and $σ> 0$ 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 Wiener process with drift are constructed under assumption that all another parameters are known. Further, we propose a certain approach for estimation of unknown parameters $x_0,μ,σ$ of the Wiener process with drift by use the values $(z^{(1)}_k)_{k \in N}$ and $(z^{(2)}_k)_{k \in N}$ being the results of observations on the $2k$-th and $2k+1$-th trajectories of the Wiener process with drift at moments $t_1$ and $t_2$ , respectively.

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Estimation of the parameters of the Ornstein-Uhlenbeck's stochastic process

It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-θt} + μ(1-e^{-θt}) + σ\int_0^t e^{-θ(t-s)} dW_s$, where $x_0 \in R$, $θ>0$, $ μ\in R$ and $σ> 0$ 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 parameter of the Ornstein-Uhlenbeck's stochastic process is constructed under assumption that all another parameters are known.

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Calculation of Improper Integrals by Using Uniformly Distributed Sequences

We present the proof of a certain modified version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$β$engeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in $(0,1)$) sequences $S_f \subset(0,1)^{\infty}$ which strictly contains the set of sequences of the form $(\{αn\})_{n \in {\bf N}}$ with irrational number $α$ and for which $\ell_1^{\infty}(S_f)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure $\ell_1$ in $(0,1)$.

math.CA↗

Calculation of Lebesgue Integrals by Using Uniformly Distributed Sequences in $(0,1)$

We present modified proof of a certain version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$β$engeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in $(0,1)$) sequences $S_f \subset(0,1)^{\infty}$ which strictly contains the set of sequences of the form $(\{αn\})_{n \in {\bf N}}$ with irrational number $α$ and for which $\ell_1^{\infty}(S_f)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure $\ell_1$ in $(0,1)$.

math.FA↗

On a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions

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 the higher order in two variables with initial condition whose coefficients are real-valued simple step functions

math.CA↗

On a linear non-homogeneous ordinary differential equation of the higher order whose coefficients are real-valued simple step functions

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 differential equation of the higher order whose coefficients are real-valued simple functions.

math.CA↗

Under Collatz conjecture the Collatz mapping has no an asymptotic mixing property $\pmod{3}$

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 the form $3m+1$ is equal to the number $\frac{3}{5}+ \frac{(-1)^{n+1}}{15 \times 2^{2(n-1)}}. $ By using these formulas it is proved that under Collatz conjecture the Collatz mapping has no an asymptotic mixing property $\pmod{3}$. It is constructed also an example of a real-valued function on the cartesian product $N^2$ of the set of all natural numbers $N$ such that an equality its repeated integrals (with respect to Banach measure in $N$) implies that Collatz conjecture fails. In addition, it is demonstrated that Collatz conjecture fails for supernatural numbers.

math.PR↗

On objective and strong objective consistent estimates of unknown parameters for statistical structures in a Polish group admitting an invariant metric

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. An example of such a weakly separated statistical structure is constructed for which a question asking "{\it whether there exists a consistent estimate of an unknown parameter}" is not solvable within the theory $(ZF)~\&~(DC)$. A question asking "{\it whether there exists an objective consistent estimate of an unknown parameter for any statistical structure in a non-locally compact Polish group with an invariant metric when subjective one exists}" is answered positively when there exists at least one such a parameter the pre-image of which under this subjective estimate is a prevalent. These results extend recent results of authors. Some examples of objective and strong objective consistent estimates in a compact Polish group $\{0; 1\}^N$ are considered in this paper.

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