Transactions of A Razmadze Mathematical Institute最新文献

The test of homogeneity is constructed by using kernel-type estimators of a distribution density. The limit power of the constructed test is found for close Pitman-type alternatives. The constructed test is compared with Pearson’s $Xu$-square test.

We study a nonparametric Bayesian approach to linear inverse problems under discrete observations. We use the discrete Fourier transform to convert our model into a truncated Gaussian sequence model, that is closely related to the classical Gaussian sequence model. Upon placing the truncated series prior on the unknown parameter, we show that as the number of observations $n\to \infty ,$ the corresponding posterior distribution contracts around the true parameter at a rate depending on the smoothness of the true parameter and the prior, and the ill-posedness degree of the problem. Correct combinations of these values lead to optimal posterior contraction rates (up to logarithmic factors). Similarly, the frequentist coverage of Bayesian credible sets is shown to be dependent on a combination of smoothness of the true parameter and the prior, and the ill-posedness of the problem. Oversmoothing priors lead to zero coverage, while undersmoothing priors produce highly conservative results. Finally, we illustrate our theoretical results by numerical examples.

The class of functions is known as invex function (invariant convex) in the literature and the name derives from the fact that the convex like property of such functions remains invariant under all diffeomorphisms of $R^n$ into $R^n.$ A noteworthy result here is that the class of invex functions is precisely the class of differentiable functions whose stationary points are global minimizers. We revisit some of the important results obtained by Hanson and Martin and extend them to constrained minimization problems with equality constraints in addition to inequality constraints. We address some conditions by which a function is invex. We propose a result to solve pseudo-invex programming problem with the help of an equivalent programming problem.

The effects of viscous dissipation and heat source on MHD flow and heat transfer from a warm, electrically conducting fluid to melting surface parallel to a constant free stream are investigated numerically. This model constitutes highly non-linear governing equations which are transformed using similarity variables and are then solved by fourth order Runge–Kutta scheme along with shooting method. The influence of the various interesting parameters on the velocity and temperature fields within the boundary layer is discussed and explained graphically. It is noticed that the melting phenomenon rises the skin friction coefficient and declines the Nusselt number at the solid interface.

Jleli and Samet (2018) introduced a new metric space and named it as $F$-space. In this paper we consider the notion of $\alpha$-$\psi$-contraction in the setting of $F$-metric spaces. We present some fixed point and coupled fixed point results in the generalized setting. Moreover, our purpose in this paper is to concerned with the solution of nonlinear neutral differential equation $x^{\prime }\left(t\right)=-a\left(t\right)x\left(t\right)+b\left(t\right)g\left(x(t-r\left(t\right))\right)+c\left(t\right)x^{\prime }(t-r\left(t\right))$with unbounded delay using fixed point theory in $F$-metric space.

New nonparametric procedure for estimating the probability density function of a positive random variable is suggested. Asymptotic expressions of the bias term and Mean Squared Error are derived. By means of graphical illustrations and evaluating the Average of $L_2$-errors we conducted comparisons of the finite sample performance of proposed estimate with the one based on kernel density method.

In development of stochastic analysis in a Banach space one of the main problem is to establish the existence of the stochastic integral from predictable Banach space valued (operator valued) random process. In the problem of representation of the Wiener functional as a stochastic integral we are faced with an inverse problem: we have the stochastic integral as a Banach space valued random element and we are looking for a suitable predictable integrand process. There are positive results only for a narrow class of Banach spaces with special geometry (UMD Banach spaces). We consider this problem in a general Banach space for a Gaussian functional.

In the present paper, we establish general representations of continuous linear functionals, which play important roles in Functional Analysis, of the absolute weighted spaces which have recently been introduced in Sarıgöl (2016, 2011), and also determine their norms. Further making use of this we give adjoint operators of matrix mappings defined on these spaces.

One of the most widely studied biological systems with excitable behavior is neural communication by nerve cells via electrical signaling. The Fitzhugh–Nagumo equation is a simplification of the Hodgin–Huxley model (Hodgin and Huxley, 1952) [24] for the membrane potential of a nerve axon. In this paper we developed a three time-level implicit method by using tension spline function. The resulting equations are solved by a tri-diagonal solver. We described the mathematical formulation procedure in detail. The stability of the presented method is investigated. Results of numerical experiments verify the theoretical behavior of the orders of convergence.

The reduced Burau representation is a natural action of the braid group $B_n$ on the first homology group $H_1({\tilde{D}}_n;Z)$ of a suitable infinite cyclic covering space ${\tilde{D}}_n$ of the $n$-punctured disc $D_n$. It is known that the Burau representation is faithful for $n\le 3$and that it is not faithful for $n\ge 5$. We use forks and noodles homological techniques and Bokut–Vesnin generators to analyze the problem for $n=4$. We present a Conjecture implying faithfulness and a Lemma explaining the implication. We give some arguments suggesting why we expect the Conjecture to be true. Also, we give some geometrically calculated examples and information about data gathered using a C++ program.