Transactions of A Razmadze Mathematical Institute最新文献

In this paper, we study the iterative method of Aitken type for solving the non-linear equations, in which the interpolation nodes are controlled by variant of Newton method or by a general method of order $p.$ By combining such methods with a generalized secant method, it is shown that the order of convergence can be increased to as high as desired and also in the limiting case efficiency of the method is 2. Several numerical examples are provided in support of the theoretical results.

In this paper, we recall Ostrowski’s inequality, Hadamard’s inequality and the definition of $log$-convex functions. We also mention an useful integral identity in the first part of our study. The second part of our study includes new results. We prove new generalizations for $log$-convex functions. Several new Ostrowski type inequalities have been established and some special cases have been given by choosing $h=0$ or $x=\frac{a+b}{2}.$

In the presented work, some procedures, usually used in modern algorithms of unconstrained optimization, are added to Polyak’s heavy ball method. Namely, periodical restarts, which guarantees monotonic decrease of the objective function along successive iterates, while restarts involve updating of the step size on the base of line search method.

For smooth objective functions, the Heavy Ball (briefly HB) and Modified Heavy Ball (briefly MHB) algorithms are described along with the problem of simplifying the form of used line-search algorithm (without changing its content). MHB and the set of test functions are implemented in C++. The set of test functions contains 44 functions, taken from Cuter/st. Solver CG_DESCENT-C-6.8 was used for MHB benchmarking. Test-functions and other materials, related to benchmarking, are uploaded to GitHub: https://github.com/kobage/.

In case of smooth and convex objective function, the convergence analysis is concentrated on reducing transformations and their orbits. A concept of reducing transformation allows us to investigate algebraic structure of convergent methods.

In the present paper the linear theory of viscoelasticity for Kelvin–Voigt materials with double porosity is considered. Some basic results on the solutions of the quasi-static and steady vibrations equations are obtained. Indeed, the fundamental solutions of the systems of equations of quasi-static and steady vibrations are constructed by elementary functions and their basic properties are established. Green’s formulae and the integral representation of regular solution in the considered theory are obtained. Finally, a wide class of the internal boundary value problems of quasi-static and steady vibrations is formulated and on the basis of Green’s formulae the uniqueness theorems for classical solutions of these problems are proved.

For $q\in (0,1)$, the $q$-deformation of the square white noise Lie algebra is introduced using the $q$-calculus. A representation of this Lie algebra is given, using the $q$-derivative (or Jackson derivative) and the multiplication operator. The free square white noise Lie algebra is defined. Moreover, its representation on the Hardy space is given.

In this paper, we study certain classes of nested fractional boundary value problems including both of the Riemann–Liouville and Caputo fractional derivatives. In addition, since we will use the signed-power operators ${\varphi }_{\nu }z≔|z{|}^{\nu -1}z,\nu \in (0,\infty )$ in the governing equations, so our desired boundary value problems possess half-linear nature. Our investigation theoretically reaches so called Lyapunov inequalities of the considered nested fractional boundary value problems, while in viewpoint of applicability using the obtained Lyapunov inequalities we establish some qualitative behavior criteria for nested fractional boundary value problems such as a disconjugacy criterion that will also be used to establish nonexistence results, upper bound estimation for maximum number of zeros of the nontrivial solutions and distance between consecutive zeros of the oscillatory solutions. Also, considering corresponding nested fractional eigenvalue problems we find spreading interval of the eigenvalues.

The Wiener–Hopf factorization theorem for rational matrices is proved with respect to very general contours using purely algebraic method.

In the paper, the authors significantly and meaningfully simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first and second kinds.

Each conjugacy class of actions of the triangle group $\Delta (3,n,k)$ over the projective line $PL\left(F_q\right)$ can be represented by a coset diagram $D(\theta ,q)$, where $\theta \in F_q$ and $q$ is a prime number. In this paper, we have considered conjugacy classes which arise from the actions of $\Delta (3,n,k)=〈r,s:r^3=s^n={\left(rs\right)}^k=1〉$ over $PL\left(F_q\right)$, where $F_q$ is finite field. The points of $PL\left(F_q\right)$ are the elements of $F_q$ together with the additional point $\infty .$

In this paper an anisotropic grand Sobolev–Morrey spaces are introduced. With the help of integral representation we study differential and differential-difference properties of functions from these spaces.