Differential Inequalities and Univalent Functions

Let ${\mathcal M}$ be the class of analytic functions in the unit disk \ID with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition \left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. Functions in $\mathcal{M}$ are known to be univalent in \ID. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\mathcal{M}$ are shown to be starlike in \ID. However we conjecture that functions in $\mathcal{M}$ are not necessarily starlike, as apparently supported by other examples.Comment: 10 pages; To appear in Lobachevskii Journal of Mathematic

On the probability of finding marked connected components using quantum walks

Finding a marked vertex in a graph can be a complicated task when using quantum walks. Recent results show that for two or more adjacent marked vertices search by quantum walk with Grover's coin may have no speed-up over classical exhaustive search. In this paper, we analyze the probability of finding a marked vertex for a set of connected components of marked vertices. We prove two upper bounds on the probability of finding a marked vertex and sketch further research directions.Comment: 13 pages. To appear at Lobachevskii Journal of Mathematic

On tomographic representation on the plane of the space of Schwartz operators and its dual

It is shown that the set of optical quantum tomograms can be provided with the topology of Frechet space. In such a case the conjugate space will consist of symbols of quantum observables including all polynomials of the position and momentum operators.Comment: 9 page

Symmetric blind information reconciliation and hash-function-based verification for quantum key distribution

We consider an information reconciliation protocol for quantum key distribution (QKD). In order to correct down the error rate, we suggest a method, which is based on symmetric blind information reconciliation for the low-density parity-check (LDPC) codes. We develop a subsequent verification protocol with the use of $\epsilon$-universal hash functions, which allows verifying the identity between the keys with a certain probability.Comment: 4 pages; 1 figure; published versio

On generalization of Sierpiński gasket in Lobachevskii plane

© 2017, Pleiades Publishing, Ltd. We construct an analogue of Sierpiński gasket in Lobachevskii plane by means of iterated function system with maps from a transformation group of this space. The investigation of a new family of attractors and a Mandelbrot set associated with it reveals higher capacity of Lobachevskii geometry compared to that of Euclid

Meromorphization of M. I. Kinder’s Formula Via the Change of Contours

© 2018, Pleiades Publishing, Ltd. Parametrical families of the exterior inverse boundary value problems going back to well-known R. B. Salimov’s book became a plentiful source of new statements and methods in the study of the above problems. Critical points of conformal radii acting as the free parameters of such problems show interesting interrelations between their parametrical dynamics and geometric behavior. M.I. Kinder’s formula connecting the numbers of local maxima and saddles of a conformal radius is generalized here on the case when the derivative of the mapping function has zeros and poles in the unit disk and on its boundary

The Problem of Projecting the Origin of Euclidean Space onto the Convex Polyhedron

© 2018, Pleiades Publishing, Ltd. This paper is aimed at presenting a systematic exposition of the existing now different formulations for the problem of projection of the origin of the Euclidean space onto the convex polyhedron (PPOCP). We have concentrated on the convex polyhedron given as a convex hull of finitely many vectors of the space. We investigated the reduction of the projection program to the problems of quadratic programming, maximin, linear complementarity, and nonnegative least squares. Such reduction justifies the opportunity of utilizing a much more broad spectrum of powerful tools of mathematical programming for solving the PPOCP. The paper’s goal is to draw the attention of a wide range of research at the different formulations of the projection problem

Mixed solutions of monotone iterative technique for hybrid fractional differential equations

In this present work we concern with mathematical modelling of biological experiments. The fractional hybrid iterative differential equations are suitable for mathematical modelling of biology and also interesting equations since the structure are rich with particular properties. The solution technique is based on the Dhage fixed point theorem that describes the mixed solutions by monotone iterative technique in the nonlinear analysis. In this method we combine two solutions, namely, lower and upper solutions. It is shown an approximate result for the hybrid fractional differential equations in the closed assembly formed by the lower and upper solutions