Volume 69 , issue 1 ( 2017 ) | back |

Composition operators and their products on $L^{2}(\Sigma)$ | 1$-$11 |

**Abstract**

The paper gives measure-theoretic characterizations of classical properties of composition operators and their products on $L^2(\Sigma)$ such as complex symmetric and semi-Kato type operators.

**Keywords:** Kato operator; conditional expectation; composition operator; multiplication operator;
polar decomposition; closed range.

**MSC:** 47B20, 47B38

The computer modelling of gluing flat images algorithms | 12$-$22 |

**Abstract**

In this paper one of the important tasks of modern computer geometry is considered: creating effective algorithms for gluing different flat images of the same object. Images are obtained by central projection from different points of view. We use numerical simulation for comparison of three known algorithms for gluing---simple linear algorithm, normalized linear algorithm and direct algorithm. In each case stability to perturbations of the initial data and speed of calculations were estimated. The results confirm hypothesis of G.V. Nosovskií\ and E.S. Skripka that the direct algorithm proposed in their work [Error estimation for the direct algorithm of projective mapping calculation in multiple view geometry, Proceedings of the Conference ``Contemporary Geometry and Related Topics'', Belgrade, Serbia-Montenegro, June 26--July 2, 2005, Faculty of Mathematics, University of Belgrade, 2006, pp.~399--408] is the most accurate and fast one.

**Keywords:** gluing flat images; numerical simulation; linear algorithms;
the hypothesis of Nosovskií\ and Skripka.

**MSC:** 94A08

Fixed point results for $(\varphi,\psi)$-contractions in metric spaces endowed with a graph and applications | 23$-$38 |

**Abstract**

In the present work, we introduce the concepts of $(G,\varphi,\psi)$-contraction and $(G,\varphi,\psi)$-graphic contraction defined on metric spaces endowed with a graph $G$ and we show that these two types of contractions generalize a large number of contractions. Subsequently, we investigate various results which assure the existence and uniqueness of fixed points for such mappings. According to the applications of our main results, we further obtain a fixed point theorem for cyclic operators and an existence theorem for the solution of a nonlinear integral equation. Moreover, some illustrative examples are provided to demonstrate our obtained results.

**Keywords:** Fixed point; connected graph; Picard operator; $(\varphi,\psi)$-type contraction.

**MSC:** 47H10, 05C40, 54H25

Some fixed point theorems on S-metric spaces | 39$-$52 |

**Abstract**

In this paper, we present some contractive mappings and prove new generalized fixed point theorems on $S$-metric spaces. Also we define the notion of a cluster point and investigate fixed points of self-mappings using cluster points on $S$-metric spaces. We obtain new generalizations of the classical Nemytskii-Edelstein and Ćirić's fixed point theorems for continuous self-mappings of compact $S$-metric spaces.

**Keywords:** $S$-metric space; fixed point theorem; $C_{S}$-mapping; $L_{S}$-mapping; cluster point.

**MSC:** 54E35, 54E40, 54E45, 54E50

Some constructions of graphs with integral spectrum | 53$-$64 |

**Abstract**

A graph $G$ is said to be an integral graph if all the eigenvalues of the adjacency matrix of $G$ are integers. A natural question to ask is which graphs are integral. In general, characterizing integral graphs seems to be a difficult task. In this paper, we define some graph operations on ordered triple of graphs. We compute their spectrum and, as an application, we give some new methods to construct infinite families of integral graphs starting with either an arbitrary integral graph or integral regular graph. Also, we present some new infinite families of integral graphs by applying our graph operations to some standard graphs like complete graphs, complete bipartite graphs etc.

**Keywords:** Integral graphs; Kronecker product.

**MSC:** 05C50

Necessity of parameter randomization in quantum contract signing | 65$-$74 |

**Abstract**

We present a proof that randomization is necessary in quantum contract signing protocol of Paunković, Bauda and Mateus. We prove that for any fixed value of the protocol parameter $\alpha$, for large $N$ the probability of cheating can be as high as $25\%$, where $N$ is the number of messages exchanged between the parties, and thus without randomization the protocol is not fair.

**Keywords:** contract signing protocol; parameter randomization; qubit.

**MSC:** 81P94, 94A60