Common fixed points of multivalued $F$-contractions on metric spaces with a directed graph

Author: Mujahid Abbas / Monther R. Alfuraidan / Talat Nazir

Abstract: In this paper, we establish the existence of common fixed points of multivalued $F$-contraction mappings on a metric space endowed with a graph. An example is presented to support the results proved herein. Our results unify, generalize and complement various known comparable results in the literature. PDF |

On $\alpha$-nonexpansive mappings in Banach spaces

Author: David Ariza-Ruiz / Carlos Hernandez Linares / Enrique Llorens-Fuster / Elena Moreno-Galvez

Abstract: In 2011 Aoyama and Kohsaka introduced the $\alpha$-nonexpansive mappings.
Here we present a further study about them and their relationships with other classes
of generalized nonexpansive mappings. PDF |

On the structure of periodic complex Horadam orbits

Author: Ovidiu D. Bagdasar / Peter J. Larcombe / Ashiq Anjum

Abstract: Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat's spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures.
Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane.
Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multi-symmetric patterns can be recovered for selected parameter values. Some applications are also suggested. PDF |

On contour representation of two dimensional patterns

Author: I. T. Banu-Demergian / G. Stefanescu

Abstract: Two-dimensional patterns are used in many research areas in computer science, ranging from image processing to specification and verification of complex software systems (via scenarios). The contribution of this paper is twofold. First, we present the basis of a new formal representation of two-dimensional patterns based on contours and their compositions. Then, we present efficient algorithms to verify correctness of the contour-representation. Finally, we briefly discuss possible applications, in particular using them as a basic instrument in developing software tools for handling two dimensional words. PDF |

Systems of knowledge representation based on stratified graphs. Application in Natural Language Generation

Author: Daniela Dănciulescu / Mihaela Colho

Abstract: The concept of stratified graph introduces some method of knowledge representation
(see [ÈšÄƒndÄƒreanu, N., Knowledge representation by labeled stratified graphs, Proc. 8th World Multi-Conference on Systemics, Cybernetics and Informatics, 5 (2004), 345--350; ÈšÄƒndÄƒreanu, N., Proving the Existence of Labelled Stratified Graphs, An. Univ. Craiova Ser. Mat. Inform., 27 (2000), 81--92]). The inference process developed for this method uses the paths of the stratified graphs, an order between the elementary arcs of a path and some results of universal algebras. The order is defined by considering a structured path instead of a regular path.
In this paper we define the concept of system of knowledge representation as a tuple of the following components: a stratified graph $\mathcal G$, a partial algebra $Y$ of real objects, an embedding mapping (an injective mapping that embeds the nodes of $\mathcal G$ into objects of $Y$) and a set of algorithms such that each of them can combine two objects of $Y$ to get some other object of $Y$. We define also the concept of inference process performed by a system of knowledge processing in which the interpretation of the symbolic elements is defined by means of natural language constructions. In this manner we obtained a mechanism for texts generation in a natural language (for this approach, Romanian). PDF |

Second order differential equations with an irregular singularity at the origin and a large parameter: convergent and asymptotic expansions

Author: Chelo Ferreira / Jose L. Lopez / Ester Perez Sinusia

Abstract: We consider the second order linear differential equation
$$
y''=\left[{\Lambda^2\over t^\alpha}+g(t)\right]y,
$$
where $\Lambda$ is a large complex parameter and $g$ is a continuous function. In previous works we have considered the case $\alpha\in(-\infty,2]$ and designed a convergent and asymptotic method for the solution of the corresponding initial value problem with data at $t=0$. In this paper we complete the research initiated in those works and analyze the remaining case $\alpha\in(2,\infty)$. We use here the same fixed point technique; the main difference is that for $\alpha\in(2,\infty)$ the convergence of the method requires that the initial datum is given at a point different from the origin; for convenience we choose the point at the infinity. We obtain a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large $\Lambda$ (not of Poincar\'e-type) of the solution of the problem. The generalization to non-linear problems is straightforward. An application to a quantum mechanical problem is given as an illustration. PDF |

A note on some positive linear operators associated with the Hermite polynomials

Author: Grațyna Krech

Abstract: In this paper we give direct approximation theorems and the Voronovskaya type asymptotic formula for certain linear operators associated with the Hermite polynomials. These operators extend the well-known Sz\'{a}sz-Mirakjan operators. PDF |

A characterization of cone-convex vector-valued functions

Author: Daishi Kuroiwa / Nicolae Popovici / Matteo Rocca

Abstract: An interesting result in convex analysis, established by J.-P. Crouzeix in 1977, states that a real-valued function defined on a linear space is convex if and only if
each function obtained from it by adding a linear functional is quasiconvex. The aim of this paper is to extend this result for vector-valued functions taking values in
a partially ordered linear space. PDF |

Abstract: The main results of this paper give a connection between strong Jensen convexity and strong convexity
type inequalities. We are also looking for the optimal Takagi type function of strong convexity.
Finally a connection will be proved between the Jensen error term and an useful error function. PDF |

A new look on the truncated pentagonal number theorem

Author: Mircea Merca

Abstract: Two new infinite families of inequalities are given in this paper for the partition function $p(n)$, using the truncated pentagonal number theorem. PDF |

On the Stancu type bivariate approximation formula

Author: Dan Miclăuș

Abstract: In the present paper we establish the form of remainder term associated to the bivariate approximation formula for Stancu type operators, using bivariate divided differences. We also shall establish an upper bound estimation for the remainder term, in the case when approximated function fulfills some given properties. PDF |

Abstract: For certain classes of analytic functions in the open unit disk $U$, we study some convexity properties for a new general integral operator. Several corollaries of the main results are also considered. PDF |

Abstract: In this paper we consider an integral operator for analytic functions in the open unit disk and we derive the order of convexity for this integral operator, on certain classes of univalent functions. PDF |

Abstract: Consider a commutative diagram of bounded linear operators between Banach spaces
with exact rows.
In what ways are the spectral and local spectral properties of $B$ related to those of the pairs of operators $A$ and $C$?
In this paper, we give our answers to this general question using tools from local spectral theory. PDF |