Volume 2 (2017): Issue 1 (January 2017)

This paper is a small review of Chebyshev’s method. The geometric interpretation as a generalization of Newton’s method is derived. Using this interpretation its global convergence is proved. Some dynamical properties are studied. As a higher order method, they are more complicated than in Newton’s method. Finally, some applications are revisited pointing out the advantages of Chebyshev’s method with respect Newton’s method.

Let G be a graph, and let k, r be nonnegative integers with k ≥ 2. A k-factor of G is a spanning subgraph F of G such that d_F(x) = k for each x ∈ V (G), where d_F(x) denotes the degree of x in F. For S ⊆ V (G), N_G(S) = ∪_x∊SN_G(x). The binding number of G is defined by bind(G)=min{|NG(S)||S|:∅≠S⊂V(G),NG(S)≠V(G)}$\begin{array}{} (G) = {\rm{min }}\{ \frac{{|{N_G}(S)|}}{{|S|}}:\emptyset \ne S \subset V(G),{N_G}(S) \ne V(G)\} \end{array}$. In this paper, we obtain a binding number and neighborhood condition for a graph to have a k-factor excluding a given r-factor. This result is an extension of the previous results.

Let E_β (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as

χαβ(G)=Σv1,v2⋅⋅⋅vβ+1∈Eβ(G)(dG(v1)dG(v2)...dG(vβ+1))α.$$\begin{array}{} \displaystyle ^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}. \end{array}$$

The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC₄C₈[p,q], tadpole graphs, wheel graphs and ladder graphs.

In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by

{ut−γ(t)Δut−Δu=gϵ(t,u), in Ωu=0, on ∂Ω.$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\ {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\ \end{array}\right. \end{array}$$

in a smooth bounded domain Ω ⊂ ℝⁿ, n ⩾ 3, where the terms g_ε are a small perturbation, in some sense, of a function f that depends only on u.

In this paper, we study 3–total edge product cordial (3–TEPC) labeling which is a variant of edge product cordial labeling. We discuss Web, Helm, Ladder and Gear graphs in this context of 3–TEPC labeling. We also discuss 3–TEPC labeling of some particular examples with corona graph.

In this paper we provide a method to prove the existence of weak solutions for a type of non-autonomous nonlocal reaction-diffusion equations. Due to the presence of the nonlocal operator in the diffusion term, we cannot apply the Monotonicity Method directly. To use it, we build an auxiliary problem with linear diffusion and later, through iterations and compactness arguments, we show the existence of solutions for the nonlocal problem.

In this paper, we compute first Zagreb index (coindex), second Zagreb index (coindex), third Zagreb index, first hyper-Zagreb index, atom-bond connectivity index, fourth atom-bond connectivity index, sum connectivity index, Randić connectivity index, augmented Zagreb index, Sanskruti index, geometric-arithmetic connectivity index and fifth geometric-arithmetic connectivity index of the line graphs of Banana tree graph and Firecracker graph.

Each year a large number of new diseases were found worldwide, which requires the development of new drugs to cure these diseases. In this process, researchers need to do a lot of work to test the effectiveness of new drugs and side effects. Due to the intrinsic connection between the characteristics of compound and its molecular structure, methods of pharmaceutical theory are widely used in the analysis of the features of the drug. By calculating the chemical indices of drug molecular structure, scientists could learn the chemistry and pharmacy characteristics of the corresponding drugs. In this paper, from the theoretical perspective, we state the following conclusions: (1) the exact expression of generalized degree distance for starlike tree is determined; (2) the eccentricity related indices of hetrofunctional dendrimer are discussed. The results obtained have broad application prospects in the pharmaceutical sciences.

In information science and other engineering applications, ontology plays an irreplaceable role to find the intrinsic semantic link between concepts and to determine the similarity score returned to the user. Ontology mapping aims to excavate the intrinsic semantic relationship between concepts from different ontologies, and the essence of these applications is similarity computation. In this article, we propose the new ontology sparse vector approximation algorithms based on the affine transformation tricks. By means of these techniques, we study the equivalent form of ontology dual problem and determine its feasible set. The simulation experiments imply that our new proposed ontology algorithm has high efficiency and accuracy in ontology similarity computation and ontology mapping in biology, chemical and related fields.

P. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm to square complex matrices with respect to consimilarity transformations A↦SAS¯−1$\begin{array}{} \displaystyle A \mapsto SA{\bar S^{ - 1}} \end{array}$ and to pairs of m × n complex matrices with respect to transformations (A,B)↦(SAR,SAR¯)$\begin{array}{} \displaystyle (A,B) \mapsto (SAR,SB\bar R) \end{array}$, in which S and R are nonsingular matrices.

In QSAR/QSPR study, topological indices are utilized to guess the bioactivity of chemical compounds. In this paper, we study the QSPR analysis of certain graph theocratical matrices and their corresponding energy. Our study reveals some important results which helps to characterize the useful topological indices based on their predicting power.

The results of single television observations of Perseid meteor shower in 2016 are presented. The Perseid shower occurs from 17 July to 24 August, peaking on or around August 12 every year. In 2016, the peak of the Perseids was Night of Aug 11 to the morning of Aug 12. The meteor video observations in Egypt are carried out at The National Researcher Institute of Astronomy and Geophysics (NRIAG). The system consists of TV - cameras Watec -902H Ultimate with the lens DV10x8SA-1 (8-80 mm (10x)) capable of recording the rapid motion of meteors entering the Earth atmosphere.

We consider a chemotaxis-reaction-diffusion system that models the dynamics of colonies of Bacillus subtilis on thin agar plates. The system of equations was proposed by Leyva et al. [14], based on a previous non-chemotactic model by Kawasaki and collaborators [9], which reproduces the dense branching patterns observed experimentally in the semi-solid agar, low-nutrient regime. Numerical simulations show that, when the chemotactic sensitivity toward nutrients is increased, the morphology of the colony changes from a dense branched pattern to a uniform envelope that propagates outward. Here, we provide a quantitative argument that explains this change in morphology. This result is based on energy estimates on the spectral equations for perturbations around the envelope front, suggesting the suppression of colony branching as a result of the stabilizing effect of the increasing chemotactic signal.

Recently, nanostructures have opened new dimensions in industry, electronics, and pharmaceutical and biological therapeutics. The topological indices are numerical tendencies that often depict quantitative structural activity/property/toxicity relationships and correlate certain physico-chemical properties such as boiling point, stability, and strain energy, of respective nanomaterial. In this article, we established closed forms of various degree-based topological indices of semi-total line graph of 2D-lattice, nanotube and nanotorus of TUC₄C₈[r, s].

We report the results of spectral and photometric observations of DQ Her. Available spectra from International Ultraviolet Explorer (IUE), Hubble Space Telescope Faint Object Spectrograph (HST - FOS) and CCD photometry of one night were used. Some profiles revealing the variations of some spectral lines at different times are presented. There is variation with time for photometric observations and the brightness of DQ Her is changed from 14 mag. to 17.7 mag with clear display of burst. The ultraviolet luminosity for emitting region is in the range of (1.9x10³¹erg s⁻¹). The mass accretion rate is in the range of (3.5x10⁻¹²M_⊙yr⁻¹). The line flux modulations at different times can be explained in terms of the accretion curtain model for intermediate polars, Kim & Beuermann (1996).

In this paper further result on odd mean labeling is discussed. We prove that the two star G = K_1,m ∧ K_1,n is an odd mean graph if and only if |m − n| ≤ 3. The condition for a graph to be odd mean is that p ≤ q + 1, where p and q stands for the number of the vertices and edges in the graph respectively.

We present a prototype task-parallel algorithm for the solution of hierarchical symmetric positive definite linear systems via the ℋ-Cholesky factorization that builds upon the parallel programming standards and associated runtimes for OpenMP and OmpSs. In contrast with previous efforts, our proposal decouples the numerical aspects of the linear algebra operation from the complexities associated with high performance computing. Our experiments make an exhaustive analysis of the efficiency attained by different parallelization approaches that exploit either task-parallelism or loop-parallelism via a runtime. Alternatively, we also evaluate a solution that leverages multi-threaded parallelism via the parallel implementation of the Basic Linear Algebra Subroutines (BLAS) in Intel MKL.

In this paper the randomized Cauchy-Euler differential equation is studied. With this aim, from a statistical point of view, both the first and second probability density functions of the solution stochastic process are computed. Then, the main statistical functions, namely, the mean, the variance and the covariance functions are determined as well. The study includes the computation of the first and second probability density functions of the regular-singular infinite point via an adequate mapping transforming the problem about the origin. The study is strongly based upon the Random Variable Transformation technique along with some results that have been recently published by some of authors to the random homogeneous linear second-order differential equation. Finally, an illustrative example is shown.

Determination of the reactor kinetic characteristics is very important for the design and development of a new reactor system. In this sense, the computation of lambda modes associated to a nuclear power reactor has interest since these modes can be used to analyze the reactor criticality and to develop modal methods to analyze transient situations in the reactor. In this paper, the lambda problem has been discretized using a high order finite element method to obtain a generalized algebraic eigenvalue problem. A multilevel method is proposed to solve this generalized eigenvalue problem combining a hierarchy of meshes with a Modified Block Newton method. The Krylov-Schur method is used to compare the efficiency of the multilevel method solving several benchmark problems.

Numerical computation of fractional order derivatives and integrals can be done by applying many approaches. They include Riemann-Liouville and Caputo formulas. Both formulas include fractional order integration and integer order differentiation but in a different order: fractional order integration can be preceded by integer order differentiation or it can be done in the reverse order. In the paper we apply both formulas for FOD computation and investigate if an application of a particular order have any advantages, e.g. resulting in higher accuracy. The outcome of the experiment is that by applying adequately selected formula of FOD computation according to the shape of a particular function, the accuracy of computation can often be increased. Application of Riemann-Liouville formula enables successful FOD computation for functions which n^th derivative does not exist.

We extend Theorem 1.1 of [J. Math. Anal. Appl. 435 (2016), 1738-1752] to show the uniqueness of large solutions for the system of (1) in star-shaped domains. This result is due to the maximum principle for cooperative systems of [J. López-Gómez and M. Molina-Meyer, Diff. Int. Eq. 7 (1994), 383-398], which allows us to establish the uniqueness without invoking to the blow-up rates of the solutions.

This paper is concerned on the distribution of a homogeneous isotropic elastic medium with diffusion under the effect of Three-phase-lag model. Normal mode analysis is used to express the exact expressions for temperature, displacements and stresses functions. Comparisons are made in the absence and presence of diffusion with some theories like Three-phase-lag and GNIII.

The algebraic structure of the linear system appears in solving fractional order Poisson’s equation by Haar wavelet collocation approach is considered. The fractional derivative is described in the Caputo sense. Comparison with the classical integer case as a limiting process is illustrated. Numerical comparison is made between the solution using the Haar wavelet method and the finite difference method. The results confirms the accuracy for the Haar wavelet method.

For a (molecular) graph, the first Zagreb index is equal to the sum of squares of the degrees of vertices, and the F-index is equal to the sum of cubes of the degrees of vertices. In this paper, we introduce sixty four new operations on graphs and study the first Zagreb index and F-index of the resulting graphs.

It is still a challenge to improve the efficiency and effectiveness of image denoising and enhancement methods. There exists denoising and enhancement methods that are able to improve visual quality of images. This is usually obtained by removing noise while sharpening details and improving edges contrast. Smoothing refers to the case of denoising when noise follows a Gaussian distribution.

Both operations, smoothing noise and sharpening, have an opposite nature. Therefore, there are few approaches that simultaneously respond to both goals. We will review these methods and we will also provide a detailed study of the state-of-the-art methods that attack both problems in colour images, separately.