Journal of Mathematics The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. A New Modified Three-Term Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence Wed, 13 Sep 2017 00:00:00 +0000 A new modified three-term conjugate gradient (CG) method is shown for solving the large scale optimization problems. The idea relates to the famous Polak-Ribière-Polyak (PRP) formula. As the numerator of PRP plays a vital role in numerical result and not having the jamming issue, PRP method is not globally convergent. So, for the new three-term CG method, the idea is to use the PRP numerator and combine it with any good CG formula’s denominator that performs well. The new modification of three-term CG method possesses the sufficient descent condition independent of any line search. The novelty is that by using the Wolfe Powell line search the new modification possesses global convergence properties with convex and nonconvex functions. Numerical computation with the Wolfe Powell line search by using the standard test function of optimization shows the efficiency and robustness of the new modification. Bakhtawar Baluch, Zabidin Salleh, Ahmad Alhawarat, and U. A. M. Roslan Copyright © 2017 Bakhtawar Baluch et al. All rights reserved. Recent Trends in Computational and Theoretical Aspects in Differential and Difference Equations Mon, 11 Sep 2017 07:50:48 +0000 Ram Jiwari, Mehmet Emir Koksal, and Haydar Akca Copyright © 2017 Ram Jiwari et al. All rights reserved. Inverse Commutativity Conditions for Second-Order Linear Time-Varying Systems Sun, 27 Aug 2017 00:00:00 +0000 The necessary and sufficient conditions where a second-order linear time-varying system is commutative with another system of the same type have been given in the literature for both zero initial states and nonzero initial states. These conditions are mainly expressed in terms of the coefficients of the differential equation describing system . In this contribution, the inverse conditions expressed in terms of the coefficients of the differential equation describing system have been derived and shown to be of the same form of the original equations appearing in the literature. Mehmet Emir Koksal Copyright © 2017 Mehmet Emir Koksal. All rights reserved. About Fixed Points in CAT(0) Spaces under a Combined Structure of Two Self-Mappings Wed, 19 Jul 2017 09:46:44 +0000 This paper investigates some fixed point-related questions including the sequence boundedness and convergence properties of mappings defined in spaces, which are parameterized by a scalar , where :   are nonexpansive Lipschitz-continuous mappings and is a metric space which is a space. Manuel De la Sen Copyright © 2017 Manuel De la Sen. All rights reserved. The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation Mon, 17 Jul 2017 09:04:11 +0000 The improved generalized tanh-coth method is used in nonlinear sixth-order solitary wave equation. This method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. The new exact solutions consisted of trigonometric functions solutions, hyperbolic functions solutions, exponential functions solutions, and rational functions solutions. The numerical results were obtained with the aid of Maple. M. Torvattanabun, J. Simmapim, D. Saennuad, and T. Somaumchan Copyright © 2017 M. Torvattanabun et al. All rights reserved. Some Properties of Serre Subcategories in the Graded Local Cohomology Modules Mon, 03 Jul 2017 00:00:00 +0000 Let be a standard homogeneous Noetherian ring with local base ring and let be a finitely generated graded -module. Let be the th local cohomology module of with respect to . Let be a Serre subcategory of the category of -modules and let be a nonnegative integer. In this paper, if then we investigate some conditions under which the -modules and are in for all . Also, we prove that if , then the graded -module is in for all . Finally, we prove that if is the biggest integer such that , then for all . Feysal Hassani and Rasul Rasuli Copyright © 2017 Feysal Hassani and Rasul Rasuli. All rights reserved. Orthogonal Symmetries and Reflections in Banach Spaces Wed, 14 Jun 2017 00:00:00 +0000 Let be a Banach space. We introduce a concept of orthogonal symmetry and reflection in . We then establish its relation with the concept of best approximation and investigate its implication on the shape of the unit ball of the Banach space by considering sections over subspaces. The results are then applied to the space of continuous functions on a compact set . We obtain some nontrivial symmetries of the unit ball of . We also show that, under natural symmetry conditions, every odd function is orthogonal to every even function in . We conclude with some suggestions for further investigations. Ali Jaballah and Fathi B. Saidi Copyright © 2017 Ali Jaballah and Fathi B. Saidi. All rights reserved. Properties of -Primal Graded Ideals Sun, 04 Jun 2017 07:12:32 +0000 Let be a commutative graded ring with unity . A proper graded ideal of is a graded ideal of such that . Let be any function, where denotes the set of all proper graded ideals of . A homogeneous element is -prime to if where is a homogeneous element in ; then . An element is -prime to if at least one component of is -prime to . Therefore, is not -prime to if each component of is not -prime to . We denote by the set of all elements in that are not -prime to . We define to be -primal if the set (if ) or (if ) forms a graded ideal of . In the work by Jaber, 2016, the author studied the generalization of primal superideals over a commutative super-ring with unity. In this paper we generalize the work by Jaber, 2016, to the graded case and we study more properties about this generalization. Ameer Jaber Copyright © 2017 Ameer Jaber. All rights reserved. Interaction of Traveling Curved Fronts in Bistable Reaction-Diffusion Equations in Sun, 07 May 2017 00:00:00 +0000 We consider the interaction of traveling curved fronts in bistable reaction-diffusion equations in two-dimensional spaces. We first characterize the growth of the traveling curved fronts at infinity; then by constructing appropriate subsolutions and supersolutions, we prove that the solution of the Cauchy problem converges to a pair of diverging traveling curved fronts in under appropriate initial conditions. Nai-Wei Liu Copyright © 2017 Nai-Wei Liu. All rights reserved. Toric Geometry of the Regular Convex Polyhedra Thu, 23 Mar 2017 00:00:00 +0000 We describe symplectic and complex toric spaces associated with the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational, and the regular icosahedron is neither simple nor rational. We remark that the last two cases cannot be treated via standard toric geometry. Fiammetta Battaglia and Elisa Prato Copyright © 2017 Fiammetta Battaglia and Elisa Prato. All rights reserved. Chromatic Numbers of Suborbital Graphs for the Modular Group and the Extended Modular Group Sun, 05 Mar 2017 09:44:49 +0000 This research studies the chromatic numbers of the suborbital graphs for the modular group and the extended modular group. We verify that the chromatic numbers of the graphs are or . The forest conditions of the graphs for the extended modular group are also described in this paper. Wanchai Tapanyo and Pradthana Jaipong Copyright © 2017 Wanchai Tapanyo and Pradthana Jaipong. All rights reserved. Non-Newtonian Comment of Lebesgue Measure in Real Numbers Tue, 28 Feb 2017 00:00:00 +0000 We would like to generalize to non-Newtonian real numbers the usual Lebesgue measure in real numbers. For this purpose, we introduce the Lebesgue measure on open and closed sets in non-Newtonian sense and examine their basic properties. Cenap Duyar and Birsen Sağır Copyright © 2017 Cenap Duyar and Birsen Sağır. All rights reserved. An Interesting Property of a Class of Circulant Graphs Mon, 27 Feb 2017 00:00:00 +0000 Suppose that and are two Cayley graphs on the cyclic additive group , where is an even integer, , , and are the inverse-closed subsets of . In this paper, it is shown that is a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum of . Finally, we show that if and is an even integer, then the adjacency matrix spectrum of is , , , (we write multiplicities as exponents). Seyed Morteza Mirafzal and Ali Zafari Copyright © 2017 Seyed Morteza Mirafzal and Ali Zafari. All rights reserved. A Mixed Discontinuous Galerkin Approximation of Time Dependent Convection Diffusion Optimal Control Problem Tue, 21 Feb 2017 00:00:00 +0000 In this paper, we investigate a mixed discontinuous Galerkin approximation of time dependent convection diffusion optimal control problem with control constraints based on the combination of a mixed finite element method for the elliptic part and a discontinuous Galerkin method for the hyperbolic part of the state equation. The control variable is approximated by variational discretization approach. A priori error estimates of the state, adjoint state, and control are derived for both semidiscrete scheme and fully discrete scheme. Numerical example is given to show the effectiveness of the numerical scheme. Qingjin Xu and Zhaojie Zhou Copyright © 2017 Qingjin Xu and Zhaojie Zhou. All rights reserved. The Mean Value for Infinite Volume Measures, Infinite Products, and Heuristic Infinite Dimensional Lebesgue Measures Mon, 30 Jan 2017 00:00:00 +0000 One of the goals of this article is to describe a setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure and of the concentration property on metric measured spaces, inspired from classical examples of means. In some cases, we get a linear extension of the limit at infinity. Then, the mean value on an infinite product is defined, first for cylindrical functions and secondly taking the uniform limit. Finally, the mean value for the heuristic Lebesgue measure on a separable infinite dimensional topological vector space (e.g., on a Hilbert space) is defined. This last object, which is not the classical infinite dimensional Lebesgue measure but its “normalized” version, is shown to be invariant under translation, scaling, and restriction. Jean-Pierre Magnot Copyright © 2017 Jean-Pierre Magnot. All rights reserved. A Generalization of the Krätzel Function and Its Applications Thu, 26 Jan 2017 00:00:00 +0000 In this paper, we introduce new functions as a generalization of the Krätzel function. We investigate recurrence relations, Mellin transform, fractional derivatives, and integral of the function . We show that the function is the solution of differential equations of fractional order. Neşe Dernek, Ahmet Dernek, and Osman Yürekli Copyright © 2017 Neşe Dernek et al. All rights reserved. Differential Calculus on -Graded Manifolds Tue, 17 Jan 2017 06:52:23 +0000 The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over -graded commutative rings and on -graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on -graded manifolds. We follow the notion of an -graded manifold as a local-ringed space whose body is a smooth manifold . A key point is that the graded derivation module of the structure ring of graded functions on an -graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body . Accordingly, the Chevalley–Eilenberg differential calculus on an -graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on -graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of -graded bundles. G. Sardanashvily and W. Wachowski Copyright © 2017 G. Sardanashvily and W. Wachowski. All rights reserved. Approximate Solution of Perturbed Volterra-Fredholm Integrodifferential Equations by Chebyshev-Galerkin Method Thu, 12 Jan 2017 09:31:33 +0000 In this work, we obtain the approximate solution for the integrodifferential equations by adding perturbation terms to the right hand side of integrodifferential equation and then solve the resulting equation using Chebyshev-Galerkin method. Details of the method are presented and some numerical results along with absolute errors are given to clarify the method. Where necessary, we made comparison with the results obtained previously in the literature. The results obtained reveal the accuracy of the method presented in this study. K. Issa and F. Salehi Copyright © 2017 K. Issa and F. Salehi. All rights reserved. Different Characterizations of Large Submodules of QTAG-Modules Tue, 03 Jan 2017 07:07:55 +0000 A module over an associative ring with unity is a -module if every finitely generated submodule of any homomorphic image of is a direct sum of uniserial modules. The study of large submodules and its fascinating properties makes the theory of QTAG-modules more interesting. A fully invariant submodule of is large in if , for every basic submodule of The impetus of these efforts lies in the fact that the rings are almost restriction-free. This motivates us to find the necessary and sufficient conditions for a submodule of a QTAG-module to be large and characterize them. Also, we investigate some properties of large submodules shared by -modules, summable modules, -summable modules, and so on. Fahad Sikander, Alveera Mehdi, and Sabah A. R. K. Naji Copyright © 2017 Fahad Sikander et al. All rights reserved. Mathematical Analysis of a Reactive Viscous Flow through a Channel Filled with a Porous Medium Tue, 27 Dec 2016 09:50:46 +0000 An investigation has been carried out to study entropy generation in a viscous, incompressible, and reactive fluid flowing steadily through a channel with porous materials. Approximate solutions for both velocity and temperature fields are obtained by using a rapidly convergent Adomian decomposition method (ADM). These solutions are then used to determine the heat irreversibility and Bejan number of the problem. Variations of other important fluid parameters are conducted, presented graphically, and discussed. Samuel O. Adesanya, J. A. Falade, J. C. Ukaegbu, and K. S. Adekeye Copyright © 2016 Samuel O. Adesanya et al. All rights reserved. Fuzzy Soft Compact Topological Spaces Sun, 25 Dec 2016 13:27:25 +0000 In this paper, we have studied compactness in fuzzy soft topological spaces which is a generalization of the corresponding concept by R. Lowen in the case of fuzzy topological spaces. Several basic desirable results have been established. In particular, we have proved the counterparts of Alexander’s subbase lemma and Tychonoff theorem for fuzzy soft topological spaces. Seema Mishra and Rekha Srivastava Copyright © 2016 Seema Mishra and Rekha Srivastava. All rights reserved. Graphs Generated by Measures Mon, 19 Dec 2016 09:32:32 +0000 In this paper, a graph is assigned to any probability measure on the -algebra of Borel sets of a topological space. Using this construction, it is proved that given any number (finite or infinite) there exists a nonregular graph such that its clique, chromatic, and dominating number equals . A. Assari and M. Rahimi Copyright © 2016 A. Assari and M. Rahimi. All rights reserved. Classifying Quadratic Forms Over in Three Variables Tue, 13 Dec 2016 08:40:35 +0000 The quadratic forms in three variables over the field are classified. Some remarks are made about the group of equivalences of the quadratic forms. Gerard Thompson Copyright © 2016 Gerard Thompson. All rights reserved. A Generalized Hermite-Hadamard Inequality for Coordinated Convex Function and Some Associated Mappings Tue, 13 Dec 2016 07:28:35 +0000 We have discussed the generalization of Hermite-Hadamard inequality introduced by Lupaş for convex functions on coordinates defined in a rectangle from the plane. Also we define that mappings are related to it and their properties are discussed. Atiq Ur Rehman, Gulam Farid, and Sidra Malik Copyright © 2016 Atiq Ur Rehman et al. All rights reserved. Some Relations between Isologic and Varietal Perfect Groups Mon, 12 Dec 2016 14:50:17 +0000 In 1940, Hall introduced the notion of -isologism, with respect to a given variety of groups . In the present article, we study the concepts of -perfect groups, -subgroup, and -quotient irreducible groups, with respect to a given variety of groups . Also we prove and obtain some results. Shokufeh Lotfi and S. Mostafa Taheri Copyright © 2016 Shokufeh Lotfi and S. Mostafa Taheri. All rights reserved. Differentiation Theory over Infinite-Dimensional Banach Spaces Thu, 08 Dec 2016 10:25:18 +0000 We study, for any positive integer and for any subset of , the Banach space of the bounded real sequences and a measure over that generalizes the -dimensional Lebesgue one. Moreover, we expose a differentiation theory for the functions defined over this space. The main result of our paper is a change of variables’ formula for the integration of the measurable real functions on . This change of variables is defined by some infinite-dimensional functions with properties that generalize the analogous ones of the standard finite-dimensional diffeomorphisms. Claudio Asci Copyright © 2016 Claudio Asci. All rights reserved. Sectional and Ricci Curvature for Three-Dimensional Lie Groups Sun, 04 Dec 2016 11:09:39 +0000 Formulas for the Riemann and Ricci curvature tensors of an invariant metric on a Lie group are determined. The results are applied to a systematic study of the curvature properties of invariant metrics on three-dimensional Lie groups. In each case the metric is reduced by using the automorphism group of the associated Lie algebra. In particular, the maximum and minimum values of the sectional curvature function are determined. Gerard Thompson and Giriraj Bhattarai Copyright © 2016 Gerard Thompson and Giriraj Bhattarai. All rights reserved. Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces Mon, 28 Nov 2016 13:31:02 +0000 We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry. Arnon Ploymukda and Pattrawut Chansangiam Copyright © 2016 Arnon Ploymukda and Pattrawut Chansangiam. All rights reserved. Presic Type Fixed Point Theorem for Four Maps in Metric Spaces Tue, 22 Nov 2016 06:27:48 +0000 We obtained a Presic type fixed point theorem for two pairs of jointly -weakly compatible maps in metric spaces. We also have given an example to illustrate our main theorem. K. P. R. Rao, Sk. Sadik, and S. Manro Copyright © 2016 K. P. R. Rao et al. All rights reserved. Multiple-Term Refinements of Young Type Inequalities Wed, 16 Nov 2016 13:34:34 +0000 Recently, a multiple-term refinement of Young’s inequality has been proved. In this paper, we show its reverse refinement. Moreover, we will present multiple-term refinements of Young’s inequality involving Kantorovich constants. Finally, we will apply scalar inequalities to operators. Daeshik Choi Copyright © 2016 Daeshik Choi. All rights reserved.