Department of Mathematics

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Fundamental ideas and mathematical basis of ontology learning algorithm
(Journal of Intelligent & Fuzzy Systems(Preprint), 2018) Zhu, Linli; Hua, Gang; Sohail Zafar; Pan, Yu
As a data utility and aided tool, ontology has been widely used in many areas of the computer. Owing to its great efficiency, ontologies have also been introduced into various engineering disciplines. In this paper, we present the fundamental ideas of how to deal with similarity measuring problem in ontology learning algorithms. The mathematical basis of ontology learning algorithms is also introduced from a statistical learning theory point of view. Finally, we present two ontology learning algorithms in multi-dividing setting and ontology sparse vector learning setting, respectively.
Approximations for soft fuzzy rough sets.
(Scientific publications of the state University of NOVI Pazar, Scientific, 2016) Tabasam Rashid
In this paper, we introduce a modified soft fuzzy rough set model. The lower and upper approximation operators are presented and their related properties are investigated. It is shown that these new models of approximations are finer than already known in the literature.
Existence of solutions to second order nonlinear coupled system with nonlinear coupled boundary conditions
(Electronic Journal of Differential Equations, 2015) Naseer Ahmad Asif; Imran Talib
In this article, study the existence of solutions for the second-order nonlinear coupled system of ordinary differential equations u 00(t) = f(t, v(t)), t ∈ [0, 1], v 00(t) = g(t, u(t)), t ∈ [0, 1], with nonlinear coupled boundary conditions φ(u(0), v(0), u(1), v(1), u0 (0), v0 (0)) = (0, 0), ψ(u(0), v(0), u(1), v(1), u0 (1), v0 (1)) = (0, 0), where f, g : [0, 1] × R → R and φ, ψ : R6 → R2 are continuous functions. Our main tools are coupled lower and upper solutions, Arzela-Ascoli theorem, and Schauder’s fixed point theorem. The results presented in this article extend those in [1, 3, 15].
Edge version of harmonic index and harmonic polynomial of some classes of graphs.
(Journal of Applied mathematics and Informatics, Lifecscience Global., 2017) Sohail Zafar; Rabia Nazir; Muhammad Shoaib Sardar; Zohaib Zahid
In this paper we define the edge version of harmonic index and harmonic polynomial of a graph G. We computed explicit formulas for the edge version of harmonic index and harmonic polynomial of many well known classes of graphs.
Solution of 7 bar tress model using derivative free methods.
(Proceedings of the Pakistan Academy of Sciences, 2015) Muhammad Saeed; Muhammad Farhan Tabassum
The focus of this research is to formulate optimization model of 7-bar trusses along with stress, stability and deflection constraints. The derivative free methods are used for the optimization of engineering design problems. These methods are basically designed for unconstrained optimization problems. In formulated optimization truss problems the constraints are handled by using exterior penalty functions. The results of the truss optimization model are obtained by using MATLAB which demonstrate the effectiveness and applicability of these derivative free methods.
Modified Adomian Decomposition Method and Homotopy Perturbation Method for Higher-Order Singular Boundary Value Problems
(Science International, 2015) Tabassum, Muhammad Farhan
In this paper, we presented effective numerical methods for the approximate solution of nonlinear higher order Boundary Value Problems (BVPs). We proposed a reliable modification of the Adomian Decomposition Method (MADM) that will accelerate the rapid convergence of the series solution. He,s polynomials are also used to overcome the difficult calculation. We also proposed the Homotopy Perturbation Method (HPM) which is highly accurate in witch only a few terms are required to obtain accurate computable solution. The validity of the methods is verified through illustrative examples. The results obtained demonstrate the accuracy and efficiency of the proposed methods.
New Mathematical Models of N-Queens Problem and its solution by a Derivative-Free Method of Optimization
(Science International, 2015) Ali, Javaid; Muhammad Saeed; Tabassum, Muhammad Farhan
Most of the optimization methods do not inherit convergence proofs. One of the measures to rank them is their potential to solve challenging problems specially formulated for this purpose. In this paper the problem involves two main issues. Firstly we present new formulations of the N queens’ configuration problem as optimization problems and secondly we modify a derivative free method so that it may be able to find the optimal configuration of the chessboard.
Use of modified equation to examine the stability of upstream differencing scheme for initial value problems
(Science International,27(3), 2015) Muhammad Farhan Tabassum
In this paper we investigate the quantitative behavior of a wide range of numerical methods for solving linear partial differential equations [PDE’s]. In order to study the properties of the numerical solutions, such as accuracy, consistency, and stability, we use the method of modified equation, which is an effective approach. To determine the necessary and sufficient conditions for computing the stability, we use a truncated version of modified equation which helps us in a better way to look into the nature of dispersive as well as dissipative errors. The Wave Equation arises in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems and in simple modals for gas dynamics, whose solution involves the method of characteristics. Accuracy and stability of Upstream Scheme is checked by using Modified Differential Equations [MDE’s].
Odd Graceful Labeling of Acyclic Graphs.
(American Journal of Applied Mathematics, 2015-06-09) Riasat, Ayesha
Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q− 1} such that, when each edge xy is assigned the label |f (x)− f (y)| , the resulting edge labels are {1, 3, 5, . . . , 2q− 1} and f is called an odd graceful labeling of G. Motivated by the work of Z. Gao [6] in which he studied the odd graceful labeling of union of any number of paths and union of any number of stars, we have determined odd graceful labeling for some other union of graphs. In this paper we formulate odd-graceful labeling for disjoint unions of graphs consisting of generalized combs, stars, bistars and paths.
Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation
(American Journal of Applied Mathematics, 2015-06-12) Asjad, M. Imran
In this paper the velocity field and the adequate shear stress corresponding to the rotational flow of an Oldroyd-B fluid, between two infinite coaxial circular cylinders, are determined by applying the finite Hankel transforms. The motion is produced by the inner cylinder that, at time t = 0+ , is subject to a time-dependent rotational shear stress. The solutions that have been obtained are presented under series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting case of general solutions. Finally, the influence of the pertinent parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations
Group decision making using comparative linguistic expression based on hesitant intuitionistic fuzzy sets.
(Applications & Applied Mathematics, 2015) Tabasam Rashid
We introduce a method for aggregation of experts’ opinions given in the form of comparative linguistic expression. An algorithmic form of technique for order preference is proposed for group decision making. A simple example is given by using this method for the selection of the best alternative as well as ranking the alternatives from the best to the worst.
A geometric aggregation operator for decision making
(Journal of Computer Science, 2015) Tabasam Rashid
We proposed an aggregation operator which is used to aggregate decision makers’ opinions in group decision making process. First, a Choquet integral-based distance between generalized interval-valued trapezoidal fuzzy numbers is defined. Then combining the generalized intervalvalued trapezoidal fuzzy number aggregation operator with Choquet integral-based distance, an extension of technique for order preference by similarity to ideal solution method is developed to deal with a multi-criteria generalized intervalvalued trapezoidal fuzzy number group decision making problems, where inter-dependent or interactive characteristics among criteria preference of decision makers are also considered. Finally, an illustrative example is provided to elaborate the proposed method.
Fate of Electromagnetic Field on the Cracking of PSR J1614-2230 in Quadratic Regime.
(Advances in High Energy Physics, 2015-09-07) Mardan, Syed Ali; Rehman, Muhammad Aziz ur
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3 . We study the cracking of compact object PSR J1614-2230 in quadratic regime with electromagnetic field. For this purpose, we develop a general formalism to determine the cracking of charged compact objects. We apply local density perturbations to hydrostatic equilibrium equation as well as physical variables involved in the model. We plot the force distribution function against radius of the star with different parametric values of model both with and without charge. It is found that PSR J1614-2230 remains stable (no cracking) corresponding to different values of parameters when charge is zero, while it exhibits cracking (unstable) when charge is introduced. We conclude that stability region increases as amount of charge increases.
Cracking of compact objects with electromagnetic field.
(Astrophysics and Space Science, 2015) Syed Ali Mardan; Muhammad Aziz Ur Rehman
In this paper, we investigate the role of electromagnetic field on the stability regions of charged selfgravitating compact objects by using the concept of cracking. For this purpose, we have applied local density perturbation scheme to the hydrostatic equilibrium equation as well as on physical parameters involved in the model. In particular, we have examined the cracking of charged compact objects like PSR J1614-2230, PSR J1903+327, Vela X-1, SMC X-1 and Cen X-3 with different values of charge. We conclude that these objects exhibit cracking and stability regions decreases with the increase of charge.
Self-gravitating spherically symmetric fluid models in Brans–Dicke gravity.
(General Relativity and Gravitation., 2015-08-06) Manzoor, Rubab
This paper is devoted to study self-gravitating spherically symmetric fluid models in Brans–Dicke gravity. We formulate a set of equations which govern the dynamics of evolving gravitating fluids through Weyl tensor, shear tensor, expansion scalar, anisotropy, energy inhomogeneity, dissipation as well as scalar field. We also discuss some particular cases according to different dynamical conditions. It is concluded that fluid models for regular distribution of scalar field are consistent with general relativity and models due to irregular distribution of scalar field deviate from theory of general relativity.
Initial ideal of binomial edge ideal in degree 2.
(Novi Sad J. Math, 2015) Zohaib Zahid
We study the initial ideal of binomial edge ideal in degree 2 ([in<(JG)]2), associated to a graph G. We computed dimension, depth, Castelnuovo-Mumford regularity, Hilbert function and Betti numbers of [in<(JG)]2 for some classes of graphs.
Spanning Simplicial Complexes of Uni-Cyclic Graphs.
(Algebra Colloquium, 2015) Kashif, Agha
In this paper, we introduce the concept of spanning simplicial complexes ∆s(G) associated to a simple finite connected graph G. We give the characterization of all spanning trees of the uni-cyclic graph Un,m. In particular, we give the formula for computing the Hilbert series and h-vector of the Stanley Riesner ring k ∆s(Un,m) . Finally, we prove that the spanning simplicial complex ∆s(Un,m) is shifted hence ∆s(Un,m) is shellable. Key words : Primary Decomposition, Hilbert Series, f-vectors, h-vectors, spanning Trees
Positive Solutions of a Singular System with Two Point Coupled Boundary Conditions
(American Journal of Applied Mathematics, 2015-06-12) Asif, Naseer Ahmad
In this paper, we study the existence of positive solutions to a system of nonlinear differential equations subject to two-point coupled boundary conditions. Further, the nonlinearities are allowed to be singular with respect to first order derivatives. An example is included to show the applicability of our result.
Existence of solutions to a second order coupled system with nonlinear coupled boundary conditions
(American Journal of Applied Mathematics, 2015) Naseer Ahmad Asif; Imran Talib
In this article, study the existence of solutions for the second-order nonlinear coupled system of ordinary differential equations u 00(t) = f(t, v(t)), t ∈ [0, 1], v 00(t) = g(t, u(t)), t ∈ [0, 1], with nonlinear coupled boundary conditions φ(u(0), v(0), u(1), v(1), u0 (0), v0 (0)) = (0, 0), ψ(u(0), v(0), u(1), v(1), u0 (1), v0 (1)) = (0, 0), where f, g : [0, 1] × R → R and φ, ψ : R6 → R2 are continuous functions. Our main tools are coupled lower and upper solutions, Arzela-Ascoli theorem, and Schauder’s fixed point theorem. The results presented in this article extend those in [1, 3, 15].
Effect of heat sink/source on peristaltic flow of jeffrey fluid through a symmetric channel.
(Alexandria Engineering Journal, Elsevier, 2015) M. Rehman; Aun Haider
t The peristaltic flow and heat transfer through a symmetric channel in the presence of heat sink/source parameter have been analyzed in this paper. It also deals with the effect of the natural convection coefficient in the momentum equation. Low Reynolds number and small wave number approximation are used to convert the non-linear partial differential equations into the non-linear ordinary differential equations. In order to solve the governing model, perturbation method has been chosen by taking a (material parameter) as a small parameter. Expressions have been obtained for temperature, velocity, stream function, pressure rise and frictional forces. The features of the flow characteristics are analyzed by plotting graphs and the results are discussed in details. It has been observed that velocity increases with an increase of a (material parameter). The peristaltic pumping and in the copumping region the pumping rate decreases by increasing the value of a (material parameter). The size of the trapped bolus decreases by increasing the value of a (material parame

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