VFAST Transactions on Mathematics
https://vfast.org/journals/index.php/VTM
<p>The <strong><em>VFAST Transactions on Mathematics</em></strong> is a yearly peer-reviewed scientific journal published by the VFAST-Research Platform. It was established in 2013 and covers the areas of Pure and Applied Mathematics and its applications. <span>The goal of this journal is to provide an international platform to Mathematicians all over the world to promote, share, and discuss various new issues and development in the field of Mathematics and its application.</span></p><p>ISSN: 2309-0022 (Online), 2411-6343 (Print)</p><p>Editor in Chief: Dr. Muhammad Arif, Associate Professor, Abdul Wali Khan University Mardan</p><p><strong><span style="text-decoration: underline;"><span class="csspropertycolor" style="box-sizing: inherit; color: red; font-family: Consolas, Menlo, 'courier new', monospace; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff; 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src="data:image/png;base64,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" alt="" /></p><p><span>This work is licensed under a </span><a href="http://creativecommons.org/licenses/by/3.0/" rel="license" target="_blank">Creative Commons Attribution License</a> CC BY</p><p> </p>Power Hamy Mean Operators for managing Cubic Linguistic Spherical Fuzzy Sets and their Applications
https://vfast.org/journals/index.php/VTM/article/view/1147
<p>In modern social administrative economic activities, we are facing a considerable amount of multi-attribute group decision making problems. The methods and theory related to this method are very useful in the field of particular disciplines as well as in operational research, and a lot of achievements have been described. Obviously the real world is full of uncertainties and classical set theory cannot be used to describe different phenomena such as beauty, intelligence, height (tallness) and age etc. This thing leads mathematicians to develop the notion of fuzzy sets. Later Zadeh introduced the concept of membership and non-membership degree. Definitely human opinion about a phenomenon may be unidirectional or multi-directional, that’s why Atanossov proposed the concept of another advance type of fuzzy sets, which is known as intuitionistic fuzzy sets. His concept is based on a degree of membership and degree of non-membership with a exquisite that their sum must not exceed 1. In our work we introduced cubic linguistic spherical fuzzy sets. Then, we proposed the fundamental operation law for CLSFVs and a series of their average operators (AOs), such as the (cubic linguistic spherical fuzzy power average), (cubic linguistic spherical fuzzy power weighted average), (cubic linguistic spherical fuzzy power hamy mean) and (cubic linguistic spherical fuzzy power weighted hamy mean) operators, was developed by combining the power average and hamy mean operators in cubic linguistic spherical fuzzy environment. Also we described some specific desirable properties of all these operators. In addition, we suggested a new MAGDM method.</p><p><strong> </strong></p>Tasawar AbbasFaisal MumtazZamir HussainRehan Zafar2022-11-282022-11-2810Assessment of the COVID-19 Pandemic's Impact on Gasoline Prices in Pakistan
https://vfast.org/journals/index.php/VTM/article/view/1153
<p>Abstract The COVID-19 virus is a pandemic that, from the outset, alters its appearance and symptoms. It has aggressively spread around the world. The COVID-19-induced fear and uncertainty are disrupting the global economy and exacerbating ﬁnancial market volatility. The most impacted countries were the United States, the United Kingdom, India, and Pakistan. The continuing COVID-19 situation is both a public health and economic concern on a worldwide. This research aims at how the spread of the COVID-19 has affected the cost of gasoline,<br />diesel, and liqueﬁed petroleum gas (LPG). Every week, statistics on COVID-19 instances and pricing are collected. The data was analyzed using the ARDL model and the Bound test to determine the short and long-term association between COVID-19 and prices. The Autoregressive distributive lag model ﬁndings reveal that conﬁrmed and mortality cases impact fuel, diesel, and LPG prices.</p>Muhammad BilalMuhammad AamirSaleem AbdullahNoor MahmoodUmair KhalilNida KhalidMaqbool AhmedMuhammad NaeemShakoor MuhammadLaiba Sultan Dar2022-11-172022-11-1710Analysis of Entropy-dependent solitons in ETG-driven magneto plasma by Variational Iteration Method
https://vfast.org/journals/index.php/VTM/article/view/1079
<p>Nowadays, the discovery of the link between entropy and plasma density and temperature opens up new avenues for mathematicians and researchers to examine alternative plasma models in terms of entropy. The linear dispersion relation and KdV equation are created after the entropy drift is taken into account in the ETG mode. In addition, the Variational Iteration Technique (VIM) is used to determine the problem’s analytical solutions. Involving the Langrange multiplier also helps to accelerate computation and lower its cost. Then, it is shown that in ETG mode, entropy impacts both the breadth and amplitude of rarefactive solitons, as well as the impact of inhomogeneity drift and the magnetic ﬁeld on the conﬁguration of solitons. In this instance, however, only the rarefactive solitons are present. Since introducing entropy to the system might alter the previous plasma ﬁndings, this study is new. Finally, the current approach will be applied to the entropy-based in magnetically restrained plasmas, solitary waves can be observed. The graphical ﬁndings are also provided using Maple-18.</p>Tasawar AbbasEhsan Ul HaqAmbreen AyubAsadullah Dawood2022-11-142022-11-1410Applying the Crank-Nicolson’s scheme to discretize the laplacian operator in polar gird system on extendable stencil 5-points to 9-points including symmetric and asymmetric lines.
https://vfast.org/journals/index.php/VTM/article/view/1151
Abstract Numerous techniques exist for solving and describing the Partial differential equation’s mathematical and computational model. The Laplacian operator is one of the most effective techniques for solving linear and nonlinear partial differential equations. It is quick, and researchers use it frequently because of its modern technique and high accuracy in results. The Crank-Nicolson (CN) scheme in the Cartesian coordinate system has been discussed in this research work. Using this method, a numerical approximation scheme in Cartesian coordinate<br />system has been discretized on a 5 point stencil, extendable to nine points. The Tailor Series was used to discretize this scheme on 5-point stencils, which will be used in FORTRAN code for numerical approximation and can be visualized in OPEDX software. The Nicolson scheme is a ﬁnite difference scheme used to solve partial differential equations such as heat, wave, and diffusion equations in both 1-D and 2-D. Because of his extendable stencil, it will create accuracy and stability in the novel results of the scheme. These extendable stencils will reduce the error of the scheme and will assist researchers in ﬁnding novel results by solving ODES and PDES using the CN method.Rabnawaz MallahWajid Ahmed SiyalSaira AslamMuhammad Suleman SialInayatullah Soomro2022-11-122022-11-1210Analytical Solution of Transient Flow of Fractional Oldroyd-B Fluid between Oscillating Cylinders
https://vfast.org/journals/index.php/VTM/article/view/1184
<p>This paper investigates the fractional Oldroyd-B ﬂuid ﬂow across two interminable coaxial cylinders, where the ﬂuid’s motion is generated by the oscillatory movement of cylinders and the oscillating pressure gradient. The proﬁle of velocity and shear stress of the ﬂow is derived with the assistance of Caputo fractional derivative utilizing an analytical technique, ﬁnite Hankel transform and Laplace transforms. The semi-analytical solution is then displayed as generalized functions of G and R satisfying fundamental constitutive equations and all initial and boundary conditions. To validate the results, some limitations have been imposed on the determined equations and the results have been contrasted against previous results. Moreover, the inﬂuence of various parameters on the ﬂow of fractionalized Oldroyd-B ﬂuid is investigated and depicted graphically.</p>Khadija ShaikhFozia ShaikhRahim Bux KhokharK.N. Memon2022-11-122022-11-1210Delta Perturbation Method for Couette-Poiseuille flows in Third grade fluids
https://vfast.org/journals/index.php/VTM/article/view/1179
<p>This work uses the Delta Perturbation Method (DPM) to theoretically evaluate the steady plane Couette-Poiseuille flowbetween two parallel plates for third-grade fluid.That'sa kind of perturbation approach and was deliveredwith the aid of Bender and his colleagues in the 1980s. Utilizing DPM, analytical solutions have been found from the governing continuity and momentum equations subject to the necessary boundary conditions. In this proposed model, the Newtonian solution is obtained through the substitution. It is possible to measure the velocity field, temperature distribution, volumetric flow rate, and average velocity of the fluid flow. We derived that the third-grade fluid's velocity will change in response to an increasing material constant from the visual and table representations of the impacts of different parameters on the velocity and temperature profiles.The suggested model additionally mentions temperature distribution losses with increases in thermal conductivity and rises as a result of increases of dynamic viscosity , constant parameters and and material constant. Here we have also find out that temperature distribution and velocity profile enhance with higher magnitude of pressure gradient</p>Kamran Nazir MemonAhsan MushtaqueFozia ShaikhAA GhotoA. M. Siddiqui2022-11-122022-11-1210Image Driven Isotropic Diffusivity and Complementary Regularization Approach for Image Denoising Problem
https://vfast.org/journals/index.php/VTM/article/view/1186
We present the idea of image driven isotropic diffusivity along with complementary regularization for image denoising problem. The method is based on the optimization of a quadratic function in L2 norm. The minimization<br />of the energy functional leads to the Partial Differential equation (PDE)-based problem. We are looking for a steady state solution of equivalent time dependent problem. We discretize the problem with standard ﬁnite differences. The steady-state numerical solution of the time dependent problem leads to the iterative procedure, which allow to compute a regularized version of the solution as a denoised image. We have applied our designed model on synthetic as well as real images. The numerous experiments have been conducted to analyse the performance of the method for the different choices of scaling parameters. From the quality of the obtained results and comparative study it is observed that the proposed model performs well as compared to well existing methods.Memoona PirzadaKhuda Bux AmurMuzaffar Bashir ArainRajab Ali Malookani2022-06-302022-06-3010Algebra Based Encryption and Decryption Algorithms
https://vfast.org/journals/index.php/VTM/article/view/1188
In this study an encryption/decryption algorithm is proposed and developed. The developed algorithm is based on the idea that any given plain text can be encrypted like a block cipher with a combination of three encryption keys k1, K2, ...KN that use any value between N = 1, 2, 3, n, . Then the cipher values can be used to make the blocks of alphabets containing only A, B, C, D, E, F, G, H, I, J, each block is separated by a space. The steps of algorithm could also be reversible for decryption of the cipher text. A MATLAB code is written to implement the algorithm and tested different input messages.Secret message consists of website<br />link and bank account details. The specialty of the algorithm is that it can be ﬂexibly used to encrypt and decrypt the secret messages containing not only English alphabets but also those messages containing the numbers, punctuations, elementary mathematics operations and the special characters. The performance of the algorithm is evaluated in terms of computational time, memory usage. From the analysis it is found that the proposed algorithm is faster in terms of execution time as compared to the modern algorithm which makes the algorithm computationally<br />secure. The proposed research particularly contributes as the addition of knowledge in the ﬁeld of cryptography and generally to the information security; consequently can be beneﬁcial to the society.<br />3Shakeel Ahmed KambohSuhail Aslam KhaskheliAbbas Ali GhotoSunny Kumar AassoriMuzaffar Bashir Arain2022-06-302022-06-30103D Finite Difference Formulation and Simulation of EHD Ion-Drag Model
https://vfast.org/journals/index.php/VTM/article/view/1189
This paper presents the simulation of electrohydrodynamically driven micropump obtained by using 3D ﬁnite difference method. EHD governing equations are discretized and then explicitly deﬁned for output parameters. A 3D prototype of ion-drag micropump with symmetric electrodes is modeled and simulated for the velocity, the pressure, electric potential and electric ﬁeld. The objective of this study was to evaluate the results obtained by ﬁnite difference method (FDM) with the results obtained by a ﬁnite element method (FEM) based<br />simulation package COMSOL Multiphysics. The comparison reveals that the numerical simulation results obtained by both the methods are appreciably close<br />to each other. The simulation results are also compared with the existing ex- perimental data and it was found that there are not high discrepancies between simulation and experimental results. The paper concludes that in case of regular geometries of ion-drag micropump the FDM is easy to implement and provides more control on different parameters involved in the simulation as compared to built-in ﬁnite element method based package.Shakeel Ahmed KambohSakina KambohAbbas GhotoFozia ShaikhNawab ahmedKamran Nazir Memon2022-06-302022-06-3010Exact solution of some nonlinear PDE$ {s} $ with conformable fractional derivative
https://vfast.org/journals/index.php/VTM/article/view/1028
In this article, the reduced differential transform method is improved to solve some nonlinear conformable time fractional partial differential equations. The solutions obtained by the purposed methd to solve the Korteweg-de Vries equation, k(m,n) for m=n=2, and Wu-Zhang (2+1)-dimensional dispersive long wave equation, show that the method is very effective.Alireza Mohammadpour2022-04-222022-04-2210Upper Bound of the Third Hankel Determinant for a Subclass of Analytic Functions Subordinate to Cosine Function
https://vfast.org/journals/index.php/VTM/article/view/392
<p>In this paper, we define a new subclass of analytic functions involving the cosine functions. For this function class, we obtain the upper bound of the third Hankel determinant.</p>Khurshid AhmadSerkan AraciMirajul HaqBilal Khan2022-04-162022-04-1610Upper Bound of the Third Hankel Determinant for a Subclass of Multivalent Functions Associated with the Bernoulli Lemniscate
https://vfast.org/journals/index.php/VTM/article/view/1022
<p>Let RL; SL and CL represent the families of multivalent bounded turning, multivalent starlike, and multivalent convex functions that are subordinated with Bernoulli lemniscate in the open unit disk E = {z : |z|<1}. In this particular paper, our goal is to find the upper bounds of Hankel’s third order determinant for the above-mentioned families.</p>Khalil UllahJihad YounisKhurshid AhmadA ManickamBilal KhanMirajul Haq2021-12-312021-12-3110A Study of Fourth Hankel Determinant of Certain Analytic Function
https://vfast.org/journals/index.php/VTM/article/view/1019
<p>The main motive of this paper is to nd an upper bound of the fourth Hankel determinant H 4;1 (f) for a subclass S; with hyperbolic domain.</p>Neelam KhanNazar KhanBasem Aref FrasinMirajul HaqBilal Khan2021-12-312021-12-3110Structures of Digraphs Arizing from Lambert Type Maps
https://vfast.org/journals/index.php/VTM/article/view/1021
<p>The Well-known function W e<sup>W</sup> is called Lambert Map. This map has been viewed by many researchers for finding the approximate solutions of exponential function especially in numerical analysis. Later, it has been incorporated in number theory for finding integral solutions of exponential congruences under a fixed modulus. Instead of We<sup>W</sup>, we use the function W<sup>2 </sup>e <sup>W</sup> and call this function as Discrete Lambert Type Function (DLTF). In this work, we produce graphs using DLTF, and discuss their structures. We show that the digraphs over DLTF satisfy many structures as these have been followed for using We<sup>W</sup>. It would be of great interest, if these results could be generalized for We<sup>W</sup>for all integers n, in the future.</p>Tayyiba SabahatSufyan AsifAsif Abd ur Rehman2021-12-312021-12-3110Upper Bound Sequences of Rotationally Symmetric Triangular Prism Constructed as Halin Graph Using Local Fractional Metric Dimension
https://vfast.org/journals/index.php/VTM/article/view/1020
<p>In this paper, we consider rotationally symmetric traingular planar network with possible planar symmetries. We ﬁnd local fractional metric dimension of planar symmetries. The objective is to search sequences of local fractional metric dimension of triangular prism planar networks by joining diﬀerent copies. We propose and prove generalized formulas of all sequences for local fractinal metric<br />dimension over triangular prism.</p>Mamoona FarooqAsif Abd ul RehmanM. Khalid MahmoodDaud Ahmad2021-12-312021-12-3110On Fixed Points of Digraphs Over Lambert Type Map
https://vfast.org/journals/index.php/VTM/article/view/1023
<p>Define f(y)=y<sup>2</sup>h<sup>y</sup> where h belongs to (Z/mZ), the Discrete Lambert Type Map (DLTM). For a set of vertices and edges over DLTM, diagraphs are obtained in which the vertices are from a whole range of residues modulo a fixed integer s, and edges are obtained when f (y) ≡ v(mods<sup>K</sup>) is solvable in Y and in terms of Diophantine equation as well. In this paper we proposed new results for for fixed point of digraphs over DLTM.</p>Tayyiba SabahatSufyan AsifAsif Abd ur Rehman2021-12-312021-12-3110Upper Bound of the Third Hankel Determinant for a Subclass of Analytic Functions Connected with Sine Functions
https://vfast.org/journals/index.php/VTM/article/view/273
Let RL sin and SL represents the families of multivalent bounded<br />turning and multivalent starlike functions that are subordinate with sine function in the open unit disk E = fz : jzj < 1g : For these families our aim is to nd the bounds of Hankel determinant of order three. Further, the estimate of third order Hankel determinant for the family SL sin in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions. Also we discuss H (3) determinant for the above mentioned families. 2 sin<br /><br />Khalil UllahQaiser KhanKhurshid AhmadA. ManickamMirajul HaqBilal Khan2020-12-312020-12-3110A Study of Third Hankel Determinant for Certain Subclasses of Analytic Function
https://vfast.org/journals/index.php/VTM/article/view/366
<p>Recently the Hankel determinant problems got attractions of many well-known authors. Third Hankel determinant problems were determined for different subclasses of analytic functions. Here in our present investigation, we define certain new subclasses of analytic functions and then we obtain the upper bonds for the third Hankel determinant.</p>Hifsa BibiBilal KhanKhurshid AhmadG. MurugusundaramoorthyJihad YounisMirajul Haq2020-12-312020-12-3110Implementation of Discrete Search Optimization Algorithms in Parallel and Their Performance Analysis
https://vfast.org/journals/index.php/VTM/article/view/576
The present paper discusses the implementation of the discrete search optimization techniques on a parallel platform (SGI-Altix 450 shared memory 64 processors system). We show that the combination of Asynchronous Parallel Iterative Deepening and Parallel Window Search technique tends to give more challenging speedups, memory consumption, and efficiency with less resource consumption as compared to the rest of the techniques. In general 5 techniques are compared with Parallel Asynchronous Window Search technique among which includes Depth First Search, Parallel Iterative Deepening, Parallel A*, Parallel Window Search and Parallel Asynchronous Iterative Deepening A*.Muhammad Hanif DuradMohammad ZulqurnainAnila UsmanIdrees Ahmad2020-11-182020-11-1810Stability Analysis of SEIVHR Epidemic Model with Saturated Incidence Rate
https://vfast.org/journals/index.php/VTM/article/view/303
The aim of this paper, to analyze stability analysis of SEIVHR epidemic model with a generalized non-linear incidence rate that spread in the host population horizontally. First, we formulate the model and find its basic reproduction number. Two equilibrium exists, namely; the disease-free and endemic equilibrium. The disease-free equilibrium is stable both locally and globally when the threshold quantity less than unity. The endemic equilibrium is locally and globally asymptotically stable when the threshold exceeds unity. Finally, we show some numerical results for the proposed model.Muhammad Ali KhanSehra .2020-03-182020-03-1810More Modular Arithmetic in Five Regular Partitions by Jacobi Triple Product Formula
https://vfast.org/journals/index.php/VTM/article/view/426
In a paper, Calkin et al., Divisibility properties of the 5-regular and 13-regular partition functions, Integers 8 (2008), [#A60], authors have discussed interesting properties for 5-regular partition functions of integers. In the continuation of this paper, we have obtained and conjectured various interesting results. In this note, we use nothing more than Jacobi's triple product identity to obtain results for 5-Regular Partitions that are stronger than those obtained by Calkin and his collaborators. The motivation for this paper is an observation that some generating functions of 5-Regular partitions are congruent to functions related to the Ramanujan's Q-series.Aneela .M. Khalid MahmoodDaud AhmadShahbaz Ali2019-12-312019-12-3110Parallelization of Encryption Algorithms using MPI
https://vfast.org/journals/index.php/VTM/article/view/577
<em>Encryption is a basic technique to achieve data confidentiality. A number of algorithms including RSA, DES, Blowfish, and AES have been parallelized using MPI and have been employed for practical file encryption. Unified performance analysis of all these algorithms has been presented using two ECB and counter modes. On basis of experimental results, some guidelines have been suggested for the end-user to select the appropriate algorithm for achieving enhanced speed up</em>Muhammad Hanif DuradAhmad RazaAli AsadMuhammad Naveed Akhtar2019-12-312019-12-3110Unified Computational Analysis of Conventional Numerical Methods for Time Dependant Heat Equation
https://vfast.org/journals/index.php/VTM/article/view/579
<em>The objective of this paper is to perform a unified error and computation time analysis of conventional numerical methods for solving the heat equation. The numerical techniques employed include Forward Difference, Backward Difference, Crank Nicolson, Alternate Directions Scheme, and DuFort-Frankel methods for the time-dependent heat equation. The heat equation has been implemented to 1-D, 2-D, and 3-D problems, both for the Neumann and Dirichlet boundary conditions. On basis of experimental results, theoretical justifications have been provided regarding the performance of each method.</em>Muhammad Naveed AkhtarMuhammad Hanif DuradAnila UsmanIrfan ul Haq2019-12-312019-12-3110Extension of Sufficient Conditions for P–Valent Starlike and Convex Function
https://vfast.org/journals/index.php/VTM/article/view/418
In this article, we consider a subclass of p-valent analytic functions which generalize the classes of p-valent starlike and p-valent convex functions of complex order. For this class, certain simple sufficiency criteria are obtained and further some applications to the generalized Alexander integral operator is also given. Several known results appear as special cases of our workM ArifMuhammad AyazMuhammad Asad2019-04-092019-04-0910Probabilistic Picture Graph
https://vfast.org/journals/index.php/VTM/article/view/394
The probabilistic graphs is an ecient mathematical tool to deal with uncertain real life problems. Probabilistic picture graph is an extension of probabilistic graphs corresponding to a given joint probability distribution. It can work very eciently in uncertain scenarios which involve more answers to these type, yes and no. In this work. we redene and introduce the idea of probabilistic graphs and probabilistic picture graphs. Some types of probabilistic picture graphs such as a strong probabilistic picture graph, complete probabilistic picture graph and complement of probabilistic picture graph are introduced and some properties are also described. The six operations namely cartesian product, composition, union, direct product, lexicographical product and strong product on probabilistic picture graph are also dened. Finally, we describe the utility of probabilistic graphs by revisiting application in a joint venture.Sidra YousafAsif Abd ur Rehman2018-12-172018-12-1710Mitigation of the effect of Standard Networks Attacks in SSL Encrypted Traffic by Encrypted Traffic Analysis
https://vfast.org/journals/index.php/VTM/article/view/578
<span>With increased use of encryption, cyber threat landscape is changing. For general public this transition shifts to more private and safer internet experiences, but at the same time it is a serious concern for security personnel now. For them it hinders control over the traffic moving on their network and poses difficulty in traffic analysis and management. Security personals are interested in knowing how the network is being accessed, whether or not that traffic contains some malware and is safe enough and compliant with your organization’s policies. This project is not about decrypting the encrypted content of the packet’s payload as it will highly degrade network performance plus some advanced encryption algorithms like AES are assumed to be perfect. So the aim of this project is to analyze encrypted traffic and find out some interesting patterns without the need for bulk decryption. The analysis will be based on flow based features and metadata. Encrypted Traffic Analytics maintains the integrity of the encrypted flow and doesn’t affect the privacy of users.</span>Muhammad HamadMuhammad Hanif DuradMuhammad Yousaf2018-12-172018-12-1710Teleparallel Homothetic Symmetry of Special Axially Symmetric Static Spacetime
https://vfast.org/journals/index.php/VTM/article/view/160
<em>In this paper we are interested to find teleparallel proper homothetic vector fields over the Lorentzian manifold of special axially symmetric static spacetime. For the purpose we applied teleparallel Lie derivative to the metric for homothetic equations and obtained ten coupled non linear differential equations. These equations are then solved for each possibility of the metric functions and it comes out that only in one case teleparallel proper homothetic vector fields exist for the special choice of metric functions. The spacetime admit eight dimensional teleparallel homothetic vector fields in which one is proper teleparallel homothetic vector and remaining seven are teleparallel Killing vector fields</em>Amjad Ali2018-10-152018-10-1510CALCULATION OF SHEAR STRENGTH PROBABILITY OF GEOGRIDS USED IN HIGHWAY EMBANKMENTS
https://vfast.org/journals/index.php/VTM/article/view/274
<span>In this study; experimental measurement data were performed for probability analysis. Geogrids are crucial reinforced material for civil engineering applications such as highway base/subbase reinforcement, railway ballast reinforcement and retaining walls. In this study, the cumulative distribution function was formed to estimate the probability of collapse risk of the highways. Probability density functions were calculated with the help of lognormal mean and lognormal standard deviation values of highway displacement points. The cumulative distribution functions were generalized and the probability of the damage was shown. With the results of this work; damage possibility can be estimated for any highway reinforced with geogrid which has same features such as 30x30mm, 40x40mm and 50x50mm etc.</span>Tuba KutukAli Gurbuz2018-01-122018-01-1210Mobile Cloud Computing: Operational Issues
https://vfast.org/journals/index.php/VTM/article/view/1030
2017-12-312017-12-3110Delayed SIR model with latency and saturated incidence rate
https://vfast.org/journals/index.php/VTM/article/view/138
In this paper, we derive and analyse a delayed $SI$ model with saturated incidence rate and <br />latent or infectious period $\tau$. We prove local stability of the system's steady states in the <br />absence and the presence of the time delay. We discover the the disease-free steady state is locally <br />asymptotically stable if $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$. While, the endemic steady <br />state is always stable for any parameter values and for all $\tau\ge0$.Muhammad Abdullahi YauH.S NdakwoMuktari Garba2017-07-262017-07-2610On Cycle-Super magic Labeling of Some Families of Graphs
https://vfast.org/journals/index.php/VTM/article/view/1029
In this work we have formulated the result of H-magic labeling of some families of graphs which included wheel graph, fan graph, ladder graph and triangular ladder. The edges of these graphs were subdivided into equal nodes and we formulated total magic and supermagic labeling of these families of graphs. A labeling of a graph is basically assigning positive integers to the edges and vertices and H-magic labeling is define as a bijection is called -magic labeling of if there exists a positive integer -called the magic sum, such that for any subgraph of isomorphic to , the sum is equal to the magic sum, . A graph is -magic if it admits an -magic labeling. In addition, if -magic labeling has the property that , then the graph is -supermagic. .Imtiaz Riaz2017-04-222017-04-2210Numerical and Analtyical Solutions of a Model of Population Dynamics
https://vfast.org/journals/index.php/VTM/article/view/271
<span>The study presents numerical and approximate analytical approximations to a model of population dynamics with unbounded mortality function. </span><br /><span>The mathematical model involves a nonlocal boundary condion. A </span><br /><span>fi nite difierence method is implemented for the numerical solution while </span><br /><span>the homotopy analysis method (HAM) is applied to obtain the approximate </span><br /><span>series solution. The HAM solution contains an auxiliary parameter </span><br /><span>which provides a convenient way of controlling the convergence region of </span><br /><span>series solution. Results are presented for typical test problem provided in </span><br /><span>literature. Comparison of the results of both methods show validity and </span><br /><span>eficiency of the methods.</span>Sidrah Ahmad2017-01-122017-01-1210On Banach Lattice Algebras
https://vfast.org/journals/index.php/VTM/article/view/304
<p>In this paper, we investigate a characterization of a Banach lattice<br />algebra with a unit to be represented as an AM-f-algebra. We also consider, for a locally compact Hausdor topological space L and a Banach lattice space A, the identification of Cb(L;Z(A)) with the center Z(C0(L;A)) of C0(L;A).</p>Rusen YilmazYilmaz Altun2016-12-262016-12-2610Fundamental Theorem Of Functions
https://vfast.org/journals/index.php/VTM/article/view/336
<p class="Default">From the fundamental theorem of homomorphisms, it is well known that any homomorphism of groups (or rings or modules or vector spaces and of general universal algebras) can be decomposed as a composition of a monomorphism and an epimorphism. This result can also be extended to general functions defined on abstract sets; that is, any function can be expressed as a composition of an injection and a surjection. The main theorem in this paper called ‘Fundamental Theorem of Functions’ provides the uniqueness of such a decomposition of functions as a composition of an injection and a surjection. The uniqueness in this theorem is proved upto the level of associates by introducing the notion of an associate of functions.</p>Gezahagne Mulat Addis2016-12-252016-12-2510Analysis of SIR Epidemic Models
https://vfast.org/journals/index.php/VTM/article/view/1027
<p>Modeling epidemics has played a vital role in predicting the impact of a disease in a population and the ability to overcome it. The objective of this paper is to introduce the concept of agent based modeling classification for SIR epidemic model, suggest a new discrete-time agent based SIR model and formally document Gillespie’s Algorithm based SIR epidemic model. Furthermore a computational analysis of chemical master equation and Gillespie’s algorithm based SIR models<br />has been performed showing that the use of latter results in considerable reduction<br />in execution time</p>Muhammad Hanif DuradMuhammad Naveed Akhtar2016-12-212016-12-2110VALUATION ON A FILTERED MODULE
https://vfast.org/journals/index.php/VTM/article/view/150
In this paper we show if R is a filtered ring and M a filtered<br />R module then we can define a valuation on a module for M. Then<br />we show that we can find an skeleton of valuation on M, and we prove some properties such that derived form it for a filtered module.Mohammad Hassan Anjom SHoaMohammad Hossein Hosseini2015-12-262015-12-2610Analysis of Ellis Fluid in Wire Coating
https://vfast.org/journals/index.php/VTM/article/view/393
<em><span style="font-size: 10.0pt; line-height: 115%; font-family: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">The focus of the present work is concerned with the study of wire coating analysis using Ellis fluid. Using the binomial series method, closed form solution is obtained and further in the closed form of solution, we discussed a special case for the parameter </span></em><!--[if gte msEquation 12]><m:oMath><span style='font-size:10.0pt;line-height:115%;font-family:"Cambria Math","serif"; mso-fareast-font-family:"Times New Roman";mso-fareast-theme-font:minor-fareast; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; mso-ansi-language:EN-US;mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><m:r><i style='mso-bidi-font-style:normal'>α</i></m:r></span></m:oMath><![endif]--><!--[if !msEquation]--><!--[endif]--><em><span style="font-size: 10.0pt; line-height: 115%; font-family: 'Times New Roman','serif'; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">, and obtained the expressions for axial velocity, shear stress, average velocity, volume flow rate and temperature distribution. Thickness of the coated wire and force on the total wire are also calculated. The effect of different parameters of interest on the variation of the velocity profile, volume flow rate, shear stress, radius of coated wire and force on the total wire are discussed and presented graphically.</span></em>Muhammad AyazTaza GulSaeed IslamMuhammad Altaf2015-12-252015-12-2510Thermal and Airflow Characteristics of Transitional Spaces in a Traditional Urban Fabric Case Study of a Covered Walkway in Timimoun (South West of Algeria)
https://vfast.org/journals/index.php/VTM/article/view/360
The research presented in this article deals with the analysis of thermal and airflow patterns in a traditional compact urban fabric called „Ksar‟ located in the city of Timimoun in the South western part of the Algerian Sahara. It focuses on a particular typology of urban transitional spaces which is “a covered walkway ”. The aim of this research is to study the thermal and airflow characteristics of these particular spaces which affect the comfort of pedestrians and influence the ambiantal conditions of neighboring houses as well. The methodology conducted in this study consists principally in applying a computational fluid dynamics model to simulate the thermal airflow behavior of a transitional space. The computational fluid dynamics software, Fluent 6.3.26 is used for the simulation of the air flow and the temperature field traversing the streets. The results indicate that under peak Summer conditions, weak and strongly fluctuating air movements characterize the walkway and for perpendicular flow, the temperature generally decreases with an increase in velocity. Areas near the passage entrance are more affected by adverse wind conditions than other areas remote from it.Khelifi LamiaBensalem RafikSemmar Djaffar2015-12-252015-12-2510Killing Symmetry in Special Axially Symmetric Static Spacetimes in Teleparalel Theory of Gravitation
https://vfast.org/journals/index.php/VTM/article/view/384
In this paper we are searching for teleparallel Killing vector fields of special axially symmetric static spacetimes in teleparallel theory of gravitation by using direct integration and algebraic techniques. After thorough investigations, the whole problem is divided into three cases under different constraints. Two of the said cases give contradiction, while one of them gives the solution of the system comprising the killing equations in the form of killing vector fields. The dimension of the killing symmetry in this case is 10.Tariq Zaman Khan2015-12-252015-12-2510A General Class Of Estimators For Finite Population Mean In The Presence Of Non-Response When Using The Second Raw Moments
https://vfast.org/journals/index.php/VTM/article/view/129
<p><em>In this paper, a general class of family of estimators for estimation of finite population mean is proposed under non-response, by using information on second<strong> </strong>raw moments. Properties of some ratio, product, modified ratio and modified product type estimators, which are members of a suggested class of estimators, are studied. It is shown that a suggested class of estimators performs better than the usual ratio and product type estimators as well as regression and other considered estimators. A numerical study is carried out to support a suggested class of estimators.<strong></strong></em></p>MANZOOR KHANJAVID SHABBIR2015-12-152015-12-1510