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    Application of Dempster-Schafer Method in Family-Based Association Studies
    (Ieee Computer Soc, 2013) Rajabli, Farid; Goktas, Unal; Inan, Gul
    In experiments designed for family-based association studies, methods such as transmission disequilibrium test require large number of trios to identify single-nucleotide polymorphisms associated with the disease. However, unavailability of a large number of trios is the Achilles' heel of many complex diseases, especially for late-onset diseases. In this paper, we propose a novel approach to this problem by means of the Dempster-Shafer method. The simulation studies show that the Dempster-Shafer method has a promising overall performance, in identifying single-nucleotide polymorphisms in the correct association class, as it has 90 percent accuracy even with 60 trios.
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    Application of perturbation-iteration method to Lotka-Volterra equations
    (Elsevier Science Inc, 2016) Aksoy, Yigit; Goktas, Unal; Pakdemirli, Mehmet; Dolapci, Ihsan Timucin
    Perturbation-iteration method is generalized for systems of first order differential equations. Approximate solutions of Lotka-Volterra systems are obtained using the method. Comparisons of our results with each other and with numerical solutions are given. The method is implemented in Mathematica, a major computer algebra system. The package PerturbationIteration.m automatically carries out the tedious calculations of the method. (C) 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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    Dempster-Schafer Method in Family Based Association Studies
    (Turgut Ozal Univ, 2012) Rajabli, Farid; Goktas, Unal; Inan, Gul
    In experiments designed for family-based association studies, methods such as transmission disequilibrium test (TDT) require large number of trios to identify single nucleotide polymorphisms (SNPs) associated with the disease. However, unavailability of large number of trios is the Achilles' heel of many complex disease studies, especially for the late-onset diseases since parents might have passed away or unavailable. This problem motivated us to look for new approaches that require smaller number of trios to detect associations in family based studies. Aiming this, we propose using Dempster-Shafer method of evidence to detect the associations in family-based studies and show that this approach has a good performance in identifying associations with moderate to small sized samples.
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    Scaling invariant Lax pairs of nonlinear evolution equations
    (Taylor & Francis Ltd, 2012) Hickman, Mark; Hereman, Willy; Larue, Jennifer; Goktas, Unal
    A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called a Lax pair. Two equivalent representations are presented. The first uses a pair of differential operators which leads to a higher order linear system for the auxiliary function. The second uses a pair of matrices which leads to a first-order linear system. In this article, we present a method, which is easily implemented in MAPLE or MATHEMATICA, to compute an operator Lax pair for a set of PDEs. In the operator representation, the determining equations for the Lax pair split into a set of kinematic constraints which are independent of the original equation and a set of dynamical equations which depend on it. The kinematic constraints can be solved generically. We assume that the operators have a scaling symmetry. The dynamical equations are then reduced to a set of nonlinear algebraic equations. This approach is illustrated with well-known examples from soliton theory. In particular, it is applied to a three parameter class of fifth-order Korteweg-de Vries (KdV)-like evolution equations which includes the Lax fifth-order KdV, Sawada-Kotera and Kaup-Kuperschmidt equations. A second Lax pair was found for the Sawada-Kotera equation.
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    Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential Difference Equations
    (Springer-Verlag Berlin, 2012) Goktas, Unal; Hereman, Willy
    Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in Mathematica. The codes INVARIANTSSYMMETRIES.M and DDERECURSIONOPERATOR.M can aid researchers interested in properties of nonlinear DDEs.
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    Symbolic computation of the roots of nonlinear algebraic equations using perturbation theory
    (Turgut Ozal Univ, 2012) Ustali, Gurhan; Goktas, Unal; Pakdemirli, Mehmet
    Two algorithms, one for estimating the magnitudes of the roots of a polynomial equation before actually solving it, and one for computing the roots of nonlinear algebraic equations using perturbation theory, are presented. The algorithms are illustrated on an example. The Mathematica package PerturbationIterationFindRootAlgorithman,m carries out all the steps of these algorithms automatically. The package is briefly discussed.