Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Citation - WoS: 3
Citation - Scopus: 3
A1-L10 Phase Boundaries and Anisotropy Via Multiple-Order Theory for an Fcc Alloy
(European Mathematical Society Publishing House, 2003) Tanoğlu, Gamze; Braun, Richard J.; Cahn, John W.; McFadden, Geoffrey B.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an fcc lattice is studied as a function of their orientation. Using a recently developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L12 and L10) such as occur in the Cu-Au system, we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases.We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L10 ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find its interfacial energy, γ as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn-Hoffman "-vector" formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is "transversely isotropic" with respect to the tetragonal axis of the L10 phase. We verify the model's consistency with the Gibbs adsorption equation.
Citation - WoS: 7
Citation - Scopus: 5
Abelian Chern-Simons Vortices and Holomorphic Burgers Hierarchy
(Pleiades Publishing, 2007-07) Pashaev, Oktay; Gürkan, Zeynep Nilhan; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We consider the Abelian Chern-Simons gauge field theory in 2+1 dimensions and its relation to the holomorphic Burgers hierarchy. We show that the relation between the complex potential and the complex gauge field as in incompressible and irrotational hydrodynamics has the meaning of the analytic Cole-Hopf transformation, linearizing the Burgers hierarchy and transforming it into the holomorphic Schrödinger hierarchy. The motion of planar vortices in Chern-Simons theory, which appear as pole singularities of the gauge field, then corresponds to the motion of zeros of the hierarchy. We use boost transformations of the complex Galilei group of the hierarchy to construct a rich set of exact solutions describing the integrable dynamics of planar vortices and vortex lattices in terms of generalized Kampe de Feriet and Hermite polynomials. We apply the results to the holomorphic reduction of the Ishimori model and the corresponding hierarchy, describing the dynamics of magnetic vortices and the corresponding lattices in terms of complexified Calogero-Moser models. We find corrections (in terms of Airy functions) to the two-vortex dynamics from the Moyal space-time noncommutativity.
Citation - WoS: 17
Citation - Scopus: 18
The Ablowitz-Ladik Lattice System by Means of the Extended (g' / G)-Expansion Method
(Elsevier Ltd., 2010-07) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We analyzed the Ablowitz-Ladik lattice system by using the extended (G′ / G)-expansion method. Further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained. We observed that some previously known results can be recovered by assigning special values to the arbitrary parameters. © 2010 Elsevier Inc. All rights reserved.
Absolute Co-Supplement and Absolute Co-Coclosed Modules
(Hacettepe Üniversitesi, 2013) Tütüncü, Derya Keskin; Toksoy, Sultan Eylem; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
A module M is called an absolute co-coclosed (absolute co-supplement) module if whenever M ≅ T/X the submodule X of T is a coclosed (supplement) submodule of T. Rings for which all modules are absolute co-coclosed (absolute co-supplement) are precisely determined. We also investigate the rings whose (finitely generated) absolute co-supplement modules are projective. We show that a commutative domain R is a Dedekind domain if and only if every submodule of an absolute co-supplement R-module is absolute co-supplement. We also prove that the class Coclosed of all short exact sequences 0→A→B→C→0 such that A is a coclosed submodule of B is a proper class and every extension of an absolute co-coclosed module by an absolute co-coclosed module is absolute co-coclosed.
Citation - WoS: 13
Citation - Scopus: 13
Absolutely S-Pure Modules and Neat-Flat Modules
(Taylor and Francis Ltd., 2015-02) Büyükaşık, Engin; Durğun, Yılmaz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Let R be a ring with an identity element. We prove that R is right Kasch if and only if injective hull of every simple right R-modules is neat-flat if and only if every absolutely pure right R-module is neat-flat. A commutative ring R is hereditary and noetherian if and only if every absolutely s-pure R-module is injective and R is nonsingular. If every simple right R-module is finitely presented, then (1)R R is absolutely s-pure if and only if R is right Kasch and (2) R is a right (Formula presented.) -CS ring if and only if every pure injective neat-flat right R-module is projective if and only if every absolutely s-pure left R-module is injective and R is right perfect. We also study enveloping and covering properties of absolutely s-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized. © 2015 Taylor & Francis Group, LLC.
Akışkan-yapı etkileşimi problemlerinde birleşik sayısal/asimtotik algoritmalar: baraj yıkımı ile oluşan akış ve diğer uygulamalar
(TÜBİTAK - Türkiye Bilimsel ve Teknolojik Araştırma Kurumu, 2014) Korobkin, Alexander; Iafrati, Alessandro; Yılmaz, Oğuz; Neslitürk, Ali İhsan; Çiçek, Barış; Kaya, Adem; Isıdıcı, Damla; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Baraj güvenliği ile ilgili ilk yasa acil durumlarda uygulanmak üzere Fransa’da 1968 yılında çıkmıştır. Günümüzde daha büyük barajların yapılması ile daha yeni düzenlemeler getirilmiştir. Baraj güvenliği acil eylem planı şunları içerir; potansiyel risklerin tespiti, bu riskleri önleyecek önlemlerin alınması, yerel yönetimlerin acil durumlardaki sorumluluk tanımı ve bilginin halka iletilmesi. Günümüzde baraj operatörlerinin acil eylem planı ışığında baraj yapılmadan önce risk değerlendirme çalışması yapması gerekmektedir. Baraj yıkıldıktan sonraki ilk 15 dakika içinde selin ulaşamadığı en yakın güvenli bölgeyi ve selin ulaşabileceği en uzak alanın tesbiti bu risk değerlendirmeleri içerisindedir.
Citation - WoS: 13
Citation - Scopus: 15
Analysis of a Corner Layer Problem in Anisotropic Interfaces
(Southwest Missouri State University, 2006-03) Alikakos, N. D.; Bates, P. W.; Cahn, J. W.; Fife, P. C.; Fusco, G.; Tanoğlu, Gamze; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered Cu3Au state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number 0(1/ε), of critical eigenvalues, where 1/ε ≫ 1 is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues λn = ε2/3 μn with μn ∼ Cn2/3, as n → +∞. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.
Citation - WoS: 21
Citation - Scopus: 21
An Analytic Approach To a Class of Fractional Differential-Difference Equations of Rational Type Via Symbolic Computation
(John Wiley and Sons Inc., 2015-01) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Fractional derivatives are powerful tools in solving the problems of science and engineering. In this paper, an analytical algorithm for solving fractional differential-difference equations in the sense of Jumarie's modified Riemann-Liouville derivative has been described and demonstrated. The algorithm has been tested against time-fractional differentialdifference equations of rational type via symbolic computation. Three examples are given to elucidate the solution procedure. Our analyses lead to closed form exact solutions in terms of hyperbolic, trigonometric, and rational functions, which might be subject to some adequate physical interpretations in the future. Copyright © 2013 JohnWiley & Sons, Ltd.
Citation - WoS: 14
Citation - Scopus: 16
Analytic Investigation of a Reaction-Diffusion Brusselator Model With the Time-Space Fractional Derivative
(Walter de Gruyter GmbH, 2014-04) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
It is well known that many models in nonlinear science are described by fractional differential equations in which an unknown function appears under the operation of a derivative of fractional order. In this study, we propose a reaction-diffusion Brusselator model from the viewpoint of the Jumarie's modified Riemann-Liouville fractional derivative. Based on the (G'/G)-expansion method, various kinds of exact solutions are obtained. Our results could be used as a starting point for numerical procedures as well.
Citation - WoS: 16
Citation - Scopus: 16
Analytic Investigation of the (2 + 1)-Dimensional Schwarzian Korteweg–de Vries Equation for Traveling Wave Solutions
(Elsevier Ltd., 2011-02-15) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
By means of the two distinct methods, the Exp-function method and the extended (G0/G)-expansion method, we successfully performed an analytic study on the (2 + 1)-dimensional Schwarzian Korteweg–de Vries equation. We exhibited its further closed form traveling wave solutions which reduce to solitary and periodic waves. New rational solutions are also revealed.
Citation - WoS: 26
Citation - Scopus: 27
Analytic Solutions To Nonlinear Differential-Difference Equations by Means of the Extended (g'/g)-expansion Method
(IOP Publishing Ltd., 2010-10) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
In this paper, a discrete extension of the (G′/G)-expansion method is applied to a relativistic Toda lattice system and a discrete nonlinear Schrödinger equation in order to obtain discrete traveling wave solutions. Closed form solutions with more arbitrary parameters, which reduce to solitary and periodic waves, are exhibited. New rational solutions are also obtained. The method is straightforward and concise, and its applications in physical sciences are promising. © 2010 IOP Publishing Ltd.
Citation - WoS: 77
Citation - Scopus: 110
Analytic Study on Two Nonlinear Evolution Equations by Using the (g'/g)-expansion Method
(Elsevier Ltd., 2009-03) Aslan, İsmail; Öziş, Turgut; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The validity and reliability of the so-called (G′/G)-expansion method is tested by applying it to two nonlinear evolutionary equations. Solutions in more general forms are obtained. When the parameters are taken as special values, it is observed that the previously known solutions can be recovered. New rational function solutions are also presented. Being concise and less restrictive, the method can be applied to many nonlinear partial differential equations.
Citation - WoS: 21
Anonymity and One-Way Authentication in Key Exchange Protocols
(Springer Verlag, 2013-05) Goldberg, Ian; Stebila, Douglas; Ustaoğlu, Berkant; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Key establishment is a crucial cryptographic primitive for building secure communication channels between two parties in a network. It has been studied extensively in theory and widely deployed in practice. In the research literature a typical protocol in the public-key setting aims for key secrecy and mutual authentication. However, there are many important practical scenarios where mutual authentication is undesirable, such as in anonymity networks like Tor, or is difficult to achieve due to insufficient public-key infrastructure at the user level, as is the case on the Internet today. In this work we are concerned with the scenario where two parties establish a private shared session key, but only one party authenticates to the other; in fact, the unauthenticated party may wish to have strong anonymity guarantees. We present a desirable set of security, authentication, and anonymity goals for this setting and develop a model which captures these properties. Our approach allows for clients to choose among different levels of authentication. We also describe an attack on a previous protocol of Øverlier and Syverson, and present a new, efficient key exchange protocol that provides one-way authentication and anonymity. © 2012 Springer Science+Business Media, LLC.
Citation - WoS: 1
Citation - Scopus: 2
Apollonius Representation and Complex Geometry of Entangled Qubit States
(IOP Publishing, 2019) Parlakgörür, Tuğçe; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
A representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant on the circles as well. Similar results are obtained also for n- tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant on Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. The bipolar and the Cassini representations for qubit state are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves.
The Application of a Finite Difference Method To a Dynamical Interface Problem
(Acad. Publications, 2003) Tanoğlu, Gamze; Ağıroğlu, İzzet Onur; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
A multiple-order-parameter model for Cu-Au system on a face cubic centered lattice was recently developed in the presence of anisotropy. In that model, three order parameters (non-conserved) and one concentration order parameter (conserved), which has been taken as a constant, were considered. Later on, the model has been extended, so that, concentration has been taken as a variable. It has been seen that two models were in a good agreement near critical temperature since the non-conserved order parameter behaves like a constant near critical temperature in both models.
Citation - WoS: 1
Citation - Scopus: 2
Application of the Division Theorem To Nonlinear Physical Models for Constructing Traveling Waves
(Politechnica University of Bucharest, 2013) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We extend the so-called first integral method, which is based on the division theorem, to the Sharma-Tasso-Olver equation and the (2+1)-dimensional modified Boussinesq equation. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Traveling wave solutions are constructed through the established first integrals.
Citation - WoS: 9
Citation - Scopus: 10
Application of the Exp-Function Method To Nonlinear Lattice Differential Equations for Multi-Wave and Rational Solutions
(John Wiley and Sons Inc., 2011-09) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
In this paper, we extend the basic Exp-function method to nonlinear lattice differential equations for constructing multi-wave and rational solutions for the first time. We consider a differential-difference analogue of the Korteweg-de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant.
Citation - WoS: 3
Citation - Scopus: 2
Application of the Exp-Function Method To the (2+1)-Dimensional Boiti-Leon Equation Using Symbolic Computation
(Taylor and Francis Ltd., 2011-03) Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Locate full-text(opens in a new window)|Full Text(opens in a new window)|View at Publisher| Export | Download | Add to List | More... International Journal of Computer Mathematics Volume 88, Issue 4, March 2011, Pages 747-761 Application of the Exp-function method to the (2+1)-dimensional Boiti-Leon-Pempinelli equation using symbolic computation (Article) Aslan, I. Department of Mathematics, Izmir Institute of Technology, Urla, Izmir 35430, Turkey View references (47) Abstract This paper deals with the so-called Exp-function method for studying a particular nonlinear partial differential equation (PDE): the (2+1)-dimensional Boiti-Leon-Pempinelli equation. The method is constructive and can be carried out in a computer with the aid of a computer algebra system. The obtained generalized solitary wave solutions contain more arbitrary parameters compared with the earlier works, and thus, they are wider. This means that our method is effective and powerful for constructing exact and explicit analytic solutions to nonlinear PDEs.
Citation - WoS: 29
Citation - Scopus: 48
Application of the G' / G-Expansion Method To Kawahara Type Equations Using Symbolic Computation
(Elsevier Ltd., 2010-06) Öziş, Turgut; Aslan, İsmail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
In this paper, Kawahara type equations are selected to illustrate the effectiveness and simplicity of the G′ / G-expansion method. With the aid of a symbolic computation system, three types of more general traveling wave solutions (including hyperbolic functions, trigonometric functions and rational functions) with free parameters are constructed. Solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters, involved in the traveling waves, as special values. © 2010 Elsevier Inc. All rights reserved.
An Application With Webmathematica
(Springer Verlag, 2003) Ufuktepe, Ünal; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
There have been many technological dawns in the last 30 years, during which the desktop computer and the Internet have been developed. The importance of Internet in education, particularly using its Web is a well-recognized fact. A wealth of resources and techniques now exist which serve as a source both for exciting examples of new teaching practices, as well as easily accessible methods for adoption into various formats of teaching and learning. Internet technology allow teachers and students keep up with their minds. It let them try their ideas as soon as they come up with them. Generally, students appreciate the convenience, choice, and flexibility that an online courses offers. Instructional designers value the standardized framework and flexibility. WebMathematica is a web-based technology developed by Wolfram Research that allows the generation of dynamic web content with Mathematica. With this technology, the distance education students should be able to explore and experiment with the mathematical concepts. In this paper we will elucidate the pedagogical issues in the application of Hamiltonian systems in the webMathematica for the distance learning environment and the shape of the future "classroom" as well as relevant educational strategies towards improving mathematics education.

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