Mathematics / Matematik

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

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Can Cpt Be Violated Through Extended Time Reversal?
(World Scientific Publishing, 2001) Erdem, Recai; Ufuktepe, Ünal; 04.05. Department of Pyhsics; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We consider the implications of the extension of time reversal through Wigner types and group extensions. We clarify its physical content and apply the results in a toy model. Finally we point out the possibility of violation of CPT in this framework.
Citation - Scopus: 2
Inequalities for the Vibrating Clamped Plate Problem
(TUBITAK, 2001) Mchale, K. P.; Ufuktepe, Ünal; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We study the eigenvalues of the vibrating clamped plate problem. We have made improvements on the bounds of the ratios of the eigenvalues of the biharmonic operator (clamped plate) using the methods of Payne, Polya, and Weinberger. The difference in our proof lies mainly with the trial functions and the orthogonality arguments. While Payne, Polya, and Weinberger and Hile and Yeh project away components along u1, u2,...,uk to meet the orthogonality conditions, we use a translation/rotation argument to meet these conditions.
Citation - WoS: 6
Citation - Scopus: 6
Soliton Resonances, Black Holes and Madelung Fluid
(Taylor and Francis Ltd., 2001-02) Pashaev, Oktay; Lee, Jyh Hao; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The reaction-diffusion system realizing a particular gauge fixing condition of the Jackiw-Teitelboim gravity is represented as a coupled pair of Burgers equations with positive and negative viscosity. For acoustic metric in the Madelung fluid representation the space-time points where dispersion change the sign correspond to the event horizon, while shock soliton solutions to the black holes, creating under collision the resonance states.
Citation - WoS: 2
Citation - Scopus: 2
Self-Dual Chern-Simons Solitons and Quantum Potential
(Taylor and Francis Ltd., 2001-02) Pashaev, Oktay; Lee, Jyh Hao; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
An influence of the quantum potential on the Chern-Simons solitons leads to quantization of the statistical parameter κ = me 2/g, and the quantum potential strength s = 1 - m 2. A new type of exponentially localized Chern-Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found.
Citation - WoS: 17
Citation - Scopus: 17
Self-Dual Vortices in Chern-Simons Hydrodynamics
(Pleiades Publishing, 2001-06) Lee, Jyh Hao; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie-Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant ℏ for the deformed strength 1 - ℏ2 of the quantum potential and to the pair of diffusion-antidiffusion equations for the strength 1 + ℏ2. Specifying the gauge field as the Abelian Chern-Simons (CS) one in 2+1 dimensions interacting with the nonlinear Schrödinger (NLS) field (the Jackiw-Pi model), we represent the theory as a planar Madelung fluid, where the CS Gauss law has the simple physical meaning of creation of the local vorticity for the fluid flow. For the static flow when the velocity of the center-of-mass motion (the classical velocity) is equal to the quantum velocity (generated by the quantum potential velocity of the internal motion), the fluid admits an N-vortex solution. Applying a gauge transformation of the Auberson-Sabatier type to the phase of the vortex wave function, we show that deformation parameter ℏ, the CS coupling constant, and the quantum potential strength are quantized. We discuss reductions of the model to 1+1 dimensions leading to modified NLS and DNLS equations with resonance soliton interactions.
Citation - Scopus: 1
Inequalities for Buckling of a Clamped Plate
(Taylor and Francis Ltd., 2002) Ufuktepe, Ünal; Mchale, K. P.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
We study the eigenvalue problems for the buckling of a clamped plate. The previous upper bound on low eigenvalues due to Payne, Pólya, and Weinberger, and Rile and Yeh are reviewed. Using methods similar to those used in bounding ratios of eigenvalues of the membrance problem, bounds for ratios of eigenvalues are found for the buckling of a clamped piate
On the Relativistic Supersymmetric Quantum Mechanics
(Springer Verlag, 2002) Mir-Kasimov, Rufat M.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The present paper is devoted to the one-dimensional relativistic supersymmetric quantum mechanics (RSUSYQM). A short formulation of RSUSYQM is given. We show that RSUSYQM is a q-deformed non-relativistic SUSYQM. Two simple examples are given.
Citation - WoS: 9
Citation - Scopus: 10
Relation Between Relativistic and Non-Relativistic Quantum Mechanics as Integral Transformation
(Springer Verlag, 2002-04) Mir-Kasimov, Rufat M.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
A formulation of quantum mechanics (QM) in the relativistic configurational space (RCS) is considered. A transformation connecting the non-relativistic QM and relativistic QM (RQM) has been found in an explicit form. This transformation is a direct generalization of the Kontorovich-Lebedev transformation. It is shown also that RCS gives an example of non-commutative geometry over the commutative algebra of functions.
Citation - WoS: 10
Citation - Scopus: 10
Special Precovers in Cotorsion Theories
(Cambridge University Press, 2002-06) Akıncı, Karen D.; Alizade, Rafail; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
A cotorsion theory is defined as a pair of classes Ext-orthogonal to each other. We give a hereditary condition (HC) which is satisfied by the (flat, cotorsion) cotorsion theory and give properties satisfied by arbitrary cotorsion theories with an HC. Given a cotorsion theory with an HC, we consider the class of all modules having a special precover with respect to the first class in the cotorsion theory and show that this class is closed under extensions. We then raise the question of whether this class is resolving or coresolving.
Citation - WoS: 15
Citation - Scopus: 13
Black Holes and Solitons of the Quantized Dispersionless Nls and Dnls Equations
(Cambridge University Press, 2002-07) Pashaev, Oktay; Lee, Jyh Hao; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The classical dynamics of non-relativistic particles are described by the Schrödinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrödinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary, values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.
Citation - WoS: 90
Citation - Scopus: 90
Resonance Solitons as Black Holes in Madelung Fluid
(World Scientific Publishing Co. Pte Ltd, 2002-08) Pashaev, Oktay; Lee, Jyh Hao; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Envelope solitons of the Nonlinear Schrödinger equation (NLS) under quantum potential's influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s < 1, the model is equivalent to the usual NLS with rescaled coupling constant, while for s > 1, to the reaction-diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1 + 1)-dimensional Jackiw-Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space-time metric. The space-time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.
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.
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: 2
Citation - Scopus: 1
Basic Calculus on Time Scale With Mathematica
(Springer Verlag, 2003) Yantır, Ahmet; Ufuktepe, Ünal; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Mathematical modeling of time dependent systems are always interesting for applied mathematicians. First continuous and then discrete mathematical modeling are built during the mathematical development from ancient to the modern times. By the discovery of the time scales, the problem of irregular controlling of time dependent systems is solved in 1990's. In this paper, we explain the derivative of functions on time scales and the solutions of some basic calculus problems by using Mathematica. © Springer-Verlag Berlin Heidelberg 2003.
Holomorphic Realization of Non-Commutative Space-Time and Gauge Invariance
(IOP Publishing, 2003) Mir-Kasimov, Rufat M.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The realization of the Poincare Lie algebra in terms of noncommutative differential calculus over the commutative algebra of functions is considered. The algebra of functions is defined on the spectrum of the unitary irreducible representations of the De Sitter group. Corresponding space-time carries the noncommutative geometry. Gauge invariance principle consistent with this noncommutative space is considered.
Derivative and Integration on Time Scale With Mathematica
(Imperial College Press, 2003) Yantır, Ahmet; 01. Izmir Institute of Technology
Mathematical modelling of time dependent systems is always interesting for applied mathematicians. First continuous and then discrete mathematical models were built in the mathematical development from ancient to modem times. With the discovery of time scale, the problem of irregular systems was solved in the 1990s. In this paper we explain the derivative and integral of functions of time scales and the solution of some basic calculus problems using Mathematica.
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.
Partial Differential Equations With Webmathematica
(Imperial College Press, 2003) Ufuktepe, Ünal; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The growing popularity of the internet, and the increasing number of computers connected to it, make it an ideal framework for remote education. Many disciplines are rethinking their traditional philosophies and techniques to adapt to the new technologies. Web-based education is an effective framework for such learning, which simplifies theory understanding, encourages learning by discovery and experimentation and undoubtedly makes the learning process more pleasant. There is a need for adequate tools to help in the elaboration of courses that might make it possible to express all the possibilities offered by www teaching. webMathematica is a web-based technology developed by Wolfram Research that allows the generation of dynamic web content with Mathematica. With this technology, distance education students should be able to explore and experiment with mathematical concepts. In this paper we present a sample lecture for Partial Differential Equations in webMathematica for the distance learning environment.
Citation - WoS: 15
Citation - Scopus: 16
The Nearly-Optimal Petrov-Galerkin Method for Convection-Diffusion Problems
(Elsevier Ltd., 2003-06) Neslitürk, Ali İhsan; Harari, Isaac; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The nearly-optimal Petrov-Galerkin (NOPG) method is employed to improve finite element computation of convection-dominated transport phenomena. The design of the NOPG method for convection-diffusion is based on consideration of the advective limit. Nonetheless, the resulting method is applicable to the entire admissible range of problem parameters. An investigation of the stability properties of this method leads to a coercivity inequality. The convergence features of the NOPG method for convection-diffusion are studied in an error analysis that is based on the stability estimates. The proposed method compares favorably to the performance of an established technique on several numerical tests.
Citation - WoS: 10
Citation - Scopus: 9
Q-Deformed and C-Deformed Harmonic Oscillators
(Yukawa Institute for Theoretical Physics, 2003-10) Sogami, Ikuo S.; Koizumi, Kouzou; Mir-Kasimov, Rufat M.; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
Hamilton functions of classical deformed oscillators (c-deformed oscillators) are derived from Hamiltonians of g-deformed oscillators of the Macfarlane and Dubna types. A new scale parameter, lq, with the dimension of length, is introduced to relate a dimensionless parameter characterizing the deformation with the natural length of the harmonic oscillator. Contraction from q-deformed oscillators to c-deformed oscillators is accomplished by keeping lq finite while taking the limit ℏ → 0. The c-deformed Hamilton functions for both types of oscillators are found to be invariant under discrete translations: the step of the translation for the Dubna oscillator is half of that for the Macfarlane oscillator. The c-deformed oscillator of the Macfarlane type has propagating solutions in addition to localized ones. Reinvestigation of the g-deformed oscillator carried out in the light of these findings for the c-deformed systems proves that the g-deformed systems are invariant under the same translation symmetries as the c-deformed systems and have propagating waves of the Bloch type

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