Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Conference Object Holomorphic Realization of Non-Commutative Space-Time and Gauge Invariance(IOP Publishing, 2003) Mir-Kasimov, Rufat M.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.Conference Object Measure on Time Scales With Mathematica(Springer Verlag, 2006) Ufuktepe, Ünal; Yantır, AhmetIn this paper we study the Lebesgue Delta-measure on time scales. We refer to [3, 4] for the main notions and facts from the general measure and Lebesgue Delta integral theory. The objective of this paper is to show how the main concepts of Mathematica can be applied to fundamentals of Lebesgue Delta- and Lebesgue Delta- measure on an arbitrary time scale and also on a discrete time scale whose rule is given by the reader. As the time scale theory is investigated in two parts, by means of alpha and rho operators, we named the measures on time scales by the set function DMeasure and NMeasure respectively for arbitrary time scales.Conference Object Citation - Scopus: 1Hipokampüsün El ve Atlas Tabanlı Otomatik Bölütlenmesinin Hacimsel Olarak Karşılaştırılması(Institute of Electrical and Electronics Engineers Inc., 2009) Kutucu, Hakan; Eker, Çağdaş; Kitiş, Ömer; Gönül, Ali SaffetHigh-resolution Magnetic resonance imaging (MRI) is helpful in diagnosing diseases such as schizophrenia, alzheimer, dementia etc. Brain segmentation is an important preprocess in medical imaging applications. In this study we compare atlas based segmentation and manual segmentation of hippocampus for volumetric measures. A statistically difference was obtained between automatic and manual measurement. We conclude that contemporary techniques are not adequate to obtain sensitive data in some barin structures such as hippocampus core.Article Citation - WoS: 3Citation - Scopus: 3A1-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.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.Article Citation - WoS: 81Citation - Scopus: 79The Resonant Nonlinear Schrödinger Equation in Cold Plasma Physics. Application of Bäcklund-Darboux Transformations and Superposition Principles(Cambridge University Press, 2007) Lee, Jiunhung; Pashaev, Oktay; Rogers, Colin; Schief, W. K.A system of nonlinear equations governing the transmission of uni-axial waves in a cold collisionless plasma subject to a transverse magnetic field is reduced to the recently proposed resonant nonlinear Schrödinger (RNLS) equation. This integrable variant of the standard nonlinear Schrödinger equation admits novel nonlinear superposition principles associated with Bäcklund-Darboux transformations. These are used here, in particular, to construct analytic descriptions of the interaction of solitonic magnetoacoustic waves propagating through the plasma.Article Citation - WoS: 11Citation - Scopus: 10A Stabilizing Subgrid for Convection-Diffusion Problem(World Scientific Publishing Co. Pte Ltd, 2006) Neslitürk, Ali İhsanA stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection-diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.Article An Application With Webmathematica(Springer Verlag, 2003) Ufuktepe, ÜnalThere 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.Article Citation - WoS: 2Citation - Scopus: 1Basic Calculus on Time Scale With Mathematica(Springer Verlag, 2003) Yantır, Ahmet; Ufuktepe, ÜnalMathematical 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.Article Citation - WoS: 30Citation - Scopus: 33The Initial Stage of Dam-Break Flow(Springer Verlag, 2009) Korobkin, Alexandre; Yılmaz, OğuzThe liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy-Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.Article Citation - WoS: 89Obtaining a Secure and Efficient Key Agreement Protocol From (h)mqv and Naxos (extended Version)(International Association for Cryptologic Research, 2009) Ustaoğlu, BerkantLaMacchia, Lauter and Mityagin recently presented a strong security definition for authenticated key agreement strengthening the well-known Canetti-Krawczyk definition. They also described a protocol, called NAXOS, that enjoys a simple security proof in the new model. Compared to MQV and HMQV, NAXOS is less efficient and cannot be readily modified to obtain a one-pass protocol. On the other hand MQV does not have a security proof, and the HMQV security proof is extremely complicated. This paper proposes a new authenticated key agreement protocol, called CMQV (`Combined' MQV), which incorporates design principles from MQV, HMQV and NAXOS. The new protocol achieves the efficiency of HMQV and admits a natural one-pass variant. Moreover, we present a simple and intuitive proof that CMQV is secure in the LaMacchia-Lauter-Mityagin model.