Mathematics / Matematik

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  • Conference Object
    Measure on Time Scales With Mathematica
    (Springer Verlag, 2006) Ufuktepe, Ünal; Yantır, Ahmet
    In 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.
  • Article
    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.
    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
    An Application With Webmathematica
    (Springer Verlag, 2003) Ufuktepe, Ünal
    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.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 1
    Basic Calculus on Time Scale With Mathematica
    (Springer Verlag, 2003) Yantır, Ahmet; Ufuktepe, Ünal
    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.
  • Conference Object
    Citation - WoS: 34
    On the Importance of Public-Key Validation in the Mqv and Hmqv Key Agreement Protocols
    (Springer Verlag, 2006) Menezes, Alfred; Ustaoğlu, Berkant
    HMQV is a hashed variant of the MQV key agreement protocol proposed by Krawczyk at CRYPTO 2005. In this paper, we present some attacks on HMQV and MQV that are successful if public keys are not properly validated. In particular, we present an attack on the two-pass HMQV protocol that does not require knowledge of the victim's ephemeral private keys. The attacks illustrate the importance of performing some form of public-key validation in Diffie-Hellman key agreement protocols, and furthermore highlight the dangers of relying on security proofs for discrete-logarithm protocols where a concrete representation for the underlying group is not specified.
  • Conference Object
    Citation - WoS: 4
    Reusing Static Keys in Key Agreement Protocols
    (Springer Verlag, 2009) Chatterjee, Sanjit; Menezes, Alfred; Ustaoğlu, Berkant
    Contrary to conventional cryptographic wisdom, the NIST SP 800-56A standard explicitly allows the use of a static key pair in more than one of the key establishment protocols described in the standard. In this paper, we give examples of key establishment protocols that are individually secure, but which are insecure when static key pairs are reused in two of the protocols. We also propose an enhancement of the extended Canetti-Krawczyk security model and definition for the situation where static public keys are reused in two or more key agreement protocols. © 2009 Springer-Verlag.
  • Conference Object
    Citation - WoS: 28
    Strongly Secure Authenticated Key Exchange Without Naxos' Approach
    (Springer Verlag, 2009) Kim, Minkyu; Fujioka, Atsushi; Ustaoğlu, Berkant
    LaMacchia, Lauter and Mityagin [15] proposed the extended Canetti-Krawczyk (eCK) model and an AKE protocol, called NAXOS. Unlike previous security models, the adversary in the eCK model is allowed to obtain ephemeral secret information related to the test session, which makes the security proof difficult. To overcome this NAXOS combines an ephemeral private key x with a static private key a to generate an ephemeral public key X; more precisely X∈=∈g H(x,a). As a result, no one is able to query the discrete logarithm of X without knowing both the ephemeral and static private keys. In other words, the discrete logarithm of an ephemeral public key, which is typically the ephemeral secret, is hidden via an additional random oracle. In this paper, we show that it is possible to construct eCK-secure protocol without the NAXOS' approach by proposing two eCK-secure protocols. One is secure under the GDH assumption and the other under the CDH assumption; their efficiency and security assurances are comparable to the well-known HMQV [12] protocol. Furthermore, they are at least as secure as protocols that use the NAXOS' approach but unlike them and HMQV, the use of the random oracle is minimized and restricted to the key derivation function. © 2009 Springer-Verlag Berlin Heidelberg.
  • Conference Object
    Citation - WoS: 9
    Towards Denial-Of Key Agreement Protocols
    (Springer Verlag, 2009) Stebila, Douglas; Ustaoğlu, Berkant
    Denial of service resilience is an important practical consideration for key agreement protocols in any hostile environment such as the Internet. There are well-known models that consider the security of key agreement protocols, but denial of service resilience is not considered as part of these models. Many protocols have been argued to be denial-of-service-resilient, only to be subsequently broken or shown ineffective. In this work we propose a formal definition of denial of service resilience, a model for secure authenticated key agreement, and show how security and denial of service resilience can be considered in a common framework, with a particular focus on client puzzles. The model accommodates a variety of techniques for achieving denial of service resilience, and we describe one such technique by exhibiting a denial-of-service-resilient secure authenticated key agreement protocol. Our approach addresses the correct integration of denial of service countermeasures with the key agreement protocol to prevent hijacking attacks that would otherwise render the countermeasures irrelevant. © 2009 Springer Berlin Heidelberg.
  • Conference Object
    Citation - WoS: 19
    Comparing the Pre- and Post-Specified Peer Models for Key Agreement
    (Springer Verlag, 2008) Menezes, Alfred; Ustaoğlu, Berkant
    In the pre-specified peer model for key agreement, it is assumed that a party knows the identifier of its intended communicating peer when it commences a protocol run. On the other hand, a party in the post-specified peer model for key agreement does not know the identifier of its communicating peer at the outset, but learns the identifier during the protocol run. In this paper we compare the security assurances provided by the Canetti-Krawczyk security definitions for key agreement in the pre- and post-specified peer models. We give examples of protocols that are secure in one model but insecure in the other. We also enhance the Canetti-Krawczyk security models and definitions to encompass a class of protocols that are executable and secure in both the pre- and post-specified peer models. © 2008 Springer-Verlag Berlin Heidelberg.
  • Article
    Citation - WoS: 32
    Citation - Scopus: 37
    Symbolic Computation and Construction of New Exact Traveling Wave Solutions To Fitzhugh-Nagumo and Klein-Gordon Equations
    (Walter de Gruyter GmbH, 2009) Öziş, Turgut; Aslan, İsmail
    With the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called (G'/G)-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed.