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Explicit Examples (explicit + example)

Distribution by Scientific Domains
Mathematics and Statistics57%

Selected Abstracts

An explicit example of a noncrossed product division algebra

MATHEMATISCHE NACHRICHTEN, Issue 1 2004
Timo Hanke
Abstract The paper presents an explicit example of a noncrossed product division algebra of index and exponent 8 over the field ,(s)(t). It is an iterated twisted function field in two variables D(x, ,)(y, , ) over a quaternion division algebra D which is defined over the number field ,(,3,,,7). The automorphisms , and , are computed by solving relative norm equations in extensions of number fields. The example is explicit in the sense that its structure constants are known. Moreover, it is pointed out that the same arguments also yield another example, this time over the field ,((s))((t)), given by an iterated twisted Laurent series ring D((x, ,))((y, , )) over the same quaternion division algebra D. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Compactifications of the heterotic string with unitary bundles

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 11 2006
T. Weigand
Abstract We describe a large new class of four-dimensional supersymmetric string vacua defined as compactifications of the E8 × E8 and the SO(32) heterotic string on smooth Calabi-Yau threefolds with unitary gauge bundles and heterotic five-branes. The conventional gauge symmetry breaking via Wilson lines is replaced by the embedding of non-flat line bundles into the ten-dimensional gauge group, thus opening up the way for phenomenologically interesting string compactifications on simply connected manifolds. After a detailed analysis of the four-dimensional effective theory we exemplify the general framework by means of a couple of explicit examples involving the spectral cover construction of stable holomorphic bundles. As for the SO(32) heterotic string, the resulting vacua can be viewed, in the S-dual Type I picture, as a generalisation of magnetized D9/D5-brane models. In the case of the E8 × E8 string, we find a natural way to construct realistic MSSM-like models, either directly or via a flipped SU(5) GUT scenario. [source]

TERM STRUCTURES OF IMPLIED VOLATILITIES: ABSENCE OF ARBITRAGE AND EXISTENCE RESULTS

MATHEMATICAL FINANCE, Issue 1 2008
Martin Schweizer
This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h. We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h. For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h, namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions. [source]

Mean-Variance Hedging for Stochastic Volatility Models

MATHEMATICAL FINANCE, Issue 2 2000
Francesca Biagini
In this paper we discuss the tractability of stochastic volatility models for pricing and hedging options with the mean-variance hedging approach. We characterize the variance-optimal measure as the solution of an equation between Doléans exponentials; explicit examples include both models where volatility solves a diffusion equation and models where it follows a jump process. We further discuss the closedness of the space of strategies. [source]

Some observations on the l2 convergence of the additive Schwarz preconditioned GMRES method

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2002
Xiao-Chuan Cai
Abstract Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H1 norm). Very little progress has been made in the theoretical understanding of the l2 behaviour of this very successful algorithm. To add to the difficulty in developing a full l2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l2 cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat,Elman,Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad,Schultz theory, is bounded from both above and below by constants multiplied by h,1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l2 convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd. [source]

A risk model driven by Lévy processes

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 2 2003
Manuel Morales
Abstract We present a general risk model where the aggregate claims, as well as the premium function, evolve by jumps. This is achieved by incorporating a Lévy process into the model. This seeks to account for the discrete nature of claims and asset prices. We give several explicit examples of Lévy processes that can be used to drive a risk model. This allows us to incorporate aggregate claims and premium fluctuations in the same process. We discuss important features of such processes and their relevance to risk modeling. We also extend classical results on ruin probabilities to this model. Copyright © 2003 John Wiley & Sons, Ltd. [source]