An asymptotic simplex method for parametric linear programming

A comparison of parametric and non-parametric distance functions: With application to European railways

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension 3 using the framework of hypercomplex function theory. This first article on hand deals with generalized Fourier and Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in ℝ3 and their series coefficients can be explicitly (one-to-one) linked with each other. Finally, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented.

In this chapter we use the Cauchy theory to study functions that are holomorphic on an annulus, and analytic functions with isolated singularities. We describe a classification for isolated singularities. Functions that are holomorphic on an annulus have Laurent series expansions, an analogue of power series expansions for holomorphic functions on discs. Holomorphic functions with a finite number of isolated singularities in a domain can be integrated using the residue theorem, an analogue of the Cauchy integral formula. We discuss local properties of these functions. The study of zeros and poles of meromorphic functions leads to a theorem of Rouché that connects the number of zeros and poles to an integral. The theorem is not only aesthetically pleasing in its own right but also allows us to give alternate proofs of many important results. In the penultimate section of this chapter we illustrate the use of complex function theory in the evaluation of real definite integrals. The appendix discusses Cauchy principal values, a way to integrate functions with certain singularities.KeywordsCauchy TheoryHolomorphic FunctionReal Definite IntegralsLaurent Series ExpansionNonconstant Meromorphic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The effect of surface and interface elasticity in the analysis of the Saint–Venant torsion problem of an eccentrically two-phase fcc circular nanorod is considered; description of the behavior of such a small structure via usual classical theories cease to hold. In this work, the problem is formulated in the context of the surface/interface elasticity. For a rigorous solution of the proposed problem, conformal mapping with a Laurent series expansion are employed together. The numerical results well illustrate that the torsional rigidity and stress distribution corresponding to such nanosized structural elements are significantly affected by the size. In order to employ surface and interface elasticity, several key properties such as surface energy, surface stresses, and surface elastic constants of several fcc materials as well as interface properties of the noncoherent fcc bicrystals are derived in terms of Rafii-Tabar and Sutton interatomic potential function. For determination of the surface/interface parameters a molecular dynamics program, which uses the above-mentioned potential function, is developed. The calculated surface and interface properties are in reasonable agreement with the corresponding results in literature. Some applications of the given results can be contemplated in the design of micro-/nano-electromechanical systems.

Stress concentration around interacting circular holes: a comparison between theory and experiments

Green's functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack

The integrand-level methods for the reduction of scattering amplitudes are powerful techniques for the analysis and the computation of loop integrals, which have already been successfully applied and automated at one-loop. Moreover, some very interesting progress has recently been made towards the higher-loop extension of such techniques. In this presentation, we review the basics principles of integrand-reduction methods within a coherent framework we developed, which can be applied to any integrand at any number of loops and is based on simple concepts of algebraic geometry, such as multivariate polynomial division. We particularly focus on semianalytic and algebraic techniques, such as the Laurent series expansion which we exploited to improve the one-loop reduction with the library NINJA, and the multi-loop divide-and-conquer approach which can always be used to find the integrand decomposition of any Feynman graph in a finite number of algebraic operations.

Fractional‐order circuits find a widespread use in different engineering applications. The problem of realising fractional‐order circuits has been discussed by several authors, however, it is far from being solved. Realising fractional‐order resistorless passive network with three element orders is been studied. At first, this study extends the two‐variable reactance matrix synthesis method to three‐variable case, and then proposes a synthesis method of fractional‐order reactance matrix with three element orders by variable substitution. The process in above methods mainly involves variable substitution, decomposition of three‐variable reactance matrix, extraction of unit inductors, Laurent series expansion, spectral factorisation of two‐variable positive semidefinite Hermitian matrix and synthesis of univariable reactance matrix. Then the above‐mentioned synthesis process is illustrated by two examples.

On the elliptical inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric field

A new self-contained approach based on the Drazin generalized inverse is used to derive many basic results in discrete time, finite state Markov decision processes. A product form representation for the transition matrix of a stationary policy gives new derivations of the average reward evaluation equations, Laurent series expansions, as well as the finite test for Blackwell optimality. This representation also suggests new computational methods.

Asymptotics of some three-species models

In this chapter we shall relate our results to some classical functional problems that were studied by Caratheodory and Schur and some related moment problems. A survey can be found in [AKH]. In this context F(z) will have a Laurent series expansion $$F(z) = \sum\nolimits_{ - \infty }^\infty {{f_k}{z^k}} $$ in the neighborhood of |z|= p = 1 and $${f_{ - k}} = {{\bar f}_k},k\, \in \,\mathbb{Z}$$ .This means that our meromorphic function F(z) will take real and positive values on the unit circle. The symmetry in the problem that is caused by $${f_{ - k}} = {{\bar f}_k},k\, \in \,\mathbb{Z}$$ will essentially reduce the complexity of the problem to half the complexity for the general case. The quantities with and without hat will contain the same information, so that we can drop one of them. Many of the algorithms that we have seen before will reduce to well known classical algorithms.

On an Identity of Ramanujan Based on The Hypergeometric Series2F1([formula omitted], [formula omitted]; [formula omitted];x)

The particular concern of this paper is the construction of a confidence region with pointwise asymptotically correct size for the true value of a parameter of interest based on the generalized Anderson-Rubin (GAR) statistic when the moment variance matrix is singular. The large sample behaviour of the GAR statistic is analysed using a Laurent series expansion around the points of moment variance singularity. Under a condition termed first order moment singularity the GAR statistic is shown to possess a limiting chi-square distribution on parameter sequences converging to the true parameter value. Violation, however, of this condition renders the GAR statistic unbounded asymptotically. The paper details an appropriate discretisation of the parameter space to implement a feasible GAR-based confidence region that contains the true parameter value with pointwise asymptotically correct size. Simulation evidence is provided that demonstrates the efficacy of the GAR-based approach to moment-based inference described in this paper.

This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.