Generalization of the BTK Theory to the Case of Finite Quasiparticle Lifetimes

A generalization to the BTK theory is developed based on the fact that the quasiparticle lifetime is finite as a result of the damping caused by the interactions. For this purpose, appropriate self-energy expressions and wave functions are inserted into the strong coupling version of the Bogoliubov equations and subsequently, the coherence factors are computed. By applying the suitable boundary conditions to the case of a normal-superconducting interface, the probability current densities for the Andreev reflection, the normal reflection, the transmission without branch crossing and the transmission with branch crossing are determined. Accordingly the electric current and the differential conductance curves are calculated numerically for Nb, Pb, and Pb0.9Bi0.1 alloy. The generalization of the BTK theory by including the phenomenological damping parameter is critically examined. The observed differences between our approach and the phenomenological approach are investigated by the numerical analysis.

Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions

We present a generalization of the finite volume evolution Galerkin scheme [M. Lukacova-Medvid'ova,J. Saibertov'a, G. Warnecke, Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comp. Phys. (2002) 183 533-562; M. Luacova-Medvid'ova, K.W. Morton, G. Warnecke, Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems, SIAM J. Sci. Comput. (2004) 26 1-30] for hyperbolic systems with spatially varying flux functions. Our goal is to develop a genuinely multi-dimensional numerical scheme for wave propagation problems in a heterogeneous media. We illustrate our methodology for acoustic waves in a heterogeneous medium but the results can be generalized to more complex systems. The finite volume evolution Galerkin (FVEG) method is a predictor-corrector method combining the finite volume corrector step with the evolutionary predictor step. In order to evolve fluxes along the cell interfaces we use multi-dimensional approximate evolution operator. The latter is constructed using the theory of bicharacteristics under the assumption of spatially dependent wave speeds. To approximate heterogeneous medium a staggered grid approach is used. Several numerical experiments for wave propagation with continuous as well as discontinuous wave speeds confirm the robustness and reliability of the new FVEG scheme.

Heat transfer in laminar flow of non-Newtonian fluids in ducts of elliptical section

Laminar forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumption used in this work is a laminar flow of a power flow inside elliptical tube, under a boundary condition of first kind with constant physical properties and negligible axial heat diffusion (high Peclet number). To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm-Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number and the average Nusselt number for various cross-section aspect ratios. (C) 2006 Elsevier. SAS. All rights reserved.

Thermally developing forced convection of non-Newtonian fluids inside elliptical ducts

Laminar-forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumptions used in this work are a non-Newtonian fluid, laminar flow, constant physical properties, and negligible axial heat diffusion (high Peclet number). Most of the previous research in elliptical ducts deal mainly with aspects of fully developed laminar flow forced convection, such as velocity profile, maximum velocity, pressure drop, and heat transfer quantities. In this work, we examine heat transfer in a hydrodynamically developed, thermally developing laminar forced convection flow of fluid inside an elliptical tube under a second kind of a boundary condition. To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm-Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform, where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number, and the average Nusselt number for various cross-section aspect ratios.

Thermomechanical generalization of pin joints in finite plates

The understanding of thermoelastic behaviour of joints is significant in order to ensure the integrity of large and complex structures exposed to a thermal environment, particularly in fields such as aerospace and nuclear engineering. Thermomechanical generalization of partial contact behaviour of a pin joint under combined in-plane mechanical loading and on-axis unidirectional heat flow has already been established by the authors for the analytically simpler domains of large plates. This paper successfully extends the on-going investigation to a single pin in a finite rectangular isotropic plate as a two-dimensional abstraction from a practical situation of a multipin fastener joint. The finite element method is used to analyse the joint problem under on-axis thermomechanical loading and unified load-contact relationships are established for a class of loading conditions.

Spontaneous emission lifetime distribution of infinite line antennas in two-dimensional photonic crystals with finite size

The authors calculate the lifetime distribution functions of spontaneous emission from infinite line antennas embedded in two-dimensional disordered photonic crystals with finite size. The calculations indicate the coexistence of both accelerated and inhibited decay processes in disordered photonic crystals with finite size. The decay behavior of the spontaneous emission from infinite line antennas changes significantly by varying factors such as the line antennas' positions in the disordered photonic crystal, the shape of the crystal, the filling fraction, and the dielectric constant. Moreover, the authors analyze the effect of the degree of disorder on spontaneous emission. (c) 2007 American Institute of Physics.

Recognition on Finite Set of Events: Bayesian Analysis of Generalization Ability and Classification Tree Pruning

The problem of recognition on finite set of events is considered. The generalization ability of classifiers for this problem is studied within the Bayesian approach. The method for non-uniform prior distribution specification on recognition tasks is suggested. It takes into account the assumed degree of intersection between classes. The results of the analysis are applied for pruning of classification trees.

Comparison of eight published static finite element models of the intact lumbar spine : predictive power of models improves when combined together

Finite element (FE) model studies have made important contributions to our understanding of functional biomechanics of the lumbar spine. However, if a model is used to answer clinical and biomechanical questions over a certain population, their inherently large inter-subject variability has to be considered. Current FE model studies, however, generally account only for a single distinct spinal geometry with one set of material properties. This raises questions concerning their predictive power, their range of results and on their agreement with in vitro and in vivo values. Eight well-established FE models of the lumbar spine (L1-5) of different research centres around the globe were subjected to pure and combined loading modes and compared to in vitro and in vivo measurements for intervertebral rotations, disc pressures and facet joint forces. Under pure moment loading, the predicted L1-5 rotations of almost all models fell within the reported in vitro ranges, and their median values differed on average by only 2° for flexion-extension, 1° for lateral bending and 5° for axial rotation. Predicted median facet joint forces and disc pressures were also in good agreement with published median in vitro values. However, the ranges of predictions were larger and exceeded those reported in vitro, especially for the facet joint forces. For all combined loading modes, except for flexion, predicted median segmental intervertebral rotations and disc pressures were in good agreement with measured in vivo values. In light of high inter-subject variability, the generalization of results of a single model to a population remains a concern. This study demonstrated that the pooled median of individual model results, similar to a probabilistic approach, can be used as an improved predictive tool in order to estimate the response of the lumbar spine.

A New Finite Element Method for Strain Gradient Theories and Applications to Fracture Analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J2 deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.

A finite-element analysis of viscoelastically damped sandwich plates

A finite element analysis associated with an asymptotic solution method for the harmonic flexural vibration of viscoelastically damped unsymmetrical sandwich plates is given. The element formulation is based on generalization of the discrete Kirchhoff theory (DKT) element formulation. The results obtained with the first order approximation of the asymptotic solution presented here are the same as those obtained by means of the modal strain energy (MSE) method. By taking more terms of the asymptotic solution, with successive calculations and use of the Padé approximants method, accuracy can be improved. The finite element computation has been verified by comparison with an analytical exact solution for rectangular plates with simply supported edges. Results for the same plates with clamped edges are also presented.

Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

Combinatorial properties of finite geometric lattices

Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k) ≥ w(1) for k = 2, 3, ..., n-1. Second, w(k) = w(1) if and only if k = n-1 and L is modular. Several corollaries concerning the "matching" of points and dual points are derived from these theorems.

Both theorems can be regarded as a generalization of a theorem of de Bruijn and Erdös concerning ʎ= 1 designs. The second can also be considered as the converse to a special case of Dilworth's theorem on finite modular lattices.

These results are related to two conjectures due to G. -C. Rota. The "unimodality" conjecture states that the w(k)'s form a unimodal sequence. The "Sperner" conjecture states that a set of non-comparable elements in L has cardinality at most max/k {w(k)}. In this thesis, a counterexample to the Sperner conjecture is exhibited.

Lifetime distribution of spontaneous emission from line antennas in two-dimensional quasi-periodic photonic crystals

We investigate the lifetime distribution functions of spontaneous emission from line antennas embedded in finite-size two-dimensional 12-fold quasi-periodic photonic crystals. Our calculations indicate that two-dimensional quasi-periodic crystals lead to the coexistence of both accelerated and inhibited decay processes. The decay behaviors of line antennas are drastically changed as the locations of the antennas are varied from the center to the edge in quasi-periodic photonic crystals and the location of transition frequency is varied.

On equations for regular languages, finite automata, and sequential networks

We consider systems of equations of the form where A is the underlying alphabet, the Xi are variables, the Pi,a are boolean functions in the variables Xi, and each δi is either the empty word or the empty set. The symbols υ and denote concatenation and union of languages over A. We show that any such system has a unique solution which, moreover, is regular. These equations correspond to a type of automation, called boolean automation, which is a generalization of a nondeterministic automation. The equations are then used to determine the language accepted by a sequential network; they are obtainable directly from the network.

On the Relationship Between Generalization Error, Hypothesis Complexity, and Sample Complexity for Radial Basis Functions

In this paper, we bound the generalization error of a class of Radial Basis Function networks, for certain well defined function learning tasks, in terms of the number of parameters and number of examples. We show that the total generalization error is partly due to the insufficient representational capacity of the network (because of its finite size) and partly due to insufficient information about the target function (because of finite number of samples). We make several observations about generalization error which are valid irrespective of the approximation scheme. Our result also sheds light on ways to choose an appropriate network architecture for a particular problem.