The Basis Properties of Some Systems of Exponential Functions,Cosines, and Sines

We consider some systems of exponential functions, cosines, and sines with complex-valued coefficients and establish a necessary and sufficient condition for completeness and minimality of these systems in Lebesgue spaces.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Kurzweil-Henstock type integral on zero-dimensional group and some of its applications

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case. Supported by RFFI-08-01-00669.     

Picone type identities and definiteness of quadratic functionals on time scales

In this paper we derive a new sufficient condition for the nonnegativity of time scale quadratic functionals associated to time scale symplectic systems. To establish this result, a new global Picone formula is derived. Another proof of a special case of the result is shown to be obtained via a Sturmian comparison technique. Furthermore, we derive several new Picone type identities which, in particular, do not impose a certain delta-differentiability assumption, and we survey known ones from the literature. The results in this paper complete our earlier work on the definiteness of a time scale quadratic functional in terms of its corresponding time scale symplectic system.     

Completeness of the Root Function System of a Nonlocal Problem in Lp

We obtain sufficient conditions for the completeness of the root function system for nonlocal elliptic problems in the Sobolev spaces W_p ^l(Q).     

A Turbulent-Wake Estimation Using Radial Basis Function Neural Networks

An estimation of the fluctuations in the passive-tracer concentration for the turbulent wake behind an airfoil is presented. The estimation is based on experimental modelling using Radial Basis Function Neural Networks. For the experiment the fluctuations of the concentration in the turbulent wake were recorded with a visualization method. The records of the concentration in the selected regions of the turbulent wake were used as the input and output regions for the training and estimation with neural networks. The uncertainty of the estimation increased with increasing distance between the input and the output regions. The power spectra, the spatial correlation functions and the profiles of the concentration were calculated from the measured and estimated fluctuations of the concentration. The measured and estimated concentration power spectra were in reasonable agreement. The measured and estimated spatial correlation functions and the profiles of the concentration showed a similar agreement.     

A Class of Incomplete Orthogonal Factorization Methods. II: Implementation and Results

We present, implement and test several incomplete QR factorization methods based on Givens rotations for sparse square and rectangular matrices. For square systems, the approximate QR factors are used as right-preconditioners for GMRES, and their performance is compared to standard ILU techniques. For rectangular matrices corresponding to linear least-squares problems, the approximate R factor is used as a right-preconditioner for CGLS. A comprehensive discussion is given about the uses, advantages and shortcomings of the preconditioners. AMS subject classification (2000) 65F10, 65F25, 65F50.Received May 2002. Revised October 2004. Communicated by Åke Björck.     

Recent developments in error estimates for scattered-data interpolation via radial basis functions

Error estimates for scattered data interpolation by shifts of a positive definite function for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. However, apart from special cases where data is gridded, these interpolation estimates do not apply when the target functions generating the data are outside of the associated RKHS, and in fact until very recently no estimates were known in such situations. In this paper, we review these estimates in cases where the underlying space is Rⁿ and the positive definite functions are radial basis functions (RBFs). AMS subject classification 41A25, 41A05, 41A63, 42B35Research supported by grant DMS-0204449 from the National Science Foundation.