A generalized jacobi theta function

The delta function initial condition solution v^*(x,t;y) at x = y ≥ 0 of the generalized Feller equation is used to define a generalized Jacobi Theta function \documentclass{article}\pagestyle{empty}\begin{document}$ \Theta (x,t) = \upsilon *(x,t;0) + 2\sum\limits_{n = 1}^\infty {v*(x,t;y_n)} $\end{document} for a sufficiently rapidly increasing and unbounded positive sequence {y_y}. It is shown that Θ(x,t) is analytic in each variable in certain regions of the complex x and t planes and that it is a solution of the generalized Feller equation. For those parameters for which this equation reduces to the heat equation, Θ(x,t) reduces to the third Jacobi Theta function.     

Capillary gas chromatography/mass spectrometry with chemical ionization and negative ion detection for confirmation of identity of patulin in apple juice

Gas chromatography/mass spectrometry (GC/MS) with negative ion chemical ionization permits detection of underivatized patulin in apple juice extracts while minimizing co-extractive responses. The technique has been used with a variety of capillary columns in quadrupole, ion trap, and magnetic sector GC/MS instruments to confirm presumptive findings of patulin in apple juice at concentrations ranging from 68 to 3700 micrograms/L. The demonstrated ability to use any of these 3 mass spectrometers and several capillary columns to confirm the identity of patulin are significant strengths of the technique.     

Weak Solutions of Fluid–Solid Interaction Problems

This paper is concerned with various variational formulations for the fluid–solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Gårding's inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.     

A technique for the indexed assessment of fumes generated by high-power laser processing

Information presented by other workers on fumes generated during high-power CO₂ laser processing tends in most cases to be empirical and qualitative in nature, which has limited the significance of this data to real industrial applications. This paper reviews information on the nature of laser processing fumes and proposes a model for indexing the risk associated with the fumes from both plastics and metal materials processing. The index allows the risks from a specific laser processing system to be assessed and classified into one of three groups. Classification of the risk associated with particular fumes enables appropriate choices about the extraction and filtration system to be made. Application of this model highlights the risks involved in processing stainless steel, owing to the formation of Cr(vi). It also highlights the problems associated with processing Kevlar, polycarbonate, and polyamide.     

Biorthogonal sequences of solutions of the generalized feller equation

The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.     

Secondary loss generation by gas turbine support struts

Short cylindrical struts are commonly employed to carry services across the annular flow passages of gas turbines and to provide mechanical support. Velocity variations along the span of the strut will be large and secondary flow becomes important. For bluff bodies, boundary layer separation tends to be fixed close to the maximum thickness of the strut, or any sharp edges, so that secondary flow effects have only a minor influence on wake formation. In the case of more streamlined shapes, the effect of Reynolds number and freestream turbulence level on boundary layer growth are much more significant. Moreover, the secondary flows generated by the interaction between the strut cross-section and the end-wall boundary layers may influence the position of separation, thus changing the distributions of pressure on the strut surface and in the wake. These modifications lead to large variations in the total drag force experienced by the strut. A recent wind tunnel investigation is described in which wake pressure measurements have been used to determine the additional losses produced by the secondary flow generation. Experiments have been performed on isolated struts for both circular and streamlined cross-section over a range of Reynolds number, aspect ratio and thickness-to-chord ratio. A principal finding is that the results for the streamlined struts may be reduced to a correlation which embraces the effects of cross-sectional geometry as well as the end-wall boundary layer thickness, the Reynolds and the Mach numbers.     

On the approximate solution of elliptic,self adjoint boundary value problems

Summary Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.     

On the uniform continuity of metric projections in Banach spaces

Let M be a convex Chebyshev subset of a uniformly convex and uniformly smooth Banach space. It is proved that the metric projection P_M of X onto M is uniformly continuous on every bounded subset of X. Moreover, a global and explicit estimate on the modulus of continuity of the metric projection is obtained.