• Volume 113, Issue 4

November 2003,   pages  355-464

• 𝐿1-Convergence of Complex Double Fourier Series

It is proved that the complex double Fourier series of an integrable function 𝑓(𝑥, 𝑦) with coefficients {𝑐𝑗𝑘} satisfying certain conditions, will converge in 𝐿1-norm. The conditions used here are the combinations of Tauberian condition of Hardy–Karamata kind and its limiting case. This paper extends the result of Bray [1] to complex double Fourier series.

• Pfister Involutions

The question of the existence of an analogue, in the framework of central simple algebras with involution, of the notion of Pfister form is raised. In particular, algebras with orthogonal involution which split as a tensor product of quaternion algebras with involution are studied. It is proven that, up to degree 16, over any extension over which the algebra splits, the involution is adjoint to a Pfister form. Moreover, cohomological invariants of those algebras with involution are discussed.

• Triebel–Lizorkin Space Estimates for Multilinear Operators of Sublinear Operators

In this paper, we obtain the continuity for some multilinear operators related to certain non-convolution operators on the Triebel–Lizorkin space. The operators include Littlewood–Paley operator and Marcinkiewicz operator.

• Stability Estimates for ℎ-𝑝 Spectral Element Methods for General Elliptic Problems on Curvilinear Domains

In this paper we show that the ℎ-𝑝 spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska–Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming ℎ-𝑝 spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

• A Probabilistic Approach to Second Order Variational Inequalities with Bilateral Constraints

We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of a unique viscosity solution of these variational inequalities and give a stochastic representation to this solution. As an application, we study a stochastic game with stopping times and show the existence of a saddle point equilibrium.

• Uniform Stability of Damped Nonlinear Vibrations of an Elastic String

Here we are concerned about uniform stability of damped nonlinear transverse vibrations of an elastic string fixed at its two ends. The vibrations governed by nonlinear integro-differential equation of Kirchoff type, is shown to possess energy uniformly bounded by exponentially decaying function of time. The result is achieved by considering an energy-like Lyapunov functional for the system.

• On Howard's Conjecture in Heterogeneous Shear Flow Problem

Howard's conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a mathematically rigorous fashion for plane parallel heterogeneous shear flows with negligible buoyancy force $g𝛽 \ll 1$ (Miles J W, J. Fluid Mech. 10 (1961) 496–508), where 𝛽 is the basic heterogeneity distribution function).

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