Segal-Bargmann Transform and Paley-Wiener Theorems on Motion Groups

We study the Segal-Bargmann transform on a motion group R-n v K, where K is a compact subgroup of SO(n) A characterization of the Poisson integrals associated to the Laplacian on R-n x K is given We also establish a Paley-Wiener type theorem using complexified representations

Nonnegative integral solution of linear equations

A method to obtain a nonnegative integral solution of a system of linear equations, if such a solution exists is given. The method writes linear equations as an integer programming problem and then solves the problem using a combination of artificial basis technique and a method of integer forms.

Existence Theorems for Generalized Hammerstein Equations

In this paper we obtain existence theorems for generalized Hammerstein-type equations K(u)Nu + u = 0, where for each u in the dual X* of a real reflexive Banach space X, K(u): X -- X* is a bounded linear map and N: X* - X is any map (possibly nonlinear). The method we adopt is totally different from the methods adopted so far in solving these equations. Our results in the reflexive spacegeneralize corresponding results of Petry and Schillings.

On Paley-Wiener Theorems for the Heisenberg Group

In this paper we prove two Paley-Wiener-type theorems for the Heisenberg group. One is for the group Fourier transform which is the analogue of the classical Paley-Wiener theorem. The other one is for the spectral projections associated to the sub-Laplacian

Analogues of the Wiener-Tauberian and Schwartz theorems for radial functions on symmetric spaces

We prove a Wiener Tauberian theorem for the L-1 spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz-type theorem for complex groups. As a corollary we obtain a Wiener Tauberian type result for compactly supported distributions.

A bicharacteristic formulation of the ideal MHD equations

On a characteristic surface Omega of a hyperbolic system of first-order equations in multi-dimensions (x, t), there exits a compatibility condition which is in the form of a transport equation along a bicharacteristic on Omega. This result can be interpreted also as a transport equation along rays of the wavefront Omega(t) in x-space associated with Omega. For a system of quasi-linear equations, the ray equations (which has two distinct parts) and the transport equation form a coupled system of underdetermined equations. As an example of this bicharacteristic formulation, we consider two-dimensional unsteady flow of an ideal magnetohydrodynamics gas with a plane aligned magnetic field. For any mode of propagation in this two-dimensional flow, there are three ray equations: two for the spatial coordinates x and y and one for the ray diffraction. In spite of little longer calculations, the final four equations (three ray equations and one transport equation) for the fast magneto-acoustic wave are simple and elegant and cannot be derived in these simple forms by use of a computer program like REDUCE.

Points-to Analysis as a System of Linear Equations

We propose a novel formulation of the points-to analysis as a system of linear equations. With this, the efficiency of the points-to analysis can be significantly improved by leveraging the advances in solution procedures for solving the systems of linear equations. However, such a formulation is non-trivial and becomes challenging due to various facts, namely, multiple pointer indirections, address-of operators and multiple assignments to the same variable. Further, the problem is exacerbated by the need to keep the transformed equations linear. Despite this, we successfully model all the pointer operations. We propose a novel inclusion-based context-sensitive points-to analysis algorithm based on prime factorization, which can model all the pointer operations. Experimental evaluation on SPEC 2000 benchmarks and two large open source programs reveals that our approach is competitive to the state-of-the-art algorithms. With an average memory requirement of mere 21MB, our context-sensitive points-to analysis algorithm analyzes each benchmark in 55 seconds on an average.

Exact-Solutions Of Linear Partial-Differential Equations With Variable-Coefficients

Backlund transformations relating the solutions of linear PDE with variable coefficients to those of PDE with constant coefficients are found, generalizing the study of Varley and Seymour [2]. Auto-Backlund transformations are also determined. To facilitate the generation of new solutions via Backlund transformation, explicit solutions of both classes of the PDE just mentioned are found using invariance properties of these equations and other methods. Some of these solutions are new.

Exact free-surface flows for shallow-water equations .1. - the incompressible case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.

PVU and wave-particle splitting schemes for Euler equations of gas dynamics

A new way of flux-splitting, termed as the wave-particle splitting is presented for developing upwind methods for solving Euler equations of gas dynamics. Based on this splitting, two new upwind methods termed as Acoustic Flux Vector Splitting (AFVS) and Acoustic Flux Difference Splitting (AFDS) methods are developed. A new Boltzmann scheme, which closely resembles the wave-particle splitting, is developed using the kinetic theory of gases. This method, termed as Peculiar Velocity based Upwind (PVU) method, uses the concept of peculiar velocity for upwinding. A special feature of all these methods that the unidirectional and multidirectional parts of the flux vector are treated separately. Extensive computations done using these schemes demonstrate the soundness of the ideas.

A characteristic based numerical method with tracking for nonlinear wave equations

Many physical problems can be modeled by scalar, first-order, nonlinear, hyperbolic, partial differential equations (PDEs). The solutions to these PDEs often contain shock and rarefaction waves, where the solution becomes discontinuous or has a discontinuous derivative. One can encounter difficulties using traditional finite difference methods to solve these equations. In this paper, we introduce a numerical method for solving first-order scalar wave equations. The method involves solving ordinary differential equations (ODEs) to advance the solution along the characteristics and to propagate the characteristics in time. Shocks are created when characteristics cross, and the shocks are then propagated by applying analytical jump conditions. New characteristics are inserted in spreading rarefaction fans. New characteristics are also inserted when values on adjacent characteristics lie on opposite sides of an inflection point of a nonconvex flux function, Solutions along characteristics are propagated using a standard fourth-order Runge-Kutta ODE solver. Shocks waves are kept perfectly sharp. In addition, shock locations and velocities are determined without analyzing smeared profiles or taking numerical derivatives. In order to test the numerical method, we study analytically a particular class of nonlinear hyperbolic PDEs, deriving closed form solutions for certain special initial data. We also find bounded, smooth, self-similar solutions using group theoretic methods. The numerical method is validated against these analytical results. In addition, we compare the errors in our method with those using the Lax-Wendroff method for both convex and nonconvex flux functions. Finally, we apply the method to solve a PDE with a convex flux function describing the development of a thin liquid film on a horizontally rotating disk and a PDE with a nonconvex flux function, arising in a problem concerning flow in an underground reservoir.

A note on a class of singular integro-differential equations

A simplified analysis is employed to handle a class of singular integro-differential equations for their solutions

Exact free-surface flows for shallow-water equations .2. the compressible case

Exact free surface flows with shear in a compressible barotropic medium are found, extending the authors' earlier work for the incompressible medium. The barotropic medium is of finite extent in the vertical direction, while it is infinite in the horizontal direction. The ''shallow water'' equations for a compressible barotropic medium, subject to boundary conditions at the free surface and at the bottom, are solved in terms of double psi-series, Simple wave and time-dependent solutions are found; for the former the free surface is of arbitrary shape while for the latter it is a damping traveling wave in the horizontal direction, For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed. In the case of an isothermal medium, when gamma = 1, we again find simple wave and time-dependent solutions.