On the solution of the equation ut+unux+H(x, t, u)=0

We consider the equation u(t) + u(n)u(x) + H(x, t, u) = 0 and derive a transformation relating it to u(t) + u(n)u(x) = 0. Special cases of the equation appearing in applications are discussed. Initial value problems and asymptotic behaviour of the solution are studied.

Local polynomial convexity of the union of two totally-real surfaces at their intersection

We consider the following question: Let S (1) and S (2) be two smooth, totally-real surfaces in C-2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is S-1 boolean OR S-2 locally polynomially convex at the origin? If T (0) S (1) a (c) T (0) S (2) = {0}, then it is a folk result that the answer is yes. We discuss an obstruction to the presumed proof, and provide a different approach. When dim(R)(T0S1 boolean AND T0S2) = 1, we present a geometric condition under which no consistent answer to the above question exists. We then discuss conditions under which we can expect local polynomial convexity.

Some aspects of the solitary waves in a relativistic inhomogeneous plasma

In contrast to earlier observations on various solitary wave propagations, especially those bifurcated by the compressive and rarefactive solitary waves, the existence of spiky and explosive solitary waves is here believed to arise because of the presence of free and trapped electrons. So far, very few studies have been carried out to satisfactorily explain the presence of the solitary waves in space as observed by satellites. It is also attempted to highlight the probable impact on the various solitary wave propagations in a generalized multi-component, inhomogeneous plasma upon consideration of a relativistic treatment. It is expected that such a treatment will prove the existence of the solitary waves most expeditiously and exhibit the presence of chaos therein, thus giving a suitable explanation to the observations of various forms of spiky and explosive solitary waves in space-plasma. Copyright (C) 1996 Elsevier Science Ltd

The approximate parametrization of the curve of intersection of implicit surfaces

A clear definition of an approximate parametrization of the curve of intersection of (n-1) implicit surfaces in Rn is given. It is justified that marching methods yield such an approximation.

Singularity structure and chaotic behavior of the homopolar disk dynamo

We study in great detail a system of three first-order ordinary differential equations describing a homopolar disk dynamo (HDD). This system displays a large variety of behaviors, both regular and chaotic. Existence of periodic solutions is proved for certain ranges of parameters. Stability criteria for periodic solutions are given. The nonintegrability aspects of the HDD system are studied by investigating analytically the singularity structure of the system in the complex domain. Coexisting attractors (including period-doubling sequence) and coexisting strange attractors appear in some parametric regimes. The gluing of strange attractors and the ungluing of a strange attractor are also shown to occur. A period of bifurcation leading to chaos, not observed for other chaotic systems, is shown to characterize the chaotic behavior in some parametric ranges. The limiting case of the Lorenz system is also studied and is related to HDD.

Coherence of the real symmetric Hardy algebra

Let D denote the open unit disk in C centered at 0. Let H-R(infinity) denote the set of all bounded and holomorphic functions defined in D that also satisfy f(z) = <(f <(z)over bar>)over bar> for all z is an element of D. It is shown that H-R(infinity) is a coherent ring.

Solution of a hypersingular integral equation of the second kind

A straightforward analysis involving the complex function-theoretic method is employed to determine the closed-form solution of a special hypersingular integral equation of the second kind, and its known solution is recovered.

Spectral asymptotics of the Helmholtz model in fluid-solid structures

A model representing the vibrations of a coupled fluid-solid structure is considered. This structure consists of a tube bundle immersed in a slightly compressible fluid. Assuming periodic distribution of tubes, this article describes the asymptotic nature of the vibration frequencies when the number of tubes is large. Our investigation shows that classical homogenization of the problem is not sufficient for this purpose. Indeed, our end result proves that the limit spectrum consists of three parts: the macro-part which comes from homogenization, the micro-part and the boundary layer part. The last two components are new. We describe in detail both macro- and micro-parts using the so-called Bloch wave homogenization method. Copyright (C) 1999 John Wiley & Sons, Ltd.

On the solution of the problem of scattering of surface water waves by a sharp discontinuity in the surface boundary conditions

Closed-form analytical expressions are derived for the reflection and transmission coefficients for the problem of scattering of surface water waves by a sharp discontinuity in the surface-boundary-conditions, for the case of deep water. The method involves the use of the Havelock-type expansion of the velocity potential along with an analysis to solve a Carleman-type singular integral equation over a semi-infinite range. This method of solution is an alternative to the Wiener-Hopf technique used previously.

Calculation of the front part of the sonic boom signature for a Maneuvering aerofopil

The sonic boom at a large distance from its source consists of a leading shock, a trailing shock and a one parameter family of nonlinear wavefronts in between these shocks. A new ray theoretical method using a shock ray theory and a weakly nonlinear lay theory has been used to obtain the shock fronts and wavefronts respectively, for a maneuvering aerofoil in a homogeneous medium. This method introduces a one parameter family of Cauchy problems to calculate the shock and wave fronts emerging from the surface of the aerofoil. These problems are solved numerically to obtain the leading shock front and the nonlinear wavefronts emerging from the front portion of the aerofoil.

Ontheproof of the Poincare´ conjecture

In 2002, Perelman proved the Poincare conjecture, building on the work of Richard Hamilton on the Ricci flow. In this article, we sketch some of the arguments and attempt to place them in a broader dynamical context.

On the proof of the Poincare´ conjecture

In 2002, Perelman proved the Poincare conjecture, building on the work of Richard Hamilton on the Ricci flow. In this article, we sketch some of the arguments and attempt to place them in a broader dynamical context.

An analogue of the Wiener Tauberian Theorem for the Heisenberg Motion group

We show that the Wiener Tauberian property holds for the Heisenberg Motion group TnBof the same result for a wider class of groups. However, our exposition is almost self contained and the techniques used in the proof are relatively simple.

Some Aspects of the Kobayashi and Carath,odory Metrics on Pseudoconvex Domains

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carath,odory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C (2), and on convex finite type domains in C (n) using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C (2) and convex finite type domains in C (n) in terms of Euclidean parameters. Second, a version of Vitushkin's theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C (1)-isometries of the Kobayashi and Carath,odory metrics on a smoothly bounded strongly pseudoconvex domain.