H. Azad

Symmetry analysis of wave equation on static spherically symmetric spacetimes with higher symmetries

Symmetry analysis of wave equation on all static spherically symmetric spacetimes admitting maximal isometry groups G10 or G7 or G6 is carried out. Symmetry algebras of the wave equation are found and their structural information-in the sense of Iwasawa decomposition-is obtained. Joint invariants of appropriate subalgebras are utilized to obtain many exact solutions of the wave equation on static spherically symmetric spacetimes.

Polynomial solutions of differential equations

AbstractA new approach for investigating polynomial solutions of differential equations is proposed. It is based on elementary linear algebra. Any differential operator of the form L(y)=∑k=0k=Nak(x)y(k), where ak is a polynomial of degree ≤ k, over an infinite field F has all eigenvalues in F in the space of polynomials of degree at most n, for all n. If these eigenvalues are distinct, then there is a unique monic polynomial of degree n which is an eigenfunction of the operator L, for every non-negative integer n. Specializing to the real field, the potential of the method is illustrated by recovering Bochners classification of second order ODEs with polynomial coefficients and polynomial solutions, as well as cases missed by him - namely that of Romanovski polynomials, which are of recent interest in theoretical physics, and some Jacobi type polynomials. An important feature of this approach is the simplicity with which the eigenfunctions and their orthogonality and norms can be determined, resulting in significant reduction in computational complexity of such problems. 2000 MSC: 33C45; 34A05; 34A30; 34B24.

Analytic solutions of initial-boundary-value problems of transient conduction using symmetries

Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.

Quasi-potentials and Kähler–Einstein Metrics on Flag Manifolds

The proof of this result uses ideas suggested by the theory of linear algebraic groups, in particular Section 8 of Steinberg’s ‘‘Lectures on w x w x Chevalley Groups’’ St1 and Section 14 of Borel and Hirzebruch Bo]Hi . An essential ingredient is an explicit description of the differential forms w x dual to the Dynkin lines St2 ; this description goes back to Borel and w x Hirzebruch Bo]Hi, Section 14 . A special case of this result is proved and w x used by the first and the second authors in Az]Ko to prove the existence of complete Ricci-flat Kahler metrics on the complexification of Rieman̈ nian symmetric spaces of compact type. In our attempt to prove our main w x result, we eventually simplified the arguments in Bo]Hi, Section 14 .

Holomorphic principal bundles over a compact Kähler manifold

Abstract Let ρ:G→H be a homomorphism between connected reductive algebraic groups over C such that the center of the Lie algebra g is sent to the center of h . If EG is a holomorphic principal G-bundle over a compact connected Kahler manifold M, and EG is semistable (resp. polystable), then the principal H-bundle EG×GH is also semistable (resp. polystable). A G-bundle over M is polystable if and only if it admits an Einstein–Hermitian connection; this is an analog of a theorem of Uhlenbeck and Yau for G-bundles. Two different formulations of the G-bundle analog of the Harder–Narasimhan reduction have been established. The equivalence of the two formulations is a consequence of a group theoretic result.

Spherically symmetric manifolds which admit five isometries

Spherically symmetric space–times which admit a five‐dimensional group of motions are studied and it is shown that it can never be locally a maximal group of motions but globally it could be a maximal group.

Embedding algorithms and applications to differential equations

Abstract Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry.

A point symmetry based method for transforming ODEs with three-dimensional symmetry algebras to their canonical forms

We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.

Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

If L is a semisimple Lie algebra of vector fields on ℝN with a split Cartan subalgebra C, then it is proved here that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical form of L in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on ℝ3 and to determine all representations of sl(N, ℝ) as local vector fields on ℝN−1. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits sl(3, ℝ) as its symmetry algebra and for a system of two second-order ordinary differential equations that admits sl(4, ℝ) as its symmetry algebra.

The projection method in evaluating multiple integrals

Projections are fundamental to computations involving areas and volumes and are emphasized as such in all textbooks. In this note it is shown how one can work out projections in suitable planes without actually drawing the figure by using inequalities as a substitute for drawing, and their usefulness is illustrated by examples in which a drawing is either very difficult or time-consuming.