• Volume 100, Issue 1

April 1990,   pages  1-94

• Relative extrema of Legendre functions of the second kind

Let μk,n denote the relative maxima of ⋎Qn(x)⋎, the Legendre function of the second kind, ordered so that μk+1,n occurs to the left of μk,n. They by analogy with a theorem of Szegö for Legendre polynomials, μk,n+1&lt;μk,n,k=1,...,n,n=1,2,...

• Multipliers for the Weyl transform and Laguerre expansions

Let Pk denote the projection of L2(RR) onto the kth eigenspace of the operator (-δ+⋎x⋎2 andSNα=(1/ANαkN=0AN−kαPk. We study the multiplier transformTNα for the Weyl transform W defined byW(TNαf)=SnαW(f). Applications to Laguerre expansions are given.

• On the inverse Laplace transform of the product of a general class of polynomials and the multivariable H-function

In this paper we evaluate the inverse Laplace transform of$$\begin{gathered} s^{ - \eta } (s^{l_1 } + \lambda _1 )^{ - \sigma } (s^{l_2 } + \lambda _2 )^{ - \rho } \hfill \\ \times S_n^m [xs^{ - W} (S^{l_1 } + \lambda _1 )^{ - \upsilon } (S^{l_2 } + \lambda _2 )^{ - w} ]S_{n'}^{m'} [ys^{ - w'} (S^{l_1 } + \lambda _1 )^{ - \upsilon '} (S^{l_2 } + \lambda _2 )^{ - w_r } ] \hfill \\ \times H[z_1 s^{ - W_1 } (S^{l_1 } + \lambda _1 )^{ - \upsilon _1 } (S^{l_2 } + \lambda _2 )^{ - w_1 } ,...,z_r s^{ - w_r } (S^{l_1 } + \lambda _1 )^{ - \upsilon _r } (S^{l_2 } + \lambda _2 )^{ - w'} ] \hfill \\ \end{gathered}$$

Due to the general nature of the multivariable H-function involved herein, the inverse Laplace transform of the product of a large number of special functions involving one or more variables, occurring frequently in the problems of theoretical physics and engineering sciences can be obtained as simple special cases of our main findings. For the sake of illustration, we obtain here the inverse Laplace transform of a product of the Hermite polynomials, the Jacobi polynomials andr different modified Bessel functions of the second kind. A theorem obtained by Srivastava and Singh follows as a special case of our main result.

• Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT5, be the translate ofT bys inS defined byT5(x)=(Tx)5. We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the mapsT5. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μs, where μs denotes the unique vector measure corresponding to the operatorT5.

• Verschiebung and Frobenius operators

Metropolis and Rota introduced the concept of the necklace ring Nr(A) of a commutative ringA. WhenA contains Q as a subring there is a natural bijection γ:Nr(A→1+tA[]. Grothendieck has introduced a ring structure on 1+tA[t] while studyingK-theoretic Chern classes. Nr(A) comes equipped with two families of operatorsFr,Vr called the Frobenius and Verschiebung operators. Mathematicians studying formal group laws have introduced two families of operators,Fr, andVr on 1+tA[t]. Metropolis and Rota have not however tried to show that γ preserves, these operators. They transport the operators from Nr(A) to 1+tA[t] using γ. In our present paper we show that γ does preserve all these operators.

• Criterion for smoothness of Schubert varieties in Sl(n)/B

LetG=Sl(n) andB, the Borel subgroup ofG consisting of upper triangular matrices. LetwSn andX(w)=BwB(modB), the associated Schubert variety inG/B. In this paper, we give a geometric criterion for the smoothness ofX(w). This criterion admits a neat combinatorial description in terms of the permutationw.

• Vector fields and framings on isolated complete intersection singularities

A theorem is proved for deciding as to when the complex orthogonal complement of a vector field on an isolated, complete intersection germ, is a trivial vector bundle.

• A class of totally geodesic foliations of Lie groups

This paper is devoted to classifying the foliations ofG with leaves of the formgKh−1 whereG is a compact, connected and simply connected Lie group andK is a connected closed subgroup ofG such thatG/K is a rank-1 Riemannian symmetric space. In the case whenG/K=Sn, the homotopy type of space of such foliations is also given.

• On infinite-dimensional control systems with state and control constraints

In this paper we examine infinite-dimensional control systems governed by semilinear evolution equations and having both state and control constraint. We introduce the relaxed system and show that the original trajectories are dense in an appropriate function space in the relaxed ones. We also determine the dependence of the solution set on the initial conditions. Then using those results we establish necessary and sufficient conditions for optimality for some optimization problems. Finally we prove some controllability results.

• $$\tilde K'_3$$—A new triangulation ofRn—A new triangulation ofRn

This paper introduces a new triangulation ofRn. The triangulation is to be applied for solving system of nonlinear equation through a fixed point technique.

• On shock dynamics

This is in continuation of our paper On the propagation of a multi-dimensional shock of arbitrary strength’ published earlier in this journal (Srinivasan and Prasad ). We had shown in our paper that Whitham’s shock dynamics, based on intuitive arguments, cannot be relied on for flows other than those involving weak shocks and that too with uniform flow behind the shock. Whitham  refers to this as misinterpretation of his approximation and claims that his theory is not only correct but also provides a natural closure of the open system of the equations of Maslov . The main aim of this note is to refute Whitham’s claim with the help of an example and a numerical integration of a problem in gasdynamics.

• Corrections to some expressions in “On the propagation of a multi-dimensional shock of arbitrary strength”