Algebraic stacks

TOMÁS L GÓMEZ
Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
E-mail: tomas@math.tifr.res.in

This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.

Variational formulae for Fuchsian groups over families of algebraic curves

DAKSHINI BHATTACHARYYA
Last Address: The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India

We study the problem of understanding the uniformizing Fuchsian groups for a family of plane algebraic curves by determining explicit first variational formulae for the generators.

Limits of commutative triangular systems on locally compact groups

RIDDHI SHAH
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India

On a locally compact group G, if n_n^kn ® m, (k_n® ¥), for some probability measures n_n and m on G, then a sufficient condition is obtained for the set A = {n_n^m | m £ k_n} to be relatively compact; this in turn implies the embeddability of a shift of m. The condition turns out to be also necessary when G is totally disconnected. In particular, it is shown that if G is a discrete linear group over R then a shift of the limit m is embeddable. It is also shown that any infinitesimally divisible measure on a connected nilpotent real algebraic group is embeddable. 

Topological *-algebras with C*-enveloping algebras II

S J BHATT
Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar 388 120, India
E-mail: tomas@math.tifr.res.in

Universal C*-algebras C*(A) exist for certain topological *-algebras called algebras with a C*-enveloping algebra. A Frechet *-algebra A has a C*-enveloping algebra if and only if every operator representation of A maps A into bounded operators. This is proved by showing that every unbounded operator representation , continuous in the uniform topology, of a topological *-algebra A, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C* algebra E(A) of A. Given a C*-dynamical system (G, A, a), any topological *-algebra B containing C_c(G, A) as a dense *-subalgebra and contained in the crossed product C*-algebra C*(G, A, a) satisfies E(B) = C*(G, A, a). If G = R, if B is an a-invariant dense Frechet *-subalgebra of A such that E(B) = A, and if the action a on B is m-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed product S(R, B, a) satisfies E(S(R, B, a)) = C*(R, A, a). When G is a Lie group, the C^¥-elements C^¥(A), the analytic elements C^w(A) as well as the entire analytic elements C^ew(A) carry natural topologies making them algebras with a C*-enveloping algebra. Given a non-unital C*-algebra A, an inductive system of ideals I_a is constructed satisfying A = C*-ind lim I_a; and the locally convex inductive limit ind lim I_a is an m-convex algebra with the C*-enveloping algebra A and containing the Pedersen ideal K_A of A. Given generators G with weakly Banach admissible relations R, we construct universal topological *-algebra A(G, R) and show that it has a C*-enveloping algebra if and only if (G, R) is C*-admissible.

On theequisummability of Hermite and Fourier expansions

E K NARAYANAN and S THANGAVELU
Statistics and Mathematics Division, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560 059, India
Email: naru@isibang.ac.in; veluma@isibang.ac.in

We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions.

Periodic and boundary value problems for second order differential equations

NIKOLAOS S PAPAGEORGIOU and FRANCESCA PAPALINI*
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
*Department of Mathematics, University of Ancona, Via Brecce Bianche, Ancona 60131, Italy

In this paper we study second order scalar differential equations with Sturm–Liouville and periodic boundary conditions. The vector field f(t, x, y) is Caratheodory and in some instances the continuity condition on x or y is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.

Boundary controllability of integrodifferential systems in Banach spaces

K BALACHANDRAN and E R ANANDHI
Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory.