• Volume 104, Issue 2

May 1994,   pages  305-433

• Bertini theorems for ideals linked to a given ideal

We prove a generalization of Flenner’s local Bertini theorem for complete intersections. More generally, we study properties of the ‘general’ ideal linked to a given ideal. Our results imply the following. LetR be a regular local Nagata ring containing an infinite perfect fieldk, andI⊂R is an equidimensional radical ideal of heightr, such thatR/I is Cohen-Macaulay and a local complete intersection in codimension 1. Then for the ‘general’ linked idealJα, R/Jα is normal and Cohen-Macaulay.

The proofs involve a combination of the method of basic elements, applied to suitable blow ups.

• On the Ramanujan-Petersson conjecture for modular forms of half-integral weight

It is shown that a “character twist” of a growth estimate for certain weighted infinite sums of Kloosterman sums which is equivalent to the Ramanujan-Petersson conjecture for modular forms of half-integral weight, can easily be proved using Deligne’s theorem (previously the Ramanujan-Petersson conjecture for modular forms of integral weight).

• On composition of some general fractional integral operators

In the present paper we derive three interesting expressions for the composition of two most general fractional integral oprators whose kernels involve the product of a general class of polynomials and a multivariableH-function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition of fractional integral operators involving classical orthogonal polynomials and simpler special functions (involving one or more variables) which occur rather frequently in problems of mathematical physics. We have mentioned here two special cases of the first composition formula. The first involves product of a general class of polynomials and the Fox’sH-functions and is of interest in itself. The findings of Buschman [1] and Erdélyi [4] follow as simple special cases of this composition formula. The second special case involves product of the Jacobi polynomials, the Hermite polynomials and the product of two multivariableH-functions. The present study unifies and extends a large number of results lying scattered in the lierature. Its findings are general and deep.

• On the absolute matrix summability of Fourier series and some associated series

The object of the paper is to study the absolute matrix summability problem of Fourier series, conjugate series and some associated series under a new set of conditions on matrix methods, generalising many known results in the literature.

• On absolute summability factors of infinite series

In this paper using δ-quasi-monotone sequences a theorem on$$\left| {\bar N,p_n ;\delta } \right|_k$$ summability factors of infinite series, which generalizes a theorem of Bor [4] on$$\left| {\bar N,p_n } \right|_k$$ summability factors of infinite series, is proved. Also, in the special case this theorem includes a result of Mazhar [8] on |C, 1|k summability factors.

• Rearrangements of bounded variation sequences

Letbv be the set of all bounded variation sequences. In the present paper we deduce from a theorem of Mears a necessary and sufficient condition for the rearrangement (ap(k)) to be of bounded variation wheneverak)∈bv; interestingly it coincides with Pleasants’ criterion for convergence-preserving.

• A note on a generalization of Macdonald’s identities forAl andBl

Let η(q) denote the Dedekind’s η-function. Macdonald obtained identities for η(q)dim g where g is complex simple finite dimensional Lie algebra. The aim of this paper is to obtain generalization of the above identities in the case of g=Al andBl. We also get new formulas for the generating functions of the Ramanujan’s τ-function and ϕα-functions.

• Combinatorial manifolds with complementarity

A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.

We show that if a d-dimensional combinatorial manifold M with n vertices satisfies complementarity then d=0, 2, 4, 8, or 16 with n=3d/2+3 and |M| is a “manifold like a projective plane”. Arnoux and Marin had earlier proved the converse statement.

• Deformations of complex structures on Γ/SL2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL2(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)).

We also show thatG acts trivially on the coherent cohomology groupsHi(Γ/G, O) for anyi≥0.

• Differential subordinations concerning starlike functions

Denote byS* (⌕), (0≤⌕&lt;1), the family consisting of functionsf(z)=z+a2z2+...+anzn+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎&lt;1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)μ−1&lt;h(z) implies f∈S*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.

• On the structure of stable random walks

We show that the Cauchy random walk on the line, and the Gaussian random walk on the plane are similar as infinite measure preserving transformations.

• L1 (μ,X) as a complemented subspace of its bidual

We show that for a Banach spaceX, if the space ofX-valued Bochner integrable functions is complemented in some dual space, then it is complemented in the space ofX-valued countably additive, μ-continuous vector measures.

• Stresses in an elastic plate lying over a base due to strip-loading

The closed-form analytic expressions for the stresses at any point of an elastic plate coupling in different ways to a base as a result of a two-dimensional shear strip-loading are obtained. The contact between the horizontal layer and the base is either smooth-rigid or rough-rigid or welded. The variations of the shear stresses with the horizontal distance have been studied numerically. It is found that the effect of different boundary conditions on the stress field is significant and the stresses for an elastic layer lying over an elastic half-space differ considerably from those of an entire homogeneous elastic half-space.

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