Let V be a complex vector space of dimension l. Let S be the C-algebra of polynomial functions on V. Let Ders be the S-module of derivations of S and let Ωs = Homs (Ders, S) be the dual S-module of differential 1-forms. Let {ei} be a basis for V and let {xi} be the dual basis for V. Then {Di = ∂/∂xi and {dxi} are bases for Ders and Ωs as S-modules. If f ∈ S, define a map Hess (f): Ders → Ωs by

Then Hess (f) is an S-module homomorphism which does not depend on the choice of basis for V.

Let V be a complex vector space of dimension l and let G ⊂ GL(V) be a finite reflection group. Let S be the C-algebra of polynomial functions on V with its usual G-module structure (gf)(v) = f{g-1v). Let R be the subalgebra of G-invariant polynomials. By Chevalley’s theorem there exists a set ℬ = {f1, …, fl} of homogeneous polynomials such that R = C[f1, …, fl]. We call ℬ a set of basic invariants or a basic set for G. The degrees di = deg fi are uniquely determined by G. We agree to number them so that d1 ≤ … ≤ di. The map τ: V/G → C1 defined by

is a bijection. Each reflection in G fixes some hyperplane in V.

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).

In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

Here we shall prove that Painlevé’s first transcendent, a solution of the equation y″ = 6y2 + x, can not be described as any combination of solutions of first order algebraic differential equations and those of linear differential equations. This result gives an answer to the question whether the function is truely new or not.

At the early fifties A. Weil introduced [3] and algebraic approach to the theory of infinitesimal prolongations of smooth manifolds motivated by the theory of jets developed by Ch. Ehresmann on one side and also by the return to Fermat’s methods on the infinitesimal calculus of first order, that makes use of nilpotent infinitesimals.

In his book P. Lévy discussed certain permutation groups of natural numbers in connection with the theory of functional analysis. Among them the group , called the Lévy group after T. Hida, has been studied along with Hida’s theory of white noise analysis and has become very important keeping profound contact with the Lévy Laplacian which is an infinite dimensional analogue of the ordinary Laplacian.

R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the “Height zero conjecture”. It may be stated as follows:

Let d be a square-free integer. Let

and {1, ω} forms a Z-basis for the ring of integers of the quadratic field We denote by Δ and hd the discriminant and the class number of respectively.

Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13] [23] discussed the k-rational points on E of prime power order. For a prime number p, let n = n(k, p) be the least non negative integer such that

for all elliptic curves E defined over a quadratic field k ([15]).

Let k[x, y] be the ring of polynomials in two variables over a field k of characteristic zero.

If f, g ∈ k[x, y] then we write f ~ g in the case where f = ag, for some a ∈ k = k\{0}, and we denote by [f, g] the jacobian of (f, g), that is, [f, g] = fxgy - fygx

Let I be a Cohen-Macaulay ideal of grade g > 0 in a local Gorenstein ring (R, m) with residue class field k. An R-ideal J is said to be linked to I with respect to the regular sequence α = α1 …, αg ⊂ I ∩ J if J = (α): I and I = (α): J ([6]). In this paper we are concerned with the following question: how big is dimk ((α, mJ)/mJ)? Obviously this dimension is at most g, but it could be as small as 0. If it is g then the link from J to I is called a minimal link, which is in most respects the desired type of link. The only general result known in this direction is that if I is Gorenstein, then dimk {(α, mJ)/mJ) = g unless both I and J are complete intersections (see [1], Proposition 5.2). We are able to generalize this fact to the case where (R/I)p is Gorenstein for all prime ideals p in R/I with dim (R/J)p ≤ 4; however we have to assume that I is generically a complete intersection ideal, and that R is a complete intersection (Theorem 2.3). Without the assumption on R we prove that if I is generically a complete intersection, and if for a fixed integer r the type of (R/I)p is at most r for all prime ideals p in R/I with dim (R/I)P ≤ (r + 1)2, then dimk ((α, mJ/mJ)) ≥ g — r (Proposition 2.1). If r = 1, i.e. if R/I is Gorenstein in codimension 4, then this estimate shows the dimension is at least g — 1. Theorem 2.3 can also be interpreted to yield a strong upper bound for the codimension of the non-Gorenstein-locus of certain perfect ideals: Let R be a regular local ring. Let I be an R-ideal which is generically a complete intersection, and assume that I is in the even linkage class of a Gorenstein ideal (i.e., there exists a sequence of links I ~ I1 ~ I2 ~ … ~ I2n with I2n a Gorenstein ideal); then I is a Gorenstein ideal provided that {R/I)p is Gorenstein for all prime ideals p of R/I with dim (R/I)p ≤ 4 (Corollary 3.1).