C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K0(X)), construit à l'aide des -modules localement libres sur un schéma X[14]. La somme directe et le produit tensoriel de modules font de K0(X) un anneau, et les opérations de puissances extérieures lui fournissent une structure supplémentaire, que Grothendieck appelle λ-anneau. Un λ-anneau est muni d'une filtration décroissante, la γ-filtration, et un des principaux résultats de Grothendieck est que, si X est lisse sur un corps, le groupe

est isomorphe, à la torsion près, au groupe de Chow CH1(X) des cycles de codimension i sur X, modulo l'équivalence linéaire.

The problem which motivated the writing of this paper is that of finding structure behind the decomposition of the sl3 representation spaces V* ⊗ W = Hom(V, W) for finite dimensional irreducible sl3-modules V and W. For sl2 this extends the classical Clebsch-Gordon problem. The question has been considered for sl3 in a computational way in [5]. In this paper we build a conceptual algebraic framework going beyond the enveloping algebra of sl3.

For each dominant integral weight α let Vα be an irreducible representation of sl3 of highest weight α. It is well known that, for weights α, μ, λ, the multiplicity of Vλ in Hom(Vα, Vα+μ) is bounded by the multiplicity of μ in Vλ, with equality for generic α. This suggests the possibility of a single construction of highest weight vectors of weight X in Hom(Vα, Vα+μ) which is valid for all a.

The classical Hankel transform of a conventional function ϕ on (0, ∞) defined formally by

was extended by Zemanian [21-23] to certain generalized functions of one dimension. Koh [9, 10] extended the work of [21] to n-dimensions, and that of [22] to arbitrary real values of μ. Motivated from the work of Gelfand and Shilov [6], Lee [11] introduced spaces of type Hμ and studied their Hankel transforms. The results of Lee [11] and Zemanian [21] are special cases of recent results obtained by the author and Pandey [14]. The aforesaid extensions are accomplished by using the so-called adjoint method of extending integral transforms to generalized functions. Dube and Pandey [2], Pathak and Pandey [15, 16] applied a more direct method, the so-called kernel method, for extending the Hankel and other related transforms.

Let R = k[X1, …, Xn] with k a field, and let I ⊂ R be a homogeneous ideal. The algebra R/I is said to have a pure resolution if its homogeneous minimal resolution has the form

Some of the known examples of pure resolutions include the coordinate rings of: the tangent cone of a minimally elliptic singularity or a rational surface singularity [15], a variety defined by generic maximal Pfaffians [2], a variety defined by maximal minors of a generic matrix [3], a variety defined by the submaximal minors of a generic square matrix [6], and certain of the Segre-Veronese varieties [1].

If I is in addition Cohen-Macaulay, then Herzog and Kühl have shown that the betti numbers bi are completely determined by the twists di.

Let be an infinite regular cardinal. A poset L is called an -lattice if and only if for all XL satisfying 0 < |X| < m, ∧ X and ∨ X exist.

This paper is a part of a sequence of papers, [5], [6], [7], [8], developing the theory of -lattices. For a survey of some of these results, see [9].

The -lattice is described in [6]; γ denotes the zero and γ′ the unit of . In particular, formulas for -joins and meets are given. (We repeat the essentials of this description in Section 4.)

In [6] we proved the theorem stated below. Our proof was based on characterization of (the free -lattice on P) due to [1]; as a result, our proof was very computational.

In this paper we study integral operators with domain a Banach function space Lρ1 and range another Banach function space Lρ2 or the space L0 of all measurable functions. Recall that a linear operator T from Lρ1 into L0 is called an integral operator if there exists a μ × v-measurable function T(x, y) on X × Y such that

Such an integral operator is called a Carleman integral operator if for almost every x ∊ X the function

is an element of the associate space L′ρ1, i.e.,

The main objective of this paper is to find useful q-extensions of Feldheim's [6] bilinear formula for Jacobi polynomials, namely,

1.1

where the Appel function F4 is defined by

1.2

α1, α2, ρ are arbitrary complex parameters such that the series on both sides of (1.1) are convergent, and

1.3

is the Jacobi polynomial of degree k, (a)k being the usual shifted factorial.

In 1958, Hasse [10, p. 236], in connection with his work on the 2n-th power character of 2 in the cyclotomic field Q(exp(2πi/2n)), proved that for every prime p ≡ 1 (mod 16) the pair of equations

is always solvable in integers x, u, v, w. Later in 1972 Giudici, Muskat, and Robinson [7, p. 388] showed in their work on Brewer's character sums that Hasse's system is also solvable for primes p ≡ 7 (mod 16). Moreover they also showed [7, p. 345] that for primes p ≡ 1 (mod 5) the pair of equations

is solvable in integers x, u, v, w. In this paper we consider a pair of diophantine equations (involving a prime p and an integer m) which includes, the above two systems as the special cases when m = 2 and m = 5. The system is then used to give criteria for m to be a biquadratic residue modulo p.

In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q2) can be partitioned into (q2n++ 1 + 1)/(q + l)-caps. Moreover, these caps were shown to constitute the “large points” of a PG(2n, q) in a natural way. In [3] a similar argument was used to show that once two disjoint (n – l)-subspaces are removed from PG(2n, q2), the remaining points can be partitioned into (q2n – 1)/(q2 – l)-caps.

The purpose of this paper is to give a short proof of the results found in [2], and then use the technique developed to partition PG(2n, q) into (qn + l)-caps for n even and q any prime-power. Moreover, these caps can be treated in a natural way as the “large points” of a PG(n – 1, q).

Let E be a Banach space, A a subset of its dual E*.x0 ∊ A is said to be a w*Gδ point of A if there are xn ∊ E and scalars γn, n = 1,2, 3 … such that

Denote by w*Gδ{A} the set of all w*Gδ points of A. If S is a semigroup of maps on E* and K ⊂ E*, denote by

i.e., the set of points x* in the w*closure of K which are fixed points of S (i.e., sx* = x* for each s in S}. An operator will mean a bounded linear map on a Banach space and Co B will denote the convex hull of B ⊂ E.

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

A character of a finite group G is monomial if it is induced from a linear (degree one) character of a subgroup of G. A group G is an M-group if all its complex irreducible characters (the set Irr(G)) are monomial.

In [1], Dade gave an example of an M-group with a normal subgroup which is itself not an M-group. In his group G, the supersolvable residual N is an extra special 2-group and G/N is supersolvable of even order. Moreover, the prime 2 is used in such a way that no analogous construction is possible in the case that |N| or |G:N| is odd. This led Isaacs in [8] and Dade in [2] to consider the effect of certain “oddness“ hypotheses in the study of monomial characters.

Our main results are in the same spirit. Although our techniques seem to require a restrictive assumption on the supersolvable residual of the groups we consider, our theorems provide more evidence that under fairly general circumstances normal subgroups of M-groups should be M-groups.

For X a metric continuum, 2X denotes the hyper space of all nonempty subcompacta, with the topology induced by the Hausdorff metric H, and C(X) ⊂ 2X the hyperspace of subcontinua. These hyperspaces are continua, in fact are arcwise-connected, since there exist order arcs between each hyperspace element and the element X. They also have trivial shape, i.e., maps of the hyperspaces into ANRs are homotopic to constant maps. For a detailed discussion of these and other general hyperspace properties, we refer the reader to Nadler's monograph [4].

The question of hyperspace contractibility was first considered by Wojdyslawski [8], who showed that 2X and C(X) are contractible if X is locally connected. Kelley [2] gave a more general condition (now called property K) which is sufficient, but not necessary, for hyperspace contractibility. The continuum X has property K if for every there exists δ > 0 such that, for every pair of points x, y with d(x, y) < δ and every subcontinuum M containing x, there exists a subcontinuum N containing y with .

For us an “algebra” is a finitary “universal algebra” in the sense of G. Birkhoff [9]. We are concerned in this paper with algebras whose endomorphisms are determined by small subsets. For example, an algebra A is rigid (in the strong sense) if the only endomorphism on A is the identity idA. In this case, the empty set determines the endomorphism set E(A). We place the property of rigidity at the bottom rung of a cardinal-indexed ladder of properties as follows. Given a cardinal number κ, an algebra A is minimally free over a set of cardinality κ (κ-free for short) if there is a subset X ⊂ A of cardinality κ such that every function f:X → A extends to a unique endomorphism ϕ ∊ E(A). (It is clear that A is rigid if and only if A is 0-free.) Members of X will be called counters; and we will be interested in how badly counters can fail to generate the algebra.

We actually treat double orthogonal series in detail, simply for the sake of brevity in notations. Multiple orthogonal series will be shortly indicated in the concluding Section 8.

Let (X, , μ) be an arbitrary positive measure space and {ϕjk(x):i, k = 0, 1, …} an orthonormal system defined on X. We consider the double orthogonal series

(1.1)

where {ϕik:i, k = 0, 1, …} is a double sequence of real numbers (coefficients), for which

(1.2)

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the form

For example, the solution of the initial value problem

is given by u(x, t) = Ttf(x) where

Peral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only if

From the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.

First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:

(1.1)

where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).

Let H3 be the Heisenberg group in three dimensions, Δ the fundamental subelliptic laplacian on H3 (see Section 1 for notations and definitions) and U be an open subset of H3 If φ is a continuous function on the boundary ∂U of U, the Dirichlet problem is thus,

(1)

In [3], p. 104, it was asserted by the first author that, when dU is regular (see Section 1 for this definition), the problem (1) has a solution continuous on D and a probabilistic formula was given. In [3], we prove that our probabilistic formula gives a solution of the so called “martingale problem” associated to Δ on U (see [5] for this notion). But it appears that the connection between the solution in the martingale problem sense and the true solution is not at all clear in the subelliptic case; in particular it is not obvious at all that the probabilistic formula is a C2 function.