Let Y be a complex Banach space, U an open subset of Y, f a mapping of U into Y. Then f is said to be complex analytic if for each pair of elements x and y of Y with x in U, the function f(x + ξy) of the single complex variable ξ is analytic in ξ on some neighbourhood of the origin.

In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αTƒ ‖q ≦ C‖ |x|αƒ ‖p, where Tƒ denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.

In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω∼ will denote a measurable function on Rn × Rn such that for each x ∈ Rn, Ω∼(x, .) is locally integrable and homogeneous of degree 0.

Let $\mathfrak{g}$ be a compact simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$, and the sum of any $d$ non-trivial orbits has non-empty interior. The number $d$ was later reduced to the rank of the Lie algebra (or rank +1 in the case of type ${{A}_{n}}$). More recently, the minimal integer $k\,=\,k\left( X \right)$ such that the $k$-fold convolution of the orbital measure supported on the orbit generated by $X$ is an absolutely continuous measure was calculated for each $X\,\in \,\mathfrak{g}$.

In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie algebras. We characterize the tuples $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{L}} \right)$, with ${{X}_{i}}\,\in \,\mathfrak{g}$, which have the property that the convolution of the $L$-orbital measures supported on the orbits generated by the ${{X}_{i}}$ is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of $\mathfrak{g}$ and the structure of the annihilating roots of the ${{X}_{i}}$. Such a characterization was previously known only for type ${{A}_{n}}$.

Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange‘s theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies $\mathbf{k}$ of a Hopf monoid $\mathbf{h}$ to be a Hopf submonoid: the quotient of any one of the generating series of $\mathbf{h}$ by the corresponding generating series of $\mathbf{k}$ must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopf monoid in the form of certain polynomial inequalities and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.

We define Schwartz functions, tempered functions, and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.

In [3], the perfect rings of Bass [1] were characterized in terms of torsions in the following way:

A ring R is right perfect if and only if every (hereditary) torsion in the categoryMod Rof all left R-modules is fundamental (i.e. generated by some minimal torsions) and closed under taking direct products; as a consequence, the number of all torsions inMod Ris finite and equal to 2n for a natural n.

Here, we present a simple description of those rings R which allow only two (trivial) torsions, viz. 0 and Mod R (and thus, are right perfect by [3]). Finite direct sums of these rings represent a natural generalization of completely reducible (i.e. artinian semisimple) rings (cf. Theorem 2) and we shall call them for that matter π-reducible rings. They can also be characterized in terms of their idempotent two-sided ideals.

We consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied.

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

In Planar sublattices of a free lattice, I [8] we verify Jonsson's conjecture for finite planar lattices; in particular we obtain a characterization of finite planar sublattices of a free lattice among all finite lattices. In the present paper we use arguments of a quite different flavour to obtain another characterization. Let

be the family of lattices illustrated in Figures 1, 2, 3, and 4. Our goal is to prove the following theorem: a finite lattice is a planar sublattice of a free lattice if andonly if it does not have a member of as a sublattice.

Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit, connected, reductive classical group $G$ defined over a non-Archimedean field and $A$ is the standard intertwining operator attached to a tempered representation of $G$ induced from $M$ . In this paper we determine all the cases in which Lie$(N)$ is prehomogeneous under $\text{Ad}\left( m \right)$ when $N$ is non-abelian, and give necessary and sufficient conditions for $A$ to have a pole at $0$.

In Part I of this paper we present a theory of Padé-approximants for Laurent series, and discuss their relation to orthogonal polynomials. For earlier results in this direction we may refer to (1 ; 7; 8). It is also indicated how this theory can be extended, for example, to matrix polynomials.

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on ${{\omega }_{1}}$, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of ${{\omega }_{1}}$.

Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.

In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.

One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, then

and note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

Let A be a commutative ring, and let Proj A[t0, ti]. By a vector bundle on X we mean a locally free sheaf of finite rank on X. Set t = t1/to. Then X is made up of two affine pieces U1 = Spec A[t], and U2 = Spec A[t-1]. Let P(R) denote the category of finitely generated projective modules over the ring R. The category of vector bundles on X is equivalent to the category of triples (P1,f1, P2), where P1 ∊ 𝓅 (A[t]), P2 ∊ 𝓅(A[t-1]), and

is an A[t, t-1] -isomorphism. In [2], the category of vector bundles on is denned directly in this manner, without first defining (so that one could work over a non-commutative ring). We prove that if A is a Krull ring (or a Noetherian ring with connected spec) of dimension > 0, then there is an indecomposable vector bundle of rank n on X, for every positive integer n.

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced by

The first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:

(1) R is a left perfect and every weakly projective right R-module is weakly injective.

(2) R is a direct sum of matrix rings over local QF-rings.

(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.

(4) R is right artinian ring and every weakly injective right R-module is weakly projective.

(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series

1

where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.

In (2), Fitzpatrick and Ford defined a class of spaces which they called order totally paracompact. This class of spaces is significant since the order totally paracompact metric spaces constitute the largest known class of spaces for which the small and large inductive dimensions coincide. In this paper we shall consider a class of spaces containing the class of order totally paracompact spaces; we shall call these spaces order paracompact. We shall establish the relationship of this class to the more familiar classes of spaces and obtain an invariance theorem for order paracompact spaces analogous to theorems obtained for various other classes.

In what follows, no separation axioms are assumed unless specifically mentioned. Thus, for example, a Lindelöf space is not taken to be Hausdorff without making the further assumption. Therefore, a Lindelöf space is not necessarily paracompact.

Motivated by works on extension sets in standard domains, we introduce a notion of the Carathéodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a class of two-dimensional submanifolds in the tridisc that not only turns out to be Carathéodory but also provides examples of two-dimensional domains for which the celebrated Lempert Theorem holds. Additionally, a recently introduced notion of universal sets for the Carathéodory extremal problem is studied and new results on domains admitting (no) finite universal sets are given.