For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, then

is a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundles

where Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.

Let G be a group of finite order g, A a splitting field of G of characteristic p (which may be 0) and R = AG the group algebra of G over A. In [2], the author studied some of the properties of the Grothendieck ring K(R) of the category of all finitely generated R-modules, and derived a number of consequences. This paper continues the study carried out in [2]. The study is concerned with the structure and irreducible representations of K(R). The ring K(R) is proved to be semisimple and the primitive idempotents of K(R) are explicitly constructed. If the ring K(R) is identified with the ‘algebra of representations', then Robinson's idempotent [3; 4; 5] follow from our description as a special case.

Bruck and Kleinfeld [3] proved that any alternative ring with characteristic prime to 2 must satisfy the identity

where the associator (x,y,z) is defined by (x, y, z) = (xy)z – x(yz) and . Linearization of the identity (x2, y, z) = 2x · (x, y, z) yields for characteristic prime to 2 an equivalent identity

(1)

Using the right alternative law (x, y, z) = –(y, x, z) and the flexible law (x, y, z) = –(z, y, x) which is satisfied in any alternative ring we obtain

(2)

and

(3)

This paper sets forth conditions sufficient that the generalized Lototsky method (F, dn) be regular and Mercerian. If the dn's are real and of constant sign, then the conditions are also necessary. Moreover, it follows that if f is a polynomial, then under the same conditions the method (f,dn) is equivalent to the Sonnenschein method generated by f . Various related results are also given.

Definition 2.1. Let f be a nonconstant function holomorphic on the closed unit disk and let be a complex sequence with f (l) + dn ≠ 0. Suppose

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold, Mn, in Euclidean space, which are sufficient to prove that Mn has a transverse field (see 1.1 and 1.2, definitions).

In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifold Mn.

Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).

After the research for this paper was completed, a paper of Davis (2) came to our attention.

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p-torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

We investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$, where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of ${{f}^{\left( j \right)}}$ for $j=1,2,3,4$.

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.

Let ${{\mathbf{F}}_{q}}[T]$ be the ring of polynomials over the finite field of $q$ elements and $Y$ a large integer. We say a polynomial in ${{\mathbf{F}}_{q}}[T]$ is $Y$-smooth if all of its irreducible factors are of degree at most $Y$. We show that a ternary additive equation $a\,+\,b\,=\,c$ over $Y$-smooth polynomials has many solutions. As an application, if $S$ is the set of first $s$ primes in ${{\mathbf{F}}_{q}}[T]$ and $s$ is large, we prove that the $S$-unit equation $u\,+\,v\,=\,1$ has at least $\text{exp}\left( {{s}^{1/6-\in }}\,\text{log}\,\text{q} \right)$ solutions.

The following theorem was proved by Paley and Wiener (4, p. 70; 1, p. 136).

Theorem 1. If f(z) is a canonical product of order 1 with real zeros, and f(0) = 1, the conditions

and

are equivalent. n(r) denotes the number of zeros of absolute value not exceeding r.

Instead of assuming the zeros to be all real Pfluger assumed that the zeros are close to the real axis and proved the following theorem (5 or 1, p. 143).

All rings considered here are commutative containing at least two elements, but may not have identity. A ring R is said to be selfinjective if R as an R-module is injective. A ring R is said to be pre-selfinjective if every proper homomorphic image of R is self-injective [9]. Study of pre-self-injective rings was initiated by Levy [10], who established a characterization of Noetherian pre-self-injective rings with identity in terms of other well-known types of rings. Recently Klatt and Levy [9] have characterized all pre-self-injective rings with identity. In this paper we are mainly interested in Noetherian rings. For the sake of convenience we shall call a pre-self-injective ring an (I)-ring. A ring R will be said to be a (PMI)-ring if for each proper prime ideal P with P2 ≠ 0, the ring R/P2 is self-injective. Clearly, an (I)-ring is a (PMI)-ring.

Let L/K be a finite Galois extension of local fields which are finite extensions of ℚp, the field of p-adic numbers. Let Gal(L/K) = G, and 𝔒L and ℤp be the rings of integers in L and ℚp, respectively. And let 𝔓L denote the maximal ideal of 𝔒L. We determine, explicitly in terms of specific indecomposable ℤp[G]-modules, the ℤp[G]-module structure of 𝔒L and 𝔓L, for L, a composite of two arithmetically disjoint, ramified cyclic extensions of K, one of which is only weakly ramified in the sense of Erez [6].

All topological spaces under discussion are assumed to be Tychonoff.

For any topological space X let τ(X) denote the topology of X. If X ᑕ Y then a function κ : τ(X) ⟶ τ(Y) is called an extender provided that κ(U) ∩ X = U for all U ∊ τ(X). In addition, X is said to be Kn-embedded in Y (cf. [3]) provided there is an extender κ : τ(X) ⟶ τ(Y) such that

The extender κ is called a Kn-function (cf. [3]).

Eric van Douwen has asked whether there is a space X with a subspace Z which is Ki-embedded but not K0-embedded. The aim of this note is to answer this question.

The chief purpose of this paper is to find all pairs (G, θ) where G is a finite special p-group, and θ is an automorphism of G acting trivially on the Frattini subgroup and irreducibly on the Frattini quotient. This problem arises in the context of describing finite groups having an abelian maximal subgroup. In fact, we solve a more general problem for a wider class of p-groups, which we call special F-groups, where F is a finite field of characteristic p. We point out that if p is odd, then an F-group has exponent p. On the other hand, every special 2-group is also a special GF(2)-group.

The purpose of this note is to investigate the properties of a mapping α, of a Boolean algebra A into itself, which satisfies the functional equation α(p·aq) = ap·aq, where the multiplication is infimum. This is the so-called averaging identity of Kampé de Fériét. Garrett Birkhoff has noted (2) the occurrence of this identity both in the statistical theory of turbulence and in mathematical logic. That this is more than coincidence is shown by the existence of an explicit connection between the notion of quantification in logic and certain non-linear "extremal operators" in function algebras (16). Certain linear operators are associated in a natural way with these extremal operators.

Le problème de la construction d'enveloppes d'holomorphie globales ou d'enveloppes polynomialement convexes n'a été que très peu abordé et peu d'exemples explicites sont connus. Il semble donc qu'il soit intéressant de se limiter d'abord au calcul d'enveloppes d'ensembles homogènes, par l'action d'un groupe agissant par transformations holomorphes. C'est le cas des sphères unité de Cn, des polytores par exemple. En s'inspirant des actions de groupes discutés par Ehrenpreis [2] ou par Hua [3], nous considérons ici l'action des groupes SU(2) et U(2) sur C3 identifié à l'ensemble des matrices symétriques complexes 2 × 2. L'action de SU(2) sur C3 est en fait l'action canonique de SU(2) sur une famille de quadriques dégénérant vers la quadrique singulière lorsque A tends vers 0. L'étude des orbites est équivalente à l'étude des équations de Cauchy Riemann sur SU(2), invariantes à gauche. Le spectre de l'algèbre des fonctions annulant une telle équation sur SU(2), qui sont invariantes par l'application antipodale est alors un domaine d'holomorphie de l'une de ces quadriques qui est d'ailleurs topologiquement non trivial.

This study of r-separating sets was originally motivated by the maximum-flow-minimum-cut theorem of finite networks [1 ; 2]. In working toward a continuous version of the maximum-flow-minimum-cut theorem from the usual discrete version, one is led directly to the notion of r-connectedness of a set as defined below. This notion of r-connectedness also has a very simple intuitive interpretation. Intuitively speaking, a set of points in the plane is r-connected if a person starting at any one point of the set is able to reach any other point of the set by jumping from point to point within the set, but never jumping a distance exceeding r in any one jump.

The Orlicz-Pettis type theorems are essentially statements about the equivalence of strong and weak subseries convergence of series in topological groups. The corresponding problem of the equivalence of subfamily summability for summable families was examined for instance in [2] and (16). We give here another uncountable generalization of the Orlicz-Pettis theorem in the sense that we assume less than subfamily summability and obtain straightforward generalizations of the classical case, which include or imply in a trivial manner—at least for polar topologies on groups—all known theorems “from weak to strong σ-additivity”.

Let Bn denote the unit ball in Cn with boundary S. We will be concerned with spaces of holomorphic functions on Bn and will use much of the notation and terminology found in W. Rudin's book [16]. Thus, if f is holomorphic in Bn and has homogeneous polynomial expansion

the radial derivative of f is given by