This paper studies a class of singular fibrations, called self-crossing boundary fibrations, which play an important role in semitoric and generalized complex geometry. These singular fibrations can be conveniently described using the language of Lie algebroids. We will show how these fibrations arise from (nonfree) torus actions, and how to use them to construct and better understand self-crossing stable generalized complex four-manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these types of fibrations (a so-called nodal trade).

The existence of nonzero fixed points of positive contractions in L1 spaces has received considerable attention in recent years. In 1966, Dean and Sucheston [1] and independently, Neveu [5] showed that a positive contraction has a strictly positive invariant function if and only if for any measurable subset A with positive measure, where 1 is the constant function of value one.

Let C(I) denote the linear space of continuous functions from the compact interval I = [a, b] into n-dimensional real arithmetic space Rn, and let C′(I) be the subspace of continuously differentiable functions on I. A general boundary value problem for a first-order system of n ordinary differential equations on I is given by

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

In a recent important paper of Goldschmidt [3], all finite simple groups were determined in which a non-trivial abelian 2-subgroup controls 2-fusion. Our purpose here is to present a straightforward application of this deep result to the following general question: If p is a prime and G is a finite group of order not divisible by p which admits an automorphism σ of order pn, what conditions on the fixed point subgroup CG(σ) will ensure that G is solvable?

We study the $\text{K}$-homology of the rotation algebras ${{A}_{\theta }}$ using the six-term cyclic sequence for the $\text{K}$-homology of a crossed product by $Z$. In the case that $\theta $ is irrational, we use Pimsner and Voiculescu's work on $\text{AF}$-embeddings of the ${{A}_{\theta }}$ to search for the missing generator of the even $\text{K}$-homology.

Let G be a group and N a trivial G-module. We say an element ξ ∊ H2 (G, N) is of degree ≦ n if a 2-cocycle representative of ξ is a polynomial 2-cocycle of degree ≦ n [1], Let PnH2(G, N) denote the subgroup of H2(G, N) consisting of elements with degree ≦ n. Then we have a nitration

of H2(G, N). We say that the degree of H2(G, N) is ≦ n if PnH2(G, N) = H2(G, N).

Let X be a given complex matrix of order n. If f(z) is analytic at the eigenvalues of X, one may define the primary matrix function f(X) with stem function f(z) by using any of several well-known methods: for instance, canonical forms, power series, or interpolating polynomials [9].

We introduce a wide subclass $\mathcal{F}\left( X,\,\omega \right)$ of quasi-plurisubharmonic functions in a compact Kähler manifold, on which the complex Monge-Ampère operator is well defined and the convergence theorem is valid. We also prove that $\mathcal{F}\left( X,\,\omega \right)$ is a convex cone and includes all quasi-plurisubharmonic functions that are in the Cegrell class.

Let us say that a function denned in the open unit disk D has the Montel property if the set of those points eiθ on the unit circle C where the radial limit exists coincides with the set where the angular limit exists. By a classical theorem of Montel (4), every bounded holomorphic function has this property. Meromorphic functions omitting at least three values and, more generally, the normal functions recently introduced by Lehto and Virtanen (3) also enjoy the Montel property (also see 1).

We are concerned with establishing sufficiency theorems for minima of simple integrals of the parametric type in a class of curves with variable end points and satisfying isoperimetric side conditions. The results which are obtained involve no explicit assumptions of normality. Such results can be derived by transforming our problem to a problem of Bolza and using the latest developments in the theory of that problem. More recently [6] an indirect method of proof has been published. Our object is to present a direct method of proof without transformation of the problem which is based upon a generalization of the classical theory of fields.

If H is a closed subgroup of a locally compact group G, with G/H having finite G-invariant measure, then, as observed by Atle Selberg [8], for any neighborhood U of the identity in G and any element g in G, there is an integer n > 0 such that gn is in U·H·U. A subgroup satisfying this latter condition is said to be an S-sub group, or satisfies property (S). If G is a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: If H is a closed S-subgroup of G, then G/H is compact. Property (S) has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).

Fix an elliptic curve $E/\mathbf{Q}$and assume the Riemann Hypothesis for the $L$-function $L({{E}_{D}},\,s)$ for every quadratic twist ${{E}_{D}}$ of $E$ by $D\,\in \,\mathbf{Z}$. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of ${{E}_{D}}$. We derive from this an upper bound for the density of low-lying zeros of $L({{E}_{D}},\,s)$ that is compatible with the randommatrixmodels of Katz and Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbf{R}$, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of ${{E}_{D}}$ are less than $f(D)$ for almost all $D$.

Let VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = ﹛α ; α < μ﹜.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l∞(X) such that π o = idl∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l∞(X). Similar results are also obtained for other quotient spaces.

Using the methods developed in (2 and 3), in this paper we study some properties of the configuration of generators and points of a cone in an w-dimensional finite projective space. The configuration of secants and external points of a quadric in a finite plane of even characteristic is also studied. I t is shown that these configurations lead to several series of partially balanced incomplete block (PBIB) designs. PBIB designs are defined in Bose and Shimamoto (1). A PBIB design with m associate classes is an arrangement of v treatments in b blocks such that.

In this paper we consider the problem of determining when an algebra of formal power series over a commutative ring R is the homomorphic image of a reduced incidence algebra P(R, ∽). The question of when two such algebras are isomorphic is answered in Section 8 of [1]. A slight generalization of their notion of full binomial type is introduced here.

Section 1 contains background material together writh a summary of the results of [1]. In Section 2 we present the desired characterization, and to conclude an application appears in Section 3. In Section 3 the tools of Section 2 are used to derive an equation of R. W. Robinson and R. P. Stanley which counts labelled, acyclic digraphs.

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be written

in a notation explained below.

Let $p$ be an odd prime number, and let $F$ be a field of characteristic not $p$ and not containing the group ${{\mu }_{p}}$ of $p$-th roots of unity. We consider cyclic $p$-algebras over $F$ by descent from $L\,=\,F\left( {{\mu }_{p}} \right)$. We generalize a theorem of Albert by showing that if ${{\mu }_{{{p}^{n}}}}\,\subseteq \,L$, then a division algebra $D$ of degree ${{p}^{n}}$ over $F$ is a cyclic algebra if and only if there is $d\,\in \,D$ with ${{d}^{{{P}^{n}}}}\,\in \,F\,-\,{{F}^{P}}$. Let $F(p)$ be the maximal $p$-extension of $F$. We show that $F(p)$ has a noncyclic algebra of degree $p$ if and only if a certain eigencomponent of the $p$-torsion of $\text{Br(F(p)(}{{\mu }_{p}}\text{))}$ is nontrivial. To get a better understanding of $F(p)$, we consider the valuations on $F(p)$ with residue characteristic not $p$, and determine what residue fields and value groups can occur. Our results support the conjecture that the $p$ torsion in $\text{Br}(F(p))$ is always trivial.

We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional case and linearly compact in general. As an example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show that the closure of $\left[ L,\,L \right]$ in $\mathcal{L}(G)$ is algebraic in $\mathcal{L}(G)$.

We also discuss the Hopf algebra of representative functions $H(L)$ of a residually finite dimensional Lie algebra $L$. As an example, we show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$ is connected, then the canonical Hopf algebra morphism from $K\left[ G \right]$ into $H(L)$ is injective if and only if $L$ is algebraically dense in $\mathcal{L}(G)$.

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.