We verify a generalization of (3.3) from [Lê73] proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using [Zariski68, Lipman83, Oh93, Gau88] for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

This paper studies the $K$-theoretic invariants of the crossed product ${{C}^{*}}$-algebras associated with an important family of homeomorphisms of the tori ${{\mathbb{T}}^{n}}$ called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products whose corresponding $n\,\times \,n$ integer matrices are unipotent of maximal degree always have the same rank ${{a}_{n}}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\mathfrak{s}\mathfrak{l}\left( 2,\,\mathbb{C} \right)$, we show that, remarkably, ${{a}_{n}}$ has a combinatorial significance. For example, every ${{a}_{2n+1}}$ is just the number of ways that 0 can be represented as a sum of integers between –$n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence $\{{{a}_{n}}\}$ is given. Finally, we describe the order structure of the ${{K}_{0}}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

The following paper defines a new type of ordering of higher level on a commutative ring. This definition allows the set of all orderings of level n to be given a topology which we show is consistent with the topology of the real spectrum.

We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.

The Lefschetz theorem on hyperplane sections, as proved by Andreotti and Frankel (1), depends upon the following result.

THEOREM. If M is a non-singular affine algebraic variety of real dimension 2k of complex n-space, then

This theorem, which is interesting in itself, has been strengthened by Milnor (7), who showed that M has the homotopy type of a k-dimensional CW-complex.

In this paper we generalize the above theorem in two directions. First, we replace complex n-space by some other complete simply connected Riemannian manifold which either has non-positive curvature or is a compact symmetric space. Secondly, we allow M and to be quasi-Kâhlerian (see below) instead of Kählerian.

We first introduce some notation. Let M and be C∞ Riemannian manifolds with M isometrically immersed in . Denote by 〈, 〉 the metric tensor of either M or .

In this paper weak Perron integrals are characterized as n-dimensional interval functions F which are additive, differentiable almost everywhere in the weak sense and which satisfy a new continuity condition concerning the singular set. Before, only one-dimensional Perron integrals were characterized by the theorem of Hake- Alexendrov-Looman, and analogous results for strong Perron integrals (which are best analyzed, but more restrictive) are not available in higher dimensions yet. In order to formulate our continuity condition we introduce an outer measure μ by means of a new weak variation of F which is required to vanish on all null sets. The same condition is also necessary and sufficient for the integral of the weak derivative to yield the original interval function. This “fundamental theorem” is split into two fundamental inequalities of very general nature which contain additional singular terms involving our variation. These inequalities are very useful also for Lebesgue integrals.

In (5),I considered two-term conditions in π-exact couples, of which the exact couple of Federer (7) is an example. Let M(X, Y)be the space of all maps from X to Y with the compact-open topology. Our aim in this paper is to construct a π-exact couple , where Xis a finite-dimensional (in the sense of Lebesgue) metric space and , a certain (rather large) class of spaces. Specifically, is the class of all topological spaces Xwhich possess the following property (P).

(P) Let Y be a (possibly infinite) simplicial complex. There exists x0 ∈ X and y0 ∊ Y such that [X, x0]≃ [Y, y0].

In § 5 it will be seen that contains all CW complexes and all metric absolute neighbourhood retracts (ANR)s.

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by

(1)

where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying

(2)

where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by

(3)

(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)

We shall call a linear function on the elements of a ring R skew-symmetric if it vanishes whenever at least two of the variables are equal. Here we shall study rings R of characteristic not 2 which satisfy the following two identities:

1

(2)is skew-symmetric.

Both of these identities hold in alternative rings. The fact that F(w, x, y, z) is skew-symmetric in alternative rings is an important tool in the study of such rings. It is also obvious that both identities hold in commutative rings. But unlike other recent generalizations of commutative and alternative rings it turns out that there exist simple, finite dimensional algebras of degree two which are neither alternative nor commutative and satisfy (1) and (2).

Heilbronn [6] proved that for any ϵ > 0 there exists C(ϵ) such that for any real θ and N ≧ 1 there is an integer x satisfying

where ||α|| denotes the difference between α and the nearest integer, taken positively. Danicic [2] obtained an analogous result for the fractional parts of θxk and in 1967 Davenport [4] generalized Heilbronn's result to polynomials of degree with no constant term. The last condition is essential, for if there is a constant term then no analogous result can hold (see Koksma [7, Kap. 6 SatzlO]).

Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold $M$. Let $u:\,D\,\to \,M$ be a minimal immersion from a disk $D$ with $u(\partial D)\,\subset \,L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n\,+\,\mu (E,\,F)$, where $\mu (E,\,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p\,\in \,\partial D;$; if $M=\mathbb{C}{{P}^{n}}$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n\,+\,\mu (E,\,F)$ if $u$ is unstable.

In this paper we consider two closely related problems concerning a complete locally m-convex (LMC) algebra A with identity. Let a be a fixed element of A, and let P(a) be the smallest closed subalgebra containing a and 1. If B is any subalgebra containing a and 1, we let σ(a; B) denote the spectrum of a as an element of B. (I) Describe the set σ(a; P(a)) in terms of σ(a; A). (II) Give necessary and sufficient conditions in order that σ (a; B) = σ(a; A) for every closed subalgebra B of A which contains a and 1.

The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*.