Elliptic units of global function fields were first studied by D. Hayes in the case that deg ∞ is assumed to be 1, and he obtained some class number formulas using elliptic units. We generalize Hayes’ results to the case that deg ∞ is arbitrary.

A space X is a D-space if, for every neighborhood assignment f there is a closed discrete set D such that f(D) = X. In this paper we give some necessary conditions and some sufficient conditions for a resolution of a topological space to be a D-space. In particular, if a space X is resolved at each x ∊ X into a D-space Y x by continuous mappings fx : X − {x} → Yx , then the resolution is a D-space if and only if {x} × Bd(Yx ) is a D-space.

This paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

For the perturbed Hill’s equation on the line,

we study the behaviour of the matrix m-function at the spectral gap endpoints. In particular, we extend the result of Hinton, Klaus and Shaw that En , a gap endpoint, is a half-bound state (HBS) if and only if (E − En)½m(E) approaches a nonzero constant as E → En , to the present case.

If A and B are disjoint ideals on ω, there is a tower preserving σ-centered forcing which introduces a subset of ω which meets every infinite member of A in an infinite set and is almost disjoint fromeverymember of B. We can then produce a model in which all compact separable radial spaces are Fréchet, thus answering a question of P. Nyikos. The question of the existence of compact ccc radial spaces which are not Fréchet was first asked by Chertanov (see [Arh78]).

In dealing with the spectral synthesis property for a plane curve with nonzero curvature, a key step is to have a uniform L ∞ estimate for some smoothing operators related to the curve. In this paper, we will show that the same L ∞ estimate holds true for a plane curve that may have zero curvature.

Suppose A is a unital C*-algebra and m:A → B(X) is unital bounded algebra homomorphism where B(X) is the algebra of all operators on a Banach space X. When X is a Hilbert space, a problem of Kadison [9] asks whether m is similar to a *-homomorphism. Haagerup [5] has shown that the answer is positive when m(A) has a cyclic vector or whenever m is completely bounded. We use this to show m(A) is reflexive (Alg Lat m(A) = m(A)−sot ) whenever X is a Hilbert space. Our main result is that whenever A is a separable GCR C*-algebra and X is a reflexive Banach space, then m(A) is reflexive.

A new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case.

Let 𝔾 p,q(𝔽) be the Grassmann space of all q-dimensional 𝔽-vector subspaces of 𝔽 p, where 𝔽 stands for ℝ, ℂ or ℍ (the quaternions). Here 𝔾 p,q(𝔽) is regarded as a real algebraic variety. The paper investigates which C∞ maps from a nonsingular real algebraic variety X into 𝔾 p,q(𝔽) can be approximated, in the C ∞ compact-open topology, by real algebraic morphisms.

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

The purpose of this paper is to present a short, self-contained proof of Euler’s relation. The ingredients of this proof are (i) the principle of inclusion and exclusion of combinatorics and (ii) the Euler characteristic; a development of the Euler characteristic is included.

We show that the multiplier space , where X is BMOA, VMOA, B, B0 or disk algebra A. We give the multipliers from , we also give the multipliers from , C 0, BMOA, and Hp .

This paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af -matrix is also discussed.