Codimension one spheres in Rn with double tangent balls

In contrast to the situation in $\text{R}$³, where a 2-sphere with double tangent balls at each point must be tamely embedded in $\text{R}$³, there exist wild (n?1)-spheres in $\text{R}$ⁿ for n>3 with this same geometric property. However, if the sphere Σ is tame moduio a subset X that lies in a polyhedron P that is tame in Σ, the dimension of P is less than n?2, n>4, and Σ has double tangent balls over X, then Σ must be tame in $\text{R}$ⁿ. Also if the tangent balls extend over P and are pairwise congruent, the dimensional restriction on P can be dropped. Examples are given to support the necessity of the hypotheses of the included theorems.     

A maximum principle in Rn + 1

In this paper we establish maximum principles of the Cauchy problem for hyperbolic equations in $\text{R}$³ and $\text{R}$^{n + 1}(n ? 2). Our maximum principles generalize the results of Weinberger [5], and Sather [3, 4] for a class of equations such that the coefficients can be allowed to depend upon t, as well, in {x₁, x₂, t}-space and {x₁, x₂,…, x_n, t}-space. Throughout this paper, the influence of the work of Douglis [1] is apparent. See [2].     

k-dimensional skeletoids in Rn and the Hilbert cube

In their paper on pseudo-boundaries and pseudo-interiors R. Geoghegan and R.R. Summerhill construct k-dimensional pseudo-boundaries in $\text{R}$ⁿ, where they used West's notion of a pseudo-boundary, rather than Toruńczyk's. In this paper we construct pseudo-boundaries in the sense of Toruńczyk (skeletoids) in $\text{R}$ⁿ and use this result to find k-dimensional skeletoids in the Hilbert cube.     

Riemann-Lebesgue subsets of Rn and representations which vanish at infinity

The main theorem of this paper is that if χ is a character of a connected closed normal subgroup of a connected Lie group, then every matrix element of the induced representation U^χ vanishes at infinity modulo the kernel of that representation. As a consequence, it is shown that every faithful irreducible unitary representation of a connected motion group vanishes at infinity. In the course of the development a generalization of the classical Riemann-Lebesgue lemma is proved. Suppose $\text{M}$ is an analytic submanifold of $\text{R}$ⁿ which is not contained in any proper hyperplane. Then the Fourier transform of any measure, which is concentrated on $\text{M}$ and which is absolutely continuous with respect to the “Lebesgue” measure on $\text{M}$, vanishes at infinity.     

On the spectral synthesis problem for hypersurfaces of RN

Let F¹(Rⁿ) denote the Fourier algebra on $\text{R}$ⁿ, and $\text{D}$($\text{R}$ⁿ) the space of test functions on $\text{R}$ⁿ. A closed subset E of $\text{R}$ⁿ is said to be of spectral synthesis if the only closed ideal J in F¹(Rⁿ) which has E as its hull

$\text{h(J)={x ? R}{}^{\mathrm{n}}\text{:f(x)=0}\text{for all}\text{f ? J}}$

is the ideal

$\text{k(E)={f?F}{}^1\text{(R}{}^{\mathrm{n}}\text{):f(E)=0}}$

. We consider sufficiently regular compact subsets of smooth submanifolds of $\text{R}$ⁿ with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra $\text{(k(E)∩D(R}{}^{\mathrm{n}}\text{))}{}^{\mathrm{?}}\text{j(E)}$, where j(E) denotes the smallest closed ideal in F¹(Rⁿ) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k(E)∩D(Rⁿ) is dense in k(E). Together this solves the synthesis problem for such sets.     

A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.     

Spaces N ∪ R and their dimensions

About spaces N∪$\text{R}$ (see [2, Exercise 5I]), the following are proved: (1) dim $\text{N∪}\text{R}\text{= dim β(N∪}\text{R}\text{)?N∪}\text{R}\text{,(2)if|β(N∪}\text{R}\text{)?N∪}\text{R}\text{|<2}{}^{\mathrm{?}}{}^{\mathrm{o}}$, then no real-valued continuous fu ction on N∪$\text{R}$ is onto (and hence, dim $\text{N∪}\text{R}\text{=0}$), (3) any compact metric space without isolated points is homeomorphic to some $\text{β(N∪}\text{R}\text{)?N∪}\text{R}$ and (4)there are spaces X,X₁ and X₂ of the form N∪$\text{R}$ such that X=X₁∪X₂,X₂andX₂ are zero sets of X, and dim X=n, dimX₁=dimX₂=0, where n=1,2,… or ∞.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

Properties of Hida processes on R2. 2. Prediction and interpolation problems for processes on R2

The properties of N-Hida processes Part 1 (B. Prum, 1984, J. Multivar. Anal.15, 336–360) are studied when the indices set is $\text{R}$². First, the past of a point (s, t) of $\text{R}$² is extended to $\text{G}$_st = σ{γ_uv, u ≤ s or v ≤ t}. The dimension of the linear space generated by the conditional expectations of an N-Hida process γ_z when z goes over a p × q lattice is bounded by N(p + q ? 1). The same problem is then considered when the expectations are taken conditionally to the field generated by the process outside of a rectangle, and the bound of the dimension of the linear space generated on a lattice is also given. Special attention is devoted to the case when γ_z is a combination of strong martingales.     

On R∞ and Q∞-manifolds

We give a characterization of manifolds modeled on $\text{R}{}^{\mathrm{\infty }}\text{= dir lim}$ or $\text{R}{}^{\mathrm{n}}\text{Q}{}^{\mathrm{\infty }}\text{=dir lim Q}{}^{\mathrm{n}}$, where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.     

Large time behaviour of the solutions of a semilinear parabolic equation in RN

The asymptotic behaviour as t tends to +∞ of the solution of $\text{(?u}\text{?t)}\text{? Δu + u¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{= 0}$ in $\text{R}$^N × $\text{R}$⁺, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of $\text{(N + 2)}\text{N ? p}$ and also on the rate of decay of the admissible initial data u(0, x) as $\text{¦x¦}$ tends to +∞.     

An analogue of the thom isomorphism for crossed products of a C1 algebra by an action of R

In this paper we show the existence and uniqueness of a natural isomorphism ø^j_α of K_j(A) with K_j+1(A ?_α$\text{R}$), j ? $\text{Z}$/2 where (A, $\text{R}$, α) is a $\text{C1}$ dynamical $\text{R}$-system, K is the functor of topological K theory and A ?_α$\text{R}$ is the crossed product of A by the action of $\text{R}$. The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace $\text{}\text{?}$gt, one has $\text{}\text{?}\text{gtø}{}^1{}_{\mathrm{\alpha }}\text{[u] = (}\text{1}\text{2}\text{iπ) τ(δ(u)u1)}$ for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C(S³) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple $\text{C1}$ algebra with no nontrivial idempotent.     

On the ∂-Neumann problem for generalized functions

Using the new theory of generalized functions developed by one of the authors the $\text{?}$ equation in $\text{C}$ⁿ is studied. In particular it is proven that if G is any generalized function on $\text{C}$ (in the above sense) then there is a generalized function S on $\text{C}$ such that $\text{?S}\text{?}\text{z}\text{?}\text{= G}$. Several other results are proven valid in polydiscs of $\text{C}$ⁿ, for which differential forms whose coefficients are generalized functions are introduced.     

Rickart C1 algebras,II

The theory of inner-outer factorization in the Hardy spaces H^p in the unit disc $\text{D}$ is well known and has many applications. It does not carry over to the spaces H^p on the polydisc $\text{D}$ⁿ or the ball $\text{B}$ⁿ when n > 1. However, for Lumer's Hardy spaces (LH)^p on any simply connected complex analytic manifold, we introduce the notions of internal and external functions and prove that every f? (LH)^p has a factorization f = I_ε × E_ε, where I_ε is internal and E_ε is external, and E_ε? (LH)^p?ε, for any ε > 0. The factorization is not unique and an example of Rudin shows that the ε is needed, at least when $\text{p =}\text{2}\text{m}$, where m is an integer.     

On some properties of convolution operators in K′1 and S′

Let $\text{H}$′ be either the space $\text{K}$′₁ of distributions of exponential growth or the space $\text{S}$′ of tempered distributions, and let $\text{O}$′_C($\text{H}$′:$\text{H}$′) be the space of convolution operators in $\text{H}$′. In each case $\text{H}$′ is the dual of a space $\text{H}$ of C^∞-functions which are in $\text{O}$′_C($\text{H}$′:$\text{H}$′). We establish necessary and sufficient conditions on the Fourier transform $\text{S}\text{?}$ of ? of S ? $\text{O}$′_C($\text{H}$′:$\text{H}$′) in order that every distribution u? $\text{O}$′_C($\text{H}$′:$\text{H}$′) with S1u?$\text{H}$ be in $\text{H}$. If $\text{H}$′ = $\text{K}$′₁, the condition is equivalent to S×H′¹=H′¹.     

Integrals and derivatives of regularly varying functions in Rd and domains of attraction of stable distributions II.

A theorem on regularly varying functions in $\text{R}$² is proved and applied to domains of attraction of stable laws with index 1 ? α ? 2. We also present a theory of Π-variation in $\text{R}$². Unlike the situation in $\text{R}$¹ the latter is not connected with domain of attraction theory. The situation in $\text{R}$^d (d > 1) is more complicated but not essentially different; for simplicity we limit ourselves to $\text{R}$². This article complements de Haan and Resnick (1979) where the situation for 0 < α < 1 was considered.     

Equivariant embeddings of principal Z-bundles into complex vector bundles

Let π: E→X be a principal $\text{Z}$_n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and $\text{zε}\text{Z}{}_{\mathrm{n}}\text{?S}{}^1\text{?}\text{Z}$. We compute the primary obstruction $\text{cεH}{}^{\mathrm{2m}}\text{(X;}\text{Z}\text{)}$ to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle ε^m. We use these criteria for a discussion of the classification of the equivalence classes of principal $\text{Z}$-bundles that admit equivariant embeddings into ε^m. Finally, we offer an example of a principal $\text{Z}$-bundle that admit an ordinary but not an equivariant embedding into ε¹.     

ALPS: Matrices with nonpositive off-diagonal entries

The purpose of this paper is to characterize and interrelate various degrees of stability and semipositivity for real square matrices having nonpositive off-diagonal entries. The major classes considered are the sets of diagonally stable, stable, and semipositive matrices, denoted respectively by $\text{A}$, $\text{L}$, and $\text{S}$. The conditions defining these classes are weakened, and the resulting classes are examined. Their relationship to the classes of real matrices $\text{P}$ and $\text{P}$₀, whose off-diagonal entries are nonpositive and whose principal minors are respectively all positive and all nonnegative, is also included.     

Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p [X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_p[X]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.     

Embedding EnG in Euclidean space

It is shown that if G is an arbitrary upper semicontinuous decomposition of Eⁿ for which π(N_G embeds in S^m for some m?3, then the decomposition space $\text{E}{}^{\mathrm{n}}\text{G}$ embeds as a closed subset of E^n+m+1. The proof consists of constructing a cell-like upper semicontinuous decomposition G? of E^n+m+1 which intersects Eⁿ to yield precisely G and using Edwards' Cell-Like Approximation Theorem to show that G? is shrinkable. As an immediate corollary, $\text{E}{}^{\mathrm{n}}\text{G}$ embeds in E^n+2k+2 whenever G is an arbitrary k-dimensional upper semicontinuous decomposition of Eⁿ. This is an improvement of (n?1)-dimensions over the corresponding dimension theoretic result and examples due to Daverman show that this result is sharp in case n is odd and off by no more than one dimension in case n is even.