Wild double tangent ball embeddings of spheres in En

It has been known for years that a 2-sphere Σ in E³ must be flatly embedded if it has double tangent balls on opposite sides of Σ at each of its points. However, when the double tangent balls are not required to be on opposite sides of Σ, pathological embeddings exist. This paper details the allowable wild embeddings of spheres having these indiscrete double tangent balls and discusses the higher dimensional analogues.     

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.     

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.     

Retraction spaces and the homotopy metric

Let X be a finite-dimensional compactum. Let $\text{R}$(X) and $\text{N}$(X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2^X_h be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let R^X_h be the subspace of 2^X_h consisting of the ANR's in X that are retracts of X.We show that $\text{N}$(S^m) is simply-connected for m > 1. We show that if X is an ANR and A₀?R^X_h, then lim_i→∞A_i=A₀ in 2^X_h if and only if for every retraction r₀ of X onto A₀ there are, for almost all i, retractions r_i of X onto A_i such that lim_i→∞r_i=r_o in $\text{R}$(X). We show that if X is an ANR, then the local connectedness of $\text{R}$(X) implies that of R^X_h. We prove that $\text{R}$(M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR.     

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 ∞.     

Generalized level crossings and tangencies of a random field with smooth sample functions

Tangencies and level crossings of a random field X:$\text{R}$^m₊×Ω→$\text{R}$ⁿ (which is not necessarily Gaussian) are studied under the assumption that almost every sample path is continuously differentiable. If n=m and if the random field has uniformly bounded sample derivatives and uniformly bounded densities for the distributions of the X_l, then for a compact K?$\text{R}$^m₊ and any fixed level, the restriction to K of almost every sample path has no tangencies to the level and at most finitely many crossings. The case of n≠m is also examined. Some generic properties, which hold for a residual set of random fields, are analyzed. Proofs involve the concepts of regularity and transversality from differential topology.     

The Euclidean distance construction of order homomorphisms

The Euclidean distance technique of Arrow and Hahn is used to construct an upper semicontinuous order homomorphism (partial utility function) from (X, ≻) to ($\text{R}$, >), where X is a closed, convex subset of $\text{R}$^N and ≻ is a continuous strict partial order on X. It is also shown that the order homomorphism is upper semicontinuous as a function on $\text{X×}\text{P}\text{(X)}$, where$\text{P}\text{(X)}$ is the set of continuous strict partial orders on X, taken with the topology of closed convergence.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,II

Let L = ∑_{j = 1}^mX_j² be sum of squares of vector fields in $\text{R}$ⁿ satisfying a Hörmander condition of order 2: span{X_j, [X_i, X_j]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X₁,…, X_m are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator $\text{?}{}^2\text{?x}{}^2\text{+ x}{}^2\text{?}{}^2\text{?t}{}^2$ in $\text{R}$². (b) The real part of the Kohn Laplacian on the Heisenberg group $\text{∑}{}_{\mathrm{j\; ?\; 1}}{}^{\mathrm{n}}\text{(}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2\text{+ (}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2$ in $\text{R}$^{2n + 1}. In contrast to non-characteristic points, C^∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.     

Ultimate boundedness results for a certain system of third-order non-linear differential equations

We present in this paper ultimate boundedness results for a third-order system of non-linear differential equations of the form $\text{·X + A}\text{X}\text{?}\text{+ B/.X + H(X) = P(t, X, /.X,}\text{X}\text{?}\text{)}$. where A, B are constant symmetric n × n matrices and X, H(X), P(t, X, X, X) are real n-vectors with H:R_n→R_n and P: RXR_nXR_nXR_nXR_n → XR_ncontinuous in their respective arguments. Our results give an n-dimensional analogue of an earlier result of Ezeilo in [1] and extend other earlier results for the case in which we do not necessarily require that H(X) be differentiable.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

Approximation by rational modules in Sobolev and Lipschitz norms

Let X?C be compact, 0>n∈Z, and g a continuous function on X. Let R(n,g,X) be the rational module consisting of the functions on X of the type r₀ + r₁g + ··· + r_ngⁿ, where r_j is a rational function with poles off X, 0 ? j ? n. It is shown that if X is nowhere dense, g is sufficiently smooth, and $\text{}\text{?}\text{t6g(z) ≠ 0, z ∈ X}$, then the restriction to X of each function in C∈(C) is approximable in the Lip(n ? 1, X)-norm, n ? 2, by functions in R(n, g, X). Also dealt with are approximation problems in Sobolev norms by more general types of rational modules.     

Geometric Lipschitz spaces and applications to complex function theory and nilpotent groups

It is known that a function on $\text{R}$ⁿ which can be well approximated by polynomials, in the mean over Euclidean balls, is Lipschitz smooth in the usual sense. In this paper an analogous theorem is proved in which $\text{R}$ⁿ is replaced by a set X, the averages over balls are replaced by a family of sublinear operators satisfying certain axioms, and the polynomials are replaced by a class of functions having certain regularity properties with respect to the averaging operators. Applications are given to function theory on domains in $\text{C}$ⁿ, to nilpotent Lie groups, and to the classical Euclidean case. The first application provides a characterization of the duals of Hardy spaces on the ball in $\text{C}$ⁿ.     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

A transfer spectral sequence for fixed point free involutions with an application to stunted real projective spaces

We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with $\text{E}{}^1{}_1\text{? H}{}^1\text{(X;}\text{Z}{}_2\text{)}$ and $\text{E}{}^1\text{∞ = 0}$. As an application we prove that for any n > 0 there is a natural number k(n) such that if n > k(n) and X is a homotopy $\text{R}\text{P}{}^{\mathrm{n+k}}\text{R}\text{P}{}^{\mathrm{n}}$, then X will not admit a fixed point free involution.     

Minimum number of edges in graphs that are both P2- and Pi-connected

A simple graph with n vertices is called P_i-connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P₂- and P_i-connected are obtained. Namely if $\text{i?}\text{1}\text{2}\text{(n+1)}$, then |E(G)|?((4i?5)/(2i?2))(n?1), and if $\text{i >}\text{1}\text{2}\text{(n+ 1)}$, then |E(G)|?2(n?1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

The behavior on fundamental group of a free pro-homotopy equivalence

We study the question: given a morphism $\text{?{(X}{}_{\mathrm{n}}\text{, x}{}_{\mathrm{n}}\text{)}→{(Y}{}_{\mathrm{n}}\text{, y}{}_{\mathrm{<mtext>n</mtext>}}\text{)}}$ in the category pro-(Poi nted. Homotopy) where the domain and range are inverse sequences of well-pointed CW complexes, and given that ? induces an isomorphism {X_n}→{Y_n} in pro-(Homotopy), what additional hypotheses force ? to be an isomorphism in pro-(Pointed Homotopy)? Conjecture. If the dimensions of the Y_n's are bounded, then ? is an isomorphism in pro-(Pointed Homotopy). We first prove the special case of this conjecture in which dim Y_n?d<∞ for all n, and $\text{lim}\text{{H}{}_{\mathrm{d}}\text{Y}{}_{\mathrm{n}}\text{}≠0}$, Y_n being the universal cover of Y_n. Then we deal with the general case: we show that there are certain elements of each π₁Y_n with the properties: (i) these elements commute if and only if ? is an isomorphism in pro-(Pointed Homotopy); (ii) if dim Y_n?d<∞ for all n, then powers of these elements commute. While (i) and (ii) are not incompatible, this result puts severe restrictions on the nature of any possible counter-example to the conjecture.Our method also gives pro-homotopy analogues of the well-known fact that if a K(π, 1) is N-dimensional, then π is torsion-free and contains no abelian subgroup of rank>N. The latter theorems apply to inverse sequences {Y_n} of CW complexes where dim Y_n is finite but not necessarily bounded, hence in particular to infinite-dimensional shape-aspherical compacta.