The periodicity in stable equivariant surgery

The classical surgery theory (see [5] and [23]) computes the structure set S_m (M, rel ?) of manifolds homotopy equivalent to M relative to the boundary. Siebenmann showed that in topological category, the structure set is 4-periodic: S_m(M, rel ?) ? S_m+4(M × D⁴, rel ?) up to a copy of ?; see [12]. Cappell and Weinberger gave a geometric interpretation of this periodicity in [8]. By using Weinberger's stratified surgery theory (see [24]), we extend this to an equivariant periodicity result for topological manifolds with homotopically stratified actions by compact Lie groups, with D⁴ replaced by the unit ball of certain group representations. In particular, if G is an odd order group acting on a topological manifold M, then the equivariant stable structure sets satisfy S (M, rel ?) ? S(M × D(?⁴ ? ?G), rel ?) up to copies of ?. © 1993 John Wiley & Sons, Inc.     

A generalization of Plantholt's theorem

Let K denote the complete graph K_2n+1 with each edge replicated r times and let χ′(G) denote the chromatic index of a multigraph G. A multigraph G is critical if χ′(G) > χ′(G/e) for each edge e of G. Let S be a set of sn – 1 edges of K. We show that, for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r – s for all e ∈ E(G/S). Plantholt [M. Plantholt, The chromatic index of graphs with a spanning star. J. Graph Theory 5 (1981) 5–13] proved this result in the case when r = 1.     

Z-cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5

When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all v ∈ T = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all v ∈ T there exists a Z-cyclic TWh(vⁿ) for all n ≥ 1, and (2) if v_i ∈ T, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?_i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.     

Subelliptic Estimates for a Class of Degenerate Elliptic Integro-Differential Operators

In this paper we prove subelliptic estimates for operators of the form Δ_x + λ² (x)S in ?^N = ? × ?, where the operator S is an elliptic integro - differential operator in ?^N and λ is a nonnegative Lipschitz continuous function.     

Ulm Classification of Analytic Equivalence Relations in Generic Universes

We prove that if every real belongs to a set generic extension of L, then every Σ equivalence relation E on reals either admits a Δ₁ reduction to the equality on the set 2^{< ω1} of all countable binary sequences, or the Vitali equivalence E₀ continuously embeds in E. The proofs are based on a topology generated by OD sets.     

The determining number of a Cartesian product

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G=G□?□G is the prime factor decomposition of a connected graph then Det(G)=max{Det(G)}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q_n)=?log₂n?+1 which matches the lower bound, and that Det(K)=?log₃(2n+1)?+1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(Hⁿ)=Θ(logn). © 2009 Wiley Periodicals, Inc. J Graph Theory     

Compact homomorphisms of regular banach algebras

Let A be a complex commutative Banach algebra and let M_A be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points ?₁, ?₂ ∈ M_A, there is an element a ∈ A for which , and ‖a‖ ? C, where is the Gelfand transform of a ∈ A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators II

This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type Ψ with χ χ ? ? and 0 ≤ γ ≤ 1 in the weighted function spaces B (?ⁿ, w(x)) and F (?ⁿ, w(x)) treated in [17]. Furthermore, we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b₂(x) b(x, D) b₁(x), where b₁(x) and b₂(x) are appropriate functions and b(x, D) ? Ψ. Finally, on the basis of the Birman-Schwinger principle, we deal with the “negative spectrum” (bound states) of related symmetric operators in L₂.     

Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form

We consider a second‐order differential operator A( x )=?∑?_ia_ij( x )?_j+ ∑?_j(b_j( x )·)+c( x ) on ?^d, on a bounded domain D with Dirichlet boundary conditions on ?D, under mild assumptions on the coefficients of the diffusion tensor a_ij. The object is to construct monotone numerical schemes to approximate the solution of the problem A( x )u( x )=µ( x ), x ∈D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness introducing function spaces needed to prove convergence results. Then, we define non‐standard stencils on grid‐knots that lead to extended discretization schemes by matrices possessing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W‐convergence in detail, and mention convergence in C and L₁ spaces. We conclude by a numerical example illustrating the schemes and convergence results. Copyright © 2008 John Wiley & Sons, Ltd.     

Subgraphs smaller than the girth

For graphs A, B, let () denote the number of subsets of nodes of A for which the induced subgraph is B. If G and H both have girth > k, and if () = () for every k-node tree T, then for every k-node forest F, () = (). Say the spread of a tree is the number of nodes in a longest path. If G is regular of degree d, on n nodes, with girth > k, and if F is a forest of total spread ≤k, then the value of () depends only on n and d.     

On determinantal representation for the generalized inverse A and its applications

In this paper, first we establish a determinantal representation for the group inverse A_g of a square matrix A. Based on this, a determinantal representation for the generalized inverse A is presented. As an application, we give a determinantal formula for the unique solution of the general restricted linear system: Ax=b(x ∈ T, b ∈ AT and dim(AT)=dim(T)), which reduces to the common Cramer rule if A is non‐singular. These results extend our earlier work. Copyright © 2006 John Wiley & Sons, Ltd.     

On Σ1-definable Functions Provably Total in I ∏

We prove the following theorem: Let φ(x) be a formula in the language of the theory PA^? of discretely ordered commutative rings with unit of the form ?yφ′(x,y) with φ′ and let ∈ Δ₀ and let fφ: ? → ? such that f_φ(x) = y iff φ′(x,y) & (?z < y) φ′(x,z). If I ∏ ∈(?x ≥ 0), φ then there exists a natural number K such that I ∏ ? ?y?x(x > y ? ?φ(x) < x^K). Here I ∏₁^? denotes the theory PA^? plus the scheme of induction for formulas φ(x) of the form ?yφ′(x,y) (with φ′) with φ′ ∈ Δ₀.     

A Reuniformization for Contractive Mappings in Uniform Spaces

Let X be a complete uniform HAUSDORFF space with a uniformity generated by a saturated family of pseudometrics ?? = {?_α(x, y): α ? A} and let T: X → X be a continuous mapping. The paper contains necessary and sufficient conditions for the existence of a new family of pseudometrics ??*={?*(x, y): α*?A*} generated the same topology such that T is contractive with respect to ??*.     

ANALYTIC COMPLETENESS THEOREM FOR ABSOLUTELY CONTINUOUS BIPROBABILITY MODELS

Hoover [2] proved a completeness theorem for the logic L(∫)??. The aim of this paper is to prove a similar completeness theorem with respect to product measurable biprobability models for a logic L(∫₁, ∫₂) with two integral operators. We prove: If T is a ∑₁ definable theory on ?? (a countable admissible set and ω ∈) and consistent with the axioms of L(∫₁, ∫₂), then there is an analytic absolutely continuous biprobability model in which every sentence in T is satified.     

Designs from projective planes and PBD bases

The essential theme of this article is the exploitation of known configurations in projective planes in order to construct pairwise balanced designs. Among the results shown are (q² + 1)/2 ∈ B((q + 1)/2, 2) for all odd prime powers q; the resolution of 31 of the Mullin and Stinson's open cases in the spectrum of B(P₇, 1), where P₇ is odd prime powers ≥ 7; some constructions showing the inessential nature of values in E, where E are PBD generating sets containing values equal to 1 mod(a) for 5 ≤ a ≤ 7; some constructions with block sizes being prime powers ≥ 8, yielding 7 MOLS of order 158 and 9 MOLS of order 254; and some designs on 28, 38, 42, 56, 63, 68, 72, and 156 points with interesting block sizes. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 341–374, 1999     

The red‐shift effect and radiation decay on black hole spacetimes

We consider solutions to the linear wave equation □_g? = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface Σ, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of ? may contain the bifurcate event horizon.) It is shown that the energy flux F(??) of the solution (as measured by a strictly timelike T? that asymptotically matches the static Killing field) through arbitrary achronal subsets ?? of the black hole exterior region satisfies the bound F(??) ≤ C E(v + u), where v and u denote the infimum of the Eddington‐Finkelstein advanced and retarded time of ??, v₊ denotes max{1, v}, and u₊ denotes max{1, u}, where C is a constant depending only on the parameter M, and E depends on a suitable norm of the solution on the hypersurface t ? u + v = 1. (The bound applies in particular to subsets ?? of the event horizon or null infinity.) It is also shown that ? satisfies the pointwise decay estimate |?| ≤ C Ev in the entire exterior region, and the estimates |r?| ≤ C_R?E(1 + |u|)^?1/2 and |r^1/2?| ≤ C_R?Eu in the region {r ≥ R?} ∩ J⁺(Σ) for any R? > 2M. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red‐shift effect. The results in particular give an independent proof of the classical result |?| ≥ C E of Kay and Wald without recourse to the discrete isometries of spacetime. © 2009 Wiley Periodicals, Inc.     

Entropy Numbers in Weighted Function Spaces and Eigenvalue Distributions of Some Degenerate Pseudodifferential Operators I

In this paper we study weighted function spaces of type B(?ⁿ, Q(x)) and F(?ⁿ, Q(x)), where Q(x) is a weight function of at most polynomial growth. Of special interest are the weight functions Q(x) = (1 + |x|²)^α/2 with α ? ?. The main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

On smallest triangles

Pick n points independently at random in ?², according to a prescribed probability measure μ, and let Δ ≤ Δ ≤ … be the areas of the () triangles thus formed, in nondecreasing order. If μ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n³Δ : i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(μ). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that if μ is the uniform probability measure on the region S, then κ(μ) ≤ 2/|S|, where |S| denotes the area of S. Equality holds in that κ(μ) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitányi. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 206–223, 2003