A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph

For a graph G, we define σ₂(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k of length at most four such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k. And show if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k, V(C ₁) ∪...∪ V(C _k ) = V(G), and |C _i | ≤ 4 for all 1 ≤ i ≤ k − 1. The condition of degree sum σ₂(G) ≥ n + k − 1 is sharp. Received: December 20, 2006. Final version received: December 12, 2007.     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.     

Triangular finite elements of HCT type and classC ρ

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C ^ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ₃ of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC ¹ cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.     

THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS

51.IntroductionandNotationsLetD=(V,E)beadigraphandL(D)denotethesetofcyclelengthsofD.ForuEVandintegeri21,letfo(u):={vEVIthereedestsadirectedwalkoflengthifromutov}.WedelveRo(u):={u}.Letu,vEV.IfN (v)=N (v)andN--(v)=N--(v),thenwecanvacopyofu.LotDbeaprimitivedigraphand7(D)denotetheexponentofD.In1950,H.WielandtI61foundthat7(D)5(n--1)' 1andshowedthatthereisapiquedigraphthatattainsthisbound.In1964,A.L.DulmageandN.S.Mendelsohn[2]ObservedthattherearegapsintheexponentsetEd={ry(D)IDEPD.}…     

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

An infeasible-interior-point potential-reduction algorithm for linear programming

n . The method is based on Rockafellar’s proximal point algorithm and a cutting-plane technique. At each step, we use an approximate proximal point p^a(x_k) of x_k to define a v_k∈∂_εkf(p^a(x_k)) with ε_k≤α∥v_k∥, where α is a constant. The method monitors the reduction in the value of ∥v_k∥ to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of ∥v_k∥. Without the differentiability of f, the method converges globally and the rate of convergence is Q-linear. Superlinear convergence is also discussed to extend the characterization result of Dennis and Moré. Numerical results show the good performance of the method. Received October 3, 1995 / Revised version received August 20, 1998 Published online January 20, 1999     

An asymptotic property of gap series. II

An upper bound estimate in the law of the iterated logarithm for Σf(n _k ω) where n_k+1∫n_k≧ 1 + ck ^-α (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n _k} is assumed. As an application, the law of the iterated logarithm is proved when {n _k} is the arrangement in increasing order of the set B(τ)={₁ ⁱ ₁...q_τ ⁱ _τ|i₁,...,i_τ∈N ₀}, where τ≧ 2, N ₀=NU{0}, and q ₁,...,q _τ are integers greater than 1 and relatively prime to each others. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

The second centralizer of a Bernoulli shift is just its powers

If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T ⁱ |i ∈ Z}. The proof is carried out by constructing an action of Z², {T ₁ ⁱ °T ₂ ⁱ |i, j ∈ Z}, whereT ₁ is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T ₁,T ₂ ^i} ={T ₁ ⁱ °T ₂ ^k l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.     

Accuracy of several multidimensional refinable distributions

Compactly supported distributions f₁,..., f_r on ℝ^d are fefinable if each f_i is a finite linear combination of the rescaled and translated distributions f_j(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝^d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σ_k∈Λ c_k f(Ax−k), where Λ is a finite subset of Γ, the c_k are r×r matrices, and f=(f₁,...,f_r)^T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced from linear combinations of translates of f₁,...,f_r along the lattice Γ. We determine the accuracy p from the matrices c_k. Moreover, we determine explicitly the coefficients y_α,i(k) such that x^α=Σ _i=1 ^r Σ_k∈Γy_α,i(k) fi(x+k). These coefficients are multivariate polynomials y_α,i(x) of degree |α| evaluated at lattice points k∈Γ.