Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

Raising approximation order of refinable vector by increasing multiplicity

An algorithm is presented for raising an approximation order of any given orthogonal multiscaling function with the dilation factor a. Let φ(x) = [φ1(x),φ2(x),…,φr(x)]T be an orthogonal multiscaling function with the dilation factor a and the approximation order m. We can construct a new orthogonal multiscaling function φnew(x) = [ φT(x). f3r 1(x),φr 2(x),…,φr s(x)}T with the approximation order m L(L ∈ Z ). In other words, we raise the approximation order of multiscaling function φ(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function φ(x) = [φ1 (x), φ2(x), …, φr(x)]T is symmetric, then the new orthogonal multiscaling function φnew(x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given.     

A critical oscillation constant as a variable of time scales for half-linear dynamic equations

We present criteria of Hille-Nehari type for the half-linear dynamic equation (r(t)Φ(y ^Δ))^Δ+p(t)Φ(y ^σ ) = 0 on time scales. As a particular important case we get that there is a a (sharp) critical constant which may be different from what is known from the continuous case, and its value depends on the graininess of a time scale and on the coefficient r. As applications we state criteria for strong (non)oscillation, examine generalized Euler type equations, and establish criteria of Kneser type. Examples from q-calculus, a Hardy type inequality with weights, and further possibilities for study are presented as well. Our results unify and extend many existing results from special cases, and are new even in the well-studied discrete case.     

Integral inequalities with weights for maxiamal operators

Let μ be a measure on the upper half-space R ₊ ⁿ⁺¹ , and v a weight onR ⁿ, we give a characterization for the pair (v, μ) such that ∥M(fv)∥_{L Θ (μ}) ⩽ c ∥f∥_{L Θ (μ}), where Φ is an N-function satisfying Δ₂ condition andMf(x,t), is the maximal function onR ₊ ⁿ⁺¹ , which was introduced by Ruiz, F. and Torrea, J.. Supported by NSFC.     

Augmentation quotients for complex representation rings of dihedral groups

Denote by D _m the dihedral group of order 2m. Let ℛ(D _m ) be its complex representation ring, and let Δ(D _m ) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δ ⁿ (D _m )/Δ ⁿ⁺¹(D _m ) for each positive integer n.     

On some new generalized difference sequence spaces defined by a sequence of moduli

In this paper, we define the new generalized difference sequence spaces [V, λ, F, p, q]₀(Δ _v ^m ), [V, λ, F, p, q]₁(Δ _v ^m ) and [V, λ, F, p, q]_∞(Δ _v ^m ). We also study some inclusion relations between these spaces.     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

E Φ andh Φ fail to beM-ideals inL Φ andl Φ in the case of the Orlicz norm

It is well known that in the case of the Luxemburg normE ^Φ (resp.h ^Φ) is anM ideal inL ^Φ (resp.l ^Φ), see [1], [9], [15] and [6]; [17] and [18]. It is proved in this paper that in the case of the Orlicz normE ^Φ (resph ^Φ) is anM-ideal inL ^Φ (resp.l ^Φ) iff Φ satisfies the suitable Δ₂ or Φ^*(a(Φ^*)), where a(Φ^*) is linear on the interval [0,u]} and Φ^* denotes the function complementary to Φ in the sense of Young. It is also proved that any linear continuous regular (i.e. order continuous) functional ξ_υ overE ^Φ (resp.h ^Φ) generated byv∈ L(h ^Φ*) (resp.v∈ L(h ^Φ*)) which attains its norm on the unit sphereS(E ^Φ) (resp.S(h ^Φ)), has a unique norm-preserving extension toL ^Φ (resp.l ^Φ). Finally, it is proved thatL ^Φ (resp.l ^Φ) has the property that any linear continuous regular functional ξ_υ overE ^Φ (resp.h ^Φ) has a unique norm-preserving extension toL ^Φ (resp.l ^Φ) iff Φ orE ^Φ satisfies the suitable Δ₂ and in the second case Φ^* attains the value 1.     

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

A Five Color Zero-Sum Generalization

Let g_zs(m, 2k) (g_zs(m, 2k+1)) be the minimal integer such that for any coloring Δ of the integers from 1, . . . , g_zs(m, 2k) by (the integers from 1 to g_zs(m, 2k+1) by ) there exist integers such that 1. there exists j_x such that Δ(x_i) ∈ for each i and ∑_i=1^m Δ(x_i) = 0 mod m (or Δ(x_i)=∞ for each i); 2. there exists j_y such that Δ(y_i) ∈ for each i and ∑_i=1^m Δ(y_i) = 0 mod m (or Δ(y_i)=∞ for each i); and 1. 2(x_m−x₁)≤y_m−x₁. In this note we show g_zs(m, 2)=5m−4 for m≥2, g_zs(m, 3)=7m+−6 for m≥4, g_zs(m, 4)=10m−9 for m≥3, and g_zs(m, 5)=13m−2 for m≥2. Supported by NSF grant DMS 0097317     

Clifford martingale Φ-Equivalence betweenS(f) andf*

TheL ²-norm equivalence between a Clifford martingalef and its square functionS(f) plays an important role in the proof of theL ²-boundedness of Cauchy integral operators on Lipschitz graphs and the CliffordT(b) Theorem [2, 4]. This note generalises the result to the Φ-equivalence between the maximal functionf* andS(f), where Φ is a nondecreasing and continuous function fromIR ⁺ toIR ⁺, of the moderate growth Φ(2u)≤C ₁Φ(u) and satisfies Φ(0)=0.     

The Linear 2-Arboricity of Planar Graphs

 Let G be a planar graph with maximum degree Δ and girth g. The linear 2-arboricity la ₂(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la ₂(G)≤⌈(Δ+1)/2⌉+12; (2) la ₂(G)≤⌈(Δ+1)/2⌉+6 if g≥4; (3) la ₂(G)≤⌈(Δ+1)/2⌉+2 if g≥5; (4) la ₂(G)≤⌈(Δ+1)/2⌉+1 if g≥7. Received: June 28, 2001 Final version received: May 17, 2002 Acknowledgments. This work was done while the second and third authors were visiting the Institute of Mathematics, Academia Sinica, Taipei. The financial support provided by the Institute is greatly appreciated.     

On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements

Some criterions in order thatl ¹ embeds complementably inE ^Φ(μ) and inL ^Φ(μ) are given. It is also proved that every idealL inL ^Φ(μ) such thatI _Φ(x/‖x‖Φ)=1 for anyxεL/{0} is contained inE ^Φ(μ).     

Construction of multiwavelets with high approximation order and symmetry

In this paper, based on existing symmetric multiwavelets, we give an explicit algorithm for constructing multiwavelets with high approximation order and symmetry. Concretely, suppose Φ(x) := (φ1(x), . . . , φr(x))T is a symmetric refinable function vectors with approximation order m. For an arbitrary nonnegative integer n, a new symmetric refinable function vector Φnew(x) := (φn1 ew(x), . . . , φrn ew(x))T with approximation order m + n can be constructed through the algorithm mentioned above. Additionally,...     

Mollification formulas and implicit smoothing

This paper develops some mollification formulas involving convolutions between popular radial basis function (RBF) basic functions Φ, and suitable mollifiers. Polyharmonic splines, scaled Bessel kernels (Matern functions) and compactly supported basic functions are considered. A typical result is that in ℛ^d the convolution of |{•}|^β and (•²+c ²)^−(β+2d)/2 is the generalized multiquadric (•²+c ²)^β/2 up to a multiplicative constant. The constant depends on c>0, β, where ℜ(β)>−d, and d. An application which motivated the development of the formulas is a technique called implicit smoothing. This computationally efficient technique smooths a previously obtained RBF fit by replacing the basic function Φ with a smoother version Ψ during evaluation.     

On the structure of the augmentation quotient group for some nonabelian 2-groups

Let G be a finite nonabelian group, ℤG its associated integral group ring, and Δ(G) its augmentation ideal. For the semidihedral group and another nonabelian 2-group the problem of their augmentation ideals and quotient groups Q _n (G) = Δ ⁿ (G)/Δ ⁿ⁺¹(G) is deal with. An explicit basis for the augmentation ideal is obtained, so that the structure of its quotient groups can be determined.     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

On the Riesz means of coefficients of <Emphasis Type="Italic">m</Emphasis>th symmetric power <Emphasis Type="Italic">L</Emphasis>-functions

Let c _n be the Fourier coefficients of L(sym ^m   f, s), and Δρ(x; sym ^m   f) be the error term in the asymptotic formula for ∑ _n≪x c _n . In this paper, we study the Riesz means of Δρ(x; sym ^m   f) and obtain a truncated Voronoi-type formula under the hypothesis Nice(m, f).     

Some results on distance two labelling of outerplanar graphs

Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following results： （1） χ（G^2） = 7 if △= 6; （2） λ（G） ≤ △ ＋5 if △ ≥ 4, and ）,（G）≤ 7 if △ = 3; and （3） there is an outerplanar graph G with △ = 4 such that ）λ（G） = 7. These improve some known results on the distance two labelling of outerplanar graphs.