List Point Arboricity of Dense Graphs

Let G be a simple graph. The point arboricity ρ(G) of G is defined as the minimum number of subsets in a partition of the point set of G so that each subset induces an acyclic subgraph. The list point arboricity ρ _l (G) is the minimum k so that there is an acyclic L-coloring for any list assignment L of G which |L(v)| ≥ k. So ρ(G) ≤ ρ _l (G) for any graph G. Xue and Wu proved that the list point arboricity of bipartite graphs can be arbitrarily large. As an analogue to the well-known theorem of Ohba for list chromatic number, we obtain ρ _l (G + K _n ) = ρ(G + K _n ) for any fixed graph G when n is sufficiently large. As a consequence, if ρ(G) is close enough to half of the number of vertices in G, then ρ _l (G) = ρ(G). Particularly, we determine that , where K _2(n) is the complete n-partite graph with each partite set containing exactly two vertices. We also conjecture that for a graph G with n vertices, if then ρ _l (G) = ρ(G). Research supported by NSFC (No.10601044) and XJEDU2006S05.     

Estimates for parametric Marcinkiewicz integrals in BMO and Campanato spaces

In this paper, the authors consider the behaviors of a class of parametric Marcinkiewicz integrals μ _Ω ^ρ , μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ on BMO(ℝ ⁿ ) and Campanato spaces with complex parameter ρ and the kernel Ω in Llog⁺ L(S ⁿ⁻¹). Here μ _Ω,λ ^*,ρ and μ _Ω,S ^ρ are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g _λ *-function and the Lusin area function S, respectively. Under certain weak regularity condition on Ω, the authors prove that if f belongs to BMO(ℝ ⁿ ) or to a certain Campanato space, then [μ _Ω,λ ^*,ρ (f)]², [μ _Ω,S ^ρ (f)]² and [μ _Ω ^ρ (f)]² are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.     

A note on convergent polynomials of interpolation

In this note we define a sequence {L_n(f;x)} of interpolatory polynomials based on a system x_n={x_kn, k=1,2,…n} of nodes to be a sequence of QLIP if for every f(x)∈C[−1,1], L_n(f; x) tends uniformly to f(x) and ρ_n=1+o(1) as n→∞, where ρ_n is the ratio of the number of points in x_n, at which L_n(f;x) coincides with f(x), and the degree of L_n(f;x). Two sequences of QLIP are constructed, one of which is based on a Bernstein process and the other the Freud-Sharma's construction.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

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).     

Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent

We prove the boundedness of the maximal operator Mr in the spaces L^p（·）（Г,p） with variable exponent p（t） and power weight p on an arbitrary Carleson curve under the assumption that p（t） satisfies the log-condition on Г. We prove also weighted Sobolev type L^p（·）（Г, p） → L^q（·）（Г, p）-theorem for potential operators on Carleson curves.     

Hypersingular parameterized Marcinkiewicz integrals with variable kernels on Sobolev and Hardy-Sobolev spaces

Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Equivalent Characterizations for Boundedness of Maximal Singular Integrals on ax+b-Groups

Let (S,d,ρ) be the affine group ℝ ⁿ ⋉ℝ⁺ endowed with the left-invariant Riemannian metric d and the right Haar measure ρ, which is of exponential growth at infinity. In this paper, for any linear operator T on (S,d,ρ) associated with a kernel K satisfying certain integral size condition and H?rmander’s condition, the authors prove that the following four statements regarding the corresponding maximal singular integral T ^∗ are equivalent: T ^∗ is bounded from L_c^￥L_{c}^{\infty} to BMO, T ^∗ is bounded on L ^p for all p∈(1,∞), T ^∗ is bounded on L ^p for some p∈(1,∞) and T ^∗ is bounded from L ¹ to L ^1,∞. As applications of these results, for spectral multipliers of a distinguished Laplacian on (S,d,ρ) satisfying certain Mihlin-H?rmander type condition, the authors obtain that their maximal singular integrals are bounded from L_c^￥L_{c}^{\infty} to BMO, from L ¹ to L ^1,∞, and on L ^p for all p∈(1,∞).     

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu(m)bTu(n)ag}m,n∈p on local field are investigated. Accurately, that the canonical dual of frame {Eu(m)bTu(n)ag}m,n∈p also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu(m)bTu(n)ag}m,n∈p to be a frame for L2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L2(K) are established. An example is finally given.     

Finite dimensional spectral inverse problem for a bundle of hermitian quadratic forms

The paper is devoted to recovering the coefficients of a pair of Hermitian quadratic forms c(x, x) and (x, x) in a special basis, in which the matrix of the form c(x, x) is tridiagonal and the matrix of the form m(x, x) is diagonal. The form c(x, x) is positive definite, and the form m(x, x) is nondegenerate, but is not positive difinite in contrast with a well-known case. The data of the inverse problem include the spectrum λ₁...,λ_n of the bundle II_λ and the set of numbers ρ₁...,ρ_n connected with the main normalized elements of the bundle. A procedure for solving the inverse problem is described. The characteristic conditions for λ₁...,λ_n; are found that provide its solution. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 33–36, 1990. Translated by T. N. Surkova.     

Cohomological growth rates and Kazhdan-Lusztig polynomials

In a previous work, the authors established various bounds for the dimensions of degree n cohomology and Ext-groups, for irreducible modules of semisimple algebraic groups G (in positive characteristic p) and (Lusztig) quantum groups U _ζ (at roots of unity ζ). These bounds depend only on the root system, and not on the characteristic p or the size of the root of unity ζ. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system ϕ. For quantum groups U _ζ with a fixed ϕ, we show the sequence {max _{L irred} dim H ⁿ (U _ζ , L)} _n has polynomial growth independent of ζ. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that {log max _{L irred} dim H ⁿ (G,L)} has polynomial growth ≤ 3 for any fixed prime p (and ≤ 4 if p is allowed to vary with n). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov and Lubotzky [13].     

Low-dimensional representations of Aut (F 2)

LetF ₂ be the free group of rank two, and Φ₂ its automorphism group. We consider the problem of describing the representations of Φ₂ of degreen for small values ofn. Our main result is the classification (up to equivalence) of all indecomposable representations ρ of Φ₂ of degreen≤4 such that ρ(F ₂) ≠ 1. There are only finitely many such representations, and in all them ρ(F ₂) is solvable. This is no longer true in higher dimensions. Already forn=6 there exists a 1-parameter family of irreducible nonequivalent representations of Φ₂ such that ρ(F ₂) contains a free non-abelian subgroup. We also obtain some new 4-dimensional representations of the braid groupB ₄ which are indecomposable and reducible at the same time. It would be interesting to find some applications of these representations. Supported in part by the NSERC Grant A-5285 Supported in part by an NSERC grant     

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.     

Operators on the Lattice of Pseudovarieties of Finite Semigroups

L (F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [U ^ρ, U _ρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of U ^ρ, bases of pseudoidentities for U ^ρ, instances of commutativity (U ^ρ)^σ = U ^σ)^ρ, as well as the semigroups generated by certain pairs of such operators.     

Sausages are good packings

LetB ^d be thed-dimensional unit ball and, for an integern, letC _n ={x ¹,...,x ⁿ } be a packing set forB ^d , i.e.,|x ⁱ −x ^j |≥2, 1≤i<j≤n. We show that for every a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(C _n )+ρB ^d )≥V(conv(S _n )+ρB ^d ), whereS _n is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove that, for every convex bodyK and ρ<1/32d ⁻²,V(conv(C _n )+ρK)≥V(conv(S _n )+ρK), whereC _n is a packing set with respect toK andS _n is a minimal “sausage” arrangement ofK, holds.     

On Minimum Stars and Maximum Matchings

We discuss worst-case bounds on the ratio of maximum matching and minimum median values for finite point sets. In particular, we consider ``minimum stars,' which are defined by a center chosen from the given point set, such that the total geometric distance L <sub>S</sub> to all the points in the set is minimized. If the center point is not required to be an element of the set (i.e., the center may be a Steiner point), we get a ``minimum Steiner star' of total length L _SS . As a consequence of triangle inequality, the total length L _M of a maximum matching is a lower bound for the length L _SS of a minimum Steiner star, which makes the worst-case value ρ(SS,M) of the value L _SS /L _M interesting in the context of optimal communication networks. The ratio also appears as the duality gap in an integer programming formulation of a location problem by Tamir and Mitchell. In this paper we show that for a finite set that consists of an even number of points in the plane and Euclidean distances, the worst-case ratio ρ(S,M) cannot exceed . This proves a conjecture of Suri, who gave an example where this bound is achieved. For the case of Euclidean distances in two and three dimensions, we also prove upper and lower bounds for the worst-case value ρ(S,SS) of the ratio L _S /L _SS , and for the worst-case value ρ(S,M) of the ratio L _S /L _M . We give tight upper bounds for the case where distances are measured according to the Manhattan metric: we show that in three-dimensional space, ρ(SS,M) is bounded by 3/2, while in two-dimensional space L _SS =L _M , extending some independent observations by Tamir and Mitchell. Finally, we show that ρ(S,SS) is 3/2 in the two-dimensional case, and 5/3 in the three-dimensional case. Received January 1, 1999, and in revised form July 15, 1999.     

Smolyak's algorithm for weighted <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of multivariate functions with bounded <Emphasis Type="Italic">r</Emphasis>th mixed derivatives over ℝ<Superscript>d</Superscript>

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on . We consider the space ℱ_r,d which consists of multivariate functions whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.