Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Some Notes on Signed Edge Domination in Graphs

The closed neighborhood N_G[e] of an edge e in a graph G is the set consisting of e and of all edges having an end-vertex in common with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge dominating function of G. The signed edge domination number γ_s^′(G) of G is defined as . Recently, Xu proved that γ_s^′(G) ≥ |V(G)| − |E(G)| for all graphs G without isolated vertices. In this paper we first characterize all simple connected graphs G for which γ_s^′(G) = |V(G)| − |E(G)|. This answers Problem 4.2 of [4]. Then we classify all simple connected graphs G with precisely k cycles and γ_s^′(G) = 1 − k, 2 − k. A. Khodkar: Research supported by a Faculty Research Grant, University of West Georgia. Send offprint requests to: Abdollah Khodkar.     

Existence of positive solutions for a singular <Emphasis Type="Italic">p</Emphasis>-Laplacian differential equation

In this paper, we are concerned with the existence of positive solutions for a singular p-Laplacian differential equation
（φp（u＇））＇＋β/r φp（u＇）-γ ｜u＇｜^p/u ＋ g（r）=0,0〈r〈1,
subject to the Dirichlet boundary conditions： u（0） = u（1） =0, where φp（s） = ｜sl^P-2s,p 〉 2,β 〉0, γ〉（p-1）/p （β ＋ 1）, and g（r） ∈ C^1 [0, 1] with g（r） 〉 0 for all τ ∈ [0, 1]. We use the method of elliptic regularization, by carrying out two limit processes, to get a positive solution.     

Optimal strong parity edge-coloring of complete graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let (G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring). Always (G) ≥ p(G) ≥ χ′(G). We prove that (K _n ) = 2^⌈lgn⌉ − 1 for all n. The optimal strong parity edge-coloring of K _n is unique when n is a power of 2, and the optimal colorings are completely described for all n. Partially supported by NSF grant CCR 0093348. Work supported in part by the NSA under Award No. MDA904-03-1-0037.     

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

Some notes on the adams spectral sequence

In the beginning of 1980's Cohen, R. proved thath0b(1k)=B(1,k)=B(1,k)⊗ζ₁ survives toE _∞ in the Adams spectral sequence. Later, Cohen, R. and Goerss, P. proved that is a permanent cycle. And they are represented by ζ _k ,ηj respectively. Here the author proved: Theorem 2: Let 2≤s≤p−1,j≥3, then β _s η _j ≠0. Theorem 3: For 2≤s≤p−1,k≥2, β _s ζ _k ≠0 in the stable homotopy groups of spheres. As a remark, we get in . Supported by Doctoral Program Fundation.     

Riesz multiwavelet bases generated by vector refinement equation

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L ₂(ℝ ^s ). Suppose ψ = (ψ¹,..., ψ ^r ) ^T and are two compactly supported vectors of functions in the Sobolev space (H ^μ(ℝ ^s )) ^r for some μ > 0. We provide a characterization for the sequences {ψ _jk ^l : l = 1,...,r, j ε ℤ, k ε ℤ ^s } and to form two Riesz sequences for L ₂(ℝ ^s ), where ψ _jk ^l = m ^j/2ψ ^l (M ^j ·−k) and , M is an s × s integer matrix such that lim _n→∞ M ⁻ⁿ = 0 and m = |detM|. Furthermore, let ϕ = (ϕ¹,...,ϕ ^r ) ^T and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψ^ν = (ψ^ν1,...,ψ^νr ) ^T and , ν = 1,..., m − 1 such that two sequences {ψ _jk ^νl : ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ ^s } and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ ^s } form two Riesz multiwavelet bases for ^L ₂(ℝ ^s ). The bracket product [f, g] of two vectors of functions f, g in (L ₂(ℝ ^s )) ^r is an indispensable tool for our characterization. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)     

Functoriality of automorphic <Emphasis Type="Italic">L</Emphasis>-functions through their zeros

Let E be a Galois extension of ℚ of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π₁)…L(s, π _k ), where π _j , j = 1,…, k, are automorphic cuspidal representations of , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal _E/ℚ, then L(s, π) must equal a single L-function attached to a cuspidal representation of and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ℚ. As E is not assumed to be solvable over ℚ, our results are beyond the scope of the current theory of base change and automorphic induction. Our results are unconditional when m,m ₁,…,m _k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases. The first author was supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the National Security Agency (Grant No. H98230-06-1-0075). The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein     

Categoricity,amalgamation, and tameness

Theorem: For each 2 ≤ k < ω there is an -sentence ϕ_k such that (1) ϕ_k is categorical in μ if μ≤ℵ_k−2; (2) ϕ_k is not ℵ_k−2-Galois stable (3) ϕ_k is not categorical in any μ with μ>ℵ_k−2; (4) ϕ_k has the disjoint amalgamation property (5) For k > 2 (a) ϕ_k is (ℵ₀, ℵ_k−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵ_k−3; (b) ϕ_k is ℵ_m-Galois stable for m ≤ k − 3 (c) ϕ_k is not (ℵ_k−3, ℵ_k−2). The first author is partially supported by NSF grant DMS-0500841.     

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.     

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

On signed distance-k-domination in graphs

The signed distance-k-domination number of a graph is a certain variant of the signed domination number. If v is a vertex of a graph G, the open k-neighborhood of v, denoted by N _k (v), is the set N _k (v) = {u: u ≠ v and d(u, v) ⩽ k}. N _k [v] = N _k (v) ⋃ {v} is the closed k-neighborhood of v. A function f: V → {−1, 1} is a signed distance-k-dominating function of G, if for every vertex . The signed distance-k-domination number, denoted by γ _k,s (G), is the minimum weight of a signed distance-k-dominating function on G. The values of γ _2,s (G) are found for graphs with small diameter, paths, circuits. At the end it is proved that γ _2,s (T) is not bounded from below in general for any tree T.     

A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

In this paper we consider the problem of bounding the Betti numbers, b _i (S), of a semi-algebraic set S⊂ℝ ^k defined by polynomial inequalities P ₁≥0,…,P _s ≥0, where P _i ∈ℝ[X ₁,…,X _k ], s<k, and deg (P _i )≤2, for 1≤i≤s. We prove that for 0≤i≤k−1,

Transversals of additive Latin squares

LetA={a ₁, …,a _k} andB={b ₁, …,b _k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S _ksuch that the sums α _i +b _i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤ _p ) ^a and in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order. Supported by Hungarian research grants OTKA F030822 and T029759. Supported by the Catalan Research Council under grant 1998SGR00119. Partially supported by the Hungarian Research Foundation (OTKA), grant no. T029132.     

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood N _G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values , taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γ _st ′(G). Obviously, γ _st ′(G) is defined only for graphs G which have no connected components isomorphic to K ₂. In this paper we present some lower bounds for γ _st ′(G). In particular, we prove that γ _st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ _st ′(T). Research supported by a Faculty Research Grant, University of West Georgia.     

Omega subgroups of powerful <Emphasis Type="Italic">p</Emphasis>-groups

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ω_i(G) ≤ p ⁱ for odd p, and expΩ_i(G) ≤ 2 ⁱ⁺¹ for p = 2; (ii) the index |G: G ^{p i}| coincides with the number of elements of G of order at most p ⁱ. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPV05/99.     

Sets of rigged paths with Virasoro characters

Let {M _r,s ^(p,p′)}_{1≤r≤p−1,1≤s≤p′−1} be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ _r=1 ^p−1 M _r,s ^(p,p′) in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form , where φ _−n ^(r′,r):M _r,s ^(p,p′)→M _r′,s ^(p,p′) are the Fourier components of the (2,1)-primary field and |r ₀,s〉 is the highest weight vector of . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p−1 M _r,1^(p,p′). We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.      

Detection of Some Elements in the Stable Homotopy Groups of Spheres

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π＊S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres πp^nq＋2pq＋q-3S which isof order p and is represented by kohn ∈ ExtA^3,P^nq＋2pq＋q（Zp,Zp） in the Adams spectral sequence, wherep 〉 5 is an odd prime, n ≥3 and q = 2（p-1）. In the course of the proof, a new family of homotopy elements in πp^nq＋（p＋1）q-1V（1） which is represented by β＊i＇＊i＊（hn） ∈ ExtA^2,pnq＋（p＋1）q＋1 （H^＊V（1）, Zp） in the Adams sequence is detected.     

Some applications of coincidence theory

Summary As a first application we compute the Lefschetz coincidence number of maps between manifolds whose rational singular cohomologyH* has a simple system of generators. For the second let be anH-manifold with multiplicationm. Define for ,m ₂ (x,x)=m(x,x) andm _k (x)=m(x,m _k−1 (x)), for allk>2. All roots of equationm _k (x)=m ₈ (x),k>s such thatm _k (x)=m ₃ (x) butm _i (x)≠m _j (x) for allk>i>j,s≥j≥0 andk−s does not dividei−j, split into a finite number of equivalence classes. We compute precisely the numbers of classes such that their members satisfy the above property. This article was processed by the author using the Springer-Verlag T_EX macro package 1991.     

带振荡因子的粗双曲奇异积分算子

The singular integral operator J Ω,α, and the Marcinkiewicz integral operator (~μ)Ω,α are studied. The kernels of the operators behave like |y|-n-α(α＞0) near the origin, and contain an oscillating factor ei|y|-β(β＞0) and a distribution Ω on the unit sphere Sn-1 It is proved that, if Ω is in the Hardy space Hr (Sn-1) with 0＜r= (n-1)/(n-1 )(＞0), and satisfies certain cancellation condition,then J Ω,α and uΩ,α extend the bounded operator from Sobolev space Lpγ to Lebesgue space Lp for some p. The result improves and extends some known results.