泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

Asymptotic Determination of the Last Packing Number of Quadruples

A 3-(n,4,1) packing design consists of an n-element set X and a collection of 4-element subsets of X, called blocks, such that every 3-element subset of X is contained in at most one block. The packing number of quadruples d(3,4,n) denotes the number of blocks in a maximum 3-(n,4,1) packing design, which is also the maximum number A(n,4,4) of codewords in a code of length n, constant weight 4, and minimum Hamming distance 4. In this paper the last packing number A(n,4,4) for n≡ 5(mod 6) is shown to be equal to Johnson bound with 21 undecided values n=6k+5, k∈{m: m is odd , 3≤ m≤ 35, m≠ 17,21}∪ {45,47,75,77,79,159}. AMS Classification:05B40, 94B25     

The old subvariety ofJ 0(pq) and the Eisenstein kernel in Jacobians

Whenp, q are distinct odd primes, and γ:J ₀(p)²×J ₀(q)²→J ₀(pq) is the natural map defined by the degeneracy maps, Ribet [10] determined the odd part of the kernel of γ. We study the 2-primary part of this kernel through its intersection with the Eisenstein kernelJ ₀(p)[I _p )²×J ₀(q)[I _q ]². We determine this intersection forp≢1 mod 16,q≢1 mod 16, and also produce new elements of ker γ wheneverp≡9 mod 16 orq≡9 mod 16. These sharpen Ribet's results in [10].     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.     

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

On a method for interpolating functions on chaotic nets

Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x| ^m form odd,K(x)=|x| ^m In |x| form even (x∈ℝ ⁿ ),P is the set of real polynomials inn variables of total degree ≤m/2, andx ₁,...,x _N ∈ℝ ⁿ . We construct a function of the form

Vector-valued Hardy inequalities andB-convexity

Inequalities of the form for allf∈H ¹, where {m _k } are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterizeB-convexity. It is also shown that for 1<q<∞ and 0<α<∞ the spaceX=H(1,q,γa) consisting of analytic functions on the unit disc such that satisfies the previous inequality for vector valued functions inH ¹ (X), defined as the space ofX-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {m _k }={2^k} but not for {m _k }={k ^a } for any α ∈ N. The author has been partially supported by the Spanish DGICYT, Proyecto PB95-0291.     

Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

Tracking and invariance problems for some classes of discrete delay systems

Consider the system with perturbation g _k ∈ ℝ ⁿ and output z _k = Cx _k . Here, A _k ,A _k ^(s) ∈ ℝ ^{n × n} , B _k ⁽¹⁾ ∈ ℝ ^{n × p} , B _k ⁽²⁾ ∈ ℝ ^{n × m} , C ∈ ℝ ^{p × n} . We construct a special Lyapunov-Krasovskii functional in order to synthesize controls u _k ⁽¹⁾ and u _k ⁽²⁾ for which the following properties are satisfied:

The Existence of BIB Designs

Abstract Given any positive integers k≥ 3 and λ, let c(k, λ) denote the smallest integer such that v∈B(k, λ) for every integer v≥c(k, λ) that satisfies the congruences λv(v− 1) ≡ 0(mod k(k− 1)) and λ(v− 1) ≡ 0(mod k− 1). In this article we make an improvement on the bound of c(k, λ) provided by Chang in [4] and prove that . In particular, . Supported by NSFC Grant No. 19701002 and Huo Yingdong Foundation     

K 1,p k-factorization of complete bipartite graphs

Abstract. In this paper, it is shown that a sufficient condition for the existence of a     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Further Results on the Existence of Splitting BIBDs and Application to Authentication Codes

Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λ v(v−1)≡0 (mod (c ² u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes.     

P 3-factorization of complete multipartite graphs

In this note it is shown that a necessary and sufficient condition for the existence of a P3-factorizatlon of complete multipartite graph λK, is (1) m≥3, (2) mn≡0(mod 3) and (3)λ(m-1)n≡0(mod 4).     

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     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

On vertex-coloring edge-weighting of graphs

A k-edge-weighting w of a graph G is an assignment of an integer weight, w(e) ∈ {1,…,k}, to each edge e. An edge-weighting naturally induces a vertex coloring c by defining c(u) = Σ _e∋u w(e) for every u ∈ V (G). A k-edge-weighting of a graph G is vertex-coloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). When k ≡ 2 (mod 4) and k ⩾ 6, we prove that if G is k-colorable and 2-connected, δ(G) ⩾ k − 1, then G admits a vertex-coloring k-edge-weighting. We also obtain several sufficient conditions for graphs to be vertex-coloring k-edge-weighting.