On some special codes over Fp + uFp + uFp

We define a new map between codes over F_p + uF_p + u²F_p and F_p which is different to that defined in [2]. It is proved that the image of the linear cyclic code over the commutative ring F_p + uF_p + u²F_p with length n under this map is a distance-invariant quasi-cyclic code of index p² with length p²n over F_p. Moreover, it is proved that, if (n, p) = 1, then every code with length p²n over F_p which is the image of a linear (1 − u²)-cyclic code with length n over F_p + uF_p + u²F_p under this map is permutation equivalent to a quasi-cyclic code of index p².     

Integral complete 4-partite graphs

A graph G is called integral if all the roots of the characteristic polynomial P(G;x) are integers. In the paper the first known integral complete 4-partite graph K_p1,p2,p3,p4, where p₁<p₂<p₃<p₄, is constructed.     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

On density of interpolation points,a Kadec-type theorem,and Saff's principle of contamination inL p-approximation

Letf ∈L _p (I) and denote byB _n,p (f) the polynomial of bestL _p-approximation tof of degreen (1<p<∞,I=[?1,1], the norm is weightedL _p-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman forp=2 we show that for every 1<p<∞ andf ∈L _p (I) (not a polynomial) points of sign change of the error functionf-B _n,p (f) are dense inI asn→∞.     

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp₁(±x)Φp₂(±xp₁)?Φpr(±xp₁p₂?pr−1), where N=p₁p₂?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α²pβ−1 with odd prime p or separable polynomials of any odd degree.     

Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphs K_p1,p2,…,pr=K_{a1·p1,a2·p2,…,as·ps} with s=3,4. We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s=4, we give a positive answer to a question of Wang et al. [Integral complete r-partite graphs, Discrete Math. 283 (2004) 231-241]. The problem of the existence of integral complete multipartite graphs K_{a1·p1,a2·p2,…,as·ps} with arbitrarily large number s remains open.     

On a conjecture by B. Grünbaum

Denote by p_k(M) or υ_k(M) the number of k-gonal faces or k-valent of the convex 3-polytope M, respectively. Completely solving a problem by B. Grünbaum, the following theorem is proved: Given sequences of nonnegative integers p = (p₃, p₄,…p_m), υ = (υ₃, υ₄,…,υ_n) satisfying ∑_k?3(6−k)p_k + 2∑_k?3(3−k)υ_k = 12, there exists a convex 3-polytope M with p_k(M) = p_k for all k ≠ 6 and υ_k for all k ≠ 3 if and only if for the sequences p, υ the following does not hold: ∑p_i = 0 for i odd and ∑υ_i = 1 for i ? 0 (mod 3).     

Multilinear Calderón-Zygmund operators on the product of Lebesgue spaces with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rd, the author proves that for the multilinear Calderón-Zygmund operator, its boundedness from the product of Hardy space H¹(μ)×H¹(μ) into L1/2(μ) implies its boundedness from the product of Lebesgue spaces Lp₁(μ)×Lp₂(μ) into Lp(μ) with 1<p₁,p₂<∞ and p satisfying 1/p=1/p₁+1/p₂.     

On boundedness of discrete multilinear singular integral operators

Let m(ξ,η) be a measurable locally bounded function defined in R². Let 1?p₁,q₁,p₂,q₂<∞ such that pi=1 implies qi=∞. Let also 0<p₃,q₃<∞ and 1/p=1/p₁+1/p₂−1/p₃. We prove the following transference result: the operator

     

Line graph eigenvalues and line energy of caterpillars

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d?3 and n?2(d-1). Let p=[p₁,p₂,…,p_d-1] with p₁?1,p₂?1,…,p_d-1?1 such that

p₁+p₂+?+p_d-1=n-d+1.     

A sharp nonasymptotic bound and phase diagram of L 1/2 regularization

We derive a sharp nonasymptotic bound of parameter estimation of the L1/2 regularization.The bound shows that the solutions of the L1/2 regularization can achieve a loss within logarithmic factor of an ideal mean squared error and therefore underlies the feasibility and effectiveness of the L1/2regularization.Interestingly,when applied to compressive sensing,the L1/2 regularization scheme has exhibited a very promising capability of completed recovery from a much less sampling information.As compared with the Lp(0 p 1) penalty,it is appeared that the L1/2 penalty can always yield the most sparse solution among all the Lp penalty when 1/2 ≤ p 1,and when 0 p 1/2,the Lp penalty exhibits the similar properties as the L1/2 penalty.This suggests that the L1/2 regularization scheme can be accepted as the best and therefore the representative of all the Lp(0 p 1) regularization schemes.     

Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball

In this paper we study the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the type Δ_pu+f(r,u)=0 in the unit ball B, u=0 on ∂B. Here Δ_p denotes the p Laplace operator Δ_p=div(|∇u|^p−2∇u), p>1. The main ideas rely on the Maximum Principle and an implicit function theorem that we derive in a suitable weighted space. This space is essential to deal with the case p≠2.     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

Uniqueness of linear combinations (mod p)

In an earlier paper [1] one of us examined the question: Given a prime p what is the maximal number n=f(p) in any set of residues {a₁, a₂, …, a_n} (mod p) so that at least one of the differences a_i−a_j is incongruent to all other differences a_i′−a_j′?It was determined that the answer is of the order of magnitude log p. That is, there exist constants 0<c₁<c₂ so that c₁ log p<f(p)<c₂ log p for all p.In this note we examine a natural generalization, where instead of differences we examine the set of linear expressions with prescribed coefficients.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

p-Adic dynamical systems of Chebyshev polynomials

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ? _p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T _p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ? _p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T _p in the field ? _p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T _n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.     

Torsion points on the jacobian of a Fermat quotient

Let p≥5 be a prime, ζ a primitive pth root of unity and λ=1−ζ. For 1≤s≤p−2, the smooth projective model C_p,s of the affine curve v^p=u^s(1−u) is a curve of genus (p−1)/2 whose jacobian J_p,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ). In 1981, Greenberg determined the field of rationality of the p-torsion subgroup of J_p,s and moreover he proved that the λ³-torsion points of J_p,s are all rational over Q(ζ). In this paper we determine quite explicitly the λ³-torsion points of J_p,1 for p=5 and p=7, as well as some further p-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for pth powers.     

Fine and Wilf words for any periods

Let w = w₁ … w_n be a word of maximal length n, and with a maximal number of distinct letters for this length, such that w has periods p₁, …, p_n but not period gcd(p₁,…,p_r). We provide a fast algorithm to compute n and w. We show that w is uniquely determined apart from isomorphism and that it is a palindrome. Furthermore we give lower and upper bounds for n as explicit functions of p₁, …p_r. For r = 2 the exact value of n is due to Fine and Wilf. In case the number of distinct letters in the extremal word equals r a formula for n had been given by Castelli, Mignosi and Restivo in case r = 3 and by Justin if r > 3.     

Fonctions lipschitziennes et espaces de Sobolev fractionnaires

We consider a semigroup of Markovian and symmetric operators to which we associate fractional Sobolev spaces D^α_p (0 < α < 1 and 1 < p < ∞) defined as domains of fractional powers (−A_p)^α/2, where A_p is the generator of the semigroup in L^p. We show under rather general assumptions that Lipschitz continuous functions operate by composition on D^α_p if p ≥ 2. This holds in particular in the case of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.