Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

Bounds for Covering Codes over Large Alphabets

Let K_q(n,R) denote the minimum number of codewords in any q-ary code of length n and covering radius R. We collect lower and upper bounds for K_q(n,R) where 6 ≤ q ≤ 21 and R ≤ 3. For q ≤ 10, we consider lengths n ≤ 10, and for q ≥ 11, we consider n ≤ 8. This extends earlier results, which have been tabulated for 2 ≤ q ≤ 5. We survey known bounds and obtain some new results as well, also for s-surjective codes, which are closely related to covering codes and utilized in some of the constructions.AMS Classification: 94B75, 94B25, 94B65Gerzson Kéri - Supported in part by the Hungarian National Research Fund, Grant No. OTKA-T029572.Patric R. J. Östergård - Supported in part by the Academy of Finland, Grants No. 100500 and No. 202315.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

On the Erd s-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

A link between Ramsey numbers for stars and matchings and the Erd s-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e (K_2m−1)→{0, 1,…, m−1} is a mapping of the edges of the complete graph on 2m−1 vertices into {0, 1,…, m−1}, then there exists a star K_1,m in K_2m−1 with edges e₁, e₂,…, e_m such that c(e₁)+c(e₂)++c(e_m)≡0 (mod m). Theorem 8. Let m be an integer. If c : e(K^r_(r+1)m−1)→{0, 1,…, m−1} is a mapping of all the r-subsets of an (r+1)m−1 element set S into {0, 1,…, m−1}, then there are m pairwise disjoint r-subsets Z₁, Z₂,…, Z_m of S such that c(Z₁)+c(Z₂)++c(Z_m)≡0 (mod m).     

Decycling Hypercubes

The decycling number, (G), of a graph G is the least number of vertices of G whose deletion results in an induced subgraph without any cycles. Improved bounds are obtained for the decycling number (Q _n ) of the hypercube Q _n . Further, it is shown that (Q _n )=2 ^{n –1}–A(n,4) if and only if Q _n has a minimum decycling set that consists of pairwise non-adjacent vertices, where A(n,4) denotes the size of a maximum binary code of length n and minimum Hamming distance 4. Research supported by NSERC     

On reducibility of n-ary quasigroups

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order |Σ| if in the relation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. Q is permutably reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n-k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup S is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an n-ary quasigroup Q belongs to {3,…,n-3}, then Q is permutably reducible.     

Two New Classes of Difference Families

A Z-cyclic triplewhist tournament for 4n+1 players, or briefly a TWh(4n+1), is equivalent to a n-set {(a_i, b_i, c_i, d_i) | i=1, …, n} of quadruples partitioning Z_4n+1−{0} with the property that ⁿ_i=1 {±(a_i−c_i), ±(b_i−d_i)}=ⁿ_i=1 {±(a_i−b_i), ±(c_i−d_i)}=ⁿ_i=1 {±(a_i−d_i), ±(b_i−c_i)}=Z_4n+1−{0}. The existence problem for Z-cyclic TWh(p)'s with p a prime has been solved for p1 (mod 16). I. Anderson et al. (1995, Discrete Math.138, 31–41) treated the case of p≡5 (mod 8) while Y. S. Liaw (1996, J. Combin. Des.4, 219–233) and G. McNay (1996, Utilitas Math.49, 191–201) treated the case of p≡9 (mod 16). In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes p≡1 (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all≡1 (mod 4) and distinct from 5, 13, and 17.     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

Optimal binary covering codes of length 2j

The minimum size of a binary covering code of length n and covering radius r is denoted by K(n,r), and codes of this length are called optimal. For j > 0 and n = 2^j, it is known that K(n,1) = 2 · K(n?1,1) = 2^{n ? j}. Say that two binary words of length n form a duo if the Hamming distance between them is 1 or 2. In this paper, it is shown that each optimal binary covering code of length n = 2^j, j > 0, and covering radius 1 is the union of duos in just one way, and that the closed neighborhoods of the duos form a tiling of the set of binary words of length n. Methods of constructing such optimal codes from optimal covering codes of length n ? 1 (that is, perfect single‐error‐correcting codes) are discussed. The paper ends with the construction of an optimal covering code of length 16 that does not contain an extension of any optimal covering code of length 15. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Weak Asymptotics for the Generating Polynomials of the Stirling Numbers of the Second Kind

For the horizontal generating functions P_n(z)=∑ⁿ_k=1 S(n, k) z^k of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Q_n(z)=P_n(nz) there are two main results: an oscillating asymptotic for z(−e, 0) and a uniform asymptotic on every compact subset of \[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.     

Greedily Finding a Dense Subgraph

Given an n-vertex graph with nonnegative edge weights and a positive integer k ≤ n, our goal is to find a k-vertex subgraph with the maximum weight. We study the following greedy algorithm for this problem: repeatedly remove a vertex with the minimum weighted-degree in the currently remaining graph, until exactly k vertices are left. We derive tight bounds on the worst case approximation ratio R of this greedy algorithm: (1/2 + n/2k)² − O(n ^− 1/3) ≤ R ≤ (1/2 + n/2k)² + O(1/n) for k in the range n/3 ≤ k ≤ n and 2(n/k − 1) − O(1/k) ≤ R ≤ 2(n/k − 1) + O(n/k²) for k < n/3. For k = n/2, for example, these bounds are 9/4 ± O(1/n), improving on naive lower and upper bounds of 2 and 4, respectively. The upper bound for general k compares well with currently the best (and much more complicated) approximation algorithm based on semidefinite programming.     

Universal entire functions with gap power series

Let be the family of all compact sets in which have connected complement. For K ε M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior.Suppose that {z_n} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of ₀.If Q has density Δ(Q) = 1 then there exists a universal entire function with lacunary power series

1. (1) (z) = ε^∞_{v = 0 v}Z^v, _v = 0 for v Q, which has for all K ε M the following properties simultaneously

2. (2) the sequence {(Z + Z_n)} is dense in A(K)

3. (3) the sequence { (ZZ_n)} is dense in A(K) if 0 K.

Also a converse result is proved: If is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δ_max(Q) = 1.     

Transformations preserving normality and Wishart-ness

Let R^n×p, (n), Gl(p) and ₊(p) denote respectively the set of n×p matrices, the set of n×n orthogonal matrices, the set of p×p nonsingular matrices and the set of p × p positive definite matrices. In this paper, it is first shown that a bijective and bimeasurable transformation (BBT) g on R^p≡R^p×1 preserving the multivariate normality of N_p(μ, Σ) for fixed μ=μ₁, μ₂ (μ₁≠μ₂) and for all Σ ₊(p) is of the form g(x)=Ax+b a.e. for some (A, b)Gl(p)×R^p. Second, a BBT g on R^n×p preserving the form for certain 's and all Σ ₊(p) is shown to be of the form g(x)=QxA+E a.e. for some (Q, A, E) (n)×Gl(p)×R^n×p. Third, a BBT h on ₊(p) preserving the Wishart-ness of W_p(Σ, m) (m≥p) for all Σ ₊(p) is shown to be of the form h(w)=A′wA a.e. for some AGl(p). Fourth, a BBT k(x, w)=(k₁(x, w), k₂(x, w)) on R^n×p× ₊(p) which preserves the form of for certain 's and all Σ ₊(p) is shown to be of the form k(x, w)=(QxA+E, A′wA) a.e. for some (Q, A, E) (n)×Gl(p)×R^n×p.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H₁H_n in an ideal incompressible fluid on a rotating sphere is analytically studied (H_n is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M₋ⁿ, M₊ⁿ, H_n and M₀ⁿ−H_n depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M₋ⁿ and M₊ⁿ due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace H_n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H_n, the factor norm controls the perturbation part orthogonal to H_n. It is shown that in the set M₋ⁿ (χ(t)<n(n+1)), any nonzonal RH wave of subspace H₁H_n (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M₀ⁿ−H_n. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M₊ⁿ of small-scale perturbations (χ(t)>n(n+1)) is still open problem.     

Oscillations of Higher Order Neutral Differential Equations with Variable Coefficients

We obtain sufficient conditions for the oscillation of all solutions of the higher order neutral differential equation dⁿ/d^m[y(t) + P(t) y(t - μ)] + Q(t) y(t ?σ) = 0, t ≧ t₀ where n ≧ 1, P ? C[t₀, ∞), R ], Q ? C[t₀, ∞), R ] and τ, μ ? R ⁺. Our results extend and improve several known results in the literature.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Error bounds for least squares approximation by polynomials

Let f ε Cⁿ⁺¹[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials q_k associated with a distribution dα on [−1, 1]. It is shown that if q_n+1/q_n max(q_n+1(1)/q_n(1), −q_n+1(−1)/q_n(−1)), then f − H[f] f^{n + 1} · q_n+1/q_{n + 1}^{(n + 1)}, where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)^α (1 + t)^β dt, α, β > −1, the condition on q_n+1/q_n is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.     

Second-order linearity of the general signed-rank statistic

Let X₁,…, X_n be i.i.d. random variables symmetric about zero. Let R_i(t) be the rank of |X_i − tn^−1/2| among |X₁ − tn^−1/2|,…, |X_n − tn^−1/2| and T_n(t) = Σ_{i = 1}ⁿφ((n + 1)⁻¹R_i(t))sign(X_i − tn^−1/2). We show that there exists a sequence of random variables V_n such that sup_{0 ≤ t ≤ 1} |T_n(t) − T_n(0) − tV_n| → 0 in probability, as n → ∞. V_n is asymptotically normal.     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature

We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of constant mean curvature in ann-dimensional sphereB _R, 1>R>R ₀ ⁽ⁿ⁾ , (R ₀ ⁽ⁿ⁾ being a constant depending only onn), without imposing boundary conditions or bounds of any sort.

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Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB _R, 1>R>R ₀ ⁽ⁿ⁾, (R ₀ ⁽ⁿ⁾ essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.