More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.     

Periodicities of partition functions and stirling numbers modulo p

If p(n, k) is the number of partitions of n into parts ≤k, then the sequence {p(k, k), p(k + 1, k),…} is periodic modulo a prime p. We find the minimum period Q = Q(k, p) of this sequence. More generally, we find the minimum period, modulo p, of {p(n; T)}_{n ≥ 0}, the number of partitions of n whose parts all lie in a fixed finite set T of positive integers. We find the minimum period, modulo p, of {S(k, k), S(k + 1, k),…}, where these are the Stirling numbers of the second kind. Some related congruences are proved. The methods involve the use of cyclotomic polynomials over Z_p[x].     

Uniform and mean approximation by certain weighted polynomials,with applications

LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa _n =a _n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP _n(x) of degree at mostn, the sup norm ofP _n(x)W(a _n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p _n (x)W(a _n x)} ₁ ^∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p _n (x)W(a _n x)} ₁ ^∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL _p analogue for 0W(x)=exp(?|x| ^α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| ^p exp(?|x| ^α ),α > 0,p > ?1.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

Sums over numbers with restricted prime factors

For k a non-negative integer, let P_k(n) denote the kth largest prime factor of n where P₀(n) = +∞ and if the number of prime factors of n is less than k, then P_k(n) = 1. We shall study the asymptotic behavior of the sum Ψ_k(x, y; g) = Σ_{1 ≤ n ≤ x, Pk(n) ≤ y}g(n), where g(n) is an arithmetic function satisfying certain general conditions regarding its behavior on primes. The special case where g(n) = μ(n), the Möbius function, is discussed as an application.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

Stirling polynomials

An investigation is made of the polynomials f_k(n) = S(n + k, n) and g_k(n) = (?1)^ks(n, n ? k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 ? x)^2k+1Σ_n=0^∞f_k(n)xⁿ analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.     

The general divisor problem

Let Δ_k(x) = Δ(a₁, …, a_k; x) be the error term in the asymptotic formula for the summatory function of the general divisor function d(a₁, …, a_k; n), which is generated by ζ(a₁s) … ζ(a_ks) (1 ≤ a₁ ≤ … ≤ a_k are fixed integers). Precise omega results for the mean square integral ∫₁^x Δ_k²(x) dx are given, with applications to some specific divisor problems.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

On a class of similarity solutions of the porous media equation,III

It is shown how existence questions for general multiparameter eigenvalue problems can be treated quite simply using degree theory. The equations to be solved are W_n(λ)x_n = 0 ≠ x_n, n = 1, 2,…, k, where λ ? $\text{R}$^k and each W_n(λ) is a self-adjoint linear operator on a Hilbert space H_n. The W_n, which may be unbounded, depend continuously on λ in a suitable sense. A coercivity condition for large ∥ λ ∥ is used, and is shown to be equivalent, in the “linear” case, to a standard determinantal definiteness condition.     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.     

Oscillations of the system of linear nonautonomous difference equations

We consider the linear nonautonomous system of difference equations x_n+1−x_n+P(n)x_n−k=0, n=0,1,2,… , where k∈Z, P(n)∈R^rxr. We obtain sufficient conditions for the system to be oscillatory. The conditions based on the eigenvalues of the matrix coefficients of the system.     

Равносходимость разложений в кратный ряд и интеграл фурье, «прямоугольные частичные суммы» которых рассматриваются по некоторой подпоследовательности

In this paper we investigate the problem of the equiconvergence on T ^N = [-π, π) ^N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L _p (T ^N ) and g ∈ L _p (? ^N ), where p > 1, N ≥ 3, g(x) = f(x) on T ^N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– _n (x; f) and J _α(x; g), respectively, have indices n ∈ ? ^N and α ∈ ? ^N (n _j = [α _j ], j = 1,...,N, [t] is the integer part of t ∈ ?¹), in those certain components are the elements of “lacunary sequences”.     

On the Hamilton-Jacobi equation and infimal convolution in the framework of Sobolev-functions

We study the regularity properties of the Hamilton-Jacobi flow equation and infimal convolution in the case where the initial datum function is continuous and lies in a given Sobolev-space W ^1,p (? ⁿ ). We prove that under suitable assumptions it holds for solutions w(x, t) that D _x w(·, t) → Du(·) in L ^p (? ⁿ ) as t → 0. Moreover, we construct examples showing that our results are essentially optimal.     

On shortest crucial words avoiding abelian powers

Let k≥2 be an integer. An abeliankth power is a word of the form X₁X₂?X_k where X_i is a permutation of X₁ for 2≤i≤k. A word W is said to be crucial with respect to abelian kth powers if W avoids abelian kth powers, but Wx ends with an abelian kth power for any letter x occurring in W.Evdokimov and Kitaev (2004) [2] have shown that the shortest length of a crucial word on n letters avoiding abelian squares is 4n−7 for n≥3. Furthermore, Glen et al. (2009) [3] proved that this length for abelian cubes is 9n−13 for n≥5. They have also conjectured that for any k≥4 and sufficiently large n, the shortest length of a crucial word on n letters avoiding abelian kth powers, denoted by ?_k(n), is k²n−(k²+k+1). This is currently the best known upper bound for ?_k(n), and the best known lower bound, provided in Glen et al., is 3kn−(4k+1) for n≥5 and k≥4. In this note, we improve this lower bound by proving that for n≥2k−1, ?_k(n)≥k²n−(2k³−3k²+k+1); thus showing that the aforementioned conjecture is true asymptotically (up to a constant term) for growing n.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.