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1.
In this paper we continue our investigation on “Extremal problems under dimension constraint” introduced in [2]. Let E(n, k) be the set of (0,1)-vectors in ? n with k one's. Given 1 ≤ m, w ≤ n let X ? E(n, m) satisfy span (X) ∩ E(n, w) = ?. How big can |X| be? This is the main problem studied in this paper. We solve this problem for all parameters 1 ≤ m, w ≤ n and n > n 0(m, w). 相似文献
2.
《Journal de Mathématiques Pures et Appliquées》1999,78(2):177-202
Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions Wmi+ni(z)Rmi, ni(z)i = 0∞, where Rmi, ni(z) = Pmi(z)/Qni(z) with deg Pmi ≤ mi and deg Qni ≤ ni for all i ≥ 0 and where mi + ni → ∞ as i → ∞ such that lim mi/[mi + ni] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included. 相似文献
3.
Albert T Lundell 《Journal of Number Theory》1978,10(1):35-54
In this paper we calculate which prime powers ps divide Δn, m = g.c.d.{k! S(n, k)|m ≤ k ≤ n} for s < p. Here S(n, k) is a Stirling number of the second kind. 相似文献
4.
Craig M Cordes 《Journal of Number Theory》1973,5(6):537-540
Let A = (aij) be an n × m matrix with aij ∈ K, a field of characteristic not 2, where Σi=1naij2 = e, 1 ≤ j ≤ m, and Σi=1naijaij′ = 0 for j ≠ j′. The question then is when is it possible to extend A, by adding columns, to obtain a matrix with orthogonal columns of the same norm. The question is answered for n ? 7 ≤ m ≤ n as well as for more general cases. Complete solutions are given for global and local fields, the answer depending on what congruence class modulo 4 n belongs to and how few squares are needed to sum to e. 相似文献
5.
We obtain a finite-dimensional Perron effect of change of values λ 1 ≤ … ≤ λ n < 0 of all arbitrarily specified negative characteristic exponents of the n-dimensional system of linear approximation with infinitely differentiable bounded coefficients to arbitrarily specified, arranged in ascending order, values β k ≥ λ k , k = 1, …, n, of characteristic exponents of all nontrivial solutions of an n-dimensional nonlinear differential system with an infinitely differentiable perturbation of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. Each value β k is realized by all nontrivial solutions of the perturbed system issuing from the difference R k |R k?1 of embedded subspaces R 1 ? R 2 ? … ? R n . 相似文献
6.
F. M. Korkmasov 《Vestnik St. Petersburg University: Mathematics》2007,40(2):138-151
It is shown that if P m α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ mn α,β (x, y)} m,n=0 r = {P m α,β (x)P n α,β (y)} m, n=0 r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω N×N = ?ub;(x i , y i ) i,j=0 N , where x i and y i are the zeros of the Jacobi polynomial P n α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]2, we construct the discrete partial Fourier-Jacobi sums of the rectangular type S m, n, N α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S m, n, N α,β ∥ of the discrete sums S m, n, N α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) q + 1/2), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S m, n, N α,β (f; x, y) are considered. 相似文献
7.
In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators. 相似文献
8.
A. V. Men’shov 《Siberian Mathematical Journal》2014,55(3):440-450
We study the solvability of random systems of equations on the free abelian group ? m of rank m. Denote by SAT(? m , k, n) and \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) the sets of all systems of n equations of k unknowns in ? m satisfiable in ? m and ? m respectively. We prove that the asymptotic density \(\rho \left( {SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)} \right)\) of the set \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) equals 1 for n ≤ k and 0 for n > k. As regards, SAT(? m , k, n) for n < k, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between (Π j=k?n+1 k ζ(j))?1 and \(\left( {\tfrac{{\zeta (k + m)}} {{\zeta (k)}}} \right)^n\) , where ξ(s) is the Riemann zeta function. For n ≤ k, a connection is established between the asymptotic density of SAT(? m , k, n) and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of SAT(? m , n, n). We prove that ρ(SAT(? m , k, n)) = 0 for n > k. 相似文献
9.
For a double array of independent random elements {Vmn,m ≥ 1,n ≥ 1} in a real separable Banach space,conditions are provided under which the weak and strong laws of large numbers for the double sums mi=1 nj=1Vij,m ≥ 1,n ≥ 1 are equivalent.Both the identically distributed and the nonidentically distributed cases are treated.In the main theorems,no assumptions are made concerning the geometry of the underlying Banach space.These theorems are applied to obtain Kolmogorov,Brunk–Chung,and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in Rademacher type p(1 ≤ p ≤ 2) Banach spaces. 相似文献
10.
Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes Bpqsm(Tk) with respect to the multiple trigonometric system T(k) in the metric of Lr(Tk) for a number of relations between the parameters s, p, q, r, and m (s = (s1,..., sn) ∈ R+n, 1 ≤ p, q, r ≤ ∞, m = (m1,..., mn) ∈ Nn, k = m1 +... + mn). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates. 相似文献
11.
Linda Lesniak-Foster 《Journal of Combinatorial Theory, Series B》1977,22(3):263-273
With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers m ≥ 2, the mth iterated line graph Lm(G) of G is defined to be L(Lm-1(G)). A graph G of order p ≥ 3 is n-Hamiltonian, 0 ≤ n ≤ p ? 3, if the removal of any k vertices, 0 ≤ k ≤ n, results in a Hamiltonian graph. It is shown that if G is a connected graph with δ(G) ≥ 3, where δ(G) denotes the minimum degree of G, then L2(G) is (δ(G) ? 3)-Hamiltonian. Furthermore, if G is 2-connected and δ(G) ≥ 4, then L2(G) is (2δ(G) ? 4)-Hamiltonian. For a connected graph G which is neither a path, a cycle, nor the graph K(1, 3) and for any positive integer n, the existence of an integer k such that Lm(G) is n-Hamiltonian for every m ≥ k is exhibited. Then, for the special case n = 1, bounds on (and, in some cases, the exact value of) the smallest such integer k are determined for various classes of graphs. 相似文献
12.
Let n(k, l,m), k ≤ l ≤ m, be the smallest integer such that any finite planar point set which has at least n(k, l,m) points in general position, contains an empty convex k-hole, an empty convex l-hole and an empty convex m-hole, in which the three holes are pairwise disjoint. In this article, we prove that n(4, 4, 5) ≤ 16. 相似文献
13.
Fix integers m, n such that 1 ≤ m ≤ n ? 3. Let X ? Pn be an integral non-degenerate m-dimensional variety. Assume either char(K) = 0 or char(K) > deg(X). Here we prove that all general 0-dimensional sections of X containing a tangent vector to a smooth point of X are protectively equivalent if and only if n ? m + 1 ≤ deg(X) ≤ n ? m + 2. 相似文献
14.
Order-sharp estimates are established for the best N-term approximations of functions in the classes $B_{pq}^{sm} (\mathbb{T}^k )$ and $L_{pq}^{sm} (\mathbb{T}^k )$ of Nikol’skii-Besov and Lizorkin-Triebel types with respect to the multiple system of Meyer wavelets in the metric of $L_r (\mathbb{T}^k )$ for various relations between the parameters s, p, q, r, and m (s = (s 1, ..., s n ) ∈ ? + n , 1 ≤ p, q, r ≤ ∞, m = (m 1, ..., m n ) ∈ ? n , and k = m 1 + ... + m n ). The proof of upper estimates is based on variants of the so-called greedy algorithms. 相似文献
15.
V. N. Karpushkin 《Moscow University Mathematics Bulletin》2014,69(3):128-131
Exact order uniform estimates of oscillatory integrals with monomial phase are obtained. These results are close to the hypothesis of V. I. Arnold on uniform estimates of oscillatory integrals. Namely, for absolute values of oscillatory integrals, estimates of order ρ ?1/k ln n?1 τ uniform with respect to phase and amplitude are derived for every sufficiently small perturbation of phase (i.e., the monomial \(x_1^{m_1 } \ldots x_n^{m_n }\) , m j ≤ k, 1 ≤ k, is perturbed by monomials \(x_1^{s_1 } \ldots x_n^{s_n }\) , where s j ≤ k, 1 ≤ j ≤ n), and for each amplitude φ ∈ C 0 2 (R n ), n > 0. In the case |m| < nk the upper uniform estimate with the same perturbation and the same amplitude has the order τ ?1/k ln n?2 τ. The estimate by order τ ?1/k ln n?2 τ was proved in the case when the amplitude vanishes at the origin. In the case k = 1, a uniform estimate of order τ ?1ln n?2 τ is valid. This implies a uniform estimate for a polynomial phase. Previously the upper estimate of an oscillatory integral by the value 32 n τ ?1/k ln n?1(τ+2) was known for the amplitude being the characteristic function of a cube and the same phase. 相似文献
16.
Patrick Morton 《Journal of Number Theory》1980,12(1):122-127
A basis of eigenvectors is given for the matrix = (e2πimn/q), (1 ≤ m, n ≤ q). The eigenvectors arise from the characters on the reduced residue class group (mod q). 相似文献
17.
Horst Alzer 《Periodica Mathematica Hungarica》2013,67(2):243-249
We prove that the inequality $\pi ^2 \left( m \right) + \pi ^2 \left( n \right) \leqslant \tfrac{5} {4}\pi ^2 \left( {m + n} \right)$ holds for all integers m, n ≥ 2. The constant factor 5/4 is sharp. This complements a result of Panaitopol, who showed in 2001 that ½ π 2(m+ n) ≤ π 2(m) + π 2(n) is valid for all m, n ≥ 2. Here, as usual, π(n) denotes the number of primes not exceeding n. 相似文献
18.
For a connected simple graph G, the eccentricity ec(v) of a vertex v in G is the distance from v to a vertex farthest from v, and d(v) denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξc(G), is defined as v∈V(G)d(v)ec(v). In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(n ≤ m ≤ n + 4), and propose a conjecture on the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(m ≥ n + 5). 相似文献
19.
In their papers (Technical Report CS-TR 50, University of Central Florida, 1980; J. Combin. Theory Ser. B32 (1982), 90–94) Brigham and Dutton introduce the notion of (n : i)-chromatic numbers of a graph, a generalization of Stahl's nth chromatic numbers (J. Combin. Theory Ser. B20, (1976), 185–203). The (n : i)-chromatic number of a graph G, denoted by χni(G), is the smallest integer m such that each vertex of G can be colored with a set of n colors in {1, 2,…, m} in such a way that any two adjacent vertices have exactly i colors in common. Brigham and Dutton conjecture at the end of loc cit that for all integers n and i with 0 ≤ i ≤ n ? 1, and for every graph G, χni+1(G) ≤ χni(G). We prove this conjecture in some special cases and disprove it in the general case. 相似文献
20.
For Pm ∈ ?[z1, …, zn], homogeneous of degree m we investigate when the graph of Pm in ?n+1 satisfies the Phragmén-Lindelöf condition PL(?n+1, log), or equivalently, when the operator $i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)$ admits a continuous solution operator on C∞(?n+1). This is shown to happen if the varieties V+- ? {z ∈ ?n: Pm(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V+- and u(x) ≤ 0 on V+- ∩ ?n also satisfies u(z) A¦ Im z¦ on V+-. Necessary as well as sufficient conditions for V+- to satisfy (SPL) are derived and several examples are given. 相似文献