共查询到20条相似文献,搜索用时 109 毫秒
1.
Let fk(n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f2t ? 1(n) ? f2t(n + 1) ? with equalities holding iff there exists a Steiner-system S(t, 2t ? 1, n). The bounds are approximately best possile for k ? 6 and of correct order of magnitude for k >/ 7, as well, even if the corresponding Steiner-systems do not exist.Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i.e., the nonuniform case). 相似文献
2.
Robert Donaghey 《Journal of Combinatorial Theory, Series A》1976,21(2):155-163
This paper treats the class of sequences {an} that satisfy the recurrence relation between the odd and even terms of {an} that involves the coefficients of tan(t), namely A combinatorial setting is then provided to elucidate the appearance of the tangent coefficients in this equation. 相似文献
3.
Let Ω be a simply connected domain in the complex plane, and , the space of functions which are defined and analytic on , if K is the operator on elements defined in terms of the kernels ki(t, s, a1, …, an) in by is the identity operator on , then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on defined in terms of a kernel w(t, s, a1, …, an) in by Wu = ∝antw(t, s, a1, …, an) u(s, a1, …, an) ds. ΠW is the operator; ΠWu = ∝an ? 1w(t, s, a1, …, an) u(s, a1, …, an) ds. ΠK is the operator; ΠKu = ∑i = 1n ? 1 ∝aitki(t, s, a1, …, an) ds + ∝an ? 1tkn(t, s, a1, …, an) u(s, a1, …, an) ds. The operator M is of the form m(t, a1, …, an)I, where and maps elements of into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator on functions u in , by . A determinant of the operator is defined as an element of . This is mapped into by setting an + 1 = t to give m(t, a1, …, an). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels ki defining K are separable in the sense that ki(t, s, a1, …, an) = Pi(t, a1, …, an) Qi(s, a1, …, an), where Pi is a 1 × pi matrix and Qi is a pi × 1 matrix, each with elements in , explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case. 相似文献
4.
Tom Brylawski 《Discrete Mathematics》1977,18(3):243-252
In “The Slimmest Geometric Lattices” (Trans. Amer. Math. Soc.). Dowling and Wilson showed that if G is a combinatorial geometry of rank r(G) = n, and if X(G) = Σμ(0, x)λr ? r(x) = Σ (?1)r ? kWkλk is the characteristic polynomial of G, then Thus γ(G) ? 2r ? 1 (n+2), where γ(G) = Σwk. In this paper we sharpen these lower bounds for connected geometries: If G is connected, r(G) ? 3, and n(G) ? 2 ((r, n) ≠ (4,3)), then |μ| ? (r? 1)n; and γ ? (2r ? 1 ? 1)(2n + 2). These bounds are all achieved for the parallel connection of an r-point circuit and an (n + 1)point line. If G is any series-parallel network, , and then . Further, if β is the Crapo invariant, then β(G) ? max(1, n ? r + 2). This lower bound is achieved by the parallel connection of a line and a maximal size series-parallel network. 相似文献
5.
M.M Dodson 《Journal of Number Theory》1973,5(4):287-292
Let θ(k, pn) be the least s such that the congruence (mod pn) has a nontrivial solution. It is shown that if k is sufficiently large and divisible by p but not by p ? 1, then . We also obtain the average order of θ(k), the least s such that the above congruence has a nontrivial solution for every prime p and every positive integer n. 相似文献
6.
Paul Erdős Norbert Sauer Jonathan Schaer Joel Spencer 《Journal of Combinatorial Theory, Series B》1975,18(2):180-183
The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that . 相似文献
7.
Tomas Schonbek 《Journal of Differential Equations》1985,56(2):290-296
New and more elementary proofs are given of two results due to W. Littman: (1) Let . The estimate cannot hold for all u?C0∞(Q), Q a cube in , some constant C. (2) Let n ? 2, p ≠ 2. The estimate cannot hold for all C∞ solutions of the wave equation □u = 0 in ; all t ?; some function C: → . 相似文献
8.
R.E. Stong 《Topology and its Applications》1982,13(1):103-113
This note calculates the height of the first Stiefel-Whitney class in the cohomology of the real Grassmannians and determines the length of the longest nontrivial cup-product in with k?4. 相似文献
9.
Shlomo Moran 《Journal of Combinatorial Theory, Series B》1984,37(2):113-141
Let V be a set of n points in Rk. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) lk(n) are the extremal values of l(V) defined by lk(n)=max{l(V)|V∈Vnk}, where Vnk={V|V?Rk,|V|=n, d(V)=1}. A set V∈Vnk is “longest” if l(V)=lk(n). In this paper, first some geometrical properties of longest sets in R2 are studied which are used to obtain l2(n) for small n′s, and then asymptotic bounds on lk(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δk(n)=max{δ(V)|V∈Vnk}. It is easily observed that . Hence, exists. It is shown that for all , and hence, for all . For k=2, this implies that , which generalizes an observation of Fejes-Toth that . It is also shown that . The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that . 相似文献
10.
Joel M Cohen 《Journal of Functional Analysis》1982,48(3):301-309
Let G be a group and g1,…, gt a set of generators. There are approximately (2t ? 1)n reduced words in g1,…, gt, of length ?n. Let be the number of those which represent 1G. We show that exists. Clearly 1 ? γ ? 2t ? 1. is the cogrowth. 0 ? η ? 1. In fact . The entropic dimension of G is shown to be 1 ? η. It is then proved that d(G) = 1 if and only if G is free on g1,…, gt and d(G) = 0 if and only if G is amenable. 相似文献
11.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
12.
We calculate some size Ramsey numbers involving stars. For example we prove that for t ? k ? 2 and n sufficiently large the size Ramsey number. 相似文献
13.
Let be a Dirichlet form in , where Ω is an open subset of n, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let k be a Dirichlet form on some k-dimensional submanifold of Ω. The paper is devoted to the study of the closability of the forms E with domain and defined by: ki where 1 ? kp < ? < n, and where , gki denote restrictions of ?, g in to . Conditions are given for E to be closable if, for each i = 1,…, p, one has ki = n ? i. Other conditions are given for E to be nonclosable if, for some i, ki < n ? i. 相似文献
14.
Norman Levinson 《Journal of Mathematical Analysis and Applications》1973,43(1):123-127
For s = σ + it, σ > 1, and integer k ? 1. In a previous paper, Ω results for , where obtained in an elementary way by choosing N = N(k, c) so that the size of the single term dk(N) gave Ω results slightly better than existing ones. Here the method will be shown to give the same results for Re ζ(c + it). 相似文献
15.
Larry W. Cusick 《Topology and its Applications》1985,21(1):9-18
We show that if X is a finite CW-complex admitting a fixed point free involution then there is a singly graded spectral sequence with and . As an application we prove that for any n > 0 there is a natural number k(n) such that if n > k(n) and X is a homotopy , then X will not admit a fixed point free involution. 相似文献
16.
Juan C. Peral 《Journal of Functional Analysis》1980,36(1):114-145
Let u(x, t) be the solution of utt ? Δxu = 0 with initial conditions . Consider the linear operator . (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in Lp if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for . (b) If n = 2k ? 1, the result is valid for . This result are sharp in the sense that for p such that we prove the existence of in such a way that . Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers and finally we get that the convolution against the kernel is bounded in H1. 相似文献
17.
J Bustoz 《Journal of Mathematical Analysis and Applications》1981,79(1):71-79
It is known that the classical orthogonal polynomials satisfy inequalities of the form Un2(x) ? Un + 1(x) Un ? 1(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if Pnλ(x) and Lnα(x) are the ultraspherical and Laguerre polynomials and . We also prove the inequality is a positive constant depending on α and β. 相似文献
18.
Ludwig Arnold 《Linear algebra and its applications》1976,13(3):185-199
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let , put (bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally (empirical distribution function of the eigenvalues of . Suppose (i) , (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then ,where W is absolutely continuous with (semicircle) density 相似文献
19.
D Gaier 《Journal of Mathematical Analysis and Applications》1979,70(1):236-239
Let ? be defined on Tr and have an absolutely convergent Fourier series . Set . In this paper the problem of determining the limit of , as n → ∞, is studied. 相似文献
20.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献