Let X1, X2, … be a sequence of independent and identically distributed random variables with mean zero such that the common distribution function belongs to the domain of attraction of a stable law Gα,β with 1<α<2 and β=1 or α=2. If Sn=X1+…Xn and N(ξ)=min{k:Sk>ξ}, ξ>0, then it is shown that , 0<t<1, converges weakly under the Skorohod J1-topology to a stable subordinator of index , where B1(n) depends on the norming constant for Sn. 相似文献
Let A be an n-square normal matrix over , and Qm, n be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α,β∈Qm, n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…,m} write z.sfnc;α∩β|=k if there exists a rearrangement of 1,…,m, say i1,…,ik, ik+1,…,im, such that α(ij)=β(ij), j=1,…,k, and {α(ik+1),…,α(im)};∩{β(ik+1),…,β(im)}=ø. Let be the group of n-square unitary matrices. Define the nonnegative number , where |α∩β|=k. Theorem 1 establishes a bound for ?k(A), 0?k<m?1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n?2m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that . 相似文献
If is a form of odd degree k with real coefficients in s variables where s ≥ c1(k), then there are integers x1,… xs not all zero, with |(x1,… xs)| < 1. 相似文献
The length ln of a longest common subsequence before time n sequences (B11, B12, …) (B21, B22, …) is the cardinality of the largest increasing set of pairs of integers {(j1α, j2α)} such that ?1?α?ln, (B1j1α=B2j2α). If B1 and B2 are independent random sequences with co-ordinates i.i.d. uniform on {1, 2, …, k}, it follows from Kingman's subadditive ergodic theorem that the ratio ln/n converges to a constant ck a.s. A method of deriving lower bounds for the constants ck is given, the bounds obtained improving known lower bounds, for k>2. The rate of decrease of ck with k is shown to be no faster than 1/√k, contrasting with P{B1i?B2i}=1/k. Finally, an alternative method of deriving lower bounds is given and used to improve the lower bound for c2. 相似文献
Let k be an odd positive integer. Davenport and Lewis have shown that the equations with integer coefficients, have a nontrivial solution in integers x1,…, xN provided that Here it is shown that for any ? > 0 and k > k0(?) the equations have a nontrivial solution provided that 相似文献
Let k, λ, and υ be positive integers. A perfect cyclic design in the class PD(υ, k, λ) consists of a pair (Q, B) where Q is a set with |Q| = υ and B is a collection of cyclically ordered k-subsets of Q such that every ordered pair of elements of Q are t apart in exactly λ of the blocks for t = 1, 2, 3,…, k?1. To clarify matters the block [a1, a2, …, ak] has cyclic order a1 < a2 < a3 … < ak < a1 and ai and ai+1 are said to be t apart in the block where i + t is taken mod k. In this paper we are interested only in the cases where λ = 1 and υ ≡ 1 mod k. Such a design has blocks. If the blocks can be partitioned into υ sets containing pairwise disjoint blocks the design is said to be resolvable, and any such partitioning of the blocks is said to be a resolution. Any set of pairwise disjoint blocks together with a singleton consisting of the only element not in one of the blocks is called a parallel class. Any resolution of a design yields υ parallel classes. We denote by RPD(υ, k, 1) the class of all resolvable perfect cyclic designs with parameters υ, k, and 1. Associated with any resolvable perfect cyclic design is an orthogonal array with k + 1 columns and υ rows with an interesting conjugacy property. Also a design in the class RPD(υ, k, 1) is constructed for all sufficiently large υ with υ ≡ 1 mod k. 相似文献
Let A be an n × n normal matrix over , and Qm, n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Qm, n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i1,…, ik, ik+1,…, im, such that α(ij) = β(ij), i = 1,…, k, and {α(ik+1),…, α(im) } ∩ {β(ik+1),…, β(im) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let n be the group of n × n unitary matrices. Define the nonnegative number where | α ∩ β| = k. It is proved that Let A be semidefinite hermitian. We conjecture that ρ0(A) ? ρ1(A) ? ··· ? ρm(A). These inequalities have been tested by machine calculations. 相似文献
Let Rk(n) denote the number of ways of representing the integers not exceeding n as the sum of k members of a given sequence of nonnegative integers. Suppose that , and R. C. Vaughan has shown that the relation Rk(n) = G(n) + o(nδ log?ξn) as n → +∞ is impossible when G(n) is a linear combination of powers of n and the dominant term of G(n) is cnβ, c > 0. P. T. Bateman, for the case k = 2, has shown that similar results can be obtained when G(n) is a convex or concave function. In this paper, we combine the ideas of Vaughan and Bateman to extend the theorems stated above to functions whose fractional differences are of one sign for large n. Vaughan's theorem is included in ours, and in the case we show that a better choice of parameter improves Vaughan's result by enabling us to drop the power of log n from the estimate of the error term. 相似文献
Let a1 < a2 < … be a sequence of positive integers such that no ak is a sum of distinct other terms. Erdös conjectured that if a1 ≥ n, then , where, ?n → 0 as n → ∞. This result, which is the best possible, is established in this paper. 相似文献
Let n ? k ? t be positive integers, and let Ω be a set of n elements. Let C(n, k, t) denote the number of k-tuples of Ω in a minimal system of k-tuples such that every t-tuple is contained in at least one k-tuple of the system. C(n, k, t) has been determined in all cases for which [W. H. Mills, Ars Combinatoria8 (1979), 199–315]. C(n, k, t) is determined in the case . 相似文献
A successivity in a linear order is a pair of elements with no other elements between them. A recursive linear order with recursive successivities is recursively categorical if every recursive linear order with recursive successivities isomorphic to is in fact recursively isomorphic to . We characterize those recursive linear orders with recursive successivities that are recursively categorical as precisely those with order type k1+g1+k2+g2+…+gn-1+kn where each kn is a finite order type, non-empty for i?{2,…,n-1} and each gi is an order type from among {ω,ω*,ω+ω*}∪{k·η:k<ω}. 相似文献
The following results are proved: Let A = (aij) be an n × n complex matrix, n ? 2, and let k be a fixed integer, 1 ? k ? n ? 1.(1) If there exists a monotonic G-function f = (f1,…,fn) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have then the rank of A is ? n ? k + 1. (2) If A is irreducible and if there exists a G-function f = (f1,…,fn) such that for every subset of S of {1,…,n} consisting of k + 1 elements we have then the rank of A is ? n ? k + 1 if k ? 2, n ? 3; it is ? n ? 1 if k = 1. 相似文献
Let S be a set of n elements, and k a fixed positive integer . Katona's problem is to determine the smallest integer m for which there exists a family = {A1, …, Am} of subsets of S with the following property: |i| ? k (i = 1, …, m), and for any ordered pair xi, xi ∈ S (i ≠ j) there is A1 ∈ such that xi ∈ A1, xj ? A1. It is given in this note that . 相似文献
Let T be a rooted tree structure with n nodes a1,…,an. A function f: {a1,…,an} into {1 < ? < k} is called monotone if whenever ai is a son of aj, then f(ai) ≥ f(aj). The average number of monotone bijections is determined for several classes of tree structures. If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ??nn? (n → ∞) are obtained for several classes of tree structures. 相似文献
Let X = [1, n] be a finite set of cardinality n and let be a family of k-subsets of X. Suppose that any two members of intersect in at least t elements and for some given positive constant c, every element of X is contained in less than c || members of . How large || can be and which are the extremal families were problems posed by Erdös, Rothschild, and Szemerédi. In this paper we answer some of these questions for n > n0(k, c). One of the results is the following: let . Then whenever is an extremal family we can find a 7-3 Steiner system such that consists exactly of those k-subsets of X which contain some member of . 相似文献
Let be a UHF-algebra of Glimm type n∞, and {αg: g?G} a strongly continuous group of 1-automorphisms of product type on , for G compact. Let α be the C1-subalgebra of fixed elements of . We show that any extremal normalized trace on α arises as the restriction of a symmetric product state ? on of the form ? = ?k?1 ω. As an example we classify the extremal traces on α for the case G = SU(n), αg = ?k ? 1 Ad(g). 相似文献
Let Δ(α + β) = |Hλ2?r+1| where Hr is the complete symmetric function in (α1 + β1), (α2 + β2), …, (αn + βn). It is proved that Δ(α + β) ? Δ(α) + Δ(β). This inequality is generalised for certain symmetric functions defined by Littlewood. Let . Then we prove that Ω(α + β) ? Ω(α) + Ω(β). Here λ1, λ2, λ3, …, λn is a partition such that λn > λn?1 > ··· > λ2 > λ1. 相似文献
It is shown that if satisfies , where σk(A) denotes the sum of all kth order subpermanent of A, then Per[λJn+(1?λ)A] is strictly decreasing in the interval 0<λ<1. 相似文献
Let k1, k2,…, kn be given integers, 1 ? k1 ? k2 ? … ? kn, and let S be the set of vectors x = (x1,…, xn) with integral coefficients satisfying 0 ? xi ? ki, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities xi ? yi, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k1 + k2 + … + kn and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h1, h2,…, hn) which satisfy h1 + h2 + … + hn = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if , if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if . 相似文献
Let k, l denote positive integers with (k, l) = 1. Denote by p(k, l) the least prime p ≡ l(mod k). Let P(k) be the maximum value of p(k, l) for all l. We show , where γ is Euler's constant and ? is Euler's function. We also show for almost all k. 相似文献