共查询到20条相似文献,搜索用时 15 毫秒
1.
G.F Clements 《Journal of Combinatorial Theory, Series A》1977,22(3):368-371
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
2.
Cai Mao-cheng 《Discrete Mathematics》1984,48(1):121-123
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
3.
Jorge L.C Sanz Thomas S Huang 《Journal of Mathematical Analysis and Applications》1984,104(1):302-308
In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z1…,zn), zi ? , i=1, …, n, is uniquely determined from the magnitude of f(x1…,xn): | f(x1…,xn)|, xi ? , i=1,…, n, except for (1) linear shifts: i(α1z1+…+αn2n+β), β, αi?, i=1,…, n; and (2) conjugation: . 相似文献
4.
Frank Markham Brown 《Discrete Applied Mathematics》1982,4(2):87-96
Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation over a finite Boolean algebra B may be expressed as the union of a collection of subsets of Bn, each of the form {(x1, …, xn) | ai≤xi≤bi, ai?B, bi?B, i = 1, …, n}. We refer to such subsets of Bn as segments and to the collection as a segmental cover of K. We show that is consistent if and only if ? can be expressed by one of a class of sum-of-products expressions which we call segmental formulas. The object of this paper is to relate segmental covers of K to segmental formulas for ?. 相似文献
5.
Daniel J. Madden 《Journal of Number Theory》1978,10(3):303-323
If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation generates a tower of extensions through where . In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki?1(yi); yip ? yi = Bi, where, as a divisor in Ki?1, Bi has the form . In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants. 相似文献
6.
Lee K. Jones 《Discrete Mathematics》1976,15(1):107-108
Nous donnons une généralisation et une démonstration très courte d'un théorème de Kleitman qui dit: Pour toute paire d'idéaux , (β) d'éléments dans le produit cartésien de k ensembles totalement ordonnés P = [1, 2, … n1] ? … ? [1, 2, … nk], nous avons (). ( ou en langage probabiliste . 相似文献
7.
G.F Clements 《Journal of Combinatorial Theory, Series A》1984,37(1):91-97
Let kn ? kn?1 ? … ? k1 be positive integers and let () denote the coefficient of xi in . For given integers l, m, where 1 ? l ? kn + kn?1 + … + k1 and , it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which . Moreover, any m l-subsets of a multiset with ki elements of type i, i = 1, 2,…, n, will contain at least different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k1 = 1, the numbers () are binomial coefficients and the result is the Kruskal-Katona theorem. 相似文献
8.
G.F. Clements 《Journal of Combinatorial Theory, Series A》1978,25(2):153-162
Let M be a finite set consisting of ki elements of type i, i = 1, 2,…, n and let S denote the set of subsets of M or, equivalently, the set of all vectors x = (x1, x2,…,xn) with integral coefficients xi satisfying 0 ? xi ? ki, i = 1, 2,…, n. An antichain is a subset of S in which there is no pair of distinct vectors x and y such that x is contained in y (that is, there is no pair of distinct vectors x and y such that the inequalities xi ? yi, i = 1, 2,…, n all hold). Let denote the number of vectors in S which are contained in at least one vector in and let , the number of basic elements in . For given m we give procedures for calculating min and min , where the minima are taken over all m-element antichains in S. 相似文献
9.
Let D be an (m,n;k;λ1,λ2)-group divisible difference set (GDDS) of a group G, written additively, relative to H, i.e. D is a k-element subset of G, H is a normal subgroup of G of index m and order n and for every nonzero element g of G,?{(d1,d2)?,d1,d2?D,d1?d2=g}? is equal to λ1 if g is in H, and equal to λ2 if g is not in H. Let H1,H2,…,Hm be distinct cosets of H in G and Si=D∩Hi for all i=1,2,…,m. Some properties of S1,S2,…,Sm are studied here. Table 1 shows all possible cardinalities of Si's when the order of G is not greater than 50 and not a prime. A matrix characterization of cyclic GDDS's with λ1=0 implies that there exists a cyclic affine plane of even order, say n, only if n is divisible by 4 and there exists a cyclic (n?1,n?1,n?1)-difference set. 相似文献
10.
R.-D Reiss 《Journal of multivariate analysis》1981,11(3):386-399
Let n denote the sample size, and let ri ∈ {1,…,n} fulfill the conditions ri ? ri?1 ≥ 5 for i = 1,…,k. It is proved that the joint normalized distribution of the order statistics Zri:n, i = 1,…,k, is independent of the underlying probability measure up to a remainder term of order . A counterexample shows that, as far as central order statistics are concerned, this remainder term is not of the order if ri ? ri?1 = 1 for i = 2,…,k. 相似文献
11.
Hans-Dietrich O.F Gronau 《Journal of Combinatorial Theory, Series A》1980,29(3):370-375
Let 1 ? k1 ? k2 ? … ? kn be integers and let S denote the set of all vectors x = (x1, x2, …, xn) with integral coordinates satisfying 0 ? xi ? ki, i = 1, 2, …, n. The complement of x is (k1 ? x1, k2 ? x2, …, kn ? xn) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities xi ? yi, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given. 相似文献
12.
G.F. Clements 《Discrete Mathematics》1977,18(3):253-264
Let k1 ? k2? ? ? kn be given positive integers and let S denote the set of vectors x = (x1, x2, … ,xn) with integer components satisfying 0 ? x1 ? kni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l. 相似文献
13.
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
14.
The Dirichlet integral provides a formula for the volume over the k-dimensional simplex ω={x1,…,xk: xi?0, i=1,…,k, s?∑k1xi?T}. This integral was extended by Liouville. The present paper provides a matrix analog where now the region becomes , where now each Vi is a p×p symmetric matrix and A?B means that A?B is positive semidefinite. 相似文献
15.
A p-cover of is a family of subsets such that ∪ Si = n and |Si ∩ Si| ? p for i ≠ j. We prove that for fixed p, the number of p-cover of n is O(np+1logn). 相似文献
16.
Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If n denotes the symmetric group on {1,2,…,n} then we define the projection by the formula , where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If , then x1?x2? … ?xn denotes the member of Tn(V) such that for each y1 ,2,…,yn in V, and x1·x2… xn denotes . If B? Sn(V) and there exists , such that B = x1·x2…xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C. 相似文献
17.
Stanley J Benkoski 《Journal of Number Theory》1976,8(2):218-223
If r, k are positive integers, then denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r = 1. An explicit formula for is derived and it is shown that .If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2…psas; pi ∈ S and ai ≥ 0 for all i} and denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xi ≤ n and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for are derived and it is shown that . 相似文献
18.
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
19.
20.
Christer Borell 《Journal of Mathematical Analysis and Applications》1973,41(2):300-312
We study certain functionals and obtain an inverse Hölder inequality for n functions f1a1,…,fnan (fk concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f1,…,fn, where .Finally we give an inverse Minkowski inequality for concave functions. 相似文献