Suppose x1, x2,…, is a sequence of vectors in Rk, 6Xn6⩽1, where 6(x1,…,xk)6 = maxj|xj|. An algorithm is given for choosing a corresponding sequence ε1, ε2,…, of numbers, εn = ±1, so that 6ε1x1+ … +εnxn6 remains small. 相似文献
For integral? m?2, let x1,…, xm be any unit vectors in Rn, the real Euclidean space of n dimensions. We obtain an upper bound for the quantity mini≠j|xi-xj| which, though not as simple, is uniformly sharper than one recently obtained by the author. The result has application to the so-called maximum-dispersal problem, an open problem recently popularized by Klee. 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 Xj (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y1,…,Ym) and X′ = (X1,…,Xn) be independently distributed, and A = (ajk) be an n × n random coefficient matrix with ajk = ajk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ~ N(O,I) if and only if X ~ N(O,I). provided that certain conditions defined in terms of the ajk are satisfied. The task of this paper is to delete the identical assumption on X1,…,Xn and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered. 相似文献
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
Let G be an abelian group, let s be a sequence of terms s1, s2, …, sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let $$W \odot S = \left\{ {w_1 s_1 + \cdots + w_n s_n :w_i a term of W,w_i \ne w_j for i \ne j} \right\},$$ which is a particular kind of weighted restricted sumset. We show that |W ⊙ S| ≥ min{|G| ? 1, n}, that W ⊙ S = G if n ≥ |G| + 1, and also characterize all sequences S of length |G| with W ⊙ S ≠ G. This result then allows us to characterize when a linear equation $$a_1 x_1 + \cdots + a_r x_r \equiv \alpha mod n,$$ where α, a1, …, ar ∈ ? are given, has a solution (x1, …, xr) ∈ ?r modulo n with all xi distinct modulo n. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G \cong C_{n_1 } \oplus C_{n_2 }$ (where n1 |n2 and n2 ≥ 3) having k distinct terms, for any k ε [3, min{n1 + 1, exp(G)}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence. 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
For 1 ⩽ k ⩽ n, let V(n, k) denote the set of n(n − 1) … (n − k + 1) sequences of k distinct elements from {1, …, n}. Let us define the graph Γ(n, k) on the vertex set V(n, k) by joining two sequences if they differ at exactly one place. We investigate the chromatic number and another related parameter of these graphs. We give a sharp answer for some infinite families, using the theory of sharply transitive permutation groups. The problems discussed are related to a question of Henkin, Monk and Tarski in mathematical logic. 相似文献
Let 1?k1?k2?…?kn be integers and let S denote the set of all vectors x = (x1, …, xn with integral coordinates satisfying 0?xi?ki, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of ki elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities xi?yi, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k1 + k2 + …kn. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k1-x1, k2-x2, …, kn -xn). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and does not exceed the number of vectors in with first coordinate different from k1, then . 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
If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.Pn will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. Pm × Pn will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(Pm × Pn) = min(m, n). 相似文献
Let Ω be a domain in Rn and T = ∑j,k = 1n(?j ? ibj(x)) ajk(x)(?k ? ibk(x)), where the ajk and the bj are real valued functions in , and the matrix (ajk(x)) is symmetric and positive definite for every . If T0 is the same as T but with bj = 0, j = 1,…, n, and if u and Tu are in , then T. Kato has established the distributional inequality ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T ? q on Rn. In this paper we shall obtain Kato's inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rn. 相似文献
In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x1,x2,…,xk?nk there exist k distinct integers j1,j2,…,jk so that xi= aii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k). 相似文献
Let X1, X2, …, Xm be finite sets. The present paper is concerned with the m2 ? m intersection numbers |Xi ∩ Xj| (i ≠ j). We prove several theorems on families of sets with the same prescribed intersection numbers. We state here one of our conclusions that requires no further terminology. Let T1, T2, …, Tm be finite sets and let m ? 3. We assume that each of the elements in the set union T1 ∪ T2 ∪ … ∪ Tm occurs in at least two of the subsets T1, T2, …, Tm. We further assume that every pair of sets Ti and Tj (i ≠ j) intersect in at most one element and that for every such pair of sets there exists exactly one set Tk (k ≠ i, k ≠ j) such that Tk intersects both Ti and Tj. Then it follows that the integer m = 2m′ + 1 is odd and apart from the labeling of sets and elements there exist exactly m′ + 1 such families of sets. The unique family with the minimal number of elements is {1}, {2}, …, {m′}, {1}, {2}, …, {m′}, {1, 2, …, m′}. 相似文献
It is shown that, whenever m1, m2,…, mn are natural numbers such that the pairwise greatest common divisors, dij=(mi, mj), i≠j are distinct and different from 1, then there exist integers a1, a2,…,an such that the solution sets of the congruences x≡i (modmi), i= 1,2,…,n are disjoint. 相似文献
Let Rij be a given set of μi× μj matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[Fij], i,j=1,…, n, such that Fij=Rij for |i?j|?m, F admits either a left or right block triangular factorization, and (F?1)ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given Rij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied. 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
For a set of positive and relative prime integers A = {a1…,ak}, let Γ(A) denote the set of integers of the form a1x1+…+akxk with each xj ≥ 0. Let g(A) (respectively, n(A) and s(A)) denote the largest integer (respectively, the number of integers and sum of integers) not in Γ(A). Let S*(A) denote the set of all positive integers n not in Γ(A) such that n + Γ(A) \ {0} ? Γ((A)\{0}. We determine g(A), n(A), s(A), and S*(A) when A = {a, b, c} with a | (b + c). 相似文献
Ek(x2,…, xn) is defined by Ek(a2,…, an) = 1 if and only if ∑i=2nai = k. We determine the periods of sequences generated by the shift registers with the feedback functions x1 + Ek(x2,…, xn) and x1 + Ek(x2,…, xn) + Ek+1(x2,…, xn) over the field GF(2). 相似文献