共查询到20条相似文献,搜索用时 15 毫秒
1.
Paul Binding 《Journal of Mathematical Analysis and Applications》1984,102(1):233-243
The multiparameter eigenvalue problem Wm(λ) xm = xm, , m = 1,…, k, where /gl /gE k, xm is a nonzero element of the separable Hilbert space Hm, and Tm and Vmn are compact symmetric is studied. Various properties, including existence and uniqueness, of λ = λi ? k for which the imth greatest eigenvalue of Wm(λi) equals one are proved. “Right definiteness” is assumed, which means positivity of the determinant with (m, n)th entry (ym, Vmnym) for all nonzero ym?Hm, m = 1 … k. This gives a “Klein oscillation theorem” for systems of an o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors xmi is also given, thereby connecting the abstract oscillation theory with a result of Atkinson. 相似文献
2.
Elmer K Hayashi 《Journal of Number Theory》1981,13(2):176-191
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
3.
For a sequence A = {Ak} of finite subsets of N we introduce: , , where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation constitutes a finite semi-group N∪ (semi-group N∩) (group ). For N∪, N∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for analogues of Rohrbach inequality: , where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: , où A(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations , un semi-groupe fini N∪, N∩ ou un groupe N1 respectivement. Pour N∪, N∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N∪, les analogues de l'inégalité de Rohrbach: , où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj. 相似文献
4.
James F. Lynch 《Discrete Mathematics》1981,33(3):281-287
Two sets of sets, C0 and C1, are said to be visually equivalent if there is a 1-1 mapping m from C0 onto C1 such that for every S, T?C0, S ∩ T=0 if and only if m(S)∩ m(T)=0 and S?T if and only if m(S)?m(T). We find estimates for V(k), the number of equivalence classes of this relation on sets of k sets, for finite and infinite k. Our main results are that for finite k, , where α and β are approximately 0.7255 and 2.5323 respectively, and there is a set N of cardinality such that there are V(k) visually distinct sets of k subsets of N. 相似文献
5.
Some techniques for the study of the algebraic curve C(A) which generates the numerical range W(A) of an n×n matrix A as its convex hull are developed. These enable one to give an explicit point equation of C(A) and a formula for the curvature of C(A) at a boundary point of W(A). Applied to the case of a nonnegative matrix A, a simple relation is found between the curvature of the function Φ(A)=p((1?α)A+ αAT) (pbeingthePerronroot) at and the curvature of W(A) at the Perron root of . A connection with 2-dimensional pencils of Hermitian matrices is mentioned and a conjecture formulated. 相似文献
6.
L.E. Trotter 《Discrete Mathematics》1975,12(4):373-388
We examine a family of graphs called webs. For integers n ? 2 and , the web W(n, k) has vertices Vn = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?Vn (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement. 相似文献
7.
Christopher Bingham 《Journal of multivariate analysis》1974,4(2):210-223
Define coefficients (κλ) by Cλ(Ip + Z)/Cλ(Ip) = Σk=0l Σ?∈k (?λ) Cκ(Z)/Cκ(Ip), where the Cλ's are zonal polynomials in p by p matrices. It is shown that C?(Z) etr(Z)/k! = Σl=k∞ Σλ∈l (?λ) Cλ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (?λ), ? ∈ k, k ≤ 3. Several identities involving the (?λ)'s are derived. These are used to derive explicit expressions for coefficients of in expansions of P(Z), for all monomials P(Z) in sj = tr Zj of degree k ≤ 5. 相似文献
8.
Let a complex pxn matrix A be partitioned as A′=(A′1,A′2,…,A′k). Denote by ?(A), A′, and A? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A(14) be an inner inverse of A such that A(14)A is a Hermitian matrix. Let B=(A(14)1,A(14)2,…,Ak(14)) and .Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A(14) or for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition is no longer true. 相似文献
9.
Let V be an n-dimentional unitary space with inner product (·,·) and S the set {x∈V:(x, x)=1}. For any A∈Hom(V, V) and with ∣q∣?1, we define . If q=1, then W(A:q) is just the classical numerical range {(Ax, x):x∈S}, the convexity of which is well known. Another generalization of the numerical range is the C-numerical range, which is defined to be the set where C∈Hom(V, V). In this note, we prove that W(A:q) is always convex and that WC(A) is convex for all A if rank C=1 or n=2. 相似文献
10.
A.H Zemanian 《Journal of Mathematical Analysis and Applications》1976,55(2):394-406
The subject of this work is the voltage and current transfer ratios of three-terminal networks having no mutual coupling and whose impedances are analytic functions taking their values in an abelian self-adjoint algebra of bounded linear operators on a Hilbert space. Each such value is also assumed to be invertible. It is shown that these ratios have the form [I + A(ζ)]?1, where, for each ζ in a sufficiently small open cone in the right-half complex plane with apex at the origin and the real axis as its bisector, the numerical range of A(ζ) is contained in a compact subset of the open right-half plane. This implies that the ratios are strictly contractive for each ζ in the cone. The angle of the cone is , where k is the number of internal nodes of a certain “surrogate” network; this result is best possible. For two-terminal-pair networks the ratios are shown to be strictly contractive for each ζ in a similar cone with angle . 相似文献
11.
K. Inoue 《Journal of multivariate analysis》1976,6(2):295-308
We consider two Gaussian measures P1 and P2 on (C(G), ) with zero expectations and covariance functions R1(x, y) and R2(x, y) respectively, where Rν(x, y) is the Green's function of the Dirichlet problem for some uniformly strongly elliptic differential operator A(ν) of order , on a bounded domain G in d (ν = 1, 2). It is shown that if the order of A(2) ? A(1) is at most , then P1 and P2 are equivalent, while if the order is greater than , then P1 and P2 are not always equivalent. 相似文献
12.
Michael Klemm 《Journal of Combinatorial Theory, Series A》1984,36(3):364-372
Let d be the minimum distance of an (n, k) code , invariant under an abelian group acting transitively on the basis of the ambient space over a field F with char F × n. Assume that contains the repetition code, that dim( ∏ ⊥) = k ? 1 and that the supports of the minimal weight vectors of form a 2-design. Then d2 ? d + 1 ? n with equality if and only if the design is a projective plane of order d ? 1. The case d2 ? d + 1 = n can often be excluded with Hall's multiplier theorem on projective planes, a theorem which follows easily from the tools developed in this paper Moreover, if d2 ? d + 1 > n and F = GF(2) then (d ? 1)2 ? n. Examples are the generalized quadratic residue codes. 相似文献
13.
We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B1(t),…,Bp(t)) U-1, t ∈ ?, and if each matrix function Bk has the polynomial form then A itself has the polynomial form , where , dk being the degree of the minimal polynomial of the matrix Ck, for every k ∈ {1,…,p}. 相似文献
14.
Young Han Choe 《Journal of Mathematical Analysis and Applications》1985,106(2):293-320
A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C0-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)?1 exists and the family {(λ ? β)k(λI ? A)?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has limk → z (I ? k?1tA)?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C0-semigroup by an operator which is an element of is obtained. 相似文献
15.
Given an integer k>0, our main result states that the sequence of orders of the groups (respectively, of the groups ) is Cesàro equivalent as n→∞ to the sequence C1(k)nk2?1 (respectively, C2(k)nk2), where the coefficients C1(k) and C2(k) depend only on k; we give explicit formulas for C1(k) and C2(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to . We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
16.
David Chillingworth 《Journal of Functional Analysis》1980,35(2):251-278
Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which ; here we write for . Say ? is of standard type if at all critical points of ?c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function fo?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in n. 相似文献
17.
Harald K. Wimmer 《Linear algebra and its applications》1975,10(2):139-146
The matrix equation , W ?0, is studied. In the case A1H+HA = W[H?A1HA = W], the controllability matrix of (A1,W) is used to determine the number of eigenvalues of A on the imaginary axis [on the unit circle]. As an application a result of Pták on the critical exponent of the spectral norm is obtained. Estimates for the eigenvalues of A satisfying fH(A) = M are given. 相似文献
18.
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
19.
A lower (upper) bound is given for the distribution of each dj, j = k + 1, …, p (j = 1, …, s), the jth latent root of AB?1, where A and B are independent noncentral and central Wishart matrices having with rank and Wp(n, Σ), respectively. Similar bound are also given for the distributions of noncentral means and canonical correlations. The results are applied to obtain lower bounds for the null distributions of some multivariate test statistics in Tintner's model, MANOVA and canonical analysis. 相似文献
20.
Stephen M. Paneitz 《Journal of Functional Analysis》1981,41(3):315-326
Let Sp() be the symplectic group for a complex Hibert space . Its Lie algebra sp() contains an open invariant convex cone C0; each element of C0 commutes with a unique sympletic complex structure. The Cayley transform : X∈ sp()→(I + X)1∈ Sp() is analyzed and compared with the exponential mapping. As an application we consider equations of the form is strongly continuous, and show that if ∝?∞∞ ∥A(t)∥ dt < 2 and ∝? t8∞A(t) dt?C0, the (scattering) operator , where St′(t) is the solution such that St′(t′) = I, is in the range of restricted to C0. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp() to a unitary operator. 相似文献