共查询到20条相似文献,搜索用时 31 毫秒
1.
Let A be an arbitrary n×n matrix, partitioned so that if A=[Aij], then all submatrices Aii are square. If x is a positive vector, it is well-known that , where , contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion Gi(x) is replaced by the union of circular disks centered at the eigenvalues of Aii. 相似文献
2.
Yasuhiro Takeuchi Norihiko Adachi 《Journal of Mathematical Analysis and Applications》1981,79(1):141-162
This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) , for every q = (q1,…, qn)T?Rn. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f1(x),…,fn(x))T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R+0n. 相似文献
3.
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
4.
Let A be an n×n integral matrix with determinant D>0, and let P(A) be the n-parallelepiped determined by the columns {Ai}ni=1 of A, Let L be the set of integral vectors in P(A), and let G(A) be the subset of L consisting of vectors whose coefficients xi satisfy 0?xi<1. We show that G(A), equipped with addition modulo 1 on the coefficients xi, is an Abelian group of order D, whose invariant factors are the invariant factors of the integral matrix A. We give a formula for |L|, and show that |L| is not a similarity invariant. 相似文献
5.
Let be a polynomial in the variables x1,…,xp with nonnegative real coefficients which sum to one, let A1,…,Ap be stochastic matrices, and let be the stochastic matrix which is obtained from ? by substituting the Kronecker product of An11,…,Anppfor each term Xn11·?·Xnpp. In this paper, we present necessary and sufficient conditions for the Cesàro limit of the sequence of the powers of to be equal to the Kronecker product of the Cesàro limits associated with each of A1,…,Ap. These conditions show that the equality of these two matrices depends only on the number of ergodic sets under and?or the cyclic structure of the ergodic sets under A1,…,Ap, respectively. As a special case of these results, we obtain necessary and sufficient conditions for the interchangeability of the Kronecker product and the Cesàro limit operator. 相似文献
7.
Let A be an infinite sequence of positive integers a1 < a2 <… and put , . In Part I, it was proved that . In this paper, this theorem is sharpened by estimating DA(x) in terms of fA(x). It is shown that limx→+∞sup DA(x) exp(?c1(logfA(x))2) = +∞ and that this assertion is not true if c1 is replaced by a large constant c2. 相似文献
8.
C.J.K Batty 《Journal of Functional Analysis》1984,57(3):233-243
Let (A, G, α) be a C1-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in and a finite constant κ such that whenever xi lies in a spectral subspace , where . This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G. 相似文献
9.
We show that the fourth order form in five variables, , is nonnegative, but cannot be written as a sum of squares of quadratic forms. 相似文献
10.
R.J. Williams 《Advances in Applied Mathematics》1985,6(1):1-3
Let {Xt, t ≥ 0} be Brownian motion in d (d ≥ 1). Let D be a bounded domain in d with C2 boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” Ex{exp(∝0τDq(Xt)dt)1A(XτD)}, where τD is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any ), is the unique solution in of the Schrödinger boundary value problem . 相似文献
11.
Norbert Wielens 《Journal of Functional Analysis》1985,61(1):98-115
A necessary and sufficient condition is given for the generalized Schrödinger operator to be essentially self-adjoint in , under general assumptions on ? and for arbitrary domains Ω in n. In particular, if ? is strictly positive and locally Lipschitz continuous on , then A is essentially. self-adjoint. Examples of non-essential self-adjointness and a complete discussion of the one-dimensional case are also given. These results have applications to the problem of the essential self-adjointness of quantum Hamiltonians and to the uniqueness problem of Markov processes. 相似文献
12.
Let be the Clifford algebra constructed over a quadratic n-dimensional real vector space with orthogonal basis {e1,…, en}, and e0 be the identity of . Furthermore, let Mk(Ω;) be the set of -valued functions defined in an open subset Ω of Rm+1 (1 ? m ? n) which satisfy Dkf = 0 in Ω, where D is the generalized Cauchy-Riemann operator and k? N. The aim of this paper is to characterize the dual and bidual of Mk(Ω;). It is proved that, if Mk(Ω;) is provided with the topology of uniform compact convergence, then its strong dual is topologically isomorphic to an inductive limit space of Fréchet modules, which in its turn admits Mk(Ω;) as its dual. In this way, classical results about the spaces of holomorphic functions and analytic functionals are generalized. 相似文献
13.
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
14.
Milton Rosenberg 《Journal of multivariate analysis》1978,8(2):295-316
Let p, q be arbitrary parameter sets, and let be a Hilbert space. We say that x = (xi)i?q, xi ? , is a bounded operator-forming vector (?Fq) if the Gram matrix 〈x, x〉 = [(xi, xj)]i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on , the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from to . Then exists a linear operator ǎ from (the Banach space) Fq to Fp on (A) = {x:x ? Fq, is p × q bounded on } such that y = ǎx satisfies yj?σ(x) = {space spanned by the xi}, 〈y, x〉 = A〈x, x〉 and . This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes. 相似文献
15.
A series of inequalities are developed relating the spectral radius ?(A ° B) of the Schur product A ° B of two nonnegative matrices A and B with those of ?(A ° A) and ?(B ° B) yielding . As a corollary it is proved that the spectral radius of the Schur powers ?r = ?(A[r]), A[r] = A ° A °?°A (r factors) satisfies is decreasing while is increasing, the latter provided A is a stochastic matrix. The entropy of a finite stationary Markov chain is identified with . A number of majorization comparisons for the spectral radius of Schur powers is given. 相似文献
16.
David S. Jerison 《Journal of Functional Analysis》1981,43(2):224-257
Let L = ∑j = 1mXj2 be sum of squares of vector fields in n satisfying a Hörmander condition of order 2: span{Xj, [Xi, Xj]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X1,…, Xm are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator in 2. (b) The real part of the Kohn Laplacian on the Heisenberg group in 2n + 1. In contrast to non-characteristic points, C∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x. 相似文献
17.
For a class of subsets of a set X, let V() be the smallest n such that no n-element set F?X has all its subsets of the form A ∩ F, A ∈ . The condition V() <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ? D for some C, D∈, then if and only if is linearly ordered by inclusion. If is of the form , i=1,2,…,n}, where each is linearly ordered by inclusion, then . If H is an (n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and is the collection of all sets {x: f(x)>0} for f in H, then . 相似文献
18.
M. Neumann 《Linear algebra and its applications》1976,14(1):41-51
In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A?Cm × nr. The splitting A = M ? N is called subproper if R(A) ? R(M) and . Consider the iteration . We characterize the convergence of this scheme to a solution of the linear system. When A?Rm×nr, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution. 相似文献
19.
I Herbst 《Journal of Functional Analysis》1982,48(2):224-251
Let , with ? a normalized Gaussian. Suppose ≠ 0 and that has no eigenfunctions in L2(3N. If H1ψ = μψ with μ < infσess(H1), then (ψ, e?itHψ) decays exponentially at a rate governed by the positions of the resonances of H. 相似文献
20.
Let λ1 and λN be, respectively, the greatest and smallest eigenvalues of an N×N hermitian matrix H=(hij), and x=(x1,x2,…,xN) with (x,x)=1. Then, it is known that (1) λ1?(x,Hx)?λN and (2) if, in addition, H is positive definite, . Assuming that y=(y1,y2,…, yN) and |yi|?1, i=1,2,…,N, it is shown in this paper that these inequalities remain true if H and H?1 are, respectively, replaced by the Hadamard products and , where M(y) is a matrix defined by . Subsequently, these results are extended to improve the spectral bounds of . 相似文献