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1.
If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map is convex. These yield the concavity of the map (0<p?1) (Lieb's theorem) and the convexity of the map (0<p?1), as well as the convexity of the map .These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices. 相似文献
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3.
Let denote the class of hidden Z-matrices, i.e., if and only if there exist Z-matrices X and Y such that the following two conditions are satisfied: Let P denote the class of real square matrices having positive principal minors. The class has arisen recently as a generalization of the class of Z-matrices [9,23,24]. In this paper, we explore various matrix-theoretic aspects of the class . 相似文献
4.
Let A be an n×n complex matrix. For a suitable subspace of Cn the Schur compression A and the (generalized) Schur complement A/ are defined. If A is written in the form according to the decomposition and if B is invertible, then and The commutativity rule for Schur complements is proved: This unifies Crabtree and Haynsworth's quotient formula for (classical) Schur complements and Anderson's commutativity rule for shorted operators. Further, the absorption rule for Schur compressions is proved: . 相似文献
5.
In this Note we consider nonnegative solutions for the nonlinear equation in , where is the so called Pucci operator and the ei are the eigenvalues of M et Λ?λ>0. We prove that if u satisfies the decreasing estimate for some β satisfying (β?1)(p?1)>2+α then u is radial. In a second time we prove that if and u is a nonnegative radial solution of (1), u(x)=g(r), such that g″ changes sign at most once, then u is zero. To cite this article: I. Birindelli, F. Demengel, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 相似文献
6.
R.J Cook 《Journal of Number Theory》1983,17(1):80-92
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
7.
M Jungerman 《Journal of Combinatorial Theory, Series B》1979,26(2):154-158
The nonorientable genus of K4(n) is shown to satisfy: , . 相似文献
8.
In this Note we present some results on the existence of radially symmetric solutions for the nonlinear elliptic equation
(1)
Here N?3, p>1 and denotes the Pucci's extremal operators with parameters 0<λ?Λ. The goal is to describe the solution set as function of the parameter p. We find critical exponents , that satisfy: (i) If then there is no nontrivial solution of (). (ii) If then there is a unique fast decaying solution of (). (iii) If then there is a unique pseudo-slow decaying solution to (). (iv) If pp+<p then there is a unique slow decaying solution to (). Similar results are obtained for the operator . To cite this article: P.L. Felmer, A. Quaas, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 909–914. 相似文献
9.
Thomas G. Kurtz 《Stochastic Processes and their Applications》1978,6(3):223-240
A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form where , the Y1 are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies Under very general conditions limN→∞XN(t)=X(t) a.s. The process XN(t) is compared to the diffusion processes given by and Under conditions satisfied by most of the applied probability models, it is shown that XN,ZN and V can be constructed on the same sample space in such a way that and 相似文献
10.
This paper considers canonical forms for the similarity action of Gl(n) on : , Those canonical forms are obtained as an application of a more general method to select canonical elements Mc in the orbits of a matrix group G acting on a set of matrices . We define a total order (?) on , different from the lexicographic order l? [0l?x ? x <0, but and consider normalized -elements with a minimal number of parameters: It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method. 相似文献
11.
If s1(A) ? ? ? sm(A) are the singular values of A ? Mm,n(C), and if 1 ?k ?m ? and p ? 1, then is a unitarily invariant norm. In this paper a complete determination of the extreme points on the corresponding unit spheres is accomplished in all cases, enabling the isometries with respect to Φp,k to be determined in the case p = 1. This removes the restriction m = n in an earlier paper of the author and Marcus. 相似文献
12.
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
13.
Stanisław Lewanowicz 《Journal of Computational and Applied Mathematics》1979,5(3):193-206
In this paper we are constructing a recurrence relation of the form for integrals (called modified moments) in which Ck(λ) is the k-th Gegenbauer polynomial of order , and f is a function satisfying the differential equation of order n, where p0, p1, …, pn ? 0 are polynomials, and mk〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense. 相似文献
14.
Let G be the metacyclic group of order pq given by where p is an odd prime, q ≥ 2 a divisor of p ? 1, and where j belongs to the exponent q mod p. Let V denote the group of units of augmentation 1 in the integral group ring G of G. In this paper it is proved that the number of conjugacy classes of elements of order p in V is where ν, μ0, and H are suitably defined numbers. 相似文献
15.
William E Strawderman 《Journal of multivariate analysis》1974,4(3):255-264
Families of minimax estimators are found for the location parameters of a p-variate distribution of the form , where G(·) is a known c.d.f. on (0, ∞), p ≥ 3 and the loss is sum of squared errors. The estimators are of the form where 0 ≤ a ≤ 2, r(X′X) is nondecreasing, and is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s. 相似文献
16.
Let A denote a decomposable symmetric complex valued n-linear function on Cm. We prove , where · denotes the symmetric product and ? the tensor product. As a consequence we have per , where M is a positive semidefinite Hermitian matrix and per denotes the permanent function. A sufficient condition for equality in the matrix inequality is that M is a nonnegative diagonal matrix. 相似文献
17.
Ludwig Arnold 《Linear algebra and its applications》1976,13(3):185-199
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let , put (bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally (empirical distribution function of the eigenvalues of . Suppose (i) , (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then ,where W is absolutely continuous with (semicircle) density 相似文献
18.
A.M Fink 《Journal of Mathematical Analysis and Applications》1982,90(1):251-258
Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetNn (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g(n)(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofNnis. Ifg(x) ∈ Nnis such that, thenfor
1
. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4). 相似文献
19.
Homogenization in open sets with holes 总被引:1,自引:0,他引:1
Doina Cioranescu Jeannine Saint Jean Paulin 《Journal of Mathematical Analysis and Applications》1979,71(2):590-607
Let Qr be a cylindrical bar with r cylindrical cavities having generators parallel to those of Qr. Let Ω be the cross-section of the bar, Ω1 the cross-section of the domain occupied by the material and the cross- sections of the cavities: . The study of the elastic torsion of this bar leads to the following problem [see 2., 3., 267–320)]: (1) where μ is the shear modulus of the material, α is the angle of twist and represents the stress function. In this paper the problem (1) with an increasing number of holes which are distributed periodically is considered. One would like to know if has a limit , and if so, the equation satisfied by this limit. This is an “homogenization” problem — the heterogeneous bar Qr is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Qr. A more general problem will be studied and the case of elastic torsion will be obtained as an application. The proof uses the energy method [see Lions (Collège de France, 1975–1977), Tartar (Collège de France, 1977)] and extension theorems. A related problem is the homogenization of a perforated plate [cf. Duvaut (to appear)]. 相似文献
20.
John R Dickerson Paul A Erickson 《Journal of Mathematical Analysis and Applications》1974,47(3):513-519
The behavior of an infinite sequence of ordinary differential equations of the form: , , where Xn(t) is a vector valued function of R+, is studied in spaces of infinite sequences of vectors. In particular, sufficient conditions for asymptotic stability of this sequence of linear equations are established and applied to the stability analysis of a string of vehicles with a simple form of automatic control. 相似文献