共查询到20条相似文献,搜索用时 31 毫秒
1.
Robb J Muirhead 《Journal of multivariate analysis》1974,4(3):341-346
The generalized binomial coefficients (κλ) are defined by , where the Ck(R) are the zonal polynomials of the m × m matrix R. In this paper some simple expressions are derived which allow straightforward calculation of a large number of these coefficients. 相似文献
2.
F Thaine 《Journal of Number Theory》1982,15(3):304-317
We study properties of the polynomials φk(X) which appear in the formal development Πk ? 0n (a + bXk)rk = Σk ≥ 0φk(X) ar ? kbk, where rk ∈ and r = Σrk. this permits us to obtain the coefficients of all cyclotomic polynomials. Then we use these properties to expand the cyclotomic numbers Gr(ξ) = Πk = 1p ? 1 (a + bξk)kr, where p is a prime, ξ is a primitive pth root of 1, a, b ∈ and 1 ≤ r ≤ p ? 3, modulo powers of ξ ? 1 (until (ξ ? 1)2(p ? 1) ? r). This gives more information than the usual logarithmic derivative. Suppose that . Let . We prove that Gr(ξ) ≡ cp mod p(ξ ? 1)2 for some c ∈ , if and only if Σk = 1p ? 1kp ? 2 ? rmk ≡ 0 (mod p). We hope to show in this work that this result is useful in the study of the first case of Fermat's last theorem. 相似文献
3.
Pierrette Cassou-Noguès 《Journal of Number Theory》1982,14(1):32-64
In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σn?1rP(n)?s ξn, where P ∈ + [X1,…,Xr] and ξn = ξ1n1 … ξrnr, with ξi ∈ , such that |ξi| = 1 and ξi ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over by the ξi and the coefficients of P. If, there exists a number field K, containing the ξi, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a -adic function Z(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k). 相似文献
4.
Sylvia Halász 《Journal of Combinatorial Theory, Series A》1984,37(1):85-90
Given a convex domain C and a positive integer k, inscribe k nonoverlapping convex domains into C, all of them similar to C. Denote by f(k) the maximal sum of their circumferences. In this paper it is shown, that for C square, parallelogram or triangle (1) the first increase of f(k) after k = l2 occurs not later than at k = l2 + 2, (2) constructions can be given, where the following lower bounds are attained for f(k) = f(l2 + j): where c denotes the circumference of C. 相似文献
5.
The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: and , respectively, where ?L and CM stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λsI + Σs?1j=0λjLj and M(λ) = λtI + Σt7minus;1j=0λjMj. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r2l × r2l[l = max(s, t)] and enjoys some symmetry properties. 相似文献
6.
Suppose r = (r1, …, rM), rj ? 0, γkj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γk · r = ∑jγkjrj. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑j = 1Naje?λγj · r, where each aj is a nonzero real number. More specifically, if , we study the dependence of . This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when are given. Several examples illustrate the complicated nature of . The results have immediate implication to the theory of stability for difference equations x(t) ? ∑k = 1MAkx(t ? rk) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere. 相似文献
7.
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
8.
William H Barker 《Journal of Functional Analysis》1975,20(3):179-207
Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by p(G)(0<p?2), (ii) the K-biinvariant elements in p(G) by p(G), (iii) the positive definite (zonal) spherical functions by , and (iv) the spherical transform on p(G) by ? → gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto l(G), (ii) there exists a unique measure μ of polynomial growth on such that T[ψ]=∫pψdμ for all ψ?I1(G) (iii) all measures μ of polynomial growth on are obtained in this way, and (iv) T may be extended to a particular p(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of . These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan. 相似文献
9.
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
10.
We define the function jνκ for all real κ > 0 as follows: for κ = 1, 2, … the jνκ denotes the kth positive zero of the Bessel function Jν(z) of first kind and for k ? 1 < κ < k, jνκ denotes the kth positive zero of the cylinder Bessel function Cν(z) = cos αJν(z) ? sin αYν(Z) with α = (k ? ν)π (see [2]), where Yν(x) is the Bessel function of second kind. We introduce the function ι(x) for x > ? 1, . and we prove, among other things, the inequality . Moreover, we find the first three terms of the asymptotic expansion of ι(x), for large values of x and other properties of this function. 相似文献
11.
Ronald Evans 《Journal of Number Theory》1977,9(1):61-62
Let P(X) be a homogeneous polynomial in X = (x, y), Q(X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q(X). Let A(m) = {N ∈ × : Q(N) = m}. It is shown that if ΣN∈A(m)P(N) = 0 for each m = 1, 2, 3,… then ΣU∈GP(UX) ≡ 0. 相似文献
12.
13.
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
14.
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
15.
Given operator polynomials A(λ) and B(λ), one of which is monic, conditions are given for the existence of operator polynomials C(λ) and D(λ) such that A(λ)C(λ) + B(λ)D(λ) = I for every . A special case will give a characterization of controlla- bility of an infinite-dimensional linear control system. 相似文献
16.
Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number such that there exist different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(pm, ? 2) = (pm ? 2)!, R(pm + 1, ? 3) = (pm ? 2)!, , R(k, ? 0) = k, R(pm, ? 1) = pm(pm ? 1), R(pm + 1, ? 2) = (pm + 1)pm(pm ? 1). The exact value of R(k, ? λ) is determined whenever k ? k0(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k0(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for . 相似文献
17.
This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λn(T)) of a p-absolutely summing operator, p ? 2, satisfy This solves a problem of A. Pietsch. We give applications of this to integral operators in Lp-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π2(n), P = (p1, …, pn) and show that for T ∈ Π2(n), 2 = (2, …, 2) ∈ Nn, the eigenvalues of T satisfy More generally, we show that for T ∈ Πp(n), P = (p1, …, pn), pi ? 2, the eigenvalues are absolutely p-summable, We also consider the distribution of eigenvalues of p-nuclear operators on Lr-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely , 0 < p < ∞, where αn denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l1 then X must be a Hilbert space. 相似文献
18.
Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L2(k) are analyzed in terms of the elementary generator, , for n even. Integral operators are defined using the fundamental solutions pn(x, t) to ut = Au and using real polynomials ql,…, qk on m by the formula, for q = (ql,…, qk), m. It is determined when, strongly on L2(k), . If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form. 相似文献
19.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
20.
M.M. Robertson 《Journal of Number Theory》1977,9(2):258-270
For 1 ≦ l ≦ j, let l = ?h=1q(l){alh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the alh are positive integers such that j > 1, al1 < … < alq(2) ≦ M, and let l′ = l ∪ {0}. Let p(n : ) be the number of partitions of n = (n1,…,nj) where, for 1 ≦ l ≦ j, the lth component of each part belongs to l and let p1(n : ) be the number of partitions of n into different parts where again the lth component of each part belongs to l. Asymptotic formulas are obtained for p(n : ), p1(n : ) where all but one nl tend to infinity much more rapidly than that nl, and asymptotic formulas are also obtained for p(n : ′), p1(n ; ′), where one nl tends to infinity much more rapidly than every other nl. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the nl tend to infinity at approximately the same rate. 相似文献