共查询到20条相似文献,搜索用时 62 毫秒
1.
Boguslaw Tomaszewski 《Journal of Functional Analysis》1984,55(1):63-67
It is shown, for n ? m ? 1, that there exist inner maps Φ: Bn → Bm with boundary values such that . where σn and σm are the Haar measures on ?Bn and ?Bm, respectively, and A ? Bn is an arbitrary Borel set. 相似文献
2.
Let H and K be symmetric linear operators on a C1-algebra with domains D(H) and D(K). H is defined to be strongly K-local if implies for A?D(H) ∩ D(K) and ω in the state space of , and H is completely strongly K-local if implies for A ∈ D(H) ∩ D(K) and Ω in the state of , and H is cpmpletely strongly K-local if is -local on U?Mn for all n ? 1, where n is the identity on the n × n matrices Mn. If is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of 1-automorphisms of with generator δ0 and δ is a derivation which commutes with τ and is completely strongly δ0-local then δ generates a group α of 1-automorphisms of . Various characterizations of α are given and the particular case of periodic τ is discussed. 相似文献
3.
Alan McIntosh 《Journal of Functional Analysis》1978,30(2):264-275
It is shown that there is a closed symmetric derivation δ of a C1-algebra with dense domain (δ), an element A = A1 ?(δ), and a C1-function such that (A)?(δ). Some estimates are derived for , where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in (H) which depends linearly on t, while is not differentiable. It is also shown that there exists (H) which is not C1-self-adjoint even though it satisfies for all t ? 相似文献
4.
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
5.
6.
L.B Richmond 《Journal of Number Theory》1976,8(4):390-396
Asymptotic results are obtained for pA(k)(n), the kth difference of the function pA(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that thereby verifying for these A a conjecture of Bateman and Erdös. 相似文献
7.
Let Fn denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=ANxN+ ?+ A1x+A0?Fn[x]. A function is called a right polynomial function iff there exists an A(x)?Fn[x] such that for every B?Fn. This paper obtains unique representations for and determines the number of right polynomial functions. 相似文献
8.
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
9.
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
10.
R.A. Maller 《Stochastic Processes and their Applications》1978,8(2):171-179
Let Xi be iidrv's and Sn=X1+X2+…+Xn. When EX21<+∞, by the law of the iterated logarithm for some constants αn. Thus the r.v. is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX21<+∞ that EYh<+∞ if and only if for 0<h<1 and δ> , whereas whenever h>0 and . 相似文献
11.
Robert S Strichartz 《Journal of Functional Analysis》1982,49(1):91-127
The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L1(Rn) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on Rn + 1 homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x0 = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε1) Sing(ψ, ε2)f = ?2π2(∝ φ)(∝ ψ)f + Sing(A, ε1, + ε2)f, where (with notation ). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol , where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity. 相似文献
12.
Explicit and asymptotic solutions are presented to the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + min1 ? t ? n(αM(t) + βM(n + 1 ? t)) for the cases (1) α + β < 1, is rational, and g(n) = δnI. (2) α + β > 1, min(α, β) > 1, is rational, and (a) g(n) = δn1, (b) g(n) = 1. The general form of this recurrence was studied extensively by Fredman and Knuth [J. Math. Anal. Appl.48 (1974), 534–559], who showed, without actually solving the recurrence, that in the above cases , where γ is defined by α?γ + β?γ = 1, and that does not exist. Using similar techniques, the recurrence M(1) = g(1), M(n + 1) = g(n + 1) + max1 ? t ? n(αM(t) + βM(n + 1 ? t)) is also investigated for the special case α = β < 1 and g(n) = 1 if n is odd = 0 if n is even. 相似文献
13.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献
14.
Let (m?n) denote the linear space of all m × n complex or real matrices according as = or . Let c=(c1,…,cm)≠0 be such that c1???cm?0. The c-spectral norm of a matrix A?m×n is the quantity . where σ1(A)???σm(A) are the singular values of A. Let d=(d1,…,dm)≠0, where d1???dm?0. We consider the linear isometries between the normed spaces and , and prove that they are dual transformations of the linear operators which map (d) onto (c), where . 相似文献
15.
Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If n denotes the symmetric group on {1,2,…,n} then we define the projection by the formula , where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If , then x1?x2? … ?xn denotes the member of Tn(V) such that for each y1 ,2,…,yn in V, and x1·x2… xn denotes . If B? Sn(V) and there exists , such that B = x1·x2…xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C. 相似文献
16.
Richard Askey Deborah Tepper Haimo 《Journal of Mathematical Analysis and Applications》1977,59(1):119-129
We study degeneration for ? → + 0 of the two-point boundary value problems , and convergence of the operators T?+ and T?? on 2(?1, 1) connected with them, T?±u := τ?±u for all for all . Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real ∞-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T?+ and of T?? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a 1 and b c 0 and in which we can compute the limits exactly. We show that (T?+ ? λ)?1 converges for ? → +0 strongly to (T0+ ? λ)?1 if . In an analogous way, we define the operator T?+, n (n ? in the Sobolev space H0?n(? 1, 1) as a restriction of τ?+ and prove strong convergence of (T+?,n ? λ)?1 for ? → +0 in this space of distributions if . With aid of the maximum principle we infer from this that, if h?1, the solution of τ?+u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? if ? ?.Finally we prove by duality that the solution of τ??u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?. 相似文献
17.
Shlomo Moran 《Journal of Combinatorial Theory, Series B》1984,37(2):113-141
Let V be a set of n points in Rk. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) lk(n) are the extremal values of l(V) defined by lk(n)=max{l(V)|V∈Vnk}, where Vnk={V|V?Rk,|V|=n, d(V)=1}. A set V∈Vnk is “longest” if l(V)=lk(n). In this paper, first some geometrical properties of longest sets in R2 are studied which are used to obtain l2(n) for small n′s, and then asymptotic bounds on lk(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δk(n)=max{δ(V)|V∈Vnk}. It is easily observed that . Hence, exists. It is shown that for all , and hence, for all . For k=2, this implies that , which generalizes an observation of Fejes-Toth that . It is also shown that . The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that . 相似文献
18.
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
19.
We consider unbounded derivations in C1-algebras commuting with compact groups of 1-automorphisms. A closed 1-derivation δ in a C1-algebra is said to be a generator if there exists a strongly continuous one-parameter subgroup t∈→τ(t)? Aut() such that . If δ is known to commute with a compact abelian action α:G→Aut(), and if δ(a) = 0 for all a in the fixed point algebra α of the action G, then we show that δ is necessarily a generator. Moreover, in any faithful G-covariant representation, there is a commutative operator field γ ∈ ? → v(γ) such that is possibly unbounded but affiliated with the center of {α}″, and etδ(x) = xetv(γ) for all x in the Arveson spectral subspace α(γ). In particular, if is the CAR algebra over an infinite-dimensional Hilbert space and α is the gauge group, then any such derivation δ is a scalar multiple of the generator of the gauge group. 相似文献
20.
Sidney I. Resnick 《Stochastic Processes and their Applications》1973,1(1):67-82
{Xn,n?1} are i.i.d. random variables with continuous d.f. F(x). Xj is a record value of this sequence if Xj>max{X1,…,Xj?1}. Consider the sequence of such record values {XLn,n?1}. Set R(x)=-log(1?F(x)). There exist Bn > 0 such that . in probability (i.p.) iff i.p. iff → ∞ as x→∞ for all k>1. Similar criteria hold for the existence of constants An such that XLn?An → 0 i.p. Limiting record value distributions are of the form N(-log(-logG(x))) where G(·) is an extreme value distribution and N(·) is the standard normal distribution. Domain of attraction criteria for each of the three types of limit laws can be derived by appealing to a duality theorem relating the limiting record value distributions to the extreme value distributions. Repeated use is made of the following lemma: If , then XLn=Y0+…+Yn where the Yj's are i.i.d. and . 相似文献