共查询到20条相似文献,搜索用时 62 毫秒
1.
According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where , (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials. 相似文献
2.
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
3.
n independent adiabatic invariants in involution are found for a slowly varying Hamiltonian system of order 2n × 2n. The Hamiltonian system considered is , where A(t) is a 2n × 2n real matrix with distinct, pure imaginary eigen values for each t? [?∞, ∞], and , for all j > 0. The adiabatic invariants Is(u, t), s = 1,…, n are expressed in terms of the eigen vectors of A(t). Approximate solutions for the system to arbitrary order of ? are obtained uniformly for t? [?∞, ∞]. 相似文献
4.
J.J.A.M Brands 《Journal of Mathematical Analysis and Applications》1978,63(1):54-64
This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ1, τ2 be nonnegative continuous functions on (0, ∞) such that τ1 ? τ2 is a bounded function on [1, ∞) and t ? τ1(t) → ∞ if t → ∞. Then is oscillatory if and only if is oscillatory. 相似文献
5.
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
6.
Ming-Po Chen Cheh-Chih Yeh Cheng-Shu Yu 《Journal of Mathematical Analysis and Applications》1977,59(2):211-215
For nonlinear retarded differential equations and the sufficient conditions are given on fi, pi, Fi, and h under which every bounded nonoscillatory solution of () or () tends to zero as t → ∞. 相似文献
7.
Thomas G Kurtz 《Journal of Functional Analysis》1976,23(2):135-144
For each t ? 0, let A(t) generate a contraction semigroup on a Banach space L. Suppose the solution of ut = ?A(t)u is given by an evolution operator V?(t, s). Conditions are given under which converges strongly as ? → 0 to a semigroup T(t) generated by the closure of .This result is applied to the following situation: Let B generate a contraction group S(t) and the closure of ?A + B generate a contraction semigroup S?(t). Conditions are given under which converges strongly to a semigroup generated by the closure of . This work was motivated by and generalizes a result of Pinsky and Ellis for the linearized Boltzmann Equation. 相似文献
8.
Joel M Cohen 《Journal of Functional Analysis》1982,48(3):301-309
Let G be a group and g1,…, gt a set of generators. There are approximately (2t ? 1)n reduced words in g1,…, gt, of length ?n. Let be the number of those which represent 1G. We show that exists. Clearly 1 ? γ ? 2t ? 1. is the cogrowth. 0 ? η ? 1. In fact . The entropic dimension of G is shown to be 1 ? η. It is then proved that d(G) = 1 if and only if G is free on g1,…, gt and d(G) = 0 if and only if G is amenable. 相似文献
9.
Charles J Monlezun 《Journal of Mathematical Analysis and Applications》1974,47(1):133-152
A theory of scattering for the time dependent evolution equations (1) is developed. The wave operators are defined in terms of the evolution operators Uj(t, s), which govern (1). The scattering operator remains unitary. Sufficient conditions for existence and completeness of the wave operators are obtained; these are the main results. General properties, such as the chain rule and various intertwining relations, are also established. Applications include potential scattering (H0(t) = ?Δ, Δ denoting the Laplacian, and H1(t) = ?Δ + q(t, ·)) and scattering for second-order differential operators with coefficients constant in the spatial variable (). 相似文献
10.
Let , where P is a distribution with P(0)=0. Then is a non-decreasing function of k, and Πk is a non-increasing function of k. 相似文献
11.
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
12.
For fixed p (0 ≤ p ≤ 1), let {L0, R0} = {0, 1} and X1 be a uniform random variable over {L0, R0}. With probability p let {L1, R1} = {L0, X1} or = {X1, R0} according as ; with probability 1 ? p let {L1, R1} = {X1, R0} or = {L0, X1} according as , and let X2 be a uniform random variable over {L1, R1}. For n ≥ 2, with probability p let {Ln, Rn} = {Ln ? 1, Xn} or = {Xn, Rn ? 1} according as , with probability 1 ? p let {Ln, Rn} = {Xn, Rn ? 1} or = {Ln ? 1, Xn} according as , and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n ≥ 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n → ∞. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 ? y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability). 相似文献
13.
Jerome A Goldstein James T Sandefur 《Journal of Mathematical Analysis and Applications》1979,67(1):58-74
Let H be a self-adjoint operator on a complex Hilbert space . The solution of the abstract Schrödinger equation is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥2 is independent of t. When does the energy break up into different kinds of energy E = ∑j = 1NEj(t) which become asymptotically equipartitioned ? (That is, for all j and all data u(0).) The “classical” case is the abstract wave equation self-adjoint on 1. This becomes a Schrödinger equation in a Hilbert space (essentially is two copies of 1), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C1-algebra approach to quantum field theory and statistical mechanics. (2) Let A1,…, AN be commuting self-adjoint operators with N = 2 or 4. Then the equation admits equipartition of energy if and only if exp(it(Aj ? Ak)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k. 相似文献
14.
M.K Sundareshan M.A.L Thathachar 《Journal of Mathematical Analysis and Applications》1973,42(3):674-683
Sufficient conditions are given for the L2-stability of a class of feedback systems consisting of a linear operator G and a nonlinear gain function, either odd monotone or restricted by a power-law, in cascade, in a negative feedback loop. The criterion takes the form of a frequency-domain inequality, Re[1 + Z(jω)] G(jω) ? δ > 0 ?ω? (?∞, +∞), where Z(jω) is given by, Z(jω) = β[Y1(jω) + Y2(jω)] + (1 ? β)[Y3(jω) ? Y3(?jω)], with 0 ? β ? 1 and the functions y1(·), y2(·) and y3(·) satisfying the time-domain inequalities, , y2(·) = 0, t > 0 and ? > 0, and , c2 being a constant depending on the order of the power-law restricting the nonlinear function. The criterion is derived using Zames' passive operator theory and is shown to be more general than the existing criteria. 相似文献
15.
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
16.
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
17.
Eng-Bin Lim 《Journal of Differential Equations》1978,30(1):49-53
The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x0, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution of degree k (cf. [1]) for a given positive integer k. Furthermore, , where ¦ · ¦ is any norm in Rn. 相似文献
18.
Min Ming Tang 《Journal of Mathematical Analysis and Applications》1977,57(2):368-381
In this paper we study the behavior of solutions of some quasilinear parabolic equations of the form as t → ∞. In particular, the solutions of these equations will decay to zero as t → ∞ in the L∞ norm. 相似文献
19.
The system is investigated, where x and y are scalar functions of time (t ? 0), and n space variables , and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution , which is bounded for r ? 0 and satisfies x(0) >x0, y(0) > y0, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique. 相似文献
20.
David W. Bange 《Journal of Differential Equations》1975,17(1):61-72
This paper treats the quasilinear, parabolic boundary value problem u(0, t) = ?1(t); u(l, t) = ?2(t) on an infinite strip with the functions being periodic in t. The major theorem of the paper gives sufficient conditions on for this problem to have a periodic solution u(x, t) which may be constructed by successive approximations with an integral operator. Some corollaries to this theorem offer more explicit conditions on and indicate a method for determining the initial estimate at which the iteration may begin. 相似文献