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1.
Consider the renewal equation in the form (1) , where is a probability density on [0, ∞) and limt → ∞g(t) = g0. Asymptotic solutions of (1) are given in the case when f(t) has no expectation, i.e., . These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation. 相似文献
2.
Let = (1, 2, …, n)′ be the least-squares estimator of θ = (θ1, θ2, …, θn)′ by the realization of the process y(t) = Σk = 1nθkfk(t) + ξ(t) on the interval T = [a, b] with f = (f1, f2, …, fn)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and Ds-optimality are used as criteria for choosing f in X. A-, D-, and Ds-optimal models can be constructed explicitly by means of r. 相似文献
3.
David Terman 《Journal of Differential Equations》1983,47(3):406-443
We consider the pure initial value problem for the system of equations , the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here , where H is the Heaviside step function and . This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if limt→∞ s(t) = ∞. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as . 相似文献
4.
M.Z Nashed 《Journal of Mathematical Analysis and Applications》1976,53(2):359-366
Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let be the linear integral operator induced by the kernel K(s, t) on the space 2[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6Kx?y6 in the 2-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝01K(si, t) x(t) dt = y(si), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating (where 2 is the generalized inverse of ), without recourse to the normal equation , that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind. 相似文献
5.
《Advances in Mathematics》1985,56(3):238-282
Let n be the Lie algebra gln(C), let S(n) be the symmetric algebra of n, and let T(n) be the tensor algebra of n. In a recent paper, R. K. Gupta studied certain sequences of representations R∞ = (Rn)∞n = 1, where Rn is a representation of n. These sequences have the property that every irreducible representation occurring in S(n) is in exactly one of these sequences. Fixing f, she considers s(R∞, f) which is the limit on n of the multiplicity of Rn in Sf(n), the fth-graded piece of S(n). She and R. P. Stanley independently showed that the limit s(R∞, f) exists and is given by an amazingly elegant formula. They call s(R∞, f) the stable multiplicity of Rn in Sf(n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R∞ for all of the classical Lie algebras n are studied, and a simple formula for the stable multiplicity m(R)∞, ψ, f, ∞) of Rn in the ψ-isotypic component of Tf(n), where ψ is any irreducible character of the symmetric group tSf, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)∞, ψ, f, ∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T(n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S(n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of Tf(n) projects onto Kostant's decomposition of Sf(n). 相似文献
6.
Ronald E Bruck 《Journal of Mathematical Analysis and Applications》1980,76(1):159-173
We show that if u is a bounded solution on + of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈Lloc2(+;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t) 0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in and that a smoothing effect takes place. 相似文献
7.
We obtain asymptotic estimates for the quantity r = log P[Tf[rang]t] as t → ∞ where Tf = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d2(·)/dx2 and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫t0 λ1 (f(s) ds(1 + o(1)) where λ1(f) is the smallest eigenvalue for the process killed at ±f. 相似文献
8.
M.P Heble 《Journal of Mathematical Analysis and Applications》1983,93(2):363-384
Given a cocycle a(t) of a unitary group {U1}, ?∞ < t < ∞, on a Hilbert space , such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;t0Usxds + β(t) for a unique x ? , β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as if existing. For a stationary diffusion process on 1, with Ω1, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on , based on the paths restricted to the time interval [0, 1]. It is shown that is continuous and bounded on . The shift τt, defines a unitary representation {Ut}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, has a C∞ spectral density function f;. It is then shown that σ2({Ut}, I) = f;(O). 相似文献
9.
Ilkka Karasalo 《Journal of Mathematical Analysis and Applications》1975,51(3):516-538
The 2-norm of the infinite vector of the terms of the Taylor series of an analytic function is used to measure the “unsmoothness” of the function. The sets of solutions to the scalar differential equations y′(t) = λy(t) + f(t) and y′(t) = q(t)y(t) + f(t) are analyzed with respect to this norm. A number of results on the particular solution with minimum norm are given. 相似文献
10.
The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) Lnx(t) + δq(t) f(x[g1(t)],…, x[gm(t)]) = 0, where δ = ± 1 and L0x(t) = x(t), Lkx(t) = ak(t)(Lk ? 1x(t))., , is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y.(t) + c1(t) f(y[g1(t)],…, y[gm(t)]) = 0 and (an ? 1(t)z.(t)). ? c2(t) f(z[g1(t)],…, z[gm(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos. 相似文献
11.
Joseph Blum 《Journal of Combinatorial Theory, Series A》1974,17(2):156-161
Let f(n) denote the number of square permutations in the symmetric group Sn. This paper proves a conjecture that f(2k + 1) = (2k + 1)f(2k) and provides efficient procedures for the computation of f(n). The behavior of f(n) as n → ∞ is investigated and an asymptotic result obtained which shows that f(n) ~ 2Knne?n, where . 相似文献
12.
Hui-Hsiung Kuo 《Journal of Functional Analysis》1976,21(1):63-75
Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator is defined by f(x) = ?lim∈←0{E[f((τ∈ξ))] ? f(x)}/E[τ∈ξ, where τx? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that f(x) = ?trace D2f(x)+〈Df(x),x〉 for a large class of functions f. Let rt(x, dy) be the transition probabilities of U(t). The λ-potential Gλf, λ > 0, and normalized potential Rf of f are defined by Gλf(X) = ∫0∞e?λtrtf(x) dt and Rf(x) = ∫0∞ [rtf(x) ? rtf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D2Gλf(x) ? 〈DGλf(x), x〉 = ?f(x) + λGλf(x) and trace D2Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫Bf(y)p1(dy), where p1 is the Wiener measure in B with parameter 1. Some approximation theorems are also proved. 相似文献
13.
Let ø(t) (t ∈ n) be a retarded, Lorentz-invariant function which satisfies, in addition, condition (c). We call “R” the family of such functions. Let f(z) be the Laplace transform of ø(t) ∈ . We prove (Theorem 1) that f(z) can be expressed as a K-transform (formula (I, 2; 1)). We apply this formula to evaluate several Laplace transforms. We show that it affords simple proofs of important known results. Formula (I, 2; 1) is an effective complement to L. Schwartz' method of evaluating Fourier transforms via Laplace transforms (“Théorie des distributions,” p. 264, Hermann, Paris, 1966). We think this is the most useful application of our formula. 相似文献
14.
Robert Donaghey 《Journal of Combinatorial Theory, Series A》1979,27(3):360-364
It is shown that the compositional inverse of either of two transformations of a given series can be determined from the compositional inverse of the series. Specifically, if t · f(t) and t · g(t) are compositional inverses, then so are t · fk(t) and , where fk(t) is the kth Euler transformation of f(t) and . 相似文献
15.
Mitsuhiro Nakao 《Journal of Differential Equations》1982,44(1):63-81
A simple result concerning integral inequalities enables us to give an alternative proof of Waltman's theorem: limt → ∞ ∝t0a(s) ds = ∞ implies oscillation of the second order nonlinear equation y″(t) + a(t) f(y(t)) = 0; to prove an analog of Wintner's theorem that relates the nonoscillation of the second order nonlinear equations to the existence of solutions of some integral equations, assuming that limt → ∞ ∝t0a(s) ds exists; and to give an alternative proof and to extend a result of Butler. An often used condition on the coefficient a(t) is given a more familiar equivalent form and an oscillation criterion involving this condition is established. 相似文献
16.
Simeon M. Berman 《Stochastic Processes and their Applications》1983,15(3):213-238
Let X(t) be the trigonometric polynomial Σkj=0aj(Utcosjt+Vjsinjt), –∞< t<∞, where the coefficients Ut and Vt are random variables and aj is real. Suppose that these random variables have a joint distribution which is invariant under all orthogonal transformations of R2k–2. Then X(t) is stationary but not necessarily Gaussian. Put Lt(u) = Lebesgue measure {s: 0?s?t, X(s) > u}, and M(t) = max{X(s): 0?s?t}. Limit theorems for Lt(u) and for u→∞ are obtained under the hypothesis that the distribution of the random norm (Σkj=0(U2j+V2j))1 2 belongs to the domain of attraction of the extreme value distribution exp{ e–2}. The results are also extended to the random Fourier series (k=∞). 相似文献
17.
Arthur Lubin 《Journal of Functional Analysis》1974,17(4):388-394
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral = ⊕L2(vt) dm(t) and the operator on , where e(s, t) = exp ∫st ∫Tdvλ(θ) dm(λ). Let μt be the measure defined by for all continuous ?, and let ?t(z) = exp[?∫ (eiθ + z)(eiθ ? z)?1dμt(gq)]. Call {vt} regular iff for all for 1 a.e. 相似文献
18.
Bernt Øksendal 《Journal of Functional Analysis》1980,36(1):72-87
Let Jωx(t) = x + ∝0tbω(s) ds, where bω is planar Brownian motion starting at 0. A Wiener-type criterion is proved for the process Jωx(t): Let K be a compact plane set and let x?K. Then if ∑ 2nM1(An(x)?K) < ∞ (where and M1 denotes one-dimensional Hausdorff content), the process Jωx(t) stays within K for a positive period of time t, a.s. In particular, this applies to almost all x with respect to area in the nowhere dense “Swiss Cheese” sets. The method is based on general potential theory for Markov processes. 相似文献
19.
Robert Donaghey 《Journal of Combinatorial Theory, Series A》1980,28(1):111-114
For a formal power series with nonnegative integer coefficients, the compositional inverse of is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of hi colors. Since the compositional inverse of the Euler transformation of f(t) is the star transformation [[g(t)]?1 ? 1]?1 of g(t), [2], it follows that the Euler transformation of f(t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of hi colors for i > 1, and h1 ? 1 colors for i = 1. 相似文献
20.
Thomas G Kurtz 《Journal of Functional Analysis》1973,12(1):55-67
Let U(t) and S(t) be strongly continuous contraction semigroups on a Banach space L with infinitesimal operators A and B, respectively. Suppose the closure of A + αB generates a semigroup Tα(t). The behavior of Tα(t) as α goes to infinity is examined. In particular, suppose S(t) converges strongly to P. If the closure of PA generates a semigroup T(t) on (P), then Tα(t) goes to T(t) on (P). If PA = 0 and if BVf = ?f for fε(P), conditions are given that imply Tα(αt) converges on (P) to a semigroup generated by the closure of PAVA.The results are used to obtain new and known limit theorems for random evolutions, which in turn give approximation theorems for diffusion processes. 相似文献