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1.
Let M∞ be the supremum of a random walk drifting to -∞ which is generated by the partial sums of a sequence of independent identically distributed random variables with a common distribution F. We prove that the moment generating function E exp( sM∞) is a rational function if and only if the function ∫ 0∞ exp( sx) F(d x) is rational. 相似文献
2.
Suppose { k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and { cj} ∞j=0, { dj} ∞j=0 are sequences of real numbers such that Σ jc2j < ∞, Σ jd2j < ∞. Then, under appropriate moment conditions on { k, −∞ < k < ∞}, yk Σ ∞j=0cjk-j, zk Σ ∞j=0djk-j exist almost surely and in
4 and the question of Gaussian approximation to S[t]Σ [t]k=1 ( yk zk − E{ yk zk}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on k for “time” k ≤ 0, weaken the stationarity assumptions on { k, −∞ < k < ∞}, and improve the summability conditions on { cj} ∞j=0, { dj} ∞j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let dj = cj-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation. 相似文献
3.
The work highlights within the framework of E. Mach’s “Denkökonomie” the crucial role played by the electron-volt unit system in uncovering terce relationships in the various fundamental interactions. In particular it is shown that a great deal of insight and analytical simplicity is gained from the number coincidence between the E(∞) dimensionless inverse fine structure constant and the mass of a postulated expectation π-meson mπ=(mπ±+mπ0)/2 when measured in Mega-electron-volt. This line of thought could be continued to the k-meson and relate it to the exceptional Lie group of super strings and unification namely E8 E8. We show further that mπ could be viewed as a two quark building block relating the old Yukawa theory to an extended quark model based on E-infinity Cantorian spacetime. 相似文献
4.
The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ i = 1n (ƒ i( xi + λ gi( xi)), under simple constraints Σ i = 1n xi = M, li≤ xi≤ ui and xi: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and li and ui are given lower and upper bounds on xi. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O( G√ Mlog 2 n + n log n + n log( M/ n)), where G = Σ i = 1n ui − Σ i = 1n li and an evaluation of ƒ i(·) or gi(·) is assumed to be done in constant time. 相似文献
5.
We consider a new problem, the Kth best valued assignment problem. Given a bipartite graph G and a cost vector w on its edge set, this is the problem of finding a perfect matching Mk in G such that there exist perfect matchings M1,…, MK−1 satisfying w( M1) < < w( MK−1) < w( MK), and w( MK) < w( M) for all perfect matchings M with w( M) ≠ w( M1),…, w( MK). Here w( M) denotes the sum of costs of edges in M. In this paper, we propose two algorithms for solving this problem and verify the efficiency of our algorithms by our preliminary computational experiments. 相似文献
6.
A random graph Gn( x) is constructed on independent random points U1,…, Un distributed uniformly on [0,1] d, d1, in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0< x<1. The connectivity distance cn, the smallest x for which Gn( x) is connected, is shown to satisfy For d2, the random graph Gn( x) behaves like a d-dimensional version of the random graphs of Erdös and Rényi, despite the fact that its edges are not independent: cn/ dn→1, a.s., as n→∞, where dn is the largest nearest-neighbor link, the smallest x for which Gn( x) has no isolated vertices. 相似文献
7.
Let C be a planar region. Choose n points p1,,p nI.I.D. from the uniform distribution over C. Let MCn be the number of these points that are maximal. If C is convex it is known that either E( MCn)=Θ(√ n)> or E( MCn)=O( log n). In this paper we will show that, for general C, there is very little that can be said, a-priori, about E( MCn). More specifically we will show that if g is a member of a large class of functions then there is always a region C such that E( MCn)=Θ( g( n)). This class contains, for example, all monotically increasing functions of the form g( n)= nlnβn, where 0<<1 and β0. This class also contains nondecreasing functions like g( n)= ln*n. The results in this paper remain valid in higher dimensions. 相似文献
8.
While the theory of relativity was formulated in real spacetime geometry, the exact formulation of quantum mechanics is in a mathematical construction called Hilbert space. For this reason transferring a solution of Einstein’s field equation to a quantum gravity Hilbert space is far of being a trivial problem. On the other hand (∞) spacetime which is assumed to be real is applicable to both, relativity theory and quantum mechanics. Consequently, one may expect that a solution of Einstein’s equation could be interpreted more smoothly at the quantum resolution using the Cantorian (∞) theory. In the present paper we will attempt to implement the above strategy to study the Eguchi–Hanson gravitational instanton solution and its interpretation by ‘t Hooft in the context of quantum gravity Hilbert space as an event and a possible solitonic “extended” particle. Subsequently we do not only reproduce the result of ‘t Hooft but also find the mass of a fundamental “exotic” symplictic-transfinite particle m1.8 MeV as well as the mass Mx and M (Planck) which are believed to determine the GUT and the total unification of all fundamental interactions respectively. This may be seen as a further confirmation to an argument which we put forward in various previous publications in favour of an alternative mass acquisition mechanism based on unification and duality considerations. Thus even in case that we never find the Higgs particle experimentally, the standard model would remain substantially intact as we can appeal to tunnelling and unification arguments to explain the mass. In fact a minority opinion at present is that finding the Higgs particle is not a final conclusive argument since one could ask further how the Higgs particle came to its mass which necessitates a second Higgs field. By contrast the present argument could be viewed as an ultimate theory based on the existence of a “super” force, beyond which nothing else exists. 相似文献
9.
Consider the first-order neutral nonlinear difference equation of the form , where τ > 0, σ i ≥ 0 ( i = 1, 2,…, m) are integers, { pn} and { qn} are nonnegative sequences. We obtain new criteria for the oscillation of the above equation without the restrictions Σ n=0∞ qn = ∞ or Σ n=0∞ nqn Σ j=n∞ qj = ∞ commonly used in the literature. 相似文献
10.
We consider the nonlinear parabolic equation ut = ( k( u) ux) x + b( u) x, where u = u( x, t, x ε R1, t > 0; k( u) ≥ 0, b( u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ( t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation ut = ( unux) x + ( uβ) x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations. 相似文献
11.
Let π i : Ei→ M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections si : M→ Ei. Suppose that U is an open neighborhood of s1( M) in E1 and F : U→ E2 a smooth embedding so that π 2Fs1 : M→ M is homotopic to a diffeomorphism f. We show that if k>[( n+1)/2]+1 then E1 and the induced bundle f*E2 are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [( n+1)/2]+1< kn we give examples of nonisomorphic oriented bundles E1 and E2 of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E1 and f*E2 are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[( n+1)/2]+1 and M is a homotopy n-sphere. 相似文献
12.
Let B be a separable Banach space. The following is one of the results proved in this paper. The Banach space B is of cotype p if and only if 1. dn, n 1, has no subsequence converging in probability, and 2. ∑n 1|an|p < ∞ whenever ∑n 1andn converges almost surely are equivalent for every sequence dn, n 1, of symmetric independent random elements taking values in B.
Author Keywords: Bounded in probability; convergence in probability; cotype; uniform tightness condition 相似文献
13.
If
are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n( n+1)/2, where n=dim H. For any two maximal nests
there exists a basis { f1, f2,…, fn} of H and a permutation π such that
and
where Mi= span{ f1, f2,…, fi} and Ni= span{ fπ(1), fπ(2),…, fπ(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π( j)= n− j+1,1 jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras. 相似文献
14.
If a˜cardinal κ 1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ 0 and its new cofinality, ρ, is less than κ 0, then, under some additional assumptions, each cardinal λ>κ 1 less than cc( P(κ 1)/[κ 1] <κ1) is collapsed to κ 0 as well. If in addition N= M[ f], where f : ρ→κ 1 is an unbounded mapping, then N is a˜|λ|=κ 0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba. 相似文献
15.
In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate 1 p < ∞, for every polynomial P of degree at most n + rn1/3, where W2 is a Freud weight such as exp(−¦ x¦ ), > 1, λ jn are the Christoffel numbers, xjn are the zeros of the orthonormal polynomials for the weight W2, and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m( n) if m( n) = n + ξ nn1/3, where ξ n → ∞ as n → ∞. Previous estimates could sum only over those xjn with ¦ xjn¦ σ x1n, some fixed 0 < σ < 1. 相似文献
16.
The slow growing hierarchy is commonly defined as follows: G0( x) = 0, Gx−1( x) := Gx( x) + 1 and Gλ( x) := Gλ[x]( x) where λ< 0 is a limit and ·[·]: 0∩ Lim × ω → 0 is a given assignment of fundamental sequences for the limits below 0. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy ( Gx) x<0 is slow growing, i.e. each Gx is majorized by a Kalmar elementary recursive function. It is shown in this paper that the slow growing hierarchy (Gx)x<0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by Gx for some <0. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences. The paper is essentially self-contained. Only little knowledge about ordinals less than 0 — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed. 相似文献
17.
In 1996, J.C. Bermond, T. Kodate, S. Perennes and N. Marlin conjectured that the set Fσ of fixed points of some complete rotation σ of the toroidal mesh TM( p) k is not separating (that is Fσ does not disconnect TM( p) k). They also conjectured that the set Fω of fixed points of any complete rotation ω of any Cayley digraph is not separating. In this paper, we prove the first conjecture and disprove the second one. 相似文献
18.
Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal IΔ in the polynomial ring A = k[ x1, …, xn], and its quotient k[Δ] = A/ IΔ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ * which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim k Tor iA ( k[Δ], k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ *. As corollaries, we prove that IΔ has a linear resolution as A-module if and only if Δ * is Cohen-Macaulay over k, and show how to compute the Betti numbers dim k Tor iA ( k[Δ], k) in some cases where Δ * is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed. 相似文献
19.
Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ 1. We consider the power method, i.e. that of choosing a vector v0 and setting vk = Akv0; then the Rayleigh quotients Rk = ( Avk, vk)/( vk, vk) usually converge to λ 1 as k → ∞ (here ( u, v) denotes their inner product). In this paper we give two methods for determining how close Rk is to λ 1. They are both based on a bound on λ 1 − Rk involving the difference of two consecutive Rayleigh quotients and a quantity ω k. While we do not know how to directly calculate ω k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ 1 − Rk which is proportional to (λ 2/λ 1) 2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ). 相似文献
20.
Oscillation criteria for the second-order half-linear differential equation [r(t)|ξ′(t)|−1 ξ′(t)]′ + p(t)|ξ(t)|−1ξ(t)=0, t t0 are established, where > 0 is a constant and
exists for t [ t0, ∞). We apply these results to the following equation: where
, D = ( D1,…, DN), Ω a = x
N : |x| ≥ a} is an exterior domain, and c C([a, ∞),
), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant. 相似文献
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