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961.
962.
We consider the existence of distributional (or L
2
) solutions of the matrix refinement equation
where P is an r×r matrix with trigonometric polynomial entries.
One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional
solution if and only if the matrix P
(0) has an eigenvalue of the form 2
n
, . A characterization of the existence of L
2
-solutions of the above matrix refinement equation in terms of the mask is also given.
A concept of L
2
-weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions
of a matrix refinement equation, we characterize this weak stability in terms of the mask.
August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997. 相似文献
963.
The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes.
A class of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation
width = which measures the worst-case approximation error over all functions by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρ
n
(W
r,d
p
, L
q
) , both being a constant factor of n
-r/d
, for a Sobolev class W
r,d
p
, . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of W
r,d
p
by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators.
March 12, 1997. Dates revised: August 26, 1997, October 24, 1997, March 16, 1998, June 15, 1998. Date accepted: June 25, 1998. 相似文献
964.
For a wide class of local martingales (M
t
) there is a default function, which is not identically zero only when (M
t
) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability
of functions of sup
s≤t
M
s
in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales,
and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting
and last exit times.
Received: 6 August 1996 / Revised version: 27 July 1998 相似文献
965.
Nobuo Yoshida 《Probability Theory and Related Fields》1999,115(1):1-40
We consider a ferromagnetic spin system with unbounded interactions on the d-dimensional integer lattice (d > 1). Under mild assumptions on the one-body interactions (so that arbitrarily deep double wells are allowed), we prove that
if the coupling constants are small enough, then the finite volume Gibbs states satisfy the log-Sobolev inequality uniformly
in the volume and the boundary condition.
Received: 11 November 1997 / Revised version: 17 July 1998 相似文献
966.
Jean Bertoin 《Probability Theory and Related Fields》1999,114(1):97-121
We investigate the nature of the intersection of two independent regenerative sets. The approach combines Bochners subordination
and potential theory for a pair of Markov processes in duality.
Received: 21 November 1997 / Revised version: 31 August 1998 相似文献
967.
Jean-François Delmas 《Probability Theory and Related Fields》1999,114(4):505-547
We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting
behavior of the volume of the ɛ-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous
result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove
the support of X
t
is capacity-equivalent to [0, 1]2 in ℝd, d≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]4 in ℝd, d≥ 5.
Received: 7 April 1998 / Revised version: 2 October 1998 相似文献
968.
We investigate the limiting fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks
with finitely many patterns at the critical temperature 1/β
c
= 1. At the critical temperature, the measure-valued random variables given by the distribution of the appropriately scaled
order parameter under the Gibbs measure converge weakly towards a random measure which is non-Gaussian in the sense that it
is not given by a Dirac measure concentrated in a Gaussian distribution. This remains true in the case of β = β
N
→β
c
= 1 as N→∞ provided β
N
converges to β
c
= 1 fast enough, i.e., at speed ?(1/). The limiting distribution is explicitly given by its (random) density.
Received: 12 May 1998 / Revised version: 14 October 1998 相似文献
969.
J. Müller 《Constructive Approximation》1999,15(4):523-536
Taylor sections S
n
(f) of an entire function f often provide easy computable polynomial approximants of f . However, while the rate of convergence of (S
n
(f))
n
is nearly optimal on circles around the origin, this is no longer true for other plane sets as, for example, real compact
intervals. The aim of this paper is to construct for certain families of (entire) functions sequences of polynomial approximants
which are computable with essentially the same effort as Taylor sections and which have a better rate of convergence on some
parts of the plane. The resulting method may be applied, for example, to (modified) Bessel functions, to confluent hypergeometric
functions, or to parabolic cylinder functions.
October 2, 1997. Date revised: March 12, 1998. Date accepted: April 28, 1998. 相似文献
970.
Superpolynomial Lower Bounds for Monotone Span Programs 总被引:2,自引:0,他引:2
monotone span programs computing explicit functions. The best previous lower bound was by Beimel, Gál, Paterson [7]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower
bounds are based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. We prove an lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower
bounds for linear secret sharing schemes.
We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear
size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the
perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields.
Received: August 1, 1996 相似文献