The extensive study of metric spaces and their embeddings
has so far focused on embeddings that preserve pairwise distances.
A very intriguing concept introduced by Feige
allows us to quantify the extent to which larger
structures are preserved by a given embedding.
We investigate this concept, focusing on several major graph families
such as paths, trees, cubes, and expanders.
We find some similarities to the regular (pairwise) distortion,
as well as some striking differences. 相似文献
We propose a simple method to efficiently probe dynamical nonstationarity in observed time series. In a space time-index plot, the density distributions as a function of normalized time-index are V-shaped due to nonstationarity. We show that this method is workable for short data sets and typical examples are illustrated. 相似文献
We study the Ginzburg-Landau functional
for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation
of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional.
Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001 相似文献
Stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences. One application is data assimilation, i.e., the combination of model output with observations to form the best picture of the system under study. For any given quantity to be estimated, the relative weights of the model and the data will be adjusted according to estimated model and data error statistics, so implementation of any data assimilation scheme will require some assumption about errors, which are considered to be random. For dynamical models, some assumption about the evolution of errors will be needed. Stochastic models are also applied in studies of predictability.
The formal theory of stochastic processes was well developed in the last half of the twentieth century. One consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification. In some cases the rules of ordinary calculus must be modified.
The formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists. The purpose of this article is to provide an informal introduction to the relevant theory, and to point out those situations in which that theory must be applied in order to model random processes correctly. 相似文献
A one-dimensional lattice random walk in the presence ofm equally spaced traps is considered. The step length distribution is a symmetric exponential. An explicit analytic expression is obtained for the probability that the random walk will be trapped at thejth trapping site. 相似文献
We provide an approximate analysis of the transient sojourn time for a processor sharing queue with time varying arrival and
service rates, where the load can vary over time, including periods of overload. Using the same asymptotic technique as uniform
acceleration as demonstrated in [12] and [13], we obtain fluid and diffusion limits for the sojourn time of the Mt/Mt/1 processor-sharing queue. Our analysis is enabled by the introduction of a “virtual customer” which differs from the notion
of a “tagged customer” in that the former has no effect on the processing time of the other customers in the system. Our analysis
generalizes to non-exponential service and interarrival times, when the fluid and diffusion limits for the queueing process
are known. 相似文献
A focused ion beam (FIB) Moiré method is proposed to measure the in-plane deformation of object in a micrometer scale. The FIB Moiré is generated by the interference between a prepared specimen grating and FIB raster scan lines. The principle of the FIB Moiré is described. The sensitivity and accuracy of deformation measurement are discussed in detail. Several specimen gratings with 0.14 and 0.20 μm spacing are used to generate FIB Moiré patterns. The FIB Moiré method is successfully used to measure the residual deformation in a micro-electro-mechanical system structure after removing the SiO2 sacrificial layer with a 5000 lines/mm grating. The results demonstrate the feasibility of this method. 相似文献