491,585 research outputs found
Random path representation and sharp correlations asymptotics at high-temperatures
We recently introduced a robust approach to the derivation of sharp
asymptotic formula for correlation functions of statistical mechanics models in
the high-temperature regime. We describe its application to the nonperturbative
proof of Ornstein-Zernike asymptotics of 2-point functions for self-avoiding
walks, Bernoulli percolation and ferromagnetic Ising models. We then extend the
proof, in the Ising case, to arbitrary odd-odd correlation functions. We
discuss the fluctuations of connection paths (invariance principle), and relate
the variance of the limiting process to the geometry of the equidecay profiles.
Finally, we explain the relation between these results from Statistical
Mechanics and their counterparts in Quantum Field Theory
A note on convergence of the equi-energy sampler
In a recent paper `The equi-energy sampler with applications statistical
inference and statistical mechanics' [Ann. Stat. 34 (2006) 1581--1619], Kou,
Zhou & Wong have presented a new stochastic simulation method called the
equi-energy (EE) sampler. This technique is designed to simulate from a
probability measure , perhaps only known up to a normalizing constant. The
authors demonstrate that the sampler performs well in quite challenging
problems but their convergence results (Theorem 2) appear incomplete. This was
pointed out, in the discussion of the paper, by Atchad\'e & Liu (2006) who
proposed an alternative convergence proof. However, this alternative proof,
whilst theoretically correct, does not correspond to the algorithm that is
implemented. In this note we provide a new proof of convergence of the
equi-energy sampler based on the Poisson equation and on the theory developed
in Andrieu et al. (2007) for \emph{Non-Linear} Markov chain Monte Carlo (MCMC).
The objective of this note is to provide a proof of correctness of the EE
sampler when there is only one feeding chain; the general case requires a much
more technical approach than is suitable for a short note. In addition, we also
seek to highlight the difficulties associated with the analysis of this type of
algorithm and present the main techniques that may be adopted to prove the
convergence of it
Developing Mathematics Enrichment Workshops for Middle School Students: Philosophy and Sample Workshops
This paper describes our approach to organizing enrichment activities using advanced mathematics topics for diverse audiences of middle school students. We discuss our philosophy and approaches for the structure of these workshops, and then provide sample schedules and resource materials. The workshops cover activities on the following topics: Graphing Calculators; The Chaos Game; Statistical Sampling; CT Scans–the reconstruction problem; The Platonic and Archimedean solids; The Shape of Space; Symmetry; The Binary Number System and the game of NIM; Graph Theory: Proof by Counterexample
A simple proof of Born’s rule for statistical interpretation of quantum mechanics
The Born’s rule to interpret the square of wave function as the probability to get a specific value in measurement has been accepted as a postulate in foundations of quantum mechanics. Although there have been so many attempts at deriving this rule theoretically using different approaches such as frequency operator approach, many-world theory, Bayesian probability and envariance, literature shows that arguments in each of these methods are circular. In view of absence of a convincing theoretical proof, recently some researchers have carried out experiments to validate the rule up-to maximum possible accuracy using multi-order interference (Sinha et al, Science, 329, 418 [2010]). But, a convincing analytical proof of Born’s rule will make us understand the basic process responsible for exact square dependency of probability on wave function. In this paper, by generalizing the method of calculating probability in common experience into quantum mechanics, we prove the Born’s rule for statistical interpretation of wave function
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