Given the restrictions noted above ( ergodicity, linearity ) the coherence function estimates the extent to which y ( t ) may be predicted from x ( t ) by an optimum linear least squares function.
32.
The averaging brackets \ left \ langle \ ldots \ right \ rangle is assumed to be an ensemble average over phase space or, under an assumption of ergodicity, a time average of a single system.
33.
Any time-invariant operations also preserves AEP, stationarity and ergodicity and we may easily turn a stationary process to non-stationary without losing AEP by nulling out a finite number of time samples in the process.
34.
However, complex disordered systems such as a spin glass show an even more complicated form of ergodicity breaking where the properties of the thermodynamic equilibrium state seen in practice are much more difficult to predict purely by symmetry arguments.
35.
Furthermore, weak mixing ( and thus also strong mixing ) implies ergodicity : that is, every system that is weakly mixing is also ergodic ( and so one says that mixing is a " stronger " notion than ergodicity ).
36.
Furthermore, weak mixing ( and thus also strong mixing ) implies ergodicity : that is, every system that is weakly mixing is also ergodic ( and so one says that mixing is a " stronger " notion than ergodicity ).
37.
"' Umbrella sampling "'is a technique in computational physics and chemistry, used to improve sampling of a system ( or different systems ) where ergodicity is hindered by the form of the system's energy landscape.
38.
Holowinsky was awarded the SASTRA Ramanujan Prize in 2011 for his contributions to " areas of mathematics influenced by the genius Srinivasa Ramanujan ", for proving, with Kannan Soundararajan, an important case of the quantum unique ergodicity ( QUE ) conjecture.
39.
The subsequent work of interpretation of the Parisi solution by M . Mezard, G . Parisi, M . A . Virasoro and many others revealed the complex nature of a glassy low temperature phase characterized by ergodicity breaking, ultrametricity and non-selfaverageness.
40.
Ratner's theorems provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form ? \ " G ", where " G " is a Lie group and ? is a lattice in " G ".
