31,942 research outputs found
Boltzmann-Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics
Boltzmann-Gibbs random fields are defined in terms of the exponential
expression exp(-H), where H is a suitably defined energy functional of the
field states x(s). This paper presents a new Boltzmann-Gibbs model which
features local interactions in the energy functional. The interactions are
embodied in a spatial coupling function which uses smoothed kernel-function
approximations of spatial derivatives inspired from the theory of smoothed
particle hydrodynamics. A specific model for the interactions based on a
second-degree polynomial of the Laplace operator is studied. An explicit,
mesh-free expression of the spatial coupling function (precision function) is
derived for the case of the squared exponential (Gaussian) smoothing kernel.
This coupling function allows the model to seamlessly extend from discrete data
vectors to continuum fields. Connections with Gaussian Markov random fields and
the Mat\'{e}rn field with are established.Comment: 29 pages, 4 figure
Dynamics of supercooled liquids: density fluctuations and Mode Coupling Theory
We write equations of motion for density variables that are equivalent to
Newtons equations. We then propose a set of trial equations parameterised by
two unknown functions to describe the exact equations. These are chosen to best
fit the exact Newtonian equations. Following established ideas, we choose to
separate these trial functions into a set representing integrable motions of
density waves, and a set containing all effects of non-integrability. It
transpires that the static structure factor is fixed by this minimum condition
to be the solution of the Yvon-Born-Green (YBG) equation. The residual
interactions between density waves are explicitly isolated in their Newtonian
representation and expanded by choosing the dominant objects in the phase space
of the system, that can be represented by a dissipative term with memory and a
random noise. This provides a mapping between deterministic and stochastic
dynamics. Imposing the Fluctuation-Dissipation Theorem (FDT) allows us to
calculate the memory kernel. We write exactly the expression for it, following
two different routes, i.e. using explicitly Newtons equations, or instead,
their implicit form, that must be projected onto density pairs, as in the
development of the well-established Mode Coupling Theory (MCT). We compare
these two ways of proceeding, showing the necessity to enforce a new equation
of constraint for the two schemes to be consistent. Thus, while in the first
`Newtonian' representation a simple gaussian approximation for the random
process leads easily to the Mean Spherical Approximation (MSA) for the statics
and to MCT for the dynamics of the system, in the second case higher levels of
approximation are required to have a fully consistent theory
Large-time Behavior of the Solutions to Rosenau Type Approximations to the Heat Equation
In this paper we study the large-time behavior of the solution to a general
Rosenau type approximation to the heat equation, by showing that the solution
to this approximation approaches the fundamental solution of the heat equation
at a sub-optimal rate. The result is valid in particular for the central
differences scheme approximation of the heat equation, a property which to our
knowledge has never been observed before.Comment: 20 page
Separable time-causal and time-recursive spatio-temporal receptive fields
We present an improved model and theory for time-causal and time-recursive
spatio-temporal receptive fields, obtained by a combination of Gaussian
receptive fields over the spatial domain and first-order integrators or
equivalently truncated exponential filters coupled in cascade over the temporal
domain. Compared to previous spatio-temporal scale-space formulations in terms
of non-enhancement of local extrema or scale invariance, these receptive fields
are based on different scale-space axiomatics over time by ensuring
non-creation of new local extrema or zero-crossings with increasing temporal
scale. Specifically, extensions are presented about parameterizing the
intermediate temporal scale levels, analysing the resulting temporal dynamics
and transferring the theory to a discrete implementation in terms of recursive
filters over time.Comment: 12 pages, 2 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1404.203
Approximations for the boundary crossing probabilities of moving sums of normal random variables
In this paper we study approximations for boundary crossing probabilities for
the moving sums of i.i.d. normal random variables. We propose approximating a
discrete time problem with a continuous time problem allowing us to apply
developed theory for stationary Gaussian processes and to consider a number of
approximations (some well known and some not). We bring particular attention to
the strong performance of a newly developed approximation that corrects the use
of continuous time results in a discrete time setting. Results of extensive
numerical comparisons are reported. These results show that the developed
approximation is very accurate even for small window length
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