975,315 research outputs found
Scaled Boolean Algebras
Scaled Boolean algebras are a category of mathematical objects that arose
from attempts to understand why the conventional rules of probability should
hold when probabilities are construed, not as frequencies or proportions or the
like, but rather as degrees of belief in uncertain propositions. This paper
separates the study of these objects from that not-entirely-mathematical
problem that motivated them. That motivating problem is explicated in the first
section, and the application of scaled Boolean algebras to it is explained in
the last section. The intermediate sections deal only with the mathematics. It
is hoped that this isolation of the mathematics from the motivating problem
makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press,
To appear in Advances in Applied Mathematic
The Scaled Universe
It is shown that the mysterious quantum prescription of microphysics has
analogues at the scale of stars, galaxies and superclusters, the common feature
in all these cases being Brownian type fractality. These considerations are
shown to lead to pleasingly meaningful results in agreement with observed data.Comment: 8 pages, Te
Scaled Universe II
In an earlier paper we had pointed out that Quantum Mechanical type effects
are seen at different scales in the macro universe also. In this paper we
obtain a rationale for this, which lies in the picture of bound material
systems, spanning a Compton wavelength type extent, separated by much larger
and relatively much less dense distances.Comment: 4 pages, Te
Scaled-free objects II
This work creates two categories of "array-weighted sets" for the purposes of
constructing universal matrix-normed spaces and algebras. These universal
objects have the analogous universal property to the free vector space, lifting
maps completely bounded on a generation set to a completely bounded linear map
of the matrix-normed space.
Moreover, the universal matrix-normed algebra is used to prove the existence
of a free product for matrix-normed algebras using algebraic methods.Comment: 46 pages. Version 4 fixed a few minor typos. Version 3 added
matricial completion; fixed an arithmetic error in Example 3.5.10. Version 2
added a preliminaries section on weighted sets and matricial Banach spaces,
incorporating much of "Matricial Banach spaces" in summary; fixed a domain
issue in Lemma 3.3.2; simplified Examples 3.5.10 and 4.11; added more proofs
to Sections 4 and
Aging Scaled Brownian Motion
Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of
passive tracers in complex and biological systems. It is a highly
non-stationary process governed by the Langevin equation for Brownian motion,
however, with a power-law time dependence of the noise strength. Here we study
the aging properties of SBM for both unconfined and confined motion.
Specifically, we derive the ensemble and time averaged mean squared
displacements and analyze their behavior in the regimes of weak, intermediate,
and strong aging. A very rich behavior is revealed for confined aging SBM
depending on different aging times and whether the process is sub- or
superdiffusive. We demonstrate that the information on the aging factorizes
with respect to the lag time and exhibits a functional form, that is identical
to the aging behavior of scale free continuous time random walk processes.
While SBM exhibits a disparity between ensemble and time averaged observables
and is thus weakly non-ergodic, strong aging is shown to effect a convergence
of the ensemble and time averaged mean squared displacement. Finally, we derive
the density of first passage times in the semi-infinite domain that features a
crossover defined by the aging time.Comment: 10 pages, 8 figures, REVTe
Operator scaled Wiener bridges
We introduce operator scaled Wiener bridges by incorporating a matrix scaling
in the drift part of the SDE of a multidimensional Wiener bridge. A sufficient
condition for the bridge property of the SDE solution is derived in terms of
the eigenvalues of the scaling matrix. We analyze the asymptotic behavior of
the bridges and briefly discuss the question whether the scaling matrix
determines uniquely the law of the corresponding bridge.Comment: 21 page
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