128 research outputs found
Topological complexity of the relative closure of a semi-Pfaffian couple
Gabrielov introduced the notion of relative closure of a Pfaffian couple as
an alternative construction of the o-minimal structure generated by
Khovanskii's Pfaffian functions. In this paper, use the notion of format (or
complexity) of a Pfaffian couple to derive explicit upper-bounds for the
homology of its relative closure.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti
numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve
Quasi-exactly solvable quartic: elementary integrals and asymptotics
We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where
p, h and P are polynomials in one variable. For the case when h is an odd cubic
polynomial, we found an interesting identity which is used to describe the
spectral locus. We also establish some asymptotic properties of the QES
spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references,
corrected misprint
Reverse Detection of Short-Term Earthquake Precursors
We introduce a new approach to short-term earthquake prediction based on the
concept of selforganization of seismically active fault networks. That approach
is named "Reverse Detection of Precursors" (RDP), since it considers precursors
in reverse order of their appearance. This makes it possible to detect
precursors undetectable by direct analysis. Possible mechanisms underlying RDP
are outlined. RDP is described with a concrete example: we consider as
short-term precursors the newly introduced chains of earthquakes reflecting the
rise of an earthquake correlation range; and detect (retrospectively) such
chains a few months before two prominent Californian earthquakes - Landers,
1992, M = 7.6, and Hector Mine, 1999, M = 7.3, with one false alarm. Similar
results (described elsewhere) are obtained by RDP for 21 more strong
earthquakes in California (M >= 6.4), Japan (M >= 7.0) and the Eastern
Mediterranean (M >= 6.5). Validation of the RDP approach requires, as always,
prediction in advance for which this study sets up a base. We have the first
case of advance prediction; it was reported before Tokachi-oki earthquake (near
Hokkaido island, Japan), Sept. 25, 2003, M = 8.1. RDP has potentially important
applications to other precursors and to prediction of other critical phenomena
besides earthquakes. In particular, it might vindicate some short-term
precursors, previously rejected as giving too many false alarms.Comment: 17 pages, 5 figure
Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Dynamical Systems
In this paper we study a class of dynamical systems defined by Pfaffian maps. It is a sub-class of o-minimal dynamical systems which capture rich
continuous dynamics and yet can be studied using finite bisimulations.
The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors; see e.g. Brihaye et al (2004), Davoren (1999), Lafferriere et al (2000).
The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done by Korovina et al (2004) where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained.
The bounds provide a basis for designing efficient algorithms for computing
bisimulations, solving reachability and motion planning problems
Quasi-exactly solvable quartic: real algebraic spectral locus
We describe the real quasi-exactly solvable spectral locus of the
PT-symmetric quartic using the Nevanlinna parametrization.Comment: 17 pages, 11 figure
Spherical Quadrilaterals with Three Non-integer Angles
A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that one corner of a quadrilateral is integer (i.e., its angle is a multiple of π) while the angles at its other three corners are not multiples of π. The problem is equivalent to classification of Heun's equations with real parameters and unitary monodromy, with the trivial monodromy at one of its four singular point
Universality of Cluster Dynamics
We have studied the kinetics of cluster formation for dynamical systems of
dimensions up to interacting through elastic collisions or coalescence.
These systems could serve as possible models for gas kinetics, polymerization
and self-assembly. In the case of elastic collisions, we found that the cluster
size probability distribution undergoes a phase transition at a critical time
which can be predicted from the average time between collisions. This enables
forecasting of rare events based on limited statistical sampling of the
collision dynamics over short time windows. The analysis was extended to
L-normed spaces () to allow for some amount of
interpenetration or volume exclusion. The results for the elastic collisions
are consistent with previously published low-dimensional results in that a
power law is observed for the empirical cluster size distribution at the
critical time. We found that the same power law also exists for all dimensions
, 2D L norms, and even for coalescing collisions in 2D. This
broad universality in behavior may be indicative of a more fundamental process
governing the growth of clusters
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