92 research outputs found
Coupled Nonlinear Schr\"{o}dinger equation and Toda equation (the Root of Integrability)
We consider the relation between the discrete coupled nonlinear
Schr\"{o}dinger equation and Toda equation. Introducing complex times we can
show the intergability of the discrete coupled nonlinear Schr\"{o}dinger
equation. In the same way we can show the integrability in coupled case of dark
and bright equations. Using this method we obtain several integrable equations.Comment: 11 pages, LateX, to apper in J. Phys. Soc. Jpn. Vol. 66, No
Digital herders and phase transition in a voting model
In this paper, we discuss a voting model with two candidates, C_1 and C_2. We
set two types of voters--herders and independents. The voting of independent
voters is based on their fundamental values; on the other hand, the voting of
herders is based on the number of votes. Herders always select the majority of
the previous votes, which is visible to them. We call them digital herders.
We can accurately calculate the distribution of votes for special cases. When
r>=3, we find that a phase transition occurs at the upper limit of t, where t
is the discrete time (or number of votes). As the fraction of herders
increases, the model features a phase transition beyond which a state where
most voters make the correct choice coexists with one where most of them are
wrong. On the other hand, when r<3, there is no phase transition. In this case,
the herders' performance is the same as that of the independent voters.
Finally, we recognize the behavior of human beings by conducting simple
experiments.Comment: 26 pages, 10 figure
The Davey Stewartson system and the B\"{a}cklund Transformations
We consider the (coupled) Davey-Stewartson (DS) system and its B\"{a}cklund
transformations (BT). Relations among the DS system, the double
Kadomtsev-Petviashvili (KP) system and the Ablowitz-Ladik hierarchy (ALH) are
established. The DS hierarchy and the double KP system are equivalent. The ALH
is the BT of the DS system in a certain reduction. {From} the BT of coupled DS
system we can obtain new coupled derivative nonlinear Schr\"{o}dinger
equations.Comment: 13 pages, LaTe
Correlated Binomial Models and Correlation Structures
We discuss a general method to construct correlated binomial distributions by
imposing several consistent relations on the joint probability function. We
obtain self-consistency relations for the conditional correlations and
conditional probabilities. The beta-binomial distribution is derived by a
strong symmetric assumption on the conditional correlations. Our derivation
clarifies the 'correlation' structure of the beta-binomial distribution. It is
also possible to study the correlation structures of other probability
distributions of exchangeable (homogeneous) correlated Bernoulli random
variables. We study some distribution functions and discuss their behaviors in
terms of their correlation structures.Comment: 12 pages, 7 figure
Evaluation of Tranche in Securitization and Long-range Ising Model
This econophysics work studies the long-range Ising model of a finite system
with spins and the exchange interaction and the external
field as a modely for homogeneous credit portfolio of assets with default
probability and default correlation . Based on the discussion
on the phase diagram, we develop a perturbative calculation method for
the model and obtain explicit expressions for and the
normalization factor in terms of the model parameters and . The
effect of the default correlation on the probabilities
for defaults and on the cumulative distribution
function are discussed. The latter means the average loss rate
of the``tranche'' (layered structure) of the securities (e.g. CDO), which are
synthesized from a pool of many assets. We show that the expected loss rate of
the subordinated tranche decreases with and that of the senior
tranche increases linearly, which are important in their pricing and ratings.Comment: 21 pages, 9 figure
Hungry Volterra equation, multi boson KP hierarchy and Two Matrix Models
We consider the hungry Volterra hierarchy from the view point of the multi
boson KP hierarchy. We construct the hungry Volterra equation as the
B\"{a}cklund transformations (BT) which are not the ordinary ones. We call them
``fractional '' BT. We also study the relations between the (discrete time)
hungry Volterra equation and two matrix models. From this point of view we
study the reduction from (discrete time) 2d Toda lattice to the (discrete time)
hungry Volterra equation.Comment: 13 pages, LaTe
Orthonormal Polynomials on the Unit Circle and Spatially Discrete Painlev\'e II Equation
We consider the polynomials orthonormal with respect to the weight on the unit circle in the complex plane. The leading coefficient
is found to satisfy a difference-differential (spatially discrete)
equation which is further proved to approach a third order differential
equation by double scaling. The third order differential equation is equivalent
to the Painlev\'e II equation. The leading coefficient and second leading
coefficient of can be expressed asymptotically in terms of the
Painlev\'e II function.Comment: 16 page
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