395 research outputs found
Random equations in aerodynamics
Literature was reviewed to identify aerodynamic models which might be treated by probablistic methods. The numerical solution of some integral equations that arise in aerodynamical problems were investigated. On the basis of the numerical studies a qualitative theory of random integral equations was developed to provide information on the behavior of the solutions of these equations (in particular, boundary and asymptotic behavior, and stability) and their statistical properties without actually obtaining explicit solutions of the equations
Computational solution of random equations
Issued as Progress reports no. [1-3], and Final report, Project no. G-37-63
On the generalized semigroup relation in the strong operator topology
AbstractLet be a Banach space, and let –() denote the Banach algebra of endomorphisms of . A one-parameter family of operator-valued functions {S(t), t∈R+}, where S(t):R+→–(), is said to be a generalized semigroup of operators on if (1) S(s+t)−−S(s)S(t)=F(s, t), s, t∈R+, (2) S(s)S(t)=S(t)S(s), (3) S(O)=I, where F(s,t): R+× ×R+ → –(). Solutions of Eq. (1) in the vmiform operator topology were considered in an earlier paper of the authors (Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 1170–1174). In this paper the authors investigate the analytical properties of Eq. (1) in the strong operator topology, when it is assumed that the perturbation family F(s, t) is bounded relative to the family S(t). A representation of S(t) is given; and, as an example, a generalized semigroup of translations on C[0, ∞] is considered
Condensation of the roots of real random polynomials on the real axis
We introduce a family of real random polynomials of degree n whose
coefficients a_k are symmetric independent Gaussian variables with variance
= e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly
the mean number of real roots for large n. As \alpha is varied, one finds
three different phases. First, for 0 \leq \alpha \sim
(\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase
where grows algebraically with a continuously varying exponent,
\sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for
\alpha > 2, one finds a third phase where \sim n. This family of real
random polynomials thus exhibits a condensation of their roots on the real line
in the sense that, for large n, a finite fraction of their roots /n are
real. This condensation occurs via a localization of the real roots around the
values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure
Finite-size scaling of directed percolation in the steady state
Recently, considerable progress has been made in understanding finite-size
scaling in equilibrium systems. Here, we study finite-size scaling in
non-equilibrium systems at the instance of directed percolation (DP), which has
become the paradigm of non-equilibrium phase transitions into absorbing states,
above, at and below the upper critical dimension. We investigate the
finite-size scaling behavior of DP analytically and numerically by considering
its steady state generated by a homogeneous constant external source on a
d-dimensional hypercube of finite edge length L with periodic boundary
conditions near the bulk critical point. In particular, we study the order
parameter and its higher moments using renormalized field theory. We derive
finite-size scaling forms of the moments in a one-loop calculation. Moreover,
we introduce and calculate a ratio of the order parameter moments that plays a
similar role in the analysis of finite size scaling in absorbing nonequilibrium
processes as the famous Binder cumulant in equilibrium systems and that, in
particular, provides a new signature of the DP universality class. To
complement our analytical work, we perform Monte Carlo simulations which
confirm our analytical results.Comment: 21 pages, 6 figure
QED cascades induced by circularly polarized laser fields
The results of Monte-Carlo simulations of electron-positron-photon cascades
initiated by slow electrons in circularly polarized fields of ultra-high
strength are presented and discussed. Our results confirm previous qualitative
estimations [A.M. Fedotov, et al., PRL 105, 080402 (2010)] of the formation of
cascades. This sort of cascades has revealed the new property of the
restoration of energy and dynamical quantum parameter due to the acceleration
of electrons and positrons by the field and may become a dominating feature of
laser-matter interactions at ultra-high intensities. Our approach incorporates
radiation friction acting on individual electrons and positrons.Comment: 13 pages, 10 figure
Distribution of roots of random real generalized polynomials
The average density of zeros for monic generalized polynomials,
, with real holomorphic and
real Gaussian coefficients is expressed in terms of correlation functions of
the values of the polynomial and its derivative. We obtain compact expressions
for both the regular component (generated by the complex roots) and the
singular one (real roots) of the average density of roots. The density of the
regular component goes to zero in the vicinity of the real axis like
. We present the low and high disorder asymptotic
behaviors. Then we particularize to the large limit of the average density
of complex roots of monic algebraic polynomials of the form with real independent, identically distributed
Gaussian coefficients having zero mean and dispersion . The average density tends to a simple, {\em universal}
function of and in the domain where nearly all the roots are located for
large .Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed
tarfile (.66MB) containing 8 Postscript figures is available by e-mail from
[email protected]
A new transform for solving the noisy complex exponentials approximation problem
The problem of estimating a complex measure made up by a linear combination
of Dirac distributions centered on points of the complex plane from a finite
number of its complex moments affected by additive i.i.d. Gaussian noise is
considered. A random measure is defined whose expectation approximates the
unknown measure under suitable conditions. An estimator of the approximating
measure is then proposed as well as a new discrete transform of the noisy
moments that allows to compute an estimate of the unknown measure. A small
simulation study is also performed to experimentally check the goodness of the
approximations.Comment: 42 pages, 5 figure
Two-Body Random Ensembles: From Nuclear Spectra to Random Polynomials
The two-body random ensemble (TBRE) for a many-body bosonic theory is mapped
to a problem of random polynomials on the unit interval. In this way one can
understand the predominance of 0+ ground states, and analytic expressions can
be derived for distributions of lowest eigenvalues, energy gaps, density of
states and so forth. Recently studied nuclear spectroscopic properties are
addressed.Comment: 8 pages, 4 figures. To appear in Physical Review Letter
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