1,318 research outputs found
Hill's Equation with Random Forcing Parameters: Determination of Growth Rates through Random Matrices
This paper derives expressions for the growth rates for the random 2 x 2
matrices that result from solutions to the random Hill's equation. The
parameters that appear in Hill's equation include the forcing strength and
oscillation frequency. The development of the solutions to this periodic
differential equation can be described by a discrete map, where the matrix
elements are given by the principal solutions for each cycle. Variations in the
forcing strength and oscillation frequency lead to matrix elements that vary
from cycle to cycle. This paper presents an analysis of the growth rates
including cases where all of the cycles are highly unstable, where some cycles
are near the stability border, and where the map would be stable in the absence
of fluctuations. For all of these regimes, we provide expressions for the
growth rates of the matrices that describe the solutions.Comment: 22 pages, 3 figure
Singularity results for functional equations driven by linear fractional transformations
We consider functional equations driven by linear fractional transformations,
which are special cases of de Rham's functional equations. We consider
Hausdorff dimension of the measure whose distribution function is the solution.
We give a necessary and sufficient condition for singularity. We also show that
they have a relationship with stationary measures.Comment: 14 pages, Title changed, to appear in Journal of Theoretical
Probabilit
The Future of Intergenerational Relations in Aging Societies
As the pressure mounts to reduce the public costs of supporting rapidly aging societies, responsibility for supporting elderly people will increasingly fall on their family members. This essay explores the familyâs capacity to respond to these growing challenges. In particular, we examine how family change and growing inequality pose special problems in developed nations, especially the United States. This essay mentions a series of studies supported by the MacArthur Foundation Research Network on an Aging Society that aim to examine the future of intergenerational exchange. We focus particularly on adults who have dependent and young-adult children and who must also care for elderly parents, a fraction of the population that will grow substantially in the coming twenty-five years
Ray splitting in paraxial optical cavities
We present a numerical investigation of the ray dynamics in a paraxial
optical cavity when a ray splitting mechanism is present. The cavity is a
conventional two-mirror stable resonator and the ray splitting is achieved by
inserting an optical beam splitter perpendicular to the cavity axis. We show
that depending on the position of the beam splitter the optical resonator can
become unstable and the ray dynamics displays a positive Lyapunov exponent.Comment: 13 pages, 7 figures, 1 tabl
Exact Lyapunov Exponent for Infinite Products of Random Matrices
In this work, we give a rigorous explicit formula for the Lyapunov exponent
for some binary infinite products of random real matrices. All
these products are constructed using only two types of matrices, and ,
which are chosen according to a stochastic process. The matrix is singular,
namely its determinant is zero. This formula is derived by using a particular
decomposition for the matrix , which allows us to write the Lyapunov
exponent as a sum of convergent series. Finally, we show with an example that
the Lyapunov exponent is a discontinuous function of the given parameter.Comment: 1 pages, CPT-93/P.2974,late
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
Recurrence and algorithmic information
In this paper we initiate a somewhat detailed investigation of the
relationships between quantitative recurrence indicators and algorithmic
complexity of orbits in weakly chaotic dynamical systems. We mainly focus on
examples.Comment: 26 pages, no figure
Quenched Averages for self-avoiding walks and polygons on deterministic fractals
We study rooted self avoiding polygons and self avoiding walks on
deterministic fractal lattices of finite ramification index. Different sites on
such lattices are not equivalent, and the number of rooted open walks W_n(S),
and rooted self-avoiding polygons P_n(S) of n steps depend on the root S. We
use exact recursion equations on the fractal to determine the generating
functions for P_n(S), and W_n(S) for an arbitrary point S on the lattice. These
are used to compute the averages and over different positions of S. We find that the connectivity constant
, and the radius of gyration exponent are the same for the annealed
and quenched averages. However, , and , where the exponents
and take values different from the annealed case. These
are expressed as the Lyapunov exponents of random product of finite-dimensional
matrices. For the 3-simplex lattice, our numerical estimation gives ; and , to be
compared with the annealed values and .Comment: 17 pages, 10 figures, submitted to Journal of Statistical Physic
âSwappingâ Families: Serial Parenting and Economic Support for Children
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71408/1/j.1741-3737.2000.00111.x.pd
Helicopter Parents and Landing Pad Kids: Intense Parental Support of Grown Children
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/92448/1/j.1741-3737.2012.00987.x.pd
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