1,287 research outputs found

### A nilpotent IP polynomial multiple recurrence theorem

We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and
McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important
tools in our proof include a generalization of Leibman's result that polynomial
mappings into a nilpotent group form a group and a multiparameter version of
the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate

### 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

### How do random Fibonacci sequences grow?

We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and
for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} =
|F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is independent and
either + with probability $p$ or - with probability $1-p$ ($0<p\le 1$). Our
main result is that the exponential growth of $F_n$ for $0<p\le 1$ (linear
case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by
$\int_0^\infty \log x d\nu_\alpha (x),$ where $\alpha$ is an explicit
function of $p$ depending on the case we consider, and $\nu_\alpha$ is an
explicit probability distribution on \RR_+ defined inductively on
Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent
is not an analytic function of $p$, since we prove that it is equal to zero for
$0<p\le1/3$. We also give some results about the variations of the largest
Lyapunov exponent, and provide a formula for its derivative

### Do We Invest Less Time in Children? Trends in Parental Time in Selected Industrialized Countries Since the 1960\u27s

This paper examines trends in parental time in selected industrialized countries since the 1960s using time-use survey data. Despite the time pressures to which today’s families are confronted, parents appear to be devoting more time to children than they did some 40 years ago. Results also suggest a decrease in the differences between fathers and mothers in time devoted to children. Mothers continue to devote more time to childcare than fathers, but the gender gap has been reduced. These results are observed in several countries and therefore suggest a large global trend towards an increase in parental time investment with their children

### Ramsey numbers and adiabatic quantum computing

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In
fact, for the two-color Ramsey numbers $R(m,n)$ with $m,n\geq 3$, only nine are
currently known. We present a quantum algorithm for the computation of the
Ramsey numbers $R(m,n)$. We show how the computation of $R(m,n)$ can be mapped
to a combinatorial optimization problem whose solution can be found using
adiabatic quantum evolution. We numerically simulate this adiabatic quantum
algorithm and show that it correctly determines the Ramsey numbers R(3,3) and
R(2,s) for $5\leq s\leq 7$. We then discuss the algorithm's experimental
implementation, and close by showing that Ramsey number computation belongs to
the quantum complexity class QMA.Comment: 4 pages, 1 table, no figures, published versio

### 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

### Growth and Structure of Stochastic Sequences

We introduce a class of stochastic integer sequences. In these sequences,
every element is a sum of two previous elements, at least one of which is
chosen randomly. The interplay between randomness and memory underlying these
sequences leads to a wide variety of behaviors ranging from stretched
exponential to log-normal to algebraic growth. Interestingly, the set of all
possible sequence values has an intricate structure.Comment: 4 pages, 4 figure

### 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 $2\times 2$ real matrices. All
these products are constructed using only two types of matrices, $A$ and $B$,
which are chosen according to a stochastic process. The matrix $A$ is singular,
namely its determinant is zero. This formula is derived by using a particular
decomposition for the matrix $B$, 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

### 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

- …