6,057 research outputs found
Growth rate for the expected value of a generalized random Fibonacci sequence
A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/-
g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n.
We generalize these sequences to the case when the coin is unbalanced (denoting
by p the probability of a +), and the recurrence relation is of the form g_n =
|\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that
the expected value of g_n grows exponentially fast. When \lambda = \lambda_k =
2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n
grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression
for the growth rate. The involved methods extend (and correct) those introduced
in a previous paper by the second author
Finite-state Markov Chains obey Benford's Law
A sequence of real numbers (x_n) is Benford if the significands, i.e. the
fraction parts in the floating-point representation of (x_n) are distributed
logarithmically. Similarly, a discrete-time irreducible and aperiodic
finite-state Markov chain with probability transition matrix P and limiting
matrix P* is Benford if every component of both sequences of matrices (P^n -
P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that
established Benford behavior both for Newton's method and for
finite-dimensional linear maps, via the classical theories of uniform
distribution modulo 1 and Perron-Frobenius, this paper derives a simple
sufficient condition (nonresonant) guaranteeing that P, or the Markov chain
associated with it, is Benford. This result in turn is used to show that almost
all Markov chains are Benford, in the sense that if the transition
probabilities are chosen independently and continuously, then the resulting
Markov chain is Benford with probability one. Concrete examples illustrate the
various cases that arise, and the theory is complemented with several
simulations and potential applications.Comment: 31 pages, no figure
Near-optimal perfectly matched layers for indefinite Helmholtz problems
A new construction of an absorbing boundary condition for indefinite
Helmholtz problems on unbounded domains is presented. This construction is
based on a near-best uniform rational interpolant of the inverse square root
function on the union of a negative and positive real interval, designed with
the help of a classical result by Zolotarev. Using Krein's interpretation of a
Stieltjes continued fraction, this interpolant can be converted into a
three-term finite difference discretization of a perfectly matched layer (PML)
which converges exponentially fast in the number of grid points. The
convergence rate is asymptotically optimal for both propagative and evanescent
wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201
- …