1,050 research outputs found
Hausdorff dimension of boundaries of self-affine tiles in R^n
We present a new method to calculate the Hausdorff dimension of a certain
class of fractals: boundaries of self-affine tiles. Among the interesting
aspects are that even if the affine contraction underlying the iterated
function system is not conjugated to a similarity we obtain an upper- and
lower-bounds for its Hausdorff dimension. In fact, we obtain the exact value
for the dimension if the moduli of the eigenvalues of the underlying affine
contraction are all equal (this includes Jordan blocks). The tiles we discuss
play an important role in the theory of wavelets. We calculate the dimension
for a number of examples
A New Method for Multi-Bit and Qudit Transfer Based on Commensurate Waveguide Arrays
The faithful state transfer is an important requirement in the construction
of classical and quantum computers. While the high-speed transfer is realized
by optical-fibre interconnects, its implementation in integrated optical
circuits is affected by cross-talk. The cross-talk between densely packed
optical waveguides limits the transfer fidelity and distorts the signal in each
channel, thus severely impeding the parallel transfer of states such as
classical registers, multiple qubits and qudits. Here, we leverage on the
suitably engineered cross-talk between waveguides to achieve the parallel
transfer on optical chip. Waveguide coupling coefficients are designed to yield
commensurate eigenvalues of the array and hence, periodic revivals of the input
state. While, in general, polynomially complex, the inverse eigenvalue problem
permits analytic solutions for small number of waveguides. We present exact
solutions for arrays of up to nine waveguides and use them to design realistic
buses for multi-(qu)bit and qudit transfer. Advantages and limitations of the
proposed solution are discussed in the context of available fabrication
techniques
Transients in the Synchronization of Oscillator Arrays
The purpose of this note is threefold. First we state a few conjectures that
allow us to rigorously derive a theory which is asymptotic in N (the number of
agents) that describes transients in large arrays of (identical) linear damped
harmonic oscillators in R with completely decentralized nearest neighbor
interaction. We then use the theory to establish that in a certain range of the
parameters transients grow linearly in the number of agents (and faster outside
that range). Finally, in the regime where this linear growth occurs we give the
constant of proportionality as a function of the signal velocities (see [3]) in
each of the two directions. As corollaries we show that symmetric interactions
are far from optimal and that all these results independent of (reasonable)
boundary conditions.Comment: 11 pages, 4 figure
Stability Conditions for Coupled Autonomous Vehicles Formations
In this paper, we give necessary conditions for stability of coupled
autonomous vehicles in R. We focus on linear arrays with decentralized
vehicles, where each vehicle interacts with only a few of its neighbors. We
obtain explicit expressions for necessary conditions for stability in the cases
that a system consists of a periodic arrangement of two or three different
types of vehicles, i.e. configurations as follows: ...2-1-2-1 or
...3-2-1-3-2-1. Previous literature indicated that the (necessary) condition
for stability in the case of a single vehicle type (...1-1-1) held that the
first moment of certain coefficients of the interactions between vehicles has
to be zero. Here, we show that that does not generalize. Instead, the
(necessary) condition in the cases considered is that the first moment plus a
nonlinear correction term must be zero
On Rank Driven Dynamical Systems
We investigate a class of models related to the Bak-Sneppen model, initially
proposed to study evolution. The BS model is extremely simple and yet captures
some forms of "complex behavior" such as self-organized criticality that is
often observed in physical and biological systems.
In this model, random fitnesses in are associated to agents located
at the vertices of a graph . Their fitnesses are ranked from worst (0) to
best (1). At every time-step the agent with the worst fitness and some others
\emph{with a priori given rank probabilities} are replaced by new agents with
random fitnesses. We consider two cases: The \emph{exogenous case} where the
new fitnesses are taken from an a priori fixed distribution, and the
\emph{endogenous case} where the new fitnesses are taken from the current
distribution as it evolves.
We approximate the dynamics by making a simplifying independence assumption.
We use Order Statistics and Dynamical Systems to define a \emph{rank-driven
dynamical system} that approximates the evolution of the \emph{distribution} of
the fitnesses in these rank-driven models, as well as in the Bak-Sneppen model.
For this simplified model we can find the limiting marginal distribution as a
function of the initial conditions. Agreement with experimental results of the
BS model is excellent.Comment: 12 gigures, 20 page
Transients of platoons with asymmetric and different Laplacians
We consider an asymmetric control of platoons of identical vehicles with
nearest-neighbor interaction. Recent results show that if the vehicle uses
different asymmetries for position and velocity errors, the platoon has a short
transient and low overshoots. In this paper we investigate the properties of
vehicles with friction. To achieve consensus, an integral part is added to the
controller, making the vehicle a third-order system. We show that the
parameters can be chosen so that the platoon behaves as a wave equation with
different wave velocities. Simulations suggest that our system has a better
performance than other nearest-neighbor scenarios. Moreover, an
optimization-based procedure is used to find the controller properties
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