1,098 research outputs found
The mobile Boolean model: an overview and further results
This paper offers an overview of the mobile Boolean stochastic geometric
model which is a time-dependent version of the ordinary Boolean model in a
Euclidean space of dimension . The main question asked is that of obtaining
the law of the detection time of a fixed set. We give various ways of thinking
about this which result into some general formulas. The formulas are solvable
in some special cases, such the inertial and Brownian mobile Boolean models. In
the latter case, we obtain some expressions for the distribution of the
detection time of a ball, when the dimension is odd and asymptotics when
is even. Finally, we pose some questions for future research.Comment: 19 page
A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity
We give a short proof of the Cauchy-Binet determinantal formula using
multilinear algebra by first generalizing it to an identity {\em not} involving
determinants. By extending the formula to abstract Hilbert spaces we obtain, as
a corollary, a generalization of the classical Parseval identity.Comment: 9 pages, 2 diagram
Stationary flows and uniqueness of invariant measures
In this short paper, we consider a quadruple ,where is a -algebra of subsets of , and is
a measurable bijection from into itself that preserves the measure
. For each , we consider the measure obtained by taking
cycles (excursions) of iterates of from . We then derive a relation
for that involves the forward and backward hitting times of by the
trajectory at a point .
Although classical in appearance, its use in obtaining uniqueness of invariant
measures of various stochastic models seems to be new. We apply the concept to
countable Markov chains and Harris processes
Pattern classification using a linear associative memory
Pattern classification is a very important image processing task. A typical pattern classification algorithm can be broken into two parts; first, the pattern features are extracted and, second, these features are compared with a stored set of reference features until a match is found. In the second part, usually one of the several clustering algorithms or similarity measures is applied. In this paper, a new application of linear associative memory (LAM) to pattern classification problems is introduced. Here, the clustering algorithms or similarity measures are replaced by a LAM matrix multiplication. With a LAM, the reference features need not be separately stored. Since the second part of most classification algorithms is similar, a LAM standardizes the many clustering algorithms and also allows for a standard digital hardware implementation. Computer simulations on regular textures using a feature extraction algorithm achieved a high percentage of successful classification. In addition, this classification is independent of topological transformations
Convergence to the Tracy-Widom distribution for longest paths in a directed random graph
We consider a directed graph on the 2-dimensional integer lattice, placing a
directed edge from vertex to , whenever ,
, with probability , independently for each such pair of
vertices. Let denote the maximum length of all paths contained in an
rectangle. We show that there is a positive exponent , such
that, if , as , then a properly centered/rescaled
version of converges weakly to the Tracy-Widom distribution. A
generalization to graphs with non-constant probabilities is also discussed.Comment: 20 pages, 2 figure
A spin quantum bit with ferromagnetic contacts for circuit QED
We theoretically propose a scheme for a spin quantum bit based on a double
quantum dot contacted to ferromagnetic elements. Interface exchange effects
enable an all electric manipulation of the spin and a switchable strong
coupling to a superconducting coplanar waveguide cavity. Our setup does not
rely on any specific band structure and can in principle be realized with many
different types of nanoconductors. This allows to envision on-chip single spin
manipulation and read-out using cavity QED techniques
Iterating Brownian motions, ad libitum
Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over
the whole real line such that B_i(0)=0. We consider the nth iterated Brownian
motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of
processes (W_n) do not converge in a functional sense, we prove that the
finite-dimensional marginals converge. As a consequence, we deduce that the
random occupation measures of W_n converge towards a random probability measure
\mu_\infty. We then prove that \mu_\infty almost surely has a continuous
density which must be thought of as the local time process of the infinite
iteration of independent Brownian motions
A note on the convergence of renewal and regenerative processes to a Brownian bridge
The standard functional central limit theorem for a renewal process with
finite mean and variance, results in a Brownian motion limit. This note shows
how to obtain a Brownian bridge process by a direct procedure that does not
involve conditioning. Several examples are also considered.Comment: 7 page
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