10,243 research outputs found
Detecting Simultaneous Integer Relations for Several Real Vectors
An algorithm which either finds an nonzero integer vector for
given real -dimensional vectors such
that or proves that no such integer vector with
norm less than a given bound exists is presented in this paper. The cost of the
algorithm is at most exact arithmetic
operations in dimension and the least Euclidean norm of such
integer vectors. It matches the best complexity upper bound known for this
problem. Experimental data show that the algorithm is better than an already
existing algorithm in the literature. In application, the algorithm is used to
get a complete method for finding the minimal polynomial of an unknown complex
algebraic number from its approximation, which runs even faster than the
corresponding \emph{Maple} built-in function.Comment: 10 page
Quantized Detector Networks: A review of recent developments
QDN (quantized detector networks) is a description of quantum processes in
which the principal focus is on observers and their apparatus, rather than on
states of SUOs (systems under observation). It is a realization of Heisenberg's
original instrumentalist approach to quantum physics and can deal with time
dependent apparatus, multiple observers and inter-frame physics. QDN is most
naturally expressed in the mathematical language of quantum computation, a
language ideally suited to describe quantum experiments as processes of
information exchange between observers and their apparatus. Examples in quantum
optics are given, showing how the formalism deals with quantum interference,
non-locality and entanglement. Particle decays, relativity and non-linearity in
quantum mechanics are discussed.Comment: 59 pages, 14 figures, to be published in Int. J. Mod. Phys.
Geodesic continued fractions and LLL
We discuss a proposal for a continued fraction-like algorithm to determine
simultaneous rational approximations to real numbers
. It combines an algorithm of Hermite and Lagarias
with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with
parameter as . The new idea in this paper is that checking
the LLL-conditions consists of solving linear equations in
For differential equations with r parameters, 2r+1 experiments are enough for identification
Given a set of differential equations whose description involves unknown
parameters, such as reaction constants in chemical kinetics, and supposing that
one may at any time measure the values of some of the variables and possibly
apply external inputs to help excite the system, how many experiments are
sufficient in order to obtain all the information that is potentially available
about the parameters? This paper shows that the best possible answer (assuming
exact measurements) is 2r+1 experiments, where r is the number of parameters.Comment: This is a minor revision of the previously submitted report; a couple
of typos have been fixed, and some comments and two new references have been
added. Please see http://www.math.rutgers.edu/~sontag for related wor
On optimal completions of incomplete pairwise comparison matrices
An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper
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