64,209 research outputs found
On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions
The implementation of global optimization algorithms, using the arithmetic of
infinity, is considered. A relatively simple version of implementation is
proposed for the algorithms that possess the introduced property of strong
homogeneity. It is shown that the P-algorithm and the one-step Bayesian
algorithm are strongly homogeneous.Comment: 11 pages, 1 figur
The Olympic medals ranks, lexicographic ordering and numerical infinities
Several ways used to rank countries with respect to medals won during
Olympic Games are discussed. In particular, it is shown that the unofficial
rank used by the Olympic Committee is the only rank that does not allow
one to use a numerical counter for ranking – this rank uses the lexicographic
ordering to rank countries: one gold medal is more precious than any number
of silver medals and one silver medal is more precious than any number of
bronze medals. How can we quantify what do these words, more precious,
mean? Can we introduce a counter that for any possible number of medals
would allow us to compute a numerical rank of a country using the number
of gold, silver, and bronze medals in such a way that the higher resulting
number would put the country in the higher position in the rank? Here we
show that it is impossible to solve this problem using the positional numeral
system with any finite base. Then we demonstrate that this problem can be
easily solved by applying numerical computations with recently developed
actual infinite numbers. These computations can be done on a new kind of
a computer – the recently patented Infinity Computer. Its working software
prototype is described briefly and examples of computations are given. It is
shown that the new way of counting can be used in all situations where the
lexicographic ordering is required
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area
The Koch snowflake is one of the first fractals that were mathematically
described. It is interesting because it has an infinite perimeter in the limit
but its limit area is finite. In this paper, a recently proposed computational
methodology allowing one to execute numerical computations with infinities
and infinitesimals is applied to study the Koch snowflake at infinity. Numerical
computations with actual infinite and infinitesimal numbers can be
executed on the Infinity Computer being a new supercomputer patented in
USA and EU. It is revealed in the paper that at infinity the snowflake is not
unique, i.e., different snowflakes can be distinguished for different infinite
numbers of steps executed during the process of their generation. It is then
shown that for any given infinite number n of steps it becomes possible to
calculate the exact infinite number, Nn, of sides of the snowflake, the exact
infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn,
of the Koch snowflake as the result of multiplication of the infinite Nn by
the infinitesimal Ln. It is established that for different infinite n and k the
infinite perimeters Pn and Pk are also different and the difference can be infinite.
It is shown that the finite areas An and Ak of the snowflakes can be
also calculated exactly (up to infinitesimals) for different infinite n and k and
the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed
starting from different initial conditions are also studied and their
quantitative characteristics at infinity are computed
Autocalibration with the Minimum Number of Cameras with Known Pixel Shape
In 3D reconstruction, the recovery of the calibration parameters of the
cameras is paramount since it provides metric information about the observed
scene, e.g., measures of angles and ratios of distances. Autocalibration
enables the estimation of the camera parameters without using a calibration
device, but by enforcing simple constraints on the camera parameters. In the
absence of information about the internal camera parameters such as the focal
length and the principal point, the knowledge of the camera pixel shape is
usually the only available constraint. Given a projective reconstruction of a
rigid scene, we address the problem of the autocalibration of a minimal set of
cameras with known pixel shape and otherwise arbitrarily varying intrinsic and
extrinsic parameters. We propose an algorithm that only requires 5 cameras (the
theoretical minimum), thus halving the number of cameras required by previous
algorithms based on the same constraint. To this purpose, we introduce as our
basic geometric tool the six-line conic variety (SLCV), consisting in the set
of planes intersecting six given lines of 3D space in points of a conic. We
show that the set of solutions of the Euclidean upgrading problem for three
cameras with known pixel shape can be parameterized in a computationally
efficient way. This parameterization is then used to solve autocalibration from
five or more cameras, reducing the three-dimensional search space to a
two-dimensional one. We provide experiments with real images showing the good
performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi
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