18 research outputs found
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
Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming
This paper deals with an analysis of the Conjugate Gradient (CG) method (Hestenes and Stiefel in J Res Nat Bur Stand 49:409-436, 1952), in the presence of degenerates on indefinite linear systems. Several approaches have been proposed in the literature to issue the latter drawback in optimization frameworks, including reformulating the original linear system or recurring to approximately solving it. All the proposed alternatives seem to rely on algebraic considerations, and basically pursue the idea of improving numerical efficiency. In this regard, here we sketch two separate analyses for the possible CG degeneracy. First, we start detailing a more standard algebraic viewpoint of the problem, suggested by planar methods. Then, another algebraic perspective is detailed, relying on a novel recently proposed theory, which includes an additional number, namely grossone. The use of grossone allows to work numerically with infinities and infinitesimals. The results obtained using the two proposed approaches perfectly match, showing that grossone may represent a fruitful and promising tool to be exploited within Nonlinear Programming
Lower and Upper Estimates of the Quantity of Algebraic Numbers
It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using â‘ -based infinite numbers is applied to measure the set A (where the number â‘ is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the â‘ -based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in â‘ -based numbers