36,841 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 difficulty of prime factorization is a consequence of the positional numeral system
The importance of the prime factorization problem is very well known
(e.g., many security protocols are based on the impossibility of a fast factorization
of integers on traditional computers). It is necessary from a number k
to establish two primes a and b giving k = a · b. Usually, k is written in a positional
numeral system. However, there exists a variety of numeral systems
that can be used to represent numbers. Is it true that the prime factorization is
difficult in any numeral system? In this paper, a numeral system with partial
carrying is described. It is shown that this system contains numerals allowing
one to reduce the problem of prime factorization to solving [K/2] − 1
systems of equations, where K is the number of digits in k (the concept of
digit in this system is more complex than the traditional one) and [u] is the
integer part of u. Thus, it is shown that the difficulty of prime factorization is
not in the problem itself but in the fact that the positional numeral system is
used traditionally to represent numbers participating in the prime factorization.
Obviously, this does not mean that P=NP since it is not known whether
it is possible to re-write a number given in the traditional positional numeral
system to the new one in a polynomial time
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
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