84 research outputs found
Higher order numerical differentiation on the Infinity Computer
There exist many applications where it is necessary to approximate
numerically derivatives of a function which is given by a computer procedure.
In particular, all the fields of optimization have a special interest in such a
kind of information. In this paper, a new way to do this is presented for a new
kind of a computer -- the Infinity Computer -- able to work numerically with
finite, infinite, and infinitesimal numbers. It is proved that the Infinity
Computer is able to calculate values of derivatives of a higher order for a
wide class of functions represented by computer procedures. It is shown that
the ability to compute derivatives of arbitrary order automatically and
accurate to working precision is an intrinsic property of the Infinity Computer
related to its way of functioning. Numerical examples illustrating the new
concepts and numerical tools are given.Comment: 12 pages, no figure
Solving ordinary differential equations on the Infinity Computer by working with infinitesimals numerically
There exists a huge number of numerical methods that iteratively construct
approximations to the solution of an ordinary differential equation
(ODE) starting from an initial value and using a
finite approximation step that influences the accuracy of the obtained
approximation. In this paper, a new framework for solving ODEs is presented for
a new kind of a computer -- the Infinity Computer (it has been patented and its
working prototype exists). The new computer is able to work numerically with
finite, infinite, and infinitesimal numbers giving so the possibility to use
different infinitesimals numerically and, in particular, to take advantage of
infinitesimal values of . To show the potential of the new framework a
number of results is established. It is proved that the Infinity Computer is
able to calculate derivatives of the solution and to reconstruct its
Taylor expansion of a desired order numerically without finding the respective
derivatives analytically (or symbolically) by the successive derivation of the
ODE as it is usually done when the Taylor method is applied. Methods using
approximations of derivatives obtained thanks to infinitesimals are discussed
and a technique for an automatic control of rounding errors is introduced.
Numerical examples are given.Comment: 25 pages, 1 figure, 3 table
Counting systems and the First Hilbert problem
The First Hilbert problem is studied in this paper by applying two
instruments: a new methodology distinguishing between mathematical objects and
mathematical languages used to describe these objects; and a new numeral system
allowing one to express different infinite numbers and to use these numbers for
measuring infinite sets. Several counting systems are taken into consideration.
It is emphasized in the paper that different mathematical languages can
describe mathematical objects (in particular, sets and the number of their
elements) with different accuracies. The traditional and the new approaches are
compared and discussed.Comment: 16 pages, no figure
Numerical computations and mathematical modelling with infinite and infinitesimal numbers
Traditional computers work with finite numbers. Situations where the usage of
infinite or infinitesimal quantities is required are studied mainly
theoretically. In this paper, a recently introduced computational methodology
(that is not related to the non-standard analysis) is used to work with finite,
infinite, and infinitesimal numbers \textit{numerically}. This can be done on a
new kind of a computer - the Infinity Computer - able to work with all these
types of numbers. The new computational tools both give possibilities to
execute computations of a new type and open new horizons for creating new
mathematical models where a computational usage of infinite and/or
infinitesimal numbers can be useful. A number of numerical examples showing the
potential of the new approach and dealing with divergent series, limits,
probability theory, linear algebra, and calculation of volumes of objects
consisting of parts of different dimensions are given.Comment: 20 pages, 3 figure
On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function
The Riemann Hypothesis has been of central interest to mathematicians for a
long time and many unsuccessful attempts have been made to either prove or
disprove it. Since the Riemann zeta function is defined as a sum of the
infinite number of items, in this paper, we look at the Riemann Hypothesis
using a new applied approach to infinity allowing one to easily execute
numerical computations with various infinite and infinitesimal numbers in
accordance with the principle `The part is less than the whole' observed in the
physical world around us. The new approach allows one to work with functions
and derivatives that can assume not only finite but also infinite and
infinitesimal values and this possibility is used to study properties of the
Riemann zeta function and the Dirichlet eta function. A new computational
approach allowing one to evaluate these functions at certain points is
proposed. Numerical examples are given. It is emphasized that different
mathematical languages can be used to describe mathematical objects with
different accuracies. The traditional and the new approaches are compared with
respect to their application to the Riemann zeta function and the Dirichlet eta
function. The accuracy of the obtained results is discussed in detail.Comment: 27 pages, 1 figur
A new applied approach for executing computations with infinite and infinitesimal quantities
A new computational methodology for executing calculations with infinite and
infinitesimal quantities is described in this paper. It is based on the
principle `The part is less than the whole' introduced by Ancient Greeks and
applied to all numbers (finite, infinite, and infinitesimal) and to all sets
and processes (finite and infinite). It is shown that it becomes possible to
write down finite, infinite, and infinitesimal numbers by a finite number of
symbols as particular cases of a unique framework. The new methodology has
allowed us to introduce the Infinity Computer working with such numbers (its
simulator has already been realized). Examples dealing with divergent series,
infinite sets, and limits are given.Comment: 30 pages, 2 figure
Generation of symmetric exponential sums
In this paper, a new method for generation of infinite series of symmetric
identities written for exponential sums in real numbers is proposed. Such
systems have numerous applications in theory of numbers, chaos theory,
algorithmic complexity, dynamic systems, etc. Properties of generated
identities are studied. Relations of the introduced method for generation of
symmetric exponential sums to the Morse-Hedlund sequence and to the theory of
magic squares are established
Using blinking fractals for mathematical modeling of processes of growth in biological systems
Many biological processes and objects can be described by fractals. The paper
uses a new type of objects - blinking fractals - that are not covered by
traditional theories considering dynamics of self-similarity processes. It is
shown that both traditional and blinking fractals can be successfully studied
by a recent approach allowing one to work numerically with infinite and
infinitesimal numbers. It is shown that blinking fractals can be applied for
modeling complex processes of growth of biological systems including their
season changes. The new approach allows one to give various quantitative
characteristics of the obtained blinking fractals models of biological systems.Comment: 19 pages, 12 figure
Parallel Information Algorithm with Local Tuning for Solving Multidimensional GO Problems
In this paper we propose a new parallel algorithm for solving global
optimization (GO) multidimensional problems. The method unifies two powerful
approaches for accelerating the search: parallel computations and local tuning
on the behavior of the objective function. We establish convergence conditions
for the algorithm and theoretically show that the usage of local information
during the global search permits to accelerate solving the problem
significantly. Results of numerical experiments executed with 100 test
functions are also reported.Comment: 11 page
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~ of steps it becomes possible to calculate the
exact infinite number, , of sides of the snowflake, the exact
infinitesimal length, , of each side and the exact infinite perimeter,
, of the Koch snowflake as the result of multiplication of the infinite
by the infinitesimal . It is established that for different infinite
and the infinite perimeters and are also different and the
difference can be infinite. It is shown that the finite areas and
of the snowflakes can be also calculated exactly (up to infinitesimals) for
different infinite and and the difference results to be
infinitesimal. Finally, snowflakes constructed starting from different initial
conditions are also studied and their quantitative characteristics at infinity
are computed.Comment: 16 pages, 1 figur
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