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

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 $y(x)$ of an ordinary differential equation (ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a finite approximation step $h$ 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 $h$. 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 $y(x)$ 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

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

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

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~$n$ of steps it becomes possible to calculate the exact infinite number, $N_n$, of sides of the snowflake, the exact infinitesimal length, $L_n$, of each side and the exact infinite perimeter, $P_n$, of the Koch snowflake as the result of multiplication of the infinite $N_n$ by the infinitesimal $L_n$. It is established that for different infinite $n$ and $k$ the infinite perimeters $P_n$ and $P_k$ are also different and the difference can be infinite. It is shown that the finite areas $A_n$ and $A_k$ of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite $n$ and $k$ and the difference $A_n - A_k$ 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

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