Abstract: We propose the numerical method defined by
xn+1 = xn − λ[f(xn − μh(xn))/]f'(xn) , n ∈ N,
and determine the control parameter λ and μ to converge cubically. In addition, we derive the asymptotic error constant. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-precision computability.
Abstract: This paper has, as its point of departure, the foundational
axiomatic theory of E. De Giorgi (1996, Scuola Normale
Superiore di Pisa, Preprints di Matematica 26, 1), based on two
primitive notions of quality and relation. With the introduction of
a unary relation, we develop a system totally based on the sole
primitive notion of relation. Such a modification enables a definition
of the concept of dynamic unary relation. In this way we construct a
simple language capable to express other well known theories such
as Robinson-s arithmetic or a piece of a theory of concatenation. A
key role in this system plays an abstract relation designated by “( )",
which can be interpreted in different ways, but in this paper we will
focus on the case when we can perform computations and obtain
results.
Abstract: In this paper a functional interpretation of quantum
theory (QT) with emphasis on quantum field theory (QFT) is proposed.
Besides the usual statements on relations between a functions
initial state and final state, a functional interpretation also contains
a description of the dynamic evolution of the function. That is, it
describes how things function. The proposed functional interpretation
of QT/QFT has been developed in the context of the author-s work
towards a computer model of QT with the goal of supporting
the largest possible scope of QT concepts. In the course of this
work, the author encountered a number of problems inherent in the
translation of quantum physics into a computer program. He came
to the conclusion that the goal of supporting the major QT concepts
can only be satisfied, if the present model of QT is supplemented
by a "functional interpretation" of QT/QFT. The paper describes a
proposal for that
Abstract: A computer model of Quantum Theory (QT) has been
developed by the author. Major goal of the computer model was
support and demonstration of an as large as possible scope of QT.
This includes simulations for the major QT (Gedanken-) experiments
such as, for example, the famous double-slit experiment.
Besides the anticipated difficulties with (1) transforming exacting
mathematics into a computer program, two further types of problems
showed up, namely (2) areas where QT provides a complete mathematical
formalism, but when it comes to concrete applications the
equations are not solvable at all, or only with extremely high effort;
(3) QT rules which are formulated in natural language and which do
not seem to be translatable to precise mathematical expressions, nor
to a computer program.
The paper lists problems in all three categories and describes also
the possible solutions or circumventions developed for the computer
model.