3,839 research outputs found
On the statistical distribution of prime numbers, a view from where the distribution of prime numbers is not erratic
Currently there is no known efficient formula for primes. Besides that, prime
numbers have great importance in e.g., information technology such as
public-key cryptography, and their position and possible or impossible
functional generation among the natural numbers is an ancient dilemma. The
properties of the functions 2ab+a+b in the domain of natural numbers are
introduced, analyzed, and exhibited to illustrate how these single out all the
prime numbers from the full set of odd numbers. The characterization of odd
primes vs. odd non-primes can be done with 2ab+a+b among the odd natural
numbers as an analogue to the other, well known type of fundamental
characterization for irrational and rational numbers among the real numbers.
The prime number theorem, twin primes and erratic nature of primes, are also
commented upon with respect to selection, as well as with the Fermat and Euler
numbers as examples.Comment: 27 pages, 4 tables, 3 figure
Mathematical Foundations of Consciousness
We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical
primitives. The Anti-foundation Axiom plays a significant role in our
development, since among other of its features, its replacement for the Axiom
of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic
interpretations. These interpretations also depend on such allied notions for
sets as pictures, graphs, decorations, labelings and various mappings that we
use. A syntax and semantics of operators acting on sets is developed. Such
features enable construction of a theory of non-well-founded sets that we use
to frame mathematical foundations of consciousness. To do this we introduce a
supplementary axiomatic system that characterizes experience and consciousness
as primitives. The new axioms proceed through characterization of so- called
consciousness operators. The Russell operator plays a central role and is shown
to be one example of a consciousness operator. Neural networks supply striking
examples of non-well-founded graphs the decorations of which generate
associated sets, each with a Platonic aspect. Employing our foundations, we
show how the supervening of consciousness on its neural correlates in the brain
enables the framing of a theory of consciousness by applying appropriate
consciousness operators to the generated sets in question
Searching the solution space in constructive geometric constraint solving with genetic algorithms
Geometric problems defined by constraints have an exponential number
of solution instances in the number of geometric elements involved.
Generally, the user is only interested in one instance such that
besides fulfilling the geometric constraints, exhibits some additional
properties.
Selecting a solution instance amounts to selecting a given root every
time the geometric constraint solver needs to compute the zeros of a
multi valuated function. The problem of selecting a given root is
known as the Root Identification Problem.
In this paper we present a new technique to solve the root
identification problem. The technique is based on an automatic search
in the space of solutions performed by a genetic algorithm. The user
specifies the solution of interest by defining a set of additional
constraints on the geometric elements which drive the search of the
genetic algorithm. The method is extended with a sequential niche
technique to compute multiple solutions. A number of case studies
illustrate the performance of the method.Postprint (published version
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