Authors: Cristian Dumitrescu
Comments: 11 Pages. I hope that the interested reader will implement and test the algorithm (contact information included in the article).
In this article I describe an efficient, randomized algorithm (section 4) that solves the 3- SAT problem (known to be NP complete) with high probability, and a bit of the history of the problem under consideration. In the last section I present an interesting application, based on an idea that belongs to Godel.
Category: General Mathematics
Authors: Luis Sancho
Comments: 125 Pages.
According to General Systems Sciences the Universe and all its parts are fractal super-organisms. They can therefore be explained with mathematical, fractal languages and biological, organic laws.
Yet the mathematics of fractal systems and the causal logic of super-organisms differ from classic mathematics and logic. Most sciences are described with Euclidean mathematics using the Cartesian->Galilean->Einsteinian abstraction of a single space-time continuum; while they use Aristotelian Logic based in a single arrow of time or ‘will of the Universe’ – called energy or entropy in physics.
Fractal, organic systems however display a fractal geometry, as they extend through several scales of size and organization that form a ‘5th dimension’ of space-time, in which microcosmic entities (particles, cells) become parts of bigger ‘whole’ superorganisms, which again become smaller parts of bigger super-organisms, from particles to galaxies, from atoms to socio-biological systems.
To describe those scales and its causal logic flows of energy and information we need to evolve the present formalism of mathematics, redefining in non-euclidean fractal terms the 4 Euclidean postulates of mathematics (definition of point, line, plane and equality), departing departing from a new concept of ‘a point with breadth’, or fractal point that grows in size and display internal parts when we come closer to it.
This new geometry defines a topological Universe composed of ∞ organic systems, each of one displaying the 3 topologies of a 4D space-times:
An energetic spherical/planar membrane that separates the point from reality, an informative, hyperbolic, ‘zero point’ and an intermediate toroid, cyclical volume that transfers and trans-forms energy and information between those 2 E⇔I poles.
It also requires a more complex logic to explain it. Since such systems display in dynamic terms 3 arrows of causality and time, one for each of those topologies. There is therefore an arrow of creation of information and an arrow of creation of energy that converge, creating and reproducing all the complementary, fractal systems of the infinite, scalar universe.
In this paper we formalize the Non Aristotelian logic and Non Euclidean topology of all universal, organic systems, giving multiple examples of species of nature that obey the laws of the Non-AE complex Universe and its formalism, which should become the foundation of a new paradigm of science that improves according to the Correspondence Principle the Euclidean, Mechanistic, Simplex description of reality of the previous scientific paradigm.
Category: General Mathematics