International Journal of Scientific Engineering and Technology

Sundaram Ramchandran ¹

Affiliations
1 SUNY Binghamton, Binghamton, NY, US

Abstract

This article looks at the issues relating to and arising out of self similarity of scale with links to "fractal" geometry and more generally investigates the possibility of finding the limit of a dynamical process given an initial generator/set of generators as also the inverse problem of finding the initial image that results in a particular attractor given the process. The latter could also be related to finding the point at which the image vanishes (including the possibility of oscillatory phenomenon). This problem is viewed through the "Operator paradigm" and links to recursive reflective phenomenon such as consciousness are examined. The above is also linked to the "Satisfiability problem" in computer science with links to Godel's theorem in mathematical logic. At a more constructive level, methods of constructing more general shapes out of the basic shapes such as "Seiperenski's Triangle" through basic operations (including diffraction) are examined with the possibility of lifting the results obtained to functions defined on the above shapes. Some paradoxes relating to scale which emerge in this process such as Zeno's paradox and other related paradoxes are examined. The article concludes by examining the possibilities of applying the outlined techniques to mathematical logic/meta-mathematics itself underlining the constructive nature of mathematical proof (with possible links to the area of optimization, especially discrete and combinatorial optimization).

Keywords

Self-Similarity, Fractal Dimension, Complex Dimension, Unit Scale, Reciprocal/Frequency Space, Solitons, Conceptual Lattices, Sieperenski's Triangle, Attractor/Limit, Limit Cycle, Iterated Function Systems, Operator, Zeno's Paradox, Infinite Series, Chebyshev Polynomials, Godel's Theorem, Heisenberg's Uncertainty Principle, Truth Value.

Full Text

   

Sundaram Ramchandran ¹

Affiliations
1 SUNY Binghamton, Binghamton, NY, US

Abstract

For modulating signals, the compact (see note 1) representation of the signals assumes importance to simplify the design of circuits and in the context of design of systems that are robust. This article examines the nature of numbers and goes on to survey the methods of representing algebraic irrational numbers. It also briefly examines the methods of physically realizing the same, especially in the context of frequency modulation techniques, speech processing and other applications as also to issues of sampling in DSP.

Keywords

Irrational, Transcendental, Polynomial, Roots, Phi, Imaginary, Complex, Recursive, Dynamical System, Chaos, Hysteresis, Base, Fields, Rings, Algebraic Structure.

Full Text