Dark Heart Fractal Software and Research Papers
Software: The papers listed below come with a suite of Mac XCode applications with full source code, and additional Matlab files.
The most up-to-date downloadable releases of the major XCode applications, which have been tested for both Tiger and Snow Leopard are as follows:
NOTE: Dark Heart and Riemann Zeta Viewers are now bundled together!!
- Riemann Zeta Viewer Latest Version 1.7.1 with Mellin transform smoothing: Application - Source - RZ Flight Manual
Includes movie generation, c-scripting, and many types of abstract L-function.
- MATLAB Toolbox with XCode viewers: Download - Manual
- Dark Heart Viewer Latest Version 1.4: Application - Source - DH Flight Manual
- Wave Function Method Viewer: Application - Source - WF Flight Manual
- Modified Inverse Iteration Viewer: Application - Source
DHViewer and RZViewer are the twin primary applications covering most of the research examples. RZViewer deals with all the weird analogues of the Riemann zeta function, while the twin application DHViewer examines a wide variety of rational and transcendental functions. This means that the widest variety of complex functions are explored including some of the most difficult ones to model.
The Wave Function method is a method I invented, which performs effective inverse iteration by forward mapping the domain and colouring by a wave function of the eventual iterated range. As far as I know it is the only method which enables inverse iteration of functions like zeta or compex functions whose inverses cannot be solved explicitly. For comparison see the modified inverse application which works only for the standard map f(z)=z2+c.
- A Dynamical Key to the Riemann Hypothesis May 2011 This note sets out a dynamical basis for the non-trivial zeros of the Riemann zeta function being on the critical line x = 1/2. It does not prove the Riemann Hypothesis (RH), but it does give a dynamical explanation for why zeta and the various L-functions do have their non-trivial zeros on the critical line and why other closely related functions do not. It suggests RH is an additional unprovable postulate of the number system, similar to the axiom of choice, associated with the limiting behavior of the primes as n tends to infinity.
- Fractal Geography of the Zeta Function Mar 2011 The quadratic Mandelbrot set has been referred to as the most complex and beautiful object in mathematics and the Riemann Zeta function takes the prize for the most complicated and enigmatic function. Here we elucidate the spectrum of Mandelbrot and Julia sets of Zeta, to unearth the geography of its chaotic and fractal diversities, combining these two extremes into one intrepid journey into the deepest abyss of complex function space.
- Exploding the Dark Heart of Chaos: March-December 2009 An exploration of the universality of the cardioid at the centre of the Mandelbrot set extended to the diversity of complex analytic functions with Mac XCode applications and Quicktime movies illustrating each example.
- Experimental Observations on the Riemann Hypothesis and the Collatz Conjecture 22 May 2009 This paper seeks to explore whether the Riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity and to provide an experimental basis to discover some of the mathematical enigmas surrounding these conjectures.
- Exploring Quantum and Classical Chaos in the Stadium Billiard 9 July 2009 Abstract: This paper explores quantum and classical chaos in the stadium billiard using Matlab simulations to investigate the behavior of wave functions in the stadium and the corresponding classical orbits believed to underlie wave function scarring.
- The Ising Model of Spin Interactions as an Oracle of Self-Organized Criticality, Fractal Mode-Locking and Power Law Statistics in Neurodynamics Aug 2009.