Algebraic dynamics of certain gamma function values

Abstract We present significant numerical evidence, based on the entropy analysis by lumping of the binary expansion of certain values of the Gamma function, that some of these values correspond to incompressible algorithmic information. In particular, the value Γ(1/5) corresponds to a peak of non-compressibility as anticipated on a priori grounds from number-theoretic considerations. Other fundamental constants are similarly considered. This work may be viewed as an invitation for other researchers to apply information theoretic and decision theory techniques in number theory and analysis

Words and Transcendence

Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ should satisfy some of the laws that are shared by almost all numbers. There is no explicitly known example of a triple $(g,a,x)$, where $g\ge3$ is an integer, $a$ a digit in $\{0,...,g-1\}$ and $x$ a real irrational algebraic number, for which one can claim that the digit $a$ occurs infinitely often in the $g$-ary expansion of $x$. However, some progress has been made recently, thanks mainly to clever use of Schmidt's subspace theorem. We review some of these results

Nonintegrability, Chaos, and Complexity

Two-dimensional driven dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n*3 dimensions. Even driven-dissipative deterministic dynamical systems that are critical, chaotic or complex have n-1 local time-independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes. Those conserevation laws generally have either branch cuts, phase singularities, or both. The consequence of the existence of singular conservation laws for experimental data analysis, and also for the search for scale-invariant critical states via uncontrolled approximations in deterministic dynamical systems, is discussed. Finally, the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling, nonuniversal, and to some degree computationally complex

The New Science of Complexity

Deterministic chaos, and even maximum computational complexity, have been discovered within Newtonian dynamics. Economists assume that prices and price changes can also obey abstract mathematical laws of motion. Sociologists and other postmodernists advertise that physics and chemistry have outgrown their former limitations, that chaos and complexity provide new holistic paradigms for science, and that the boundaries between the hard and soft sciences, once impenetrable, have disappeared like the Berlin Wall. Three hundred years after the deaths of Galileo, Descartes, and Kepler, and the birth of Newton, reductionism appears to be on the decline, with holistic approaches to science on the upswing. We therefore examine the evidence that dynamical laws of motion may be discovered from empirical studies of chaotic or complex phenomena, and also review the foundation of reductionism in invariance principle

Multi-algorithmic Cryptography using Deterministic Chaos with Applications to Mobile Communications

In this extended paper, we present an overview of the principal issues associated with cryptography, providing historically significant examples for illustrative purposes as part of a short tutorial for readers that are not familiar with the subject matter. This is used to introduce the role that nonlinear dynamics and chaos play in the design of encryption engines which utilize different types of Iteration Function Systems (IFS). The design of such encryption engines requires that they conform to the principles associated with diffusion and confusion for generating ciphers that are of a maximum entropy type. For this reason, the role of confusion and diffusion in cryptography is discussed giving a design guide to the construction of ciphers that are based on the use of IFS. We then present the background and operating framework associated with a new product - CrypsticTM - which is based on the application of multi-algorithmic IFS to design encryption engines mounted on a USB memory stick using both disinformation and obfuscation to ‘hide’ a forensically inert application. The protocols and procedures associated with the use of this product are also briefly discussed

COMPUTER TOOLS FOR SOLVING MATHEMATICAL PROBLEMS: A REVIEW

The rapid development of digital computer hardware and software has had a dramatic influence on mathematics, and contrary. The advanced hardware and modern sophistical software such as computer visualization, symbolic computation, computerassisted proofs, multi-precision arithmetic and powerful libraries, have provided resolving many open problems, a huge very difficult mathematical problems, and discovering new patterns and relationships, far beyond a human capability. In the first part of the paper we give a short review of some typical mathematical problems solved by computer tools. In the second part we present some new original contributions, such as intriguing consequence of the presence of roundoff errors, distribution of zeros of random polynomials, dynamic study of zero-finding methods, a new three-point family of methods for solving nonlinear equations and two algorithms for the inclusion of a simple complex zero of a polynomial

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

Machine learning in astronomy

The search to find answers to the deepest questions we have about the Universe has fueled the collection of data for ever larger volumes of our cosmos. The field of supernova cosmology, for example, is seeing continuous development with upcoming surveys set to produce a vast amount of data that will require new statistical inference and machine learning techniques for processing and analysis. Distinguishing between real objects and artefacts is one of the first steps in any transient science pipeline and, currently, is still carried out by humans - often leading to hand scanners having to sort hundreds or thousands of images per night. This is a time-consuming activity introducing human biases that are extremely hard to characterise. To succeed in the objectives of future transient surveys, the successful substitution of human hand scanners with machine learning techniques for the purpose of this artefact-transient classification therefore represents a vital frontier. In this thesis we test various machine learning algorithms and show that many of them can match the human hand scanner performance in classifying transient difference g, r and i-band imaging data from the SDSS-II SN Survey into real objects and artefacts. Using principal component analysis and linear discriminant analysis, we construct a grand total of 56 feature sets with which to train, optimise and test a Minimum Error Classifier (MEC), a naive Bayes classifier, a k-Nearest Neighbours (kNN) algorithm, a Support Vector Machine (SVM) and the SkyNet artificial neural network