Sequence transformations in proofs of irrationality of some fundamental constants

Transformation of number sequences (convergence acceleration) is one of the classical chapters of numerical analysis. These algorithms are used both for solution of practical problems and for the development of more advanced numerical methods. At the same time, numerical methods have found numerous applications in the number theory. One of the classical problems of number theory is the proof of irrationality of some fundamental constants, where the high rate of convergence of sequences of rational numbers plays a crucial role. However, as far as we know, the applications of (classical) convergence acceleration algorithms to the proofs of irrationality do not exist. This study is an attempt to fill this gap and to draw attention to this direction of research.

Visual Characterization of Misclassified Class C GPCRs through Manifold-based Machine Learning Methods

G-protein-coupled receptors are cell membrane proteins of great interest in biology and pharmacology. Previous analysis of Class C of these receptors has revealed the existence of an upper boundary on the accuracy that can be achieved in the classification of their standard subtypes from the unaligned transformation of their primary sequences. To further investigate this apparent boundary, the focus of the analysis in this paper is placed on receptor sequences that were previously misclassified using supervised learning methods. In our experiments, these sequences are visualized using a nonlinear dimensionality reduction technique and phylogenetic trees. They are subsequently characterized against the rest of the data and, particularly, against the rest of cases of their own subtype. This exploratory visualization should help us to discriminate between different types of misclassification and to build hypotheses about database quality problems and the extent to which GPCR sequence transformations limit subtype discriminability. The reported experiments provide a proof of concept for the proposed method.

The Serial Collaborator: A Meta-Pianist for Real-Time Tonal and Non-Tonal Music Generation

Serial music, which is mainly non-tonal, superimposes compositional freedom onto an unusually rigorous process of pitch-sequence transformations based on ‘tone rows’: a row is usually a sequence of notes using each of the 12 chromatic pitches once. Compositional freedom comprises forming chords from the sequences, and in multi-strand music, also in simultaneously presenting different segments of pitch-sequences. The present project coded a real-time serial music composer for automatic or interactive music performance. This Serial Keyboardist Collaborator can perform keyboard music which is impossible for a human to realize. Surprisingly, it was also useful in making more tonal music based on the same rigorous pitch-sequence generation.

Double sequence transformations that guarantee a given rate of p-convergence

In this paper the following sequence space is presented. Let [t] be a positive double sequence and define the sequence space ?''(t) = {complex sequences x : xk,l = O(tk,l)}. The set of geometrically dominated double sequences is defined as G'' = U r,s?(0,1) G(r, s) where G(r, s) = {complex sequences x : x k,l = O(rk sl)} for each r, s in the interval (0, 1). Using this definition, four dimensional matrix characterizations of l?,?, c'', and c0'' into G'' and into ?''(t) are presented. In addition to these definitions and characterizations it should be noted that this ensure a rate of converges of at least as fast as [t]. Other natural implications will also be presented.