Related papers: Symbolic dynamical scales: modes, orbitals, and tr…
A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^{-1}) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer,…
In this article, we study the behaviour of discrete one-dimensional dynamical systems associated to functions on finite sets. We formalise the global orbit pattern formed by all the periodic orbits (gop) as the ordered set of periods when…
A set of basic notes, or `scale', forms the basis of music. Scales are specific to specific genre of music. In this second article of the series we explore the development of various scales associated with western classical music, arguably…
Symbolic dynamics is a coarse-grained description of dynamics. By taking into account the ``geometry'' of the dynamics, it can be cast into a powerful tool for practitioners in nonlinear science. Detailed symbolic dynamics can be developed…
Periodic orbit theory allows calculations of long time properties of chaotic systems from traces, dynamical zeta functions and spectral determinants of deterministic evolution operators, which are in turn evaluated in terms of periodic…
Dynamic systems have found their use in sound synthesis as well as score synthesis. These levels can be integrated in monolithic autonomous systems in a novel approach to algorithmic composition that shares certain aesthetic motivations…
This paper focuses on the nominal durations of musical events (notes and rests) in a symbolic musical score, and on how to conveniently handle these in computer applications. We propose the usage of a temporal unit that is directly related…
Discrete interaction models for the classical harmonic oscillator are used for introducing new mathematical generalizations in the usual continuous formalism. The inverted harmonic potential and generalized discrete hyperbolic and…
A numerical technique for calculating effective actions of electromagnetic backgrounds is proposed, which is based on the string-inspired worldline formalism. As examples, we consider scalar electrodynamics in three and four dimensions to…
When designing and developing scale selection mechanisms for generating hypotheses about characteristic scales in signals, it is essential that the selected scale levels reflect the extent of the underlying structures in the signal. This…
We present a formalism for computing classically measurable quantities directly from on-shell quantum scattering amplitudes. We discuss the ingredients needed for obtaining the classical result, and show how to set up the calculation to…
Symbolic regression is a machine learning technique that can learn the governing formulas of data and thus has the potential to transform scientific discovery. However, symbolic regression is still limited in the complexity and…
Generative statistical models of chord sequences play crucial roles in music processing. To capture syntactic similarities among certain chords (e.g. in C major key, between G and G7 and between F and Dm), we study hidden Markov models and…
We demonstrate the creation of nontrivial (meta) stable states (patterns), localized, chaotic, entangled or decoherent, from the basic localized modes in various collective models arising from the quantum hierarchy described by Wigner-like…
Subword tokenization has been widely successful in text-based natural language processing (NLP) tasks with Transformer-based models. As Transformer models become increasingly popular in symbolic music-related studies, it is imperative to…
How can neural networks perform so well on compositional tasks even though they lack explicit compositional representations? We use a novel analysis technique called ROLE to show that recurrent neural networks perform well on such tasks by…
Music and language are structurally similar. Such structural similarity is often explained by generative processes. This paper describes the recent development of probabilistic generative models (PGMs) for language learning and symbol…
Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time. Several dynamical modes, such as fixed points, 2 - cycles…
This paper studies the emergence of interpretable categorical features within large language models (LLMs), analyzing their behavior across training checkpoints (time), transformer layers (space), and varying model sizes (scale). Using…
In the workspace model of machining an experimental procedure is implemented to determine the elastic behaviour of the machining system. In an other work we have proposed a static characterization of the machining system. In this paper a…