Introduction to Gestural Similarity in Music. An Application of Category Theory to the Orchestra

Mathematics, and more generally computational sciences, intervene in several aspects of music. Mathematics describes the acoustics of the sounds giving formal tools to physics, and the matter of music itself in terms of compositional structures and strategies. Mathematics can also be applied to the entire making of music, from the score to the performance, connecting compositional structures to acoustical reality of sounds. Moreover, the precise concept of gesture has a decisive role in understanding musical performance. In this paper, we apply some concepts of category theory to compare gestures of orchestral musicians, and to investigate the relationship between orchestra and conductor, as well as between listeners and conductor/orchestra. To this aim, we will introduce the concept of gestural similarity. The mathematical tools used can be applied to gesture classification, and to interdisciplinary comparisons between music and visual arts.Comment: The final version of this paper has been published by the Journal of Mathematics and Musi

Fuzzy Bigraphs: An Exercise in Fuzzy Communicating Agents

Bigraphs and their algebra is a model of concurrency. Fuzzy bigraphs are a generalization of birgraphs intended to be a model of concurrency that incorporates vagueness. More specifically, this model assumes that agents are similar, communication is not perfect, and, in general, everything is or happens to some degree.Comment: 11 pages, 3 figure

Two generalizations of Kohonen clustering

The relationship between the sequential hard c-means (SHCM), learning vector quantization (LVQ), and fuzzy c-means (FCM) clustering algorithms is discussed. LVQ and SHCM suffer from several major problems. For example, they depend heavily on initialization. If the initial values of the cluster centers are outside the convex hull of the input data, such algorithms, even if they terminate, may not produce meaningful results in terms of prototypes for cluster representation. This is due in part to the fact that they update only the winning prototype for every input vector. The impact and interaction of these two families with Kohonen's self-organizing feature mapping (SOFM), which is not a clustering method, but which often leads ideas to clustering algorithms is discussed. Then two generalizations of LVQ that are explicitly designed as clustering algorithms are presented; these algorithms are referred to as generalized LVQ = GLVQ; and fuzzy LVQ = FLVQ. Learning rules are derived to optimize an objective function whose goal is to produce 'good clusters'. GLVQ/FLVQ (may) update every node in the clustering net for each input vector. Neither GLVQ nor FLVQ depends upon a choice for the update neighborhood or learning rate distribution - these are taken care of automatically. Segmentation of a gray tone image is used as a typical application of these algorithms to illustrate the performance of GLVQ/FLVQ

A theory of structural model validity in simulation.

During the last decennia, the practice of simulation has become increasingly popular among many system analysts, model builders and general scientists for the purpose of studying complex systems that surpass the operability of analytical solution techniques. As a consequence of the pragmatic orientation of simulation, a vital stage for a successful application is the issue of validating a constructed simulation model. Employing the model as an effective instrument for assessing the benefit of structural changes or for predicting future observations makes validation an essential part of any productive simulation study. The diversity of the employment field of simulation however brings about that there exists an irrefutable level of ambiguity concerning the principal subject of this validation process. Further, the literature has come up with a plethora of ad hoc validation techniques that have mostly been inherited from standard statistical analysis. It lies within the aim of this paper to reflect on the issue of validation in simulation and to present the reader with a topological parallelism of the classical philosophical polarity of objectivism versus relativism. First, we will position validation in relation to verification and accreditation and elaborate on the prime actors in validation, i.e. a conceptual model, a formal model and behaviour. Next, we will formally derive a topological interpretation of structural validation for both objectivists and relativists. As will be seen, recent advances in the domain of fuzzy topology allow for a valuable metaphor of a relativistic attitude towards modelling and structural validation. Finally, we will discuss several general types of modelling errors that may occur and examine their repercussion on the natural topological spaces of objectivists and relativists. We end this paper with a formal, topological oriented definition of structural model validity for both objectivists and relativists. The paper is concluded with summarising the most important findings and giving a direction for future research.Model; Simulation; Theory; Scientists; Processes; Statistical analysis;