Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

On some extremal position problems for graphs

The general position number of a graph $G$ is the size of the largest set of vertices $S$ such that no geodesic of $G$ contains more than two elements of $S$. The monophonic position number of a graph is defined similarly, but with `induced path' in place of `geodesic'. In this paper we investigate some extremal problems for these parameters. Firstly we discuss the problem of the smallest possible order of a graph with given general and monophonic position numbers. We then determine the asymptotic order of the largest size of a graph with given general or monophonic position number, classifying the extremal graphs with monophonic position number two. Finally we establish the possible diameters of graphs with given order and monophonic position number

Choreographing the extended agent : performance graphics for dance theater

Thesis (Ph. D.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2005.Includes bibliographical references (v. 2, leaves 448-458).The marriage of dance and interactive image has been a persistent dream over the past decades, but reality has fallen far short of potential for both technical and conceptual reasons. This thesis proposes a new approach to the problem and lays out the theoretical, technical and aesthetic framework for the innovative art form of digitally augmented human movement. I will use as example works a series of installations, digital projections and compositions each of which contains a choreographic component - either through collaboration with a choreographer directly or by the creation of artworks that automatically organize and understand purely virtual movement. These works lead up to two unprecedented collaborations with two of the greatest choreographers working today; new pieces that combine dance and interactive projected light using real-time motion capture live on stage. The existing field of"dance technology" is one with many problems. This is a domain with many practitioners, few techniques and almost no theory; a field that is generating "experimental" productions with every passing week, has literally hundreds of citable pieces and no canonical works; a field that is oddly disconnected from modern dance's history, pulled between the practical realities of the body and those of computer art, and has no influence on the prevailing digital art paradigms that it consumes.(cont.) This thesis will seek to address each of these problems: by providing techniques and a basis for "practical theory"; by building artworks with resources and people that have never previously been brought together, in theaters and in front of audiences previously inaccessible to the field; and by proving through demonstration that a profitable and important dialogue between digital art and the pioneers of modern dance can in fact occur. The methodological perspective of this thesis is that of biologically inspired, agent-based artificial intelligence, taken to a high degree of technical depth. The representations, algorithms and techniques behind such agent architectures are extended and pushed into new territory for both interactive art and artificial intelligence. In particular, this thesis ill focus on the control structures and the rendering of the extended agents' bodies, the tools for creating complex agent-based artworks in intense collaborative situations, and the creation of agent structures that can span live image and interactive sound production. Each of these parts becomes an element of what it means to "choreograph" an extended agent for live performance.Marc Downie.Ph.D

Modelling Motivic Processes in Music: A Mathematical Approach

This thesis proposes a new model for motivic analysis which, being based on the metaphor of a web or network and expanded using the mathematical field of graph theory, balances the polar concerns prevalent in analytical writing to date: those of static, out-of-time category membership and dynamic, in-time process. The concepts that constitute the model are presented in the third chapter, both as responses to a series of analytical observations (using the worked example of Beethoven’s Piano Sonata in F minor, Op. 2, No. 1), and as rigorously defined mathematical formalisms. The other chapters explore in further detail the disciplines and methodologies on which this model impinges, and serve both to motivate, and to reflect upon, its development. Chapter 1 asks what it means to make mathematical statements about music, and seeks to disentangle mathematics (as a tool or language) from science (as a method), arguing that music theory’s aims can be met by the former without presupposing its commonly assumed inextricability from the latter. Chapter 2 provides a thematic overview of the field of motivic theory and analysis, proposing four archetypal models that combine to underwrite much thought on the subject before outlining the problems inherent in a static account and the creative strategies that can be used to construct a dynamic account. Finally, Chapter 4 applies these strategies, together with Chapter 3’s model and the piece’s extensive existing scholarly literature, to the analysis of the first and last movements of Mahler’s Sixth Symphony. The central theme throughout – as it relates to mathematical modelling, music theory, and music analysis – is that of potential, invitation, openness, and dialogic engagement

Dynamical and topological tools for (modern) music analysis

Is it possible to represent the horizontal motions of the melodic strands of a contrapuntal composition, or the main ideas of a jazz standard as mathematical entities? In this work, we suggest a collection of novel models for the representation of music that are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. Here, we tackle the problem of music representation following three non-orthogonal directions. We suggest a formalisation of the concept of voice leading (the assignment of an instrument to each voice in a sequence of chords) suggesting a horizontal viewpoint on music, constituted by the simultaneous motions of superposed melodies. This formalisation naturally leads to the interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids (paths in the Euclidean space), and their main features are visualised as geometric configurations of collections of 3-dimensional strands. Thereafter, we neglect this time-related information, in order to reduce the problem to the study of vertical musical entities. The model we propose is derived from a topological interpretation of the Tonnetz (a graph commonly used in computational musicology) and the deformation of its vertices induced by a harmonic and a consonance-oriented function, respectively. The 3-dimensional shapes derived from these deformations are classified using the formalism of persistent homology. This powerful topological technique allows to compute a fingerprint of a shape, that reflects its persistent geometrical and topological properties. Furthermore, it is possible to compute a distance between these fingerprints and hence study their hierarchical organisation. This particular feature allows us to tackle the problem of automatic classification of music in an innovative way. Thus, this novel representation of music is evaluated on a collection of heterogenous musical datasets. Finally, a combination of the two aforementioned approaches is proposed. A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time. To conclude, we shall represent music as a time series of topological fingerprints, whose metric nature allows to compare pairs of time-varying shapes in both topological and in musical terms. In particular the dissimilarity scores computed by aligning such sequences shall be applied both to the analysis and classification of music

Proceedings of the 7th Sound and Music Computing Conference

Proceedings of the SMC2010 - 7th Sound and Music Computing Conference, July 21st - July 24th 2010

Modelling, Monitoring, Control and Optimization for Complex Industrial Processes

This reprint includes 22 research papers and an editorial, collected from the Special Issue "Modelling, Monitoring, Control and Optimization for Complex Industrial Processes", highlighting recent research advances and emerging research directions in complex industrial processes. This reprint aims to promote the research field and benefit the readers from both academic communities and industrial sectors