7,915 research outputs found
Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs
A graph is {\em matching-decyclable} if it has a matching such that
is acyclic. Deciding whether is matching-decyclable is an NP-complete
problem even if is 2-connected, planar, and subcubic. In this work we
present results on matching-decyclability in the following classes: Hamiltonian
subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian
subcubic graphs we show that deciding matching-decyclability is NP-complete
even if there are exactly two vertices of degree two. For chordal and
distance-hereditary graphs, we present characterizations of
matching-decyclability that lead to -time recognition algorithms
Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization
Solving the NP-hard Maximum Cut or Binary Quadratic Optimization Problem to optimality is important in many applications including Physics, Chemistry, Neuroscience, and Circuit Layout. The leading approaches based on linear/semidefinite programming require the separation of so-called odd-cycle inequalities for solving relaxations within their associated branch-and-cut frameworks. In their groundbreaking work, F. Barahona and A.R. Mahjoub have given an informal description of a polynomial-time separation procedure for the odd-cycle inequalities. Since then, the odd-cycle separation problem has broadly been considered solved. However, as we reveal, a straightforward implementation is likely to generate inequalities that are not facet-defining and have further undesired properties. Here, we present a more detailed analysis, along with enhancements to overcome the associated issues efficiently. In a corresponding experimental study, it turns out that these are worthwhile, and may speed up the solution process significantly
Tonal music theory: A psychoacoustic explanation?
From the seventeenth century to the present day, tonal harmonic music has had a number of invariant properties such as the use of specific chord progressions (cadences) to induce a sense of closure, the asymmetrical privileging of certain progressions, and the privileging of the major and minor scales.
The most widely accepted explanation has been that this is due to a process of enculturation: frequently occurring musical patterns are learned by listeners, some of whom become composers and replicate the same patterns, which go on to influence the next āgenerationā of composers, and so on. In this paper, however, I present a possible psychoacoustic explanation for some important regularities of tonal-harmonic music. The core of the model is two different measures of pitch-based distance between chords. The first is voice-leading distance; the second is spectral pitch distanceāa measure of the distance between the partials in one chord compared to those in another chord.
I propose that when a pair of triads has a higher spectral distance than another pair of triads that is voice-leading-close, the former pair is heard as an alteration of the latter pair, and seeks resolution. I explore the extent to which this model can predict the familiar tonal cadences described in music theory (including those containing tritone substitutions), and the asymmetries that are so characteristic of tonal harmony. I also show how it may be able to shed light upon the privileged status of the major and minor scales (over the modes)
Tonal prisms: iterated quantization in chromatic tonality and Ravel's 'Ondine'
The mathematics of second-order maximal evenness has far-reaching potential for application in music analysis. One of its assets is its foundation in an inherently continuous conception of pitch, a feature it shares with voice-leading geometries. This paper reformulates second-order maximal evenness as iterated quantization in voice-leading spaces, discusses the implications of viewing diatonic triads as second-order maximally even sets for the understanding of nineteenth-century modulatory schemes, and applies a second-order maximally even derivation of acoustic collections in an in-depth analysis of Ravel's āOndineā. In the interaction between these two very different applications, the paper generalizes the concepts and analytical methods associated with iterated quantization and also pursues a broader argument about the mutual dependence of mathematical music theory and music analysis.Accepted manuscrip
Fluctuating Currents in Stochastic Thermodynamics I. Gauge Invariance of Asymptotic Statistics
Stochastic Thermodynamics uses Markovian jump processes to model random
transitions between observable mesoscopic states. Physical currents are
obtained from anti-symmetric jump observables defined on the edges of the graph
representing the network of states. The asymptotic statistics of such currents
are characterized by scaled cumulants. In the present work, we use the
algebraic and topological structure of Markovian models to prove a gauge
invariance of the scaled cumulant-generating function. Exploiting this
invariance yields an efficient algorithm for practical calculations of
asymptotic averages and correlation integrals. We discuss how our approach
generalizes the Schnakenberg decomposition of the average entropy-production
rate, and how it unifies previous work. The application of our results to
concrete models is presented in an accompanying publication.Comment: PACS numbers: 05.40.-a, 05.70.Ln, 02.50.Ga, 02.10.Ox. An accompanying
pre-print "Fluctuating Currents in Stochastic Thermodynamics II. Energy
Conversion and Nonequilibrium Response in Kinesin Models" by the same authors
is available as arXiv:1504.0364
Groupoid Extensions of Mapping Class Representations for Bordered Surfaces
The mapping class group of a surface with one boundary component admits
numerous interesting representations including as a group of automorphisms of a
free group and as a group of symplectic transformations. Insofar as the mapping
class group can be identified with the fundamental group of Riemann's moduli
space, it is furthermore identified with a subgroup of the fundamental path
groupoid upon choosing a basepoint. A combinatorial model for this, the mapping
class groupoid, arises from the invariant cell decomposition of Teichm\"uller
space, whose fundamental path groupoid is called the Ptolemy groupoid. It is
natural to try to extend representations of the mapping class group to the
mapping class groupoid, i.e., construct a homomorphism from the mapping class
groupoid to the same target that extends the given representations arising from
various choices of basepoint.
Among others, we extend both aforementioned representations to the groupoid
level in this sense, where the symplectic representation is lifted both
rationally and integrally. The techniques of proof include several algorithms
involving fatgraphs and chord diagrams. The former extension is given by
explicit formulae depending upon six essential cases, and the kernel and image
of the groupoid representation are computed. Furthermore, this provides
groupoid extensions of any representation of the mapping class group that
factors through its action on the fundamental group of the surface including,
for instance, the Magnus representation and representations on the moduli
spaces of flat connections.Comment: 24 pages, 4 figures Theorem 3.6 has been strengthened, and Theorems
8.1 and 8.2 have been adde
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