153 research outputs found

    Covariogram of non-convex sets

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    The covariogram of a compact set A contained in R^n is the function that to each x in R^n associates the volume of A intersected with (A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than 9 points that have equal discrete covariogram.Comment: 15 pages, 7 figures, accepted for publication on Mathematik

    On the reconstruction of planar lattice-convex sets from the covariogram

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    A finite subset KK of Zd\mathbb{Z}^d is said to be lattice-convex if KK is the intersection of Zd\mathbb{Z}^d with a convex set. The covariogram gKg_K of K⊆ZdK\subseteq \mathbb{Z}^d is the function associating to each u \in \integer^d the cardinality of K∩(K+u)K\cap (K+u). Daurat, G\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets KK from gKg_K. We provide a partial positive answer to this problem by showing that for d=2d=2 and under mild extra assumptions, gKg_K determines KK up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.Comment: accepted in Discrete and Computational Geometr

    String Reconstruction from Substring Compositions

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    Motivated by mass-spectrometry protein sequencing, we consider a simply-stated problem of reconstructing a string from the multiset of its substring compositions. We show that all strings of length 7, one less than a prime, or one less than twice a prime, can be reconstructed uniquely up to reversal. For all other lengths we show that reconstruction is not always possible and provide sometimes-tight bounds on the largest number of strings with given substring compositions. The lower bounds are derived by combinatorial arguments and the upper bounds by algebraic considerations that precisely characterize the set of strings with the same substring compositions in terms of the factorization of bivariate polynomials. The problem can be viewed as a combinatorial simplification of the turnpike problem, and its solution may shed light on this long-standing problem as well. Using well known results on transience of multi-dimensional random walks, we also provide a reconstruction algorithm that reconstructs random strings over alphabets of size ≥4\ge4 in optimal near-quadratic time

    Temperament in Bach's Well-tempered clavier : a historical survey and a new evaluation according to dissonance theory

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    After a historical survey of temperament in Bach's Well-Tempered Clavier by Johann Sebastian Bach, an analysis of the work has been made by applying a number of historical good temperaments as well as some recent proposals. The results obtained show that the global dissonance for all preludes and fugues in major keys can be minimized using the Kirnberger II temperament. The method of analysis used for this research is based on the mathematical theories of sensory dissonance, which have been developed by authors such as Hermann Ludwig Ferdinand von Helmholtz, Harry Partch, Reinier Plomp, Willem J. M. Levelt and William A. SetharesDesprés d'una visió històrica sobre el temperament a El clavecí ben temperat de Johann Sebastian Bach, s'ha realitzat una anàlisi de l'obra aplicant divesos bons temperaments històrics a més d'algunes propostes recents. Els resultats obtinguts demostren que la dissonància global per a tots els preludis i fugues en tonalitats majors pot minimitzar-se utilitzant el temperament Kirnberger II. El mètode d'anàlisi utilitzat per a aquesta recerca està basat en les teories matemàtiques de la dissonància sensorial desenvolupades per autors com Hermann Ludwig Ferdinand von Helmholtz, Harry Partch, Reinier Plomp, Willem J. M. Levelt i William A. Sethare

    Complete Issue 10, 1994

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    Some necessary clarifications about the chords' problem and the Partial Digest Problem

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    AbstractWe state in previous paper [A. Daurat, Y. Gérard, M. Nivat, The chords’ problem, Theoret. Comput. Sci. 282(2) (2002) 319–336] that the chords’ problem can be solved in polynomial time. This result is however ambiguous and some people have been abused because the encoding of the data has not been given. The correctness of the result requires to specify the encoding of the data that we have used and to highlight the difference with the usual encoding implicitly considered in Partial Digest Problem

    Creating music in the classroom with tablet computers: An activity system analysis of two secondary school communities.

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    Tablet computers are becoming inextricably linked with innovation and change in schools. Increasingly therefore, music teachers must consider how tablet computers might influence creative musical development in their own classroom. This qualitative research into two secondary school communities aims to develop understandings about what really happens when students and a music teacher-researcher compose music in partnership with a tablet computer. A sociocultural definition of creativity, theories of Activity, and the musicking argument inform a new systemic framework which guides fieldwork. This framework becomes the unit of analysis from which the research questions and a multi-case, multimodal methodology emerge. The methodology developed here honours the situated nature of those meanings which emerge in each of the two school communities. Consequently, research findings are presented as two separate case reports. Five mixed-ability pairs are purposively sampled from each community to represent the broad range of musical experience present in that setting. A Video-enhanced, participant-observation method ensures that systemic, multimodal musicking behaviours are captured as they emerge overtime. Naturalistic group interviewing at the end of the project reveals how students’ broader musical cultures, interests and experiences influence their tablet-mediated classroom behaviour. Findings develop new understandings about how tablet-mediated creative musical action champions inclusive musicking (musical experience notwithstanding) and better connects the music classroom and its institutional requirements with students’ informal music-making practices. The systems of classroom Activity which emerge also compensate for those moments when the tablet attempts to overtly determine creative behaviour or conversely, does not do enough to ensure a creative outcome. In fact, all system dimensions (e.g. student partner/teacher/student/tablet) influence tablet- mediated action by feeding the system with musical and technological knowledge, which was also pedagogically conditioned. This musical, technological and pedagogical conditioning is mashed-up, influencing action just-in-time, according to cultural, local and personal need. A new method of visual charting is developed to ‘peer inside’ these classroom-situated systems. Colour-coded charts evidence how classroom musicians make use of and synthesize different system dimensions to find, focus and fix their creative musical ideas over time. There are also implications for research, policy and practice going forward. In terms of researching digitally-mediated creativity, a new social-cultural Activity framework is presented which encourages researchers to revise their definition of creativity itself. Such a definition would emphasise the role of cultural, local and personal constraint in creative musical development. With reference to classroom practice, this research discovers that when students partner with tablet computers, their own musical interests, experiences and desires are forwarded. Even though these desires become fused with institutional requirements, students take ownership of their learning and are found rightfully proud of their creative products. This naturalistic, community-driven form of tablet- mediated creative musical development encourages policy makers and teachers to reposition the music classroom: to reconnect it with the local community it serves
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