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Papers by Matthew Klassen

CERN European Organization for Nuclear Research - Zenodo, Jun 7, 2022

The origin of this paper comes from the collaboration of the authors on the UPISketch software pr... more The origin of this paper comes from the collaboration of the authors on the UPISketch software project (see [1]), which is a creation of the Iannis Xenakis Center and a descendant of the UPIC project of Xenakis. The software, created in 2017, allows the user to draw curves which can be interpreted as musical gestures, acting on elements such as pitch, time, and timbre. Two fundamental mathematical tools used in this process are splines, to model graphical gestures, and cellular automata, to generate musical gestures such as pitch sequences. In this paper we apply both of these techniques on the ªmicro-levelº to the timbre of waveforms, using the approach of cycle interpolation. A waveform is modeled as a sequence of cycles, and each cycle is modeled as a cubic spline curve. The B-spline coefficients of each cycle form a discrete representation, which can be manipulated through the use of cellular automata. In this way, key cycles can be generated, and can be interpolated with B-splines to form new and interesting timbres. We illustrate this generation and design of waveforms in our own software implementation with JUCE, which in turn will become part of UPISketch. Copyright: c 2022 Matthew Klassen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Algebraic points of low degree on curves of low rank

... Copyright © 1993, Matthew James Klassen ... which one of the coordinates is O. (This conjectu... more ... Copyright © 1993, Matthew James Klassen ... which one of the coordinates is O. (This conjecture is now known to follow from the conjecture of Taniyama on elliptic curves thanks to work of Ribet and Frey, and it appears to be, as of June 1993, a Theorem of Andrew Wiles.) Now ...

The purpose of this note is to provide some applications of a theorem of Faltings ([Fa1]) to smoo... more The purpose of this note is to provide some applications of a theorem of Faltings ([Fa1]) to smooth plane curves, using ideas from [A] and [AH]. Let C be a smooth projective plane curve defined by an equation of degree d

Arithmetic and geometry of the curve y3+ 1= x4

Journal Fur Die Reine Und Angewandte Mathematik, 1994

The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's c... more The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's conjecture to smooth plane curves. Let CCC be a smooth plane curve defined by an equation of degree ddd with integral coefficients. We show that for dge7d\ge 7dge7, the curve CCC has only finitely many points whose field of definition has degree led−2\le d-2led−2 over QQQ, and that for dge8d\ge 8dge8, all but finitely many points of CCC whose field of definition has degree led−1\le d-1led−1 over QQQ arise as points of intersection of rational lines through rational points of CCC.

The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's c... more The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's conjecture to smooth plane curves. Let CCC be a smooth plane curve defined by an equation of degree ddd with integral coefficients. We show that for dge7d\ge 7dge7, the curve CCC has only finitely many points whose field of definition has degree led−2\le d-2led−2 over QQQ, and that for dge8d\ge 8dge8, all but finitely many points of CCC whose field of definition has degree led−1\le d-1led−1 over QQQ arise as points of intersection of rational lines through rational points of CCC.

CERN European Organization for Nuclear Research - Zenodo, Jun 7, 2022

The origin of this paper comes from the collaboration of the authors on the UPISketch software pr... more The origin of this paper comes from the collaboration of the authors on the UPISketch software project (see [1]), which is a creation of the Iannis Xenakis Center and a descendant of the UPIC project of Xenakis. The software, created in 2017, allows the user to draw curves which can be interpreted as musical gestures, acting on elements such as pitch, time, and timbre. Two fundamental mathematical tools used in this process are splines, to model graphical gestures, and cellular automata, to generate musical gestures such as pitch sequences. In this paper we apply both of these techniques on the ªmicro-levelº to the timbre of waveforms, using the approach of cycle interpolation. A waveform is modeled as a sequence of cycles, and each cycle is modeled as a cubic spline curve. The B-spline coefficients of each cycle form a discrete representation, which can be manipulated through the use of cellular automata. In this way, key cycles can be generated, and can be interpolated with B-splines to form new and interesting timbres. We illustrate this generation and design of waveforms in our own software implementation with JUCE, which in turn will become part of UPISketch. Copyright: c 2022 Matthew Klassen et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Algebraic points of low degree on curves of low rank

... Copyright © 1993, Matthew James Klassen ... which one of the coordinates is O. (This conjectu... more ... Copyright © 1993, Matthew James Klassen ... which one of the coordinates is O. (This conjecture is now known to follow from the conjecture of Taniyama on elliptic curves thanks to work of Ribet and Frey, and it appears to be, as of June 1993, a Theorem of Andrew Wiles.) Now ...

The purpose of this note is to provide some applications of a theorem of Faltings ([Fa1]) to smoo... more The purpose of this note is to provide some applications of a theorem of Faltings ([Fa1]) to smooth plane curves, using ideas from [A] and [AH]. Let C be a smooth projective plane curve defined by an equation of degree d

Arithmetic and geometry of the curve y3+ 1= x4

Journal Fur Die Reine Und Angewandte Mathematik, 1994

The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's c... more The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's conjecture to smooth plane curves. Let CCC be a smooth plane curve defined by an equation of degree ddd with integral coefficients. We show that for dge7d\ge 7dge7, the curve CCC has only finitely many points whose field of definition has degree led−2\le d-2led−2 over QQQ, and that for dge8d\ge 8dge8, all but finitely many points of CCC whose field of definition has degree led−1\le d-1led−1 over QQQ arise as points of intersection of rational lines through rational points of CCC.

The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's c... more The purpose of this note is to provide some applications of Faltings' recent proof of S. Lang's conjecture to smooth plane curves. Let CCC be a smooth plane curve defined by an equation of degree ddd with integral coefficients. We show that for dge7d\ge 7dge7, the curve CCC has only finitely many points whose field of definition has degree led−2\le d-2led−2 over QQQ, and that for dge8d\ge 8dge8, all but finitely many points of CCC whose field of definition has degree led−1\le d-1led−1 over QQQ arise as points of intersection of rational lines through rational points of CCC.