### Algebraic Path Planning of 6R/P Manipulators [6RP_2016]

Project Lead: Jose Capco

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# Dr. Jose Capco

### Research Area

Algebraic Geometry in Kinematics## Ongoing Projects

### Algebraic Path Planning of 6R/P Manipulators [6RP_2016]

Project Duration: 11/01/2016 - 10/01/2020MoreProject Website## Publications

All 2019 - 2017 2016 - 2014 2013 - 2011 2010 - 2008 2007 - 2005 2004 - 2002 2001 - 1999 1998 - 1996 1995 - 1993 1992 - 1990 1989 - 1987 1986 - 1965 ### 2019

### Sum of Squares over Rationals

#### J. Capco, C. Scheiderer

RISC. Technical report, 2019. [url] [pdf]
### 2018

### Generalizing some Results in Field Theory for Rings

#### Jose Capco

Communications in Algebra, pp. 0-12. 2018. 0092-7872. Preprint. [pdf]
### The Number of Realizations of a Laman Graph

#### Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho

SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125. 2018. 2470-6566. [url]
### 2017

### The number of realizations of a Laman graph

#### Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

Research Institute for Symbolic Computation (RISC/JKU). Technical report, 2017. [url] [pdf]
### Computing the number of realizations of a Laman graph

#### Jose Capco, Georg Grasegger, Matteo Gallet, Christoph Koutschan, Niels Lubbes, Josef Schicho

In: Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017), Vadim Lozin (ed.), Proceedings of The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)61, pp. 207-213. 2017. ISSN 1571-0653. [url]
### 2015

### Odd Collatz Sequence and Binary Representations

#### Jose Capco

RISC Report , 2015. [pdf]

Other than my own working area, I am also interested in number theory (and its extensions: algorithmic/algebraic/analytic number theories), cryptography, real algebraic geometry (my PhD thesis), commutative algebra in general (worked a bit with Galois theory on commutative rings), graph theory, computational geometry, fractal sets (looked a bit on Julia sets from esoteric fields), elliptic curves, Lie algebra and mathematical chess problems (see Noam Elkies on enumerative chess problems).

Project Lead: Jose Capco

[Capco]

@**techreport**{RISC5884,

author = {J. Capco and C. Scheiderer},

title = {{Sum of Squares over Rationals}},

language = {english},

abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},

year = {2019},

institution = {RISC},

length = {0},

url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}

}

author = {J. Capco and C. Scheiderer},

title = {{Sum of Squares over Rationals}},

language = {english},

abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},

year = {2019},

institution = {RISC},

length = {0},

url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}

}

[Capco]

@**article**{RISC5375,

author = {Jose Capco},

title = {{Generalizing some Results in Field Theory for Rings}},

language = {english},

journal = {Communications in Algebra},

pages = {0--12},

isbn_issn = {0092-7872},

year = {2018},

note = {Preprint},

refereed = {no},

length = {13}

}

author = {Jose Capco},

title = {{Generalizing some Results in Field Theory for Rings}},

language = {english},

journal = {Communications in Algebra},

pages = {0--12},

isbn_issn = {0092-7872},

year = {2018},

note = {Preprint},

refereed = {no},

length = {13}

}

[Capco]

@**article**{RISC5700,

author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{The Number of Realizations of a Laman Graph}},

language = {english},

journal = {SIAM Journal on Applied Algebra and Geometry},

volume = {2},

number = {1},

pages = {94--125},

isbn_issn = {2470-6566},

year = {2018},

refereed = {yes},

length = {32},

url = {https://doi.org/10.1137/17M1118312}

}

author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{The Number of Realizations of a Laman Graph}},

language = {english},

journal = {SIAM Journal on Applied Algebra and Geometry},

volume = {2},

number = {1},

pages = {94--125},

isbn_issn = {2470-6566},

year = {2018},

refereed = {yes},

length = {32},

url = {https://doi.org/10.1137/17M1118312}

}

[Capco]

@**techreport**{RISC5418,

author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{The number of realizations of a Laman graph}},

language = {english},

abstract = {Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems. },

year = {2017},

institution = {Research Institute for Symbolic Computation (RISC/JKU)},

length = {42},

url = {http://www.koutschan.de/data/laman/}

}

author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{The number of realizations of a Laman graph}},

language = {english},

abstract = {Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems. },

year = {2017},

institution = {Research Institute for Symbolic Computation (RISC/JKU)},

length = {42},

url = {http://www.koutschan.de/data/laman/}

}

[Capco]

@**inproceedings**{RISC5478,

author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{Computing the number of realizations of a Laman graph}},

booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},

language = {english},

abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},

volume = {61},

pages = {207--213},

isbn_issn = {ISSN 1571-0653},

year = {2017},

editor = {Vadim Lozin},

refereed = {yes},

keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},

length = {7},

conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},

url = {http://www.koutschan.de/data/laman/}

}

author = {Jose Capco and Georg Grasegger and Matteo Gallet and Christoph Koutschan and Niels Lubbes and Josef Schicho},

title = {{Computing the number of realizations of a Laman graph}},

booktitle = {{Electronic Notes in Discrete Mathematics (Proceedings of Eurocomb 2017)}},

language = {english},

abstract = {Laman graphs model planar frameworks which are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. In a recent paper we provide a recursion formula for this number of realizations using ideas from algebraic and tropical geometry. Here, we present a concise summary of this result focusing on the main ideas and the combinatorial point of view.},

volume = {61},

pages = {207--213},

isbn_issn = {ISSN 1571-0653},

year = {2017},

editor = {Vadim Lozin},

refereed = {yes},

keywords = {Laman graph; minimally rigid graph; tropical geometry; euclidean embedding; graph realization},

length = {7},

conferencename = {The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB'17)},

url = {http://www.koutschan.de/data/laman/}

}

[Capco]

@**techreport**{RISC5321,

author = {Jose Capco},

title = {{Odd Collatz Sequence and Binary Representations}},

language = {english},

abstract = {Here we investigate the odd numbers in Collatz sequences (sequences arising from the $3n+1$ problem). We are especially interested in methods in binary number representations of the numbers in the sequence. In the first section, we show some results for odd Collatz sequences using mostly binary arithmetics. We see how some results become more obvious in binary arithmetic than in usual method of computing the Collatz sequence. In the second section of this paper we deal with some known results and show how we can use binary representation and OCS from the first section to prove some known results. We give a generalization of a result by Andaloro \cite{And2} and show a generalized sufficient condition for the Collatz conjecture to be true: If for a fixed natural number $n$ the Collatz conjecture holds for numbers congruent to $1$ modulo $2^n$ then the Collatz conjecture is true.},

year = {2015},

howpublished = {RISC Report },

length = {11},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

author = {Jose Capco},

title = {{Odd Collatz Sequence and Binary Representations}},

language = {english},

abstract = {Here we investigate the odd numbers in Collatz sequences (sequences arising from the $3n+1$ problem). We are especially interested in methods in binary number representations of the numbers in the sequence. In the first section, we show some results for odd Collatz sequences using mostly binary arithmetics. We see how some results become more obvious in binary arithmetic than in usual method of computing the Collatz sequence. In the second section of this paper we deal with some known results and show how we can use binary representation and OCS from the first section to prove some known results. We give a generalization of a result by Andaloro \cite{And2} and show a generalized sufficient condition for the Collatz conjecture to be true: If for a fixed natural number $n$ the Collatz conjecture holds for numbers congruent to $1$ modulo $2^n$ then the Collatz conjecture is true.},

year = {2015},

howpublished = {RISC Report },

length = {11},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

Phone: +43 732 2468 9921

Fax: +43 732 2468 9930

eMail: secretary@risc.jku.at

WWW: https://www.risc.jku.at

Office: Monday-Thursday 8-16h, Friday 8-12h.

Research Institute for Symbolic Computation (RISC)

Johannes Kepler University

Altenbergerstraße 69

A-4040 Linz, Austria

Research Institute for Symbolic Computation (RISC)

Schloss Hagenberg (Castle of Hagenberg)

Kirchenplatz 5b

A-4232 Hagenberg im Mühlkreis, Austria