2021
|
Lange-Hegermann, M; Robertz, D; Seiler, W M; Seiß, M Singularities of Algebraic Differential Equations Journal Article Advances in Applied Mathematics, 131 , pp. Paper No. 102266, 2021. Abstract | BibTeX @article{LangeHegermannRobertzSeilerSeiss,
title = {Singularities of Algebraic Differential Equations},
author = {Lange-Hegermann, M. and Robertz, D. and Seiler, W. M. and Seiß, M.},
year = {2021},
date = {2021-08-26},
journal = {Advances in Applied Mathematics},
volume = {131},
pages = {Paper No. 102266},
abstract = {There exists a well established differential topological theory of
singularities of ordinary differential equations. It has mainly studied
scalar equations of low order. We propose an extension of the key
concepts to arbitrary systems of ordinary or partial differential
equations. Furthermore, we show how a combination of this geometric
theory with (differential) algebraic tools allows us to make parts of the
theory algorithmic. Our three main results are firstly a proof that even
in the case of partial differential equations regular points are generic.
Secondly, we present an algorithm for the effective detection of all
singularities at a given order or, more precisely, for the determination
of a regularity decomposition. Finally, we give a rigorous definition of
a regular differential equation, a notoriously difficult notion,
ubiquitous in the geometric theory of differential equations, and show
that our algorithm extracts from each prime component a regular
differential equation. Our main tools are on the one hand the algebraic
resp.\ differential Thomas decomposition and on the other hand the
Vessiot theory of differential equations.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
There exists a well established differential topological theory of
singularities of ordinary differential equations. It has mainly studied
scalar equations of low order. We propose an extension of the key
concepts to arbitrary systems of ordinary or partial differential
equations. Furthermore, we show how a combination of this geometric
theory with (differential) algebraic tools allows us to make parts of the
theory algorithmic. Our three main results are firstly a proof that even
in the case of partial differential equations regular points are generic.
Secondly, we present an algorithm for the effective detection of all
singularities at a given order or, more precisely, for the determination
of a regularity decomposition. Finally, we give a rigorous definition of
a regular differential equation, a notoriously difficult notion,
ubiquitous in the geometric theory of differential equations, and show
that our algorithm extracts from each prime component a regular
differential equation. Our main tools are on the one hand the algebraic
resp. differential Thomas decomposition and on the other hand the
Vessiot theory of differential equations. |
Chudoba, R; Niemeyer, A; Spartali, H; Robertz, D; Plesken, W Description of the origami waterbomb cell kinematics as a basis for the design of thin-walled oricrete shells Inproceedings Behnejad, S A; Parke, G A R; Samavati, O A (Ed.): IASS2020/21-SURREY7: Annual Symposium of the International Association for Shell and Spatial Structures (IASS2020/21) and the 7th Surrey International Conference on Spatial Structure, 2021. BibTeX @inproceedings{ChudobaNiemeyerSpartaliRobertzPlesken1,
title = {Description of the origami waterbomb cell kinematics as a basis for the design of thin-walled oricrete shells},
author = {Chudoba, R. and Niemeyer, A. and Spartali, H. and Robertz, D. and Plesken, W.},
editor = {Behnejad, S. A. and Parke, G. A. R. and Samavati, O. A.},
year = {2021},
date = {2021-08-23},
booktitle = {IASS2020/21-SURREY7: Annual Symposium of the International Association for Shell and Spatial Structures (IASS2020/21) and the 7th Surrey International Conference on Spatial Structure},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Hashemi, A; Izgin, T; Robertz, D; Seiler, W M An Involutive GVW Algorithm and the Computation of Pommaret Bases Journal Article Mathematics in Computer Science, 15 (3), pp. 419-452, 2021. Abstract | BibTeX @article{HashemiIzginRobertzSeiler2,
title = {An Involutive GVW Algorithm and the Computation of Pommaret Bases},
author = {Hashemi, A. and Izgin, T. and Robertz, D. and Seiler, W. M.},
year = {2021},
date = {2021-06-07},
journal = {Mathematics in Computer Science},
volume = {15},
number = {3},
pages = {419-452},
abstract = {The GVW algorithm computes simultaneously Gröbner bases of a given ideal and of the syzygy module of the given generating set. In this work, we discuss an extension of it to involutive bases. Pommaret bases play here a special role in several respects. We distinguish between a fully involutive GVW algorithm which determines involutive bases for both the given ideal and the syzygy module and a semi-involutive version which computes for the syzygy module only an ordinary Gröbner basis. A prototype implementation of the developed algorithms in Maple is described.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The GVW algorithm computes simultaneously Gröbner bases of a given ideal and of the syzygy module of the given generating set. In this work, we discuss an extension of it to involutive bases. Pommaret bases play here a special role in several respects. We distinguish between a fully involutive GVW algorithm which determines involutive bases for both the given ideal and the syzygy module and a semi-involutive version which computes for the syzygy module only an ordinary Gröbner basis. A prototype implementation of the developed algorithms in Maple is described. |
2020
|
Brakhage, K H; Niemeyer, A C; Plesken, W; Robertz, D; Strzelczyk, A The icosahedra of edge length 1 Journal Article Journal of Algebra, 545 , pp. 4–26, 2020. Abstract | Links | BibTeX @article{BrakhageNiemeyerPleskenRobertzStrzelczyk,
title = {The icosahedra of edge length 1},
author = {Brakhage, K. H. and Niemeyer, A. C. and Plesken, W. and Robertz, D. and Strzelczyk, A.},
doi = {doi:10.1016/j.jalgebra.2019.04.028},
year = {2020},
date = {2020-01-01},
journal = {Journal of Algebra},
volume = {545},
pages = {4--26},
abstract = {Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes of these icosahedra with non-trivial automorphism group and find one curve of flexible icosahedra. Visualisations and explicit data for this paper are available under http://algebra.data.rwth-aachen. de/Icosahedra/visualplusdata.html.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes of these icosahedra with non-trivial automorphism group and find one curve of flexible icosahedra. Visualisations and explicit data for this paper are available under http://algebra.data.rwth-aachen. de/Icosahedra/visualplusdata.html. |
Lange-Hegermann, M; Robertz, D Thomas Decomposition and Nonlinear Control Systems Book Chapter Quadrat A.; Zerz, E (Ed.): Algebraic and Symbolic Computation Methods in Dynamical Systems, 9 , pp. 117-146, Springer, 2020. Abstract | Links | BibTeX @inbook{LangeHegermannRobertz2b,
title = {Thomas Decomposition and Nonlinear Control Systems},
author = {Lange-Hegermann, M. and Robertz, D.},
editor = {Quadrat, A.; Zerz, E.},
url = {https://link.springer.com/chapter/10.1007/978-3-030-38356-5_4},
year = {2020},
date = {2020-01-01},
booktitle = {Algebraic and Symbolic Computation Methods in Dynamical Systems},
volume = {9},
pages = {117-146},
publisher = {Springer},
series = {Advances in Delays and Dynamics},
abstract = {This paper applies the Thomas decomposition technique to nonlinear control
systems, in particular to the study of the dependence of the system behavior on
parameters. Thomas' algorithm is a symbolic method which splits
a given system of nonlinear partial differential equations into a finite family
of so-called simple systems which are formally integrable and define a partition
of the solution set of the original differential system. Different simple systems
of a Thomas decomposition describe different structural behavior of the control
system in general. The paper gives an introduction to the Thomas decomposition
method and shows how notions such as invertibility, observability and flat outputs
can be studied. A Maple implementation of Thomas' algorithm is used to illustrate
the techniques on explicit examples.},
keywords = {},
pubstate = {published},
tppubtype = {inbook}
}
This paper applies the Thomas decomposition technique to nonlinear control
systems, in particular to the study of the dependence of the system behavior on
parameters. Thomas' algorithm is a symbolic method which splits
a given system of nonlinear partial differential equations into a finite family
of so-called simple systems which are formally integrable and define a partition
of the solution set of the original differential system. Different simple systems
of a Thomas decomposition describe different structural behavior of the control
system in general. The paper gives an introduction to the Thomas decomposition
method and shows how notions such as invertibility, observability and flat outputs
can be studied. A Maple implementation of Thomas' algorithm is used to illustrate
the techniques on explicit examples. |
2019
|
Gerdt, V P; Lange-Hegermann, M; Robertz, D The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs Journal Article Computer Physics Communications, 234 , pp. 202-215, 2019. Abstract | Links | BibTeX @article{GerdtLangeHegermannRobertzc,
title = {The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs},
author = {Gerdt, V. P. and Lange-Hegermann, M. and Robertz, D.},
url = {https://doi.org/10.1016/j.cpc.2018.07.025},
year = {2019},
date = {2019-01-02},
journal = {Computer Physics Communications},
volume = {234},
pages = {202-215},
abstract = {We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given.
|
Gerdt, V P; Robertz, D Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems Inproceedings Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation, 15-18 July 2019, Beihang University, Beijing, China, pp. 163–170, 2019. Abstract | BibTeX @inproceedings{GerdtRobertz6,
title = {Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems},
author = {Gerdt, V. P. and Robertz, D.},
year = {2019},
date = {2019-01-01},
booktitle = {Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation, 15-18 July 2019, Beihang University, Beijing, China},
pages = {163--170},
abstract = {For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential Thomas decomposition to the input system, resulting in a partition of the solution set. We consider the output simple subsystem that contains a solution of interest. Then, for this subsystem, we suggest an algorithm for verification of s-consistency for its finite difference approximation. For this purpose we develop a difference analogue of the differential Thomas decomposition, both of which jointly allow to verify the s-consistency of the approximation. As an application of our approach, we show how to produce s-consistent difference approximations to the incompressible Navier-Stokes equations including the pressure Poisson equation.},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential Thomas decomposition to the input system, resulting in a partition of the solution set. We consider the output simple subsystem that contains a solution of interest. Then, for this subsystem, we suggest an algorithm for verification of s-consistency for its finite difference approximation. For this purpose we develop a difference analogue of the differential Thomas decomposition, both of which jointly allow to verify the s-consistency of the approximation. As an application of our approach, we show how to produce s-consistent difference approximations to the incompressible Navier-Stokes equations including the pressure Poisson equation. |
2018
|
Robertz, D Formal methods for systems of partial differential equations Journal Article Les cours du CIRM, 6 (1), pp. 1–37, 2018. Links | BibTeX @article{Robertz10,
title = {Formal methods for systems of partial differential equations},
author = {Robertz, D.},
url = {http://ccirm.cedram.org/item?id=CCIRM_2018__6_1_A3_0},
year = {2018},
date = {2018-12-31},
issuetitle = {Journ'ees Nationales de Calcul Formel (2018), Exp. No. III.},
journal = {Les cours du CIRM},
volume = {6},
number = {1},
pages = {1--37},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
2016
|
Craven, Matthew J; Robertz, Daniel A parallel evolutionary approach to solving systems of equations in polycyclic groups Journal Article Groups Complexity Cryptology, 8 (2), pp. 109–125, 2016. Abstract | Links | BibTeX @article{craven2016parallel,
title = {A parallel evolutionary approach to solving systems of equations in polycyclic groups},
author = {Matthew J Craven and Daniel Robertz},
doi = {10.1515/gcc-2016-0012},
year = {2016},
date = {2016-11-01},
journal = {Groups Complexity Cryptology},
volume = {8},
number = {2},
pages = {109--125},
publisher = {De Gruyter},
abstract = {The Anshel–Anshel–Goldfeld (AAG) key exchange protocol is based upon the multiple conjugacy problem for a finitely-presented group. The hardness in breaking this protocol relies on the supposed difficulty in solving the corresponding equations for the conjugating element in the group. Two such protocols based on polycyclic groups as a platform were recently proposed and were shown to be resistant to length-based attack. In this article we propose a parallel evolutionary approach which runs on multicore high-performance architectures. The approach is shown to be more efficient than previous attempts to break these protocols, and also more successful. Comprehensive data of experiments run with a GAP implementation are provided and compared to the results of earlier length-based attacks. These demonstrate that the proposed platform is not as secure as first thought and also show that existing measures of cryptographic complexity are not optimal. A more accurate alternative measure is suggested. Finally, a linear algebra attack for one of the protocols is introduced.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Anshel–Anshel–Goldfeld (AAG) key exchange protocol is based upon the multiple conjugacy problem for a finitely-presented group. The hardness in breaking this protocol relies on the supposed difficulty in solving the corresponding equations for the conjugating element in the group. Two such protocols based on polycyclic groups as a platform were recently proposed and were shown to be resistant to length-based attack. In this article we propose a parallel evolutionary approach which runs on multicore high-performance architectures. The approach is shown to be more efficient than previous attempts to break these protocols, and also more successful. Comprehensive data of experiments run with a GAP implementation are provided and compared to the results of earlier length-based attacks. These demonstrate that the proposed platform is not as secure as first thought and also show that existing measures of cryptographic complexity are not optimal. A more accurate alternative measure is suggested. Finally, a linear algebra attack for one of the protocols is introduced. |
Robertz, D Formal Algorithmic Elimination for PDEs Inproceedings Proceedings of the 2016 International Symposium on Symbolic and Algebraic Computation, 19-22 July 2016, Waterloo, Ontario, Canada, pp. 19-22, 2016. BibTeX @inproceedings{Robertz9b,
title = {Formal Algorithmic Elimination for PDEs},
author = {Robertz, D.},
year = {2016},
date = {2016-07-29},
booktitle = {Proceedings of the 2016 International Symposium on Symbolic and Algebraic Computation, 19-22 July 2016, Waterloo, Ontario, Canada},
pages = {19-22},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Gerdt, V P; Robertz, D Lagrangian Constraints and Differential Thomas Decomposition Journal Article Advances in Applied Mathematics, 72 , pp. 113-138, 2016. Abstract | Links | BibTeX @article{GerdtRobertz5,
title = {Lagrangian Constraints and Differential Thomas Decomposition},
author = {Gerdt, V. P. and Robertz, D.},
url = {http://dx.doi.org/10.1016/j.aam.2015.09.005},
year = {2016},
date = {2016-01-01},
journal = {Advances in Applied Mathematics},
volume = {72},
pages = {113-138},
abstract = {In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian becomes singular, then our approach allows to detect such domains and compute the relevant constraints. In doing so, we assume that the Lagrangian of a model is a differential polynomial and apply the differential Thomas decomposition algorithm to the Euler-Lagrange equations.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian becomes singular, then our approach allows to detect such domains and compute the relevant constraints. In doing so, we assume that the Lagrangian of a model is a differential polynomial and apply the differential Thomas decomposition algorithm to the Euler-Lagrange equations.
|
2015
|
Robertz, D Recent Progress in an Algebraic Analysis Approach to Linear Systems Journal Article Multidimensional Systems and Signal Processing, 26 (2), pp. 349-388, 2015. Links | BibTeX @article{Robertz7,
title = {Recent Progress in an Algebraic Analysis Approach to Linear Systems},
author = {Robertz, D.},
url = {https://doi.org/10.1007/s11045-014-0280-9},
year = {2015},
date = {2015-04-01},
journal = {Multidimensional Systems and Signal Processing},
volume = {26},
number = {2},
pages = {349-388},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
2014
|
Quadrat, A; Robertz, D A constructive study of the module structure of rings of partial differential operators Journal Article Acta Applicandae Mathematicae, 133 (1), pp. 187-234, 2014. Abstract | Links | BibTeX @article{QuadratRobertz9,
title = {A constructive study of the module structure of rings of partial differential operators},
author = {Quadrat, A. and Robertz, D.},
url = {http://dx.doi.org/10.1007/s10440-013-9864-x},
year = {2014},
date = {2014-01-01},
journal = {Acta Applicandae Mathematicae},
volume = {133},
number = {1},
pages = {187-234},
abstract = {The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras $A_n(k)$ (i.e., the rings of partial differential operators with polynomial coefficients) over a base field $k$ of characteristic zero. Using the very
simplicity of the ring $A_n(k)$, we show how to explicitly compute a unimodular element of a finitely generated left $A_n(k)$-module of rank at least two. This result is used to constructively decompose any finitely generated left $A_n(k)$-module into a direct sum of a free left $A_n(k)$-module and a left $A_n(k)$-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of $A_n(k)$ which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left $A_n(k)$-module with module of relations of rank at least two to two. In particular, any finitely generated torsion left $A_n(k)$-module can be generated by two elements and is the image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left $A_n(k)$-module of rank $r$ can be generated by $r+1$ elements but no fewer. These results are implemented in the Stafford package and their system-theoretical interpretations are given within a $D$-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result due to Caro and Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras $A_n(k)$ (i.e., the rings of partial differential operators with polynomial coefficients) over a base field $k$ of characteristic zero. Using the very
simplicity of the ring $A_n(k)$, we show how to explicitly compute a unimodular element of a finitely generated left $A_n(k)$-module of rank at least two. This result is used to constructively decompose any finitely generated left $A_n(k)$-module into a direct sum of a free left $A_n(k)$-module and a left $A_n(k)$-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of $A_n(k)$ which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left $A_n(k)$-module with module of relations of rank at least two to two. In particular, any finitely generated torsion left $A_n(k)$-module can be generated by two elements and is the image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left $A_n(k)$-module of rank $r$ can be generated by $r+1$ elements but no fewer. These results are implemented in the Stafford package and their system-theoretical interpretations are given within a $D$-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result due to Caro and Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.
|
Robertz, D Formal Algorithmic Elimination for PDEs Book Springer, Cham, 2014. BibTeX @book{Robertz6,
title = {Formal Algorithmic Elimination for PDEs},
author = {Robertz, D.},
year = {2014},
date = {2014-01-01},
volume = {2121},
publisher = {Springer},
address = {Cham},
series = {Lecture Notes in Mathematics},
keywords = {},
pubstate = {published},
tppubtype = {book}
}
|
2013
|
Lange-Hegermann, M; Robertz, D Thomas decompositions of parametric nonlinear control systems Inproceedings Proceedings of the 5th Symposium on System Structure and Control, Grenoble (France), pp. 291–296, 2013. BibTeX @inproceedings{LangeHegermannRobertz1,
title = {Thomas decompositions of parametric nonlinear control systems},
author = {Lange-Hegermann, M. and Robertz, D.},
year = {2013},
date = {2013-01-01},
booktitle = {Proceedings of the 5th Symposium on System Structure and Control, Grenoble (France)},
pages = {291--296},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Quadrat, A; Robertz, D Stafford's reduction of linear partial differential systems Inproceedings Proceedings of the 5th Symposium on System Structure and Control, Grenoble (France), pp. 309–314, 2013. BibTeX @inproceedings{QuadratRobertz8,
title = {Stafford's reduction of linear partial differential systems},
author = {Quadrat, A. and Robertz, D.},
year = {2013},
date = {2013-01-01},
booktitle = {Proceedings of the 5th Symposium on System Structure and Control, Grenoble (France)},
pages = {309--314},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Robertz, D Applying Thomas decomposition and algebraic analysis to certain nonlinear PDE systems Inproceedings Mathematisches Forschungsinstitut Oberwolfach Report No. ??/2013, pp. ???, 2013. BibTeX @inproceedings{Robertz8,
title = {Applying Thomas decomposition and algebraic analysis to certain nonlinear PDE systems},
author = {Robertz, D.},
year = {2013},
date = {2013-01-01},
booktitle = {Mathematisches Forschungsinstitut Oberwolfach Report No. ??/2013},
pages = {???},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
2012
|
Bächler, T; Gerdt, V P; Lange-Hegermann, M; Robertz, D Algorithmic Thomas Decomposition of Algebraic and Differential Systems Journal Article Journal of Symbolic Computation, 47 (10), pp. 1233–1266, 2012. Abstract | BibTeX @article{BaechlerGerdtLangeHegermannRobertz2,
title = {Algorithmic Thomas Decomposition of Algebraic and Differential Systems},
author = {Bächler, T. and Gerdt, V. P. and Lange-Hegermann, M. and Robertz, D.},
year = {2012},
date = {2012-01-01},
journal = {Journal of Symbolic Computation},
volume = {47},
number = {10},
pages = {1233--1266},
abstract = {In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these system into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and non-vanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of Bächler et al. (2010) and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these system into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and non-vanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of Bächler et al. (2010) and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms. |
Gerdt, V P; Robertz, D Computation of Difference Gröbner Bases Journal Article Computer Science Journal of Moldova, 20 (2 (59)), pp. 203–226, 2012. BibTeX @article{GerdtRobertz4,
title = {Computation of Difference Gröbner Bases},
author = {Gerdt, V. P. and Robertz, D.},
year = {2012},
date = {2012-01-01},
journal = {Computer Science Journal of Moldova},
volume = {20},
number = {2 (59)},
pages = {203--226},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
2010
|
Plesken, W; Robertz, D Linear differential elimination for analytic functions Journal Article Mathematics in Computer Science, 4 (2--3), pp. 231–242, 2010. BibTeX @article{PleskenRobertz7,
title = {Linear differential elimination for analytic functions},
author = {Plesken, W. and Robertz, D.},
year = {2010},
date = {2010-01-01},
journal = {Mathematics in Computer Science},
volume = {4},
number = {2--3},
pages = {231--242},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Quadrat, A; Robertz, D Controllability and differential flatness of linear analytic ordinary differential systems Inproceedings Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), Budapest, Hungary, 2010. BibTeX @inproceedings{QuadratRobertz7,
title = {Controllability and differential flatness of linear analytic ordinary differential systems},
author = {Quadrat, A. and Robertz, D.},
year = {2010},
date = {2010-01-01},
booktitle = {Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), Budapest, Hungary},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Gerdt, V P; Robertz, D Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification Inproceedings Watt, S M (Ed.): Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 25-28 July 2010, TU M"unchen, Germany, pp. 53–59, 2010. BibTeX @inproceedings{GerdtRobertz3,
title = {Consistency of Finite Difference Approximations for Linear PDE Systems and its Algorithmic Verification},
author = {Gerdt, V. P. and Robertz, D.},
editor = {S. M. Watt},
year = {2010},
date = {2010-01-01},
booktitle = {Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, 25-28 July 2010, TU M"unchen, Germany},
pages = {53--59},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Bächler, T; Gerdt, V P; Lange-Hegermann, M; Robertz, D Thomas Decomposition of Algebraic and Differential Systems Inproceedings Gerdt, V P; Koepf, W; Mayr, E W; Vorozhtsov, E H (Ed.): Computer Algebra in Scientific Computing, 12th International Workshop, CASC 2010, Tsakhkadzor, Armenia, pp. 31–54, Springer, 2010. BibTeX @inproceedings{BaechlerGerdtLangeHegermannRobertz1,
title = {Thomas Decomposition of Algebraic and Differential Systems},
author = {Bächler, T. and Gerdt, V. P. and Lange-Hegermann, M. and Robertz, D.},
editor = {Gerdt, V. P. and Koepf, W. and Mayr, E. W. and Vorozhtsov, E. H.},
year = {2010},
date = {2010-01-01},
booktitle = {Computer Algebra in Scientific Computing, 12th International Workshop, CASC 2010, Tsakhkadzor, Armenia},
pages = {31--54},
publisher = {Springer},
series = {Lecture Notes in Computer Science, Vol. 6244},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
2009
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Plesken, W; Robertz, D The average number of cycles Journal Article Archiv der Mathematik, 93 (5), pp. 445–449, 2009. BibTeX @article{PleskenRobertz6,
title = {The average number of cycles},
author = {Plesken, W. and Robertz, D.},
year = {2009},
date = {2009-11-01},
journal = {Archiv der Mathematik},
volume = {93},
number = {5},
pages = {445--449},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Barakat, M; Robertz, D conley: Computing connection matrices in Maple Journal Article Journal of Symbolic Computation, 44 (5), pp. 540–557, 2009. BibTeX @article{BarakatRobertz4,
title = {conley: Computing connection matrices in Maple},
author = {Barakat, M. and Robertz, D.},
year = {2009},
date = {2009-01-01},
journal = {Journal of Symbolic Computation},
volume = {44},
number = {5},
pages = {540--557},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Robertz, D Noether normalization guided by monomial cone decompositions Journal Article Journal of Symbolic Computation, 44 (10), pp. 1359–1373, 2009. BibTeX @article{Robertz3,
title = {Noether normalization guided by monomial cone decompositions},
author = {Robertz, D.},
year = {2009},
date = {2009-01-01},
journal = {Journal of Symbolic Computation},
volume = {44},
number = {10},
pages = {1359--1373},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Robertz, D The Auslander-Buchsbaum-Serre theorem Inproceedings Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2009, pp. 2750–2751, 2009. BibTeX @inproceedings{Robertz4,
title = {The Auslander-Buchsbaum-Serre theorem},
author = {Robertz, D.},
year = {2009},
date = {2009-01-01},
booktitle = {Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2009},
pages = {2750--2751},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Robertz, D Parametrizing linear systems Inproceedings Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2009, pp. 2755–2758, 2009. BibTeX @inproceedings{Robertz5,
title = {Parametrizing linear systems},
author = {Robertz, D.},
year = {2009},
date = {2009-01-01},
booktitle = {Mathematisches Forschungsinstitut Oberwolfach Report No. 50/2009},
pages = {2755--2758},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
2008
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Barakat, M; Robertz, D homalg: A meta-package for homological algebra Journal Article Journal of Algebra and Its Applications, 7 (3), pp. 299–317, 2008. BibTeX @article{BarakatRobertz1,
title = {homalg: A meta-package for homological algebra},
author = {Barakat, M. and Robertz, D.},
year = {2008},
date = {2008-01-01},
journal = {Journal of Algebra and Its Applications},
volume = {7},
number = {3},
pages = {299--317},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Plesken, W; Robertz, D Elimination for coefficients of special characteristic polynomials Journal Article Experimental Mathematics, 17 (4), pp. 499–510, 2008. BibTeX @article{PleskenRobertz5,
title = {Elimination for coefficients of special characteristic polynomials},
author = {Plesken, W. and Robertz, D.},
year = {2008},
date = {2008-01-01},
journal = {Experimental Mathematics},
volume = {17},
number = {4},
pages = {499--510},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Quadrat, A; Robertz, D Baer's extension problem for multidimensional linear systems Inproceedings Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2008), Virginia Tech, Blacksburg, Virginia (USA), 2008. BibTeX @inproceedings{QuadratRobertz6,
title = {Baer's extension problem for multidimensional linear systems},
author = {Quadrat, A. and Robertz, D.},
year = {2008},
date = {2008-01-01},
booktitle = {Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2008), Virginia Tech, Blacksburg, Virginia (USA)},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
2007
|
Chyzak, F; Quadrat, A; Robertz, D OreModules: A Symbolic Package for the Study of Multidimensional Linear Systems Inproceedings Chiasson, J; Loiseau, J -J (Ed.): Applications of Time-Delay Systems, pp. 233–264, Springer, 2007. BibTeX @inproceedings{ChyzakQuadratRobertz4,
title = {OreModules: A Symbolic Package for the Study of Multidimensional Linear Systems},
author = {Chyzak, F. and Quadrat, A. and Robertz, D.},
editor = {Chiasson, J. and Loiseau, J.-J.},
year = {2007},
date = {2007-01-01},
booktitle = {Applications of Time-Delay Systems},
pages = {233--264},
publisher = {Springer},
series = {LNCIS 352},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Robertz, D Janet Bases and Applications Inproceedings Rosenkranz, M; Wang, D (Ed.): Groebner Bases in Symbolic Analysis, pp. 139–168, de Gruyter, 2007. BibTeX @inproceedings{Robertz2,
title = {Janet Bases and Applications},
author = {Robertz, D.},
editor = {Rosenkranz, M. and Wang, D.},
year = {2007},
date = {2007-01-01},
booktitle = {Groebner Bases in Symbolic Analysis},
pages = {139--168},
publisher = {de Gruyter},
series = {Radon Series on Computational and Applied Mathematics 2},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Quadrat, A; Robertz, D Computation of bases of free modules over the Weyl algebras Journal Article Journal of Symbolic Computation, 42 (11--12), pp. 1113–1141, 2007. BibTeX @article{QuadratRobertz5,
title = {Computation of bases of free modules over the Weyl algebras},
author = {Quadrat, A. and Robertz, D.},
year = {2007},
date = {2007-01-01},
journal = {Journal of Symbolic Computation},
volume = {42},
number = {11--12},
pages = {1113--1141},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Plesken, W; Robertz, D Some Elimination Problems for Matrices Inproceedings Ganzha, V G; Mayr, E W; Vorozhtsov, E V (Ed.): Computer Algebra in Scientific Computing, 10th International Workshop, CASC 2007, Bonn, Germany, pp. 350–359, Springer, 2007. BibTeX @inproceedings{PleskenRobertz4,
title = {Some Elimination Problems for Matrices},
author = {Plesken, W. and Robertz, D.},
editor = {Ganzha, V. G. and Mayr, E. W. and Vorozhtsov, E. V.},
year = {2007},
date = {2007-01-01},
booktitle = {Computer Algebra in Scientific Computing, 10th International Workshop, CASC 2007, Bonn, Germany},
pages = {350--359},
publisher = {Springer},
series = {Lecture Notes in Computer Science, Vol. 4770},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
2006
|
Gerdt, V P; Robertz, D A Maple Package for Computing Gröbner Bases for Linear Recurrence Relations Journal Article Nuclear Instruments and Methods in Physics Research, A: Accelerators, Spectrometers, Detectors and Associated Equipment, 559 (1), pp. 215–219, 2006. BibTeX @article{GerdtRobertz1,
title = {A Maple Package for Computing Gröbner Bases for Linear Recurrence Relations},
author = {Gerdt, V. P. and Robertz, D.},
year = {2006},
date = {2006-01-01},
journal = {Nuclear Instruments and Methods in Physics Research, A: Accelerators, Spectrometers, Detectors and Associated Equipment},
volume = {559},
number = {1},
pages = {215--219},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Plesken, W; Robertz, D Representations, commutative algebra, and Hurwitz groups Journal Article Journal of Algebra, 300 (1), pp. 223-247, 2006. Links | BibTeX @article{PleskenRobertz3,
title = {Representations, commutative algebra, and Hurwitz groups},
author = {Plesken, W. and Robertz, D.},
url = {http://dx.doi.org/10.1016/j.jalgebra.2006.02.021},
year = {2006},
date = {2006-01-01},
journal = {Journal of Algebra},
volume = {300},
number = {1},
pages = {223-247},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Barakat, M; Robertz, D homalg: First steps to an abstract package for homological algebra Inproceedings Proceedings of the X meeting on computational algebra and its applications (EACA 2006), Seville (Spain), pp. 29–32, 2006. BibTeX @inproceedings{BarakatRobertz2,
title = {homalg: First steps to an abstract package for homological algebra},
author = {Barakat, M. and Robertz, D.},
year = {2006},
date = {2006-01-01},
booktitle = {Proceedings of the X meeting on computational algebra and its applications (EACA 2006), Seville (Spain)},
pages = {29--32},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Quadrat, A; Robertz, D Constructive computation of flat outputs of a class of multidimensional linear systems with variable coefficients Inproceedings Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan), pp. 583–595, 2006. BibTeX @inproceedings{QuadratRobertz3,
title = {Constructive computation of flat outputs of a class of multidimensional linear systems with variable coefficients},
author = {Quadrat, A. and Robertz, D.},
year = {2006},
date = {2006-01-01},
booktitle = {Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan)},
pages = {583--595},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Barakat, M; Robertz, D Computing invariants of multidimensional linear systems on an abstract homological level Inproceedings Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan), pp. 542–559, 2006. BibTeX @inproceedings{BarakatRobertz3,
title = {Computing invariants of multidimensional linear systems on an abstract homological level},
author = {Barakat, M. and Robertz, D.},
year = {2006},
date = {2006-01-01},
booktitle = {Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan)},
pages = {542--559},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Quadrat, A; Robertz, D On the Monge problem and multidimensional optimal control Inproceedings Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan), pp. 596–605, 2006. BibTeX @inproceedings{QuadratRobertz4,
title = {On the Monge problem and multidimensional optimal control},
author = {Quadrat, A. and Robertz, D.},
year = {2006},
date = {2006-01-01},
booktitle = {Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006), Kyoto (Japan)},
pages = {596--605},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Robertz, D Representations, commutative algebra, and Hurwitz groups (joint work with Wilhelm Plesken) Inproceedings Mathematisches Forschungsinstitut Oberwolfach Report No. 30/2006, pp. 1842–1843, 2006. BibTeX @inproceedings{Robertz0,
title = {Representations, commutative algebra, and Hurwitz groups (joint work with Wilhelm Plesken)},
author = {Robertz, D.},
year = {2006},
date = {2006-01-01},
booktitle = {Mathematisches Forschungsinstitut Oberwolfach Report No. 30/2006},
pages = {1842--1843},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Robertz, D Formal Computational Methods for Control Theory PhD Thesis RWTH Aachen, Germany, 2006. Links | BibTeX @phdthesis{Robertz1,
title = {Formal Computational Methods for Control Theory},
author = {Robertz, D.},
url = {https://publications.rwth-aachen.de/record/61055},
year = {2006},
date = {2006-01-01},
school = {RWTH Aachen, Germany},
keywords = {},
pubstate = {published},
tppubtype = {phdthesis}
}
|
Corves, B; Abel, D; Plesken, W; Harmeling, F; Robertz, D; Maschuw, J Methoden und Werkzeuge zum Entwurf mechatronischer Bewegungssysteme mit ungleichmäßig übersetzenden Getrieben Inproceedings VDI Berichte 1963, pp. 557–573, VDI-Verlag, 2006. BibTeX @inproceedings{CorvesAbelPleskenHarmelingRobertzMaschuw,
title = {Methoden und Werkzeuge zum Entwurf mechatronischer Bewegungssysteme mit ungleichmäßig übersetzenden Getrieben},
author = {Corves, B. and Abel, D. and Plesken, W. and Harmeling, F. and Robertz, D. and Maschuw, J.},
year = {2006},
date = {2006-01-01},
booktitle = {VDI Berichte 1963},
pages = {557--573},
publisher = {VDI-Verlag},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
2005
|
Chyzak, F; Quadrat, A; Robertz, D Effective algorithms for parametrizing linear control systems over Ore algebras Journal Article Applicable Algebra in Engineering, Communication and Computing, 16 (5), pp. 319-376, 2005. Links | BibTeX @article{ChyzakQuadratRobertz2,
title = {Effective algorithms for parametrizing linear control systems over Ore algebras},
author = {Chyzak, F. and Quadrat, A. and Robertz, D.},
url = {http://dx.doi.org/10.1007/s00200-005-0188-6},
year = {2005},
date = {2005-01-01},
journal = {Applicable Algebra in Engineering, Communication and Computing},
volume = {16},
number = {5},
pages = {319-376},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Plesken, W; Robertz, D Constructing Invariants for Finite Groups Journal Article Experimental Mathematics, 14 (2), pp. 175–188, 2005. BibTeX @article{PleskenRobertz1,
title = {Constructing Invariants for Finite Groups},
author = {Plesken, W. and Robertz, D.},
year = {2005},
date = {2005-01-01},
journal = {Experimental Mathematics},
volume = {14},
number = {2},
pages = {175--188},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Plesken, W; Robertz, D Janet's approach to presentations and resolutions for polynomials and linear pdes Journal Article Archiv der Mathematik, 84 (1), pp. 22–37, 2005. BibTeX @article{PleskenRobertz2,
title = {Janet's approach to presentations and resolutions for polynomials and linear pdes},
author = {Plesken, W. and Robertz, D.},
year = {2005},
date = {2005-01-01},
journal = {Archiv der Mathematik},
volume = {84},
number = {1},
pages = {22--37},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|
Quadrat, A; Robertz, D Parametrizing all solutions of uncontrollable multidimensional linear systems Inproceedings Proceedings of the 16th IFAC World Congress, Prague (Czech Republic), July 4-8, 2005, 2005. BibTeX @inproceedings{QuadratRobertz1,
title = {Parametrizing all solutions of uncontrollable multidimensional linear systems},
author = {Quadrat, A. and Robertz, D.},
year = {2005},
date = {2005-01-01},
booktitle = {Proceedings of the 16th IFAC World Congress, Prague (Czech Republic), July 4-8, 2005},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Quadrat, A; Robertz, D On the blowing-up of stably free behaviours Inproceedings Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Seville (Spain), December 12-15, 2005, pp. 1541–1546, 2005. BibTeX @inproceedings{QuadratRobertz2,
title = {On the blowing-up of stably free behaviours},
author = {Quadrat, A. and Robertz, D.},
year = {2005},
date = {2005-01-01},
booktitle = {Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Seville (Spain), December 12-15, 2005},
pages = {1541--1546},
keywords = {},
pubstate = {published},
tppubtype = {inproceedings}
}
|
Gerdt, V P; Robertz, D Computation of Gröbner Bases for Systems of Linear Difference Equations Journal Article Computeralgebra-Rundbrief, 37 , pp. 8–16, 2005. BibTeX @article{GerdtRobertz2,
title = {Computation of Gröbner Bases for Systems of Linear Difference Equations},
author = {Gerdt, V. P. and Robertz, D.},
year = {2005},
date = {2005-01-01},
journal = {Computeralgebra-Rundbrief},
volume = {37},
pages = {8--16},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
|