Singular.m is an interface package, allowing the execution of Singular functions from Mathematica notebooks, written by Manuel Kauers and Viktor Levandovskyy. ...

# Dr. Viktor Levandovskyy

### Research Area

(Noncommutative) Computer Algebra## Software

Authors: Manuel Kauers, Viktor Levandovskyy

MoreSoftware Website## Publications

### 2011

[Levandovskyy]

### On Two-generated Non-commutative Algebras Subject to the Affine Relation

#### Viktor Levandovskyy, Christoph Koutschan, Oleksandr Motsak

In: Proceedings of CASC 2011, Vladimir Gerdt, Wolfram Koepf, Ernst W. Mayr, Evgenii Vorozhtsov (ed.), Lecture Notes in Computer Science 6885, pp. 309-320. 2011. Springer, ISBN 978-3-642-23567-2. [pdf]@

author = {Viktor Levandovskyy and Christoph Koutschan and Oleksandr Motsak},

title = {{On Two-generated Non-commutative Algebras Subject to the Affine Relation}},

booktitle = {{Proceedings of CASC 2011}},

language = {english},

abstract = {We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y^m x^n in terms of standard monomials x^i y^j for many algebras of the considered type. Such formulas are used in e.g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.},

series = {Lecture Notes in Computer Science},

volume = {6885},

pages = {309--320},

publisher = {Springer},

isbn_issn = {ISBN 978-3-642-23567-2},

year = {2011},

editor = {Vladimir Gerdt and Wolfram Koepf and Ernst W. Mayr and Evgenii Vorozhtsov},

refereed = {yes},

length = {12}

}

**inproceedings**{RISC4374,author = {Viktor Levandovskyy and Christoph Koutschan and Oleksandr Motsak},

title = {{On Two-generated Non-commutative Algebras Subject to the Affine Relation}},

booktitle = {{Proceedings of CASC 2011}},

language = {english},

abstract = {We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + ax + by + c for q in K^* and a, b, c in K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y^m x^n in terms of standard monomials x^i y^j for many algebras of the considered type. Such formulas are used in e.g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.},

series = {Lecture Notes in Computer Science},

volume = {6885},

pages = {309--320},

publisher = {Springer},

isbn_issn = {ISBN 978-3-642-23567-2},

year = {2011},

editor = {Vladimir Gerdt and Wolfram Koepf and Ernst W. Mayr and Evgenii Vorozhtsov},

refereed = {yes},

length = {12}

}

### 2008

[Levandovskyy]

### Computing one of Victor Moll's irresistible integrals with computer algebra

#### Christoph Koutschan, Viktor Levandovskyy

Computer Science Journal of Moldova 16(1(46)), pp. 35-49. 2008. ISSN 1561-4042. [pdf]@

author = {Christoph Koutschan and Viktor Levandovskyy},

title = {{Computing one of Victor Moll's irresistible integrals with computer algebra}},

language = {english},

abstract = {We investigate a certain quartic integral from V. Moll's book"`Irresistible Integrals" and demonstrate how it can be solvedby computer algebra methods, namely by using non-commutative Gröbnerbases. We present recent implementations in the computer algebra systemsSingular and Mathematica.},

journal = {Computer Science Journal of Moldova},

volume = {16},

number = {1(46)},

pages = {35--49},

isbn_issn = {ISSN 1561--4042},

year = {2008},

refereed = {yes},

length = {15}

}

**article**{RISC3421,author = {Christoph Koutschan and Viktor Levandovskyy},

title = {{Computing one of Victor Moll's irresistible integrals with computer algebra}},

language = {english},

abstract = {We investigate a certain quartic integral from V. Moll's book"`Irresistible Integrals" and demonstrate how it can be solvedby computer algebra methods, namely by using non-commutative Gröbnerbases. We present recent implementations in the computer algebra systemsSingular and Mathematica.},

journal = {Computer Science Journal of Moldova},

volume = {16},

number = {1(46)},

pages = {35--49},

isbn_issn = {ISSN 1561--4042},

year = {2008},

refereed = {yes},

length = {15}

}

### 2006

[Levandovskyy]

### Intersection of Ideals with Non-commutative Subalgebras

#### V. Levandovskyy

In: Proceedings of the ISSAC 2006 Conference, Jean-Guillaume Dumas (ed.), Proceedings of ISSAC 2006, pp. 212-219. July 2006. ACM Press, 1-59593-095-7.@

author = {V. Levandovskyy},

title = {{Intersection of Ideals with Non-commutative Subalgebras}},

booktitle = {{Proceedings of the ISSAC 2006 Conference}},

language = {english},

abstract = {Computation of an intersection of a left ideal with a subalgebra, which is not fully investigateduntil now, is important for different areas of mathematics. We present an algorithm for the computation of the preimage of a left ideal under a morphism of non--commutative $GR$--algebras, and show both its abilities and limitations.The main computational tools are the elimination of variablesby means of Gr\"obner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure.},

pages = {212--219},

publisher = {ACM Press},

isbn_issn = {1-59593-095-7},

year = {2006},

month = {July},

editor = {Jean-Guillaume Dumas},

refereed = {yes},

length = {8},

conferencename = {ISSAC 2006}

}

**inproceedings**{RISC2940,author = {V. Levandovskyy},

title = {{Intersection of Ideals with Non-commutative Subalgebras}},

booktitle = {{Proceedings of the ISSAC 2006 Conference}},

language = {english},

abstract = {Computation of an intersection of a left ideal with a subalgebra, which is not fully investigateduntil now, is important for different areas of mathematics. We present an algorithm for the computation of the preimage of a left ideal under a morphism of non--commutative $GR$--algebras, and show both its abilities and limitations.The main computational tools are the elimination of variablesby means of Gr\"obner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure.},

pages = {212--219},

publisher = {ACM Press},

isbn_issn = {1-59593-095-7},

year = {2006},

month = {July},

editor = {Jean-Guillaume Dumas},

refereed = {yes},

length = {8},

conferencename = {ISSAC 2006}

}

[Levandovskyy]

### Intersection of Ideals with Non-commutative Subalgebras

#### V. Levandovskyy

J. Kepler University Linz. Technical report no. 2006-14, April 2006. the final version has been accepted for the Proceedings ISSAC 2006. SFB Report. [pdf]@

author = {V. Levandovskyy},

title = {{Intersection of Ideals with Non-commutative Subalgebras}},

language = {english},

abstract = {Computation of an intersection of a left ideal with a subalgebra, which is not fully investigateduntil now, is important for different areas of mathematics. We present an algorithm for the computation of the preimage of a left ideal under a morphism of non--commutative $GR$--algebras, and show both its abilities and limitations.The main computational tools are the elimination of variables by means of Gr\"obner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure.},

number = {2006-14},

year = {2006},

month = {April},

note = {the final version has been accepted for the Proceedings ISSAC 2006},

institution = {J. Kepler University Linz},

keywords = {Non--commutative algebra, Groebner bases, elimination, intersection with subalgebra, preimage of ideal, homomorphism of algebras, restriction},

length = {16},

type = {SFB Report}

}

**techreport**{RISC2894,author = {V. Levandovskyy},

title = {{Intersection of Ideals with Non-commutative Subalgebras}},

language = {english},

abstract = {Computation of an intersection of a left ideal with a subalgebra, which is not fully investigateduntil now, is important for different areas of mathematics. We present an algorithm for the computation of the preimage of a left ideal under a morphism of non--commutative $GR$--algebras, and show both its abilities and limitations.The main computational tools are the elimination of variables by means of Gr\"obner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure.},

number = {2006-14},

year = {2006},

month = {April},

note = {the final version has been accepted for the Proceedings ISSAC 2006},

institution = {J. Kepler University Linz},

keywords = {Non--commutative algebra, Groebner bases, elimination, intersection with subalgebra, preimage of ideal, homomorphism of algebras, restriction},

length = {16},

type = {SFB Report}

}

[Levandovskyy]

### PLURAL, a Non-commutative Extension of SINGULAR: Past, Present and Future

#### V. Levandovskyy

In: Mathematical Software - ICMS 2006, A. Iglesias, N. Takayama (ed.), Proceedings of International Congress on Mathematical Software, LNCS 4151, pp. 144-157. September 2006. 0302-9743. [pdf]@

author = {V. Levandovskyy},

title = {{PLURAL, a Non--commutative Extension of SINGULAR: Past, Present and Future}},

booktitle = {{Mathematical Software - ICMS 2006}},

language = {english},

abstract = {We describe the non--commutative extension of the computer algebra systemSINGULAR, called PLURAL. In the system, we provide rich functionality for symbolic computation within a wide class of non--commutative algebras. We discuss the computational objects of PLURAL, the implementation of main algorithms, various aspects of software engineering and numerous applications.},

series = {LNCS},

number = {4151},

pages = {144--157},

isbn_issn = {0302-9743},

year = {2006},

month = {September},

editor = {A. Iglesias and N. Takayama},

refereed = {yes},

length = {14},

conferencename = {International Congress on Mathematical Software}

}

**inproceedings**{RISC2961,author = {V. Levandovskyy},

title = {{PLURAL, a Non--commutative Extension of SINGULAR: Past, Present and Future}},

booktitle = {{Mathematical Software - ICMS 2006}},

language = {english},

abstract = {We describe the non--commutative extension of the computer algebra systemSINGULAR, called PLURAL. In the system, we provide rich functionality for symbolic computation within a wide class of non--commutative algebras. We discuss the computational objects of PLURAL, the implementation of main algorithms, various aspects of software engineering and numerous applications.},

series = {LNCS},

number = {4151},

pages = {144--157},

isbn_issn = {0302-9743},

year = {2006},

month = {September},

editor = {A. Iglesias and N. Takayama},

refereed = {yes},

length = {14},

conferencename = {International Congress on Mathematical Software}

}

[Levandovskyy]

### Algebraic systems theory and computer algebraic methods for some classes of linear control systems

#### V. Levandovskyy, E. Zerz

In: Proc. of the International Symposium on Mathematical Theory of Networks and Systems (MTNS'06), Y. Yamamoto (ed.), Proceedings of International Symposium on Mathematical Theory of Networks and Systems (MTNS'06), pp. -. July 2006. 1234-8765. [pdf]@

author = {V. Levandovskyy and E. Zerz},

title = {{Algebraic systems theory and computer algebraic methods for some classes of linear control systems}},

booktitle = {{Proc. of the International Symposium on Mathematical Theory of Networks and Systems (MTNS'06)}},

language = {english},

pages = {--},

isbn_issn = {1234-8765},

year = {2006},

month = {July},

editor = {Y. Yamamoto},

refereed = {yes},

length = {6},

conferencename = {International Symposium on Mathematical Theory of Networks and Systems (MTNS'06)}

}

**inproceedings**{RISC2962,author = {V. Levandovskyy and E. Zerz},

title = {{Algebraic systems theory and computer algebraic methods for some classes of linear control systems}},

booktitle = {{Proc. of the International Symposium on Mathematical Theory of Networks and Systems (MTNS'06)}},

language = {english},

pages = {--},

isbn_issn = {1234-8765},

year = {2006},

month = {July},

editor = {Y. Yamamoto},

refereed = {yes},

length = {6},

conferencename = {International Symposium on Mathematical Theory of Networks and Systems (MTNS'06)}

}

[Levandovskyy]

### An Interface between Mathematica and Singular

#### Manuel Kauers, Viktor Levandovskyy

SFB F013. Technical report no. 2006-29, 2006. [pdf] [ps]@

author = {Manuel Kauers and Viktor Levandovskyy},

title = {{An Interface between Mathematica and Singular}},

language = {english},

number = {2006-29},

year = {2006},

institution = {SFB F013},

length = {13}

}

**techreport**{RISC2971,author = {Manuel Kauers and Viktor Levandovskyy},

title = {{An Interface between Mathematica and Singular}},

language = {english},

number = {2006-29},

year = {2006},

institution = {SFB F013},

length = {13}

}

### 2005

[Levandovskyy]

### Computer algebraic methods for the structural analysis of linear control systems

#### V. Levandovskyy, E. Zerz

Proceedings in Applied Mathematics and Mechanics (PAMM) 5(1), pp. 717-718. 2005. WILEY-VCH, ISSN 1617-7061. DOI: 10.1002/pamm.200510333. [pdf]@

author = {V. Levandovskyy and E. Zerz},

title = {{Computer algebraic methods for the structural analysis of linear control systems}},

language = {english},

abstract = {We give an overview of the mathematical background of the {\sc Singular} control library. },

journal = {Proceedings in Applied Mathematics and Mechanics (PAMM)},

volume = {5},

number = {1},

pages = {717--718},

publisher = {WILEY-VCH},

isbn_issn = {ISSN 1617-7061},

year = {2005},

note = {DOI: 10.1002/pamm.200510333},

refereed = {yes},

keywords = {control theory, computer algebra, algebraic analysis},

length = {2}

}

**article**{RISC2898,author = {V. Levandovskyy and E. Zerz},

title = {{Computer algebraic methods for the structural analysis of linear control systems}},

language = {english},

abstract = {We give an overview of the mathematical background of the {\sc Singular} control library. },

journal = {Proceedings in Applied Mathematics and Mechanics (PAMM)},

volume = {5},

number = {1},

pages = {717--718},

publisher = {WILEY-VCH},

isbn_issn = {ISSN 1617-7061},

year = {2005},

note = {DOI: 10.1002/pamm.200510333},

refereed = {yes},

keywords = {control theory, computer algebra, algebraic analysis},

length = {2}

}