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Patera, Jiri

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Honorary professor

Faculty of Arts and Science - Department of Mathematics and Statistics

André-Aisenstadt

Courriels

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Affiliations

  • Membre Centre de recherches mathématiques
  • Membre CRM — Centre de recherches mathématiques

Research area


Student supervision Expand all Collapse all

Construction of graphene, nanotubes and polytopes using finite reflection groups Theses and supervised dissertations / 2019-10
Grabowiecka, Zofia
Abstract
The goal of this thesis is to study structures obtained from finite reflection groups. The work is contained in four published papers, one submitted article and a research paper currently in preparation, with partial results presented as a chapter of this thesis. In the first article, we present a reduction of the orbits of finite Coxeter groups to their subgroups. We use projection matrices, that is, mappings that transform the simple roots of a reflection group to the simple roots of the appropriate subgroup. The results presented in this paper focus on non-crystallographic finite reflection groups. Moreover, we use polytopes generated by the non-crystallographic group H3 to illustrate the obtained branching rules, i.e., reductions of orbits of the finite Coxeter groups. In the second article, we study polytopes with 60 vertices, generated by the non-crystallographic group H3. We use a method of decoration of the Coxeter–Dynkin diagram to describe their structure in detail, and decompose their vertices into sums of orbits of lower-dimensional symmetries. The third article compares two notations used to describe polyhedra generated by reflection groups, namely the Schläfli symbol, and the dominant point notation. Here, we present the advantages of each method, we explain the two approaches, and we illustrate them through examples. In the fourth article, we focus on graphene, i.e., a hexagonal tiling of the plane that possesses remarkable properties when the vertices are modelled with carbon atoms. In this work, we present different methods to obtain graphene from lattices and three-dimensional orbits of finite reflection groups. Moreover, a technique to colour the hexagons by a finite number of colours is provided, along with a systematic method to refine the graphene. In the fifth article, we use special functions and Fourier transforms to process data sampled on a square lattice of the Lie group SU(2) × SU(2), related to the A1 × A1 symmetry group.

Link to the Papyrus's document

Special functions of Weyl groups and their continuous and discrete orthogonality Theses and supervised dissertations / 2014-04
Motlochova, Lenka
Abstract
This thesis presents several properties and applications of four families of Weyl group orbit functions called $C$-, $S$-, $S^s$- and $S^l$-functions. These functions may be viewed as generalizations of the well-known Chebyshev polynomials. They are related to orthogonal polynomials associated with simple Lie algebras, e.g. the multivariate Jacobi and Macdonald polynomials. They have numerous remarkable properties such as continuous and discrete orthogonality. In particular, it is shown that the $S^s$- and $S^l$-functions characterized by certain parameters are mutually orthogonal with respect to a discrete measure. Their discrete orthogonality allows to deduce two types of Fourier-like discrete transforms for each simple Lie algebra with two different lengths of roots. Similarly to the Chebyshev polynomials, these four families of functions have applications in numerical integration. We obtain in this thesis various cubature formulas, for functions of several variables, arising from $C$-, $S^s$- and $S^l$-functions. We also provide a~complete description of discrete multivariate cosine transforms of types V--VIII involving the Weyl group orbit functions arising from simple Lie algebras $C_n$ and $B_n$, called antisymmetric and symmetric cosine functions. Furthermore, we study four families of multivariate Chebyshev-like orthogonal polynomials introduced via (anti)symmetric cosine functions.

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Brisure de symétrie par la réduction des groupes de Lie simples à leurs sous-groupes de Lie réductifs maximaux Theses and supervised dissertations / 2012-12
Larouche, Michelle
Abstract
In this work, we exploit properties well known for weight systems of representations to define them for individual orbits of the Weyl groups of simple Lie algebras, and we extend some of these properties to orbits of non-crystallographic Coxeter groups. Points of an orbit of a finite Coxeter group G are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found. The orbits of Weyl groups of simple Lie algebras of all types are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of the algebra. Matrices transforming points of the orbits of the algebra into points of subalgebra orbits are listed for all cases n<=8 and for many infinite series of algebra-subalgebra pairs. Numerous examples of branching rules are shown. Finally, we present a new, uniform and comprehensive description of centralizers of the maximal regular subgroups in compact simple Lie groups of all types and ranks. Explicit formulas for the action of such centralizers on irreducible representations of the simple Lie algebras are given and shown to have application to computation of the branching rules with respect to these subalgebras.

Link to the Papyrus's document

Families of orthogonal functions defined by the Weyl groups of compact Lie groups Theses and supervised dissertations / 2012-08
Hakova, Lenka
Abstract
Several families of multivariable special functions, called orbit functions, are defined in the context of Weyl groups of compact simple Lie groups/Lie algebras. These functions have been studied for almost a century now because of their relation to characters of irreducible representations of Lie algebras, their symmetries and orthogonalities. Our main interest is the description of discrete orthogonality relations and their corresponding discrete transforms which allow the applications of orbit functions in the processing of multidimensional data. This description is provided for the Weyl group of different lengths of root, in particular groups of rank 2 for so-called $E-$orbit functions and of rank 3 for all the other families of special functions.

Link to the Papyrus's document

Choix d'un associateur 2-D pour le balayage multiple et optimisation de l'estimation des pistes Theses and supervised dissertations / 2009
Moreau, Francis
Abstract
Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Link to the Papyrus's document

Décomposition des produits de fonctions d'orbites symétriques et antisymétriques des groupes de Weyl Theses and supervised dissertations / 2006
Dubois, Valérie
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Link to the Papyrus's document

Modélisation et intégration du contexte dans le cadre de la détection de cibles en imagerie radar Theses and supervised dissertations / 2006
Bonneau, Olivier
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Link to the Papyrus's document

Research projects Expand all Collapse all

Supplément COVID-19 CRSNG_Fourier transforms on multidimensional lattices and their exploitation in physics CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2020 - 2021

Deep learning based approaches for hard and soft data fusion towards better maritime domain awareness MITACS Inc. / 2020 - 2020

Hard and Soft Information Fusion to Aid Situation Understanding OODA Technologies inc. / 2018 - 2019

Fourier transforms on multidimensional lattices and their exploitation in physics CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2016 - 2023

CENTRE DE RECHERCHES MATHEMATIQUES (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2015 - 2023

(BOURSE ZOFIA GRABOWIECKA) EXPLOITATION OF SPECIAL FUNCTIONS ORTHOGONAL ON FINITE DOMAINS OF 2D AND 3D LATTICES-FOURIER EXPANSIONS OF FUNCTIONS SAMPLED ON LATTICES FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2014 - 2014

(BOURSE LENKA MOTLOCHOVA) EXPLOITATION OF SPECIAL FUNCTIONS ORTHOGONAL ON FINITE DOMAINS OF 2D AND 3D LATTICES-FOURIER EXPANSIONS OF FUNCTIONS SAMPLED ON LATTICES FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2013 - 2015

(BOURSE MARZENA SZAJEWSKA) EXPLOITATION OF SPECIAL FUNCTIONS ORTHOGONAL ON FINITE DOMAINS OF 2D AND 3D LATTICES-FOURIER EXPANSIONS OF FUNCTIONS SAMPLED ON LATTICES FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2013 - 2015

COMPUTATIONAL RESOURCES FOR RESEARCH IN MATHEMATICS AND STATISTICS CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 2013 - 2015

EXPLOITATION OF SPECIAL FUNCTIONS ORTHOGONAL ON FINITE DOMAINS OF 2D AND 3D LATTICES-FOURIER EXPANSIONS OF FUNCTIONS SAMPLED ON LATTICES Innovation, Sciences et Développement économique Canada / 2013 - 2015

(BOURSE AGNIESZKA MARIA TERESZKIEWICK QUI REMPLACE OLIVIER ROUCH POUR LE DEUXIEME STAGE) DEVELOPMENT OF ALGORITHMS AND METHODS FOR FUSION OF IMPRECISE INFORMATION FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2012 - 2013

DEVELOPMENT OF LAGORITHMS AND METHODS FOR FUSION OF IMPRECISE INFORMATION FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2012 - 2013

SYMMETRIES IN PHYSICS APPLICATIONS / 2011 - 2015

DEVELOPMENT OF ALGORITHMS AND METHODS FOR FUSION OF IMPRECISE INFORMATION / 2011 - 2013

RESEAU MITACS // DEVELOPMENT OF ALGORITHMS AND METHODS FOR FUSION OF IMPRECISE INFORMATION Secrétariat Inter-Conseil et Réseaux des centres d'excellence (RCE) / 2011 - 2013

NEW ORTHOGONAL POLYNOMIALS AND THEIR APPLICATIONS / 2011 - 2012

DEVELOPMENT OF ALGORITHMS AND METHODS FOR INFORMATION FUSION ENABLED DECISION SUPPORT / 2011 - 2011

CENTRE DE RECHERCHES MATHEMATIQUES (CRM) FRQNT/Fonds de recherche du Québec - Nature et technologies (FQRNT) / 2008 - 2016

SYMMETRIES IN PHYSICS APPLICATIONS CRSNG/Conseil de recherches en sciences naturelles et génie du Canada (CRSNG) / 1994 - 2017

Selected publications Expand all Collapse all

Gaussian cubature arising from hybrid characters of simple Lie groups

R.V. Moody, L. Motlochova, J. Patera, Gaussian cubature arising from hybrid characters of simple Lie groups , pp 20 (2014), accepted in, J. Fourier Analysis and its Applications

Polytope contractions within icosahedral symmetry

M. Bodner, J. Patera, M. Szajewska, Polytope contractions within icosahedral symmetry , pp 10 (2014), accpeted in, Canadian Journal of Physics

Breaking of icosahedral Symmetry: C60 to C70

M. Bodner, J. Patera, M. Szajewska, Breaking of icosahedral Symmetry: C60 to C70 9(3), (2014), , PLoS ONE

Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2

Lemire F., Patera J. et Szajewska M., Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2 10.1007/s10773-014-2444-7, (2014), , International Journal of Theoretical Physics

Icosahedral symmetry breaking: C60 to C78, C96 and to related nanotubes

Bodner M., Bourret E., Patera J. et Szajewska M., Icosahedral symmetry breaking: C60 to C78, C96 and to related nanotubes 6, 650-655 (2014), , Acta Crystallographica(A70)

Icosahedral symmetry breaking: C60 to C78, C96 and to related nanoutubes, Acta Crystallographica (A70)

M. Bodner, E. Bourret, J. Patera, M. Szajewska, Icosahedral symmetry breaking: C60 to C78, C96 and to related nanoutubes, Acta Crystallographica (A70) , (2014), , Foundations of Crystallography

Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2

F. Lemire, J. Patera, M. Szajewska, Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2 , (2014), 10.1007/s10773-014-2444-7, International Journal of theoretical Physics

C70, C80, C90, and carbon nanotubes by breaking of the icosahedral symmetry of C60

M. Bodner, J. Patera, M. Szajewska, C70, C80, C90, and carbon nanotubes by breaking of the icosahedral symmetry of C60 69(6), (2013), , Acta Crystallographica Section A: Foundation of Crystallography

Four families of Weyl group orbit functions of B3 and C3

L. Hakova, J. Hrivnak, J. Patera, Four families of Weyl group orbit functions of B3 and C3 54, (2013), , J. Math. Physics

On generalizing the Shmushkevich method

M. Bodner, G. Chadzitaskos, J. Patera, A. Tereszkiewicz, On generalizing the Shmushkevich method 53(5), (2013), , Acta Polytechnica

Four families of Weyl group orbit functions of $B_3$ and $C_3$

Hàkovà, Lenka, Hrivnàk, Ji\v rí et Patera, Ji\v rí, Four families of Weyl group orbit functions of $B_3$ and $C_3$ 54, 083501, 19 (2013), , J. Math. Phys.

Six types of E-functions of the Lie groups O(5) and G(2)

L. Hakova, J. Hrivnak, J. Patera, Six types of E-functions of the Lie groups O(5) and G(2) 45, (2012), , J. Phys. A: Math Theor.

On Discretization of Tori of Compact Simple Lie Groups II

J. Hrivnak, L. Motlochova, J. Patera, On Discretization of Tori of Compact Simple Lie Groups II 45, (2012), , J. Phys. A: Math. Theor.

Three-variable exponentiol functions of the alternating group

J. Hrivnak, J. Patera, S. Posta, Three-variable exponentiol functions of the alternating group 45, (2012), , J. Phys. A: Math. Theor.

Affine reflection groups for tiling applications: knot thoery and DNA

M. Boder, J. Patera, M. Peterson, Affine reflection groups for tiling applications: knot thoery and DNA 53, (2012), , J. Math. Phys.

Centralizers of maximal regular subgroups of compact simple Lie groups

M. Larouche, F.W. Lemire, J. Patera, Centralizers of maximal regular subgroups of compact simple Lie groups 44, (2011), , J. Phys. A: Math. Theor.

On E-functions of semisimple Lie algebras

J. Hrivnak, I. Kashuba, J. Patera, On E-functions of semisimple Lie algebras 44, (2011), , J. Phys. A: Math. Theor.

Branching rules for Weyl group orbits of simple Lie Algebras Bn and Dn

M. Larouche, J. Patera, Branching rules for Weyl group orbits of simple Lie Algebras Bn and Dn 44, (2011), , J. Phys. A: Math. Theor.

Orthogonal polynomials of compact simple Lie groups

Nesterenko, Maryna, Patera, Ji\v rí et Tereszkiewicz, Agnieszka, Orthogonal polynomials of compact simple Lie groups Nesterenko, Maryna and Patera, Ji\v rí and Tereszkiewicz, Agnieszka, Art. ID 969424, 23 (2011), , Int. J. Math. Math. Sci.

Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups

Moody, Robert V. et Patera, Ji\v rí, Cubature formulae for orthogonal polynomials in terms of elements of finite order of compact simple Lie groups 47, 509--535 (2011), , Adv. in Appl. Math.

Orthogonal polynomials of compact simple Lie groups. The branching rules for polynomials

M. Nesterenko, J. Patera, M. Szajewska, A. Tereszkiewicz, Orthogonal polynomials of compact simple Lie groups. The branching rules for polynomials 43, (2010), , J. Phys. A: Math. Theor.

Two-dimensional symmetric and antisymmetric generalizations of sine functions

J. Hrivnak, L. Motlochova, J. Patera, Two-dimensional symmetric and antisymmetric generalizations of sine functions 51, (2010), , J. Math. Phys.

On E-discretization of tori of compact simple Lie groups

J. Hrivnak, J. Patera, On E-discretization of tori of compact simple Lie groups 43, (2010), , J. Phys. A: Math. Thoer.

Two dimensional symmetric and antisymmetric generalizations of exponential and cosine functions

J. Hrivnak, J. Patera, Two dimensional symmetric and antisymmetric generalizations of exponential and cosine functions 51, (2010), , J. Math. Phys.

Tereszkiewicz, Orbit functions of SU(n) and Chebyshev polynomials

M. NESTERENKO, J. PATERA, Tereszkiewicz, Orbit functions of SU(n) and Chebyshev polynomials , (2010), , Group Analysis of Differential Équations and Integrable Systems, Protaras, Cyprus

Branching rules for the Weyl group orbits of the Lie algebra A_n

Larouche, M., Nesterenko, M. et Patera, J., Branching rules for the Weyl group orbits of the Lie algebra A_n 42, 485203, 15 (2009), , J. Phys. A

On discretization of tori of compact simple Lie groups

J. Hrivnak, J. Patera, On discretization of tori of compact simple Lie groups 42, 385208, 26 (2009), , J. Phys. A

Alternating group and multivariate exponential functions

Klimyk A.U. et Patera J., Alternating group and multivariate exponential functions , 233-246 (2009), , Neolithic Scots to John McKay AMS-CRM Proceedings and Lectures Notes Series

Quasicrystal models in cryptography in Geometric Methods in Physics

M. NESTERENKO, J. PATERA, Quasicrystal models in cryptography in Geometric Methods in Physics , 148-159 (2009), , APS Conference Proceedings 1191

Discretization of tori of exceptional simple Lie algebras

J. HRIVNAK, J. PATERA, Discretization of tori of exceptional simple Lie algebras , 110-115 (2009), , Geometric Methods in Physics, APS Conference Proceedings 1191

Invariants of Lie algebras via moving frame approach

V. BOYKO, J. PATERA, R. POPOVYCH, Invariants of Lie algebras via moving frame approach , 36-44 (2009), , Group Analysis of Differential Équations and Integrable Systems, Protaras, Cyprus

On orbit functions, their properties and applications

M. NESTERENKO, J. PATERA, On orbit functions, their properties and applications , 155-163 (2009), , Group Analysis of Differential Équations and Integrable Systems, Protaras, Cyprus

Image sampling with quasicrystals

Grundland, Mark, Patera, Jiri, Masàkovà, Zuzana et Dodgson, Neil A., Image sampling with quasicrystals 5, Paper 075, 23 (2008), , SIGMA Symmetry Integrability Geom. Methods Appl.

The rings of n-dimensional polytopes

Hàkovà, L., Larouche, M. et Patera, J., The rings of n-dimensional polytopes 41, 495202, 21 (2008), , J. Phys. A

Three-dimensional C-, S- and E-transforms

Nesterenko, Maryna et Patera, Jiri, Three-dimensional C-, S- and E-transforms 41, 475205, 31 (2008), , J. Phys. A

Computing with almost periodic functions

Moody, R. V., Nesterenko, M. et Patera, J., Computing with almost periodic functions 64, 654--669 (2008), , Acta Crystallogr. Sect. A

Alternating multivariate trigonometric functions and corresponding Fourier transforms

Klimyk, A. U. et Patera, J., Alternating multivariate trigonometric functions and corresponding Fourier transforms 41, 145205, 16 (2008), , J. Phys. A

Invariants of solvable Lie algebras with triangular nilradicals and diagonal nilindependent elements

Boyko, Vyacheslav, Patera, Ji\v rí et Popovych, Roman, Invariants of solvable Lie algebras with triangular nilradicals and diagonal nilindependent elements 428, 834--854 (2008), , Linear Algebra Appl.

E-orbit functions

Klimyk, Anatoliy U. et Patera, Ji\v rí, E-orbit functions 4, Paper 002, 57 (2008), , SIGMA Symmetry Integrability Geom. Methods Appl.

Edge detection algorithm based on DCT continuous extension technique

Asatryan D.G. et Patera J., Edge detection algorithm based on DCT continuous extension technique 71, 795-799 (2008), , Physics of Atomic Nuclei

Discretization of compact semisimple Lie groups

J. HRIVNAK, J. PATERA, Discretization of compact semisimple Lie groups , 196-202 (2008), , Geometric Methods in Physics, AIP Conf. Proc. 1079

Quasicrystals in cryptography, in Aspects of Network and information Security

M. NESTERENKO, J. PATERA, Quasicrystals in cryptography, in Aspects of Network and information Security , 274-282 (2008), , NATO Advanced Studies Institute on Network Security and Intrusion Detection, IOS Press, Amsterdam

Invariants of Lie algebras with fixed structure of nilradicals

Boyko, Vyacheslav, Patera, Jiri et Popovych, Roman, Invariants of Lie algebras with fixed structure of nilradicals 40, 113-130 (2007), , J. Phys. A

(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

Klimyk, A. et Patera, J., (Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms 48, 093504, 24 (2007), , J. Math. Phys.

Fine group gradings of the real forms of sl(4,C), sp(4,C), and o(4,C)

Patera, Jiri, Pelantovà, Edita et Svobodovà, Milena, Fine group gradings of the real forms of sl(4,C), sp(4,C), and o(4,C) 48, 093503, 25 (2007), , J. Math. Phys.

Discrete and continuous exponential transforms of simple Lie groups of rank 2

Kashuba, I. et Patera, J., Discrete and continuous exponential transforms of simple Lie groups of rank 2 40, 4751--4774 (2007), , J. Phys. A

Invariants of solvable Lie algebras with one nilindependent diagonal element

Boyko, Vyacheslav, Patera, Ji\v rí et Popovych, Roman, Invariants of solvable Lie algebras with one nilindependent diagonal element 40, (2007), , J. Phys. A

(Anti)symmetric multidimensional exponential functions and the corresponding Fourier transforms

Klimyk, Anatoliy et Patera, Jiri, (Anti)symmetric multidimensional exponential functions and the corresponding Fourier transforms 40, 10473-10489 (2007), , J.Phys.A

Antisymmetric orbit functions

Klimyk A.U. et Patera, J., Antisymmetric orbit functions 3, 83 (2007), , SIGMA (Symmetry,Integrability and Geometry: Methods and Applications)

The discrete SU3 transform and its continuous extension for triangular lattices

Atoyan A. et Patera, J., The discrete SU3 transform and its continuous extension for triangular lattices 57, 745-764 (2007), , J. Geom. Physics

Invariants of triangular Lie algebras

Boyko V., Patera J. et Popovych R., Invariants of triangular Lie algebras 40, (2007), , J. Phys. A: Math. Theor.

Fine gradings of the real forms of sl(4,C), sp(4,C) and o(4,C)

Patera J., Pelantova E. et Svobodova M., Fine gradings of the real forms of sl(4,C), sp(4,C) and o(4,C) 48, (2007), , J. Math. Phys.

Orthogonality within the families of C-, S-, and E-functions of any compact semisimple Lie group

Moody, Robert V. et Patera, Ji\v rí, Orthogonality within the families of C-, S-, and E-functions of any compact semisimple Lie group 2, Paper 076, 14 (2006), , SIGMA Symmetry Integrability Geom. Methods Appl.

Graded contractions of the Pauli graded sl(3,C)

Hrivnàk, J., Novotny, P., Patera, J. et Tolar, J., Graded contractions of the Pauli graded sl(3,C) 418, 498-550 (2006), , Linear Algebra Appl.

Computation of invariants of Lie algebras by means of moving frames

Boyko, Vyacheslav, Patera, Ji\v rí et Popovych, Roman, Computation of invariants of Lie algebras by means of moving frames 39, 5749-5762 (2006), , J. Phys. A

Discrete and continuous sine transform generalized to semisimple Lie groups of rank two

Patera, J. et Zaratsyan, A., Discrete and continuous sine transform generalized to semisimple Lie groups of rank two 47, 043512, 22 (2006), , J. Math. Phys.

Orbit functions

Klimyk, Anatoliy et Patera, Jiri, Orbit functions 2, Paper 006, 60 (2006), , SIGMA Symmetry Integrability Geom. Methods Appl.

Cosine transform generalized to Lie groups SU(2)xSU(2) and O(5): application to textural image processing

M. GERMAIN, J. PATERA, Cosine transform generalized to Lie groups SU(2)xSU(2) and O(5): application to textural image processing , (2006), , Canadian Conference on Electrical and Computer Engineering, IEEE Canada

Multiresolution analysis of digital images using the continuous extension of discrete group transforms

M. GERMAIN, J. PATERA, Multiresolution analysis of digital images using the continuous extension of discrete group transforms , 6-15 (2006), , SPIE 6065, Computational Imaging IV; Charles A. Bouman, Eric L.Miller, Ilya Pollak; Eds

Interpolation using cosine transforms generalized to Lie groups

J. PATERA, A. ZARATSYAN, H. ZHU, Interpolation using cosine transforms generalized to Lie groups , (2006), , SPIE 6064, Image Processing

Discrete and Continuous Cosine Transform Generalized to Lie Groups SU(3) and G(2)

Patera, J. et Zaratsyan, A., Discrete and Continuous Cosine Transform Generalized to Lie Groups SU(3) and G(2) 46, 113506, 17 (2005), , J. Math. Phys.

Compact simple Lie groups and their $C$-, $S$-, and $E$-transforms

Patera, Jiri, Compact simple Lie groups and their $C$-, $S$-, and $E$-transforms 1, Paper 025, 6 (2005), , SIGMA Symmetry Integrability Geom. Methods Appl.

Classification of Voronoi and Delone tiles of quasicrystals. III. Decagonal acceptance window of any size

Masàkovà, Z., Patera, J. et Zich, J., Classification of Voronoi and Delone tiles of quasicrystals. III. Decagonal acceptance window of any size 38, 1947-1960 (2005), , J. Phys. A

Discrete and Continuous Cosine Transform Generalized to Lie Groups SU(2)xSU(2) and O(5)

Patera, J. et Zaratsyan, A., Discrete and Continuous Cosine Transform Generalized to Lie Groups SU(2)xSU(2) and O(5) 46, 113506, 17 (2005), , J. Math. Phys.

Fourier transform method for imaging atmospheric Cherenkov telescopes

Atoyan, A., Patera, J., Sahakian, V., et Akhperjanian, A, Fourier transform method for imaging atmospheric Cherenkov telescopes 23, 79-95 (2005), , Astroparticle Phys.

Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

Atoyan, A. et Patera, J., Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization 45, 2468-2491 (2004), , J. Math. Phys.

Fine gradings of o(4,C),

Patera, Jiri, Pelantovà, Edita et Svobodovà, Milena, Fine gradings of o(4,C), 45, 2188-2198 (2004), , J. Math. Phys.

Pauli graded contractions of sl(3,C)

M. HAVLICEK, J. PATERA, E. PELANTOVA, Pauli graded contractions of sl(3,C) , 37-42 (2004), , Nonlinear Math. Phys., suppl.,11

Symmetry in Nonlinear Mathematical physics

PATERA J, Symmetry in Nonlinear Mathematical physics , 1152-1160 (2004), , Of the Nat. Acad. Sci, 30

Sharp and generating functions in group theory, in Symmetry in physics

PATERA J., R.T., Sharp and generating functions in group theory, in Symmetry in physics 34, 159-163 (2004), , CRM Proccedings and Lecture Notes

Statistics and implementation of aperiodic pseudorandom number generators

Guimond, Louis-Sébastien, Patera, Jan et Patera, Jiri, Statistics and implementation of aperiodic pseudorandom number generators 46, 295-318 (2003), , Appl. Numer. Math.

Classification of distinct Voronoi and Delone tiles in two-dimensional quasicrystals with circular acceptance window of arbitrary size

Masàkovà, Z., Patera, J. et Zich, J., Classification of distinct Voronoi and Delone tiles in two-dimensional quasicrystals with circular acceptance window of arbitrary size 36, 1895-1912 (2003), , J. Phys. A

Classification of distinct Voronoi and Delone tiles in two-dimensional quasicrystals : General Method

Masakovà , Z., Patera, J. et Zich, J., Classification of distinct Voronoi and Delone tiles in two-dimensional quasicrystals : General Method 36, 1869-1894 (2003), , J. Phys. A

Graded contractions of Jordan algebras and of their representations

Kashuba, Iryna et Patera, Ji\v rí, Graded contractions of Jordan algebras and of their representations 36, 12453-12473 (2003), , J. Phys. A

Algebraic solutions of the Neumann boundery value problems on fundamental region of a compact semisimple Lie group

PATERA J, Algebraic solutions of the Neumann boundery value problems on fundamental region of a compact semisimple Lie group , 26-31 (2003), , group theory and numerical methods, Montreal

Continuous extension of the discrete cosine transform and its applications to data processing, Proceedings of the Workshop

ATOYAN A, PATERA J, Continuous extension of the discrete cosine transform and its applications to data processing, Proceedings of the Workshop , 26-31 (2003), , Group theory and numerical methods, Montréal

Application of the continuous extension of discrete cosine transform to images taken by FLIR detectors

ATOYAN A, PATERA J, Application of the continuous extension of discrete cosine transform to images taken by FLIR detectors , (2003), , NATO ASI 2003, Data Fusion for Situation Monitoring, Incident Detection, Alert and Response Management

Application of multidimensional discrete transforms on Lie groups for image processing

AKHPERJANIAN A., ATOYAN A., PATERA J., SAHAKIAN V., Application of multidimensional discrete transforms on Lie groups for image processing , (2003), , NATO ASI 2003

Innate brain language and grammar: Implications for human language and music, in Stochastic point processes

SHAW G., BODNER M., PATERA J., Innate brain language and grammar: Implications for human language and music, in Stochastic point processes , 287-305 (2003), , Srinivasan and A. Vihayakumar, Narosa Publishing, New Delhi

The eight fine gradings of sl(4,C) and o(6,C)

Patera, Jiri, Pelantovà, Edita et Svobodovà, Milena, The eight fine gradings of sl(4,C) and o(6,C) 43, 6353-6378 (2002), , J. Math. Phys.

Affine extension of noncrystallographic Coxeter groups and quasicrystals

Patera, Jiri et Twarock, Reidun, Affine extension of noncrystallographic Coxeter groups and quasicrystals 35, 1551-1574 (2002), , J. Phys. A

Automorphisms of a finest grading of sl(n,C)

Havlícek, Miloslav, Patera, Jiri, Pelantovà, Edita et Tolar, Jiri, Automorphisms of a finest grading of sl(n,C) 43, 1083-1094 (2002), , J. Math. Phys.

The Music-Math Connection

Bodner M., Derr C., Leng X., Patera J., Petterson M., Ticheli F., Vuong S. et Shaw g., The Music-Math Connection Number 3, 9-16 (2002), , Early Childhood Connections 8

Proving the deterministic period breaking of linear congruential generators using two tile quasicrystals

Guimond, Louis-Sébastien et Patera, Jiri, Proving the deterministic period breaking of linear congruential generators using two tile quasicrystals 71, 319-332 (electronic) (2002), , Math. Comp.

Combining random number generators using cut and project sequences

Guimond, Louis-Sébastien, Patera, Jiri et Patera, Jan, Combining random number generators using cut and project sequences 51, 305--311 (2001), , Czechoslovak J. Phys.

Fine grading of o(5, C), sp(4, C) and of their real forms

Patera, Jiri, Pelantovà, Edita et Svobodovà, Milena, Fine grading of o(5, C), sp(4, C) and of their real forms 42, 3839-3853 (2001), , J. Math. Phys.

On fine gradings and their symmetries

Masakovà A., Patera J., Pelantovà E. et Tolar, J., On fine gradings and their symmetries 51, 383-391 (2001), , Czechoslovak J. Phys.

Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals

Masakovà A., Patera J. et Pelantovà E., Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals 79, 669-687 (2001), , Can. J. Phys. (Gerhard Herzberg memorial issue)

Acceptance windows compatible with a quasicrystal fragment, in From Quasicrystals to More Complex Systems

Masakovà, Zuzana, Patera, Jiri et Pelantovà, Edita, Acceptance windows compatible with a quasicrystal fragment, in From Quasicrystals to More Complex Systems 13, 167-193 (2000), , ed. F. Axel, F. Denoyer, J.P. Gazeau, Springer, Centre de Physique des Houches

On Lie gradings. III. Gradings of the real forms of classical Lie algebras

Havlícek, Miloslav, Patera, Jiri et Pelantovà, Edita, On Lie gradings. III. Gradings of the real forms of classical Lie algebras 314, 1-47 (2000), , Linear Algebra Appl.

sl(3, C) generator matrix elements in a Pauli subgroup basis

de Guise, H., Patera, J. et Sharp, R. T., sl(3, C) generator matrix elements in a Pauli subgroup basis 41, 4860-4880 (2000), , J. Math. Phys.

The fine gradings of sl(3,C) and their symmetries

HAVLICEK M., PATERA J. & PELANTOVA E. ET TOLAR J., The fine gradings of sl(3,C) and their symmetries 1, 57-61 (2000), , Group Theor. Methods in Physics, Eds. A.N. Sissakian, G.S. Pogosyan and L.G. Mardoyan

Lattice-like properties of quasicrystal models with quadratic irrationalities, in Quantum Theory and Symmetries

MASSAKOVA Z., PATERA J. & PELANTOVA E., Lattice-like properties of quasicrystal models with quadratic irrationalities, in Quantum Theory and Symmetries , 499-509 (2000), , H.D. Doebner, V.K.Dobrev, J.-D.Hennig, W. Lucke, Word Scientific

Quasicrystal Lie algebras and their generalizations

Patera, Jiri et Twarock, Reidun, Quasicrystal Lie algebras and their generalizations 315, 241-256 (1999), , Phys. Rep.

Graded contractions of the Lie algebra e(2,1)

Patera, J., Pogosyan, G. et Winternitz, P., Graded contractions of the Lie algebra e(2,1) 32, 805-826 (1999), , J. Phys. A

Minimal distances in quasicrystals

Masàkovà, Z., Patera, J. et Pelantovà, E., Minimal distances in quasicrystals 31, 1539-1552 (1998), , J. Phys. A

Inflation centers in cut and project quasicrystals

Masàkovà, Zuzana, Patera, Jiri et Pelantovà, Edita, Inflation centers in cut and project quasicrystals 31, 1443-1453 (1998), , J. Phys. A

Non-crystallographic root systems, in Quasicrystals and Discrete Geometry

Chen L., Moody R.V. et Patera J., Non-crystallographic root systems, in Quasicrystals and Discrete Geometry 10, 135-178 (1998), , Fields Institute Monograph Series

Densities, minimal distances, and coverings of quasicrystals

Moody, R. V. et Patera, J., Densities, minimal distances, and coverings of quasicrystals 195, 613-626 (1998), , Comm. Math. Phys.

Quasicrystal Lie algebras

Patera, Jiri, Pelantovà, Edita et Twarock, Reidun, Quasicrystal Lie algebras 246, 209-213 (1998), , Phys. Lett. A

Tau wavelets in the plane

Gazeau, Jean Pierre, Patera, Jiri et Pelantovà, Edita, Tau wavelets in the plane 39, 4201-4212 (1998), , J. Math. Phys.

Selfsimilar Delone sets and cut and project quasicrystals

Masàkovà , Zuzana, Patera, Jiri et Pelantovà , Edita, Selfsimilar Delone sets and cut and project quasicrystals 31, 4927-4946 (1998), , J. Phys. A

On Liegradings II

Havlicek, Miloslav, Patera, Jiri et Pelantova, Edita, On Liegradings II 277, 97-125 (1998), , Linear Algebra Appl.

On gradings of Lie algebras and représentations, in Lie Theory and its Applications II

PATERA J. & TOLAR J., On gradings of Lie algebras and représentations, in Lie Theory and its Applications II , 109-118 (1998), , H.-D. Doebner, V.K. Dobrev, and J. Hilgert, World Scientific, Singapore

The signatures of finite dimensional representations of the de Sitter groups SO (4,1) and SO (3,2)

Grimm, S., de Montigny, M. et Patera, J., The signatures of finite dimensional representations of the de Sitter groups SO (4,1) and SO (3,2) 30, 7463-7471 (1997), , J. Phys. A

On the fine gradings of simple classical Lie algebras

Havlicek, Miloslav, Patera, Jiri et Pelantovà, Edita, On the fine gradings of simple classical Lie algebras 12, 189-194 (1997), , Internat. J. Modern Phys. A

Decomposition of tensor products of the fundamental representations of E8, in Advances in Mathematical Sciences

Grimm S. et Patera J., Decomposition of tensor products of the fundamental representations of E8, in Advances in Mathematical Sciences 11, 329-355 (1997), , CRM's 25 Years

Simple physical generation of quasicrystals

Janot C. et Patera J., Simple physical generation of quasicrystals 233, 110-114 (1997), , Phys. Lett. A

Non-crystallographic root systems and quasicrystals, Proc. NATO ASI Aperiodic Long Ranger Order

PATERA, J., Non-crystallographic root systems and quasicrystals, Proc. NATO ASI Aperiodic Long Ranger Order , 443-465 (1997), , Waterloo, Ontario, Canada

Tau-wavelets of Haar

Gazeau, J.-P. et Patera, J., Tau-wavelets of Haar 29, 4549-4559 (1996), , J. Phys. A

Dynamical generation of quasicrystals

Moody, R. V. et Patera, J., Dynamical generation of quasicrystals 36, 291-300 (1996), , Lett. Math. Phys.

Generating functions for the Coxeter group H4

Lam, C. S., Patera, J. et Sharp, R. T., Generating functions for the Coxeter group H4 29, 7705-7719 (1996), , J. Phys. A

Grading refinements in the contractions of Lie algebras and their invariants

Ait Abdelmalek, M., Leng, X., Patera, J. et Winternitz, P., Grading refinements in the contractions of Lie algebras and their invariants 29, 7519-7543 (1996), , J. Phys. A

Graded contractions of representations of orthogonal and symplectic Lie algebras with respect to their maximal parabolic subalgebras

Leng, Xiaodan et Patera, J., Graded contractions of representations of orthogonal and symplectic Lie algebras with respect to their maximal parabolic subalgebras 28, 3785-3807 (1995), , J. Phys. A

Voronoi domains and dual cells in the generalized kaleidoscope with applications to root and weight lattices

Moody, R. V. et Patera, J., Voronoi domains and dual cells in the generalized kaleidoscope with applications to root and weight lattices 47, 573-605 (1995), , Canad. J. Math.

Description of reflection generated polytopes using decorated Coxeter diagrams

Champagne B., Kjiri M., Patera J. et Sharp R.T., Description of reflection generated polytopes using decorated Coxeter diagrams 73, 566-584 (1995), , Can. J. Phys.

The pentacrystals

PATERA J., The pentacrystals , 17-31 (1995), , Les Editions de Physique, eds. F. Axel and D. Gratias, Springer

Colourings of quasicrystals

Moody, R. V. et Patera, J., Colourings of quasicrystals 72, 442-452 (1994), , Canad. J. Phys.

Graded contractions of the affine Lie algebra $A^{(1)}_1$, its representations and tensor products, and an application to the branching rule $A^{(1)}_1\supset A^{(1)}_1$

Hussin, A., King, R. C., Leng, X. et Patera, J., Graded contractions of the affine Lie algebra $A^{(1)}_1$, its representations and tensor products, and an application to the branching rule $A^{(1)}_1\supset A^{(1)}_1$ 27, 4125-4152 (1994), , J. Phys. A

Transitively differential groups of degree three

Kantor, I. L. et Patera, J., Transitively differential groups of degree three 35, 443-458 (1994), , J. Math. Phys.

Graded contractions and kinematical groups of space-time

de Montigny, M., Patera, J. et Tolar, J., Graded contractions and kinematical groups of space-time 35, 405-425 (1994), , J. Math. Phys.

Graded contractions of representations of special linear Lie algebras with respect to their maximal parabolic subalgebras

Leng, Xiaodan et Patera, J., Graded contractions of representations of special linear Lie algebras with respect to their maximal parabolic subalgebras 27, 1233-1250 (1994), , J. Phys. A

Graded contractions of Casimir operators

Bincer, A. M. et Patera, J., Graded contractions of Casimir operators 26, 5621-5628 (1993), , J. Phys. A

Quasicrystals and icosians

Moody, R. V. et Patera, J., Quasicrystals and icosians 26, 2829-2853 (1993), , J. Phys. A

Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams

Moody, R. V. et Patera, J., Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams 25, 5089-5134 (1992), , J. Phys. A

Orbit-orbit branching rules between simple low-rank algebras and equal-rank subalgebras

Gingras, F., Patera, J. et Sharp, R. T., Orbit-orbit branching rules between simple low-rank algebras and equal-rank subalgebras 33, 1618-1626 (1992), , J. Math. Phys.

Discrete and continuous graded contractions of representations of Lie algebras

Moody, R. V. et Patera, J., Discrete and continuous graded contractions of representations of Lie algebras 24, 2227-2257 (1991), , J. Phys. A

Discrete and continuous graded contractions of Lie algebras and superalgebras

de Montigny, M. et Patera, J., Discrete and continuous graded contractions of Lie algebras and superalgebras 24, 525-547 (1991), , J. Phys. A

The higher rank Virasoro algebras

Patera, J. et Zassenhaus, H., The higher rank Virasoro algebras 136, 1-14 (1991), , Comm. Math. Phys.

Graded contractions of sl(3,C)

Couture, M., Patera, J., Sharp, R. T. et Winternitz, P., Graded contractions of sl(3,C) 32, 2310-2318 (1991), , J. Math. Phys.

Affine Kac-Moody Algebras, Weight Multiplicities and Branching Rules

KASS S., MOODY R.V,PATERA J. ET SLANSKY, Affine Kac-Moody Algebras, Weight Multiplicities and Branching Rules 1 & 2, 897 (1991), , University of California Press, Berkeley

Solvable Lie algebras of dimension 4 over perfect fields

Patera, J. et Zassenhaus, H., Solvable Lie algebras of dimension 4 over perfect fields 142, 1-17 (1990), , Linear Algebra Appl.

On intersections of A1 subgroups in the exceptional simple Lie groups E6, E7, E8

Patera, J., Rodri­guez, M. et Zaoui, M., On intersections of A1 subgroups in the exceptional simple Lie groups E6, E7, E8 23, 5695-5705 (1990), , J. Phys. A

Subjoinings of affine Kac-Moody algebras

Leng, X., Patera, J. et Sharp, R. T., Subjoinings of affine Kac-Moody algebras 23, 3397-3407 (1990), , J. Phys. A

The construction of solvable Lie algebras from equidimensional nilpotent algebras

Patera, J. et Zassenhaus, H., The construction of solvable Lie algebras from equidimensional nilpotent algebras 133, 89-120 (1990), , Linear Algebra Appl.

The 785 conjugacy classes of rational elements of finite order in E8

McKay W.G., Moody R.V., Patera J. et Pianzola A., The 785 conjugacy classes of rational elements of finite order in E8 110, 79-123 (1990), , Contemporary Math.

Simplie Users Manual; Macintosh software for représentations of simple Lie algebras

McKAY W.G., PATERA J. & RAND D., Simplie Users Manual; Macintosh software for représentations of simple Lie algebras , 49 (1990), , Les publicaions CRM, Montréal

Tables of représentations of simple Lie algebras

McKAY W.G., PATERA J. & RAND D., Tables of représentations of simple Lie algebras Exceptional simple Lie algebras, 318 (1990), , Les publications CRM

Tables od dominant weight multiplicities for représentations of simple Lie algebras

McKAY W.G., PATERA J. & RAND D., Tables od dominant weight multiplicities for représentations of simple Lie algebras , 340 (1985), , Mercel Dekker, New York

Tables of dimensions, incidences, and branching rules for représentations of simple Lie algebras

McKAY, W.G., PATERA, J., Tables of dimensions, incidences, and branching rules for représentations of simple Lie algebras , 317 (1981), , Marcel Dekker, New Yord

Branching rules for représentations of simple Lie algebras

PATERA, J. SANKOFF, D., Branching rules for représentations of simple Lie algebras , 99 (1973), , Presses Université de Montréal

Awards and Recognition

  • Conseil des arts du Canada Bourse Killam, 1991

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