The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Shuhei Tsujie - One of the best experts on this subject based on the ideXlab platform.
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Canonical Systems of basic invariants for unitary reflection groups
Canadian Mathematical Bulletin, 2016Co-Authors: Norihiro Nakashima, Hiroaki Terao, Shuhei TsujieAbstract:It has been known that there exists a Canonical System for every finite real reflection group. The first and the third authors obtained an explicit formula for a Canonical System in the previous paper. In this article, we first define Canonical Systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a Canonical System for every unitary reflection group.
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A Canonical System of basic invariants of a finite reflection group
Journal of Algebra, 2014Co-Authors: Norihiro Nakashima, Shuhei TsujieAbstract:Abstract A Canonical System of basic invariants is a System of invariants satisfying a set of differential equations. The properties of a Canonical System are related to the mean value property for polytopes. In this article, we naturally identify the vector space spanned by a Canonical System of basic invariants with an invariant space determined by a fundamental antiinvariant. From this identification, we obtain explicit formulas of Canonical Systems of basic invariants. The construction of the formulas does not depend on the classification of finite irreducible reflection groups.
Valery A Gaiko - One of the best experts on this subject based on the ideXlab platform.
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bifurcation analysis of a multi parameter lienard polynomial System
IFAC-PapersOnLine, 2018Co-Authors: Valery A Gaiko, C Vuik, Huibert ReijmAbstract:Abstract In this paper, we study a multi-parameter Lienard polynomial System carrying out its global bifurcation analysis. To control the global bifurcations of limit cycle in this Systems, it is necessary to know the properties and combine the effects of all its field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the application of a Canonical System with field rotation parameters. Using this method, we present a solution of Hilbert’s Sixteenth Problem on the maximum number of limit cycles and their distribution for the Lienard polynomial System. We also conduct some numerical experiments to illustrate the obtained results.
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The geometry of limit cycle bifurcations in polynomial dynamical Systems
Conference Publications 2011, 2011Co-Authors: Valery A GaikoAbstract:In this paper, applying a Canonical System with eld rotation parameters and using geometric properties of the spirals lling the interior and exterior domains of limit cycles, we solve the problem on the maximum number of limit cycles for the classical Lienard polynomial System which is related to the solution of Smale's thirteenth problem. By means of the same geometric approach, we generalize the obtained results and solve the problem on the maximum number of limit cycles surrounding a unique singular point for an arbitrary polynomial System which is related to the solution of Hilbert's sixteenth problem on the maximum number and relative position of limit cycles for planar polynomial dynamical Systems.
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limit cycles of a quadratic System with two parallel straight line isoclines
arXiv: Dynamical Systems, 2008Co-Authors: Valery A GaikoAbstract:In this paper, a quadratic System with two parallel straight line-isoclines is considered. This System corresponds to the System of class II in the classification of Ye Yanqian. Using the field rotation parameters of the constructed Canonical System and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic System with two parallel straight line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner-Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed earlier by the author.
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limit cycle bifurcations in a quadratic System with two parallel straight line isoclines
Reports of the Department of Applied Mathematical Analysis, 2008Co-Authors: Valery A GaikoAbstract:In this paper, a quadratic System with two parallel straight line-isoclines is considered. This System corresponds to the System of class II in the classification of Ye Yanqian [13]. Using the field rotation parameters of the constructed Canonical System and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic System with two parallel straight line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a dierent way: applying the Wintner‐Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in [7].
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Limit cycles of quadratic Systems
Nonlinear Analysis: Theory Methods & Applications, 2008Co-Authors: Valery A GaikoAbstract:Abstract In this paper, the global qualitative analysis of planar quadratic dynamical Systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case of polynomial Systems is suggested. Using geometric properties of four field rotation parameters of a new Canonical System which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic System is equal to four and their only possible distribution is (3:1) [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003]. Besides, applying the Wintner–Perko termination principle for multiple limit cycles to our Canonical System, we prove in a different way that a quadratic System has at most three limit cycles around a singular point (focus) and give another proof of the same conjecture.
Norihiro Nakashima - One of the best experts on this subject based on the ideXlab platform.
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Canonical Systems of basic invariants for unitary reflection groups
Canadian Mathematical Bulletin, 2016Co-Authors: Norihiro Nakashima, Hiroaki Terao, Shuhei TsujieAbstract:It has been known that there exists a Canonical System for every finite real reflection group. The first and the third authors obtained an explicit formula for a Canonical System in the previous paper. In this article, we first define Canonical Systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a Canonical System for every unitary reflection group.
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A Canonical System of basic invariants of a finite reflection group
Journal of Algebra, 2014Co-Authors: Norihiro Nakashima, Shuhei TsujieAbstract:Abstract A Canonical System of basic invariants is a System of invariants satisfying a set of differential equations. The properties of a Canonical System are related to the mean value property for polytopes. In this article, we naturally identify the vector space spanned by a Canonical System of basic invariants with an invariant space determined by a fundamental antiinvariant. From this identification, we obtain explicit formulas of Canonical Systems of basic invariants. The construction of the formulas does not depend on the classification of finite irreducible reflection groups.
Alexander Sakhnovich - One of the best experts on this subject based on the ideXlab platform.
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On explicit inversion of a subclass of operators with D -difference kernels and Weyl theory of the corresponding Canonical Systems
Positivity, 2009Co-Authors: Alexander Sakhnovich, Alexander A. Karelin, Juan Carlos Seck-tuoh-mora, G. Perez-lechuga, M. Gonzalez-hernandezAbstract:Explicit inversion formulas for a subclass of integral operators with D-difference kernels on a finite interval are obtained. A case of the positive operators is treated in greater detail. An application to the inverse problem to recover Canonical System from a Weyl function is given.
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discrete Canonical System and non abelian toda lattice backlund darboux transformation weyl functions and explicit solutions
Mathematische Nachrichten, 2007Co-Authors: Alexander SakhnovichAbstract:A version of the iterated Backlund–Darboux transformation, where Darboux matrix takes a form of the transfer matrix function from the System theory, is constructed for the discrete Canonical System and non-Abelian Toda lattice. Results on the transformations of the Weyl functions, insertion of the eigenvalues, and construction of the bound states are obtained. A wide class of the explicit solutions is given. An application to the semi-infinite block Jacobi matrices is treated. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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Bäcklund-Darboux Transformation for Non-Isospectral Canonical System and Riemann-Hilbert Problem
Symmetry Integrability and Geometry: Methods and Applications, 2007Co-Authors: Alexander SakhnovichAbstract:A GBDT version of the Backlund-Darboux transformation is constructed for a non-isospectral Canonical System, which plays essential role in the theory of random matrix models. The corresponding Riemann-Hilbert problem is treated and some explicit formulas are obtained. A related inverse problem is formulated and solved.
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scattering problems for a Canonical System with a pseudo exponential potential
Asymptotic Analysis, 2002Co-Authors: I. Gohberg, Alexander SakhnovichAbstract:Scattering problems are solved for a Canonical differential System with a pseudo-exponential potential v. The latter means that v is defined in terms of a triple consisting of an n × n matrix β and two n × m matrices γ1 and γ2 satisfying β ∗ − β = iγ2γ ∗ 2 , via the formula v(x) = −2iγ ∗ 1 e ixα ∗ Σ(x) −1 e ixα γ, γ = −(γ1 + iγ2), where α = β − γ1γ ∗ 2 and the matrix function Σ is given by Σ(x) = In + � x 0 Λ(t)Λ(t) ∗ dt, Λ(x) = (e −ixα γ1 e ixα γ). Such a potential may not be summable. Explicit formulas are presented for the scattering function and the reflection coeffi- cient of the System, and for other functions defined in terms of the asymptotic properties of the fundamental solution of the corresponding differential System. The corresponding inverse problems are also solved explicitly. The state space method from algebraic System theory is used as a basic tool. The results extend those in (11, 2, 5, 4).
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Canonical Systems on the Line with Rational Spectral Densities: Explicit Formulas
Differential Operators and Related Topics, 2000Co-Authors: Israel Gohberg, Marinus A. Kaashoek, Alexander SakhnovichAbstract:Explicit formulas for the direct and inverse spectral problems for a Canonical System on the full line with rational spectral density are obtained via a reduction to the half line case.
Keshav Raj Acharya - One of the best experts on this subject based on the ideXlab platform.
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Remling’s theorem on Canonical Systems
Journal of Mathematical Physics, 2016Co-Authors: Keshav Raj AcharyaAbstract:In this paper, we extend the Remling’s theorem on Canonical Systems that the ω limit points of the Hamiltonian under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure of a Canonical System.
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An Alternate Proof of the De Branges Theorem on Canonical Systems
ISRN Mathematical Analysis, 2014Co-Authors: Keshav Raj AcharyaAbstract:The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by Canonical Systems, is constant on . This provides an alternative proof of the De Branges theorem that the Canonical Systems with imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a Canonical System.
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The Spectral Theory of a Canonical System
arXiv: Spectral Theory, 2012Co-Authors: Keshav Raj AcharyaAbstract:The aim of this paper is to show that, in the limit circle case, the deficiency index of a Symmetric relation induced by a Canonical System $Ju'(x)=z H(x)u(x)$ is constant on $\C$. This provides a simple proof of the limit point case for the Canonical System when trace$H \equiv 1$. To this end, we first discuss the deficiency index and spectral theory of any symmetric relation in a Hilbert space $ \mathcal H $. Then we analyze the spectrum of the relation induced by the Canonical System.
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An Alternate Proof of De Branges Theorem on Canonical Systems
arXiv: Spectral Theory, 2012Co-Authors: Keshav Raj AcharyaAbstract:The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by Canonical Systems, is constant on C. This provides an alternative proof of the De Branges theorem that the Canonical Systems with tr H(x)=1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a Canonical System.