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Nizar Demni - One of the best experts on this subject based on the ideXlab platform.
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Markov Semi-groups Associated with the Complex Unimodular Group $$Sl(2,{\mathbb {C}})$$ S
Journal of Fourier Analysis and Applications, 2019Co-Authors: Nizar DemniAbstract:In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011 ) from the restriction of a particular positive definite function on the complex unimodular group $$SL(2,{\mathbb {C}})$$ S L ( 2 , C ) to two commutative subalgebras of its universal $$C^{\star }$$ C ⋆ -algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $$-\,1$$ - 1 , and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $$Mp(4,{\mathbb {R}})$$ M p ( 4 , R ) and to the Landau Operator in the complex plane.
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Markov Semi-groups Associated with the Complex Unimodular Group $$Sl(2,{\mathbb {C}})$$ S
Journal of Fourier Analysis and Applications, 2019Co-Authors: Nizar DemniAbstract:In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011 ) from the restriction of a particular positive definite function on the complex unimodular group $$SL(2,{\mathbb {C}})$$ S L ( 2 , C ) to two commutative subalgebras of its universal $$C^{\star }$$ C ⋆ -algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $$-\,1$$ - 1 , and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $$Mp(4,{\mathbb {R}})$$ M p ( 4 , R ) and to the Landau Operator in the complex plane.
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Markov Semi-groups Associated with the Complex Unimodular Group \(Sl(2,{\mathbb {C}})\)
Journal of Fourier Analysis and Applications, 2019Co-Authors: Nizar DemniAbstract:In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group \(SL(2,{\mathbb {C}})\) to two commutative subalgebras of its universal \(C^{\star }\)-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index \(-\,1\), and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation \(Mp(4,{\mathbb {R}})\) and to the Landau Operator in the complex plane.
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Markov semi-groups associated with the complex unimodular group $Sl(2,\mathbb{C})$
arXiv: Probability, 2018Co-Authors: Nizar DemniAbstract:In this paper, we derive the explicit expressions of two Markov semi-groups constructed by P. Biane in \cite{Bia1} from the restriction of a particular positive definite function on the complex unimodular group $Sl(2,\mathbb{C})$ to two commutative subalgebras of its universal $C^{\star}$-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $-1$, and yield absolutely-convergent double series representations of the semi-group densities. In the last part of the paper, we discuss the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $Mp(4,\mathbb{R})$ and to the Landau Operator in the complex plane.
Alexandre S. Shcherbakov - One of the best experts on this subject based on the ideXlab platform.
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Qualitative analysis of ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor heterolasers with an external fiber cavity
Physics and Simulation of Optoelectronic Devices XIX, 2011Co-Authors: Alexandre S. Shcherbakov, Pedro Moreno Zarate, Joaquín Campos Acosta, Alicia Pons AglioAbstract:An advanced qualitative characterization of simultaneously existing various low-power trains of ultra-short optical pulses with an internal frequency modulation in a distributed laser system based on semiconductor heterostructure is presented. The scheme represents a hybrid cavity consisting of a single-mode heterolaser operating in the active mode-locking regime and an external long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and linear optical losses. In fact, we consider the trains of optical dissipative solitons, which appear within double balance between the second-order dispersion and cubic-law nonlinearity as well as between the active-medium gain and linear optical losses in a hybrid cavity. Moreover, we operate on specially designed modulating signals providing non-conventional composite regimes of simultaneous multi-pulse active mode-locking. As a result, the mode-locking process allows shaping regular trains of picosecond optical pulses excited by multi-pulse independent on each other sequences of periodic modulations. In so doing, we consider the arranged hybrid cavity as a combination of a quasi-linear part responsible for the active mode-locking by itself and a nonlinear part determining the regime of dissipative soliton propagation. Initially, these parts are analyzed individually, and then the primarily obtained data are coordinated with each other. Within this approach, a contribution of the appeared cubically nonlinear Ginzburg-Landau Operator is analyzed via exploiting an approximate variational procedure involving the technique of trial functions.
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Analysis of originating ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor heterolasers with an external fiber cavity
Photonics North 2010, 2010Co-Authors: Alexandre S. Shcherbakov, Pedro Moreno Zarate, Joaquín Campos Acosta, Alicia Pons Aglio, Svetlana MansurovaAbstract:We present an advanced approach to describing low-power trains of bright picosecond optical dissipative solitary pulses with an internal frequency modulation in practically important case of exploiting semiconductor heterolaser operating in near-infrared range in the active mode-locking regime. In the chosen schematic arrangement, process of the active mode-locking is caused by a hybrid nonlinear cavity consisting of this heterolaser and an external rather long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and small linear optical losses. Our analysis of shaping dissipative solitary pulses includes three principal contributions associated with the modulated gain, total optical losses, as well as with linear and nonlinear phase shifts. In fact, various trains of the non-interacting to one another optical dissipative solitons appear within simultaneous balance between the second-order dispersion and cubic-law Kerr nonlinearity as well as between active medium gain and linear optical losses in a hybrid cavity. Our specific approach makes possible taking the modulating signals providing non-conventional composite regimes of a multi-pulse active mode-locking. Within our model, a contribution of the appearing nonlinear Ginzburg-Landau Operator to the parameters of dissipative solitary pulses is described via exploiting an approximate variational procedure involving the technique of trial functions.
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Dynamics of shaping ultrashort optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external long-haul single-mode fiber cavity
Ultrafast Phenomena in Semiconductors and Nanostructure Materials XIV, 2010Co-Authors: Alexandre S. Shcherbakov, Pedro Moreno ZarateAbstract:We describe the conditions of shaping regular trains of optical dissipative solitary pulses, excited by multi-pulse sequences of periodic modulating signals, in the actively mode-locked semiconductor laser heterostructure with an external long-haul single-mode silicon fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and linear optical losses. The presented model for the analysis includes three principal contributions associated with the modulated gain, optical losses, as well as linear and nonlinear phase shifts. In fact, the trains of optical dissipative solitary pulses appear within simultaneous presenting and a balance of mutually compensating interactions between the second-order dispersion and cubic-law Kerr nonlinearity as well as between active medium gain and linear optical losses in the combined cavity. Within such a model, a contribution of the nonlinear Ginzburg-Landau Operator to shaping the parameters of optical dissipative solitary pulses is described via exploiting an approximate variational procedure involving the technique of trial functions. Finally, the results of the illustrating proof-of-principle experiments are briefly presented and discussed in terms of optical dissipative solitary pulses.
Pedro Moreno Zarate - One of the best experts on this subject based on the ideXlab platform.
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Qualitative analysis of ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor heterolasers with an external fiber cavity
Physics and Simulation of Optoelectronic Devices XIX, 2011Co-Authors: Alexandre S. Shcherbakov, Pedro Moreno Zarate, Joaquín Campos Acosta, Alicia Pons AglioAbstract:An advanced qualitative characterization of simultaneously existing various low-power trains of ultra-short optical pulses with an internal frequency modulation in a distributed laser system based on semiconductor heterostructure is presented. The scheme represents a hybrid cavity consisting of a single-mode heterolaser operating in the active mode-locking regime and an external long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and linear optical losses. In fact, we consider the trains of optical dissipative solitons, which appear within double balance between the second-order dispersion and cubic-law nonlinearity as well as between the active-medium gain and linear optical losses in a hybrid cavity. Moreover, we operate on specially designed modulating signals providing non-conventional composite regimes of simultaneous multi-pulse active mode-locking. As a result, the mode-locking process allows shaping regular trains of picosecond optical pulses excited by multi-pulse independent on each other sequences of periodic modulations. In so doing, we consider the arranged hybrid cavity as a combination of a quasi-linear part responsible for the active mode-locking by itself and a nonlinear part determining the regime of dissipative soliton propagation. Initially, these parts are analyzed individually, and then the primarily obtained data are coordinated with each other. Within this approach, a contribution of the appeared cubically nonlinear Ginzburg-Landau Operator is analyzed via exploiting an approximate variational procedure involving the technique of trial functions.
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Analysis of originating ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor heterolasers with an external fiber cavity
Photonics North 2010, 2010Co-Authors: Alexandre S. Shcherbakov, Pedro Moreno Zarate, Joaquín Campos Acosta, Alicia Pons Aglio, Svetlana MansurovaAbstract:We present an advanced approach to describing low-power trains of bright picosecond optical dissipative solitary pulses with an internal frequency modulation in practically important case of exploiting semiconductor heterolaser operating in near-infrared range in the active mode-locking regime. In the chosen schematic arrangement, process of the active mode-locking is caused by a hybrid nonlinear cavity consisting of this heterolaser and an external rather long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and small linear optical losses. Our analysis of shaping dissipative solitary pulses includes three principal contributions associated with the modulated gain, total optical losses, as well as with linear and nonlinear phase shifts. In fact, various trains of the non-interacting to one another optical dissipative solitons appear within simultaneous balance between the second-order dispersion and cubic-law Kerr nonlinearity as well as between active medium gain and linear optical losses in a hybrid cavity. Our specific approach makes possible taking the modulating signals providing non-conventional composite regimes of a multi-pulse active mode-locking. Within our model, a contribution of the appearing nonlinear Ginzburg-Landau Operator to the parameters of dissipative solitary pulses is described via exploiting an approximate variational procedure involving the technique of trial functions.
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Dynamics of shaping ultrashort optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external long-haul single-mode fiber cavity
Ultrafast Phenomena in Semiconductors and Nanostructure Materials XIV, 2010Co-Authors: Alexandre S. Shcherbakov, Pedro Moreno ZarateAbstract:We describe the conditions of shaping regular trains of optical dissipative solitary pulses, excited by multi-pulse sequences of periodic modulating signals, in the actively mode-locked semiconductor laser heterostructure with an external long-haul single-mode silicon fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and linear optical losses. The presented model for the analysis includes three principal contributions associated with the modulated gain, optical losses, as well as linear and nonlinear phase shifts. In fact, the trains of optical dissipative solitary pulses appear within simultaneous presenting and a balance of mutually compensating interactions between the second-order dispersion and cubic-law Kerr nonlinearity as well as between active medium gain and linear optical losses in the combined cavity. Within such a model, a contribution of the nonlinear Ginzburg-Landau Operator to shaping the parameters of optical dissipative solitary pulses is described via exploiting an approximate variational procedure involving the technique of trial functions. Finally, the results of the illustrating proof-of-principle experiments are briefly presented and discussed in terms of optical dissipative solitary pulses.
Demni Nizar - One of the best experts on this subject based on the ideXlab platform.
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Markov semi-groups associated with the complex unimodular group $Sl(2,\mathbb{C})$
2019Co-Authors: Demni NizarAbstract:In this paper, we derive the explicit expressions of two Markov semi-groups constructed by P. Biane in \cite{Bia1} from the restriction of a particular positive definite function on the complex unimodular group $Sl(2,\mathbb{C})$ to two commutative subalgebras of its universal $C^{\star}$-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $-1$, and yield absolutely-convergent double series representations of the semi-group densities. In the last part of the paper, we discuss the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $Mp(4,\mathbb{R})$ and to the Landau Operator in the complex plane.Comment: The intertwining Operator is derived in the case of principal serie
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Markov semi-groups associated with the complex unimodular group $Sl(2,\mathbb{C})$
Springer Verlag, 2019Co-Authors: Demni NizarAbstract:The intertwining Operator is derived in the case of principal seriesInternational audienceIn this paper, we derive the explicit expressions of two Markov semi-groups constructed by P. Biane in \cite{Bia1} from the restriction of a particular positive definite function on the complex unimodular group $Sl(2,\mathbb{C})$ to two commutative subalgebras of its universal $C^{\star}$-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index $-1$, and yield absolutely-convergent double series representations of the semi-group densities. In the last part of the paper, we discuss the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation $Mp(4,\mathbb{R})$ and to the Landau Operator in the complex plane
Anne Beaulieu - One of the best experts on this subject based on the ideXlab platform.
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the kernel of the linearized ginzburg Landau Operator
2018Co-Authors: Anne BeaulieuAbstract:We consider a linear system of ordinary differential equations from the two dimensional Ginzburg-Landau equation. We prove that this system doesn't admit globally bounded solutions, except those that come from invariance of the Ginzburg-Landau equation under the action of the group of the translations and rotations.
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some remarks on the linearized Operator about the radial solution for the ginzburg Landau equation
Nonlinear Analysis-theory Methods & Applications, 2003Co-Authors: Anne BeaulieuAbstract:Abstract We consider the linearized Operators, denoted L d,1 , of the Ginzburg–Landau Operator Δu+u(1−|u|2) in R2, about the radial solutions ud,1(x)=fd(r)eidθ, for all d⩾1. We state the correspondence between the real vector space of the bounded solutions of the equation L d,1 w=0 and the eigenvalues of the linearized Operators of the equations Δu+1/e2u(1−|u|2)=0, in B(0,1), about the radial solutions ud,e(x)=fd(r/e)eidθ, that tend to 0 as e tends to 0.
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Some remarks on the linearized Operator about the radial solution for the Ginzburg–Landau equation
Nonlinear Analysis: Theory Methods & Applications, 2003Co-Authors: Anne BeaulieuAbstract:Abstract We consider the linearized Operators, denoted L d,1 , of the Ginzburg–Landau Operator Δu+u(1−|u|2) in R2, about the radial solutions ud,1(x)=fd(r)eidθ, for all d⩾1. We state the correspondence between the real vector space of the bounded solutions of the equation L d,1 w=0 and the eigenvalues of the linearized Operators of the equations Δu+1/e2u(1−|u|2)=0, in B(0,1), about the radial solutions ud,e(x)=fd(r/e)eidθ, that tend to 0 as e tends to 0.
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Some remarks on the linearized Operator about the radial solution for the Ginzburg-Landau equation
Nonlinear Analysis: Theory Methods and Applications, 2003Co-Authors: Anne BeaulieuAbstract:We consider the linearized Operators, denoted L-d,L-1, of the Ginzburg-Landau Operator Deltau + u(1 - \u\(2)) in R-2, about the radial solutions U-d,U-1(X) = f(d)(r)e(idtheta), for all d greater than or equal to 1. We state the correspondence between the real vector space of the bounded solutions of the equation L(d,1)w=0 and the eigenvalues of the linearized Operators of the equations Deltau + 1/epsilon(2)u(1 - \u\(2)) = 0, in W B(0, 1), about the radial solutions u(d,epsilon)(x) = f(d)(r/epsilon)e(idtheta), that tend to 0 as epsilon tends to 0. (C) 2003 Published by Elsevier Ltd.
