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Andrea Malchiodi - One of the best experts on this subject based on the ideXlab platform.

  • A variational approach to Liouville Equations
    Bollettino dell'Unione Matematica Italiana, 2016
    Co-Authors: Andrea Malchiodi
    Abstract:

    We consider a class of Liouville Equations motivated by uniformization problems in a singular setting, as well as from models in Mathematical Physics. We will study existence from a variational point of view, using suitable improvements of the Moser-Trudinger inequality. These improvements allow to study the concentration properties of conformal volume, and to reduce the problem to studying explicit finite-dimensional objects.

  • A topological join construction and the Toda system on compact surfaces of arbitrary genus
    Analysis & PDE, 2015
    Co-Authors: Aleks Jevnikar, Sadok Kallel, Andrea Malchiodi
    Abstract:

    We consider a Toda system of Liouville Equations defined on a compact surface which arises as a model for non-abelian Chern-Simons vortices. For the first time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k \in \mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components.

  • an improved geometric inequality via vanishing moments with applications to singular Liouville Equations
    arXiv: Analysis of PDEs, 2012
    Co-Authors: Daniele Bartolucci, Andrea Malchiodi
    Abstract:

    We consider a class of singular Liouville Equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.

  • weighted barycentric sets and singular Liouville Equations on compact surfaces
    Journal of Functional Analysis, 2012
    Co-Authors: Alessandro Carlotto, Andrea Malchiodi
    Abstract:

    Given a closed surface, we prove a general existence result for some elliptic PDE with exponential nonlinearities and negative Dirac deltas, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional might be unbounded from below, we define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its low sublevels. As a result, the analytic problem is reduced to a topological one concerning the contractibility of this model space. To this aim, we prove a new functional inequality in the spirit of Chen and Li (1991) [11] and then employ a min–max scheme based on conical construction, jointly with the blow-up analysis in Bartolucci and Montefusco (2007) [4] (after Bartolucci and Tarantello, 2002; Brezis and Merle, 1991 and ). This study is motivated by abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature with conical singularities (generalizing a problem raised in Kazdan and Warner, 1974 [24]).

  • New Improved Moser–Trudinger Inequalities and Singular Liouville Equations on Compact Surfaces
    Geometric and Functional Analysis, 2011
    Co-Authors: Andrea Malchiodi, David Ruiz
    Abstract:

    We consider a singular Liouville equation on a compact surface, arising from the study of Chern–Simons vortices in a self-dual regime. Using new improved versions of the Moser–Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.

Shaul Mukamel - One of the best experts on this subject based on the ideXlab platform.

  • stochastic Liouville Equations for femtosecond stimulated raman spectroscopy
    arXiv: Quantum Physics, 2015
    Co-Authors: Bijay Kumar Agarwalla, Hideo Ando, Konstantin E Dorfman, Shaul Mukamel
    Abstract:

    Electron and vibrational dynamics of molecules are commonly studied by subjecting them to two interactions with a fast actinic pulse that prepares them in a nonstationary state and after a variable delay period $T$, probing them with a Raman process induced by a combination of a broadband and a narrowband pulse. This technique known as femtosecond stimulated Raman spectroscopy (FSRS) can effectively probe time resolved vibrational resonances. We show how FSRS signals can be modeled and interpreted using the stochastic Liouville Equations (SLE) originally developed for NMR lineshapes. The SLE provides a convenient simulation protocol that can describe complex dynamics due to coupling to collective coordinates at much lower cost that a full dynamical simulation. The origin of the dispersive features which appear when there is no separation of timescales between vibrational variations and dephasing is clarified.

  • stochastic Liouville Equations for femtosecond stimulated raman spectroscopy
    Journal of Chemical Physics, 2015
    Co-Authors: Bijay Kumar Agarwalla, Hideo Ando, Konstantin E Dorfman, Shaul Mukamel
    Abstract:

    Electron and vibrational dynamics of molecules are commonly studied by subjecting them to two interactions with a fast actinic pulse that prepares them in a nonstationary state and after a variable delay period T, probing them with a Raman process induced by a combination of a broadband and a narrowband pulse. This technique, known as femtosecond stimulated Raman spectroscopy (FSRS), can effectively probe time resolved vibrational resonances. We show how FSRS signals can be modeled and interpreted using the stochastic Liouville Equations (SLE), originally developed for NMR lineshapes. The SLE provide a convenient simulation protocol that can describe complex dynamics caused by coupling to collective bath coordinates at much lower cost than a full dynamical simulation. The origin of the dispersive features that appear when there is no separation of timescales between vibrational variations and the dephasing time is clarified.

Vladislav V. Kravchenko - One of the best experts on this subject based on the ideXlab platform.

  • A Neumann series of Bessel functions representation for solutions of Sturm–Liouville Equations
    Calcolo, 2018
    Co-Authors: Vladislav V. Kravchenko, Sergii M. Torba
    Abstract:

    A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville Equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $$\omega $$ the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on $$\omega $$ . A similar result is valid when $$\omega \in {\mathbb {C}}$$ belongs to a strip $$\left| \hbox {Im }\omega \right|

  • a neumann series of bessel functions representation for solutions of sturm Liouville Equations
    Calcolo, 2018
    Co-Authors: Vladislav V. Kravchenko, Sergii M. Torba
    Abstract:

    A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville Equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $$\omega $$ the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on $$\omega $$ . A similar result is valid when $$\omega \in {\mathbb {C}}$$ belongs to a strip $$\left| \hbox {Im }\omega \right| Liouville equation is developed and illustrated on a test problem.

  • a neumann series of bessel functions representation for solutions of sturm Liouville Equations
    arXiv: Classical Analysis and ODEs, 2016
    Co-Authors: Vladislav V. Kravchenko, Sergii M. Torba
    Abstract:

    A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville Equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter $\omega$ the difference between the exact solution and the approximate one (the truncated NSBF) depends on $N$ (the truncation parameter) and the coefficients of the equation and does not depend on $\omega$. A similar result is valid when $\omega\in\mathbb{C}$ belongs to a strip $|Im\omega|Liouville equation is developed and illustrated on a test problem.

  • Analytic approximation of transmutation operators and related systems of functions
    Boletín de la Sociedad Matemática Mexicana, 2016
    Co-Authors: Vladislav V. Kravchenko, Sergii M. Torba
    Abstract:

    In Kravchenko and Torba (J Comput Appl Math 275:1–26, 2015 ) a method for approximate solution of Sturm–Liouville Equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of solutions and eigendata was reduced to approximation of a primitive of the potential by a finite linear combination of certain specially constructed functions obtained from the generalized wave polynomials introduced in Khmelnytskaya et al. (J Math Anal Appl 399:191–212, 2013 ) and Kravchenko and Torba (Complex Anal Oper Theory 9:379–429, 2015 ). The method allows one to compute both lower and higher eigendata with an extreme accuracy. Since the solution of the approximation problem is the main step in the application of the method, the properties of the system of functions involved are of primary interest. In Kravchenko and Torba (J Comput Appl Math 275:1–26, 2015 ) two basic properties were established: the completeness in appropriate functional spaces and the linear independence. In this paper we present a considerably more complete study of the systems of functions. We establish their relation with another linear differential second-order equation, find out certain operations (in a sense, generalized derivatives and antiderivatives) which allow us to generate the next such function from a previous one. We obtain the uniqueness of the coefficients of expansions in terms of such functions and a corresponding generalized Taylor theorem, as well as formulas for exact expansion coefficients involving the operations mentioned above. We also construct the invertible integral operators transforming powers of the independent variable into the functions under consideration and establish their commutation relations with differential operators. We present some error bounds for the solution of the approximation problem depending on the smoothness of the potential and show that these error bounds are close to optimal in order. Also, we provide a rigorous justification of the alternative formulation of the proposed method allowing one to make use of the known initial values of the solutions at the left endpoint of the spectral problem.

  • analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems
    Journal of Computational and Applied Mathematics, 2015
    Co-Authors: Vladislav V. Kravchenko, Sergii M. Torba
    Abstract:

    A method for approximate solution of spectral problems for Sturm-Liouville Equations based on the construction of the Delsarte transmutation operators is presented. In fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials?introduced in Khmelnytskaya et?al. (2013), and Kravchenko and Torba (2014). The method allows one to compute both lower and higher eigendata with an extreme accuracy.

Y. Ricard - One of the best experts on this subject based on the ideXlab platform.

  • Time-dependent density anomalies in a stratified, viscoelastic mantle: Implications for the geoid, Earth's rotation and sea-level fluctuations
    Surveys in Geophysics, 1993
    Co-Authors: R. Sabadini, G. Spada, Y. Ricard
    Abstract:

    The effects on the ℓ = 2 geoid component and Earth's rotation due to internal mass anomalies are analyzed for a stratified viscoelastic mantle described by a Maxwell rheology. Our approach is appropriate for a simplified modeling of subduction. Sea-level fluctuations induced by long-term rotational instabilities are also considered. The displacement of the Earth's axis of rotation, called true polar wander (TPW) and the induced eustatic sea-level fluctuations, are extremely sensitive to viscosity and density stratification at the 670 km seismic discontinuity. Phase-change models for the transition zone generally allow for huge amount of TPW, except for large viscosity increases; the dominant contribution in Liouville Equations comes from a secular term that reflects the viscous behaviour of the mantle. In chemically stratified models, TPW is drastically reduced due to dynamic compensation of the mass anomalies at the upper-lower mantle interface. When the source is embedded in the upper mantle close to the chemical density jump, transient rotational modes are the leading terms in the linear Liouville Equations. Long-term rotation instabilities are valuable contributors to the third order cycles in the eustatic sea-level curves. Rates of sea-level fluctuations of the order of 0.05-0.1 mm/yr are induced by displacements of the Earth's axis of rotation compatible with paleomagnetic data. © 1993 Kluwer Academic Publishers.

  • Isostatic deformations and polar wander induced by redistribution of mass within the Earth
    Journal of Geophysical Research, 1992
    Co-Authors: Y. Ricard, R. Sabadini, G. Spada
    Abstract:

    The effects at the surface of the Earth of time-dependent mantle mass anomalies are analyzed within the framework of a viscoelastic mantle with Maxwell rheology. The implications for the Earth's rotation are developed using the linearized Liouville Equations valid for small polar displacements. The displacement of the Earth's axis of rotation is very sensitive to the viscosity profile of the mantle and to the nature of the 670-km seismic discontinuity. For chemically stratified models, true polar wander is drastically reduced as a consequence of dynamic compensation of the mass anomalies at the upper-lower mantle interface. When the source is embedded in the upper mantle in the proximity of the chemical density jump, the polar wander stops after a finite time controlled by the slowest isostatic mode. -from Authors

  • Excitation of true polar wander by subduction
    Nature, 1992
    Co-Authors: G. Spada, Y. Ricard, R. Sabadini
    Abstract:

    TRUE polar wander - the global motion of the mantle relative to the Earth's rotation axis - is known from the analysis of palaeomagnetic data, hotspot tracks and plate motions to have occurred at velocities of up to 0.5°Myr-1 since the Late Cretaceous period1-4. We address here the longstanding question of how fast episodes of true polar wander (TPW) can be excited5, by analysing the impact of the distribution and activity of subduction zones on polar motion. Using nonlinear Liouville Equations, which allow us to treat large excursions of the polar axis, we show that unrealistically fast TPW is excited by subduction episodes unless the lower mantle has a viscosity at least 10 times that of the upper mantle. This need for a viscosity increase with depth in the mantle reinforces the conclusions of previous studies on post-glacial rebound and geoid anomalies, theoretical creep laws and some preliminary results on TPW induced by density anomalies embedded in the mantle6-10. The lower viscosity in the upper mantle means that upper-mantle density anomalies are most effective in exciting TPW. Changes in the pattern of subduction through time may be responsible for both episodes of fast TPW and times of quiescence in polar motion. © 1992 Nature Publishing Group.

R. Sabadini - One of the best experts on this subject based on the ideXlab platform.

  • Time-dependent density anomalies in a stratified, viscoelastic mantle: Implications for the geoid, Earth's rotation and sea-level fluctuations
    Surveys in Geophysics, 1993
    Co-Authors: R. Sabadini, G. Spada, Y. Ricard
    Abstract:

    The effects on the ℓ = 2 geoid component and Earth's rotation due to internal mass anomalies are analyzed for a stratified viscoelastic mantle described by a Maxwell rheology. Our approach is appropriate for a simplified modeling of subduction. Sea-level fluctuations induced by long-term rotational instabilities are also considered. The displacement of the Earth's axis of rotation, called true polar wander (TPW) and the induced eustatic sea-level fluctuations, are extremely sensitive to viscosity and density stratification at the 670 km seismic discontinuity. Phase-change models for the transition zone generally allow for huge amount of TPW, except for large viscosity increases; the dominant contribution in Liouville Equations comes from a secular term that reflects the viscous behaviour of the mantle. In chemically stratified models, TPW is drastically reduced due to dynamic compensation of the mass anomalies at the upper-lower mantle interface. When the source is embedded in the upper mantle close to the chemical density jump, transient rotational modes are the leading terms in the linear Liouville Equations. Long-term rotation instabilities are valuable contributors to the third order cycles in the eustatic sea-level curves. Rates of sea-level fluctuations of the order of 0.05-0.1 mm/yr are induced by displacements of the Earth's axis of rotation compatible with paleomagnetic data. © 1993 Kluwer Academic Publishers.

  • Isostatic deformations and polar wander induced by redistribution of mass within the Earth
    Journal of Geophysical Research, 1992
    Co-Authors: Y. Ricard, R. Sabadini, G. Spada
    Abstract:

    The effects at the surface of the Earth of time-dependent mantle mass anomalies are analyzed within the framework of a viscoelastic mantle with Maxwell rheology. The implications for the Earth's rotation are developed using the linearized Liouville Equations valid for small polar displacements. The displacement of the Earth's axis of rotation is very sensitive to the viscosity profile of the mantle and to the nature of the 670-km seismic discontinuity. For chemically stratified models, true polar wander is drastically reduced as a consequence of dynamic compensation of the mass anomalies at the upper-lower mantle interface. When the source is embedded in the upper mantle in the proximity of the chemical density jump, the polar wander stops after a finite time controlled by the slowest isostatic mode. -from Authors

  • Excitation of true polar wander by subduction
    Nature, 1992
    Co-Authors: G. Spada, Y. Ricard, R. Sabadini
    Abstract:

    TRUE polar wander - the global motion of the mantle relative to the Earth's rotation axis - is known from the analysis of palaeomagnetic data, hotspot tracks and plate motions to have occurred at velocities of up to 0.5°Myr-1 since the Late Cretaceous period1-4. We address here the longstanding question of how fast episodes of true polar wander (TPW) can be excited5, by analysing the impact of the distribution and activity of subduction zones on polar motion. Using nonlinear Liouville Equations, which allow us to treat large excursions of the polar axis, we show that unrealistically fast TPW is excited by subduction episodes unless the lower mantle has a viscosity at least 10 times that of the upper mantle. This need for a viscosity increase with depth in the mantle reinforces the conclusions of previous studies on post-glacial rebound and geoid anomalies, theoretical creep laws and some preliminary results on TPW induced by density anomalies embedded in the mantle6-10. The lower viscosity in the upper mantle means that upper-mantle density anomalies are most effective in exciting TPW. Changes in the pattern of subduction through time may be responsible for both episodes of fast TPW and times of quiescence in polar motion. © 1992 Nature Publishing Group.