We prove the existence of a torus that is invariant with respect to the flow of a vector field th... more We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an "a posteriori" format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables. For more information see https://ejde.math.txstate.edu/Volumes/2020/126/abstr.html
List of Figures xvi Chapter 1. Introduction Chapter 2. Circle Maps and a Periodically Pulsating C... more List of Figures xvi Chapter 1. Introduction Chapter 2. Circle Maps and a Periodically Pulsating Cavity 2.1 Literature Review .
We compute accurately the golden critical invariant circles of several area-preserving twist maps... more We compute accurately the golden critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier
In the present paper, which is a sequel of [1], we consider the dimensional reduction of differen... more In the present paper, which is a sequel of [1], we consider the dimensional reduction of differential operators (DOs) that are invariant with respect to the action of a connected Lie group G. The action of G on vector bundles induces naturally actions of G on their sections and on the DOs between them. In [1] we constructed explicitly the reduced bundle ξ, such that the set of all its sections, C∞(ξG), is in a bijective correspondence with the set C∞(ξ) of all G-invariant sections of the original vector bundle ξ. The main goal of the present paper is, given a G-invariant DO D : C∞(ξ) → C∞(η) to construct the reduced DO D : C∞(ξG) → C∞(ηG). Our construction of D uses the geometrically natural language of jet bundles which best reveals the geometry of the DOs and reduces the manipulations with DOs to simple algebraic operations. Since ξ was constructed in [1] by restricting a certain bundle to a submanifold of its base, an essential ingredient of the dimensional reduction of a DO is t...
Semi-discrete wavelet transforms are discrete in scale, as in Mallat's multi-resolution anal... more Semi-discrete wavelet transforms are discrete in scale, as in Mallat's multi-resolution analysis, but continuous in position. The number of coefficients and algorithmic complexity then grows only as NlogN where N is the number of points (pixels) in the time-series (image). The redundancy of this representation at each scale has been exploited in denoising and data compression applications but we see it here as an asset when cumulating spatial statistics. Following Arnéodo, the wavelets are normalized in such a way that the scaling exponents of the moments of the coefficients are the same as for structure functions at all orders, at least in nonstationary/stationary-increment signals. We apply 1D and 2D semi-discrete transforms to remote sensing data on cloud structure from a variety of sources: NASA's MODerate Imaging Spectroradiometer (MODIS) on Terra and Thematic Mapper (TM) on LandSat; high-resolution cloud scenes from DOE's Multispectral Thermal Imager (MTI); and an upward-looking mm-radar at one of DOE's climate observation sites supporting the Atmospheric Radiation Measurement (ARM) Program. We show that the scale-dependence of the variance of the wavelet coefficients is always a better discriminator of transition from stationary to nonstationary behavior than conventional methods based on auto-correlation analysis, 2nd-order structure function (a.k.a. the semi-variogram), or spectral analysis. Examples of stationary behavior are (delta-correlated) instrumental noise and large-scale decorrelation of cloudiness; here wavelet coefficients decrease with increasing scale. Examples of nonstationary behavior are the predominant turbulent structure of cloud layers as well as instrumental or physical smoothing in the data; here wavelet coefficients increase with scale. In all of these regimes, we have theoretical expectations and/or empirical evidence of power-law relations for wavelet statistics with respect to scale as is expected in physical (finite-scaling) fractal signals. In particular, this implies the presence of long-range correlations in cloud structure coming from the important nonstationary regime. Finally, ramifications of this finding for cloud-radiation interaction are discussed in the context of climate modeling.
Spatial and/or temporal variabilities of clouds is of paramount importance for at least two in te... more Spatial and/or temporal variabilities of clouds is of paramount importance for at least two in tensely researched sub-problems in global and regional climate modeling: (1) cloud-radiation interaction where correlations can trigger 3D radiative transfer effects; and (2) dynamical cloud modeling where the goal is to realistically reproduce the said correlations. We propose wavelets as a simple yet powerful way of quantifying cloud variability. More precisely, we use 'semi-discrete' wavelet transforms which, at least in the present statistical applications, have advantages over both its continuous and discrete counterparts found in the bulk of the wavelet literature. With the particular choice of normalization we adopt, the scale-dependence of the variance of the wavelet coefficients (i.e,, the wavelet energy spectrum) is always a better discriminator of transition from 'stationary' to 'nonstationary' behavior than conventional methods based on auto-correlation ...
We prove the existence of a torus that is invariant with respect to the flow of a vector field th... more We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an “a posteriori” format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables.
We present photometric observations and transit solutions of the exoplanets XO-2b, HAT-P-18b and ... more We present photometric observations and transit solutions of the exoplanets XO-2b, HAT-P-18b and WASP 80b. Our solution of the XO-2b transit gave system parameters whose values are close to those of the previous studies. The solutions of the new transits of HAT-P-18b and WASP 80b differ from the previous ones by bigger stellar and planet radii. We obtained new values of the target initial epochs corresponding to slightly different periods. Our investigation reaffirmed that small telescopes can be used successfully for the study of exoplanets orbiting stars brighter than 13 mag.
We study processes that consist of deterministic evolution punctuated at random times by disturba... more We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).
International Journal of Geometric Methods in Modern Physics
We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sec... more We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sections [Formula: see text] of a vector bundle [Formula: see text] with base [Formula: see text], in the language of jet bundles. When the base of [Formula: see text] is restricted to a submanifold [Formula: see text], all information about derivatives in directions that are not tangent to [Formula: see text] is lost. To restrict [Formula: see text] to a DO [Formula: see text] acting on sections [Formula: see text] of the restricted bundle [Formula: see text] (with [Formula: see text] the natural embedding), one must choose an auxiliary DO [Formula: see text] and express the derivatives non-tangent to [Formula: see text] from the kernel of [Formula: see text]. This is equivalent to choosing a splitting of certain short exact sequence of jet bundles. A property of [Formula: see text] called formal integrability is crucial for restriction’s self-consistency. We give an explicit example illus...
Abstract The upper-bound of bulk time-averaged energy dissipation rate of body-forced plane shear... more Abstract The upper-bound of bulk time-averaged energy dissipation rate of body-forced plane shear flow with free-slip boundaries has been studied theoretically by deriving a mini-max variational problem from the Navier-Stokes equations. Rigorous bound has been obtained at high Reynolds numbers. In our work, the bound is improved by introducing a balance parameter into the mini-max problem, which includes higher-order derivatives of the velocity field in the function to be optimized. The resulting Euler-Lagrange equations are ...
We study critical invariant circles of several noble rotation numbers at the edge of breakdown fo... more We study critical invariant circles of several noble rotation numbers at the edge of breakdown for area preserving maps of the cylinder which violate the twist conditions. These circles admit essentially unique parameterizations by rotational coordinates. We present a high accuracy computation of about 10 7 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and show that, to the extent of the precision, it only depends on the tail of the continued fraction expansion.
We study the problem of the asymptotic behavior of the electromagnetic field in an optical resona... more We study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically (with d ≥ 2 incommensurate frequencies). We show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d-dimensional torus that preserves a foliation by irrational straight lines. In particular, the Jacobian of this map has (d − 1) eigenvalues equal to 1. We present rigorous and numerical results about several dynamical features of such maps. We also show how these dynamical features translate into properties for the field in the cavity. In particular, we show that when the torus map satisfies a KAM theorem-which happens for a Cantor set of positive measure of parameters-the energy of the electromagnetic field remains bounded. When the torus map is in a resonant region-which happens in open sets of parameters inside the gaps of the previous Cantor set-the energy grows exponentially.
Journal of Physics A: Mathematical and Theoretical, 2007
In the last decades, renormalization group (RG) ideas have been applied to describe universal pro... more In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disk boundaries, etc.). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators. The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit.
Journal of Optics B: Quantum and Semiclassical Optics, 2005
We study the problem of the behavior of a quantum massless scalar field in the space between two ... more We study the problem of the behavior of a quantum massless scalar field in the space between two parallel infinite perfectly conducting plates, one of them stationary, the other moving periodically. We reformulate the physical problem into a problem about the asymptotic behavior of the iterates of a map of the circle, and then apply results from theory of dynamical systems to study the properties of the map. Many of the general mathematical properties of maps of the circle translate into properties of the field in the cavity. For example, we give a complete classification of the possible resonances in the system, and show that small enough perturbations do not destroy the resonances. We use some mathematical identities to give transparent physical interpretation of the processes of creation and amplification of the quantum field due to the motion of the boundary and to elucidate the similarities and the differences between the classical and quantum fields in domains with moving boundaries.
We present a formalism for dimensional reduction based on the local properties of invariant cross... more We present a formalism for dimensional reduction based on the local properties of invariant cross-sections ("fields") and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e., to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a "non-canonical" reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space. Subj. Class.: Differential geometry; Geometrical approaches to partial differential equations; Field theory.
Journal of Geometry and Physics - J GEOM PHYSICS, 2003
We present a formalism for dimensional reduction based on the local properties of invariant cross... more We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (``fields'') and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a ``non-canonical'' reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.
We prove the existence of a torus that is invariant with respect to the flow of a vector field th... more We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an "a posteriori" format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables. For more information see https://ejde.math.txstate.edu/Volumes/2020/126/abstr.html
List of Figures xvi Chapter 1. Introduction Chapter 2. Circle Maps and a Periodically Pulsating C... more List of Figures xvi Chapter 1. Introduction Chapter 2. Circle Maps and a Periodically Pulsating Cavity 2.1 Literature Review .
We compute accurately the golden critical invariant circles of several area-preserving twist maps... more We compute accurately the golden critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier
In the present paper, which is a sequel of [1], we consider the dimensional reduction of differen... more In the present paper, which is a sequel of [1], we consider the dimensional reduction of differential operators (DOs) that are invariant with respect to the action of a connected Lie group G. The action of G on vector bundles induces naturally actions of G on their sections and on the DOs between them. In [1] we constructed explicitly the reduced bundle ξ, such that the set of all its sections, C∞(ξG), is in a bijective correspondence with the set C∞(ξ) of all G-invariant sections of the original vector bundle ξ. The main goal of the present paper is, given a G-invariant DO D : C∞(ξ) → C∞(η) to construct the reduced DO D : C∞(ξG) → C∞(ηG). Our construction of D uses the geometrically natural language of jet bundles which best reveals the geometry of the DOs and reduces the manipulations with DOs to simple algebraic operations. Since ξ was constructed in [1] by restricting a certain bundle to a submanifold of its base, an essential ingredient of the dimensional reduction of a DO is t...
Semi-discrete wavelet transforms are discrete in scale, as in Mallat's multi-resolution anal... more Semi-discrete wavelet transforms are discrete in scale, as in Mallat's multi-resolution analysis, but continuous in position. The number of coefficients and algorithmic complexity then grows only as NlogN where N is the number of points (pixels) in the time-series (image). The redundancy of this representation at each scale has been exploited in denoising and data compression applications but we see it here as an asset when cumulating spatial statistics. Following Arnéodo, the wavelets are normalized in such a way that the scaling exponents of the moments of the coefficients are the same as for structure functions at all orders, at least in nonstationary/stationary-increment signals. We apply 1D and 2D semi-discrete transforms to remote sensing data on cloud structure from a variety of sources: NASA's MODerate Imaging Spectroradiometer (MODIS) on Terra and Thematic Mapper (TM) on LandSat; high-resolution cloud scenes from DOE's Multispectral Thermal Imager (MTI); and an upward-looking mm-radar at one of DOE's climate observation sites supporting the Atmospheric Radiation Measurement (ARM) Program. We show that the scale-dependence of the variance of the wavelet coefficients is always a better discriminator of transition from stationary to nonstationary behavior than conventional methods based on auto-correlation analysis, 2nd-order structure function (a.k.a. the semi-variogram), or spectral analysis. Examples of stationary behavior are (delta-correlated) instrumental noise and large-scale decorrelation of cloudiness; here wavelet coefficients decrease with increasing scale. Examples of nonstationary behavior are the predominant turbulent structure of cloud layers as well as instrumental or physical smoothing in the data; here wavelet coefficients increase with scale. In all of these regimes, we have theoretical expectations and/or empirical evidence of power-law relations for wavelet statistics with respect to scale as is expected in physical (finite-scaling) fractal signals. In particular, this implies the presence of long-range correlations in cloud structure coming from the important nonstationary regime. Finally, ramifications of this finding for cloud-radiation interaction are discussed in the context of climate modeling.
Spatial and/or temporal variabilities of clouds is of paramount importance for at least two in te... more Spatial and/or temporal variabilities of clouds is of paramount importance for at least two in tensely researched sub-problems in global and regional climate modeling: (1) cloud-radiation interaction where correlations can trigger 3D radiative transfer effects; and (2) dynamical cloud modeling where the goal is to realistically reproduce the said correlations. We propose wavelets as a simple yet powerful way of quantifying cloud variability. More precisely, we use 'semi-discrete' wavelet transforms which, at least in the present statistical applications, have advantages over both its continuous and discrete counterparts found in the bulk of the wavelet literature. With the particular choice of normalization we adopt, the scale-dependence of the variance of the wavelet coefficients (i.e,, the wavelet energy spectrum) is always a better discriminator of transition from 'stationary' to 'nonstationary' behavior than conventional methods based on auto-correlation ...
We prove the existence of a torus that is invariant with respect to the flow of a vector field th... more We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an “a posteriori” format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables.
We present photometric observations and transit solutions of the exoplanets XO-2b, HAT-P-18b and ... more We present photometric observations and transit solutions of the exoplanets XO-2b, HAT-P-18b and WASP 80b. Our solution of the XO-2b transit gave system parameters whose values are close to those of the previous studies. The solutions of the new transits of HAT-P-18b and WASP 80b differ from the previous ones by bigger stellar and planet radii. We obtained new values of the target initial epochs corresponding to slightly different periods. Our investigation reaffirmed that small telescopes can be used successfully for the study of exoplanets orbiting stars brighter than 13 mag.
We study processes that consist of deterministic evolution punctuated at random times by disturba... more We study processes that consist of deterministic evolution punctuated at random times by disturbances with random severity; we call such processes semistochastic. Under appropriate assumptions such a process admits a unique stationary distribution. We develop a technique for establishing bounds on the rate at which the distribution of the random process approaches the stationary distribution. An important example of such a process is the dynamics of the carbon content of a forest whose deterministic growth is interrupted by natural disasters (fires, droughts, insect outbreaks, etc.).
International Journal of Geometric Methods in Modern Physics
We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sec... more We consider the restriction of a differential operator (DO) [Formula: see text] acting on the sections [Formula: see text] of a vector bundle [Formula: see text] with base [Formula: see text], in the language of jet bundles. When the base of [Formula: see text] is restricted to a submanifold [Formula: see text], all information about derivatives in directions that are not tangent to [Formula: see text] is lost. To restrict [Formula: see text] to a DO [Formula: see text] acting on sections [Formula: see text] of the restricted bundle [Formula: see text] (with [Formula: see text] the natural embedding), one must choose an auxiliary DO [Formula: see text] and express the derivatives non-tangent to [Formula: see text] from the kernel of [Formula: see text]. This is equivalent to choosing a splitting of certain short exact sequence of jet bundles. A property of [Formula: see text] called formal integrability is crucial for restriction’s self-consistency. We give an explicit example illus...
Abstract The upper-bound of bulk time-averaged energy dissipation rate of body-forced plane shear... more Abstract The upper-bound of bulk time-averaged energy dissipation rate of body-forced plane shear flow with free-slip boundaries has been studied theoretically by deriving a mini-max variational problem from the Navier-Stokes equations. Rigorous bound has been obtained at high Reynolds numbers. In our work, the bound is improved by introducing a balance parameter into the mini-max problem, which includes higher-order derivatives of the velocity field in the function to be optimized. The resulting Euler-Lagrange equations are ...
We study critical invariant circles of several noble rotation numbers at the edge of breakdown fo... more We study critical invariant circles of several noble rotation numbers at the edge of breakdown for area preserving maps of the cylinder which violate the twist conditions. These circles admit essentially unique parameterizations by rotational coordinates. We present a high accuracy computation of about 10 7 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and show that, to the extent of the precision, it only depends on the tail of the continued fraction expansion.
We study the problem of the asymptotic behavior of the electromagnetic field in an optical resona... more We study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically (with d ≥ 2 incommensurate frequencies). We show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d-dimensional torus that preserves a foliation by irrational straight lines. In particular, the Jacobian of this map has (d − 1) eigenvalues equal to 1. We present rigorous and numerical results about several dynamical features of such maps. We also show how these dynamical features translate into properties for the field in the cavity. In particular, we show that when the torus map satisfies a KAM theorem-which happens for a Cantor set of positive measure of parameters-the energy of the electromagnetic field remains bounded. When the torus map is in a resonant region-which happens in open sets of parameters inside the gaps of the previous Cantor set-the energy grows exponentially.
Journal of Physics A: Mathematical and Theoretical, 2007
In the last decades, renormalization group (RG) ideas have been applied to describe universal pro... more In the last decades, renormalization group (RG) ideas have been applied to describe universal properties of different routes to chaos (quasi-periodic, period doubling or tripling, Siegel disk boundaries, etc.). Each of the RG theories leads to universal scaling exponents which are related to the action of certain RG operators. The goal of this announcement is to show that there is a principle that organizes many of these scaling exponents. We give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations. These relations are determined by combinatorial properties of the route and become exact in an appropriate limit.
Journal of Optics B: Quantum and Semiclassical Optics, 2005
We study the problem of the behavior of a quantum massless scalar field in the space between two ... more We study the problem of the behavior of a quantum massless scalar field in the space between two parallel infinite perfectly conducting plates, one of them stationary, the other moving periodically. We reformulate the physical problem into a problem about the asymptotic behavior of the iterates of a map of the circle, and then apply results from theory of dynamical systems to study the properties of the map. Many of the general mathematical properties of maps of the circle translate into properties of the field in the cavity. For example, we give a complete classification of the possible resonances in the system, and show that small enough perturbations do not destroy the resonances. We use some mathematical identities to give transparent physical interpretation of the processes of creation and amplification of the quantum field due to the motion of the boundary and to elucidate the similarities and the differences between the classical and quantum fields in domains with moving boundaries.
We present a formalism for dimensional reduction based on the local properties of invariant cross... more We present a formalism for dimensional reduction based on the local properties of invariant cross-sections ("fields") and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e., to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a "non-canonical" reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space. Subj. Class.: Differential geometry; Geometrical approaches to partial differential equations; Field theory.
Journal of Geometry and Physics - J GEOM PHYSICS, 2003
We present a formalism for dimensional reduction based on the local properties of invariant cross... more We present a formalism for dimensional reduction based on the local properties of invariant cross-sections (``fields'') and differential operators. This formalism does not need an ansatz for the invariant fields and is convenient when the reducing group is non-compact. In the approach presented here, splittings of some exact sequences of vector bundles play a key role. In the case of invariant fields and differential operators, the invariance property leads to an explicit splitting of the corresponding sequences, i.e. to the reduced field/operator. There are also situations when the splittings do not come from invariance with respect to a group action but from some other conditions, which leads to a ``non-canonical'' reduction. In a special case, studied in detail in the second part of this article, this method provides an algorithm for construction of conformally invariant fields and differential operators in Minkowski space.
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Papers by Nikola Petrov