Lagrangian coherent structure

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Hyperbolic LCS (attracting in red and repelling in blue) and elliptic LCS in a two-dimensional turbulence simulation. (Image: Mohammad Farazmand)

Lagrangian coherent structures (LCSs) are distinguished sets of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest.[1][2][3] The type of this influence may vary, but it invariably creates a coherent trajectory pattern for which the underlying LCS serves as a theoretical centerpiece. In observations of tracer patterns in nature, one readily identifies coherent features, but it is often the underlying structure creating these features that is of interest.

Indeed, individual tracer trajectories forming coherent patterns are generally sensitive with respect to changes in their initial conditions and the system parameters. In contrast, the LCSs creating these trajectory patterns turn out to be robust and provide a simplified skeleton of the overall dynamics of the system.[3] Physical examples include floating debris, oil spills,[4] surface drifters[5][6] and chlorophyll patterns[7] in the ocean; clouds of volcanic ash[8] and spores in the atmosphere;[9] and coherent crowd patterns formed by humans[10] and animals.


While LCSs may exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows. The images below are examples of how different types of LCSs hidden in geophysical flows shape tracer patterns.



General definitions[edit]

Material surfaces[edit]

Figure 1: An invariant manifold in the extended phase space, formed by an evolving material surface.

On a phase space {\mathcal P} and over a time interval {\mathcal I}=[t_0 ,t_1] , consider a non-autonomous dynamical system defined through the flow map F^t_{t_0}\colon x_0 \mapsto x(t,t_0,x_0), mapping initial conditions x_0\in{\mathcal P} into their position x(t,t_0,x_0)\in{\mathcal P} for any time t\in{\mathcal I}. If the flow map F^t_{t_0} is a diffeomorphism for any choice of t\in {\mathcal I}, then for any smooth set {\mathcal M}(t_0) of initial conditions in {\mathcal P}, the set


{\mathcal M}=\{(x,t)\in{\mathcal P} \times {\mathcal I}\,\colon [F^t_{t_0}]^{-1}(x)\in{\mathcal M}(t_0)\}

is an invariant manifold in the extended phase space  {\mathcal P} \times {\mathcal I}. Borrowing terminology from fluid dynamics, we refer to the evolving time slice {\mathcal M}(t)= F^t_{t_0}({\mathcal M}(t_0)) of the manifold {\mathcal M} as a material surface (see Fig. 1). Since any choice of the initial condition set {\mathcal M}(t_0) yields an invariant manifold {\mathcal M}\in{\mathcal P} \times {\mathcal I}, invariant manifolds and their associated material surfaces are abundant and generally undistinguished in the extended phase space. Only few of them will act as cores of coherent trajectory patterns.

LCSs as exceptional material surfaces[edit]

In order to create a coherent pattern, a material surface {\mathcal M}(t) should exert a sustained and consistent action on nearby trajectories throughout the time interval {\mathcal I}. Examples of such action are attraction, repulsion, or shear. In principle, any well-defined mathematical property qualifies that creates coherent patterns out of randomly selected nearby initial conditions.

Most such properties can be expressed by strict inequalities. For instance, we call a material surface {\mathcal M}(t) attracting over the interval  {\mathcal I} if all small enough initial perturbations to {\mathcal M}(t_0) are carried by the flow into even smaller final perturbations to {\mathcal M}(t_1). In classical dynamical systems theory, invariant manifolds satisfying such an attraction property over infinite times are called attractors. They are not only special, but even locally unique in the phase space: no continuous family of attractors may exist.

In contrast, in dynamical systems defined over a finite time interval {\mathcal I}, strict inequalities do not define exceptional (i.e., locally unique) material surfaces. This follows from the continuity of the flow map F^t_{t_0} over {\mathcal I} . For instance, if a material surface {\mathcal M}(t) attracts all nearby trajectories over the time interval {\mathcal I}, then so will any sufficiently close other material surface.

Thus, attracting, repelling and shearing material surfaces are necessarily stacked on each other, e.g., occur in continuous families. This leads to the idea of seeking LCSs in finite-time dynamical systems as exceptional material surfaces that exhibit a coherence-inducing property more strongly than any of the neighboring material surfaces. Such LCSs, defined as extrema (or more generally, stationary surfaces) for a finite-time coherence property, will indeed serve as observed centerpieces of trajectory patterns.

Objectivity (frame-indifference) of LCSs[edit]

Assume that the phase space of the underlying dynamical system is the material configuration space of a continuum, such as a fluid or a deformable body. For instance, for a dynamical system generated by an unsteady velocity field



v=v(x,t),\qquad x\in U\subset {\mathbb R}^3,

the open set U of possible particle positions is a material configuration space. In this space, LCSs form the skeleton of the material response by the continuum, subject to external forces and constitutive properties of the material. A fundamental requirement for the self-consistent description of any material response in continuum mechanics is objectivity (material frame-indifference). Objectivity stipulates that the material response (and hence the shape and the type of LCSs framing the material response) must be invariant with respect to Euclidean changes of coordinates



x=Q(t)y+b(t),

where y\in{\mathbb R}^3 is the vector of the transformed coordinates; Q(t) is an arbitrary 3\times 3 proper orthogonal matrix representing time-dependent rotations; and b(t) is an arbitrary 3-dimensional vector representing time-dependent translations. As a consequence, any self-consistent LCS definition or criterion should be expressible in terms of quantities that are frame-invariant. For instance, the rate of strain S(x,t) and the vorticity tensor W(x,t) defined as



S(x,t)=\frac{1}{2}\left(\nabla v(x,t)+ (\nabla v(x,t))^T\right),\qquad W(x,t)=\frac{1}{2}\left(\nabla v(x,t)- (\nabla v(x,t))^T\right),

transform under Euclidean changes of frame into the quantities



{\tilde S}(y,t)=Q(t)^TS(x,t)Q(t),\qquad {\tilde W}(y,t)=Q(t)^TS(x,t)Q(t)-Q(t)^T{\dot Q}(t).

A Euclidean frame change is, therefore, equivalent to a similarity transform for S(x,t), and hence an LCS approach depending only on the eigenvalues and eigenvectors of S(x,t)[11][12] is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of W(x,t) is generally not frame-invariant.

A number of frame-independent quantities, such as \nabla v(x,t), {W}(y,t), \nabla F^t_{t_0}, as well as the averages or eigenvalues of these quantities, are routinely used in heuristic LCS detection. While such quantities may effectively mark features of the instantaneous velocity field v(x,t), the ability of these quantities to capture material mixing, transport, and coherence is limited and a priori unknown in any given frame. As an example, consider the linear unsteady fluid particle motion[3]



{\dot x}=v(x,t)=\begin{pmatrix} \sin{4t} &2+\cos{4t}\\
-2+\cos{4t}& -\sin{4t}
\end{pmatrix}x,

which is an exact solution of the two-dimensional Navier--Stokes equations. The (frame-dependent) Okubo-Weiss criterion classifies the whole domain in this flow as elliptic (vortical) because q=\frac{1}{2}( {\vert S\vert}^2-{\vert W \vert}^2)<0 holds, with \vert\,\cdot \,\vert referring to the Euclidean matrix norm. As seen in Fig. 2, however, trajectories grow exponentially along a rotating line and shrink exponentially along another rotating line.[3] In material terms, therefore, the flow is hyperbolic (saddle-type) in any frame.

Figure 2: Instantaneous streamlines and the evolution of trajectories starting from the interior of one of them in a linear solution of the Navier--Stokes equation. This dynamical system is classified as elliptic by a number of frame-dependent coherence diagnostics, such as the Okubo--Weiss criterion. (Image: Francisco Beron--Vera)

One might wonder: if Newton’s equation for particle motion and the Navier--Stokes equations for fluid motion are frame-dependent, why would one need to require frame-invariance for LCSs that characterize solutions of these equations? To see the reason, recall that the Newton and Navier-Stokes equations represent objective physical principles for material particle trajectories. These equations generate physically the same material trajectories, no matter which frame we transform the equations of motion to. Since particle trajectories generated by these equations are objective, the LCSs acting as organizing centers for the trajectories should also be objectively defined.

Specific velocity and acceleration terms in the Newton and Navier-Stokes equations do change under changes of frame. These quantities are intrinsically tied to the frame of their definition, and hence should indeed vary across frames. Their change ensures that the Newton and Navier-Stokes equations generate the same material trajectories in any two frames connected via a Euclidean transformation.

Hyperbolic LCSs[edit]

Figure 3. Attracting and repelling LCSs in the extended phase space of a two-dimensional dynamical system.

Motivated by the above discussion, the simplest way to define an attracting LCS is by requiring it to be a locally strongest attracting material surface in the extended phase space  {\mathcal P} \times {\mathcal I} (see. Fig. 3) . Similarly, a repelling LCS can be defined as a locally strongest repelling material surface. Attracting and repelling LCSs together are usually referred to as hyperbolic LCSs,[1][3] as they provide a finite-time genearalization of the classic concept of normally hyperbolic invariant manifolds in dynamical systems.

Diagnostic approach to hyperbolic LCS: Finite-time Lyapunov exponents[edit]

Heuristically, one may seek initial positions  {\mathcal M}(t_0 ) of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from  {\mathcal M}(t_0 ) grow locally at the highest rate relative to trajectories starting off of {\mathcal M}(t_0 ).[13][14] The heuristic element here is that instead of constructing a highly repelling material surface, one simply seeks points of large particle separation. Such a separation may well be due to strong shear along the set of points so identified; this set is not at all guaranteed to exert any normal repulsion on nearby trajectories.

The growth of an infinitesimal perturbation {\xi}(t) along a trajectory x(t,t_0 ,x_0) is governed by the flow map gradient \nabla F^t_{t_0}. Let \epsilon{\xi}(t_0) be a small perturbation to the initial condition x_0, with 0<\epsilon\ll 1, and with {\xi}(t_0) denoting an arbitrary unit vector in {\mathbb R}^n. This perturbation generally grows along the trajectory x(t,t_0 ,x_0) into the perturbation vector {\xi}_\epsilon(t_1;x_0)= \nabla F^{t_1}_{t_0}(x_0)\epsilon{\xi}(t_0). Then the maximum relative stretching of infinitesimal perturbations at the point x_0 can be computed as



\delta^{t_1}_{t_0}(x_0)=\lim_{\epsilon\to0}\frac{1}{\epsilon}\max_{\left|\xi(t_{0})\right|=1}\left|\xi_{\epsilon}(t_{1};x_{0})\right|=\max_{\left|\xi(t_{0})\right|=1}\sqrt{\left\langle \nabla F_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0}),\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0})\right\rangle }
=\max_{\left|\xi(t_{0})\right|=1}\sqrt{\left\langle \xi(t_{0}),\left[\nabla F_{t_{0}}^{t_{1}}(x_{0})\right]^{T}\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0})\right\rangle }
=\max_{\left|\xi(t_{0})\right|=1}\sqrt{\left\langle \xi(t_{0}),C_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0})\right\rangle },

where C^{t_1}_{t_0}=\left[ \nabla F_{t_{0}}^{t_{1}}\right]^T\nabla F_{t_{0}}^{t_{1}} denotes the right Cauchy--Green strain tensor. One then concludes[13] that the maximum relative stretching experienced along a trajectory starting from x_0 is just  \delta^{t_1}_{t_0}(x_0)=\sqrt{\lambda_n (x_0)} . As this relative stretching tends to grow rapidly, it is more convenient to work with its growth exponent  (\log{\delta^{t_1}_{t_0 }})/(t_1-t_0), which is then precisely the finite-time Lyapunov exponent (FTLE)


\mathrm{FTLE}_{t_{0}}^{t_{1}}(x_{0})=\frac{1}{2(t_{1}-t_{0})}\log\lambda_{n}(x_{0}).
Figure 4a. Attracting (red) and repelling (blue) LCSs extracted as FTLE ridges from a two-dimensional turbulence experiment (Image: Manikandan Mathur)[15]

Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or ridges) of the FTLE field.[13][16] This expectation turns out to be justified in the majority of cases: time t_0 positions of repelling LCSs are marked by ridges of \mathrm{FTLE}_{t_{0}}^{t_{1}}(x_{0}). By applying the same argument in backward time, we obtain that time t_1 positions of attracting LCSs are marked by ridges of the backward FTLE field \mathrm{FTLE}_{t_{1}}^{t_{0}}.

The classic way of computing Lyapunov exponents is solving a linear differential equation for the linearized flow map \nabla F^{t}_{t_0}(x_0). A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient.[13] For example, in a three-dimenisonal flow, we launch a trajectory x(t;t_0,x_0) from any element x_0 of a grid of initial conditions. Using the coordinate representation x=(x^1,x^2,x^3) for the evolving trajectory x(t;t_0,x_0), we approximate the gradient of the flow map as


\nabla F_{t_{0}}^{t}(x_{0})\approx
\begin{pmatrix} 
\frac{x^{1}(t;t_{0},x_{0}+\delta_{1})-x^{1}(t;t_{0},x_{0}-\delta_{1})}{\left|2\delta_{1}\right|} & \frac{x^{1}(t;t_{0},x_{0}+\delta_{2})-x^{1}(t;t_{0},x_{0}-\delta_{2})}{\left|2\delta_{2}\right|} & \frac{x^{1}(t;t_{0},x_{0}+\delta_{3})-x^{1}(t;t_{0},x_{0}-\delta_{3})}{\left|2\delta_{3}\right|}\\
\frac{x^{2}(t;t_{0},x_{0}+\delta_{1})-x^{2}(t;t_{0},x_{0}-\delta_{1})}{\left|2\delta_{1}\right|} & \frac{x^{2}(t;t_{0},x_{0}+\delta_{2})-x^{2}(t;t_{0},x_{0}-\delta_{2})}{\left|2\delta_{2}\right|} & \frac{x^{2}(t;t_{0},x_{0}+\delta_{3})-x^{2}(t;t_{0},x_{0}-\delta_{3})}{\left|2\delta_{3}\right|}\\
\frac{x^{3}(t;t_{0},x_{0}+\delta_{1})-x^{3}(t;t_{0},x_{0}-\delta_{1})}{\left|2\delta_{1}\right|} & \frac{x^{3}(t;t_{0},x_{0}+\delta_{2})-x^{3}(t;t_{0},x_{0}-\delta_{2})}{\left|2\delta_{2}\right|} & \frac{x^{3}(t;t_{0},x_{0}+\delta_{3})-x^{3}(t;t_{0},x_{0}-\delta_{3})}{\left|2\delta_{3}\right|}
\end{pmatrix}
,
Figure 4b. Attracting (red) and repelling (blue) LCSs extracted as FTLE ridges from a two-dimensional simulation of a von Karman vortex street (Image: Jens Kasten)[17]

with a small vector \delta_{i} pointing in the x^{i} coordinate direction. For two-dimensional flows, only the first 2\times 2 minor matrix of the above matrix is relevant.

Figure 5. FTLE ridges highlight both hyperbolic LCS and shearing material lines, such as the boundaries of a riverbed in a 3D model of the New River Inlet, Onslow, North Carolina (Image: Allen Sanderson).[18]

FTLE ridges have proven to be a simple and efficient tool for the visualize hyperbolic LCSs in a number of physical problems, yielding intriguing images of initial positions of hyperbolic LCSs in different applications (see, e.g., Fig. 4). However, FTLE ridges obtained over sliding time windows  [t_{0}+T,t_{1}+T] do not form material surfaces. Thus, ridges of \mathrm{FTLE}_{t_{0}+T}^{t_{1}+T}(x_{0}) under varying T cannot be used to define Lagrangian objects, such as hyperbolic LCSs. Indeed, a locally strongest repelling material surface over [t_{0},t_{1}] will generally not play the same role over [t_{0}+T,t_{1}+T] and hence its evolving position at time t_0+T will not be a ridge for \mathrm{FTLE}_{t_{0}+T}^{t_{1}+T}. Nonetheless, evolving second-derivative FTLE ridges[19] computed over sliding intervals of the form [t_{0}+T,t_{1}+T] have been identified by some authors broadly with LCSs.[19] In support of this identification. it is also often argued that the material flux over such sliding-window FTLE ridges should necessarily be small.[19][20][21]


The "FTLE ridge=LCS" identification[19] would, however, create several conceptual and mathematical issues:

  • Second-derivative FTLE ridges are necessarily straight lines and hence do not exist in physical problems.[22][23]
  • FTLE ridges computed over sliding time windows  [t_{0}+T,t_{1}+T] with a varying T are generally not Lagrangian and the flux though them may well be large [3][16]
  • FTLE ridges mark hyperbolic LCS positions, but also highlight surfaces of high shear.[16] A convoluted mixture of both types of surfaces often arises in applications (see Fig. 5 for an example).
  • There are several other types LCSs (elliptic and parabolic) beyond the hyperbolic LCSs highlighted by FTLE ridges[3]

Local variational approach to hyperbolic LCS: shrink and stretch surfaces[edit]

The local variational theory of hyperbolic LCSs builds on their original definition as strongest repelling, attracting or shearing material surfaces in the flow over the time interval [t_0,t_1].[13] At an initial point x_{0} , let n_{0} denote a unit normal to an initial material surface \mathcal{M}(t_{0}) (cf. Fig. 6). By the invariance of material lines, the tangent space T_{x_0}\mathcal{M}(t_{0}) is mapped into the tangent space of T_{x_1}\mathcal{M}(t_{1}) by the linearized flow map \nabla F_{t_{0}}^{t_{1}}(x_{0}). At the same time, the image of the normal m normal under \nabla F_{t_{0}}^{t_{1}}(x_{0}) generally does not remain normal to \mathcal{M}(t_{1}). Therefore, in addition to a normal component of length \rho(x_{0,}n_{0}), the advected normal also develops a tangential component of length \sigma(x_{0},n_{0}) (cf. Fig. 6).

Figure 6. Linearized flow geometry along an evolving material surface.

If  \rho(x_{0},n_{0})>1, then the evolving material surface \mathcal{M}(t) strictly repels nearby trajectories by the end of the time interval [t_0,t_1]. Similarly, \rho(x_{0},n_{0})<1 signals that \mathcal{M}(t) strictly attracts nearby trajectories along its normal directions. A repelling (attracting) LCS over the interval [t_0,t_1] can be defined as a material surface \mathcal{M}(t) whose net repulsion \rho(x_{0},n_{0}) is pointwise maximal (minimal) with respect to perturbations of the initial normal vector field n_{0}. As earlier, we refer to repelling and attracting LCSs collectively as hyperbolic LCSs.[13]

Solving these local extremum principles for hyperbolic LCSs in two and three dimensions yields unit normal vector fields to which hyperbolic LCSs should everywhere be tangent.[24][25][26] The existence of such normal surfaces also requires a Frobenius-type integrability condition in the three-dimensional case. All these results can be summarized as follows:[3]

Hyperbolic LCS conditions from local variational theory in dimensions n=2 and n=3
LCS Normal vector field of {\mathcal M}(t_0) for n=2,3 ODE for {\mathcal M}(t_0) for n=2 Frobenius-type PDE for {\mathcal M}(t_0) for n=3
Attracting \xi_1 (x_0)  x^{\prime}_0 =\xi_2(x_0) (stretch lines) \left\langle \nabla\times\xi_{1}(x_0),\xi_{1}(x_0)\right\rangle =0 (stretch surfaces)
Repelling \xi_n (x_0)  x^{\prime}_0 =\xi_1(x_0) (shrink lines) \left\langle \nabla\times\xi_{3}(x_0),\xi_{3}(x_0)\right\rangle =0 (shrink surfaces)

Repelling LCSs are obtained as most repelling shrink lines, starting from local maxima of  \lambda_2 (x_0). Attracting LCSs are obtained as most attracting stretch lines, starting from local minima of  \lambda_1 (x_0). These starting points serve are initial positions of exceptional saddle-type trajectories in the flow. An example of the local variational computation of a repelling LCS is shown in FIg. 7. The computational algorithm is available in LCS Tool.

Figure 7. A repelling LCS visualized as an FTLE ridge (left) and computed exactly as a shrink line (right), i.e., a solution of the ODE  x^{\prime}_0 =\xi_1(x_0) starting from a global maximum of  \lambda_2 (x_0). [24] (Image: Mohammad Farazmand)

Global variational approach to hyperbolic LCSs in two dimensions: Geodesic LCS[edit]

A general material surface experiences shear and strain in its deformation, both of which depend continuously on initial conditions by the continuity of the map  F^t_{t_0}. The averaged strain and shear within a strip of  {\mathcal O}(\epsilon)-close material lines, therefore, typically show  {\mathcal O}(\epsilon) variation within such a strip. The geodesic theory of LCSs seeks exceptionally coherent locations where this general trend fails, resulting in an order of magnitude smaller variability in shear or strain than what is normally expected across an  {\mathcal O}(\epsilon) strip. Specifically, the geodesic theory searches for LCSs as special material lines around which  {\mathcal O}(\epsilon) material strips show no  {\mathcal O}(\epsilon) variability either in the material-line averaged shear (Shearless LCSs) or in the material-line averaged strain (Strainless or Elliptic LCSs). Such LCSs turn out to be geodesics of appropriate metric tensors defined by the deformation field -- hence the name of this theory.

Shearless LCSs are found to be null-geodesics of a Lorentzian metric tensor D_{t_{0}}^{t_{1}} defined as[27]

D_{t_{0}}^{t_{1}}(x_{0})=\frac{1}{2}\left[C_{t_{0}}^{t_{1}}(x_{0})\Omega-\Omega C_{t_{0}}^{t_{1}}(x_{0})\right],\qquad \Omega=

\begin{pmatrix} 
0  & -1 \\
1 & 0 \\
\end{pmatrix}
.

Such null-geodesics can be proven to be tensorlines of the Cauchy--Green strain tensor, i.e., are tangent to the direction field formed by the strain eigenvector fields \xi_i(x_0).[27] Specifically, repelling LCSs are trajectories of x_{0}^{\prime}=\xi_{1}(x_{0}) starting from local maxima of the \lambda_2(x_0) eigenvalue field. Similarly, attracting LCSs are trajectories of x_{0}^{\prime}=\xi_{2}(x_{0}) starting from local minims of the \lambda_1(x_0) eigenvalue field. This agrees with the conclusion of the local variational theory of LCSs. The geodesic approach, however, also sheds more light on the robustness of hyperbolic LCSs: hyperbolic LCSs only prevail as stationary curves of the averaged shear functional under perturbations that leave their endpoints fixed. This is to be contrasted with parabolic LCS (see below), that are also shearless LCSs but prevail as stationary curves to the shear functional even under perturbations to their endpoints.

Elliptic LCS[edit]

Elliptc LCSs are closed and nested material surfaces that act as building blocks of the Lagrangian equivalents of vortices, i.e., rotation-dominated regions of trajectories that generally traverse the phase space without substantial stretching or folding. They mimic the behavior of Kolmogorov--Arnold--Moser (KAM) tori that form elliptic regions is Hamiltonian systems.

Diagnostic approach to elliptic LCS: Polar rotation angle[edit]

Local variational theory elliptic LCS: Polar rotation angle[edit]

A three-dimensional example of a variational computation of an elliptic LCS is shown in FIg. 8.

Figure 8. An elliptic Lagrangian Coherent Structure (or LCS, in green) and its advected position under the flow map of a chaotically forced ABC flow. Also shown in green is a circle of initial conditions placed around the LCS (on the left), advected for the same amount of time (on the right). Image: Daniel Blazevski.

Parabolic LCS[edit]

Relation to classical invariant manifolds[edit]

Classical invariant manifolds are invariant sets in the phase space {\mathcal P} of an autonomous dynamical system. In contrast, LCSs are only required to be invariant in the extended phase space. This means that even if the underlying dynamical system is autonomous, the LCSs of the system over the interval I will generally be time-dependent, acting as the evolving skeletons of evolving coherent patterns.

See also[edit]

References[edit]

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Software packages for LCS[edit]

Software packages have been developed to perform Lagrangian coherent structures computations:

Lists of software packages for Lagrangian coherent structures computations have been created by Jerrold E. Marsden[9] and Steven K. Baum.[10]

Further related papers[edit]