Tangent Map Research Articles

Adaptive Multiclass Classification for Brain Computer Interfaces

We consider the problem of multiclass adaptive classification for brain-computer interfaces and propose the use of multiclass pooled mean linear discriminant analysis (MPMLDA), a multiclass generalization of the adaptation rule introduced by Vidaurre, Kawanabe, von Bünau, Blankertz, and Müller (2010) for the binary class setting. Using publicly available EEG data sets and tangent space mapping (Barachant, Bonnet, Congedo, & Jutten, 2012) as a feature extractor, we demonstrate that MPMLDA can significantly outperform state-of-the-art multiclass static and adaptive methods. Furthermore, efficient learning rates can be achieved using data from different subjects.

Metaplectic sheets and caustic traversals in the Weyl representation

The quantum Hamiltonian generates in time a family of evolution operators. Continuity of this family holds within any choice of representation and, in particular, for the Weyl propagator, even though its simplest semiclassical approximation may develop caustic singularities. The phase jumps of the Weyl propagator across caustics have not been previously determined. The semiclassical approximation relies on individual classical trajectories together with their neighbouring tangent map. Based on the latter, one defines a continuous family of unitary tangent propagators, with an exact Weyl representation that is close to the full semiclassical approximation in an appropriate neighbourhood. The phase increment of the semiclassical Weyl propagator, as a caustic is crossed, is derived from the facts that the corresponding family of tangent operators belong to the metaplectic group and that the products of the tangent propagators are obtained from Gaussian integrals. The Weyl representation of the metaplectic group is presented here, with the correct phases determined within an intrinsic ambiguity for the overall sign. The elements that fully determine the phase increment across a particular caustic are then analysed.

GENERIC DIFFEOMORPHISMS WITH ROBUSTLY TRANSITIVE SETS

Let <TEX>${\Lambda}$</TEX> be a robustly transitive set of a diffeomorphism <TEX>$f$</TEX> on a closed <TEX>$C^{\infty}$</TEX> manifold. In this paper, we characterize hyperbolicity of <TEX>${\Lambda}$</TEX> in <TEX>$C^1$</TEX>-generic sense.

Invariant manifolds for analytic dynamical systems over ultrametric fields

We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an ultrametric Banach space over K, an analytic diffeomorphism f:M→M, and a fixed point p of f. Under suitable assumptions on the tangent map Tp(f), we construct a centre–stable manifold, a centre manifold, respectively, an a-stable manifold around p, for a given real number a∈]0,1].

GENERALIZED EXTREME VALUE DISTRIBUTION PARAMETERS AS DYNAMICAL INDICATORS OF STABILITY

We introduce a new dynamical indicator of stability based on the Extreme Value statistics showing that it provides an insight into the local stability properties of dynamical systems. The indicator performs faster than others based on the iteration of the tangent map since it requires only the evolution of the original systems and, in the chaotic regions, gives further information about the local information dimension of the attractor. A numerical validation of the method is presented through the analysis of the motions in the Standard map.

EFFICIENT INTEGRATION OF THE VARIATIONAL EQUATIONS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS: APPLICATION TO THE FERMI–PASTA–ULAM LATTICE

We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.

Monotonicity properties of harmonic maps into NPC spaces

In this paper, we study monotonicity properties of harmonic maps into general NPC spaces. In addition, we introduce the notion of Alexandrov tangent maps and state a criterion for uniqueness.

Polynomials in one variable and ranks of certain tangent maps

We study the map that sends a monic degree n complex polynomial f(x) without multiple roots to the collection of n values of its derivative at the roots of f(x). We give an answer to a question posed by Yu. S. Ilyashenko.

On the transversality of functions at the core of the transverse function approach to control

The transverse function approach to control, introduced by Morin and Samson in the early 2000s, is based on functions that are transverse to a set of vector fields in a sense formally similar to, although strictly speaking different from, the classical notion of transversality in differential topology. In this paper, a precise link is established between transversality and the functions used in the transverse function approach. It is first shown that a smooth function \({f : M \longrightarrow Q}\) is transverse to a set of vector fields which locally span a distribution D on Q if, and only if, its tangent mapping T f is transverse to D, where D is regarded as a submanifold of the tangent bundle T Q. It is further shown that each of these two conditions is equivalent to transversality of T f to D along the zero section of T M. These results are then used to rigorously state and prove that if M is compact and D is a distribution on Q, then the set of mappings of M into Q that are transverse to D is open in the strong (or “Whitney C∞-”) topology on C∞(M, Q).

Numerical integration of variational equations

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the "tangent map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic map S , while the corresponding tangent map TS is used for the integration of the variational equations. A simple and systematic technique to construct TS is also presented.

Analysis on the junctions of domain walls

In this paper, we study the local structure and the smoothness ofsingularities of free boundaries in an optimal partition problem forthe Dirichlet eigenvalues. We prove that there is a uniquehomogeneous blow up(tangent map) at each singular point in theinterior of the free boundary. As a consequence we obtain therectifiability as well as local structures of singularities.

Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy

Let O be a closed geodesic polygon in S 2 . Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2 , we evaluate the infimum Dirichlet energy, E ( H ) , for continuous tangent maps of arbitrary homotopy type H. The expression for E ( H ) involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, π 1 ( S 2 − { s 1 , … , s n } , ∗ ) . These results have applications for the theoretical modelling of nematic liquid crystal devices. To cite this article: A. Majumdar et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Generalized polar coordinates on Lie groups and numerical integrators

Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres.

Short-time linear response with reduced-rank tangent map

The recently developed short-time linear response algorithm, which predicts the response of a nonlinear chaotic forced-dissipative system to small external perturbation, yields high precision of the response prediction. However, the computation of the short-time linear response formula with the full rank tangent map can be expensive. Here, a numerical method to potentially overcome the increasing numerical complexity for large scale models with many variables by using the reduced-rank tangent map in the computation is proposed. The conditions for which the short-time linear response approximation with the reduced-rank tangent map is valid are established, and two practical situations are examined, where the response to small external perturbations is predicted for nonlinear chaotic forced-dissipative systems with different dynamical properties.

Singularities of tangent lightcone map of a timelike surface in Minkowski 4-space

In the paper, we will define tangent lightcone map, tangentlightcone curvature and tangent lightcone height function. Then we study the geometry of the timelike surfaces in Minkowski 4-space through their contact with spacelike hyperplane and give the classification of singularities of tangent lightcone map based on the Legendrian singularity theory of Arnol'd.

Unobstructedness of deformations of holomorphic maps onto Fano manifolds of Picard number 1

We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties of minimal rational tangents of the Fano manifold is finite, the key idea is to factorize the given surjective morphism, after some transformation, through a universal morphism associated to the minimal rational curves.

Defect of a unitary matrix

We analyze properties of a map f sending a unitary matrix U of size N into a doubly stochastic matrix B=f(U) defined by Bi,j=∣Ui,j∣2. For any U we define its defect, determined by the dimension of the image Df(TUU) of the space TUU tangent to the manifold of unitary matrices U at U under the tangent map Df corresponding to f. The defect of U equal to zero for a generic unitary matrix, gives an upper bound for the dimension of a smooth orbit (a manifold) stemming from U of inequivalent unitary matrices mapped into the same doubly stochastic matrix B=f(U). We demonstrate several properties of the defect and prove an explicit formula for the defect of the Fourier matrix FN of size N. In this way we obtain an upper bound for the dimension of a smooth orbit of inequivalent unitary complex Hadamard matrices stemming from FN. It is equal to zero iff N is prime and coincides with the dimension of the known orbits if N is a power of a prime. Two constructions of these orbits are presented at the end of this work.

The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects

We describe numerical tools for the stability analysis of extrasolar planetary systems. In par- ticular, we consider the relative Poincare variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO), a measure of the maximal Lyapunov exponent, that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best-fitting solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best-fitting solutions are found in dynamically active region of the phase space. The long-term stability of the system is determined by a net of low-order two-body and three-body mean mo- tion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myr of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behaviour.

Local Geometry of Planar Analytic Morphisms

The effect of analytic morphisms on plane curve singularities, via direct and inverse images, is studied in terms of certain objects attached to the morphism, its jacobian, trunk and tangent map among them.

Description of evolving anisotropy at large strains

A description of texture evolution at large strain plasticity is developed. Texture evolution is defined in terms of changes taking place in the substructure of the material. The changes in the substructure are specified by means of a tangent map defined in the same manner as for the continuum in terms of a multiplicative decomposition. It is shown that the description of the changes in the substructure can be formulated in a way completely analogous to the description of the deformation of the continuum. Within the framework of thermodynamics the evolution of the substructure, driven by its conjugated force emerging from the dissipation inequality, is studied. The performance of the model presented is studied in relation to off-axis uniaxial stress, the evolution of the substructure being compared with experimental findings. In addition, two different evolution laws for cubic material symmetry are investigated.