Tangent Map Research Articles

Geometry of infinite dimensional Cartan developments

The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer–Cartan equation on a simply connected manifold [Formula: see text] to a smooth mapping from [Formula: see text] into the Lie group. In this paper, this is generalized to infinite dimensional [Formula: see text] for infinite dimensional regular Lie groups. The Cartan development is viewed as a generalization of the evolution map of a regular Lie group. The tangent mapping of a Cartan development is identified as another Cartan development.

Critical sets of solutions of elliptic equations in periodic homogenization

AbstractIn this paper we study critical sets of solutions of second‐order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first‐order correctors, we show that the ‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε, provided that doubling indices for solutions are bounded. The key step is an estimate of “turning” of an approximate tangent map, the projection of a non‐constant solution onto the subspace of spherical harmonics of order ℓ, when the doubling index for on a sphere is trapped between and , for r between 1 and a minimal radius . This estimate is proved by using harmonic approximation successively. With a suitable L2 renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets.

Hamel's field variational integrator for simulating dynamics of thin-walled geometrically exact beams with warping effects

Using discrete variational principle, a Hamel's field variational integrator (HFVI) for simulating dynamics of thin-walled geometrically exact beams with warping effects is originally proposed. Firstly, a spacetime independent interpolation way for thin-walled beam elements is established on SE(3) × R space by using Hamel's formalism. Then, a new discretization formulation of convective velocity and convective strains on Lie algebra is proposed by using frame operators, which helps to reduce the system's nonlinearity. After that, the variations of discrete convective velocity and discrete convective strain are decoupled from the cross-section configuration variables. This way can avoid the computation of nonlinear exponential map's tangent map and reduce the computation complexity significantly. According the Hamiltonian form of proposed HFVI, the warping variables and discrete convective strains can be independently solved by using discrete compatibility equations on Lie algebra. Finally, the efficiency and accuracy of HFVI is validated by six static and dynamic examples.

Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference and the ratio of the LDs of neighboring orbits, as well as a quantity related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional symplectic standard map. We compare the characterization obtained by these LDs-based diagnostics against that achieved by the Smaller Alignment Index method of chaos detection, by recording the percentage agreement PA between the two classifications. We also study the influence of the final number of orbit iterations T, the order n of the indices, as well as the distance σ of neighboring orbits on the performance of these methods, and find appropriate T, n and σ values which allow the efficient use of the three indices as short time and computationally cheap chaos diagnostics achieving PA≳90%. Our findings clearly indicate the capability of LDs to efficiently identify chaos in systems whose phase space is difficult to visualize, due to its high dimensionality, without knowing the variational equations (tangent map) of continuous (discrete) time systems needed by traditional chaos indicators.

A blow-up formula for stationary quaternionic maps

Let ( M , J α , α = 1 , 2 , 3 ) (M, J^\alpha , \alpha =1,2,3) and ( N , J α , α = 1 , 2 , 3 ) (N, \mathcal {J}^\alpha , \alpha =1,2,3) be Hyperkähler manifolds. Suppose that u k u_k is a sequence of stationary quaternionic maps and converges weakly to u u in H 1 , 2 ( M , N ) H^{1,2}(M,N) , we derive a blow-up formula for lim k → ∞ d ( u k ∗ J α ) \lim _{k\to \infty }d(u_k^*\mathcal {J}^\alpha ) , for α = 1 , 2 , 3 \alpha =1,2,3 , in the weak sense. As a corollary, we show that the maps constructed by Chen-Li [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), pp. 375–388] and by Foscolo [J. Differential Geom. 112 (2019), pp. 79–120] cannot be tangent maps (c.f Li and Tian [Internat. Math. Res. Notices 14 (1998), pp. 735–755], Theorem 3.1) of a stationary quaternionic map satisfing d ( u ∗ J α ) = 0 d(u^*\mathcal {J}^\alpha )=0 .

Complete L∞-algebras and their homotopy theory

We analyze a model for the homotopy theory of complete filtered L∞-algebras intended for applications in algebraic and algebro-geometric deformation theory. We provide an explicit proof of an unpublished result of E. Getzler which states that the category Lieˆ∞ of such L∞-algebras and filtration-preserving ∞-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Novel applications of our approach include explicit models for homotopy pullbacks, and an analog of Whitehead's Theorem: under some mild conditions, every filtered L∞-quasi-isomorphism in Lieˆ∞ has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer–Cartan functor, which assigns a Kan simplicial set to each L∞-algebra in Lieˆ∞, is an exact functor between the respective CFOs. Finally, we provide an obstruction theory for the general problem of lifting a Maurer-Cartan element through an ∞-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map.

Multifrequency dielectric mapping of fixed mice colon tissues in cell culture media via scanning electrochemical microscopy.

Alternating current scanning electrochemical microscopy (AC-SECM) is a powerful tool for characterizing the electrochemical reactivity of surfaces. Here, perturbation in the sample is induced by the alternating current and altered local potential is measured by the SECM probe. This technique has been used to investigate many exotic a range of biological interfaces including live cells and tissues, as well as the corrosive degradation of various metallic surfaces, etc. In principle, AC-SECM imaging is derived from electrochemical impedance spectroscopy (EIS) which has been used for a century to describe interfacial and diffusive behaviour of molecules in solution or on a surface. Increasingly bioimpedance centric medical devices have become an important tool to detect evolution of tissue biochemistry. Predictive implications of measuring electrochemical changes within a tissue is one of the core concepts in developing minimally invasive and smart medical devices. In this study, cross sections of mice colon tissue were used for AC-SECM imaging. A 10 micron sized platinum probe was used for two-dimensional (2D) tan δ mapping of histological sections at a frequency of 10kHz, Thereafter, multifrequency scans were performed at 100Hz, 10kHz, 300kHz, and 900kHz. Loss tangent (tan δ) mapping of mice colon revealed microscale regions within a tissue possessing a discrete tan δ signature. This tan map may be an immediate measure of physiological conditions in biological tissues. Multifrequency scans highlight subtle changes in protein or lipid composition as a function of frequency which was recorded as loss tangent maps. Impedance profile at different frequencies could also be used to identify optimal contrast for imaging and extracting the electrochemical signature specific for a tissue and its electrolyte.

A Multi-Source Transfer Joint Matching Method for Inter-Subject Motor Imagery Decoding.

Individual differences among different subjects pose a great challenge to motor imagery (MI) decoding. Multi-source transfer learning (MSTL) is one of the most promising ways to reduce individual differences, which can utilize rich information and align the data distribution among different subjects. However, most MSTL methods in MI-BCI combine all data in the source subjects into a single mixed domain, which will ignore the effect of important samples and the large differences in multiple source subjects. To address these issues, we introduce transfer joint matching and improve it to multi-source transfer joint matching (MSTJM) and weighted MSTJM (wMSTJM). Different from previous MSTL methods in MI, our methods align the data distribution for each pair of subjects, and then integrate the results by decision fusion. Besides that, we design an inter-subject MI decoding framework to verify the effectiveness of these two MSTL algorithms. It mainly consists of three modules: covariance matrix centroid alignment in the Riemannian space, source selection in the Euclidean space after tangent space mapping to reduce negative transfer and computation overhead, and further distribution alignment by MSTJM or wMSTJM. The superiority of this framework is verified on two common public MI datasets from BCI competition IV. The average classification accuracy of the MSTJM and wMSTJ methods outperformed other state-of-the-art methods by at least 4.24% and 2.62% respectively. It's promising to advance the practical applications of MI-BCI.

Uniform profile near the point defect of Landau-de Gennes model

For the Landau-de Gennes functional on 3D domains, $$\begin{aligned} I_\varepsilon (Q,\Omega ):=\int _{\Omega }\left\{ \frac{1}{2}|\nabla Q|^2+\frac{1}{\varepsilon ^2}\left( -\frac{a^2}{2}\mathrm {tr}(Q^2)-\frac{b^2}{3}\mathrm {tr}(Q^3)+\frac{c^2}{4}[\mathrm {tr}(Q^2)]^2 \right) \right\} \,dx, \end{aligned}$$ it is well-known that under suitable boundary conditions, the global minimizer $$Q_\varepsilon $$ converges strongly in $$H^1(\Omega )$$ to a uniaxial minimizer $$Q_*=s_+(n_*\otimes n_*-\frac{1}{3}\mathrm {Id})$$ up to some subsequence $$\varepsilon _n\rightarrow \infty $$ , where $$n_*\in H^1(\Omega ,\mathbb {S}^2)$$ is a minimizing harmonic map. In this paper we further investigate the structure of $$Q_\varepsilon $$ near the core of a point defect $$x_0$$ which is a singular point of the map $$n_*$$ . The main strategy is to study the blow-up profile of $$Q_{\varepsilon _n}(x_n+\varepsilon _n y)$$ where $$\{x_n\}$$ are carefully chosen and converge to $$x_0$$ . We prove that $$Q_{\varepsilon _n}(x_n+\varepsilon _n y)$$ converges in $$C^2_{loc}(\mathbb {R}^n)$$ to a tangent map Q(x) which at infinity behaves like a “hedgehog" solution that coincides with the asymptotic profile of $$n_*$$ near $$x_0$$ . Moreover, such convergence result implies that the minimizer $$Q_{\varepsilon _n}$$ can be well approximated by the Oseen-Frank minimizer $$n_*$$ outside the $$O(\varepsilon _n)$$ neighborhood of the point defect.

Supervised and Semisupervised Manifold Embedded Knowledge Transfer in Motor Imagery-Based BCI.

A long calibration procedure limits the use in practice for a motor imagery (MI)-based brain-computer interface (BCI) system. To tackle this problem, we consider supervised and semisupervised transfer learning. However, it is a challenge for them to cope with high intersession/subject variability in the MI electroencephalographic (EEG) signals. Based on the framework of unsupervised manifold embedded knowledge transfer (MEKT), we propose a supervised MEKT algorithm (sMEKT) and a semisupervised MEKT algorithm (ssMEKT), respectively. sMEKT only has limited labelled samples from a target subject and abundant labelled samples from multiple source subjects. Compared to sMEKT, ssMEKT adds comparably more unlabelled samples from the target subject. After performing Riemannian alignment (RA) and tangent space mapping (TSM), both sMEKT and ssMEKT execute domain adaptation to shorten the differences among subjects. During domain adaptation, to make use of the available samples, two algorithms preserve the source domain discriminability, and ssMEKT preserves the geometric structure embedded in the labelled and unlabelled target domains. Moreover, to obtain a subject-specific classifier, sMEKT minimizes the joint probability distribution shift between the labelled target and source domains, whereas ssMEKT performs the joint probability distribution shift minimization between the unlabelled target domain and all labelled domains. Experimental results on two publicly available MI datasets demonstrate that our algorithms outperform the six competing algorithms, where the sizes of labelled and unlabelled target domains are variable. Especially for the target subjects with 10 labelled samples and 270/190 unlabelled samples, ssMEKT shows 5.27% and 2.69% increase in average accuracy on the two abovementioned datasets compared to the previous best semisupervised transfer learning algorithm (RA-regularized common spatial patterns-weighted adaptation regularization, RA-RCSP-wAR), respectively. Therefore, our algorithms can effectively reduce the need of labelled samples for the target subject, which is of importance for the MI-based BCI application.

A multisymplectic Lie algebra variational integrator for flexible multibody dynamics on the special Euclidean group SE (3)

A multisymplectic Lie algebra variational integrator (MLAVI) for simulating the dynamics of a flexible multibody system is proposed. The flexible bodies are described by the geometrically exact beam elements established on the special Euclidean Lie group SE(3). A lemma is also provided to demonstrate that the used discrete compatibility equation can make temporal and spatial interpolation independent. Then, the reduced Lagrangian density on the Lie algebra is derived, and the discrete dynamics of motions are established by the discretization of the covariant variational principle. Furthermore, another novel closed-form equality relation lemma on the right tangent maps of the associated exponential map and Cayley map is originally proved, which greatly helps to simplify the Jacobi matrix of the proposed MLAVI in the Newton iteration process. The accuracy and efficiency of the proposed MLAVI are validated by six static and dynamic numerical examples. Numerical results show that the proposed MLAVI can preserve the system's symplectic structure, momentum, and energy for long-time simulation.

Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric

In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the vertical rescaled metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the vertical rescaled metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along with a map between Riemannian manifolds, the target manifold is equipped with a vertical rescaled metric on its tangent bundle. Next we also discuss the harmonicity of the composition of the projection map of the tangent bundle of a Riemannian manifold with a map from this manifold into another Riemannian manifold, the source manifold being whose tangent bundle is endowed with a vertical rescaled metric. Finally, we study the harmonicity of the tangent map also the harmonicity of the identity map of the tangent bundle.

A normality criterion for families of holomorphic mappings under a condition of uniform boundedness of their tangent mappings

By introducing a Second Main Theorem of Nevanlinna theory with a strong defect relation for holomorphic curves f in Pn(C) and hyperplanes H1,…,Hq satisfying f⁎,z=0 on ∪j=1qf−1(Hj), we give a normality criterion for families of holomorphic mappings of a domain D in Cm into the projective space Pn(C) under a condition of uniform boundedness of their tangent mappings on intersections of compact subsets of D and inverse image of a finite union of hyperplanes in Pn(C) (rather than on entire compact subsets).

Normal and Tangent Maps to Frontals

The notion of frontals in Euclidean space is introduced and the normal and tangent maps to frontals are studied for both geometrical and dynamical aspects of frontals. Moreover we observe that parallels of the tangent map to a frontal curve is right equivalent to the tangent map of a frontal curve under some natural conditions.

The structure of minimal surfaces in CAT(0) spaces

We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of “branch points”. On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps. As an application, we prove Fáry–Milnor’s theorem in the CAT(0) setting.

Motor-Imagery Classification Using Riemannian Geometry with Median Absolute Deviation

Motor imagery (MI) from human brain signals can diagnose or aid specific physical activities for rehabilitation, recreation, device control, and technology assistance. It is a dynamic state in learning and practicing movement tracking when a person mentally imitates physical activity. Recently, it has been determined that a brain–computer interface (BCI) can support this kind of neurological rehabilitation or mental practice of action. In this context, MI data have been captured via non-invasive electroencephalogram (EEGs), and EEG-based BCIs are expected to become clinically and recreationally ground-breaking technology. However, determining a set of efficient and relevant features for the classification step was a challenge. In this paper, we specifically focus on feature extraction, feature selection, and classification strategies based on MI-EEG data. In an MI-based BCI domain, covariance metrics can play important roles in extracting discriminatory features from EEG datasets. To explore efficient and discriminatory features for the enhancement of MI classification, we introduced a median absolute deviation (MAD) strategy that calculates the average sample covariance matrices (SCMs) to select optimal accurate reference metrics in a tangent space mapping (TSM)-based MI-EEG. Furthermore, all data from SCM were projected using TSM according to the reference matrix that represents the featured vector. To increase performance, we reduced the dimensions and selected an optimum number of features using principal component analysis (PCA) along with an analysis of variance (ANOVA) that could classify MI tasks. Then, the selected features were used to develop linear discriminant analysis (LDA) training for classification. The benchmark datasets were considered for the evaluation and the results show that it provides better accuracy than more sophisticated methods.

Estimating Lyapunov spectrum on shape-memory alloy oscillators considering cloned dynamics and tangent map methods

Mechanical devices built with shape-memory alloys become popular due to thermomechanical coupling associated with solid-state phase transformations. The main characteristics stemming from these transformations are the intrinsic dissipation and the change of thermomechanical properties as a consequence of the phase transformations. Shape-memory alloy oscillators have a nonlinear behavior that can reach responses with different natures such as periodic, quasiperiodic, chaotic, or even hyperchaotic. A proper identification of these behaviors requires the use of dynamical tools, and among them, Lyapunov exponents are one of the most relevant. This work deals with the calculation of the Lyapunov spectrum of shape-memory alloy oscillators. One- and two-degree-of-freedom systems are of concern. Two different approaches are compared: the classical algorithm that uses a tangent space approach; and the cloned dynamics algorithm that is a Jacobian-free approach. Results show that both methods have similar results, allowing the use of cloned dynamics as an interesting alternative procedure since it avoids Jacobian calculations.

A higher-order tangent map and a conjecture on the higher Nash blowup of curves

We introduce a higher-order version of the tangent map of a morphism and find a matrix representation. We then apply this matrix to solve a conjecture by Yasuda regarding the semigroup of the higher Nash blowup of formal curves. We first show that the conjecture is true for toric curves. We conclude by exhibiting a family of non-monomial curves where the conjecture fails.

NEW APPROACHES ON DUAL SPACE

In this paper, we give how to define the basic concepts of differential geometry on Dual space. For this, dual tangent vectors that have p as dual point of application are defined. Then, the dual analytic functions defined by Dimentberg are examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps are introduced.

Impact of the Reference Point for Epithelial Thickness Measurements.

To analyze the implications of different reference points on the read-out of epithelial thickness mapping. A simulation for changing the reference point from normal-to-the-surface tangent to parallel vertical sections quantifying its effect on the read-out of epithelial thickness mapping has been developed. The simulation includes a simple modeling of corneal epithelial profiles and allows the analytical quantification of the differences in the read-out from normal-to-the-surface tangent to parallel vertical sections epithelial thickness mapping. The difference in the read-out between parallel vertical sections and normal-to-the-surface tangent epithelial thickness mapping increases for steeper corneas, but it is not largely affected by asphericity. The difference increases for thicker epithelia. The reference point for determining the readout of epithelial thickness mapping should be taken into account when interpreting output. Using conventional epithelial thickness map readings (normal-to-the surface tangent) in transepithelial ablations (representing close to parallel vertical sections) may result in induced refractive errors that can be quantified using simple theoretical simulations, because the center-to-periphery progression of the corneal epithelial profile deviates from the progression of the ablated one. Adjustments for the epithelial thickness read-out or, alternatively, for the target sphere can be easily derived. [J Refract Surg. 2020;36(2):200-207.].