Manifolds of mappings on Cartesian products

Given smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\times \cdots \times M_n\rightarrow N$$ f : M 1 × ⋯ × M n → N which are $$C^\alpha $$ C α in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $$\le \alpha _j$$ ≤ α j in the jth variable for $$j\in \{1,\ldots , n\}$$ j ∈ { 1 , … , n } , in local charts. We show that $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) admits a canonical smooth manifold structure whenever each $$M_j$$ M j is compact and N admits a local addition. The case of non-compact domains is also considered.

L1-Norm Distance Discriminant Analysis with Multiple Adaptive Graphs and Sample Reconstruction

Linear discriminant analysis (LDA) is sensitive to noise and its performance may decline greatly. Recursive discriminative subspace learning method with an L1-norm distance constraint (RDSL) formulates LDA with the maximum margin criterion and becomes robust to noise by applying L1-norm and slack variables. However, the method only considers inter-class separation and intra-class compactness and ignores the intra-class manifold structure and the global structure of data. In this paper, we present L1-norm distance discriminant analysis with multiple adaptive graphs and sample reconstruction (L1-DDA) to deal with the problem. We use multiple adaptive graphs to preserve intra-class manifold structure and simultaneously apply the sample reconstruction technique to preserve the global structure of data. Moreover, we use an alternating iterative technique to obtain projection vectors. Experimental results on three real databases demonstrate that our method obtains better classification performance than RDSL.

Manipulator Kinematics and Dynamics On Differentiable Manifolds: Part I Kinematics

Using basic tools of Euclidean space topology and differential geometry, a regular manipulator configuration space comprised of input and output coordinates and conditions that assure existence of both forward and inverse kinematic mappings is shown to be a differentiable manifold, with valuable analytical and computational properties. For effective use of the manifold structure in support of manipulator analysis and control, four categories of manipulator are treated; (1) serial manipulators in which inputs explicitly determine outputs, (2) explicit parallel manipulators in which outputs explicitly determine inputs, (3) implicit manipulators in which explicit input-output relations are not possible, and (4) compound manipulators that require use of mechanism generalized coordinates to characterize input-output relations. Basic results of differential geometry show that differentiable manifolds in each category are naturally partitioned into maximal, disjoint, path connected submanifolds in which the manipulator is singularity free, hence programmable and controllable. Model manipulators in each of the four categories are analyzed to illustrate use of the manifold structure, employing only multivariable calculus and linear algebra. Computational methods for forward and inverse kinematics and construction of ordinary differential equations of manipulator dynamics on differentiable manifolds are presented in part II of the paper, in support of manipulator control.

Critical Kirchhoff-Choquard system involving the fractional $p$-Laplacian operator and singular nonlinearities

In this paper we study a class of critical fractional $p$-Laplacian Kirchhoff-Choquard systems with singular nonlinearities and two parameters $\lambda$ and $\mu$. By discussing the Nehari manifold structure and fibering maps analysis, we establish the existence of two positive solutions for above systems when $\lambda$ and $\mu$ satisfy suitable conditions.

Neural Excursions from Low-Dimensional Manifold Structure Explain Intersubject Variation in Human Motor Learning

Humans vary greatly in their motor learning abilities, yet little is known about the neural mechanisms that underlie this variability. Recent neuroimaging and electrophysiological studies demonstrate that large-scale neural dynamics inhabit a low-dimensional subspace or manifold, and that learning is constrained by this intrinsic manifold architecture. Here we asked, using functional MRI, whether subject-level differences in neural excursion from manifold structure can explain differences in learning across participants. We had subjects perform a sensorimotor adaptation task in the MRI scanner on two consecutive days, allowing us to assess their learning performance across days, as well as continuously measure brain activity. We find that the overall neural excursion from manifold activity in both cognitive and sensorimotor brain networks is associated with differences in subjects' patterns of learning and relearning across days. These findings suggest that off-manifold activity provides an index of the relative engagement of different neural systems during learning, and that intersubject differences in patterns of learning and relearning across days are related to reconfiguration processes in cognitive and sensorimotor networks during learning.

Dimensionality Reduction of Hyperspectral Image Based on Local Constrained Manifold Structure Collaborative Preserving Embedding

Graph learning is an effective dimensionality reduction (DR) manner to analyze the intrinsic properties of high dimensional data, it has been widely used in the fields of DR for hyperspectral image (HSI) data, but they ignore the collaborative relationship between sample pairs. In this paper, a novel supervised spectral DR method called local constrained manifold structure collaborative preserving embedding (LMSCPE) was proposed for HSI classification. At first, a novel local constrained collaborative representation (CR) model is designed based on the CR theory, which can obtain more effective collaborative coefficients to characterize the relationship between samples pairs. Then, an intraclass collaborative graph and an interclass collaborative graph are constructed to enhance the intraclass compactness and the interclass separability, and a local neighborhood graph is constructed to preserve the local neighborhood structure of HSI. Finally, an optimal objective function is designed to obtain a discriminant projection matrix, and the discriminative features of various land cover types can be obtained. LMSCPE can characterize the collaborative relationship between sample pairs and explore the intrinsic geometric structure in HSI. Experiments on three benchmark HSI data sets show that the proposed LMSCPE method is superior to the state-of-the-art DR methods for HSI classification.