Manifolds

We investigate the use of the Riemannianoptimization method over the flag manifold in subspace ICA problems such as in-dependent subspace analysis (ISA) and complex ICA. In the ISA experiment, we use the Riemannian approach over the flag manifold together ...

Motivated by an application in computational geometry, we consider a novel variant of the problem of efficiently maintaining a forest of dynamic rooted trees. This variant includes an operation that merges two tree paths. In contrast to the standard problem, in which a single operation can only add or delete one arc, one merge can add and delete up to a linear number of arcs. In spite of this, we develop three different methods that need only polylogarithmic time per operation. The first method extends a solution of Farach and Thorup [1998] for the special case of paths. Each merge takes O (log 2 n ) amortized time on an n -node forest and each standard dynamic tree operation takes O (log n ) time; the latter bound is amortized, worst case, or randomized depending on the underlying data structure. For the special case that occurs in the motivating application, in which arbitrary arc deletions (cuts) do not occur, we give a method that takes O (log n ) time per operation, including m...

Musical genres are categorical labels characterizing pieces of music. Automatically classifying music into genres is gaining importance as a way to structure and organize the increasingly large numbers of music files available digitally on the web. In this work such a classification algorithm is developed and examined. The algorithm uses a vector of features based on the timbral texture of the music, and maps it into a new Euclidean space, by a non-linear method called “Diffusion Maps”, before the classification stage itself. This method allows dimensionality reduction while preserving and emphasizing the distinction between different genres. The proposed classifier classifies accurately 97% when classifying 2 musical genres, and 52% when classifying 10 musical genres. This is compared to an accuracy of 88% and 28% respectively, when classifying without the proposed mapping.

We describe an object oriented MATLAB toolbox for solving di erential equations on manifolds. The software re ects recent development within the area of geometric integration. Through the use of elements from di erential geometry, in particular Lie groups and homogeneous spaces, coordinate free formulations of numerical integrators are developed. The strict mathematical de nitions and results are well suited for implementation in an object oriented language, and, due to its simplicity, the authors have chosen MATLAB as the working environment. The basic ideas of Di Man are presented, along with particular examples that illustrate the working of and the theory behind the software package. AMS Subject Classi cation: 65L06, 34A50

It is known since the early Sixties that descriptive equa-tions of electrical circuits belong to the class of differential equations on differentiable manifolds. This result is related to the celebrated paper of Moser and Brayton [1] in 1964 where their equations for the description of ...

... Examples of such application areas arise in biological systems, image processing, medical imaging, mathematical ... of the surface PDE, and involves only the standard Cartesian differential operators. ... algorithms of Strain [25] or the public domain closest point transform of Mauch ...

The question, which shapes can be digitized without any change in the fundamental geometric and topological properties is of great importance for the reliability of image analysis algorithms, but is nevertheless unanswered for a lot of digitization schemes. While r-regularity is a sufficient criterion for shapes to be reconstructed correctly by using any regular or irregular sampling grid of certain density, necessary criteria are up to now unknown. The author proves such a necessary criterion: If you choose some sampling grid and you want a shape to be digitized correctly with any alignment of this grid, then the shape has to be a bordered 2D-manifold, i.e. its boundary has to have no junctions. This implies that any correct digitization is an extended well-composed set and thus the well known problems of defining connectivity in 2D are always due to wrong sampling or improper original shapes. This is of great importance, since extended well-composed sets have many nice topological...

A good distance metric is crucial for unsupervised learning from high-dimensional data. To learn a metric without any constraint or class label information, most unsupervised metric learning algorithms appeal to projecting observed data onto a low-dimensional manifold, where geometric relationships such as local or global pairwise distances are preserved. However, the projection may not necessarily improve the separability of the data, which is the desirable outcome of clustering. In this paper, we propose a novel ...

Let E be an ample vector bundle of rank r≥ 2 on a smooth complex projective variety X of dimension n. The aim of this paper is to describe the structure of pairs (X,E) as above whose adjoint bundles K_X+E are not nef for r=n-3. Furthermore, we give some immediate consequences of this result in adjunction theory.

A model-based control scheme is designed to regulate the cylinder air charge of a camless multicylinder engine for unthrottled operation. The controller consists of a feedforward and an adaptive feedback scheme based on a control-oriented model of the breathing process of an engine equipped with electro-hydraulic springless valvetrain. The nonlinear control scheme is designed to achieve cylinder-to-cylinder balancing, fast cycle-to-cycle response, and minimization of pumping losses. The algorithm uses conventional sensor measurements of intake manifold pressure and mass air flow to the intake manifold, and intake valve duration measurement. Closed-loop simulation results are shown for a four-cylinder engine. [S0022-0434(00)03001-X]

Let M be a smooth compact oriented Riemannian manifold, and let Δ M be the Laplace–Beltrami operator on M. Say $${0 \neq f \in \mathcal{S}(\mathbb {R}^+)}$$ , and that f (0)  =  0. For t  >  0, let K t (x, y) denote the kernel of f (t 2 Δ M ). We show that K t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t 2Δ) on $${\mathbb {R}^n}$$ . We define continuous $${\mathcal {S}}$$-wavelets on M, in such a manner that K t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous $${\mathcal {S}}$$-wavelets on M are analogous to continuous wavelets on $${\mathbb {R}^n}$$ in $${\mathcal {S}}$$ ($${\mathbb {R}^n}$$). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous $${\mathcal {S}}$$-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus $${\mathbb T^2}$$ or the sph...

In this paper, we present an interactive, visual design approach for the dimensional synthesis of planar 6R single-loop closed chains for a given rational motion using constraint manifold modification. This approach is implemented in an interactive software tool that provides mechanism designers with an intuitive way to determine the parameters of planar mechanisms, and in the process equips them with an understanding of the design process. The theoretical foundation of this work is based on representing planar displacements with planar quaternions, which can be seen as points in a special higher dimensional projective space (called the image space), and on formulating the kinematic constraints of closed chains as algebraic surfaces in the image space. Kinematic constraints under consideration limit the positions and orientation of the coupler in its workspace. In this way, a given motion of a mechanism in the Cartesian space maps to a curve in the image space that is limited to sta...

Let M be a smooth compact oriented Riemannian manifold, and let Δ_ M be the Laplace-Beltrami operator on M. Say 0 ≠ f ∈S(^+), and that f(0) = 0. For t > 0, let K_t(x,y) denote the kernel of f(t^2 Δ_ M). We show that K_t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f(t^2Δ) on ^n. We define continuous S-wavelets on M, in such a manner that K_t(x,y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on ^n in S(^n). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus ^2 or the sphere S^2, and f(s)=se^-s (the "Mexican hat" situation), we obtain two explicit approximate formulas for K_t, one to be used when t is large, and one to be used when t is s...

In this work we investigate the existence and the uniqueness of solution for a nonlinear differential equation of parabolic type on the lateral boundary Σ of a cylinder Q, cf. (1). An important part of our study is to transform this initial value problem into another one whose differential operator equation is of the type ut + a Γ udx Au − ΔΓu + u2k+1 = f on Σ, cf. (9), where k is a positive integer. The operatorA acts in Sobolev spaces on Γ, boundary of Ω. The initial value problem (9) will be studied in Section 4. Thus, we obtain the existence and the uniqueness of weak solution for (9).

This paper presents a summary of space mapping optimization application for the design of microwave devices used in satellite systems. Output multiplexers and switches are essential components of satellite systems. They are considered large scale microwave devices since they have a large number of parameters and they are very time consuming to simulate by electromagnetic simulators. Finite element EM based simulators and space-mapping optimization are combined to produce an accurate design for T-switches and ...

Let M be a smooth compact oriented Riemannian manifold, and let Δ_ M be the Laplace-Beltrami operator on M. Say 0 ≠ f ∈S(^+), and that f(0) = 0. For t > 0, let K_t(x,y) denote the kernel of f(t^2 Δ_ M). We show that K_t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f(t^2Δ) on ^n. We define continuous S-wavelets on M, in such a manner that K_t(x,y) satisfies this definition, because of its localization near the diagonal. Continuous S-wavelets on M are analogous to continuous wavelets on ^n in S(^n). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous S-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus ^2 or the sphere S^2, and f(s)=se^-s (the "Mexican hat" situation), we obtain two explicit approximate formulas for K_t, one to be used when t is large, and one to be used when t is s...

ABSTRACT In this paper, we propose a method to match multitemporal sequences of hyperspectral images using Hidden Markov Random Fields. Based on the matching of the data manifold, the algorithm matches the reflectance spectra of the classes, thus allowing the reuse of labeled examples acquired on one image to classify the other. This allows valorization of spectra collected in situ to other acquisitions than the one they were acquired for, without user supervision, prior knowledge of the class reflectance in the new domain or global information about atmospheric conditions.