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STEINBAUER, R., and M. KUNZINGER. "GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY." International Journal of Modern Physics A 17, no. 20 (2002): 2776. http://dx.doi.org/10.1142/s0217751x0201203x.
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The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed7. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.
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Zhang, Jianhai, Zhiyong Feng, Yong Su, and Meng Xing. "Bayesian Covariance Representation with Global Informative Prior for 3D Action Recognition." ACM Transactions on Multimedia Computing, Communications, and Applications 17, no. 4 (2021): 1–22. http://dx.doi.org/10.1145/3460235.
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For the merits of high-order statistics and Riemannian geometry, covariance matrix has become a generic feature representation for action recognition. An independent action can be represented by an empirical statistics over all of its pose samples. Two major problems of covariance include the following: (1) it is prone to be singular so that actions fail to be represented properly, and (2) it is short of global action/pose-aware information so that expressive and discriminative power is limited. In this article, we propose a novel Bayesian covariance representation by a prior regularization method to solve the preceding problems. Specifically, covariance is viewed as a parametric maximum likelihood estimate of Gaussian distribution over local poses from an independent action. Then, a Global Informative Prior (GIP) is generated over global poses with sufficient statistics to regularize covariance. In this way, (1) singularity is greatly relieved due to sufficient statistics, (2) global pose information of GIP makes Bayesian covariance theoretically equivalent to a saliency weighting covariance over global action poses so that discriminative characteristics of actions can be represented more clearly. Experimental results show that our Bayesian covariance with GIP efficiently improves the performance of action recognition. In some databases, it outperforms the state-of-the-art variant methods that are based on kernels, temporal-order structures, and saliency weighting attentions, among others.
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Uschmajew, André, and Bart Vandereycken. "On critical points of quadratic low-rank matrix optimization problems." IMA Journal of Numerical Analysis 40, no. 4 (2020): 2626–51. http://dx.doi.org/10.1093/imanum/drz061.
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Abstract The absence of spurious local minima in certain nonconvex low-rank matrix recovery problems has been of recent interest in computer science, machine learning and compressed sensing since it explains the convergence of some low-rank optimization methods to global optima. One such example is low-rank matrix sensing under restricted isometry properties (RIPs). It can be formulated as a minimization problem for a quadratic function on the Riemannian manifold of low-rank matrices, with a positive semidefinite Riemannian Hessian that acts almost like an identity on low-rank matrices. In this work new estimates for singular values of local minima for such problems are given, which lead to improved bounds on RIP constants to ensure absence of nonoptimal local minima and sufficiently negative curvature at all other critical points. A geometric viewpoint is taken, which is inspired by the fact that the Euclidean distance function to a rank-$k$ matrix possesses no critical points on the corresponding embedded submanifold of rank-$k$ matrices except for the single global minimum.
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BELLUCCI, STEFANO, and BHUPENDRA NATH TIWARI. "ON REAL INTRINSIC WALL CROSSINGS." International Journal of Modern Physics A 26, no. 30n31 (2011): 5171–209. http://dx.doi.org/10.1142/s0217751x11054917.
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We study moduli space stabilization of a class of BPS configurations from the perspective of the real intrinsic Riemannian geometry. Our analysis exhibits a set of implications towards the stability of the D-term potentials, defined for a set of Abelian scalar fields. In particular, we show that the nature of marginal and threshold walls of stabilities may be investigated by real geometric methods. Interestingly, we find that the leading order contributions may easily be accomplished by translations of the Fayet parameter. Specifically, we notice that the various possible linear, planar, hyperplanar and the entire moduli space stability may easily be reduced to certain polynomials in the Fayet parameter. For a set of finitely many real scalar fields, it may be further inferred that the intrinsic scalar curvature defines the global nature and range of vacuum correlations. Whereas, the underlying moduli space configuration corresponds to a noninteracting basis at the zeros of the scalar curvature, where the scalar fields become uncorrelated. The divergences of the scalar curvature provide possible phase structures, viz., wall of stability, phase transition, if any, in the chosen moduli configuration. The present analysis opens up a new avenue towards the stabilization of gauge and string moduli.
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Petersen, Peter. "Aspects of global Riemannian geometry." Bulletin of the American Mathematical Society 36, no. 03 (1999): 297–345. http://dx.doi.org/10.1090/s0273-0979-99-00787-9.
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BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.
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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explanation of the physical world, essentially depends on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, it still plays a fundamental role in non-Abelian gauge theories and in superstring theory, thanks to which a great variety of new mathematical structures has emerged. The scope of this presentation is to highlight the importance of these mathematical structures for theoretical physics. A very interesting hypothesis is that the global topological properties of the manifold's model of space–time play a major role in quantum field theory (QFT) and that, consequently, several physical quantum effects arise from the nonlocal changing metrical and topological structure of these manifold. Thus the unification of general relativity and quantum theory require some fundamental breakthrough in our understanding of the relationship between space–time and quantum process. In particular the superstring theories lead to the guess that the usual structure of space–time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections, and curvature in explaining the essential laws of nature. Kaluza–Klein theories and more remarkably superstring theory showed that space–time symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. This essentially means, in our view, that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain class of (presumably) nonsmooth manifolds. In the second part of this paper, we address the subject of topological quantum field theories (TQFTs), which constitute a remarkably important meeting ground for physicists and mathematicians. TQFTs can be used as a powerful tool to probe geometry and topology in low dimensions. Chern–Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of QFTs which can be exactly (nonperturbatively) and explicitly solved. Abelian Chern–Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link (i.e. the union of a finite number of disjoint knots). In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the well-known Jones polynomial. Powerful methods for complete analytical and nonperturbative computation of these knot and link invariants have been developed. From these invariants for unoriented and framed links in S3, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish–Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern–Simons field theory. Even perturbative analysis of Chern–Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. In Donaldson–Witten theory perturbative methods have proved their relations to Donaldson invariants. Nonperturbative methods have been applied after the work by Seiberg and Witten on N = 2 supersymmetric Yang–Mills theory. The outcome of this application is a totally unexpected relation between Donaldson invariants and a new set of topological invariants called Seiberg–Witten invariants. Not only in mathematics, Chern–Simons theories find important applications in three- and four-dimensional quantum gravity also. Work on TQFT suggests that a quantum gravity theory can be formulated in three-dimensional space–time. Attempts have been made in the last years to formulate a theory of quantum gravity in four-dimensional space–time using "spin networks" and "spin foams". More generally, the developments of TQFTs represent a sort of renaissance in the relation between geometry and physics. The most important (new) feature of present developments is that links are being established between quantum physics and topology. Maybe this link essentially rests on the fact that both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. One very interesting example is the super-symmetric quantum mechanics theory, which has a deep geometric meaning. In the Witten super-symmetric quantum mechanics theory, where the Hamiltonian is just the Hodge–Laplacian (whereas the quantum Hamiltonian corresponding to a classical particle moving on a Riemannian manifold is just the Laplace–Beltrami differential operator), differential forms are bosons or fermions depending on the parity of their degrees. Witten went to introduce a modified Hodge–Laplacian, depending on a real-valued function f. He was then able to derive the Morse theory (relating critical points of f to the Betti numbers of the manifold) by using the standard limiting procedures relating the quantum and classical theories. Super-symmetric QFTs essentially should be viewed as the differential geometry of certain infinite-dimensional manifolds, including the associated analysis (e.g. Hodge theory) and topology (e.g. Betti numbers). A further comment is that the QFTs of interest are inherently nonlinear, but the nonlinearities have a natural origin, e.g. coming from non-Abelian Lie groups. Moreover there is usually some scaling or coupling parameter in the theory which in the limit relates to the classical theory. Fundamental topological aspects of such a quantum theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit. This means that such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory. In the last decade much effort has been done to use these QFTs as a conceptual tool to suggest new mathematical results. In particular, they have led to spectacular progress in our understanding of geometry in low dimensions. It is most likely no accident that the usual QFTs can only be renormalized in (space–time) dimensions ≤4, and this is precisely the range in which difficult phenomena arise leading to deep and beautiful theories (e.g. the work of Thurston in three dimensions and Donaldson in four dimensions). It now seems clear that the way to investigate the subtleties of low-dimensional manifolds is to associate to them suitable infinite-dimensional manifolds (e.g. spaces of connections) and to study these by standard linear methods (homology, etc.). In other words we use QFT as a refined tool to study low-dimensional manifolds.
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González-Dávila, J. C., M. C. González-Dávila, and L. Vanhecke. "Invariant submanifolds in flow geometry." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, no. 3 (1997): 290–314. http://dx.doi.org/10.1017/s1446788700001026.
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AbstractWe begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symmetric space) and on the subclass of normal flow space forms. General results are derived and several examples are provided.
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Stavrinos, Panayiotis, and Sergiu I. Vacaru. "Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables." Universe 7, no. 4 (2021): 89. http://dx.doi.org/10.3390/universe7040089.
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We study new classes of generic off-diagonal and diagonal cosmological solutions for effective Einstein equations in modified gravity theories (MGTs), with modified dispersion relations (MDRs), and encoding possible violations of (local) Lorentz invariance (LIVs). Such MGTs are constructed for actions and Lagrange densities with two non-Riemannian volume forms (similar to two measure theories (TMTs)) and associated bimetric and/or biconnection geometric structures. For conventional nonholonomic 2 + 2 splitting, we can always describe such models in Finsler-like variables, which is important for elaborating geometric methods of constructing exact and parametric solutions. Examples of such Finsler two-measure formulations of general relativity (GR) and MGTs are considered for Lorentz manifolds and their (co) tangent bundles and abbreviated as FTMT. Generic off-diagonal metrics solving gravitational field equations in FTMTs are determined by generating functions, effective sources and integration constants, and characterized by nonholonomic frame torsion effects. By restricting the class of integration functions, we can extract torsionless and/or diagonal configurations and model emergent cosmological theories with square scalar curvature, R2, when the global Weyl-scale symmetry is broken via nonlinear dynamical interactions with nonholonomic constraints. In the physical Einstein–Finsler frame, the constructions involve: (i) nonlinear re-parametrization symmetries of the generating functions and effective sources; (ii) effective potentials for the scalar field with possible two flat regions, which allows for a unified description of locally anisotropic and/or isotropic early universe inflation related to acceleration cosmology and dark energy; (iii) there are “emergent universes” described by off-diagonal and diagonal solutions for certain nonholonomic phases and parametric cosmological evolution resulting in various inflationary phases; (iv) we can reproduce massive gravity effects in two-measure theories. Finally, we study a reconstructing procedure for reproducing off-diagonal FTMT and massive gravity cosmological models as effective Einstein gravity or Einstein–Finsler theories.
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Kapralov, Nikolai, Zhanna Nagornova, and Natalia Shemyakina. "Classification Methods for EEG Patterns of Imaginary Movements." Informatics and Automation 20, no. 1 (2021): 94–132. http://dx.doi.org/10.15622/ia.2021.20.1.4.
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The review focuses on the most promising methods for classifying EEG signals for non-invasive BCIs and theoretical approaches for the successful classification of EEG patterns. The paper provides an overview of articles using Riemannian geometry, deep learning methods and various options for preprocessing and "clustering" EEG signals, for example, common-spatial pattern (CSP). Among other approaches, pre-processing of EEG signals using CSP is often used, both offline and online. The combination of CSP, linear discriminant analysis, support vector machine and neural network (BPNN) made it possible to achieve 91% accuracy for binary classification with exoskeleton control as a feedback. There is very little work on the use of Riemannian geometry online and the best accuracy achieved so far for a binary classification problem is 69.3% in the work. At the same time, in offline testing, the average percentage of correct classification in the considered articles for approaches with CSP – 77.5 ± 5.8%, deep learning networks – 81.7 ± 4.7%, Riemannian geometry – 90.2 ± 6.6%. Due to nonlinear transformations, Riemannian geometry-based approaches and complex deep neural networks provide higher accuracy and better extract of useful information from raw EEG recordings rather than linear CSP transformation. However, in real-time setup, not only accuracy is important, but also a minimum time delay. Therefore, approaches using the CSP transformation and Riemannian geometry with a time delay of less than 500 ms may be in the future advantage.
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Woodward, L. M. "GLOBAL RIEMANNIAN GEOMETRY (Ellis Horwood Series: Mathematics and Its Applications)." Bulletin of the London Mathematical Society 17, no. 2 (1985): 194–96. http://dx.doi.org/10.1112/blms/17.2.194.
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Brozos-Vázquez, M., and P. Gilkey. "The global geometry of Riemannian manifolds with commuting curvature operators." Journal of Fixed Point Theory and Applications 1, no. 1 (2006): 87–96. http://dx.doi.org/10.1007/s11784-006-0001-6.
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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Mechanical Design 117, B (1995): 87–92. http://dx.doi.org/10.1115/1.2836475.
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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.
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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Vibration and Acoustics 117, B (1995): 87–92. http://dx.doi.org/10.1115/1.2838681.
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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.
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BRANDT, HOWARD E. "ASPECTS OF THE RIEMANNIAN GEOMETRY OF QUANTUM COMPUTATION." International Journal of Modern Physics B 26, no. 27n28 (2012): 1243004. http://dx.doi.org/10.1142/s0217979212430047.
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A review is given of some aspects of the Riemannian geometry of quantum computation in which the quantum evolution is represented in the tangent space manifold of the special unitary unimodular group SU(2n) for n qubits. The Riemannian right-invariant metric, connection, curvature, geodesic equation for minimal complexity quantum circuits, Jacobi equation and the lifted Jacobi equation for varying penalty parameter are reviewed. Sharpened tools for calculating the geodesic derivative are presented. The geodesic derivative may facilitate the numerical investigation of conjugate points and the global characteristics of geodesic paths in the group manifold, the determination of optimal quantum circuits for carrying out a quantum computation, and the determination of the complexity of particular quantum algorithms.
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Mikeš, Josef, Vladimir Rovenski, Sergey Stepanov, and Irina Tsyganok. "Application of the Generalized Bochner Technique to the Study of Conformally Flat Riemannian Manifolds." Mathematics 9, no. 9 (2021): 927. http://dx.doi.org/10.3390/math9090927.
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In this article, we discuss the global aspects of the geometry of locally conformally flat (complete and compact) Riemannian manifolds. In particular, the article reviews and improves some results (e.g., the conditions of compactness and degeneration into spherical or flat space forms) on the geometry “in the large" of locally conformally flat Riemannian manifolds. The results presented here were obtained using the generalized and classical Bochner technique, as well as the Ricci flow.
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Rovenski, Vladimir, Sergey Stepanov, and Irina Tsyganok. "A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings." Axioms 10, no. 4 (2021): 333. http://dx.doi.org/10.3390/axioms10040333.
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This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.
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Dunajski, Maciej. "Null Kähler Geometry and Isomonodromic Deformations." Communications in Mathematical Physics 391, no. 1 (2021): 77–105. http://dx.doi.org/10.1007/s00220-021-04270-0.
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AbstractWe construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear on the space of Bridgeland stability conditions on a Calabi–Yau threefold. Using twistor methods we show that, in dimension four—where there is a connection with dispersionless integrability—the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs.
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Bolsinov, A. V., V. S. Matveev, and A. T. Fomenko. "Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry." Sbornik: Mathematics 189, no. 10 (1998): 1441–66. http://dx.doi.org/10.1070/sm1998v189n10abeh000346.
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Marotta, Vincenzo Emilio, and Richard J. Szabo. "Algebroids, AKSZ Constructions and Doubled Geometry." Complex Manifolds 8, no. 1 (2021): 354–402. http://dx.doi.org/10.1515/coma-2020-0125.
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Abstract We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the L ∞-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature.
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Majidov, Ikhtiyor, and Taegkeun Whangbo. "Efficient Classification of Motor Imagery Electroencephalography Signals Using Deep Learning Methods." Sensors 19, no. 7 (2019): 1736. http://dx.doi.org/10.3390/s19071736.
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Single-trial motor imagery classification is a crucial aspect of brain–computer applications. Therefore, it is necessary to extract and discriminate signal features involving motor imagery movements. Riemannian geometry-based feature extraction methods are effective when designing these types of motor-imagery-based brain–computer interface applications. In the field of information theory, Riemannian geometry is mainly used with covariance matrices. Accordingly, investigations showed that if the method is used after the execution of the filterbank approach, the covariance matrix preserves the frequency and spatial information of the signal. Deep-learning methods are superior when the data availability is abundant and while there is a large number of features. The purpose of this study is to a) show how to use a single deep-learning-based classifier in conjunction with BCI (brain–computer interface) applications with the CSP (common spatial features) and the Riemannian geometry feature extraction methods in BCI applications and to b) describe one of the wrapper feature-selection algorithms, referred to as the particle swarm optimization, in combination with a decision tree algorithm. In this work, the CSP method was used for a multiclass case by using only one classifier. Additionally, a combination of power spectrum density features with covariance matrices mapped onto the tangent space of a Riemannian manifold was used. Furthermore, the particle swarm optimization method was implied to ease the training by penalizing bad features, and the moving windows method was used for augmentation. After empirical study, the convolutional neural network was adopted to classify the pre-processed data. Our proposed method improved the classification accuracy for several subjects that comprised the well-known BCI competition IV 2a dataset.
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Sakai, Hiroyuki, and Hideaki Iiduka. "Hybrid Riemannian conjugate gradient methods with global convergence properties." Computational Optimization and Applications 77, no. 3 (2020): 811–30. http://dx.doi.org/10.1007/s10589-020-00224-9.
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Bracken, Paul. "Integrable Equations and Their Evolutions Based on Intrinsic Geometry of Riemann Spaces." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–16. http://dx.doi.org/10.1155/2009/210304.
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The intrinsic geometry of surfaces and Riemannian spaces will be investigated. It is shown that many nonlinear partial differential equations with physical applications and soliton solutions can be determined from the components of the relevant metric for the space. The manifolds of interest are surfaces and higher-dimensional Riemannian spaces. Methods for specifying integrable evolutions of surfaces by means of these equations will also be presented.
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Makarenko N.G., ChoYong-beom, and Esenaliev A. B. "RIEMANNIAN METRIC FOR TEXTURE RECOGNITION." PHYSICO-MATHEMATICAL SERIES, no. 6 (December 15, 2018): 23–27. http://dx.doi.org/10.32014/2018.2518-1726.13.
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The article discusses the recognition of textures on digital images by methods of computational topology and Riemannian geometry. Topological properties of patterns are represented by segments (barcodes) obtained by filtering by the level of photometric measure. Beginning of barcode encodes level at which topological property appears (connected component and/or “hole”), and its end - level at which the property disappears. Barcodes are conveniently parameterized by coordinates of their ends in rectangular coordinate system “birth” and “death” of topological property. Such representation in form of a cloud of points on plane is called a persistence diagram (PD). In the article show that texture class recognition results are significantly better compared to other vectorization methods of PD.
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Barilari, Davide, Ugo Boscain, and Daniele Cannarsa. "On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 9. http://dx.doi.org/10.1051/cocv/2021104.
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Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
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Dipierro, Serena, Zu Gao, and Enrico Valdinoci. "Global gradient estimates for nonlinear parabolic operators." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 21. http://dx.doi.org/10.1051/cocv/2021016.
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We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.
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Noakes, Lyle. "A Global algorithm for geodesics." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 65, no. 1 (1998): 37–50. http://dx.doi.org/10.1017/s1446788700039380.
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AbstractThe problem of finding a george joinning given points x0, x1in a connected complete Riemannian manifold requires much more effort than determining a geodesic from initial data. Boundary value problems of this type are sometimes solved using shooting methods, which work best when good initial guesses are available expectually when x0, x1are nearby. Galerkin methods have their drawbacks too. The situation is much more difficult with general variational problems, which is why we focus on the Riemannian case.Our global algorithm is very simple to implement, and works well in practice, with no need for an initial guess. The proof of convergence to elementary and very carefully stated. with a view to possible generalizations latter on we have in mind the much larger class of interesting problems arising in optimal control especially from mechanical engineering.
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Aazami, Amir Babak, and Charles M. Melby-Thompson. "On the principal Ricci curvatures of a Riemannian 3-manifold." Advances in Geometry 19, no. 2 (2019): 251–62. http://dx.doi.org/10.1515/advgeom-2018-0020.
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Abstract We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.
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Ahdid, Rachid, Khaddouj Taifi, Mohamed Fakir, Said Safi, and Bouzid Manaut. "Two-Dimensional Face Recognition Methods Comparing with a Riemannian Analysis of Iso-Geodesic Curves." Journal of Electronic Commerce in Organizations 13, no. 3 (2015): 15–35. http://dx.doi.org/10.4018/jeco.2015070102.
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In this paper, the authors performed a comparative study of two-dimensional face recognition methods. This study was based on existing methods (PCA, LDA, 2DPCA, 2DLDA, SVM...) and 2D face surface analysis using a Riemannian geometry. The last system uses the representation of the image at gray level as a 2D surface in a 3D space where the third coordinate represent the intensity values of the pixels. The authors' approach is to represent the human face as a collection of closed curves, called facial curves, and apply tools from the analysis of the shape of curves using the Riemannian geometry. Their application has been tested on two well-known databases of face images ORL and YaleB. ORL data base was used to evaluate the performance of their method when the pose and sample size are varied, and the database YaleB was used to examine the performance of the system when the facial expressions and lighting are varied.
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Noakes, Lyle, and Tomasz Popiel. "Geometry for robot path planning." Robotica 25, no. 6 (2007): 691–701. http://dx.doi.org/10.1017/s0263574707003669.
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SUMMARYThere have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out.Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical constructions that imitate well-known constructions from classical approximation theory. Part III reviews a class of methods where classicalcriteriaare extended to curves in Riemannian manifolds, including several recent mathematical results that have not yet found their way into the literature.
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van der Schaft, Arjan, and Bernhard Maschke. "Geometry of Thermodynamic Processes." Entropy 20, no. 12 (2018): 925. http://dx.doi.org/10.3390/e20120925.
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Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.
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Nesterov, Yu E., and M. J. Todd. "On the Riemannian Geometry Defined by Self-Concordant Barriers and Interior-Point Methods." Foundations of Computational Mathematics 2, no. 4 (2002): 333–61. http://dx.doi.org/10.1007/s102080010032.
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Williams, Simon, Arthur George Suvorov, Zengfu Wang, and Bill Moran. "The Information Geometry of Sensor Configuration." Sensors 21, no. 16 (2021): 5265. http://dx.doi.org/10.3390/s21165265.
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In problems of parameter estimation from sensor data, the Fisher information provides a measure of the performance of the sensor; effectively, in an infinitesimal sense, how much information about the parameters can be obtained from the measurements. From the geometric viewpoint, it is a Riemannian metric on the manifold of parameters of the observed system. In this paper, we consider the case of parameterized sensors and answer the question, “How best to reconfigure a sensor (vary the parameters of the sensor) to optimize the information collected?” A change in the sensor parameters results in a corresponding change to the metric. We show that the change in information due to reconfiguration exactly corresponds to the natural metric on the infinite-dimensional space of Riemannian metrics on the parameter manifold, restricted to finite-dimensional sub-manifold determined by the sensor parameters. The distance measure on this configuration manifold is shown to provide optimal, dynamic sensor reconfiguration based on an information criterion. Geodesics on the configuration manifold are shown to optimize the information gain but only if the change is made at a certain rate. An example of configuring two bearings-only sensors to optimally locate a target is developed in detail to illustrate the mathematical machinery, with Fast Marching methods employed to efficiently calculate the geodesics and illustrate the practicality of using this approach.
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Angers, Jean-Francois, and Peter T. Kim. "Symmetry and Bayesian Function Estimation1." Calcutta Statistical Association Bulletin 56, no. 1-4 (2005): 57–80. http://dx.doi.org/10.1177/0008068320050504.
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Summary This paper develops Bayesian function estimation on compact Riemannian manifolds. The approach is to combine Bayesian methods along with aspects of spectral geometry associated with the Laplace-Beltrami operator on Riemannian manifolds. Although frequentist nonparametric function estimation in Euclidean space abound, to date, no attempt has been made with respect to Bayesian function estimation on a general Riemannian manifold. The Bayesian approach to function estimation is very natural for manifolds because one can elicit very specific prior information on the possible symmetries in question . One can then establish Bayes estimators that possess built in symmetries. A detailed analysis for the 2–sphere is provided.
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Gzyl, H., and F. Nielsen. "Geometry of the probability simplex and its connection to the maximum entropy method." Journal of Applied Mathematics, Statistics and Informatics 16, no. 1 (2020): 25–35. http://dx.doi.org/10.2478/jamsi-2020-0003.
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AbstractThe use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities.For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.
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Miah, Abu Saleh Musa, Md Abdur Rahim, and Jungpil Shin. "Motor-Imagery Classification Using Riemannian Geometry with Median Absolute Deviation." Electronics 9, no. 10 (2020): 1584. http://dx.doi.org/10.3390/electronics9101584.
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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.
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Karimova, Lyailya, Alexey Terekhov, Nikolai Makarenko, and Andrey Rybintsev. "Methods of computational topology and discrete Riemannian geometry for the analysis of arid territories." Cogent Engineering 7, no. 1 (2020): 1808340. http://dx.doi.org/10.1080/23311916.2020.1808340.
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HEBEY, EMMANUEL, FRÉDÉRIC ROBERT, and YULIANG WEN. "COMPACTNESS AND GLOBAL ESTIMATES FOR A FOURTH ORDER EQUATION OF CRITICAL SOBOLEV GROWTH ARISING FROM CONFORMAL GEOMETRY." Communications in Contemporary Mathematics 08, no. 01 (2006): 9–65. http://dx.doi.org/10.1142/s0219199706002027.
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Given (M,g) a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for fourth order critical equations like Pgu = u2♯-1, where [Formula: see text] is a Paneitz–Branson operator with constant coefficients b and c, u is required to be positive, and [Formula: see text] is critical from the Sobolev viewpoint. We prove that such equations are compact on locally conformally flat manifolds, unless b lies in some closed interval associated to the spectrum of the smooth symmetric (2,0)-tensor field involved in the definition of the geometric Paneitz–Branson operator.
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Liu, Xi, Zhengming Ma, and Guo Niu. "Mixed Region Covariance Discriminative Learning for Image Classification on Riemannian Manifolds." Mathematical Problems in Engineering 2019 (February 28, 2019): 1–11. http://dx.doi.org/10.1155/2019/1261398.
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Covariance matrices, known as symmetric positive definite (SPD) matrices, are usually regarded as points lying on Riemannian manifolds. We describe a new covariance descriptor, which could improve the discriminative learning ability of region covariance descriptor by taking into account the mean of feature vectors. Due to the specific geometry of Riemannian manifolds, classical learning methods cannot be directly used on it. In this paper, we propose a subspace projection framework for the classification task on Riemannian manifolds and give the mathematical derivation for it. It is different from the common technique used for Riemannian manifolds, which is to explicitly project the points from a Riemannian manifold onto Euclidean space based upon a linear hypothesis. Under the proposed framework, we define a Gaussian Radial Basis Function- (RBF-) based kernel with a Log-Euclidean Riemannian Metric (LERM) to embed a Riemannian manifold into a high-dimensional Reproducing Kernel Hilbert Space (RKHS) and then project it onto a subspace of the RKHS. Finally, a variant of Linear Discriminative Analyze (LDA) is recast onto the subspace. Experiments demonstrate the considerable effectiveness of the mixed region covariance descriptor and the proposed method.
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Saifutdinova, Elizaveta, Marco Congedo, Daniela Dudysova, Lenka Lhotska, Jana Koprivova, and Vaclav Gerla. "An Unsupervised Multichannel Artifact Detection Method for Sleep EEG Based on Riemannian Geometry." Sensors 19, no. 3 (2019): 602. http://dx.doi.org/10.3390/s19030602.
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In biomedical signal processing, we often face the problem of artifacts that distort the original signals. This concerns also sleep recordings, such as EEG. Artifacts may severely affect or even make impossible visual inspection, as well as automatic processing. Many proposed methods concentrate on certain artifact types. Therefore, artifact-free data are often obtained after sequential application of different methods. Moreover, single-channel approaches must be applied to all channels alternately. The aim of this study is to develop a multichannel artifact detection method for multichannel sleep EEG capable of rejecting different artifact types at once. The inspiration for the study is gained from recent advances in the field of Riemannian geometry. The method we propose is tested on real datasets. The performance of the proposed method is measured by comparing detection results with the expert labeling as a reference and evaluated against a simpler method based on Riemannian geometry that has previously been proposed, as well as against the state-of-the-art method FASTER. The obtained results prove the effectiveness of the proposed method.
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Li, Jiao-fen, Ya-qiong Wen, Xue-lin Zhou, and Kai Wang. "Effective Algorithms for Solving Trace Minimization Problem in Multivariate Statistics." Mathematical Problems in Engineering 2020 (August 10, 2020): 1–24. http://dx.doi.org/10.1155/2020/3054764.
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This paper develops two novel and fast Riemannian second-order approaches for solving a class of matrix trace minimization problems with orthogonality constraints, which is widely applied in multivariate statistical analysis. The existing majorization method is guaranteed to converge but its convergence rate is at best linear. A hybrid Riemannian Newton-type algorithm with both global and quadratic convergence is proposed firstly. A Riemannian trust-region method based on the proposed Newton method is further provided. Some numerical tests and application to the least squares fitting of the DEDICOM model and the orthonormal INDSCAL model are given to demonstrate the efficiency of the proposed methods. Comparisons with some latest Riemannian gradient-type methods and some existing Riemannian second-order algorithms in the MATLAB toolbox Manopt are also presented.
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DE MICHELI, E., G. MONTI BRAGADIN, and G. A. VIANO. "RIEMANNIAN GEOMETRICAL OPTICS: SURFACE WAVES IN DIFFRACTIVE SCATTERING." Reviews in Mathematical Physics 12, no. 06 (2000): 849–72. http://dx.doi.org/10.1142/s0129055x00000332.
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The geometrical diffraction theory, in the sense of Keller, is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the "diffracted rays", which follow from the non-uniqueness of the Cauchy problem for geodesics in a Riemannian manifold with boundary. Then, the axial caustic is here regarded as a conjugate locus, in the sense of the Riemannian geometry, and the results of the Morse theory can be applied. The methods of the algebraic topology allow us to introduce the homotopy classes of diffracted rays. These geometrical results are related to the asymptotic approximations of a solution of a boundary value problem for the reduced wave equation. In particular, we connect the results of the Morse theory to the Maslov construction, which is used to obtain the uniformization of the asymptotic approximations. Then, the border of the diffracting body is the envelope of the diffracted rays and, instead of the standard saddle point method, use is made of the procedure of Chester, Friedman and Ursell to derive the damping factors associated with the rays which propagate along the boundary. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated.
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Tian, Jian-Sheng, Wei Wang, Fei Xue, and Pei-Yong Cong. "Boundary Stabilization of the Wave Equation with Time-Varying and Nonlinear Feedback." Mathematical Problems in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/176583.
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We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying and nonlinear term. By the Riemannian geometry methods and a suitable assumption of nonlinearity and the time-varying term, we obtain the uniform decay of the energy of the system.
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Gong, Bei, and Xiaopeng Zhao. "Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/728760.
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We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.
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Li, Hao, Changsong Lin, Shupeng Wang, and Yanmei Zhang. "Stabilization of the Wave Equation with Boundary Time-Varying Delay." Advances in Mathematical Physics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/735341.
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We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.
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Frank, Philipp, Reimar Leike, and Torsten A. Enßlin. "Geometric Variational Inference." Entropy 23, no. 7 (2021): 853. http://dx.doi.org/10.3390/e23070853.
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Efficiently accessing the information contained in non-linear and high dimensional probability distributions remains a core challenge in modern statistics. Traditionally, estimators that go beyond point estimates are either categorized as Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques. While MCMC methods that utilize the geometric properties of continuous probability distributions to increase their efficiency have been proposed, VI methods rarely use the geometry. This work aims to fill this gap and proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. It is used to construct a coordinate transformation that relates the Riemannian manifold associated with the metric to Euclidean space. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation by a normal distribution. Furthermore, the algorithmic structure allows for an efficient implementation of geoVI which is demonstrated on multiple examples, ranging from low-dimensional illustrative ones to non-linear, hierarchical Bayesian inverse problems in thousands of dimensions.
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GUDMUNDSSON, SIGMUNDUR, and MARTIN SVENSSON. "Harmonic morphisms from solvable Lie groups." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 2 (2009): 389–408. http://dx.doi.org/10.1017/s0305004109002564.
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AbstractIn this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.
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Knapp, Johanna, and Maximilian Kreuzer. "Toric Methods in F-Theory Model Building." Advances in High Energy Physics 2011 (2011): 1–18. http://dx.doi.org/10.1155/2011/513436.
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We discuss recent constructions of global F-theory GUT models and explain how to make use of toric geometry to do calculations within this framework. After introducing the basic properties of global F-theory GUTs, we give a self-contained review of toric geometry and introduce all the tools that are necessary to construct and analyze global F-theory models. We will explain how to systematically obtain a large class of compact Calabi-Yau fourfolds which can support F-theory GUTs by using the software package PALP.
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Liu, Hong, Jie Li, Yongjian Wu, and Rongrong Ji. "Learning Neural Bag-of-Matrix-Summarization with Riemannian Network." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 8746–53. http://dx.doi.org/10.1609/aaai.v33i01.33018746.
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Symmetric positive defined (SPD) matrix has attracted increasing research focus in image/video analysis, which merits in capturing the Riemannian geometry in its structured 2D feature representation. However, computation in the vector space on SPD matrices cannot capture the geometric properties, which corrupts the classification performance. To this end, Riemannian based deep network has become a promising solution for SPD matrix classification, because of its excellence in performing non-linear learning over SPD matrix. Besides, Riemannian metric learning typically adopts a kNN classifier that cannot be extended to large-scale datasets, which limits its application in many time-efficient scenarios. In this paper, we propose a Bag-of-Matrix-Summarization (BoMS) method to be combined with Riemannian network, which handles the above issues towards highly efficient and scalable SPD feature representation. Our key innovation lies in the idea of summarizing data in a Riemannian geometric space instead of the vector space. First, the whole training set is compressed with a small number of matrix features to ensure high scalability. Second, given such a compressed set, a constant-length vector representation is extracted by efficiently measuring the distribution variations between the summarized data and the latent feature of the Riemannian network. Finally, the proposed BoMS descriptor is integrated into the Riemannian network, upon which the whole framework is end-to-end trained via matrix back-propagation. Experiments on four different classification tasks demonstrate the superior performance of the proposed method over the state-of-the-art methods.
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Khouja, Rima, Houssam Khalil, and Bernard Mourrain. "Riemannian Newton optimization methods for the symmetric tensor approximation problem." Linear Algebra and its Applications 637 (March 2022): 175–211. http://dx.doi.org/10.1016/j.laa.2021.12.008.
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Zhang, Yachao, Xuan Lai, Yuan Xie, Yanyun Qu, and Cuihua Li. "Geometry-Aware Discriminative Dictionary Learning for PolSAR Image Classification." Remote Sensing 13, no. 6 (2021): 1218. http://dx.doi.org/10.3390/rs13061218.
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In this paper, we propose a new discriminative dictionary learning method based on Riemann geometric perception for polarimetric synthetic aperture radar (PolSAR) image classification. We made an optimization model for geometry-aware discrimination dictionary learning in which the dictionary learning (GADDL) is generalized from Euclidian space to Riemannian manifolds, and dictionary atoms are composed of manifold data. An efficient optimization algorithm based on an alternating direction multiplier method was developed to solve the model. Experiments were implemented on three public datasets: Flevoland-1989, San Francisco and Flevoland-1991. The experimental results show that the proposed method learned a discriminative dictionary with accuracies better those of comparative methods. The convergence of the model and the robustness of the initial dictionary were also verified through experiments.