Principal Curves on Riemannian Manifolds
DEFF Research Database (Denmark)
Hauberg, Søren
2015-01-01
Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only...... in Euclidean domains. We consider extensions of Principal Component Analysis (PCA) to Riemannian manifolds. Classic Riemannian approaches seek a geodesic curve passing through the mean that optimize a criteria of interest. The requirements that the solution both is geodesic and must pass through the mean tend...
Natural Connections on Riemannian Product Manifolds
Gribacheva, Dobrinka
2011-01-01
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Minimal surfaces in Riemannian manifolds
International Nuclear Information System (INIS)
Ji Min; Wang Guangyin
1990-10-01
A multiple solution to the Plateau problem in a Riemannian manifold is established. In S n , the existence of two solutions to this problem is obtained. The Morse-Tompkins-Shiffman theorem is extended to the case when the ambient space admits no minimal sphere. (author). 20 refs
Harmonic Riemannian Maps on Locally Conformal Kaehler Manifolds
Indian Academy of Sciences (India)
We study harmonic Riemannian maps on locally conformal Kaehler manifolds ( l c K manifolds). We show that if a Riemannian holomorphic map between l c K manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we ...
Minimal Webs in Riemannian Manifolds
DEFF Research Database (Denmark)
Markvorsen, Steen
2008-01-01
For a given combinatorial graph $G$ a {\\it geometrization} $(G, g)$ of the graph is obtained by considering each edge of the graph as a $1-$dimensional manifold with an associated metric $g$. In this paper we are concerned with {\\it minimal isometric immersions} of geometrized graphs $(G, g)$ int...
Slant Riemannian maps from almost hermitian manifolds | Sahin ...
African Journals Online (AJOL)
As a generalization of holomorphic submersions, anti-invariant submersions and slant submersions, we introduce slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples, obtain the existence conditions of slant Riemannian maps and investigate harmonicity of such maps.
Harmonic Riemannian maps on locally conformal Kaehler manifolds
Indian Academy of Sciences (India)
Abstract. We study harmonic Riemannian maps on locally conformal Kaehler mani- folds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, ...
Convex functions and optimization methods on Riemannian manifolds
Udrişte, Constantin
1994-01-01
This unique monograph discusses the interaction between Riemannian geometry, convex programming, numerical analysis, dynamical systems and mathematical modelling. The book is the first account of the development of this subject as it emerged at the beginning of the 'seventies. A unified theory of convexity of functions, dynamical systems and optimization methods on Riemannian manifolds is also presented. Topics covered include geodesics and completeness of Riemannian manifolds, variations of the p-energy of a curve and Jacobi fields, convex programs on Riemannian manifolds, geometrical constructions of convex functions, flows and energies, applications of convexity, descent algorithms on Riemannian manifolds, TC and TP programs for calculations and plots, all allowing the user to explore and experiment interactively with real life problems in the language of Riemannian geometry. An appendix is devoted to convexity and completeness in Finsler manifolds. For students and researchers in such diverse fields as pu...
CMC Hypersurfaces on Riemannian and Semi-Riemannian Manifolds
International Nuclear Information System (INIS)
Perdomo, Oscar M.
2012-01-01
In this paper we generalize the explicit formulas for constant mean curvature (CMC) immersion of hypersurfaces of Euclidean spaces, spheres and hyperbolic spaces given in Perdomo (Asian J Math 14(1):73–108, 2010; Rev Colomb Mat 45(1):81–96, 2011) to provide explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-Riemannian manifolds with constant sectional curvature. In particular, we prove that every h is an element of [-1,-(2√n-1/n can be realized as the constant curvature of a complete immersion of S 1 n-1 x R in the (n + 1)-dimensional de Sitter space S 1 n+1 . We provide 3 types of immersions with CMC in the Minkowski space, 5 types of immersion with CMC in the de Sitter space and 5 types of immersion with CMC in the anti de Sitter space. At the end of the paper we analyze the families of examples that can be extended to closed hypersurfaces.
On Riemannian manifolds (Mn, g) of quasi-constant curvature
International Nuclear Information System (INIS)
Rahman, M.S.
1995-07-01
A Riemannian manifold (M n , g) of quasi-constant curvature is defined. It is shown that an (M n , g) in association with other class of manifolds gives rise, under certain conditions, to a manifold of quasi-constant curvature. Some observations on how a manifold of quasi-constant curvature accounts for a pseudo Ricci-symmetric manifold and quasi-umbilical hypersurface are made. (author). 10 refs
Riemannian multi-manifold modeling and clustering in brain networks
Slavakis, Konstantinos; Salsabilian, Shiva; Wack, David S.; Muldoon, Sarah F.; Baidoo-Williams, Henry E.; Vettel, Jean M.; Cieslak, Matthew; Grafton, Scott T.
2017-08-01
This paper introduces Riemannian multi-manifold modeling in the context of brain-network analytics: Brainnetwork time-series yield features which are modeled as points lying in or close to a union of a finite number of submanifolds within a known Riemannian manifold. Distinguishing disparate time series amounts thus to clustering multiple Riemannian submanifolds. To this end, two feature-generation schemes for brain-network time series are put forth. The first one is motivated by Granger-causality arguments and uses an auto-regressive moving average model to map low-rank linear vector subspaces, spanned by column vectors of appropriately defined observability matrices, to points into the Grassmann manifold. The second one utilizes (non-linear) dependencies among network nodes by introducing kernel-based partial correlations to generate points in the manifold of positivedefinite matrices. Based on recently developed research on clustering Riemannian submanifolds, an algorithm is provided for distinguishing time series based on their Riemannian-geometry properties. Numerical tests on time series, synthetically generated from real brain-network structural connectivity matrices, reveal that the proposed scheme outperforms classical and state-of-the-art techniques in clustering brain-network states/structures.
Introduction to global analysis minimal surfaces in riemannian manifolds
Moore, John Douglas
2017-01-01
During the last century, global analysis was one of the main sources of interaction between geometry and topology. One might argue that the core of this subject is Morse theory, according to which the critical points of a generic smooth proper function on a manifold M determine the homology of the manifold. Morse envisioned applying this idea to the calculus of variations, including the theory of periodic motion in classical mechanics, by approximating the space of loops on M by a finite-dimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinite-dimensional manifolds, and these infinite-dimensional manifolds were found useful for studying a wide variety of nonlinear PDEs. This book applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed param...
Introduction to global analysis minimal surfaces in Riemannian manifolds
Moore, John Douglas
2017-01-01
During the last century, global analysis was one of the main sources of interaction between geometry and topology. One might argue that the core of this subject is Morse theory, according to which the critical points of a generic smooth proper function on a manifold M determine the homology of the manifold. Morse envisioned applying this idea to the calculus of variations, including the theory of periodic motion in classical mechanics, by approximating the space of loops on M by a finite-dimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinite-dimensional manifolds, and these infinite-dimensional manifolds were found useful for studying a wide variety of nonlinear PDEs. This book applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed param...
Lattice Dirac fermions on a simplicial Riemannian manifold
Brower, Richard C.; Weinberg, Evan S.; Fleming, George T.; Gasbarro, Andrew D.; Raben, Timothy G.; Tan, Chung-I.
2017-06-01
The lattice Dirac equation is formulated on a simplicial complex which approximates a smooth Riemann manifold by introducing a lattice vierbein on each site and a lattice spin connection on each link. Care is taken so the construction applies to any smooth D-dimensional Riemannian manifold that permits a spin connection. It is tested numerically in 2D for the projective sphere S2 in the limit of an increasingly refined sequence of triangles. The eigenspectrum and eigenvectors are shown to converge rapidly to the exact result in the continuum limit. In addition comparison is made with the continuum Ising conformal field theory on S2. Convergence is tested for the two point, ⟨ɛ (x1)ɛ (x2)⟩, and the four point, ⟨σ (x1)ɛ (x2)ɛ (x3)σ (x4)⟩, correlators for the energy, ɛ (x )=i ψ ¯(x )ψ (x ), and twist operators, σ (x ), respectively.
Exact solutions for isometric embeddings of pseudo-Riemannian manifolds
International Nuclear Information System (INIS)
Amery, G; Moodley, J
2014-01-01
Embeddings into higher dimensions are of direct importance in the study of higher dimensional theories of our Universe, in high energy physics and in classical general relativity. Theorems have been established that guarantee the existence of local and global codimension-1 embeddings between pseudo-Riemannian manifolds, particularly for Einstein embedding spaces. A technique has been provided to determine solutions to such embeddings. However, general solutions have not yet been found and most known explicit solutions are for embedded spaces with relatively simple Ricci curvature. Motivated by this, we have considered isometric embeddings of 4-dimensional pseudo-Riemannian spacetimes into 5-dimensional Einstein manifolds. We have applied the technique to treat specific 4-dimensional cases of interest in astrophysics and cosmology (including the global monopole exterior and Vaidya-de Sitter-class solutions), and provided novel physical insights into, for example, Einstein-Gauss-Bonnet gravity. Since difficulties arise in solving the 5-dimensional equations for given 4-dimensional spaces, we have also investigated embedded spaces, which admit bulks with a particular metric form. These analyses help to provide insight to the general embedding problem
Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry
International Nuclear Information System (INIS)
Pan, Yiwen
2014-01-01
In this note we generalize the methods of http://dx.doi.org/10.1007/JHEP08(2012)141, http://dx.doi.org/10.1007/JHEP01(2013)072 and http://dx.doi.org/10.1007/JHEP05(2013)017 to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solutions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to M=S 1 ×M 4 , which leads to M being foliated by submanifolds with special properties, such as Quaternion-Kähler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S 3 or T 3 -fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N=1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation
Directory of Open Access Journals (Sweden)
Feng Qi
2014-10-01
Full Text Available The authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of Laplace operator on Riemannian manifolds.
Steiner minimal trees in small neighbourhoods of points in Riemannian manifolds
Chikin, V. M.
2017-07-01
In contrast to the Euclidean case, almost no Steiner minimal trees with concrete boundaries on Riemannian manifolds are known. A result describing the types of Steiner minimal trees on a Riemannian manifold for arbitrary small boundaries is obtained. As a consequence, it is shown that for sufficiently small regular n-gons with n≥ 7 their boundaries without a longest side are Steiner minimal trees. Bibliography: 22 titles.
Covariant Schrödinger semigroups on Riemannian manifolds
Güneysu, Batu
2017-01-01
This monograph discusses covariant Schrödinger operators and their heat semigroups on noncompact Riemannian manifolds and aims to fill a gap in the literature, given the fact that the existing literature on Schrödinger operators has mainly focused on scalar Schrödinger operators on Euclidean spaces so far. In particular, the book studies operators that act on sections of vector bundles. In addition, these operators are allowed to have unbounded potential terms, possibly with strong local singularities. The results presented here provide the first systematic study of such operators that is sufficiently general to simultaneously treat the natural operators from quantum mechanics, such as magnetic Schrödinger operators with singular electric potentials, and those from geometry, such as squares of Dirac operators that have smooth but endomorphism-valued and possibly unbounded potentials. The book is largely self-contained, making it accessible for graduate and postgraduate students alike. Since it also inc...
On the concircular curvature tensor of Riemannian manifolds
International Nuclear Information System (INIS)
Rahman, M.S.; Lal, S.
1990-06-01
Definition of the concircular curvature tensor, Z hijk , along with Z-tensor, Z ij , is given and some properties of Z hijk are described. Tensors identical with Z hijk are shown. A necessary and sufficient condition that a Riemannian V n has zero Z-tensor is found. A number of theorems on concircular symmetric space, concircular recurrent space (Z n -space) and Z n -space with zero Z-tensor are deduced. (author). 6 refs
Existence of parallel spinors on non-simply-connected Riemannian manifolds
International Nuclear Information System (INIS)
McInnes, B.
1997-04-01
It is well known, and important for applications, that Ricci-flat Riemannian manifolds of non-generic holonomy always admit a parallel [covariant constant] spinor if they are simply connected. The non-simply-connected case is much more subtle, however. We show that a parallel spinor can still be found in this case provided that the [real] dimension is not a multiple of four, and provided that the spin structure is carefully chosen. (author). 10 refs
Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature
Directory of Open Access Journals (Sweden)
Arnaldo S. Nascimento
2010-05-01
Full Text Available We prove the nonexistence of nonconstant local minimizers for a class of functionals, which typically appear in scalar two-phase field models, over smooth N-dimensional Riemannian manifolds without boundary and non-negative Ricci curvature. Conversely, for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative, we prove the existence of nonconstant local minimizers for the same class of functionals.
Geometric Scattering on Compact Riemannian Manifolds and Spectral Theory of Automorphic Functions
Brüning, J
2005-01-01
We show that the spectral properties of the Laplace--Beltrami operator on a compact Riemannian manifold with $n$ semi-lines attached to it are similar to those for a finite-volume hyperbolic manifold with $n$ cusps. Our results are further justification of the Gromov--Novikov thesis concerning relations between Hyperbolic Geometry on infinity and One-Dimensional Geometry. As an application of the corresponding results we obtain a relation between the scattering matrix on a compact Riemann surface of constant negative curvature and the Selberg zeta function for this surface.
Directory of Open Access Journals (Sweden)
Fan Yang
2015-07-01
Full Text Available Normally, polarimetric SAR classification is a high-dimensional nonlinear mapping problem. In the realm of pattern recognition, sparse representation is a very efficacious and powerful approach. As classical descriptors of polarimetric SAR, covariance and coherency matrices are Hermitian semidefinite and form a Riemannian manifold. Conventional Euclidean metrics are not suitable for a Riemannian manifold, and hence, normal sparse representation classification cannot be applied to polarimetric SAR directly. This paper proposes a new land cover classification approach for polarimetric SAR. There are two principal novelties in this paper. First, a Stein kernel on a Riemannian manifold instead of Euclidean metrics, combined with sparse representation, is employed for polarimetric SAR land cover classification. This approach is named Stein-sparse representation-based classification (SRC. Second, using simultaneous sparse representation and reasonable assumptions of the correlation of representation among different frequency bands, Stein-SRC is generalized to simultaneous Stein-SRC for multi-frequency polarimetric SAR classification. These classifiers are assessed using polarimetric SAR images from the Airborne Synthetic Aperture Radar (AIRSAR sensor of the Jet Propulsion Laboratory (JPL and the Electromagnetics Institute Synthetic Aperture Radar (EMISAR sensor of the Technical University of Denmark (DTU. Experiments on single-band and multi-band data both show that these approaches acquire more accurate classification results in comparison to many conventional and advanced classifiers.
Seeley-Gilkey coefficients for the fourth-order operators on a Riemannian manifold
International Nuclear Information System (INIS)
Gusynin, V.P.
1989-01-01
A new covariant method for computing the coefficients in the heat kernel expansion is suggested. It allows one to calculate Seeley-Gilkey coefficients for both minimal and nonminimal differential operators acting on a vector bundle over a Riemannian manifold. The coefficients for the fourth-order minimal operators in arbitrary dimension of the space are calculated. In contrast to the second-order operators the coefficients for the fourth-order (and higher) operators turn out to be essentially dependent on the space dimension. The algorithmic character of the method suggested allows one to calculate coefficients by computer using the analytical calculation system. 19 refs.; 1 fig
Renormalization Proof for Massive $\\vp_4^4$ Theory on Riemannian Manifolds
Kopper, C
2006-01-01
In this paper we present an inductive renormalizability proof for massive $\\vp_4^4$ theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite are precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show, that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counter terms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature.
Baust, Maximilian; Weinmann, Andreas; Wieczorek, Matthias; Lasser, Tobias; Storath, Martin; Navab, Nassir
2016-08-01
In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup.
3-D Human Action Recognition by Shape Analysis of Motion Trajectories on Riemannian Manifold.
Devanne, Maxime; Wannous, Hazem; Berretti, Stefano; Pala, Pietro; Daoudi, Mohamed; Del Bimbo, Alberto
2015-07-01
Recognizing human actions in 3-D video sequences is an important open problem that is currently at the heart of many research domains including surveillance, natural interfaces and rehabilitation. However, the design and development of models for action recognition that are both accurate and efficient is a challenging task due to the variability of the human pose, clothing and appearance. In this paper, we propose a new framework to extract a compact representation of a human action captured through a depth sensor, and enable accurate action recognition. The proposed solution develops on fitting a human skeleton model to acquired data so as to represent the 3-D coordinates of the joints and their change over time as a trajectory in a suitable action space. Thanks to such a 3-D joint-based framework, the proposed solution is capable to capture both the shape and the dynamics of the human body, simultaneously. The action recognition problem is then formulated as the problem of computing the similarity between the shape of trajectories in a Riemannian manifold. Classification using k-nearest neighbors is finally performed on this manifold taking advantage of Riemannian geometry in the open curve shape space. Experiments are carried out on four representative benchmarks to demonstrate the potential of the proposed solution in terms of accuracy/latency for a low-latency action recognition. Comparative results with state-of-the-art methods are reported.
A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds
Directory of Open Access Journals (Sweden)
Qiang Ru
2013-01-01
Full Text Available We study the asymptotic behavior of the parabolic Monge-Ampère equation in , in , where is a compact complete Riemannian manifold, λ is a positive real parameter, and is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.
DEFF Research Database (Denmark)
Zimmermann, Ralf
2017-01-01
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm...... converges locally and exhibits a linear rate of convergence....
Zimmermann, Ralf
2016-01-01
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear rate of convergence.
Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds
Directory of Open Access Journals (Sweden)
Sheng-lan Chen
2014-01-01
Full Text Available We introduce a class of functions called geodesic B-preinvex and geodesic B-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudo B-preinvex and geodesic quasi/pseudo B-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic B-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic B-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.
Differential calculus on the space of Steiner minimal trees in Riemannian manifolds
International Nuclear Information System (INIS)
Ivanov, A O; Tuzhilin, A A
2001-01-01
It is proved that the length of a minimal spanning tree, the length of a Steiner minimal tree, and the Steiner ratio regarded as functions of finite subsets of a connected complete Riemannian manifold have directional derivatives in all directions. The derivatives of these functions are calculated and some properties of their critical points are found. In particular, a geometric criterion for a finite set to be critical for the Steiner ratio is found. This criterion imposes essential restrictions on the geometry of the sets for which the Steiner ratio attains its minimum, that is, the sets on which the Steiner ratio of the boundary set is equal to the Steiner ratio of the ambient space
Inferring imagined speech using EEG signals: a new approach using Riemannian manifold features
Nguyen, Chuong H.; Karavas, George K.; Artemiadis, Panagiotis
2018-02-01
Objective. In this paper, we investigate the suitability of imagined speech for brain-computer interface (BCI) applications. Approach. A novel method based on covariance matrix descriptors, which lie in Riemannian manifold, and the relevance vector machines classifier is proposed. The method is applied on electroencephalographic (EEG) signals and tested in multiple subjects. Main results. The method is shown to outperform other approaches in the field with respect to accuracy and robustness. The algorithm is validated on various categories of speech, such as imagined pronunciation of vowels, short words and long words. The classification accuracy of our methodology is in all cases significantly above chance level, reaching a maximum of 70% for cases where we classify three words and 95% for cases of two words. Significance. The results reveal certain aspects that may affect the success of speech imagery classification from EEG signals, such as sound, meaning and word complexity. This can potentially extend the capability of utilizing speech imagery in future BCI applications. The dataset of speech imagery collected from total 15 subjects is also published.
Congedo, M; Rodrigues, P L C; Bouchard, F; Barachant, A; Jutten, C
2017-07-01
Riemannian geometry has been found accurate and robust for classifying multidimensional data, for instance, in brain-computer interfaces based on electroencephalography. Given a number of data points on the manifold of symmetric positive-definite matrices, it is often of interest to embed these points in a manifold of smaller dimension. This is necessary for large dimensions in order to preserve accuracy and useful in general to speed up computations. Geometry-aware methods try to accomplish this task while respecting as much as possible the geometry of the original data points. We provide a closed-form solution for this problem in a fully unsupervised setting. Through the analysis of three brain-computer interface data bases we show that our method allows substantial dimensionality reduction without affecting the classification accuracy.
A new way to interpret the Dirac equation in a non-Riemannian manifold
Energy Technology Data Exchange (ETDEWEB)
Sirley Marques-Bonham
1989-06-01
The idea of internal mass terms is shown not be an appropriate hypothesis when it is placed in connection with the components of the generalized (matrix) vierbeins being proportional to the Riemannian (gravitational) vierbeins. It would result in an undesirable canceling of the Electromagnetic and the Yang-Mills components in the generalized metric. Another hypothesis is introduced where the wave function /psi/ is Taylor expanded in a small parameter p.
Petersen, Peter
2016-01-01
Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include: a substantial addition of unique and enriching exercises scattered throughout the text; inclusion of an increased number of coordinate calculations of connection and curvature; addition of general formulas for curvature on Lie Groups and submersions; integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger; incorporation of several recent results about manifolds with posit...
International Nuclear Information System (INIS)
Bogolubov, Nikolai N. Jr.; Prykarpatsky, Anatoliy K.
2006-12-01
The differential-geometric aspects of generalized de Rham-Hodge complexes naturally related with integrable multi-dimensional differential systems of M. Gromov type, as well as the geometric structure of Chern characteristic classes are studied. Special differential invariants of the Chern type are constructed, their importance for the integrability of multi-dimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson type nonlinear strongly integrable differential system is considered, its Cartan type connection mapping and related Chern type differential invariants are analyzed. (author)
Norm of the Riemannian Curvature Tensor
Indian Academy of Sciences (India)
We consider the Riemannian functional R p ( g ) = ∫ M | R ( g ) | p d v g defined on the space of Riemannian metrics with unit volume on a closed smooth manifold where R ( g ) and d v g denote the corresponding Riemannian curvature tensor and volume form and p ∈ ( 0 , ∞ ) . First we prove that the Riemannian metrics ...
Kim, Hyunwoo J; Adluru, Nagesh; Collins, Maxwell D; Chung, Moo K; Bendlin, Barbara B; Johnson, Sterling C; Davidson, Richard J; Singh, Vikas
2014-06-23
Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., ( x i , y i ) pairs, are Euclidean is well studied. The setting where y i is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, x i ∈ R , against a manifold-valued variable, y i ∈ . We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -a manifold-valued dependent variable against multiple independent variables, i.e., f : R n → . Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL ( n )/ O ( n ) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.
Needle decompositions in Riemannian geometry
Klartag, Bo'az
2017-01-01
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.
Needle decompositions in riemannian geometry
Klartag, Bo'az
2017-01-01
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.
International Nuclear Information System (INIS)
Finkelstein, D.; Finkelstein, S.R.; Holm, C.
1986-01-01
Riemannian manifolds are but one of three ways to extrapolate from fourdimensional Minkowskian manifolds to spaces of higher dimension, and not the most plausible. If we take seriously a certain construction of time space from spinors, and replace the underlying binary spinors by N-ary hyperspinors with new ''internal'' components besides the usual two ''external'' ones, this leads to a second line, the hyperspin manifolds /sub n/ and their tangent spaces d/sub n/, different in structure and symmetry group from the Riemannian line, except that the binary spaces d 2 (Minkowski time space) and 2 (Minkowskian manifold) lie on both. d/sub n/ and /sub n/ have dimension n = N 2 . In hyperspin manifolds the energies of modes of motion multiply instead of adding their squares, and the N-ary chronometric form is not quadratic, but N-ic, with determinantal normal form. For the nine-dimensional ternary hyperspin manifold, we construct the trino, trine-Gordon, and trirac equations and their mass spectra in flat time space. It is possible that our four-dimensional time space sits in a hyperspin manifold rather than in a Kaluza-Klein Riemannian manifold. If so, then gauge quanta with spin-3 exist
Indian Academy of Sciences (India)
manifolds. Some properties of Riemannian curvature tensors and curvature scalars of. Kähler–Norden manifolds using the theory of Tachibana operators is presented. Keywords. Kähler–Norden manifold; Norden metric; twin metric; pure tensor; holo- morphic tensor. 1. Introduction. Let M2n be a Riemannian manifold with a ...
Eigenvalue pinching on spinc manifolds
Roos, Saskia
2017-02-01
We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.
Semisymmetric cubic graphs of order 16p2 16p2 16p2
Indian Academy of Sciences (India)
An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order 162. It is shown that for every odd prime , there exists a semisymmetric ...
Scattering theory for Riemannian Laplacians
DEFF Research Database (Denmark)
Ito, Kenichi; Skibsted, Erik
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another cond...... metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace-Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum....
Geometric control theory and sub-Riemannian geometry
Boscain, Ugo; Gauthier, Jean-Paul; Sarychev, Andrey; Sigalotti, Mario
2014-01-01
This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.
Indian Academy of Sciences (India)
Abstract. This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler–Norden manifolds using the theory of Tachibana operators is presented.
Nonlinear analysis on manifolds
Hebey, Emmanuel
2000-01-01
This volume offers an expanded version of lectures given at the Courant Institute on the theory of Sobolev spaces on Riemannian manifolds. "Several surprising phenomena appear when studying Sobolev spaces on manifolds," according to the author. "Questions that are elementary for Euclidean space become challenging and give rise to sophisticated mathematics, where the geometry of the manifold plays a central role." The volume is organized into nine chapters. Chapter 1 offers a brief introduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces fo
On the mean curvature of semi-Riemannian graphs in semi ...
Indian Academy of Sciences (India)
Abstract. We investigate the mean curvature of semi-Riemannian graphs in the semi-. Riemannian warped product M × f Rε, where M is a semi-Riemannian manifold, Rε is the real line R with metric εdt2 (ε = ±1), and f : M → R+ is the warping function. We obtain an integral formula for mean curvature and some results ...
Moduli spaces of riemannian metrics
Tuschmann, Wilderich
2015-01-01
This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.
Riemannian geometry in an orthogonal frame
Cartan, Elie Joseph
2001-01-01
Foreword by S S Chern. In 1926-27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan's lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. This book has many innovations, such as the n
Pilca, Mihaela
2016-09-01
Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if a complete Vaisman manifold is toric, then the associated Sasaki manifold is also toric. Conversely, a toric complete Sasaki manifold, whose Kähler cone is equipped with an appropriate compatible action, gives rise to a toric Vaisman manifold. In the special case of a strongly regular compact Vaisman manifold, we show that it is toric if and only if the corresponding Kähler quotient is toric.
Semisymmetric cubic graphs of order 16p2 16p2 16p2
Indian Academy of Sciences (India)
Abstract. An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order. 16p2. It is shown that for every odd prime p, there exists a ...
Some functional inequalities on non-reversible Finsler manifolds
Indian Academy of Sciences (India)
SHIN-ICHI OHTA
2017-11-13
0043, Japan ... The aim of this article is to put forward geometric analysis on possibly non-reversible. Finsler manifolds (in the sense of F(−v) ..... weighted Riemannian manifolds and has many geometric and analytic applications.
Roughly isometric minimal immersions into Riemannian manifolds
DEFF Research Database (Denmark)
Markvorsen, Steen
of the intrinsic combinatorial discrete Laplacian, and we will show that they share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in $N$. The intrinsic properties thus obtained may hence serve as roughly invariant descriptors for the original metric space $X$....
Absence of embedded eigenvalues for Riemannian Laplacians
DEFF Research Database (Denmark)
Ito, Kenichi; Skibsted, Erik
Schrödinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental...... on the potential, a regularity property of the domain of the Schrödinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics studied previously in the literature....
An integrality theorem for spinc manifolds
International Nuclear Information System (INIS)
Seade, J.A.
1990-04-01
A spin c manifold M n is an oriented, Riemannian manifold with an associated hermitian live bundle det(M), together with a lifting to B(spin n c ) of the classifying map of the bundle TMxU(1). We prove here an integrality theorem for spin c manifolds. 11 refs
On the mean curvature of semi-Riemannian graphs in semi ...
Indian Academy of Sciences (India)
Riemannian warped product M × f R , where is a semi-Riemannian manifold, R is the real line R with metric d t 2 ( = ± 1 ) , and f : M → R + is the warping function. We obtain an integral formula for mean curvature and some results dealing ...
Harmonic maps of finite energy for Finsler manifolds
Li, Jintang; Wang, Yiling
2018-03-01
In this paper, we study some properties of harmonic maps for Finsler manifolds. Some Liouville theorems on harmonic maps for Finsler manifolds are given. Let M be a complete simply connected Riemannian manifold with non-negative Ricci curvature and M bar be a complete Berwald manifold with non-positive flag curvature. The main purpose of this paper is to prove that there exists no non-degenerate harmonic map ϕ from M to M bar with ∫SM e(ϕ) dVSM < ∞, which generalizes the result of Schoen and Yau (1976) from Riemannian manifolds to Berwald manifolds.
Strictly convex functions on complete Finsler manifolds
Indian Academy of Sciences (India)
minimum set of a super Busemann function contains a soul of M. Clearly, a complete simply connected Riemannian manifold H of non-positive sec- tional curvature, called Hadamard manifold, has the property that the distance function to an arbitrary fixed point is strongly convex exhaustion. Also, the exponential map expp :.
The heat flows and harmonic maps from complete manifolds into generalized regular balls
International Nuclear Information System (INIS)
Li Jiayu.
1993-01-01
Let M be a complete Riemannian manifold (compact (with or without boundary) or noncompact). Let N be a complete Riemannian manifold. We generalize the existence result for harmonic maps obtained by Hildebrandt-Kaul-Widman using the heat flow method. (author). 21 refs
Classification of non-Riemannian doubled-yet-gauged spacetime
Energy Technology Data Exchange (ETDEWEB)
Morand, Kevin [Universidad Andres Bello, Departamento de Ciencias Fisicas, Santiago de Chile (Chile); Universidad Tecnica Federico Santa Maria, Centro Cientifico-Tecnologico de Valparaiso, Departamento de Fisica, Valparaiso (Chile); Park, Jeong-Hyuck [Sogang University, Department of Physics, Seoul (Korea, Republic of); Institute for Basic Science (IBS), Center for Theoretical Physics of the Universe, Seoul (Korea, Republic of)
2017-10-15
Assuming O(D,D) covariant fields as the 'fundamental' variables, double field theory can accommodate novel geometries where a Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, (n, anti n), 0 ≤ n + anti n ≤ D. Upon these backgrounds, strings become chiral and anti-chiral over n and anti n directions, respectively, while particles and strings are frozen over the n + anti n directions. In particular, we identify (0, 0) as Riemannian manifolds, (1, 0) as non-relativistic spacetime, (1, 1) as Gomis-Ooguri non-relativistic string, (D-1, 0) as ultra-relativistic Carroll geometry, and (D, 0) as Siegel's chiral string. Combined with a covariant Kaluza-Klein ansatz which we further spell, (0, 1) leads to Newton-Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as D = 10, (3, 3) may open a new scheme for the dimensional reduction from ten to four. (orig.)
Geometry of manifolds with non-negative sectional curvature
Dearricott, Owen; Kennard, Lee; Searle, Catherine; Weingart, Gregor; Ziller, Wolfgang
2014-01-01
Providing an up-to-date overview of the geometry of manifolds with non-negative sectional curvature, this volume gives a detailed account of the most recent research in the area. The lectures cover a wide range of topics such as general isometric group actions, circle actions on positively curved four manifolds, cohomogeneity one actions on Alexandrov spaces, isometric torus actions on Riemannian manifolds of maximal symmetry rank, n-Sasakian manifolds, isoparametric hypersurfaces in spheres, contact CR and CR submanifolds, Riemannian submersions and the Hopf conjecture with symmetry. Also included is an introduction to the theory of exterior differential systems.
Eisenhart, Luther Pfahler
2005-01-01
This concise text by a prominent mathematician deals chiefly with manifolds dominated by the geometry of paths. Topics include asymmetric and symmetric connections, the projective geometry of paths, and the geometry of sub-spaces. 1927 edition.
Isometric C1-immersions for pairs of Riemannian metrics
International Nuclear Information System (INIS)
D'Ambra, Giuseppina; Datta, Mahuya
2001-08-01
Let h 1 , h 2 be two Euclidean metrics on R q , and let V be a C ∞ -manifold endowed with two Riemannian metrics g 1 and g 2 . We study the existence of C 1 -immersions f:(V,g 1 ,g 2 )→(R q ,h 1 ,h 2 ) such that f*(h i )=g i for i=1,2. (author)
Some theorems on a class of harmonic manifolds
International Nuclear Information System (INIS)
Rahman, M.S.; Chen Weihuan.
1993-08-01
A class of harmonic n-manifold, denoted by HM n , is, in fact, focussed on a Riemannian manifold with harmonic curvature. A variety of results, with properties, on HM n is presented in a fair order. Harmonic manifolds are then touched upon manifolds with recurrent Ricci curvature, biRicci-recurrent curvature and recurrent conformal curvature, and, in consequence, a sequence of theorems are deduced. (author). 21 refs
On complete manifolds supporting a weighted Sobolev type inequality
International Nuclear Information System (INIS)
Adriano, Levi; Xia Changyu
2011-01-01
Highlights: → We study manifolds supporting a weighted Sobolev or log-Sobolev inequality. → We investigate manifolds of asymptotically non-negative Ricci curvature. → The constant in the weighted Sobolev inequality on complete manifolds is studied. - Abstract: This paper studies the geometric and topological properties of complete open Riemannian manifolds which support a weighted Sobolev or log-Sobolev inequality. We show that the constant in the weighted Sobolev inequality on a complete open Riemannian manifold should be bigger than or equal to the optimal one on the Euclidean space of the same dimension and that a complete open manifold of asymptotically non-negative Ricci curvature supporting a weighted Sobolev inequality must have large volume growth. We also show that a complete manifold of non-negative Ricci curvature on which the log-Sobolev inequality holds is not very far from the Euclidean space.
On complete manifolds supporting a weighted Sobolev type inequality
Energy Technology Data Exchange (ETDEWEB)
Adriano, Levi, E-mail: levi@mat.ufg.br [Instituto de Matematica e Estatistica, Universidade Federal de Goias, 74001-900 Goiania, GO (Brazil); Xia Changyu, E-mail: xia@mat.unb.br [Departamento de Matematica, Universidade de Brasilia, 70910-900 Brasilia, DF (Brazil)
2011-11-15
Highlights: > We study manifolds supporting a weighted Sobolev or log-Sobolev inequality. > We investigate manifolds of asymptotically non-negative Ricci curvature. > The constant in the weighted Sobolev inequality on complete manifolds is studied. - Abstract: This paper studies the geometric and topological properties of complete open Riemannian manifolds which support a weighted Sobolev or log-Sobolev inequality. We show that the constant in the weighted Sobolev inequality on a complete open Riemannian manifold should be bigger than or equal to the optimal one on the Euclidean space of the same dimension and that a complete open manifold of asymptotically non-negative Ricci curvature supporting a weighted Sobolev inequality must have large volume growth. We also show that a complete manifold of non-negative Ricci curvature on which the log-Sobolev inequality holds is not very far from the Euclidean space.
Submanifolds in manifolds with metric mixed 3-structures
Ianus, Stere; Ornea, Liviu; Vilcu, Gabriel Eduard
2010-01-01
Mixed 3-structures are odd-dimensional analogues of paraquaternionic structures. They appear naturally on lightlike hypersurfaces of almost paraquaternionic hermitian manifolds. We study invariant and anti-invariant submanifolds in a manifold endowed with a mixed 3-structure and a compatible (semi-Riemannian) metric. Particular attention is given to two cases of ambient space: mixed 3-Sasakian and mixed 3-cosymplectic.
An introduction to differential manifolds
Lafontaine, Jacques
2015-01-01
This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces. Its ambition is to give solid foundations. In particular, the introduction of “abstract” notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them. The book should be of interest to various readers: undergra...
Semisymmetric cubic graphs of order 16p2 16p2 16p2
Indian Academy of Sciences (India)
other words, a directed walk of length s which never includes a backtracking. A graph. X is said to be s-arc-transitive if Aut(X) is transitive on the set of s-arcs in X. In parti- cular, 0-arc-transitive means vertex-transitive, and 1-arc-transitive means arc-transitive or symmetric. The study of semisymmetric graphs was initiated by ...
Diffeomorphisms of elliptic 3-manifolds
Hong, Sungbok; McCullough, Darryl; Rubinstein, J Hyam
2012-01-01
This work concerns the diffeomorphism groups of 3-manifolds, in particular of elliptic 3-manifolds. These are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, now known to be exactly the closed 3-manifolds that have a finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to its diffeomorphism group is a homotopy equivalence. The original Smale Conjecture, for the 3-sphere, was proven by J. Cerf and A. Hatcher, and N. Ivanov proved the generalized conjecture for many of the elliptic 3-manifolds that contain a geometrically incompressible Klein bottle. The main results establish the Smale Conjecture for all elliptic 3-manifolds containing geometrically incompressible Klein bottles, and for all lens spaces L(m,q) with m at least 3. Additional results imply that for a Haken Seifert-fibered 3 manifold V, the space of Seifert fiberings has contractible components, and apart from a small...
On the stability of the Lp -norm of the Riemannian curvature tensor
Indian Academy of Sciences (India)
property of Rp for p ≥ 2 at some certain critical metrics. Before stating our results we recall a canonical decomposition of tangent space of M. From Lemma 4.57 in [3], if M is a compact Riemannian manifold, we have the orthogonal decomposition of the tangent space of M at g (which is the space S2(T. ∗. M) of symmetric.
International Nuclear Information System (INIS)
Hervik, Sigbjoern; Coley, Alan
2011-01-01
In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the S i - and N-properties, and show that if the curvature tensors of the space possess the N-property, then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null congruence. We also discuss the related Walker metrics.
The Reciprocal of the Fundamental Theorem of Riemannian Geometry
Calderon, Hector
2008-05-01
The fundamental theorem of Riemannian geometry is inverted for analytic Christoffel symbols. The inversion formula, henceforth dubbed Ricardo's formula, is obtained without ancillary assumptions and it is well suited to compute the uncertainty in the metric that arises from the uncertainty in the measurement of positions. The solution is given up to a constant conformal factor, in part, because there are no experiments that can fix such factor without probing the whole universe. Ricardo's formula excludes some pathological examples and works for manifolds of any dimension and metrics of any signature.
Bilinear Regularized Locality Preserving Learning on Riemannian Graph for Motor Imagery BCI.
Xie, Xiaofeng; Yu, Zhu Liang; Gu, Zhenghui; Zhang, Jun; Cen, Ling; Li, Yuanqing
2018-03-01
In off-line training of motor imagery-based brain-computer interfaces (BCIs), to enhance the generalization performance of the learned classifier, the local information contained in test data could be used to improve the performance of motor imagery as well. Further considering that the covariance matrices of electroencephalogram (EEG) signal lie on Riemannian manifold, in this paper, we construct a Riemannian graph to incorporate the information of training and test data into processing. The adjacency and weight in Riemannian graph are determined by the geodesic distance of Riemannian manifold. Then, a new graph embedding algorithm, called bilinear regularized locality preserving (BRLP), is derived upon the Riemannian graph for addressing the problems of high dimensionality frequently arising in BCIs. With a proposed regularization term encoding prior information of EEG channels, the BRLP could obtain more robust performance. Finally, an efficient classification algorithm based on extreme learning machine is proposed to perform on the tangent space of learned embedding. Experimental evaluations on the BCI competition and in-house data sets reveal that the proposed algorithms could obtain significantly higher performance than many competition algorithms after using same filter process.
Riemannian computing in computer vision
Srivastava, Anuj
2016-01-01
This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours). · Illustrates Riemannian computing theory on applications in computer vision, machine learning, and robotics · Emphasis on algorithmic advances that will allow re-application in other...
Osserman and conformally Osserman manifolds with warped and twisted product structure
Brozos-Vazquez, M.; Garcia-Rio, E.; Vazquez-Lorenzo, R.
2008-01-01
We characterize Osserman and conformally Osserman Riemannian manifolds with the local structure of a warped product. By means of this approach we analyze the twisted product structure and obtain, as a consequence, that the only Osserman manifolds which can be written as a twisted product are those of constant curvature.
The Hodge theory of projective manifolds
de Cataldo, Mark Andrea
2007-01-01
This book is a written-up and expanded version of eight lectures on the Hodge theory of projective manifolds. It assumes very little background and aims at describing how the theory becomes progressively richer and more beautiful as one specializes from Riemannian, to Kähler, to complex projective manifolds. Though the proof of the Hodge Theorem is omitted, its consequences - topological, geometrical and algebraic - are discussed at some length. The special properties of complex projective manifolds constitute an important body of knowledge and readers are guided through it with the help of selected exercises. Despite starting with very few prerequisites, the concluding chapter works out, in the meaningful special case of surfaces, the proof of a special property of maps between complex projective manifolds, which was discovered only quite recently.
An approach to the relativistic brachistochrone problem by sub-Riemannian geometry
Giannoni, Fabio; Piccione, Paolo; Verderesi, José A.
1997-12-01
We formulate a brachistochrone problem in Lorentzian geometry and we prove a variational principle valid for brachistochrones in stationary manifolds. This variational principle is stated in terms of geodesics in a suitable sub-Riemannian structure on M. Moreover, we prove the regularity of the solutions of our variational problem and we determine a differential equation satisfied by the brachistochrones. Some explicit examples are computed.
Geometry and physics of pseudodifferential operators on manifolds
DEFF Research Database (Denmark)
Esposito, Giampiero; Napolitano, George M.
2015-01-01
A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: existence theorem for the function that generalizes the phase; analogue of Taylor's theorem; torsion and curvature terms in the symbolic...
Dynamic graphs, community detection, and Riemannian geometry
Energy Technology Data Exchange (ETDEWEB)
Bakker, Craig; Halappanavar, Mahantesh; Visweswara Sathanur, Arun
2018-03-29
A community is a subset of a wider network where the members of that subset are more strongly connected to each other than they are to the rest of the network. In this paper, we consider the problem of identifying and tracking communities in graphs that change over time {dynamic community detection} and present a framework based on Riemannian geometry to aid in this task. Our framework currently supports several important operations such as interpolating between and averaging over graph snapshots. We compare these Riemannian methods with entry-wise linear interpolation and that the Riemannian methods are generally better suited to dynamic community detection. Next steps with the Riemannian framework include developing higher-order interpolation methods (e.g. the analogues of polynomial and spline interpolation) and a Riemannian least-squares regression method for working with noisy data.
Hempel, John
2004-01-01
A careful and systematic development of the theory of the topology of 3-manifolds, focusing on the critical role of the fundamental group in determining the topological structure of a 3-manifold … self-contained … one can learn the subject from it … would be very appropriate as a text for an advanced graduate course or as a basis for a working seminar. -Mathematical Reviews For many years, John Hempel's book has been a standard text on the topology of 3-manifolds. Even though the field has grown tremendously, the book remains one of the best and most popular introductions to the subject. The t
Statistics on Lie groups: A need to go beyond the pseudo-Riemannian framework
Miolane, Nina; Pennec, Xavier
2015-01-01
Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
Contour Propagation With Riemannian Elasticity Regularization
DEFF Research Database (Denmark)
Bjerre, Troels; Hansen, Mads Fogtmann; Sapru, W.
2011-01-01
guided corrections. This study compares manual delineations in replanning CT scans of head-and-neck patients to automatic contour propagation using deformable registration with Riemannian regularization. The potential benefit of locally assigned regularization parameters according to tissue type...
Manifolds, tensors and, forms an introduction for mathematicians and physicists
Renteln, Paul
2014-01-01
Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.
Kosinski, Antoni A
2007-01-01
The concepts of differential topology form the center of many mathematical disciplines such as differential geometry and Lie group theory. Differential Manifolds presents to advanced undergraduates and graduate students the systematic study of the topological structure of smooth manifolds. Author Antoni A. Kosinski, Professor Emeritus of Mathematics at Rutgers University, offers an accessible approach to both the h-cobordism theorem and the classification of differential structures on spheres.""How useful it is,"" noted the Bulletin of the American Mathematical Society, ""to have a single, sho
Riemannian geometry and geometric analysis
Jost, Jürgen
2017-01-01
This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research. The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the B...
International Nuclear Information System (INIS)
Chiang, Yuan-Jen.
1989-01-01
Harmonic maps between manifolds are described as the critical maps of their associated energy functionals. By using Sampson's method [Sam1], the author constructs a Sobolev's chain on a compact V-manifold and obtain Rellich's Theorem (Theorem 3.1), Sobolev's Theorem (Theorem 3.2), the regularity theorem (Theorem 3.3), the property of the eigenspaces for the Laplacian (Theorem 3.5) and the solvability of Laplacian (Theorem 3.6). Then, with these results, he constructs the Green's functions for the Laplacian on a compact V-manifold M in Proposition 4.1; and obtain an orthonormal basis for L 2 (M) formed by the eigenfunctions of the Laplacian corresponding to the eigenvalues in Proposition 4.2. He also estimates the eigenvalues and eigenfunctions of the Laplacian in Theorem 4.3, which is used to construct the heat kernel on a compact V-manifold in Proposition 5.1. Afterwards, he compares the G-invariant heat kernel functions with the G-invariant fundamental solutions of heat equations in the finite V-charts of a compact V-manifold in Theorem 6.1, and then study two integral operators associated to the heat kernel on a compact V-manifold in section 7. With all the preceding results established, in Theorem 8.3 he uses successive approximations to prove the existence of the solutions of parabolic equations on V-manifolds. Finally, he uses Theorem 8.3 to show the existence of harmonic maps from compact V-manifolds into compact Riemannian manifolds in Theorem 9.1 which extends Eells-Sampson's results [E-S
Riemannian geometric approach to chaos in SU(2) Yang-Mills theory.
Kawabe, Tetsuji; Koyanagi, Shin'ichiro
2008-03-01
Based on the Riemannian geometric approach to Hamiltonian systems with many degrees of freedom, we study a chaotic nature of the SU(2) Yang-Mills field. Particularly, we study the Lyapunov exponent of the Wu-Yang magnetic-monopole solution of the SU(2) Yang-Mills field equation by use of an analytic formula which is determined by the average Ricci curvature and its fluctuation on the Riemannian manifold. It is shown that the system is chaotic from the positive values of the Lyapunov exponent. Furthermore, we find that the energy dependence of Lyapunov exponents exhibits a crossover phenomenon. By using the linear stability analysis, we point out that this crossover is related to the instability of the monopole solution.
Nearly pseudo-Kähler manifolds and related special holonomies
Schäfer, Lars
2017-01-01
Developing and providing an overview of recent results on nearly Kähler geometry on pseudo-Riemannian manifolds, this monograph emphasizes the differences with the classical Riemannian geometry setting. The focal objects of the text are related to special holonomy and Killing spinors and have applications in high energy physics, such as supergravity and string theory. Before starting into the field, a self-contained introduction to the subject is given, aimed at students with a solid background in differential geometry. The book will therefore be accessible to masters and Ph.D. students who are beginning work on nearly Kähler geometry in pseudo-Riemannian signature, and also to non-experts interested in gaining an overview of the subject. Moreover, a number of results and techniques are provided which will be helpful for differential geometers as well as for high energy physicists interested in the mathematical background of the geometric objects they need.
Morrow, James
2006-01-01
This book, a revision and organization of lectures given by Kodaira at Stanford University in 1965-66, is an excellent, well-written introduction to the study of abstract complex (analytic) manifolds-a subject that began in the late 1940's and early 1950's. It is largely self-contained, except for some standard results about elliptic partial differential equations, for which complete references are given. -D. C. Spencer, MathSciNet The book under review is the faithful reprint of the original edition of one of the most influential textbooks in modern complex analysis and geometry. The classic
Lattes-type mappings on compact manifolds
Astola, Laura; Kangaslampi, Riikka; Peltonen, Kirsi
A uniformly quasiregular mapping acting on a compact Riemannian manifold distorts the metric by a bounded amount, independently of the number of iterates. Such maps are rational with respect to some measurable conformal structure and there is a Fatou-Julia type theory associated with the dynamical system obtained by iterating these mappings. We study a rich subclass of uniformly quasiregular mappings that can be produced using an analogy of classical Lattes' construction of chaotic rational functions acting on the extended plane bar{C} . We show that there is a plenitude of compact manifolds that support these mappings. Moreover, we find that in some cases there are alternative ways to construct this type of mapping with different Julia sets.
New spinor fields on Lorentzian 7-manifolds
Energy Technology Data Exchange (ETDEWEB)
Bonora, L. [International School for Advanced Studies (SISSA),Via Bonomea 265, 34136 Trieste (Italy); Rocha, Roldão da [Centro de Matemática, Computação e Cognição, Universidade Federal do ABC,Avenida dos Estados, 5001, Santo André (Brazil)
2016-01-21
This paper deals with the classification of spinor fields according to the bilinear covariants in 7 dimensions. The previously investigated Riemannian case is characterized by either one spinor field class, in the real case of Majorana spinors, or three non-trivial classes in the most general complex case. In this paper we show that by imposing appropriate conditions on spinor fields in 7d manifolds with Lorentzian metric, the formerly obtained obstructions for new classes of spinor fields can be circumvented. New spinor fields classes are then explicitly constructed. In particular, on 7-manifolds with asymptotically flat black hole background, these spinors can define a generalized current density which further defines a time Killing vector at the spatial infinity.
Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry
Directory of Open Access Journals (Sweden)
Hyun Seok Yang
2017-01-01
Full Text Available We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6 requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.
Rotation vectors for homeomorphisms of non-positively curved manifolds
International Nuclear Information System (INIS)
Lessa, Pablo
2011-01-01
Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation vectors are shown to exist for almost every orbit of such a dynamical system with respect to any invariant measure with compact support. The concept is then extended to flows and, as an application, it is shown how non-null rotation vectors can be used to construct a measurable semi-conjugacy between a given flow and the geodesic flow of a manifold
New complete noncompact Spin(7) manifolds
International Nuclear Information System (INIS)
Cvetic, M.; Gibbons, G.W.; Lue, H.; Pope, C.N.
2002-01-01
We construct new explicit metrics on complete noncompact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by (A 8 , is topologically R 8 and another, which we denote by B 8 , is the bundle of chiral spinors over S 4 . Unlike the previously-known complete noncompact metric of Spin(7) holonomy, which was also defined on the bundle of chiral spinors over S 4 , our new metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3 . We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L 2 -normalisable harmonic 4-form for the (A)) 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold. We use the metrics to construct new supersymmetric brane solutions in M-theory and string theory. In particular, we construct resolved fractional M2-branes involving the use of the L 2 harmonic 4-forms, and show that for each manifold there is a supersymmetric example. An intriguing feature of the new A 8 and B 8 Spin(7) metrics is that they are actually the same local solution, with the two different complete manifolds corresponding to taking the radial coordinate to be either positive or negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which by contrast do not have special holonomy. In we construct the general solution of our first-order equations for Spin(7) holonomy, and obtain further regular metrics that are complete on manifolds B 8 + and B 8 - similar to B 8
Wang, Lu; Albera, Laurent; Kachenoura, Amar; Shu, Huazhong; Senhadji, Lotfi
2014-12-01
Semi-symmetric three-way arrays are essential tools in blind source separation (BSS) particularly in independent component analysis (ICA). These arrays can be built by resorting to higher order statistics of the data. The canonical polyadic (CP) decomposition of such semi-symmetric three-way arrays allows us to identify the so-called mixing matrix, which contains the information about the intensities of some latent source signals present in the observation channels. In addition, in many applications, such as the magnetic resonance spectroscopy (MRS), the columns of the mixing matrix are viewed as relative concentrations of the spectra of the chemical components. Therefore, the two loading matrices of the three-way array, which are equal to the mixing matrix, are nonnegative. Most existing CP algorithms handle the symmetry and the nonnegativity separately. Up to now, very few of them consider both the semi-nonnegativity and the semi-symmetry structure of the three-way array. Nevertheless, like all the methods based on line search, trust region strategies, and alternating optimization, they appear to be dependent on initialization, requiring in practice a multi-initialization procedure. In order to overcome this drawback, we propose two new methods, called [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], to solve the problem of CP decomposition of semi-nonnegative semi-symmetric three-way arrays. Firstly, we rewrite the constrained optimization problem as an unconstrained one. In fact, the nonnegativity constraint of the two symmetric modes is ensured by means of a square change of variable. Secondly, a Jacobi-like optimization procedure is adopted because of its good convergence property. More precisely, the two new methods use LU and QR matrix factorizations, respectively, which consist in formulating high-dimensional optimization problems into several sequential polynomial and rational subproblems. By using both LU
Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group
Ardentov, Andrei A.; Sachkov, Yuri L.
2017-12-01
We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R+ and a discrete group of reflections Z2 × Z2 × Z2. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.
Sub-Riemannian geometry and optimal transport
Rifford, Ludovic
2014-01-01
The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.
Grassmann manifolds and the Grassmann image of submanifolds
Borisenko, A. A.; Nikolaevskii, Yu A.
1991-04-01
CONTENTS I. Introduction II. Topology of Grassmann manifolds 1. Local coordinates 2. The cell decomposition and basic topological characteristics 3. Plücker coordinates III. Riemannian geometry of Grassmann manifolds: geometric approach 1. The metric and angles between planes 2. Curvature tensor, sectional curvature, closed geodesics, the limit set 3. More about Plücker embeddings 4. G+(2,n) as a Kähler manifold IV. Grassmann manifolds as symmetric spaces 1. The structure of a symmetric space 2. Totally geodesic and totally umbilical submanifolds 3. Standard embeddings of Grassmann manifolds in Euclidean space 4. Generalization of Grassmann manifolds V. Grassmann image. Intrinsic geometry 1. Induced metric. Homothety 2. Volume of the Grassmann image 3. Grassmann image of minimal surfaces 4. Harmonicity of the Grassmann map VI. Extrinsic geometry of the Grassmann image 1. Curvature of a Grassmann manifold along the Grassmann image of a surface 2. Reconstruction of a surface from the Grassmann image 3. Second fundamental form of the Grassmann image. Surfaces with totally geodesic and totally umbilical Grassmann image VII. Notes References
Rigidity of complete noncompact bach-flat n-manifolds
Chu, Yawei; Feng, Pinghua
2012-11-01
Let (Mn,g) be a complete noncompact Bach-flat n-manifold with the positive Yamabe constant and constant scalar curvature. Assume that the L2-norm of the trace-free Riemannian curvature tensor R∘m is finite. In this paper, we prove that (Mn,g) is a constant curvature space if the L-norm of R∘m is sufficiently small. Moreover, we get a gap theorem for (Mn,g) with positive scalar curvature. This can be viewed as a generalization of our earlier results of 4-dimensional Bach-flat manifolds with constant scalar curvature R≥0 [Y.W. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011) 516-521]. Furthermore, when n>9, we derive a rigidity result for R<0.
Total Variation Regularization for Functions with Values in a Manifold
Lellmann, Jan
2013-12-01
While total variation is among the most popular regularizers for variational problems, its extension to functions with values in a manifold is an open problem. In this paper, we propose the first algorithm to solve such problems which applies to arbitrary Riemannian manifolds. The key idea is to reformulate the variational problem as a multilabel optimization problem with an infinite number of labels. This leads to a hard optimization problem which can be approximately solved using convex relaxation techniques. The framework can be easily adapted to different manifolds including spheres and three-dimensional rotations, and allows to obtain accurate solutions even with a relatively coarse discretization. With numerous examples we demonstrate that the proposed framework can be applied to variational models that incorporate chromaticity values, normal fields, or camera trajectories. © 2013 IEEE.
Spherical formulation for diagrammatic evaluations on a manifold with boundary
Tsoupros, G
2002-01-01
The mathematical formalism necessary for the diagrammatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of quantum corrections to the effective action past one-loop necessitates diagrammatic techniques. Diagrammatic evaluations and higher loop-order renormalization can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. In such a context, the stated evaluations can be accomplished through a consistent interpretation of the Feynman rules within the spherical formulation of the theory which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analysed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed.
Weyl-Euler-Lagrange Equations of Motion on Flat Manifold
Directory of Open Access Journals (Sweden)
Zeki Kasap
2015-01-01
Full Text Available This paper deals with Weyl-Euler-Lagrange equations of motion on flat manifold. It is well known that a Riemannian manifold is said to be flat if its curvature is everywhere zero. Furthermore, a flat manifold is one Euclidean space in terms of distances. Weyl introduced a metric with a conformal transformation for unified theory in 1918. Classical mechanics is one of the major subfields of mechanics. Also, one way of solving problems in classical mechanics occurs with the help of the Euler-Lagrange equations. In this study, partial differential equations have been obtained for movement of objects in space and solutions of these equations have been generated by using the symbolic Algebra software. Additionally, the improvements, obtained in this study, will be presented.
Riemannian geometry in thermodynamic fluctuation theory
International Nuclear Information System (INIS)
Ruppeiner, G.
1995-01-01
Although thermodynamic fluctuation theory originated from statistical mechanics, it may be put on a completely thermodynamic basis, in no essential need of any microscopic foundation. This review views the theory from the macroscopic perspective, emphasizing, in particular, notions of covariance and consistency, expressed naturally using the language of Riemannian geometry. Coupled with these concepts is an extension of the basic structure of thermodynamic fluctuation theory beyond the classical one of a subsystem in contact with an infinite uniform reservoir. Used here is a hierarchy of concentric subsystems, each of which samples only the thermodynamic state of the subsystem immediately larger than it. The result is a covariant thermodynamic fluctuation theory which is plausible beyond the standard second-order entropy expansion. It includes the conservation laws and is mathematically consistent when applied to fluctuations inside subsystems. Tests on known models show improvements. Perhaps most significantly, the covariant theory offers a qualitatively new tool for the study of fluctuation phenomena: the Riemannian thermodynamic curvature. The thermodynamic curvature gives, for any given thermodynamic state, a lower bound for the length scale where the classical thermodynamic fluctuation theory based on a uniform environment could conceivably hold. Straightforward computation near the critical point reveals that the curvature equals the correlation volume, a physically appealing finding. The combination of the interpretation of curvature with a well-known proportionality between the free energy and the inverse of the correlation volume yields a purely thermodynamic theory of the critical point. The scaled equation of state follows from the values of the critical exponents. The thermodynamic Riemannian metric may be put into the broader context of information theory
Some Results About Concircular and Concurrent Vector Fields On Pseudo-Kaehler Manifolds
International Nuclear Information System (INIS)
Sevinç, Sibel; Aydin Şekerci, Gülşah; Ceylan Çöken, A.
2016-01-01
Kaehler manifolds which are used in physics have a lot of application fields. In this study we only state concircular and concurrent vector field that are defined on these manifolds. A vector field on a pseudo-Riemannian manifold N is called concircular, if it satisfies ∇ X υ = μX for any vector X tangent to N , where ∇ is the Levi-Civita connection of N . Furthermore, a concircular vector field υ is called a concurrent vector field if the function μ is non-constant. So, we provide some results on submanifolds of pseudo-Kaehler manifolds with respect to a concircular vector field or a concurrent vector field. Morever, we investigate this problem for another manifolds and proof some theorems. (paper)
Renteln, Paul
2013-11-01
Preface; 1. Linear algebra; 2. Multilinear algebra; 3. Differentiation on manifolds; 4. Homotopy and de Rham cohomology; 5. Elementary homology theory; 6. Integration on manifolds; 7. Vector bundles; 8. Geometric manifolds; 9. The degree of a smooth map; Appendixes; References; Index.
Riemannian and Lorentzian flow-cut theorems
Headrick, Matthew; Hubeny, Veronika E.
2018-05-01
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut (MFMC) theorem for boundary regions, applied recently to develop a ‘bit-thread’ interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous MFMC theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth’s theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.
Transversal Dirac families in Riemannian foliations
International Nuclear Information System (INIS)
Glazebrook, J.F.; Kamber, F.W.
1991-01-01
We describe a family of differential operators parametrized by the transversal vector potentials of a Riemannian foliation relative to the Clifford algebra of the foliation. This family is non-elliptic but in certain ways behaves like a standard Dirac family in the absolute case as a result of its elliptic-like regularity properties. The analytic and topological indices of this family are defined as elements of K-theory in the parameter space. We indicate how the cohomology of the parameter space is described via suitable maps to Fredholm operators. We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument. A determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation. With respect to a generalized coupled Dirac-Yang-Mills system, we indicate how chiral anomalies are located relative to the foliation. (orig.)
Supersymmetric quantum mechanics on n-dimensional manifolds
International Nuclear Information System (INIS)
O'Connor, M.
1990-01-01
In this thesis the author investigates the properties of the supersymmetric path integral on Riemannian manifolds. Chapter 1 is a brief introduction to supersymmetric path integral can be defined as the continuum limit of a discrete supersymmetric path integral. In Chapter 3 he shows that point canonical transformations in the path integral for ordinary quantum mechanics can be performed naively provided one uses the supersymmetric path integral. Chapter 4 generalizes the results of chapter 3 to include the propagation of all the fermion sectors in supersymmetric quantum mechanics. In Chapter 5 he shows how the properties of supersymmetric quantum mechanics can be used to investigate topological quantum mechanics
Connection with torsion, parallel spinors and geometry of Spin(7) manifolds
International Nuclear Information System (INIS)
Ivanov, Stefan
2001-11-01
We show that on every Spin(7)-manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior differential. We show the vanishing and the A-circumflex genus and obtain a linear relation between Betti numbers of a compact Spin(7) manifolds which are locally but not globally conformally equivalent to a space with closed fundamental 4-form. A general solution to the Killing spinor equations is presented. (author)
Introduction to differentiable manifolds
Auslander, Louis
2009-01-01
The first book to treat manifold theory at an introductory level, this text surveys basic concepts in the modern approach to differential geometry. The first six chapters define and illustrate differentiable manifolds, and the final four chapters investigate the roles of differential structures in a variety of situations.Starting with an introduction to differentiable manifolds and their tangent spaces, the text examines Euclidean spaces, their submanifolds, and abstract manifolds. Succeeding chapters explore the tangent bundle and vector fields and discuss their association with ordinary diff
Borok, S.; Goldfarb, I.; Gol'dshtein, V.
2009-05-01
The paper concerns intrinsic low-dimensional manifold (ILDM) method suggested in [Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, combustion and flame 1992;88:239-64] for dimension reduction of models describing kinetic processes. It has been shown in a number of publications [Goldfarb I, Gol'dshtein V, Maas U. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J Appl Math 2004;69:353-74; Kaper HG, Kaper TJ, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys D 2002;165(1-2):66-93; Rhodes C, Morari M, Wiggins S. Identification of the low order manifolds: validating the algorithm of Maas and Pope. Chaos 1999;9(1):108-23] that the ILDM-method works successfully and the intrinsic low-dimensional manifolds belong to a small vicinity of invariant slow manifolds. The ILDM-method has a number of disadvantages. One of them is appearance of so-called "ghost"-manifolds, which do not have connection to the system dynamics [Borok S, Goldfarb I, Gol'dshtein V. "Ghost" ILDM - manifolds and their discrimination. In: Twentieth Annual Symposium of the Israel Section of the Combustion Institute, Beer-Sheva, Israel; 2004. p. 55-7; Borok S, Goldfarb I, Gol'dshtein V. About non-coincidence of invariant manifolds and intrinsic low-dimensional manifolds (ILDM). CNSNS 2008;71:1029-38; Borok S, Goldfarb I, Gol'dshtein V, Maas U. In: Gorban AN, Kazantzis N, Kevrekidis YG, Ottinger HC, Theodoropoulos C, editors. "Ghost" ILDM-manifolds and their identification: model reduction and coarse-graining approaches for multiscale phenomena. Berlin-Heidelberg-New York: Springer; 2006. p. 55-80; Borok S, Goldfarb I, Gol'dshtein V. On a modified version of ILDM method and its asymptotic analysis. IJPAM 2008; 44(1): 125-50; Bykov V, Goldfarb I, Gol'dshtein V, Maas U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J Appl Math 2006
Distinguished dimensions for special Riemannian geometries
Nurowski, Paweł
2008-09-01
The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions nk=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group to the subgroups Hk⊂SO(nk), with H1=SO(3), H2=SU(3), H4=Sp(3) and H8=F. This enables studies of special Riemannian geometries with structure groups Hk in dimensions nk. The necessary and sufficient conditions for the Hk geometries to admit the characteristic connection are given. As an illustration nontrivial examples of SU(3) geometries in dimension 8 admitting characteristic connection are provided. Among them are the examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections. The torsionless models for the Hk geometries have the respective symmetry groups G1=SU(3), G2=SU(3)×SU(3), G3=SU(6) and G4=E. The groups Hk and Gk constitute a part of the 'magic square' for Lie groups. The 'magic square' Lie groups suggest studies of ten other classes of special Riemannian geometries. Apart from the two exceptional cases, they have the structure groups U(3), S(U(3)×U(3)), U(6), E×SO(2), Sp(3)×SU(2), SU(6)×SU(2), SO(12)×SU(2) and E×SU(2) and should be considered in respective dimensions 12, 18, 30, 54, 28, 40, 64 and 112. The two 'exceptional' cases are: SU(2)×SU(2) geometries in dimension 8 and SO(10)×SO(2) geometries in dimension 32. The case of SU(2)×SU(2) geometry in dimension 8 is examined closer. We determine the tensor that reduces SO(8) to SU(2)×SU(2) leaving the more detailed studies of the geometries based on the magic square ideas to the forthcoming paper.
Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods.
Harandi, Mehrtash; Salzmann, Mathieu; Hartley, Richard
2018-01-01
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.
Riemannian foliations on quaternion CR-submanifolds of an almost ...
Indian Academy of Sciences (India)
The natural product of two Kählerian manifolds is also a Kählerian manifold [11] and the geometry of CR-submanifolds of Kählerian product manifolds is an interesting subject which was studied in [1] and [5]. On the other hand, the product of two quaternion Kähler manifolds does not become a quaternion Kähler manifold, ...
The construction of periodic unfolding operators on some compact Riemannian manifolds
DEFF Research Database (Denmark)
Dobberschütz, Sören; Böhm, Michael
2014-01-01
The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to struct...
Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Ondreját, Martin
2011-01-01
Roč. 36, č. 9 (2011), s. 1624-1653 ISSN 0360-5302 R&D Projects: GA ČR GA201/07/0237 Institutional research plan: CEZ:AV0Z10750506 Keywords : geometric wave equation * stochastic wave equation Subject RIV: BA - General Mathematics Impact factor: 0.894, year: 2011 http://library.utia.cas.cz/separaty/2011/SI/ondrejat-0362936.pdf
Directory of Open Access Journals (Sweden)
Stephen M. Paneitz
2008-03-01
Full Text Available This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key rôle in conformal differential geometry in dimension 4 and is now known as the Paneitz operator.
Existence and regularity of minimal surfaces on Riemannian manifolds (MN-27)
Pitts, Jon T
2014-01-01
Mathematical No/ex, 27 Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Maps between Grassmann manifolds
Indian Academy of Sciences (India)
Parameswaran Sankaran Institute of Mathematical Sciences Chennai, India sankaran@imsc.res.in Indian Academy of Sciences Platinum Jubilee Meeting Hyderabad
2009-07-02
Jul 2, 2009 ... Regarding self-maps of (complex) Grassmann manifolds the following results are well-known: Parameswaran Sankaran Institute of Mathematical Sciences Chennai, India sankaran@imsc.res.in. Indian Academy of Sciences Platinum Jubilee Meeting Hyderabad. Maps between Grassmann manifolds ...
Manifold Partition Discriminant Analysis.
Yang Zhou; Shiliang Sun
2017-04-01
We propose a novel algorithm for supervised dimensionality reduction named manifold partition discriminant analysis (MPDA). It aims to find a linear embedding space where the within-class similarity is achieved along the direction that is consistent with the local variation of the data manifold, while nearby data belonging to different classes are well separated. By partitioning the data manifold into a number of linear subspaces and utilizing the first-order Taylor expansion, MPDA explicitly parameterizes the connections of tangent spaces and represents the data manifold in a piecewise manner. While graph Laplacian methods capture only the pairwise interaction between data points, our method captures both pairwise and higher order interactions (using regional consistency) between data points. This manifold representation can help to improve the measure of within-class similarity, which further leads to improved performance of dimensionality reduction. Experimental results on multiple real-world data sets demonstrate the effectiveness of the proposed method.
Ensemble manifold regularization.
Geng, Bo; Tao, Dacheng; Xu, Chao; Yang, Linjun; Hua, Xian-Sheng
2012-06-01
We propose an automatic approximation of the intrinsic manifold for general semi-supervised learning (SSL) problems. Unfortunately, it is not trivial to define an optimization function to obtain optimal hyperparameters. Usually, cross validation is applied, but it does not necessarily scale up. Other problems derive from the suboptimality incurred by discrete grid search and the overfitting. Therefore, we develop an ensemble manifold regularization (EMR) framework to approximate the intrinsic manifold by combining several initial guesses. Algorithmically, we designed EMR carefully so it 1) learns both the composite manifold and the semi-supervised learner jointly, 2) is fully automatic for learning the intrinsic manifold hyperparameters implicitly, 3) is conditionally optimal for intrinsic manifold approximation under a mild and reasonable assumption, and 4) is scalable for a large number of candidate manifold hyperparameters, from both time and space perspectives. Furthermore, we prove the convergence property of EMR to the deterministic matrix at rate root-n. Extensive experiments over both synthetic and real data sets demonstrate the effectiveness of the proposed framework.
Non-Riemannian geometry, Born-Infeld models and trace free gravitational equations
Cirilo-Lombardo, Diego Julio
2017-12-01
Non-Riemannian generalization of the standard Born-Infeld (BI) Lagrangian is introduced and analyzed from a theory of gravitation with dynamical torsion field. The field equations derived from the proposed action lead to a trace free gravitational equation (non-Riemannian analog to the trace free equation (TFE) from Finkelstein et al., 2001; Ellis et al., 2011; Ellis, 2014) and the field equations for the torsion respectively. In this theoretical context, the fundamental constants arise all from the same geometry through geometrical invariant quantities (as from the curvature R). New results involving generation of primordial magnetic fields and the link with leptogenesis and baryogenesis are presented and possible explanations given. The physically admissible matter fields can be introduced in the model via the torsion vector hμ. Such fields include some dark matter candidates such as axion, right neutrinos and Majorana and moreover, physical observables as vorticity can be included in the same way. From a new wormhole solution in a cosmological spacetime with torsion we also show that the primordial cosmic magnetic fields can originate from hμ with the axion field (that is contained in hμ) the responsible to control the dynamics and stability of the cosmic magnetic field but not the magnetogenesis itself. As we pointed out before (Cirilo-Lombardo, 2017), the analysis of Grand Unified Theories (GUT) in the context of this model indicates that the group manifold candidates are based in SO (10), SU (5) or some exceptional groups as E (6), E (7) , etc.
Hoelder continuity of energy minimizer maps between Riemannian polyhedra
International Nuclear Information System (INIS)
Bouziane, Taoufik
2004-10-01
The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the Hoelder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete our existence theorem obtained elsewhere, and consequently we generalize completely, to the case of target polyhedra without focal points (which is a weaker geometric condition than the nonpositivity of the curvature), the Eells-Fuglede's existence and regularity theorem which is the new version of the famous Eells-Sampson's theorem. (author)
International Nuclear Information System (INIS)
Aspinwall, P.S.; Luetken, C.A.
1991-01-01
We analyze the mirror manifold hypothesis in one and three dimensions using the simplest available representations of the N = 2 superconformal algebra. The symmetries of these tensor models can be divided out to give an explicit representation of the mirror, and we give a simple group theoretical algorithm for determining which symmetries should be used. We show that the mirror of a superconformal field theory does not always have a geometrical interpretation, but when it does, deformations of complex structure of one manifold are reflected in deformations of the Kaehler form of the mirror manifold, and we show how the large radius limit of a manifold corresponds to a large complex structure limit in the mirror manifold. The mirror of the Tian-Yau three generation model is constructed both as a conformal field theory and as an algebraic variety with Euler number six. The Hodge numbers of this manifolds are fixed, but the intersection numbes are highly ambiguous, presumably reflected a rich structure of multicritical points in the moduli space of the field theory. (orig.)
On integrability of certain rank 2 sub-Riemannian structures
Czech Academy of Sciences Publication Activity Database
Kruglikov, B.S.; Vollmer, A.; Lukes-Gerakopoulos, Georgios
2017-01-01
Roč. 22, č. 5 (2017), s. 502-519 ISSN 1560-3547 R&D Projects: GA ČR(CZ) GJ17-06962Y Institutional support: RVO:67985815 Keywords : sub-Riemannian geodesic flow * Killing tensor * integral Subject RIV: BN - Astronomy, Celestial Mechanics, Astrophysics OBOR OECD: Astronomy (including astrophysics,space science) Impact factor: 1.562, year: 2016
The three-body problem and equivariant Riemannian geometry
Alvarez-Ramírez, M.; García, A.; Meléndez, J.; Reyes-Victoria, J. G.
2017-08-01
We study the planar three-body problem with 1/r2 potential using the Jacobi-Maupertuis metric, making appropriate reductions by Riemannian submersions. We give a different proof of the Gaussian curvature's sign and the completeness of the space reported by Montgomery [Ergodic Theory Dyn. Syst. 25, 921-947 (2005)]. Moreover, we characterize the geodesics contained in great circles.
On determining the isometry group of a Riemannian space
International Nuclear Information System (INIS)
Karlhede, A.; Maccallum, M.A.H.
1982-01-01
An extension of the recently discussed algorithm for deciding the equivalence problem for Riemannian metrics is presented. The extension determines the structure constants of the isometry group and enables us to obtain some information about its orbits, including the form of the Killing vectors in canonical coordinates. (author)
Tensors and Riemannian geometry with applications to differential equations
Ibragimov, Nail H
2015-01-01
This graduate textbook begins by introducing Tensors and Riemannian Spaces, and then elaborates their application in solving second-order differential equations, and ends with introducing theory of relativity and de Sitter space. Based on 40 years of teaching experience, the author compiles a well-developed collection of examples and exercises to facilitate the reader’s learning.
Inducing Weinhold's metric from Euclidean and Riemannian metrics
International Nuclear Information System (INIS)
Andresen, B.; Berry, R.S.; Ihrig, E.; Salamon, P.
1987-01-01
We show that Weinhold's metric cannot be introduced on the equation of state surface from a Euclidean metric in the ambient space of all extensive state variables, whereas it can be induced if the ambient space is assumed only to have a Riemannian metric. This metric, however, is not unique. (orig.)
Geometric calculus: a new computational tool for Riemannian geometry
International Nuclear Information System (INIS)
Moussiaux, A.; Tombal, P.
1988-01-01
We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus
Aspects of quasi-Riemannian Kaluza-Klein theory
International Nuclear Information System (INIS)
Viswanathan, K.S.; Wong, B.
1985-01-01
We consider the applications of quasi-Riemannian geometry in Kaluza-Klein theories. We find that such theories cannot be implemented for all choices of the tangent group G/sub T/ and internal space G/H for reasons of gauge invariance. Coupling of fermions to gravity poses further problems in these theories
Entire vertical graphs in Riemannian product spaces | de Lima ...
African Journals Online (AJOL)
We extend the technique developed by S.T. Yau in [21] in order to investigate the rigidity of entire vertical graphs in a Riemannian product space R × Mn, whose fiber Mn is supposed to have Ricci curvature with strict sign. In this setting, under a suitable restriction on the norm of the gradient of the function u which ...
A Random Riemannian Metric for Probabilistic Shortest-Path Tractography
DEFF Research Database (Denmark)
Hauberg, Søren; Schober, Michael; Liptrot, Matthew George
2015-01-01
of the diffusion tensor as a “random Riemannian metric”, where a geodesic is a distribution over tracts. We approximate this distribution with a Gaussian process and present a probabilistic numerics algorithm for computing the geodesic distribution. We demonstrate SPT improvements on data from the Human Connectome...
Prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold
Rovenski, Vladimir Y.; Zelenko, Leonid
2018-03-01
The mixed scalar curvature is the simplest curvature invariant of a foliated Riemannian manifold. We explore the problem of prescribing the leafwise constant mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibered manifold, we reduce the problem to solution of a nonlinear leafwise elliptic equation for the conformal factor. We are looking for its solutions that are stable stationary solutions of the associated parabolic equation. Our main tool is using of majorizing and minorizing nonlinear heat equations with constant coefficients and application of comparison theorems for solutions of Cauchy's problem for parabolic equations.
Djordjevic, A.
1982-07-08
A tool guide that makes possible the insertion of cleaning and/or inspection tools into a manifold pipe that will dislocate and extract the accumulated sediment in such manifold pipes. The tool guide basically comprises a right angled tube (or other angled tube as required) which can be inserted in a large tube and locked into a radially extending cross pipe by adjustable spacer rods and a spring-loaded cone, whereby appropriate cleaning tools can be inserted into to cross pipe for cleaning, inspection, etc.
Manifold Regularized Reinforcement Learning.
Li, Hongliang; Liu, Derong; Wang, Ding
2018-04-01
This paper introduces a novel manifold regularized reinforcement learning scheme for continuous Markov decision processes. Smooth feature representations for value function approximation can be automatically learned using the unsupervised manifold regularization method. The learned features are data-driven, and can be adapted to the geometry of the state space. Furthermore, the scheme provides a direct basis representation extension for novel samples during policy learning and control. The performance of the proposed scheme is evaluated on two benchmark control tasks, i.e., the inverted pendulum and the energy storage problem. Simulation results illustrate the concepts of the proposed scheme and show that it can obtain excellent performance.
Congedo, Marco; Barachant, Alexandre
2015-01-01
Currently the Riemannian geometry of symmetric positive definite (SPD) matrices is gaining momentum as a powerful tool in a wide range of engineering applications such as image, radar and biomedical data signal processing. If the data is not natively represented in the form of SPD matrices, typically we may summarize them in such form by estimating covariance matrices of the data. However once we manipulate such covariance matrices on the Riemannian manifold we lose the representation in the original data space. For instance, we can evaluate the geometric mean of a set of covariance matrices, but not the geometric mean of the data generating the covariance matrices, the space of interest in which the geometric mean can be interpreted. As a consequence, Riemannian information geometry is often perceived by non-experts as a "black-box" tool and this perception prevents a wider adoption in the scientific community. Hereby we show that we can overcome this limitation by constructing a special form of SPD matrix embedding both the covariance structure of the data and the data itself. Incidentally, whenever the original data can be represented in the form of a generic data matrix (not even square), this special SPD matrix enables an exhaustive and unique description of the data up to second-order statistics. This is achieved embedding the covariance structure of both the rows and columns of the data matrix, allowing naturally a wide range of possible applications and bringing us over and above just an interpretability issue. We demonstrate the method by manipulating satellite images (pansharpening) and event-related potentials (ERPs) of an electroencephalography brain-computer interface (BCI) study. The first example illustrates the effect of moving along geodesics in the original data space and the second provides a novel estimation of ERP average (geometric mean), showing that, in contrast to the usual arithmetic mean, this estimation is robust to outliers. In
Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making.
Mishra, Himanshu; Mishra, Arul
2016-01-01
Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and monetary payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.
International Nuclear Information System (INIS)
Lisboa, P.; Michael, C.
1982-01-01
We address the question of designing optimum discrete sets of points to represent numerically a continuous group manifold. We consider subsets which are extensions of the regular discrete subgroups. Applications to Monte Carlo simulation of SU(2) and SU(3) gauge theory are discussed. (orig.)
Analytic manifolds in uniform algebras
International Nuclear Information System (INIS)
Tonev, T.V.
1988-12-01
Here we extend Bear-Hile's result concerning the version of famous Bishop's theorem for one-dimensional analytic structures in two directions: for n-dimensional complex analytic manifolds, n>1, and for generalized analytic manifolds. 14 refs
Black Strings, Black Rings and State-space Manifold
Bellucci, Stefano
2011-01-01
State-space geometry is considered, for diverse three and four parameter non-spherical horizon rotating black brane configurations, in string theory and $M$-theory. We have explicitly examined the case of unit Kaluza-Klein momentum $D_1D_5P$ black strings, circular strings, small black rings and black supertubes. An investigation of the state-space pair correlation functions shows that there exist two classes of brane statistical configurations, {\\it viz.}, the first category divulges a degenerate intrinsic equilibrium basis, while the second yields a non-degenerate, curved, intrinsic Riemannian geometry. Specifically, the solutions with finitely many branes expose that the two charged rotating $D_1D_5$ black strings and three charged rotating small black rings consort real degenerate state-space manifolds. Interestingly, arbitrary valued $M_5$-dipole charged rotating circular strings and Maldacena Strominger Witten black rings exhibit non-degenerate, positively curved, comprehensively regular state-space con...
Bragman, Felix J S; McClelland, Jamie R; Jacob, Joseph; Hurst, John R; Hawkes, David J
2017-09-01
Analysis of CT scans for studying Chronic Obstructive Pulmonary Disease (COPD) is generally limited to mean scores of disease extent. However, the evolution of local pulmonary damage may vary between patients with discordant effects on lung physiology. This limits the explanatory power of mean values in clinical studies. We present local disease and deformation distributions to address this limitation. The disease distribution aims to quantify two aspects of parenchymal damage: locally diffuse/dense disease and global homogeneity/heterogeneity. The deformation distribution links parenchymal damage to local volume change. These distributions are exploited to quantify inter-patient differences. We used manifold learning to model variations of these distributions in 743 patients from the COPDGene study. We applied manifold fusion to combine distinct aspects of COPD into a single model. We demonstrated the utility of the distributions by comparing associations between learned embeddings and measures of severity. We also illustrated the potential to identify trajectories of disease progression in a manifold space of COPD.
A New Semi-Symmetric Uniﬁed Field Theory of the Classical Fields of Gravity and Electromagnetism
Directory of Open Access Journals (Sweden)
Suhendro I.
2007-10-01
Full Text Available We attempt to present a classical theoretical framework in which the gravitational and electromagnetic fields are unified as intrinsic geometric objects in the space-time manifold. For this purpose, we first present the preliminary geometric considerations dealing with the metric differential geometry of Cartan connections. The unified field theory is then developed as an extension of the general theory of relativity based on a semi- symmetric Cartan connection which is meant to be as close as possible structurally to the symmetric connection of the Einstein-Riemann space-time.
Daverman, Robert J
2007-01-01
Decomposition theory studies decompositions, or partitions, of manifolds into simple pieces, usually cell-like sets. Since its inception in 1929, the subject has become an important tool in geometric topology. The main goal of the book is to help students interested in geometric topology to bridge the gap between entry-level graduate courses and research at the frontier as well as to demonstrate interrelations of decomposition theory with other parts of geometric topology. With numerous exercises and problems, many of them quite challenging, the book continues to be strongly recommended to eve
Analysis, manifolds and physics
Choquet-Bruhat, Y
2000-01-01
Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.
Manifold Regularized Correlation Object Tracking
Hu, Hongwei; Ma, Bo; Shen, Jianbing; Shao, Ling
2017-01-01
In this paper, we propose a manifold regularized correlation tracking method with augmented samples. To make better use of the unlabeled data and the manifold structure of the sample space, a manifold regularization-based correlation filter is introduced, which aims to assign similar labels to neighbor samples. Meanwhile, the regression model is learned by exploiting the block-circulant structure of matrices resulting from the augmented translated samples over multiple base samples cropped fr...
Gradient estimates for u=ΔF(u) on manifolds and some Liouville-type theorems
Xu, Xiangjin
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: u=ΔF(u), with F(u)>0, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): u=Δ(u), p>0, and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson-Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yau's celebrated Liouville theorem for positive harmonic functions.
Hadamard States for the Klein-Gordon Equation on Lorentzian Manifolds of Bounded Geometry
Gérard, Christian; Oulghazi, Omar; Wrochna, Michał
2017-06-01
We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.
Morse theory on banach manifolds
International Nuclear Information System (INIS)
Wang, T.
1986-01-01
The Morse Theory of critical points was extended by Palais and Smale to a certain class of functions on Hilbert manifolds. However, there are many variational problems in a nonlinear setting which for technical reasons are posed not on Hilbert but on Banach manifolds of mappings. This paper introduces a concept of a multivalued gradient vector field for a function defined on a Banach manifold. Using this concept, the Morse theory is generalized to some kind of Banach manifolds. The first chapter gives a definition of nondegeneracy of critical points for a real valued function defined on a reflexive Banach manifold, and then a handle-body decomposition theorem and Morse inequalities for this manifold are obtained. The second chapter proves the existence of solutions for a differential inclusion for a so-called accretive multi-valued mapping on a Finsler manifold. The third chapter introduces a definition of nondegeneracy of critical points for a real valued function defined on a general Banach manifold and, furthermore, generalizes the Morse handle-body decomposition theorem and the Morse inequalities to the Banach manifold
Protational monopoles, non-Riemannian geometry, and quasars
Energy Technology Data Exchange (ETDEWEB)
Murphy, G.L. (Wartburg Seminary, Dubuque, IA (USA))
1983-06-01
In pursuing analogies between gravitation and electrodynamics, the question occurs as to the possible existence of a gravitational analog of the (hypothetical) magnetic monopole. This entity cannot exist in standard general relativity, but could in a theory which makes use of non-Riemannian geometry. A theory involving such ''protational monopoles'' is formulated here, and comments are made on the possible relevance of such a theory for the spectral shifts and energy releases in such objects as quasars. But it is also pointed out that the quantization of mass necessitated by protational monopoles casts doubts upon their existence.
Quantum Riemannian geometry of phase space and nonassociativity
Directory of Open Access Journals (Sweden)
Beggs Edwin J.
2017-04-01
Full Text Available Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.
Cobordism independence of Grassmann manifolds
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
Introduction. This paper is a continuation of the ongoing study of cobordism of Grassmann manifolds. Let. F denote one of the division rings R of reals, C of complex numbers, or H of quaternions. Let t = dimRF. Then the Grassmannian manifold Gk(Fn+k) is defined to be the set of all k-dimensional (left) subspaces of Fn+k.
Smooth maps of a foliated manifold in a symplectic manifold
Indian Academy of Sciences (India)
Abstract. Let M be a smooth manifold with a regular foliation F and a 2-form ω which induces closed forms on the leaves of F in the leaf topology. A smooth map f : (M, F) −→ (N,σ) in a symplectic manifold (N,σ) is called a foliated symplectic immersion if f restricts to an immersion on each leaf of the foliation and further, the.
Generalization of deformable registration in Riemannian Sobolev spaces.
Zikic, Darko; Baust, Maximilian; Kamen, Ali; Navab, Nassir
2010-01-01
In this work we discuss the generalized treatment of the deformable registration problem in Sobolev spaces. We extend previous approaches in two points: 1) by employing a general energy model which includes a regularization term, and 2) by changing the notion of distance in the Sobolev space by problem-dependent Riemannian metrics. The actual choice of the metric is such that it has a preconditioning effect on the problem, it is applicable to arbitrary similarity measures, and features a simple implementation. The experiments demonstrate an improvement in convergence and runtime by several orders of magnitude in comparison to semi-implicit gradient flows in L2. This translates to increased accuracy in practical scenarios. Furthermore, the proposed generalization establishes a theoretical link between gradient flow in Sobolev spaces and elastic registration methods.
Hamarneh, Ghassan; McIntosh, Chris; Drew, Mark S
2011-07-01
A method for visualizing manifold-valued medical image data is proposed. The method operates on images in which each pixel is assumed to be sampled from an underlying manifold. For example, each pixel may contain a high dimensional vector, such as the time activity curve (TAC) in a dynamic positron emission tomography (dPET) or a dynamic single photon emission computed tomography (dSPECT) image, or the positive semi-definite tensor in a diffusion tensor magnetic resonance image (DTMRI). A nonlinear mapping reduces the dimensionality of the pixel data to achieve two goals: distance preservation and embedding into a perceptual color space. We use multidimensional scaling distance-preserving mapping to render similar pixels (e.g., DT or TAC pixels) with perceptually similar colors. The 3D CIELAB perceptual color space is adopted as the range of the distance preserving mapping, with a final similarity transform mapping colors to a maximum gamut size. Similarity between pixels is either determined analytically as geodesics on the manifold of pixels or is approximated using manifold learning techniques. In particular, dissimilarity between DTMRI pixels is evaluated via a Log-Euclidean Riemannian metric respecting the manifold of the rank 3, second-order positive semi-definite DTs, whereas the dissimilarity between TACs is approximated via ISOMAP. We demonstrate our approach via artificial high-dimensional, manifold-valued data, as well as case studies of normal and pathological clinical brain and heart DTMRI, dPET, and dSPECT images. Our results demonstrate the effectiveness of our approach in capturing, in a perceptually meaningful way, important features in the data.
Riemannian foliations on quaternion CR-submanifolds of an almost ...
Indian Academy of Sciences (India)
Introduction. The notion of CR-submanifold, first introduced in Kähler geometry by Bejancu (see [3]), was extended in the quaternion settings by Barros, Chen and Urbano in [2]. They consider. CR-quaternion submanifolds of quaternion Kählerian manifolds as generalizations of both quaternion and totally real submanifolds.
Spinorial Characterizations of Surfaces into 3-dimensional Pseudo-Riemannian Space Forms
International Nuclear Information System (INIS)
Lawn, Marie-Amélie; Roth, Julien
2011-01-01
We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. This generalizes a recent work of the first author for spacelike immersed Lorentzian surfaces in ℝ 2,1 to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well as for spacelike and timelike immersions of surfaces of signature (0, 2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.
Manifold Regularized Correlation Object Tracking.
Hu, Hongwei; Ma, Bo; Shen, Jianbing; Shao, Ling
2018-05-01
In this paper, we propose a manifold regularized correlation tracking method with augmented samples. To make better use of the unlabeled data and the manifold structure of the sample space, a manifold regularization-based correlation filter is introduced, which aims to assign similar labels to neighbor samples. Meanwhile, the regression model is learned by exploiting the block-circulant structure of matrices resulting from the augmented translated samples over multiple base samples cropped from both target and nontarget regions. Thus, the final classifier in our method is trained with positive, negative, and unlabeled base samples, which is a semisupervised learning framework. A block optimization strategy is further introduced to learn a manifold regularization-based correlation filter for efficient online tracking. Experiments on two public tracking data sets demonstrate the superior performance of our tracker compared with the state-of-the-art tracking approaches.
Moduli space of torsional manifolds
International Nuclear Information System (INIS)
Becker, Melanie; Tseng, L.-S.; Yau, S.-T.
2007-01-01
We characterize the geometric moduli of non-Kaehler manifolds with torsion. Heterotic supersymmetric flux compactifications require that the six-dimensional internal manifold be balanced, the gauge bundle be Hermitian Yang-Mills, and also the anomaly cancellation be satisfied. We perform the linearized variation of these constraints to derive the defining equations for the local moduli. We explicitly determine the metric deformations of the smooth flux solution corresponding to a torus bundle over K3
On some hypersurfaces with time like normal bundle in pseudo Riemannian space forms
International Nuclear Information System (INIS)
Kashani, S.M.B.
1995-12-01
In this work we classify immersed hypersurfaces with constant sectional curvature in pseudo Riemannian space forms if the normal bundle is time like and the mean curvature is constant. (author). 9 refs
Spinorial characterizations of surfaces into 3-dimensional psuedo-Riemannian space forms
Lawn , Marie-Amélie; Roth , Julien
2011-01-01
9 pages; We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in $\\mathbb{R}^{2,1}$ to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well ...
Initially Approximated Quasi Equilibrium Manifold
International Nuclear Information System (INIS)
Shahzad, M.; Arif, H.; Gulistan, M.; Sajid, M.
2015-01-01
Most commonly, kinetics model reduction techniques are based on exploiting time scale separation into fast and slow reaction processes. Then, a researcher approximates the system dynamically with dimension reduction for slow ones eliminating the fast modes. The main idea behind the construction of the lower dimension manifold is based on finding its initial approximation using Quasi Equilibrium Manifold (QEM). Here, we provide an efficient numerical method, which allow us to calculate low dimensional manifolds of chemical reaction systems. This computation technique is not restricted to our specific complex problem, but it can also be applied to other reacting flows or dynamic systems provided with the condition that a large number of extra (decaying) components can be eliminated from the system. Through computational approach, we approximate low dimensional manifold for a mechanism of six chemical species to simplify complex chemical kinetics. A reduced descriptive form of slow invariant manifold is obtained from dissipative system. This method is applicable for higher dimensions and is applied over an oxidation of CO/Pt. (author)
Slow manifolds in chemical kinetics
International Nuclear Information System (INIS)
Shahzad, M.; Haq, I. U.; Sultan, F.; Wahab, A.; Faizullah, F.; Rahman, G. U.
2016-01-01
Modelling the chemical system, especially for complex and higher dimensional problems, gives an easy way to handle the ongoing reaction process with respect to time. Here, we will consider some of the newly developed computational methods commonly used for model reductions in a chemical reaction. An effective (simple) method is planned to measure the low dimensional manifold, which reduces the higher dimensional system in such a way that it may not affect the precision of the whole mechanism. The phase flow of the solution trajectories near the equilibrium point is observed while the initial approximation is measured with the spectral quasi equilibrium manifold, which starts from the equilibrium point. To make it an invariant curve, the approximated curve is first refined a certain number of times using the method of invariant grids. The other way of getting the reduced data in the low dimensional manifold is possible through the intrinsic low dimensional manifold. Then, we compare these two invariant curves given by both the methods. Finally, the idea is extended to the higher dimensional manifold, where more number of progress variables will be added. (author)
Fivebranes and 3-manifold homology
Gukov, Sergei; Putrov, Pavel; Vafa, Cumrun
2017-07-01
Motivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of various old and new homological invariants of 3-manifolds. In terms of 3d/3d correspondence, such invariants are given by the Q-cohomology of the Hilbert space of partially topologically twisted 3d N=2 theory T[ M 3] on a Riemann surface with defects. We demonstrate this by concrete and explicit calculations in the case of monopole/Heegaard Floer homology and a 3-manifold analog of Khovanov-Rozansky link homology. The latter gives a categorification of Chern-Simons partition function. Some of the new key elements include the explicit form of the S-transform and a novel connection between categorification and a previously mysterious role of Eichler integrals in Chern-Simons theory.
Generalized graph manifolds and their effective recognition
International Nuclear Information System (INIS)
Matveev, S V
1998-01-01
A generalized graph manifold is a three-dimensional manifold obtained by gluing together elementary blocks, each of which is either a Seifert manifold or contains no essential tori or annuli. By a well-known result on torus decomposition each compact three-dimensional manifold with boundary that is either empty or consists of tori has a canonical representation as a generalized graph manifold. A short simple proof of the existence of a canonical representation is presented and a (partial) algorithm for its construction is described. A simple hyperbolicity test for blocks that are not Seifert manifolds is also presented
Blowup for flat slow manifolds
DEFF Research Database (Denmark)
Kristiansen, Kristian Uldall
2017-01-01
In this paper, we present a way of extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimensions, they do appear naturally in certain settings; for example, in (a......) the regularization of piecewise smooth systems by tanh, (b) a particular aircraft landing dynamics model, and finally (c) in a model of earthquake faulting. We demonstrate the approach using a simple model system and the examples (a) and (b)....
Blowup for flat slow manifolds
Kristiansen, K. U.
2017-05-01
In this paper, we present a way of extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimensions, they do appear naturally in certain settings; for example, in (a) the regularization of piecewise smooth systems by \\tanh , (b) a particular aircraft landing dynamics model, and finally (c) in a model of earthquake faulting. We demonstrate the approach using a simple model system and the examples (a) and (b).
Matrix regularization of 4-manifolds
Trzetrzelewski, M.
2012-01-01
We consider products of two 2-manifolds such as S^2 x S^2, embedded in Euclidean space and show that the corresponding 4-volume preserving diffeomorphism algebra can be approximated by a tensor product SU(N)xSU(N) i.e. functions on a manifold are approximated by the Kronecker product of two SU(N) matrices. A regularization of the 4-sphere is also performed by constructing N^2 x N^2 matrix representations of the 4-algebra (and as a byproduct of the 3-algebra which makes the regularization of S...
Stein Manifolds and Holomorphic Mappings
Forstneric, Franc
2011-01-01
The main theme of this book is the homotopy principle for holomorphic mappings from Stein manifolds to the newly introduced class of Oka manifolds. This book contains the first complete account of Oka-Grauert theory and its modern extensions, initiated by Mikhail Gromov and developed in the last decade by the author and his collaborators. Included is the first systematic presentation of the theory of holomorphic automorphisms of complex Euclidean spaces, a survey on Stein neighborhoods, connections between the geometry of Stein surfaces and Seiberg-Witten theory, and a wide variety of applicat
Collective coordinates on symplectic manifolds
International Nuclear Information System (INIS)
Razumov, A.V.; Taranov, A.Yu.
1981-01-01
For an arbitrary Lie group of canonical transformations on a symplectic manifold collective coordinates are introduced. They describe a motion of the dynamical system as a whole under the group transformations. Some properties of Lie group of canonical transformations are considered [ru
Cayley transform on Stiefel manifolds
Macías-Virgós, Enrique; Pereira-Sáez, María José; Tanré, Daniel
2018-01-01
The Cayley transform for orthogonal groups is a well known construction with applications in real and complex analysis, linear algebra and computer science. In this work, we construct Cayley transforms on Stiefel manifolds. Applications to the Lusternik-Schnirelmann category and optimization problems are presented.
An imbedding of Lorentzian manifolds
International Nuclear Information System (INIS)
Kim, Do-Hyung
2009-01-01
A new method for imbedding a Lorentzian manifold with a non-compact Cauchy surface is presented. As an application, it is shown that any two-dimensional globally hyperbolic spacetime with a non-compact Cauchy surface can be causally isomorphically imbedded into two-dimensional Minkowski spacetime.
Obstruction theory on 8-manifolds
Czech Academy of Sciences Publication Activity Database
Čadek, M.; Crabb, M.; Vanžura, Jiří
2008-01-01
Roč. 127, č. 2 (2008), s. 167-186 ISSN 0025-2611 R&D Projects: GA ČR GA201/05/2117 Institutional research plan: CEZ:AV0Z10190503 Keywords : 8-manifolds * obstruction theory Subject RIV: BA - General Mathematics Impact factor: 0.509, year: 2008
The dynamics of slow manifolds
Verhulst, F.; Bakri, T.
2006-01-01
Invited lecture at Konferensi Nasional Matematika XIII, Semarang, 24-27 juli, 2006; to be publ. in J. Indones. Math. Soc. (2007) After reviewing a number of results from geometric singular perturbation theory, we discuss several approaches to obtain periodic solutions in a slow manifold.
Quotient of manifolds by discrete groups
International Nuclear Information System (INIS)
Ardalan, F.; Arfaei, H.
1985-09-01
Quotient of manifolds by discrete subgroups of their isometry group are considered. In particular, symmetry breaking due to the quotient structure, topological properties and harmonic analysis of the resultant manifolds are discussed and illustrated by two dimensional examples. (author)
International Nuclear Information System (INIS)
Yehia, Hamad M
2013-01-01
In this study we have formulated a theorem that generates deformations of the natural integrable conservative systems in the plane into integrable systems on Riemannian and other manifolds by introducing additional parameters into their structures. The relation of explicit solutions of the new and the original dynamics to the corresponding Jacobi (Maupertuis) geodesic flow is clarified. For illustration, we apply the result to three concrete examples of the many available integrable systems in the literature. Complementary integrals in those systems are polynomial in velocity with degrees 3, 4 and 6, respectively. As a special case of the first deformed system, a new several-parameter family of integrable mechanical systems (and geodesic flows) on S 2 is constructed. (paper)
A viewpoint on nearly conformally symmetric manifold
International Nuclear Information System (INIS)
Rahman, M.S.
1990-06-01
Some observations, with definition, on Nearly Conformally Symmetric (NCS) manifold are made. A number of theorems concerning conformal change of metric and parallel tensors on NCS manifolds are presented. It is illustrated that a manifold M = R n-1 x R + 1 , endowed with a special metric, is NCS but not of harmonic curvature. (author). 8 refs
Quantum theory of spinor field in four-dimensional Riemannian space-time
International Nuclear Information System (INIS)
Shavokhina, N.S.
1996-01-01
The review deals with the spinor field in the four-dimensional Riemannian space-time. The field beys the Dirac-Fock-Ivanenko equation. Principles of quantization of the spinor field in the Riemannian space-time are formulated which in a particular case of the plane space-time are equivalent to the canonical rules of quantization. The formulated principles are exemplified by the De Sitter space-time. The study of quantum field theory in the De Sitter space-time is interesting because it itself leads to a method of an invariant well for plane space-time. However, the study of the quantum spinor field theory in an arbitrary Riemannian space-time allows one to take into account the influence of the external gravitational field on the quantized spinor field. 60 refs
Higgs bundles and four manifolds
International Nuclear Information System (INIS)
Park, Jae-Suk.
2002-01-01
It is known that the Seiberg-Witten invariants, derived from supersymmetric Yang-Mill theories in four dimensions, do not distinguish smooth structure of certain non-simply-connected four manifolds. We propose generalizations of Donaldson-Witten and Vafa-Witten theories on a Kaehler manifold based on Higgs bundles. We showed, in particular, that the partition function of our generalized Vafa-Witten theory can be written as the sum of contributions our generalized Donaldson-Witten invariants and generalized Seiberg-Witten invariants. The resulting generalized Seiberg-Witten invariants might have, conjecturally, information on smooth structure beyond the original Seiberg-Witten invariants for non-simply-connected case
Invariance for Single Curved Manifold
Castro, Pedro Machado Manhaes de
2012-08-01
Recently, it has been shown that, for Lambert illumination model, solely scenes composed by developable objects with a very particular albedo distribution produce an (2D) image with isolines that are (almost) invariant to light direction change. In this work, we provide and investigate a more general framework, and we show that, in general, the requirement for such in variances is quite strong, and is related to the differential geometry of the objects. More precisely, it is proved that single curved manifolds, i.e., manifolds such that at each point there is at most one principal curvature direction, produce invariant is surfaces for a certain relevant family of energy functions. In the three-dimensional case, the associated energy function corresponds to the classical Lambert illumination model with albedo. This result is also extended for finite-dimensional scenes composed by single curved objects. © 2012 IEEE.
Manifold seal structure for fuel cell stack
Collins, William P.
1988-01-01
The seal between the sides of a fuel cell stack and the gas manifolds is improved by adding a mechanical interlock between the adhesive sealing strip and the abutting surface of the manifolds. The adhesive is a material which can flow to some extent when under compression, and the mechanical interlock is formed providing small openings in the portion of the manifold which abuts the adhesive strip. When the manifolds are pressed against the adhesive strips, the latter will flow into and through the manifold openings to form buttons or ribs which mechanically interlock with the manifolds. These buttons or ribs increase the bond between the manifolds and adhesive, which previously relied solely on the adhesive nature of the adhesive.
Torsions of 3-dimensional manifolds
Wurzbacher, T
2002-01-01
From the reviews: "This is an excellent exposition about abelian Reidemeister torsions for three-manifolds." ―Zentralblatt Math "This monograph contains a wealth of information many topologists will find very handy. …Many of the new points of view pioneered by Turaev are gradually becoming mainstream and are spreading beyond the pure topology world. This monograph is a timely and very useful addition to the scientific literature." ―Mathematical Reviews
Hierarchy of graph matchbox manifolds
Lukina, Olga
2011-01-01
We study a class of graph foliated spaces, or graph matchbox manifolds, initially constructed by Kenyon and Ghys. For graph foliated spaces we introduce a quantifier of dynamical complexity which we call its level. We develop the fusion construction, which allows us to associate to every two graph foliated spaces a third one which contains the former two in its closure. Although the underlying idea of the fusion is simple, it gives us a powerful tool to study graph foliated spaces. Using fusi...
Invariant Bayesian estimation on manifolds
Jermyn, Ian H.
2005-01-01
A frequent and well-founded criticism of the maximum a posteriori (MAP) and minimum mean squared error (MMSE) estimates of a continuous parameter \\gamma taking values in a differentiable manifold \\Gamma is that they are not invariant to arbitrary ``reparameterizations'' of \\Gamma. This paper clarifies the issues surrounding this problem, by pointing out the difference between coordinate invariance, which is a sine qua non for a mathematically well-defined problem, and diffeomorphism invarianc...
Effective forcing with Cantor manifolds
Kihara, Takayuki
2017-01-01
A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$, enumerates $A$. Many variants of totality and cototality have been studied in computability theory. In this note, by an effective forcing construction with strongly infinite dimensional Cantor manifolds, which can be viewed as an effectivization of Zapletal's "half-...
Construction of harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces
International Nuclear Information System (INIS)
Konderak, J.
1988-09-01
Defined here is an orthogonal multiplication for vector spaces with indefinite nondegenerate scalar product. This is then used, via the Hopf construction, to obtain harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces. Examples of harmonic maps are constructed using Clifford algebras. (author). 6 refs
On the stability of the Lp-norm of the Riemannian curvature tensor
Indian Academy of Sciences (India)
First we prove that the Riemannian metrics with non-zero constant sectional curvature are strictly stable for R p for certain values of . Then we conclude that they are strict local minimizers for R p for those values of . Finally generalizing this result we prove that product of space forms of same type and dimension are strict ...
DEFF Research Database (Denmark)
Bjerre, Troels; Hansen, Mads Fogtmann; Aznar, M.
2012-01-01
For deformable registration of computed tomography (CT) scans in image guided radiation therapy (IGRT) we apply Riemannian elasticity regularization. We explore the use of spatially varying elasticity parameters to encourage bone rigidity and local tissue volume change only in the gross tumor...
Differential geometry curves, surfaces, manifolds
Kühnel, Wolfgang
2015-01-01
This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. The second part studies the geometry of general manifolds, with particular emphasis on connections and curvature. The text is illustrated with many figures and examples. The prerequisites are undergraduate analysis and linear algebra. This new edition provides many advancements, including more figures and exercises, and-as a new feature-a good number of so
Similarity Learning of Manifold Data.
Chen, Si-Bao; Ding, Chris H Q; Luo, Bin
2015-09-01
Without constructing adjacency graph for neighborhood, we propose a method to learn similarity among sample points of manifold in Laplacian embedding (LE) based on adding constraints of linear reconstruction and least absolute shrinkage and selection operator type minimization. Two algorithms and corresponding analyses are presented to learn similarity for mix-signed and nonnegative data respectively. The similarity learning method is further extended to kernel spaces. The experiments on both synthetic and real world benchmark data sets demonstrate that the proposed LE with new similarity has better visualization and achieves higher accuracy in classification.
Cobordism independence of Grassmann manifolds
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
ν(m) divides m. Given a positive integer d, let G(d) denote the set of bordism classes of all non-bounding. Grassmannian manifolds Gk(Fn+k) having real dimension d such that k < n. The restric- tion k
Minimal genera of open 4-manifolds
Gompf, Robert E.
2013-01-01
We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with much more control of these genus functions than the compact setting seems to allow. As an application, we expand the range of 4-manifolds known to have exotic smoothings (up to ...
Pan, Han; Jing, Zhongliang; Qiao, Lingfeng; Li, Minzhe
2017-09-25
Image restoration is a difficult and challenging problem in various imaging applications. However, despite of the benefits of a single overcomplete dictionary, there are still several challenges for capturing the geometric structure of image of interest. To more accurately represent the local structures of the underlying signals, we propose a new problem formulation for sparse representation with block-orthogonal constraint. There are three contributions. First, a framework for discriminative structured dictionary learning is proposed, which leads to a smooth manifold structure and quotient search spaces. Second, an alternating minimization scheme is proposed after taking both the cost function and the constraints into account. This is achieved by iteratively alternating between updating the block structure of the dictionary defined on Grassmann manifold and sparsifying the dictionary atoms automatically. Third, Riemannian conjugate gradient is considered to track local subspaces efficiently with a convergence guarantee. Extensive experiments on various datasets demonstrate that the proposed method outperforms the state-of-the-art methods on the removal of mixed Gaussian-impulse noise.
Holomorphic bundles over elliptic manifolds
International Nuclear Information System (INIS)
Morgan, J.W.
2000-01-01
In this lecture we shall examine holomorphic bundles over compact elliptically fibered manifolds. We shall examine constructions of such bundles as well as (duality) relations between such bundles and other geometric objects, namely K3-surfaces and del Pezzo surfaces. We shall be dealing throughout with holomorphic principal bundles with structure group GC where G is a compact, simple (usually simply connected) Lie group and GC is the associated complex simple algebraic group. Of course, in the special case G = SU(n) and hence GC = SLn(C), we are considering holomorphic vector bundles with trivial determinant. In the other cases of classical groups, G SO(n) or G = Sympl(2n) we are considering holomorphic vector bundles with trivial determinant equipped with a non-degenerate symmetric, or skew symmetric pairing. In addition to these classical cases there are the finite number of exceptional groups. Amazingly enough, motivated by questions in physics, much interest centres around the group E8 and its subgroups. For these applications it does not suffice to consider only the classical groups. Thus, while often first doing the case of SU(n) or more generally of the classical groups, we shall extend our discussions to the general semi-simple group. Also, we shall spend a good deal of time considering elliptically fibered manifolds of the simplest type, namely, elliptic curves
Target manifold formation using a quadratic SDF
Hester, Charles F.; Risko, Kelly K. D.
2013-05-01
Synthetic Discriminant Function (SDF) formulation of correlation filters provides constraints for forming target subspaces for a target set. In this paper we extend the SDF formulation to include quadratic constraints and use this solution to form nonlinear manifolds in the target space. The theory for forming these manifolds will be developed and demonstrated with data.
Numerical continuation of normally hyperbolic invariant manifolds
Broer, H. W.; Hagen, A.; Vegter, G.
2007-06-01
This paper deals with the numerical continuation of invariant manifolds regardless of the restricted dynamics. Common examples of such manifolds include limit sets, codimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/centre manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds appearing in bifurcations. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. The graph transform and linear graph transform compute the perturbed manifold with its hyperbolic splitting. To globally discretize manifolds, a discrete tubular neighbourhood is used, induced by a transverse bundle composed of discrete stable and unstable bundles. This approach allows the development of the discrete graph transform/linear graph transform analogous to the usual smooth case. Convergence results are given. The discrete vector bundle construction and associated local k-plane interpolation may be of independent interest. A practical numerical implementation for solving the global equations underlying the graph transform is proposed. Relevant numerical techniques are discussed and computational tests included. An additional application is the computation of the 'slow-transient' surface of an enzyme reaction.
Strictly convex functions on complete Finsler manifolds
Indian Academy of Sciences (India)
convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss ... map expp at some point p ∈ M (and hence at every point on M) is defined on the whole tangent space Mp to M at ... The influence of the existence of convex functions on the metric and topology of under- lying manifolds has ...
Integrability conditions on Engel-type manifolds
Calin, Ovidiu; Chang, Der-Chen; Hu, Jishan
2015-09-01
We introduce the concept of Engel manifold, as a manifold that resembles locally the Engel group, and find the integrability conditions of the associated sub-elliptic system , . These are given by , . Then an explicit construction of the solution involving an integral representation is provided, which corresponds to a Poincaré-type lemma for the Engel's distribution.
Holomorphic curves in exploded manifolds: Kuranishi structure
Parker, Brett
2013-01-01
This paper constructs a Kuranishi structure for the moduli stack of holomorphic curves in exploded manifolds. To avoid some technicalities of abstract Kuranishi structures, we embed our Kuranishi structure inside a moduli stack of curves. The construction also works for the moduli stack of holomorphic curves in any compact symplectic manifold.
Harmonic manifolds with minimal horospheres are flat
Indian Academy of Sciences (India)
Abstract. In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting ...
Harmonic Manifolds with Minimal Horospheres are Flat
Indian Academy of Sciences (India)
In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting theorem.
Some comparison theorems for Kahler manifolds
Tam, Luen-Fai; Yu, Chengjie
2010-01-01
In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang .
On the manifold-mapping optimization technique
D. Echeverria (David); P.W. Hemker (Piet)
2006-01-01
textabstractIn this paper, we study in some detail the manifold-mapping optimization technique introduced in an earlier paper. Manifold mapping aims at accelerating optimal design procedures that otherwise require many evaluations of time-expensive cost functions. We give a proof of convergence for
Harmonic manifolds with minimal horospheres are flat
Indian Academy of Sciences (India)
spaces and locally rank one symmetric spaces. ... any simply connected harmonic manifold is either flat or a rank one symmetric space. .... constant functions on manifolds. The derivatives ∇. (k) σp···σp ωp can be expressed in terms of the curvature tensor and its covariant derivatives. For example, we have for v ∈ SpM,.
Classical BV theories on manifolds with boundary
Cattaneo, A.S.; Mnev, P.; Reshetikhin, N.
2014-01-01
In this paper we extend the classical BV framework to gauge theories on spacetime manifolds with boundary. In particular, we connect the BV construction in the bulk with the BFV construction on the boundary and we develop its extension to strata of higher codimension in the case of manifolds with
Directory of Open Access Journals (Sweden)
Jun Zhang
2013-12-01
Full Text Available Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, (Χ,μ. Classical information geometry prescribes, on Μθ: (i a Riemannian metric given by the Fisher information; (ii a pair of dual connections (giving rise to the family of α-connections that preserve the metric under parallel transport by their joint actions; and (iii a family of divergence functions ( α-divergence defined on Μθ x Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function expressed in the divergence function (“referential duality” and the other concerning its representation under an arbitrary monotone scaling (“representational duality”.
Differential geometry of quasi-Sasakian manifolds
International Nuclear Information System (INIS)
Kirichenko, V F; Rustanov, A R
2002-01-01
The full system of structure equations of a quasi-Sasakian structure is obtained. The structure of the main tensors on a quasi-Sasakian manifold (the Riemann-Christoffel tensor, the Ricci tensor, and other tensors) is studied on this basis. Interesting characterizations of quasi-Sasakian Einstein manifolds are obtained. Additional symmetry properties of the Riemann-Christoffel tensor are discovered and used for distinguishing a new class of CR 1 quasi-Sasakian manifolds. An exhaustive description of the local structure of manifolds in this class is given. A complete classification (up to the B-transformation of the metric) is obtained for manifolds in this class having additional properties of the isotropy kind
Static traversable wormholes in Lyra manifold
Jahromi, A. Sayahian; Moradpour, H.
At first, considering the Einstein framework, we introduce some new static traversable wormholes and study the effects of a dark energy-like source on them. Thereinafter, a brief review on Einstein field equations in Lyra manifold is presented, and we address some static traversable wormholes in the Lyra manifold which satisfy the energy conditions. It is also shown that solutions introduced in the Einstein framework may also meet the energy conditions in the Lyra manifold. Finally, we focus on vacuum Lyra manifold and find some traversable asymptotically flat wormholes. In summary, our study shows that it is theoretically possible to find a Lyra displacement vector field in a manner in which traversable wormholes satisfy the energy conditions in a Lyra manifold.
Harmonic space and quaternionic manifolds
International Nuclear Information System (INIS)
Galperin, A.; Ogievetsky, O.; Ivanov, E.
1992-10-01
A principle of harmonic analyticity underlying the quaternionic (quaternion-Kaehler) geometry is found, and the differential constraints which define this geometry are solved. To this end the original 4n-dimensional quaternionic manifold is extended to a biharmonic space. The latter includes additional harmonic coordinates associated with both the tangent local Sp(1) group and an extra rigid SU(2) group rotating the complex structures. An one-to-one correspondence is established between the quaternionic spaces and off-shell N=2 supersymmetric sigma-models coupled to N=2 supergravity. Coordinates of the analytic subspace are identified with superfields describing N=2 matter hypermultiplets and a compensating hypermultiplet of N=2 supergravity. As an illustration the potentials for the symmetric quaternionic spaces are presented. (K.A.) 22 refs
Moving Manifolds in Electromagnetic Fields
Directory of Open Access Journals (Sweden)
David V. Svintradze
2017-08-01
Full Text Available We propose dynamic non-linear equations for moving surfaces in an electromagnetic field. The field is induced by a material body with a boundary of the surface. Correspondingly the potential energy, set by the field at the boundary can be written as an addition of four-potential times four-current to a contraction of the electromagnetic tensor. Proper application of the minimal action principle to the system Lagrangian yields dynamic non-linear equations for moving three dimensional manifolds in electromagnetic fields. The equations in different conditions simplify to Maxwell equations for massless three surfaces, to Euler equations for a dynamic fluid, to magneto-hydrodynamic equations and to the Poisson-Boltzmann equation.
Function theory on symplectic manifolds
Polterovich, Leonid
2014-01-01
This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards. I like the spirit of this book. It formulates concepts clearly and explains the relationship between them. The subject matter is i...
International Nuclear Information System (INIS)
Schubert, R.
1995-05-01
We investigate the behaviour of the remainder term R(E) in the Weyl formula {nvertical stroke E n ≤E}=Vol(M).E d/2 /[(4π) d/2 Γ(d/2+1)]+R(E) for the eigenvalues E n of a Schroedinger operator on a d-dimensional compact Riemannian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d-1)/2 Θ(√E), where Θ(x) is an almost periodic function of Besicovitch class B 2 which has a limit distribution whose density is a box-shaped function. Furthermore we derive a trace formula and study higher order terms in the asymptotics of the coefficients related to the periodic orbits. The periodicity of the geodesic flow leads to a very simple structure of the trace formula which is the reason why the limit distribution can be computed explicitly. (orig.)
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available The two-point boundary value problem for second-order differential inclusions of the form ( D/ dt m ˙ ( t ∈F( t,m( t , m ˙ ( t on complete Riemannian manifolds is investigated for a couple of points, nonconjugate along at least one geodesic of Levi-Civitá connection, where D/ dt is the covariant derivative of Levi-Civitá connection and F( t,m,X is a set-valued vector with quadratic or less than quadratic growth in the third argument. Some interrelations between certain geometric characteristics, the distance between points, and the norm of right-hand side are found that guarantee solvability of the above problem for F with quadratic growth in X . It is shown that this interrelation holds for all inclusions with F having less than quadratic growth in X , and so for them the problem is solvable.
On the mean curvature of semi-Riemannian graphs in semi ...
Indian Academy of Sciences (India)
{(x,ϕ(x)) | x ∈ M}⊂ ¯M. The graph is a hypersurface of ¯M. Unlike the usual case, we generally do not make the assumption that is space-like. Some new phenomena occur in this case. The organization of this paper is as follows: In §2 we recall some notations and results about semi-Riemannian warped product. In §3 we ...
Discriminative sparse coding on multi-manifolds
Wang, J.J.-Y.
2013-09-26
Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics. However, the conventional sparse coding algorithms and their manifold-regularized variants (graph sparse coding and Laplacian sparse coding), learn codebooks and codes in an unsupervised manner and neglect class information that is available in the training set. To address this problem, we propose a novel discriminative sparse coding method based on multi-manifolds, that learns discriminative class-conditioned codebooks and sparse codes from both data feature spaces and class labels. First, the entire training set is partitioned into multiple manifolds according to the class labels. Then, we formulate the sparse coding as a manifold-manifold matching problem and learn class-conditioned codebooks and codes to maximize the manifold margins of different classes. Lastly, we present a data sample-manifold matching-based strategy to classify the unlabeled data samples. Experimental results on somatic mutations identification and breast tumor classification based on ultrasonic images demonstrate the efficacy of the proposed data representation and classification approach. 2013 The Authors. All rights reserved.
DEFF Research Database (Denmark)
Sommer, Stefan Horst; Svane, Anne Marie
2017-01-01
We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in...
Space time manifolds and contact structures
Directory of Open Access Journals (Sweden)
K. L. Duggal
1990-01-01
Full Text Available A new class of contact manifolds (carring a global non-vanishing timelike vector field is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [Gödel Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.
Reduction of locally conformal symplectic manifolds with examples of non-Kähler manifolds
Noda, Tomonori
2004-01-01
Let $(M, \\Omega)$ be a locally conformal symplectic manifold. $\\Omega$ is a non-degenerate 2-form on $M$ such that there is a closed 1-form $\\omega$, called the Lee form, satisfing $ d\\Omega=\\omega\\wedge\\Omega$. In this paper we consider Marsden-Weinstein reduction theorem which induces Jacobi-Liouville theorem as a special case. For locally conformal Kähler manifolds, this reduction theorem gives a construction of non-Kähler manifolds in general dimension.
Final design of ITER thermal shield manifold
Energy Technology Data Exchange (ETDEWEB)
Kim, Kyung-Kyu [Mecha T& S, Jinju-si 52811 (Korea, Republic of); Noh, Chang Hyun, E-mail: chnoh@nfri.re.kr [National Fusion Research Institute, Daejeon 34133 (Korea, Republic of); Kim, Yun-Kyu; Park, Sungwoo [Mecha T& S, Jinju-si 52811 (Korea, Republic of); Nam, Kwanwoo [National Fusion Research Institute, Daejeon 34133 (Korea, Republic of); Chung, Wooho [Mecha T& S, Jinju-si 52811 (Korea, Republic of); Kang, Dongkwon; Kang, Kyung-O. [National Fusion Research Institute, Daejeon 34133 (Korea, Republic of); Park, Sungmun [SFA Engineering Corporation, Hwaseong-si 10060 (Korea, Republic of); Bae, Jing Do [Korea Marine Equipment Research Institute, Busan 49111 (Korea, Republic of)
2016-11-01
Highlights: • Engineering design of thermal shield manifold is finalized. • Pipe routing, support design and flow balance are verified by analysis. • Mock-ups are fabricated to verify the design. - Abstract: The ITER thermal shield is actively cooled by 80 K pressurized helium gas. The helium coolant flows from the cold valve box to the cooling tubes on the TS panels via manifold piping. This paper describes the final design of thermal shield manifold. Pipe design to accommodate the thermal contraction considering interface with adjacent components and detailed design of support structure are presented. R&D for the pipe branch connection is carried out to find a feasible manufacturing method. Global structural behavior and structural integrity of the manifold including pipe supports are investigated by a finite element analysis based on ASME B31.3 code. Flow analyses are performed to check the flow distribution.
Branched standard spines of 3-manifolds
Benedetti, Riccardo
1997-01-01
This book provides a unified combinatorial realization of the categroies of (closed, oriented) 3-manifolds, combed 3-manifolds, framed 3-manifolds and spin 3-manifolds. In all four cases the objects of the realization are finite enhanced graphs, and only finitely many local moves have to be taken into account. These realizations are based on the notion of branched standard spine, introduced in the book as a combination of the notion of branched surface with that of standard spine. The book is intended for readers interested in low-dimensional topology, and some familiarity with the basics is assumed. A list of questions, some of which concerning relations with the theory of quantum invariants, is enclosed.
Stable harmonic maps from complete manifolds
International Nuclear Information System (INIS)
Xin, Y.L.
1986-01-01
By choosing distinguished cross-sections in the second variational formula for harmonic maps from manifolds with not too fast volume growth into certain submanifolds in the Euclidean space some Liouville type theorems have been proved in this article. (author)
Polynomial chaos representation of databases on manifolds
Energy Technology Data Exchange (ETDEWEB)
Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
2017-04-15
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
The "Parity" Anomaly On An Unorientable Manifold
Witten, Edward
2016-01-01
The "parity" anomaly -- more accurately described as an anomaly in time-reversal or reflection symmetry -- arises in certain theories of fermions coupled to gauge fields and/or gravity in a spacetime of odd dimension. This anomaly has traditionally been studied on orientable manifolds only, but recent developments involving topological superconductors have made it clear that one can get more information by asking what happens on an unorientable manifold. In this paper, we give a full descript...
megaman: Manifold Learning for Millions of Points
McQueen, James; Meila, Marina; VanderPlas, Jacob; Zhang, Zhongyue
2017-11-01
megaman is a scalable manifold learning package implemented in python. It has a front-end API designed to be familiar to scikit-learn but harnesses the C++ Fast Library for Approximate Nearest Neighbors (FLANN) and the Sparse Symmetric Positive Definite (SSPD) solver Locally Optimal Block Precodition Gradient (LOBPCG) method to scale manifold learning algorithms to large data sets. It is designed for researchers and as such caches intermediary steps and indices to allow for fast re-computation with new parameters.
Minimal contact triangulations of 3-manifolds
Datta, Basudeb; Kulkarni, Dheeraj
2016-01-01
In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted contact structures grows at most linearly with respect to the relative $d^3$ invariant. We conjecture that this bound is optimal. We also discuss, in great details, contact tria...
Layered-triangulations of 3-manifolds
Jaco, William; Rubinstein, J. Hyam
2006-01-01
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens spaces, including the classification of all normal and almost normal surfaces in these triangulations. Minimal layered-triangulations of lens spaces provide a common setting fo...
Computer calculation of Witten's 3-manifold invariant
International Nuclear Information System (INIS)
Freed, D.S.; Gompf, R.E.
1991-01-01
Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant. (orig.)
Singular reduction of Nambu-Poisson manifolds
Das, Apurba
The version of Marsden-Ratiu Poisson reduction theorem for Nambu-Poisson manifolds by a regular foliation have been studied by Ibáñez et al. In this paper, we show that this reduction procedure can be extended to the singular case. Under a suitable notion of Hamiltonian flow on the reduced space, we show that a set of Hamiltonians on a Nambu-Poisson manifold can also be reduced.
Online Manifold Regularization by Dual Ascending Procedure
Sun, Boliang; Li, Guohui; Jia, Li; Zhang, Hui
2013-01-01
We propose a novel online manifold regularization framework based on the notion of duality in constrained optimization. The Fenchel conjugate of hinge functions is a key to transfer manifold regularization from offline to online in this paper. Our algorithms are derived by gradient ascent in the dual function. For practical purpose, we propose two buffering strategies and two sparse approximations to reduce the computational complexity. Detailed experiments verify the utility of our approache...
The manifold model for space-time
International Nuclear Information System (INIS)
Heller, M.
1981-01-01
Physical processes happen on a space-time arena. It turns out that all contemporary macroscopic physical theories presuppose a common mathematical model for this arena, the so-called manifold model of space-time. The first part of study is an heuristic introduction to the concept of a smooth manifold, starting with the intuitively more clear concepts of a curve and a surface in the Euclidean space. In the second part the definitions of the Csub(infinity) manifold and of certain structures, which arise in a natural way from the manifold concept, are given. The role of the enveloping Euclidean space (i.e. of the Euclidean space appearing in the manifold definition) in these definitions is stressed. The Euclidean character of the enveloping space induces to the manifold local Euclidean (topological and differential) properties. A suggestion is made that replacing the enveloping Euclidean space by a discrete non-Euclidean space would be a correct way towards the quantization of space-time. (author)
Right-angled polyhedra and hyperbolic 3-manifolds
Vesnin, A. Yu.
2017-04-01
Hyperbolic 3-manifolds whose fundamental groups are subgroups of finite index in right-angled Coxeter groups are under consideration. The construction of such manifolds is associated with regular colourings of the faces of polyhedra and, in particular, with 4-colourings. The following questions are discussed: the structure of the set of right-angled polytopes in Lobachevskii space; examples of orientable and non-orientable manifolds, including the classical Löbell manifold constructed in 1931; connections between the Hamiltonian property of a polyhedron and the existence of hyperelliptic involutions of manifolds; the volumes and complexity of manifolds; isometry between hyperbolic manifolds constructed from 4-colourings. Bibliography: 89 titles.
Duality on Geodesics of Cartan Distributions and Sub-Riemannian Pseudo-Product Structures
Directory of Open Access Journals (Sweden)
Ishikawa Goo
2015-06-01
Full Text Available Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in (15, that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.
Online Manifold Regularization by Dual Ascending Procedure
Directory of Open Access Journals (Sweden)
Boliang Sun
2013-01-01
Full Text Available We propose a novel online manifold regularization framework based on the notion of duality in constrained optimization. The Fenchel conjugate of hinge functions is a key to transfer manifold regularization from offline to online in this paper. Our algorithms are derived by gradient ascent in the dual function. For practical purpose, we propose two buffering strategies and two sparse approximations to reduce the computational complexity. Detailed experiments verify the utility of our approaches. An important conclusion is that our online MR algorithms can handle the settings where the target hypothesis is not fixed but drifts with the sequence of examples. We also recap and draw connections to earlier works. This paper paves a way to the design and analysis of online manifold regularization algorithms.
7D supersymmetric Yang-Mills on curved manifolds
Polydorou, Konstantina; Rocén, Andreas; Zabzine, Maxim
2017-12-01
We study 7D maximally supersymmetric Yang-Mills theory on curved manifolds that admit Killing spinors. If the manifold admits at least two Killing spinors (Sasaki-Einstein manifolds) we are able to rewrite the supersymmetric theory in terms of a cohomological complex. In principle this cohomological complex makes sense for any K-contact manifold. For the case of toric Sasaki-Einstein manifolds we derive explicitly the perturbative part of the partition function and speculate about the non-perturbative part. We also briefly discuss the case of 3-Sasaki manifolds and suggest a plausible form for the full non-perturbative answer.
Approaching Moons from Resonance via Invariant Manifolds
Anderson, Rodney L.
2012-01-01
In this work, the approach phase from the final resonance of the endgame scenario in a tour design is examined within the context of invariant manifolds. Previous analyses have typically solved this problem either by using numerical techniques or by computing a catalog of suitable trajectories. The invariant manifolds of a selected set of libration orbits and unstable resonant orbits are computed here to serve as guides for desirable approach trajectories. The analysis focuses on designing an approach phase that may be tied into the final resonance in the endgame sequence while also targeting desired conditions at the moon.
Unraveling flow patterns through nonlinear manifold learning.
Tauro, Flavia; Grimaldi, Salvatore; Porfiri, Maurizio
2014-01-01
From climatology to biofluidics, the characterization of complex flows relies on computationally expensive kinematic and kinetic measurements. In addition, such big data are difficult to handle in real time, thereby hampering advancements in the area of flow control and distributed sensing. Here, we propose a novel framework for unsupervised characterization of flow patterns through nonlinear manifold learning. Specifically, we apply the isometric feature mapping (Isomap) to experimental video data of the wake past a circular cylinder from steady to turbulent flows. Without direct velocity measurements, we show that manifold topology is intrinsically related to flow regime and that Isomap global coordinates can unravel salient flow features.
Effective Field Theory on Manifolds with Boundary
Albert, Benjamin I.
In the monograph Renormalization and Effective Field Theory, Costello made two major advances in rigorous quantum field theory. Firstly, he gave an inductive position space renormalization procedure for constructing an effective field theory that is based on heat kernel regularization of the propagator. Secondly, he gave a rigorous formulation of quantum gauge theory within effective field theory that makes use of the BV formalism. In this work, we extend Costello's renormalization procedure to a class of manifolds with boundary and make preliminary steps towards extending his formulation of gauge theory to manifolds with boundary. In addition, we reorganize the presentation of the preexisting material, filling in details and strengthening the results.
Matrix regularization of embedded 4-manifolds
International Nuclear Information System (INIS)
Trzetrzelewski, Maciej
2012-01-01
We consider products of two 2-manifolds such as S 2 ×S 2 , embedded in Euclidean space and show that the corresponding 4-volume preserving diffeomorphism algebra can be approximated by a tensor product SU(N)⊗SU(N) i.e. functions on a manifold are approximated by the Kronecker product of two SU(N) matrices. A regularization of the 4-sphere is also performed by constructing N 2 ×N 2 matrix representations of the 4-algebra (and as a byproduct of the 3-algebra which makes the regularization of S 3 also possible).
Toric geometry of G2-manifolds
DEFF Research Database (Denmark)
Madsen, Thomas Bruun; Swann, Andrew Francis
We consider G2-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of T3-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz...
Foliations and the geometry of 3-manifolds
Calegari, Danny
2014-01-01
This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions.
Strictly convex functions on complete Finsler manifolds
Indian Academy of Sciences (India)
... Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 126; Issue 4. Strictly convex functions on complete Finsler manifolds. YOE ITOKAWA KATSUHIRO SHIOHAMA BANKTESHWAR TIWARI. Research Article Volume 126 Issue 4 October 2016 pp 623-627 ...
Cohomology theories on locally conformal symplectic manifolds
Czech Academy of Sciences Publication Activity Database
Le, Hong-Van; Vanžura, Jiří
2015-01-01
Roč. 19, č. 1 (2015), s. 45-82 ISSN 1093-6106 Institutional support: RVO:67985840 Keywords : locally conformal symplectic manifold * Lichnerowicz-Novikov cohomology * primitive cohomology Subject RIV: BA - General Mathematics Impact factor: 0.722, year: 2015 http://www.intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0019/0001/a003/
Four-manifolds, geometries and knots
Hillman, Jonathan A
2007-01-01
The goal of this book is to characterize algebraically the closed 4-manifolds that fibre nontrivially or admit geometries in the sense of Thurston, or which are obtained by surgery on 2-knots, and to provide a reference for the topology of such manifolds and knots. The first chapter is purely algebraic. The rest of the book may be divided into three parts: general results on homotopy and surgery (Chapters 2-6), geometries and geometric decompositions (Chapters 7-13), and 2-knots (Chapters 14-18). In many cases the Euler characteristic, fundamental group and Stiefel-Whitney classes together form a complete system of invariants for the homotopy type of such manifolds, and the possible values of the invariants can be described explicitly. The strongest results are characterizations of manifolds which fibre homotopically over S^1 or an aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to homeomorphism). As a consequence 2-knots whose groups are poly-Z are determined up to Gluck reconstruc...
The Koch curve as a smooth manifold
International Nuclear Information System (INIS)
Epstein, Marcelo; Sniatycki, Jedrzej
2008-01-01
We show that there exists a homeomorphism between the closed interval [0,1] is contained in R and the Koch curve endowed with the subset topology of R 2 . We use this homeomorphism to endow the Koch curve with the structure of a smooth manifold with boundary
A generalized construction of mirror manifolds
International Nuclear Information System (INIS)
Berglund, P.; Huebsch, T.
1993-01-01
We generalize the known method for explicit construction of mirror pairs of (2,2)-superconformal field theories, using the formalism of Landau-Ginzburg orbifolds. Geometrically, these theories are realized as Calabi-Yau hypersurfaces in weighted projective spaces. This generalization makes it possible to construct the mirror partners of many manifolds for which the mirror was not previously known. (orig.)
Indian Academy of Sciences (India)
M ISCAN and A A SALIMOV. Faculty of Arts and Science, Department of Mathematics, Ataturk University, ... This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of ..... function f , then we shall call f a holomorphic (analytic) function and g its associated function [17]. If such a function ...
Classification of framed links in 3-manifolds
Indian Academy of Sciences (India)
Classification of framed links in 3-manifolds. MATIJA CENCELJ, DUŠAN REPOVŠ and. MIKHAIL B SKOPENKOV. ∗. Institute for Mathematics, Physics and Mechanics and Faculty of Education, University of Ljubljana, P.O. Box 2964, 1001 Ljubljana, Slovenia. ∗Department of Differential Geometry, Faculty of Mechanics and ...
International Nuclear Information System (INIS)
Becker, Katrin; Becker, Melanie; Robbins, Daniel
2015-01-01
In this talk we report on recent progress in describing compactifications of string theory and M-theory on G 2 and Spin(7) manifolds. We include the infinite set of α’-corrections and describe the entire tower of massless and massive Kaluza–Klein modes resulting from such compactifications. (invited comment)
Nonsmoothable involutions on spin 4-manifolds
Indian Academy of Sciences (India)
(Math. Sci.) Vol. 121, No. 1, February 2011, pp. 37–44. c Indian Academy of Sciences. Nonsmoothable involutions on spin 4-manifolds. CHANGTAO XUE and ... A group action is said to be pseudofree if each nontrivial group element has a discrete ... For our application, we also need their equivariant handle construction.
Higher order Hessian structures on manifolds
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
the bundle of bases for the tangent spaces. When we take a connection ∇XY to be given by Koszul's definition, we see that Hessian structures and symmetric connections can be directly related to each other. Before proceeding further, we state certain results relating to higher order derivatives on manifolds. For m ∈ M, let F.
Conservative systems with ports on contact manifolds
Eberard, D.; Maschke, B.M.; van der Schaft, Arjan; Piztek, P.
In this paper we propose an extension of port Hamiltonian systems, called conservative systems with ports, which encompass systems arising from the Irreversible Thermodynamics. Firstly we lift a port Hamiltonian system from its state space manifold to the thermodynamic phase space to a contact
The structure of some classes of K-contact manifolds
Indian Academy of Sciences (India)
Abstract. We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi ...
Fluid manifold design for a solar energy storage tank
Humphries, W. R.; Hewitt, H. C.; Griggs, E. I.
1975-01-01
A design technique for a fluid manifold for use in a solar energy storage tank is given. This analytical treatment generalizes the fluid equations pertinent to manifold design, giving manifold pressures, velocities, and orifice pressure differentials in terms of appropriate fluid and manifold geometry parameters. Experimental results used to corroborate analytical predictions are presented. These data indicate that variations in discharge coefficients due to variations in orifices can cause deviations between analytical predictions and actual performance values.
Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds
Martínez-Torres, David; Miranda, Eva
2018-01-01
We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.
Hyperbolic manifolds as vacuum solutions in Kaluza-Klein theories
International Nuclear Information System (INIS)
Aref'eva, I.Ya.; Volovich, I.V.
1985-08-01
The relevance of compact hyperbolic manifolds in the context of Kaluza-Klein theories is discussed. Examples of spontaneous compactification on hyperbolic manifolds including d dimensional (d>=8) Einstein-Yang-Mills gravity and 11-dimensional supergravity are considered. Some mathematical facts about hyperbolic manifolds essential for the physical content of the theory are briefly summarized. Non-linear σ-models based on hyperbolic manifolds are discussed. (author)
Examples and counter-examples of log-symplectic manifolds
Cavalcanti, Gil R.
We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably, we show that several symplectic manifolds do not admit bona fide log-symplectic structures and several bona fide log-symplectic manifolds do not admit symplectic
Wave equations on anti self dual (ASD) manifolds
Bashingwa, Jean-Juste; Kara, A. H.
2017-11-01
In this paper, we study and perform analyses of the wave equation on some manifolds with non diagonal metric g_{ij} which are of neutral signatures. These include the invariance properties, variational symmetries and conservation laws. In the recent past, wave equations on the standard (space time) Lorentzian manifolds have been performed but not on the manifolds from metrics of neutral signatures.
Three-dimensional group manifold reductions of gravity
Linares, Román
2005-04-01
We review the three-dimensional group manifold reductions of pure Einstein gravity and we exhibit a new consistent group manifold reduction of gravity when the compactification group manifold is S3. The new reduction leads to a lower dimensional theory whose gauge group is SU(2).
Classification of third-order symmetric Lorentzian manifolds
Galaev, Anton S.
2014-01-01
Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.
Local conformal symmetry in non-Riemannian geometry and the origin of physical scales
Energy Technology Data Exchange (ETDEWEB)
De Cesare, Marco [King' s College London, Theoretical Particle Physics and Cosmology Group, Department of Physics, London (United Kingdom); Moffat, John W. [Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada); Sakellariadou, Mairi [King' s College London, Theoretical Particle Physics and Cosmology Group, Department of Physics, London (United Kingdom); Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada)
2017-09-15
We introduce an extension of the Standard Model and General Relativity built upon the principle of local conformal invariance, which represents a generalization of a previous work by Bars, Steinhardt and Turok. This is naturally realized by adopting as a geometric framework a particular class of non-Riemannian geometries, first studied by Weyl. The gravitational sector is enriched by a scalar and a vector field. The latter has a geometric origin and represents the novel feature of our approach. We argue that physical scales could emerge from a theory with no dimensionful parameters, as a result of the spontaneous breakdown of conformal and electroweak symmetries. We study the dynamics of matter fields in this modified gravity theory and show that test particles follow geodesics of the Levi-Civita connection, thus resolving an old criticism raised by Einstein against Weyl's original proposal. (orig.)
Do extended bodies move alon.o the geodesics of the Riemannian space-time
International Nuclear Information System (INIS)
Denisov, V.I.; Logunov, A.A.; Mestvirishvili, M.A.
1980-01-01
Motion of a massive self-gravitating body in the gravitational field of a distant massive source has been considered in the post-Newtonian approximation of the arbitrary metric gravitational theory. The comparison of the massive body center of mass acceleration with that of a point one, moving in Riemannian space-time, whose metrics formally is equivalent to the metrics of two moving massive bodies, makes it clear that in any metric gravitation theory, possessing energy-momentum conservation lows for matter and gravitational field, taken together, massive body does not move generally speaking along the geodesics of Riemannian space-time. Application of the obtained general formulae to the system Earth-Sun and using of the experimental results from lunar-laser-ranging has shown that the Earth during its motion along the orbit, oscillates with respect to the reference geodesic of the geometry with the period of 1 hour and the amplitude not less than 10 -2 cm, which is a post-Newtonian quantity. Therefore the deviation of the Earth motion from the geodesic may be observed in a relevant experiment, which will have a post-Newtonian accuracy. The difference in accelerations of the Earth c.m. and a prob body makes up 10 -7 in the post-Newtonian approximation from the value of the Earth acceleration. The ratio of the passive gravitational mass (defined according to Will) to the inertial mass for the Earth is not equal to unity, and differs from it by the value of approximately 10 -8
On the stability of the Lp -norm of the Riemannian curvature tensor
Indian Academy of Sciences (India)
Partial Differential Equations 5(3) (1997) 199–269. [3] Besse A L, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3). [Results in Mathematics and Related Areas (3)] (1987) (Berlin: Springer-Verlag) vol. 10. [4] Besson G, Courtois G and Gallot S, Volume et entropie minimale des espaces localement.
Topological quantum field theory and four manifolds
Marino, Marcos
2005-01-01
The present book is the first of its kind in dealing with topological quantum field theories and their applications to topological aspects of four manifolds. It is not only unique for this reason but also because it contains sufficient introductory material that it can be read by mathematicians and theoretical physicists. On the one hand, it contains a chapter dealing with topological aspects of four manifolds, on the other hand it provides a full introduction to supersymmetry. The book constitutes an essential tool for researchers interested in the basics of topological quantum field theory, since these theories are introduced in detail from a general point of view. In addition, the book describes Donaldson theory and Seiberg-Witten theory, and provides all the details that have led to the connection between these theories using topological quantum field theory. It provides a full account of Witten’s magic formula relating Donaldson and Seiberg-Witten invariants. Furthermore, the book presents some of the ...
Heterogeneous massive feature fusion on grassmannian manifold
Huang, Haichao; Liu, Hongning; Kong, Xiaoyun; Lou, Xingdan; Wang, Zepeng
2017-08-01
Two issues remain unsolved on utilizing multimodal features for pattern recognition: the missing features and the curse of dimensionality. In this paper, we address the two issues by fusing the multimodal features on the Grassmann manifold. By defining grouping constrains on multimodal features, each multimodal feature vector is grouped into a set of subspaces, and is further represented as a point on the Grassmann manifold. To deal with missing features, L2-Hausdorff distance, a metric to compare multimodal feature vectors with different number of subspaces, is computed, and a kernel matrix can be obtained accordingly. Based on the kernel matrix, two feature selection criterions, one supervised and one unsupervised, are proposed to obtain a few representative features in the kernel space. Thus, the curse of dimensionality is alleviated. Experimental results on three multimodal dataset show the proposed feature fusion can outperforms the state-of -the-art by higher accuracy.
Incremental nonlinear dimensionality reduction by manifold learning.
Law, Martin H C; Jain, Anil K
2006-03-01
Understanding the structure of multidimensional patterns, especially in unsupervised cases, is of fundamental importance in data mining, pattern recognition, and machine learning. Several algorithms have been proposed to analyze the structure of high-dimensional data based on the notion of manifold learning. These algorithms have been used to extract the intrinsic characteristics of different types of high-dimensional data by performing nonlinear dimensionality reduction. Most of these algorithms operate in a "batch" mode and cannot be efficiently applied when data are collected sequentially. In this paper, we describe an incremental version of ISOMAP, one of the key manifold learning algorithms. Our experiments on synthetic data as well as real world images demonstrate that our modified algorithm can maintain an accurate low-dimensional representation of the data in an efficient manner.
Echocardiogram enhancement using supervised manifold denoising.
Wu, Hui; Huynh, Toan T; Souvenir, Richard
2015-08-01
This paper presents data-driven methods for echocardiogram enhancement. Existing denoising algorithms typically rely on a single noise model, and do not generalize to the composite noise sources typically found in real-world echocardiograms. Our methods leverage the low-dimensional intrinsic structure of echocardiogram videos. We assume that echocardiogram images are noisy samples from an underlying manifold parametrized by cardiac motion and denoise images via back-projection onto a learned (non-linear) manifold. Our methods incorporate synchronized side information (e.g., electrocardiography), which is often collected alongside the visual data. We evaluate the proposed methods on a synthetic data set and real-world echocardiograms. Quantitative results show improved performance of our methods over recent image despeckling methods and video denoising methods, and a visual analysis of real-world data shows noticeable image enhancement, even in the challenging case of noise due to dropout artifacts. Copyright © 2015 Elsevier B.V. All rights reserved.
Dynamical systems on 2- and 3-manifolds
Grines, Viacheslav Z; Pochinka, Olga V
2016-01-01
This book provides an introduction to the topological classification of smooth structurally stable diffeomorphisms on closed orientable 2- and 3-manifolds.The topological classification is one of the main problems of the theory of dynamical systems and the results presented in this book are mostly for dynamical systems satisfying Smale's Axiom A. The main results on the topological classification of discrete dynamical systems are widely scattered among many papers and surveys. This book presents these results fluidly, systematically, and for the first time in one publication. Additionally, this book discusses the recent results on the topological classification of Axiom A diffeomorphisms focusing on the nontrivial effects of the dynamical systems on 2- and 3-manifolds. The classical methods and approaches which are considered to be promising for the further research are also discussed. < The reader needs to be familiar with the basic concepts of the qualitative theory of dynamical systems which are present...
Convex nonnegative matrix factorization with manifold regularization.
Hu, Wenjun; Choi, Kup-Sze; Wang, Peiliang; Jiang, Yunliang; Wang, Shitong
2015-03-01
Nonnegative Matrix Factorization (NMF) has been extensively applied in many areas, including computer vision, pattern recognition, text mining, and signal processing. However, nonnegative entries are usually required for the data matrix in NMF, which limits its application. Besides, while the basis and encoding vectors obtained by NMF can represent the original data in low dimension, the representations do not always reflect the intrinsic geometric structure embedded in the data. Motivated by manifold learning and Convex NMF (CNMF), we propose a novel matrix factorization method called Graph Regularized and Convex Nonnegative Matrix Factorization (GCNMF) by introducing a graph regularized term into CNMF. The proposed matrix factorization technique not only inherits the intrinsic low-dimensional manifold structure, but also allows the processing of mixed-sign data matrix. Clustering experiments on nonnegative and mixed-sign real-world data sets are conducted to demonstrate the effectiveness of the proposed method. Copyright © 2014 Elsevier Ltd. All rights reserved.
Nonsmoothable involutions on spin 4-manifolds
Indian Academy of Sciences (India)
[2] Bryan J, Seiberg-Witten theory and Z/2p actions on spin 4-manifolds, Math. Res. Lett. 5. (1998) 165–183. [3] Chen W and Kwasik S, Symmetries and exotic smooth structures on a K3 surface,. J. Topology 1(4) (2008) 923–962. [4] Edmonds A L and Ewing J H, Realizing forms and fixed point data in dimension four,.
Sasakian manifolds with purely transversal Bach tensor
Ghosh, Amalendu; Sharma, Ramesh
2017-10-01
We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature ≥2 n (2 n +1 ) , equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S3. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π1(M).
Sasaki-Einstein Manifolds and Volume Minimisation
Martelli, D; Yau, S T; Martelli, Dario; Sparks, James; Yau, Shing-Tung
2006-01-01
We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone M, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on M that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of any Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theo...
Manifold learning in machine vision and robotics
Bernstein, Alexander
2017-02-01
Smart algorithms are used in Machine vision and Robotics to organize or extract high-level information from the available data. Nowadays, Machine learning is an essential and ubiquitous tool to automate extraction patterns or regularities from data (images in Machine vision; camera, laser, and sonar sensors data in Robotics) in order to solve various subject-oriented tasks such as understanding and classification of images content, navigation of mobile autonomous robot in uncertain environments, robot manipulation in medical robotics and computer-assisted surgery, and other. Usually such data have high dimensionality, however, due to various dependencies between their components and constraints caused by physical reasons, all "feasible and usable data" occupy only a very small part in high dimensional "observation space" with smaller intrinsic dimensionality. Generally accepted model of such data is manifold model in accordance with which the data lie on or near an unknown manifold (surface) of lower dimensionality embedded in an ambient high dimensional observation space; real-world high-dimensional data obtained from "natural" sources meet, as a rule, this model. The use of Manifold learning technique in Machine vision and Robotics, which discovers a low-dimensional structure of high dimensional data and results in effective algorithms for solving of a large number of various subject-oriented tasks, is the content of the conference plenary speech some topics of which are in the paper.
Gahm, Jin Kyu; Shi, Yonggang
2018-05-01
Surface mapping methods play an important role in various brain imaging studies from tracking the maturation of adolescent brains to mapping gray matter atrophy patterns in Alzheimer's disease. Popular surface mapping approaches based on spherical registration, however, have inherent numerical limitations when severe metric distortions are present during the spherical parameterization step. In this paper, we propose a novel computational framework for intrinsic surface mapping in the Laplace-Beltrami (LB) embedding space based on Riemannian metric optimization on surfaces (RMOS). Given a diffeomorphism between two surfaces, an isometry can be defined using the pullback metric, which in turn results in identical LB embeddings from the two surfaces. The proposed RMOS approach builds upon this mathematical foundation and achieves general feature-driven surface mapping in the LB embedding space by iteratively optimizing the Riemannian metric defined on the edges of triangular meshes. At the core of our framework is an optimization engine that converts an energy function for surface mapping into a distance measure in the LB embedding space, which can be effectively optimized using gradients of the LB eigen-system with respect to the Riemannian metrics. In the experimental results, we compare the RMOS algorithm with spherical registration using large-scale brain imaging data, and show that RMOS achieves superior performance in the prediction of hippocampal subfields and cortical gyral labels, and the holistic mapping of striatal surfaces for the construction of a striatal connectivity atlas from substantia nigra. Copyright © 2018 Elsevier B.V. All rights reserved.
Laplacian embedded regression for scalable manifold regularization.
Chen, Lin; Tsang, Ivor W; Xu, Dong
2012-06-01
Semi-supervised learning (SSL), as a powerful tool to learn from a limited number of labeled data and a large number of unlabeled data, has been attracting increasing attention in the machine learning community. In particular, the manifold regularization framework has laid solid theoretical foundations for a large family of SSL algorithms, such as Laplacian support vector machine (LapSVM) and Laplacian regularized least squares (LapRLS). However, most of these algorithms are limited to small scale problems due to the high computational cost of the matrix inversion operation involved in the optimization problem. In this paper, we propose a novel framework called Laplacian embedded regression by introducing an intermediate decision variable into the manifold regularization framework. By using ∈-insensitive loss, we obtain the Laplacian embedded support vector regression (LapESVR) algorithm, which inherits the sparse solution from SVR. Also, we derive Laplacian embedded RLS (LapERLS) corresponding to RLS under the proposed framework. Both LapESVR and LapERLS possess a simpler form of a transformed kernel, which is the summation of the original kernel and a graph kernel that captures the manifold structure. The benefits of the transformed kernel are two-fold: (1) we can deal with the original kernel matrix and the graph Laplacian matrix in the graph kernel separately and (2) if the graph Laplacian matrix is sparse, we only need to perform the inverse operation for a sparse matrix, which is much more efficient when compared with that for a dense one. Inspired by kernel principal component analysis, we further propose to project the introduced decision variable into a subspace spanned by a few eigenvectors of the graph Laplacian matrix in order to better reflect the data manifold, as well as accelerate the calculation of the graph kernel, allowing our methods to efficiently and effectively cope with large scale SSL problems. Extensive experiments on both toy and real
Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds
Weeks, Jeffrey R.
2005-01-01
Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincare dodecahedral space S3/I*, the binary octahedral space S3/O*, the binary tetrahedral space S3/T*, the prism manifolds S3/D...
Homotopy classification of contact foliations on open contact manifolds
Indian Academy of Sciences (India)
64
structures on a closed manifold M, then there exists an isotopy ψt, t ∈ I, of M such that ψt : (M,ξ0) → (M,ξt) is isocontact for all t ∈ I. Remark 2.6. Gray's stability theorem is not valid on non-closed manifolds. We shall see an extension of Theorem 2.5 for such manifolds in Theorem 1.1 which is one of the main results.
Some problems of dynamical systems on three dimensional manifolds
International Nuclear Information System (INIS)
Dong Zhenxie.
1985-08-01
It is important to study the dynamical systems on 3-dimensional manifolds, its importance is showing up in its close relation with our life. Because of the complication of topological structure of Dynamical systems on 3-dimensional manifolds, generally speaking, the search for 3-dynamical systems is not easier than 2-dynamical systems. This paper is a summary of the partial result of dynamical systems on 3-dimensional manifolds. (author)
Total Generalized Variation for Manifold-valued Data
Bredies, K.; Holler, M.; Storath, M.; Weinmann, A.
2017-01-01
In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for syntheti...
CT Image Reconstruction in a Low Dimensional Manifold
Cong, Wenxiang; Wang, Ge; Yang, Qingsong; Hsieh, Jiang; Li, Jia; Lai, Rongjie
2017-01-01
Regularization methods are commonly used in X-ray CT image reconstruction. Different regularization methods reflect the characterization of different prior knowledge of images. In a recent work, a new regularization method called a low-dimensional manifold model (LDMM) is investigated to characterize the low-dimensional patch manifold structure of natural images, where the manifold dimensionality characterizes structural information of an image. In this paper, we propose a CT image reconstruc...
LCD OF AIR INTAKE MANIFOLDS PHASE 2: FORD F250 AIR INTAKE MANIFOLD
The life cycle design methodology was applied to the design analysis of three alternatives for the lower plehum of the air intake manifold for us with a 5.4L F-250 truck engine: a sand cast aluminum, a lost core molded nylon composite, and a vibration welded nylon composite. The ...
Variable area manifolds for ring mirror heat exchangers
Eng, Albert; Senterfitt, Donald R.
1988-05-01
A laser ring mirror assembly is disclosed which supports and cools an annular ring mirror of a high powered laser with a cooling manifold which has a coolant flow design which is intended to reduce thermal distortions of the ring mirror by minimizing azimuthal variations in temperature around its circumference. The cooling manifold has complementary pairs of cooling passages each of which conduct coolant in opposite flow directions. The manifold also houses adjusters which vary the depth between the annular ring mirror and each cooling, and which vary the flow area of the cooling passage to produce a control over the cooling characteristics of the cooling manifold.
Totally Contact Umbilical Lightlike Hypersurfaces of Indefinite -Manifolds
Directory of Open Access Journals (Sweden)
Rachna Rani
2013-01-01
Full Text Available We study totally contact umbilical lightlike hypersurfaces of indefinite -manifolds and prove the nonexistence of totally contact umbilical lightlike hypersurface in indefinite -space form.
Noncommutative Riemannian geometry from quantum spacetime generated by twisted Poincaré group
Aguillón, Cesar A.; Much, Albert; Rosenbaum, Marcos; Vergara, J. David
2017-11-01
We investigate a quantum geometric space in the context of what could be considered an emerging effective theory from quantum gravity. Specifically we consider a two-parameter class of twisted Poincaré algebras, from which Lie-algebraic noncommutativities of the translations are derived as well as associative star-products, deformed Riemannian geometries, Lie-algebraic twisted Minkowski spaces, and quantum effects that arise as noncommutativities. Starting from a universal differential algebra of forms based on the above-mentioned Lie-algebraic noncommutativities of the translations, we construct the noncommutative differential forms and inner and outer derivations, which are the noncommutative equivalents of the vector fields in the case of commutative differential geometry. Having established the essentials of this formalism, we construct a bimodule, which is required to be central under the action of the inner derivations in order to have well-defined contractions and from where the algebraic dependence of its coefficients is derived. This again then defines the noncommutative equivalent of the geometrical line-element in commutative differential geometry. We stress, however, that even though the components of the twisted metric are by construction symmetric in their algebra valuation, it is not so for their inverse, and thus to construct it, we made use of Gel'fand's theory of quasi-determinants, which is conceptually straightforward but computationally quite complicated beyond an algebra of 3 generators. The consequences of the noncommutativity of the Lie-algebra twisted geometry are further discussed.
The "parity" anomaly on an unorientable manifold
Witten, Edward
2016-11-01
The "parity" anomaly—more accurately described as an anomaly in time-reversal or reflection symmetry—arises in certain theories of fermions coupled to gauge fields and/or gravity in a spacetime of odd dimension. This anomaly has traditionally been studied on orientable manifolds only, but recent developments involving topological superconductors have made it clear that one can get more information by asking what happens on an unorientable manifold. In this paper, we give a full description of the "parity" anomaly for fermions coupled to gauge fields and gravity in 2 +1 dimensions on a possibly unorientable spacetime. We consider an application to topological superconductors and another application to M theory. The application to topological superconductors involves using knowledge of the "parity" anomaly as an ingredient in constructing gapped boundary states of these systems and in particular in gapping the boundary of a ν =16 system in a topologically trivial fashion. The application to M theory involves showing the consistency of the path integral of an M theory membrane on a possibly unorientable worldvolume. In the past, this has been done only in the orientable case.
Fuel rod assembly to manifold attachment
Donck, Harry A.; Veca, Anthony R.; Snyder, Jr., Harold J.
1980-01-01
A fuel element is formed with a plurality of fuel rod assemblies detachably connected to an overhead support with each of the fuel rod assemblies having a gas tight seal with the support to allow internal fission gaseous products to flow without leakage from the fuel rod assemblies into a vent manifold passageway system on the support. The upper ends of the fuel rod assemblies are located at vertically extending openings in the support and upper threaded members are threaded to the fuel rod assemblies to connect the latter to the support. The preferred threaded members are cap nuts having a dome wall encircling an upper threaded end on the fuel rod assembly and having an upper sealing surface for sealing contact with the support. Another and lower seal is achieved by abutting a sealing surface on each fuel rod assembly with the support. A deformable portion on the cap nut locks the latter against inadvertent turning off the fuel rod assembly. Orienting means on the fuel rod and support primarily locates the fuel rods azimuthally for reception of a deforming tool for the cap nut. A cross port in the fuel rod end plug discharges into a sealed annulus within the support, which serves as a circumferential chamber, connecting the manifold gas passageways in the support.
Smooth manifold structure for extreme channels
Iten, Raban; Colbeck, Roger
2018-01-01
A quantum channel from a system A of dimension dA to a system B of dimension dB is a completely positive trace-preserving map from complex dA × dA to dB × dB matrices, and the set of all such maps with Kraus rank r has the structure of a smooth manifold. We describe this set in two ways. First, as a quotient space of (a subset of) the rdB × dA dimensional Stiefel manifold. Second, as the set of all Choi-states of a fixed rank r. These two descriptions are topologically equivalent. This allows us to show that the set of all Choi-states corresponding to extreme channels from system A to system B of a fixed Kraus rank r is a smooth submanifold of dimension 2 r dAdB-dA2-r2 of the set of all Choi-states of rank r. As an application, we derive a lower bound on the number of parameters required for a quantum circuit topology to be able to approximate all extreme channels from A to B arbitrarily well.
Lagrangian descriptors of driven chemical reaction manifolds.
Craven, Galen T; Junginger, Andrej; Hernandez, Rigoberto
2017-08-01
The persistence of a transition state structure in systems driven by time-dependent environments allows the application of modern reaction rate theories to solution-phase and nonequilibrium chemical reactions. However, identifying this structure is problematic in driven systems and has been limited by theories built on series expansion about a saddle point. Recently, it has been shown that to obtain formally exact rates for reactions in thermal environments, a transition state trajectory must be constructed. Here, using optimized Lagrangian descriptors [G. T. Craven and R. Hernandez, Phys. Rev. Lett. 115, 148301 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.148301], we obtain this so-called distinguished trajectory and the associated moving reaction manifolds on model energy surfaces subject to various driving and dissipative conditions. In particular, we demonstrate that this is exact for harmonic barriers in one dimension and this verification gives impetus to the application of Lagrangian descriptor-based methods in diverse classes of chemical reactions. The development of these objects is paramount in the theory of reaction dynamics as the transition state structure and its underlying network of manifolds directly dictate reactivity and selectivity.
Projections and residues on manifolds with boundary
DEFF Research Database (Denmark)
Gaarde, Anders Borg
2008-01-01
It is a well-known result that the noncommutative residue of a pseudodifferential projection is zero on a compact manifold without boundary. Equivalently, the value of the zeta-function of P at zero, ¿¿(P, 0), is independent of ¿ for any elliptic operator P. Here ¿ denotes the angle of a ray where...... the resolvent of P has minimal growth. In this thesis, we consider the analogous questions on a compact manifold with boundary. We show that the noncommutative residue is zero for any projection in Boutet de Monvel’s calculus of pseudodifferential boundary problems. For an elliptic boundary problem {P+ + G, T...... }, with the corresponding realization B = (P + G)T, we de¿ne the sectorial projection ¿¿,¿(B) and the residue of this projection. We discuss whether this residue is always zero, through various analyses of the structure of the pro jection. The question is interesting since ¿¿(B, 0) is independent of ¿ exactly when...
Efficient orbit integration by manifold correction methods.
Fukushima, Toshio
2005-12-01
Triggered by a desire to investigate, numerically, the planetary precession through a long-term numerical integration of the solar system, we developed a new formulation of numerical integration of orbital motion named manifold correct on methods. The main trick is to rigorously retain the consistency of physical relations, such as the orbital energy, the orbital angular momentum, or the Laplace integral, of a binary subsystem. This maintenance is done by applying a correction to the integrated variables at each integration step. Typical methods of correction are certain geometric transformations, such as spatial scaling and spatial rotation, which are commonly used in the comparison of reference frames, or mathematically reasonable operations, such as modularization of angle variables into the standard domain [-pi, pi). The form of the manifold correction methods finally evolved are the orbital longitude methods, which enable us to conduct an extremely precise integration of orbital motions. In unperturbed orbits, the integration errors are suppressed at the machine epsilon level for an indefinitely long period. In perturbed cases, on the other hand, the errors initially grow in proportion to the square root of time and then increase more rapidly, the onset of which depends on the type and magnitude of the perturbations. This feature is also realized for highly eccentric orbits by applying the same idea as used in KS-regularization. In particular, the introduction of time elements greatly enhances the performance of numerical integration of KS-regularized orbits, whether the scaling is applied or not.
Manifold-Based Visual Object Counting.
Wang, Yi; Zou, Yuexian; Wang, Wenwu
2018-07-01
Visual object counting (VOC) is an emerging area in computer vision which aims to estimate the number of objects of interest in a given image or video. Recently, object density based estimation method is shown to be promising for object counting as well as rough instance localization. However, the performance of this method tends to degrade when dealing with new objects and scenes. To address this limitation, we propose a manifold-based method for visual object counting (M-VOC), based on the manifold assumption that similar image patches share similar object densities. Firstly, the local geometry of a given image patch is represented linearly by its neighbors using a predefined patch training set, and the object density of this given image patch is reconstructed by preserving the local geometry using locally linear embedding. To improve the characterization of local geometry, additional constraints such as sparsity and non-negativity are also considered via regularization, nonlinear mapping, and kernel trick. Compared with the state-of-the-art VOC methods, our proposed M-VOC methods achieve competitive performance on seven benchmark datasets. Experiments verify that the proposed M-VOC methods have several favorable properties, such as robustness to the variation in the size of training dataset and image resolution, as often encountered in real-world VOC applications.
Manifold corrections on spinning compact binaries
International Nuclear Information System (INIS)
Zhong Shuangying; Wu Xin
2010-01-01
This paper deals mainly with a discussion of three new manifold correction methods and three existing ones, which can numerically preserve or correct all integrals in the conservative post-Newtonian Hamiltonian formulation of spinning compact binaries. Two of them are listed here. One is a new momentum-position scaling scheme for complete consistency of both the total energy and the magnitude of the total angular momentum, and the other is the Nacozy's approach with least-squares correction of the four integrals including the total energy and the total angular momentum vector. The post-Newtonian contributions, the spin effects, and the classification of orbits play an important role in the effectiveness of these six manifold corrections. They are all nearly equivalent to correct the integrals at the level of the machine epsilon for the pure Kepler problem. Once the third-order post-Newtonian contributions are added to the pure orbital part, three of these corrections have only minor effects on controlling the errors of these integrals. When the spin effects are also included, the effectiveness of the Nacozy's approach becomes further weakened, and even gets useless for the chaotic case. In all cases tested, the new momentum-position scaling scheme always shows the optimal performance. It requires a little but not much expensive additional computational cost when the spin effects exist and several time-saving techniques are used. As an interesting case, the efficiency of the correction to chaotic eccentric orbits is generally better than one to quasicircular regular orbits. Besides this, the corrected fast Lyapunov indicators and Lyapunov exponents of chaotic eccentric orbits are large as compared with the uncorrected counterparts. The amplification is a true expression of the original dynamical behavior. With the aid of both the manifold correction added to a certain low-order integration algorithm as a fast and high-precision device and the fast Lyapunov
Stochastic development regression on non-linear manifolds
DEFF Research Database (Denmark)
Kühnel, Line; Sommer, Stefan Horst
2017-01-01
We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion proce...
Erratum to the paper: Compact hyperkaehler manifolds: basic results
Huybrechts, Daniel
2001-01-01
This is an Erratum to the paper: Compact hyperkaehler manifolds: basic results. (alg-geom/9705025, Inv. math. 135). We give a correct proof of the projectivity criterion for hyperkaehler manifolds. We use a recent result of Demailly and Paun math.AG/0105176.
Conformal Vector Fields on Doubly Warped Product Manifolds and Applications
Directory of Open Access Journals (Sweden)
H. K. El-Sayied
2016-01-01
Full Text Available This article aimed to study and explore conformal vector fields on doubly warped product manifolds as well as on doubly warped spacetime. Then we derive sufficient conditions for matter and Ricci collineations on doubly warped product manifolds. A special attention is paid to concurrent vector fields. Finally, Ricci solitons on doubly warped product spacetime admitting conformal vector fields are considered.
The quantum equivariant cohomology of toric manifolds through mirror symmetry
Baptista, J. M.
2008-01-01
Using mirror symmetry as described by Hori and Vafa, we compute the quantum equivariant cohomology ring of toric manifolds. This ring arises naturally in topological gauged sigma-models and is related to the Hamiltonian Gromov-Witten invariants of the target manifold.
Flow and Pressure Distribution in Fuel Cell Manifolds
DEFF Research Database (Denmark)
Lebæk, Jesper; Bang, Mads; Kær, Søren Knudsen
2010-01-01
The manifold is an essential part of the fuel cell stack. Evidently, evenly distributed reactants are a prerequisite for an efficient fuel cell stack. In this study, the cathode manifold ability to distribute air to the cells of a 70 cell stack is investigated experimentally. By means of 20...
Variable volume combustor with nested fuel manifold system
McConnaughhay, Johnie Franklin; Keener, Christopher Paul; Johnson, Thomas Edward; Ostebee, Heath Michael
2016-09-13
The present application provides a combustor for use with a gas turbine engine. The combustor may include a number of micro-mixer fuel nozzles, a fuel manifold system in communication with the micro-mixer fuel nozzles to deliver a flow of fuel thereto, and a linear actuator to maneuver the micro-mixer fuel nozzles and the fuel manifold system.
A new proof of the theorem: Harmonic manifolds with minimal ...
Indian Academy of Sciences (India)
In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting theorem.
Manifold mapping: a two-level optimization technique
Echeverría, D.; Hemker, P.W.
2008-01-01
In this paper, we analyze in some detail the manifold-mapping optimization technique introduced recently [Echeverría and Hemker in space mapping and defect correction. Comput Methods Appl Math 5(2): 107--136, 2005]. Manifold mapping aims at accelerating optimal design procedures that otherwise
Manifold mapping: a two-level optimization technique
D. Echeverria (David); P.W. Hemker (Piet)
2008-01-01
textabstractIn this paper, we analyze in some detail the manifold-mapping optimization technique introduced recently [Echeverría and Hemker in space mapping and defect correction. Comput Methods Appl Math 5(2): 107-–136, 2005]. Manifold mapping aims at accelerating optimal design procedures
The structure of some classes of K-contact manifolds
Indian Academy of Sciences (India)
metric manifold satisfying the case (1), (2) and (3) is said to be conformally symmetric. [8], ξ-conformally flat [9] and ϕ-conformally flat [3] respectively. In [8], it is proved that a conformally symmetric K-contact manifold is locally isometric to the unit sphere. In ... derivative of ϕ in the characteristic direction ξ vanishes.
Metric Relativity and the Dynamical Bridge: Highlights of Riemannian Geometry in Physics
Novello, Mario; Bittencourt, Eduardo
2015-12-01
We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric widehat {g}_{μ ν } which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associated widehat {g} geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces—beyond the uses of General Relativity that is limited only to gravitational processes—is nothing but the relativistic version of the d'Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries—one for each different body in the case of non-gravitational forces—to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical
Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics
Energy Technology Data Exchange (ETDEWEB)
Novello, Mario [Centro Brasileiro de Pesquisas Fisicas (ICRA/CBPF), Rio de Janeiro, RJ (Brazil). Instituto de Cosmologia Relatividade e Astrofisica; Bittencourt, Eduardo, E-mail: eduardo.bittencourt@icranet.org [Physics Department, La Sapienza University of Rome (Italy)
2015-12-15
We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric ĝ μ υ which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associatedĝ geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces - beyond the uses of General Relativity that is limited only to gravitational processes - is nothing but the relativistic version of the d'Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries-one for each different body in the case of non-gravitational forces-to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical Bridge formalism
Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics
International Nuclear Information System (INIS)
Novello, Mario
2015-01-01
We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric ĝ μ υ which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associatedĝ geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces - beyond the uses of General Relativity that is limited only to gravitational processes - is nothing but the relativistic version of the d'Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries-one for each different body in the case of non-gravitational forces-to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical Bridge formalism
Instantons on sine-cones over Sasakian manifolds
Bunk, Severin; Ivanova, Tatiana A.; Lechtenfeld, Olaf; Popov, Alexander D.; Sperling, Marcus
2014-09-01
We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian manifolds. It is shown that these conical Einstein manifolds are Kähler with torsion (KT) manifolds admitting Hermitian connections with totally antisymmetric torsion. Furthermore, a deformation of the metric on the sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-Kähler with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces become Calabi-Yau and hyper-Kähler conifolds, respectively. We construct gauge connections on complex vector bundles over conical KT and HKT manifolds which solve the instanton equations for Yang-Mills fields in higher dimensions.
Model Transport: Towards Scalable Transfer Learning on Manifolds
DEFF Research Database (Denmark)
Freifeld, Oren; Hauberg, Søren; Black, Michael J.
2014-01-01
We consider the intersection of two research fields: transfer learning and statistics on manifolds. In particular, we consider, for manifold-valued data, transfer learning of tangent-space models such as Gaussians distributions, PCA, regression, or classifiers. Though one would hope to simply use...... ordinary Rn-transfer learning ideas, the manifold structure prevents it. We overcome this by basing our method on inner-product-preserving parallel transport, a well-known tool widely used in other problems of statistics on manifolds in computer vision. At first, this straightforward idea seems to suffer...... “commutes” with learning. Consequently, our compact framework, applicable to a large class of manifolds, is not restricted by the size of either the training or test sets. We demonstrate the approach by transferring PCA and logistic-regression models of real-world data involving 3D shapes and image...
Contact manifolds, Lagrangian Grassmannians and PDEs
Directory of Open Access Journals (Sweden)
Eshkobilov Olimjon
2018-02-01
Full Text Available In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.. As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.
Geometric solitons of Hamiltonian flows on manifolds
Energy Technology Data Exchange (ETDEWEB)
Song, Chong, E-mail: songchong@xmu.edu.cn [School of Mathematical Sciences, Xiamen University, Xiamen 361005 (China); Sun, Xiaowei, E-mail: sunxw@cufe.edu.cn [School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081 (China); Wang, Youde, E-mail: wyd@math.ac.cn [Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 (China)
2013-12-15
It is well-known that the LIE (Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of isometries of the domain and the target space, respectively. With this insight, we propose the new concept of geometric solitons of Hamiltonian flows on manifolds, such as geometric Schrödinger flows and KdV flows for maps. Moreover, we give several examples of geometric solitons of the Schrödinger flow and geometric KdV flow, including magnetic curves as geometric Schrödinger solitons and explicit geometric KdV solitons on surfaces of revolution.
Evolutionary global optimization, manifolds and applications
Aguiar e Oliveira Junior, Hime
2016-01-01
This book presents powerful techniques for solving global optimization problems on manifolds by means of evolutionary algorithms, and shows in practice how these techniques can be applied to solve real-world problems. It describes recent findings and well-known key facts in general and differential topology, revisiting them all in the context of application to current optimization problems. Special emphasis is put on game theory problems. Here, these problems are reformulated as constrained global optimization tasks and solved with the help of Fuzzy ASA. In addition, more abstract examples, including minimizations of well-known functions, are also included. Although the Fuzzy ASA approach has been chosen as the main optimizing paradigm, the book suggests that other metaheuristic methods could be used as well. Some of them are introduced, together with their advantages and disadvantages. Readers should possess some knowledge of linear algebra, and of basic concepts of numerical analysis and probability theory....
Manifold Adaptive Label Propagation for Face Clustering.
Pei, Xiaobing; Lyu, Zehua; Chen, Changqing; Chen, Chuanbo
2015-08-01
In this paper, a novel label propagation (LP) method is presented, called the manifold adaptive label propagation (MALP) method, which is to extend original LP by integrating sparse representation constraint into regularization framework of LP method. Similar to most LP, first of all, MALP also finds graph edges from given data and gives weights to the graph edges. Our goal is to find graph weights matrix adaptively. The key advantage of our approach is that MALP simultaneously finds graph weights matrix and predicts the label of unlabeled data. This paper also derives efficient algorithm to solve the proposed problem. Extensions of our MALP in kernel space and robust version are presented. The proposed method has been applied to the problem of semi-supervised face clustering using the well-known ORL, Yale, extended YaleB, and PIE datasets. Our experimental evaluations show the effectiveness of our method.
Consistent Pauli reduction on group manifolds
Baguet, A.; Pope, C. N.; Samtleben, H.
2016-01-01
We prove an old conjecture by Duff, Nilsson, Pope and Warner asserting that the NSsbnd NS sector of supergravity (and more general the bosonic string) allows for a consistent Pauli reduction on any d-dimensional group manifold G, keeping the full set of gauge bosons of the G × G isometry group of the bi-invariant metric on G. The main tool of the construction is a particular generalised Scherk-Schwarz reduction ansatz in double field theory which we explicitly construct in terms of the group's Killing vectors. Examples include the consistent reduction from ten dimensions on S3 ×S3 and on similar product spaces. The construction is another example of globally geometric non-toroidal compactifications inducing non-geometric fluxes.
Tensors and Manifolds With Applications to Physics (2nd edn)
International Nuclear Information System (INIS)
Dray, T
2005-01-01
On the one hand, this is an excellent introduction for mathematicians to the differential geometry underlying general relativity. On the other hand, this is definitely a book for mathematicians. The book's greatest strength is its clear, precise presentation of the basic ideas in differential geometry, combined with equally clear and precise applications to theoretical physics, notably general relativity. But the book's precision is also its greatest weakness; this is not an easy book to read for non-mathematicians, who may not appreciate the notational complexity, some of which is nonstandard. The present edition is very similar to the original, published in 1992. In addition to minor revisions and clarifications of the material, there is now a brief introduction to fibre bundles, and a (very) brief discussion of the gauge theory description of fundamental particles. The index to the symbols used is also a more complete than in the past, but without the descriptive material present in the previous edition. The bulk of the book consists of a careful introduction to tensors and their properties. Tensors are introduced first as linear maps on vector spaces, and only later generalized to tensor fields on manifolds. The differentiation and integration of differential forms is discussed in detail, including Stokes' theorem, Lie differentiation and Hodge duality, and connections, curvature and torsion. To this point, Wasserman's text can be viewed as an expanded version of Bishop and Goldberg's classic text, one major difference being Wasserman's inclusion of the pseudo-Riemannian case from the beginning (in particular, when discussing Hodge duality). Whether one prefers Wasserman's approach to Bishop and Goldberg's is largely a matter of taste: Wasserman's treatment is both more complete and more precise, making it easier to check calculations in detail, but occasionally more difficult to remember what one is calculating. An instructor using this text would be well
Tensors and Manifolds With Applications to Physics (2nd edn)
Energy Technology Data Exchange (ETDEWEB)
Dray, T [Oregon State University (United States)
2005-10-21
On the one hand, this is an excellent introduction for mathematicians to the differential geometry underlying general relativity. On the other hand, this is definitely a book for mathematicians. The book's greatest strength is its clear, precise presentation of the basic ideas in differential geometry, combined with equally clear and precise applications to theoretical physics, notably general relativity. But the book's precision is also its greatest weakness; this is not an easy book to read for non-mathematicians, who may not appreciate the notational complexity, some of which is nonstandard. The present edition is very similar to the original, published in 1992. In addition to minor revisions and clarifications of the material, there is now a brief introduction to fibre bundles, and a (very) brief discussion of the gauge theory description of fundamental particles. The index to the symbols used is also a more complete than in the past, but without the descriptive material present in the previous edition. The bulk of the book consists of a careful introduction to tensors and their properties. Tensors are introduced first as linear maps on vector spaces, and only later generalized to tensor fields on manifolds. The differentiation and integration of differential forms is discussed in detail, including Stokes' theorem, Lie differentiation and Hodge duality, and connections, curvature and torsion. To this point, Wasserman's text can be viewed as an expanded version of Bishop and Goldberg's classic text, one major difference being Wasserman's inclusion of the pseudo-Riemannian case from the beginning (in particular, when discussing Hodge duality). Whether one prefers Wasserman's approach to Bishop and Goldberg's is largely a matter of taste: Wasserman's treatment is both more complete and more precise, making it easier to check calculations in detail, but occasionally more difficult to remember what one is calculating. An
Stochastic development regression on non-linear manifolds
DEFF Research Database (Denmark)
Kühnel, Line; Sommer, Stefan Horst
2017-01-01
We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion...... processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded...
Geometric transitions, flops and non-Kahler manifolds: I
International Nuclear Information System (INIS)
Becker, Melanie; Dasgupta, Keshav; Knauf, Anke; Tatar, Radu
2004-01-01
We construct a duality cycle which provides a complete supergravity description of geometric transitions in type II theories via a flop in M-theory. This cycle connects the different supergravity descriptions before and after the geometric transitions. Our construction reproduces many of the known phenomena studied earlier in the literature and allows us to describe some new and interesting aspects in a simple and elegant fashion. A precise supergravity description of new torsional manifolds that appear on the type IIA side with branes and fluxes and the corresponding geometric transition are obtained. A local description of new G2 manifolds that are circle fibrations over non-Kahler manifolds is presented
Scientific data interpolation with low dimensional manifold model
International Nuclear Information System (INIS)
Zhu, Wei; Wang, Bao; Barnard, Richard C.; Hauck, Cory D.
2017-01-01
Here, we propose to apply a low dimensional manifold model to scientific data interpolation from regular and irregular samplings with a significant amount of missing information. The low dimensionality of the patch manifold for general scientific data sets has been used as a regularizer in a variational formulation. The problem is solved via alternating minimization with respect to the manifold and the data set, and the Laplace–Beltrami operator in the Euler–Lagrange equation is discretized using the weighted graph Laplacian. Various scientific data sets from different fields of study are used to illustrate the performance of the proposed algorithm on data compression and interpolation from both regular and irregular samplings.
Scientific data interpolation with low dimensional manifold model
Zhu, Wei; Wang, Bao; Barnard, Richard; Hauck, Cory D.; Jenko, Frank; Osher, Stanley
2018-01-01
We propose to apply a low dimensional manifold model to scientific data interpolation from regular and irregular samplings with a significant amount of missing information. The low dimensionality of the patch manifold for general scientific data sets has been used as a regularizer in a variational formulation. The problem is solved via alternating minimization with respect to the manifold and the data set, and the Laplace-Beltrami operator in the Euler-Lagrange equation is discretized using the weighted graph Laplacian. Various scientific data sets from different fields of study are used to illustrate the performance of the proposed algorithm on data compression and interpolation from both regular and irregular samplings.
Solution path for manifold regularized semisupervised classification.
Wang, Gang; Wang, Fei; Chen, Tao; Yeung, Dit-Yan; Lochovsky, Frederick H
2012-04-01
Traditional learning algorithms use only labeled data for training. However, labeled examples are often difficult or time consuming to obtain since they require substantial human labeling efforts. On the other hand, unlabeled data are often relatively easy to collect. Semisupervised learning addresses this problem by using large quantities of unlabeled data with labeled data to build better learning algorithms. In this paper, we use the manifold regularization approach to formulate the semisupervised learning problem where a regularization framework which balances a tradeoff between loss and penalty is established. We investigate different implementations of the loss function and identify the methods which have the least computational expense. The regularization hyperparameter, which determines the balance between loss and penalty, is crucial to model selection. Accordingly, we derive an algorithm that can fit the entire path of solutions for every value of the hyperparameter. Its computational complexity after preprocessing is quadratic only in the number of labeled examples rather than the total number of labeled and unlabeled examples.
Quantization of a symplectic manifold associated to a manifold with projective structure
International Nuclear Information System (INIS)
Biswas, Indranil
2009-01-01
Let X be a complex manifold equipped with a projective structure P. There is a holomorphic principal C*-bundle L P ' over X associated with P. We show that the holomorphic cotangent bundle of the total space of L P ' equipped with the Liouville symplectic form has a canonical deformation quantization. This generalizes the construction in the work of and Ben-Zvi and Biswas [''A quantization on Riemann surfaces with projective structure,'' Lett. Math. Phys. 54, 73 (2000)] done under the assumption that dim C X=1.
Spatial context driven manifold learning for hyperspectral image classification
CSIR Research Space (South Africa)
Zhang, Y
2014-06-01
Full Text Available Department of Electrical and Computer Engineering, University of Houston. 2 Meraka Institute, Council for Scientific and Industrial Research, South Africa. 3 School of Civil Engineering, Purdue University, US. Abstract Manifold learning techniques have...
Manifold learning based feature extraction for classification of hyperspectral data
CSIR Research Space (South Africa)
Lunga, D
2014-01-01
Full Text Available of Electrical and Computer Engineering, University of Houston. 3. Schools of Civil Engineering and Electrical and Computer Engineering, Purdue University. Interest in manifold learning for representing the topology of large, high dimensional nonlinear data sets...
46 CFR 153.285 - Valving for cargo pump manifolds.
2010-10-01
... SHIPS CARRYING BULK LIQUID, LIQUEFIED GAS, OR COMPRESSED GAS HAZARDOUS MATERIALS Design and Equipment Piping Systems and Cargo Handling Equipment § 153.285 Valving for cargo pump manifolds. (a) When cargo...
Supervised learning for neural manifold using spatiotemporal brain activity.
Kuo, Po-Chih; Chen, Yong-Sheng; Chen, Li-Fen
2015-12-01
Determining the means by which perceived stimuli are compactly represented in the human brain is a difficult task. This study aimed to develop techniques for the construction of the neural manifold as a representation of visual stimuli. We propose a supervised locally linear embedding method to construct the embedded manifold from brain activity, taking into account similarities between corresponding stimuli. In our experiments, photographic portraits were used as visual stimuli and brain activity was calculated from magnetoencephalographic data using a source localization method. The results of 10 × 10-fold cross-validation revealed a strong correlation between manifolds of brain activity and the orientation of faces in the presented images, suggesting that high-level information related to image content can be revealed in the brain responses represented in the manifold. Our experiments demonstrate that the proposed method is applicable to investigation into the inherent patterns of brain activity.
Some functional inequalities on non-reversible Finsler manifolds
Indian Academy of Sciences (India)
SHIN-ICHI OHTA
2017-11-13
). Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the -calculus of Bakry et al (2014), we show the dimensional versions of the Poincaré–Lichnerowicz inequality, ...
Example-driven manifold priors for image deconvolution.
Ni, Jie; Turaga, Pavan; Patel, Vishal M; Chellappa, Rama
2011-11-01
Image restoration methods that exploit prior information about images to be estimated have been extensively studied, typically using the Bayesian framework. In this paper, we consider the role of prior knowledge of the object class in the form of a patch manifold to address the deconvolution problem. Specifically, we incorporate unlabeled image data of the object class, say natural images, in the form of a patch-manifold prior for the object class. The manifold prior is implicitly estimated from the given unlabeled data. We show how the patch-manifold prior effectively exploits the available sample class data for regularizing the deblurring problem. Furthermore, we derive a generalized cross-validation (GCV) function to automatically determine the regularization parameter at each iteration without explicitly knowing the noise variance. Extensive experiments show that this method performs better than many competitive image deconvolution methods.
Manopt, a Matlab toolbox for optimization on manifolds
Boumal, Nicolas; Mishra, Bamdev; Absil, P. -A.; Sepulchre, Rodolphe
2013-01-01
Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design effcient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, in...
Ideal triangulations of 3-manifolds II: taut and angle structures
Kang, Ensil; Rubinstein, J. Hyam
2005-01-01
This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program SNAPPEA of Weeks and the program SNAP of Coulson, Goodman, Hodgson and Neumann. Casson has also written a program to find hyperbolic structures on such 3-manifolds, by solving T...
Reduction of Nambu-Poisson Manifolds by Regular Distributions
Das, Apurba
2018-03-01
The version of Marsden-Ratiu reduction theorem for Nambu-Poisson manifolds by a regular distribution has been studied by Ibáñez et al. In this paper we show that the reduction is always ensured unless the distribution is zero. Next we extend the more general Falceto-Zambon Poisson reduction theorem for Nambu-Poisson manifolds. Finally, we define gauge transformations of Nambu-Poisson structures and show that these transformations commute with the reduction procedure.
Integrated high pressure manifold for thermoplastic microfluidic devices
Aghvami, S. Ali; Fraden, Seth
2017-11-01
We introduce an integrated tubing manifold for thermoplastic microfluidic chips that tolerates high pressure. In contrast to easy tubing in PDMS microfluidic devices, tube connection has been challenging for plastic microfluidics. Our integrated manifold connection tolerates 360 psi while conventional PDMS connections fail at 50 psi. Important design considerations are incorporation of a quick-connect, leak-free and high-pressure manifold for the inlets and outlets on the lid and registration marks that allow the precise alignment of the inlets and outlets. In our method, devices are comprised of two molded pieces joined together to create a sealed device. The first piece contains the microfluidic features and the second contains the inlet and outlet manifold, a frame for rigidity and a viewing window. The mold for the lid with integrated manifold is CNC milled from aluminium. A cone shape PDMS component which acts as an O-ring, seals the connection between molded manifold and tubing. The lid piece with integrated inlet and outlets will be a standard piece and can be used for different chips and designs. Sealing the thermoplastic device is accomplished by timed immersion of the lid in a mixture of volatile and non-volatile solvents followed by application of heat and pressure.
Dimensionality reduction of collective motion by principal manifolds
Gajamannage, Kelum; Butail, Sachit; Porfiri, Maurizio; Bollt, Erik M.
2015-01-01
While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods is not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the mapping from the original high-dimensional space to the embedding space. Here, we propose an alternative approach that demands a two-dimensional embedding which topologically summarizes the high-dimensional data. In this sense, our approach is closely related to the construction of one-dimensional principal curves that minimize orthogonal error to data points subject to smoothness constraints. Specifically, we construct a two-dimensional principal manifold directly in the high-dimensional space using cubic smoothing splines, and define the embedding coordinates in terms of geodesic distances. Thus, the mapping from the high-dimensional data to the manifold is defined in terms of local coordinates. Through representative examples, we show that compared to existing nonlinear dimensionality reduction methods, the principal manifold retains the original structure even in noisy and sparse datasets. The principal manifold finding algorithm is applied to configurations obtained from a dynamical system of multiple agents simulating a complex maneuver called predator mobbing, and the resulting two-dimensional embedding is compared with that of a well-established nonlinear dimensionality reduction method.
Semisupervised Support Vector Machines With Tangent Space Intrinsic Manifold Regularization.
Sun, Shiliang; Xie, Xijiong
2016-09-01
Semisupervised learning has been an active research topic in machine learning and data mining. One main reason is that labeling examples is expensive and time-consuming, while there are large numbers of unlabeled examples available in many practical problems. So far, Laplacian regularization has been widely used in semisupervised learning. In this paper, we propose a new regularization method called tangent space intrinsic manifold regularization. It is intrinsic to data manifold and favors linear functions on the manifold. Fundamental elements involved in the formulation of the regularization are local tangent space representations, which are estimated by local principal component analysis, and the connections that relate adjacent tangent spaces. Simultaneously, we explore its application to semisupervised classification and propose two new learning algorithms called tangent space intrinsic manifold regularized support vector machines (TiSVMs) and tangent space intrinsic manifold regularized twin SVMs (TiTSVMs). They effectively integrate the tangent space intrinsic manifold regularization consideration. The optimization of TiSVMs can be solved by a standard quadratic programming, while the optimization of TiTSVMs can be solved by a pair of standard quadratic programmings. The experimental results of semisupervised classification problems show the effectiveness of the proposed semisupervised learning algorithms.
STAR CLUSTERS, GALAXIES, AND THE FUNDAMENTAL MANIFOLD
International Nuclear Information System (INIS)
Zaritsky, Dennis; Zabludoff, Ann I.; Gonzalez, Anthony H.
2011-01-01
We explore whether global observed properties, specifically half-light radii, mean surface brightness, and integrated stellar kinematics, suffice to unambiguously differentiate galaxies from star clusters, which presumably formed differently and lack dark matter halos. We find that star clusters lie on the galaxy scaling relationship referred to as the fundamental manifold (FM), on the extension of a sequence of compact galaxies, and so conclude that there is no simple way to differentiate star clusters from ultracompact galaxies. By extending the validity of the FM over a larger range of parameter space and a wider set of objects, we demonstrate that the physics that constrains the resulting baryon and dark matter distributions in stellar systems is more general than previously appreciated. The generality of the FM implies (1) that the stellar spatial distribution and kinematics of one type of stellar system do not arise solely from a process particular to that set of systems, such as violent relaxation for elliptical galaxies, but are instead the result of an interplay of all processes responsible for the generic settling of baryons in gravitational potential wells, (2) that the physics of how baryons settle is independent of whether the system is embedded within a dark matter halo, and (3) that peculiar initial conditions at formation or stochastic events during evolution do not ultimately disturb the overall regularity of baryonic settling. We also utilize the relatively simple nature of star clusters to relate deviations from the FM to the age of the stellar population and find that stellar population models systematically and significantly overpredict the mass-to-light ratios of old, metal-rich clusters. We present an empirical calibration of stellar population mass-to-light ratios with age and color. Finally, we use the FM to estimate velocity dispersions for the low surface brightness, outer halo clusters that lack such measurements.
The world problem: on the computability of the topology of 4-manifolds
vanMeter, J. R.
2005-01-01
Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing machine with an arbitrary input can be encoded into the topology of a 4-manifold, such that the 4-manifold is homeomorphic to a certain other 4-manifold if and only if the corresponding Turing machine halts on the associated input. Physical implications are briefly discussed.
International Nuclear Information System (INIS)
Audretsch, J.; Gaehler, F.; Straumann, N.
1984-01-01
Previous axiomatic approaches to general relativity which led to a Weylian structure of space-time are supplemented by a physical condition which implies the existence of a preferred pseudo-Riemannian structure. It is stipulated that the trajectories of the short wave limit of classical massive fields agree with the geodesics of the Weyl connection and it is shown that this is equivalent to the vanishing of the covariant derivative of a ''mass function'' of nontrivial Weyl type.This in turn is proven to be equivalent to the existence of a preferred metric of the conformal structure such that the Weyl connection is reducible to a connection of the bundle of orthonormal frames belonging to this distinguished metric. (orig.)
Algebras and manifolds: Differential, difference, simplicial and quantum
International Nuclear Information System (INIS)
Finkelstein, D.; Rodriguez, E.
1986-01-01
Generalized manifolds and Clifford algebras depict the world at levels of resolution ranging from the classical macroscopic to the quantum microscopic. The coarsest picture is a differential manifold and algebra (dm), direct integral of familiar local Clifford algebras of spin operators in curved time-space. Next is a finite difference manifold (Δm) of Regge calculus. This is a subalgebra of the third, a Minkowskian simplicial manifold (Σm). The most detailed description is the quantum manifold (Qm), whose algebra is the free Clifford algebra S of quantum set theory. We surmise that each Σm is a classical 'condensation' of a Qm. Quantum simplices have both integer and half-integer spins in their spectrum. A quantum set theory of nature requires a series of reductions leading from the Qm and a world descriptor W up through the intermediate Σm and Δm to a dm and an action principle. What may be a new algebraic language for topology, classical or quantum, is a by-product of the work. (orig.)
Hierarchical discriminant manifold learning for dimensionality reduction and image classification
Chen, Weihai; Zhao, Changchen; Ding, Kai; Wu, Xingming; Chen, Peter C. Y.
2015-09-01
In the field of image classification, it has been a trend that in order to deliver a reliable classification performance, the feature extraction model becomes increasingly more complicated, leading to a high dimensionality of image representations. This, in turn, demands greater computation resources for image classification. Thus, it is desirable to apply dimensionality reduction (DR) methods for image classification. It is necessary to apply DR methods to relieve the computational burden as well as to improve the classification accuracy. However, traditional DR methods are not compatible with modern feature extraction methods. A framework that combines manifold learning based DR and feature extraction in a deeper way for image classification is proposed. A multiscale cell representation is extracted from the spatial pyramid to satisfy the locality constraints for a manifold learning method. A spectral weighted mean filtering is proposed to eliminate noise in the feature space. A hierarchical discriminant manifold learning is proposed which incorporates both category label and image scale information to guide the DR process. Finally, the image representation is generated by concatenating dimensionality reduced cell representations from the same image. Extensive experiments are conducted to test the proposed algorithm on both scene and object recognition datasets in comparison with several well-established and state-of-the-art methods with respect to classification precision and computational time. The results verify the effectiveness of incorporating manifold learning in the feature extraction procedure and imply that the multiscale cell representations may be distributed on a manifold.
Quasi-Newton Exploration of Implicitly Constrained Manifolds
Tang, Chengcheng
2011-08-01
A family of methods for the efficient update of second order approximations of a constraint manifold is proposed in this thesis. The concept of such a constraint manifold corresponds to an abstract space prescribed by implicit nonlinear constraints, which can be a set of objects satisfying certain desired properties. This concept has a variety of applications, and it has been successfully introduced to fabrication-aware architectural design as a shape space consisting of all the implementable designs. The local approximation of such a manifold can be first order, in the tangent space, or second order, in the osculating surface, with higher precision. For a nonlinearly constrained manifold with rather high dimension and codimension, the computation of second order approximants (osculants) is time consuming. In this thesis, a type of so-called quasi-Newton manifold exploration methods which approximate the new osculants by updating the ones of a neighbor point by 1st-order information is introduced. The procedures are discussed in detail and the examples implemented to visually verify the methods are illustrated.
Enhanced manifold regularization for semi-supervised classification.
Gan, Haitao; Luo, Zhizeng; Fan, Yingle; Sang, Nong
2016-06-01
Manifold regularization (MR) has become one of the most widely used approaches in the semi-supervised learning field. It has shown superiority by exploiting the local manifold structure of both labeled and unlabeled data. The manifold structure is modeled by constructing a Laplacian graph and then incorporated in learning through a smoothness regularization term. Hence the labels of labeled and unlabeled data vary smoothly along the geodesics on the manifold. However, MR has ignored the discriminative ability of the labeled and unlabeled data. To address the problem, we propose an enhanced MR framework for semi-supervised classification in which the local discriminative information of the labeled and unlabeled data is explicitly exploited. To make full use of labeled data, we firstly employ a semi-supervised clustering method to discover the underlying data space structure of the whole dataset. Then we construct a local discrimination graph to model the discriminative information of labeled and unlabeled data according to the discovered intrinsic structure. Therefore, the data points that may be from different clusters, though similar on the manifold, are enforced far away from each other. Finally, the discrimination graph is incorporated into the MR framework. In particular, we utilize semi-supervised fuzzy c-means and Laplacian regularized Kernel minimum squared error for semi-supervised clustering and classification, respectively. Experimental results on several benchmark datasets and face recognition demonstrate the effectiveness of our proposed method.
Trajectory design using periapse maps and invariant manifolds
Haapala, Amanda F.
The invariant manifolds associated with periodic orbits in the vicinity of the collinear libration points in the planar CR3BP have been previously demonstrated as mechanisms for transport. Trajectories that pass between adjoining regions within the zero-velocity curves pass through the invariant manifold tubes. In particular, the invariant manifolds associated with the unstable L1 and L2 periodic libration point orbits may be exploited to construct transit orbits between the interior and exterior regions associated with the zero-velocity curves. In this investigation, periapse Poincare maps are used to display the manifolds and to distinguish regions of escape and, conversely, regions of long-term capture. Manifold periapse structures are employed as a design tool to construct planar trajectories with predetermined characteristics. The strategies that are developed are demonstrated by producing planar trajectories with predetermined behaviors, namely, long-term capture orbits and transit trajectories, as well as heteroclinic and homoclinic connections. Additionally, path approximations are generated for four Jupiter family comets that experience temporary satellite capture. Periapse Poincare maps are also employed to design three-dimensional transit trajectories in the spatial circular restricted three-body problem.
Light transport on path-space manifolds
Jakob, Wenzel Alban
-stepping limitations of the theory, they often suffer from unusably slow convergence; improvements to this situation have been hampered by the lack of a thorough theoretical understanding. We address these problems by developing a new theory of path-space light transport which, for the first time, cleanly incorporates specular scattering into the standard framework. Most of the results obtained in the analysis of the ideally smooth case can also be generalized to rendering of glossy materials and volumetric scattering so that this dissertation also provides a powerful new set of tools for dealing with them. The basis of our approach is that each specular material interaction locally collapses the dimension of the space of light paths so that all relevant paths lie on a submanifold of path space. We analyze the high-dimensional differential geometry of this submanifold and use the resulting information to construct an algorithm that is able to "walk" around on it using a simple and efficient equation-solving iteration. This manifold walking algorithm then constitutes the key operation of a new type of Markov Chain Monte Carlo (MCMC) rendering method that computes lighting through very general families of paths that can involve arbitrary combinations of specular, near-specular, glossy, and diffuse surface interactions as well as isotropic or highly anisotropic volume scattering. We demonstrate our implementation on a range of challenging scenes and evaluate it against previous methods.
Differential forms and the Wodzicki residue for manifolds with boundary
Wang, Yong
2006-05-01
In [A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris,1994), 15-36, Internat Press, Cambridge, MA, 1995], Connes found a conformal invariant using Wodzicki's 1-density and computed it in the case of 4-dimensional manifold without boundary. In [W. J. Ugalde, Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361], Ugalde generalized the Connes' result to n-dimensional manifold without boundary. In this paper, we generalize the results of [A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris,1994), 15-36, Internat Press, Cambridge, MA, 1995] and [W. J. Ugalde, Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361] to the case of manifolds with boundary.
Schoen manifold with line bundles as resolved magnetized orbifolds
Energy Technology Data Exchange (ETDEWEB)
Groot Nibbelink, Stefan [Muenchen Univ. (Germany). Arnold Sommerfeld Center for Theoretical Physics; Vaudrevange, Patrick K.S. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-12-15
We give an alternative description of the Schoen manifold as the blow-up of a Z{sub 2} x Z{sub 2} orbifold in which one Z{sub 2} factor acts as a roto-translation. Since for this orbifold the fixed tori are only identified in pairs but not orbifolded, four-dimensional chirality can never be obtained using standard techniques alone. However, chirality is recovered when its tori become magnetized. To exemplify this, we construct an SU(5) GUT on the Schoen manifold with Abelian gauge fluxes, which becomes an MSSM with three generations after an appropriate Wilson line is associated to its freely acting involution. We reproduce this model as a standard orbifold CFT of the (partially) blown down Schoen manifold with a magnetic flux. Finally, in analogy to a proposal for non-perturbative heterotic models by Aldazabal et al. we suggest modifications to the heterotic orbifold spectrum formulae in the presence of magnetized tori.
Adaptive Sampling for Nonlinear Dimensionality Reduction Based on Manifold Learning
DEFF Research Database (Denmark)
Franz, Thomas; Zimmermann, Ralf; Goertz, Stefan
2017-01-01
We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space that is approxi......We make use of the non-intrusive dimensionality reduction method Isomap in order to emulate nonlinear parametric flow problems that are governed by the Reynolds-averaged Navier-Stokes equations. Isomap is a manifold learning approach that provides a low-dimensional embedding space...... that is approximately isometric to the manifold that is assumed to be formed by the high-fidelity Navier-Stokes flow solutions under smooth variations of the inflow conditions. The focus of the work at hand is the adaptive construction and refinement of the Isomap emulator: We exploit the non-Euclidean Isomap metric...
Multiscale singular value manifold for rotating machinery fault diagnosis
Energy Technology Data Exchange (ETDEWEB)
Feng, Yi; Lu, BaoChun; Zhang, Deng Feng [School of Mechanical Engineering, Nanjing University of Science and Technology,Nanjing (United States)
2017-01-15
Time-frequency distribution of vibration signal can be considered as an image that contains more information than signal in time domain. Manifold learning is a novel theory for image recognition that can be also applied to rotating machinery fault pattern recognition based on time-frequency distributions. However, the vibration signal of rotating machinery in fault condition contains cyclical transient impulses with different phrases which are detrimental to image recognition for time-frequency distribution. To eliminate the effects of phase differences and extract the inherent features of time-frequency distributions, a multiscale singular value manifold method is proposed. The obtained low-dimensional multiscale singular value manifold features can reveal the differences of different fault patterns and they are applicable to classification and diagnosis. Experimental verification proves that the performance of the proposed method is superior in rotating machinery fault diagnosis.
Unimodularity criteria for Poisson structures on foliated manifolds
Pedroza, Andrés; Velasco-Barreras, Eduardo; Vorobiev, Yury
2018-03-01
We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.
Integrable G2 Structures on 7-dimensional 3-Sasakian Manifolds
Directory of Open Access Journals (Sweden)
Nülifer ÖZDEMİR
2017-02-01
Full Text Available It is known that there exist canonical and nearly parallel $G_2$ structures on 7-dimensional 3-Sasakian manifolds. In this paper, we investigate the existence of $G_2$ structures which are neither canonical nor nearly parallel. We obtain eight new $G_2$ structures on 7-dimensional 3-Sasakian manifolds which are of general type according to the classification of $G_2$ structures by Fernandez and Gray. Then by deforming the metric determined by the $G_2$ structure, we give integrable $G_2$ structures. On a manifold with integrable $G_2$ structure, there exists a uniquely determined metric covariant derivative with anti-symetric torsion. We write torsion tensors corresponding to metric covariant derivatives with skew-symmetric torsion. In addition, we investigate some properties of torsion tensors.
Postoperative 3D spine reconstruction by navigating partitioning manifolds
Energy Technology Data Exchange (ETDEWEB)
Kadoury, Samuel, E-mail: samuel.kadoury@polymtl.ca [Department of Computer and Software Engineering, Ecole Polytechnique Montreal, Montréal, Québec H3C 3A7 (Canada); Labelle, Hubert, E-mail: hubert.labelle@recherche-ste-justine.qc.ca; Parent, Stefan, E-mail: stefan.parent@umontreal.ca [CHU Sainte-Justine Hospital Research Center, Montréal, Québec H3T 1C5 (Canada)
2016-03-15
Purpose: The postoperative evaluation of scoliosis patients undergoing corrective treatment is an important task to assess the strategy of the spinal surgery. Using accurate 3D geometric models of the patient’s spine is essential to measure longitudinal changes in the patient’s anatomy. On the other hand, reconstructing the spine in 3D from postoperative radiographs is a challenging problem due to the presence of instrumentation (metallic rods and screws) occluding vertebrae on the spine. Methods: This paper describes the reconstruction problem by searching for the optimal model within a manifold space of articulated spines learned from a training dataset of pathological cases who underwent surgery. The manifold structure is implemented based on a multilevel manifold ensemble to structure the data, incorporating connections between nodes within a single manifold, in addition to connections between different multilevel manifolds, representing subregions with similar characteristics. Results: The reconstruction pipeline was evaluated on x-ray datasets from both preoperative patients and patients with spinal surgery. By comparing the method to ground-truth models, a 3D reconstruction accuracy of 2.24 ± 0.90 mm was obtained from 30 postoperative scoliotic patients, while handling patients with highly deformed spines. Conclusions: This paper illustrates how this manifold model can accurately identify similar spine models by navigating in the low-dimensional space, as well as computing nonlinear charts within local neighborhoods of the embedded space during the testing phase. This technique allows postoperative follow-ups of spinal surgery using personalized 3D spine models and assess surgical strategies for spinal deformities.
Postoperative 3D spine reconstruction by navigating partitioning manifolds
International Nuclear Information System (INIS)
Kadoury, Samuel; Labelle, Hubert; Parent, Stefan
2016-01-01
Purpose: The postoperative evaluation of scoliosis patients undergoing corrective treatment is an important task to assess the strategy of the spinal surgery. Using accurate 3D geometric models of the patient’s spine is essential to measure longitudinal changes in the patient’s anatomy. On the other hand, reconstructing the spine in 3D from postoperative radiographs is a challenging problem due to the presence of instrumentation (metallic rods and screws) occluding vertebrae on the spine. Methods: This paper describes the reconstruction problem by searching for the optimal model within a manifold space of articulated spines learned from a training dataset of pathological cases who underwent surgery. The manifold structure is implemented based on a multilevel manifold ensemble to structure the data, incorporating connections between nodes within a single manifold, in addition to connections between different multilevel manifolds, representing subregions with similar characteristics. Results: The reconstruction pipeline was evaluated on x-ray datasets from both preoperative patients and patients with spinal surgery. By comparing the method to ground-truth models, a 3D reconstruction accuracy of 2.24 ± 0.90 mm was obtained from 30 postoperative scoliotic patients, while handling patients with highly deformed spines. Conclusions: This paper illustrates how this manifold model can accurately identify similar spine models by navigating in the low-dimensional space, as well as computing nonlinear charts within local neighborhoods of the embedded space during the testing phase. This technique allows postoperative follow-ups of spinal surgery using personalized 3D spine models and assess surgical strategies for spinal deformities
Spectral invariants of operators of Dirac type on partitioned manifolds
DEFF Research Database (Denmark)
Booss-Bavnbek, Bernhelm; Bleecker, D.
2004-01-01
We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds...... with boundary. We emphasize various (occasionally overlooked) aspects of rigorous definitions and explain the quite different stability properties. Moreover, we utilize the heat equation approach in various settings and show how these topological and spectral invariants are mutually related in the study...
Gauge groups and topological invariants of vacuum manifolds
International Nuclear Information System (INIS)
Golo, V.L.; Monastyrsky, M.I.
1978-01-01
The paper is concerned with topological properties of the vacuum manifolds in the theories with the broken gauge symmetry for the groups of the type SO(k) x U(n), SO(k) x SO(p) x U(r). For the Ginsburg-Landau theory of the superfluid 3 He the gauge transformations are discussed. They provide the means to indicate all possible types of the vacuum manifolds, which are likely to correspond to distinct phases of the superfluid 3 He. Conditions on the existence of the minimums of the Ginsburg-Landau functional are discussed
The Persistence of a Slow Manifold with Bifurcation
DEFF Research Database (Denmark)
Kristiansen, Kristian Uldall; Palmer, P.; Robert, M.
2012-01-01
his paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated...... by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifold's normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation....
Distributed mean curvature on a discrete manifold for Regge calculus
International Nuclear Information System (INIS)
Conboye, Rory; Miller, Warner A; Ray, Shannon
2015-01-01
The integrated mean curvature of a simplicial manifold is well understood in both Regge Calculus and Discrete Differential Geometry. However, a well motivated pointwise definition of curvature requires a careful choice of the volume over which to uniformly distribute the local integrated curvature. We show that hybrid cells formed using both the simplicial lattice and its circumcentric dual emerge as a remarkably natural structure for the distribution of this local integrated curvature. These hybrid cells form a complete tessellation of the simplicial manifold, contain a geometric orthonormal basis, and are also shown to give a pointwise mean curvature with a natural interpretation as the fractional rate of change of the normal vector. (paper)
Spontaneous compactification and Ricci-flat manifolds with torsion
International Nuclear Information System (INIS)
McInnes, B.
1985-06-01
The Freund-Rubin mechanism is based on the equation Rsub(ik)=lambdagsub(ik) (where lambda>0), which, via Myers' Theorem, implies ''spontaneous'' compactification. The difficulties connected with the cosmological constant in this approach can be resolved if torsion is introduced and lambda set equal to zero, but then compactification ''by hand'' is necessary, since the equation Rsub(ik)=0 can be satisfied both on compact and on non-compact manifolds. In this paper we discuss the global geometry of Ricci-flat manifolds with torsion, and suggest ways of restoring the ''spontaneity'' of the compactification. (author)
Orientifolds of type IIA strings on Calabi-Yau manifolds
Indian Academy of Sciences (India)
The advent of D-branes has led to a better understanding of dualities involving strong coupling limits. In particular, Ж = 1 compactifications of the heterotic string (on Calabi-Yau manifolds) are no longer the only string theories of phe- nomenological interest. One such class is furnished by M-theory compactifications.
Nonparametric Bayes Classification and Hypothesis Testing on Manifolds
Bhattacharya, Abhishek; Dunson, David
2012-01-01
Our first focus is prediction of a categorical response variable using features that lie on a general manifold. For example, the manifold may correspond to the surface of a hypersphere. We propose a general kernel mixture model for the joint distribution of the response and predictors, with the kernel expressed in product form and dependence induced through the unknown mixing measure. We provide simple sufficient conditions for large support and weak and strong posterior consistency in estimating both the joint distribution of the response and predictors and the conditional distribution of the response. Focusing on a Dirichlet process prior for the mixing measure, these conditions hold using von Mises-Fisher kernels when the manifold is the unit hypersphere. In this case, Bayesian methods are developed for efficient posterior computation using slice sampling. Next we develop Bayesian nonparametric methods for testing whether there is a difference in distributions between groups of observations on the manifold having unknown densities. We prove consistency of the Bayes factor and develop efficient computational methods for its calculation. The proposed classification and testing methods are evaluated using simulation examples and applied to spherical data applications. PMID:22754028
Sampling from Determinantal Point Processes for Scalable Manifold Learning.
Wachinger, Christian; Golland, Polina
2015-01-01
High computational costs of manifold learning prohibit its application for large datasets. A common strategy to overcome this problem is to perform dimensionality reduction on selected landmarks and to successively embed the entire dataset with the Nyström method. The two main challenges that arise are: (i) the landmarks selected in non-Euclidean geometries must result in a low reconstruction error, (ii) the graph constructed from sparsely sampled landmarks must approximate the manifold well. We propose to sample the landmarks from determinantal distributions on non-Euclidean spaces. Since current determinantal sampling algorithms have the same complexity as those for manifold learning, we present an efficient approximation with linear complexity. Further, we recover the local geometry after the sparsification by assigning each landmark a local covariance matrix, estimated from the original point set. The resulting neighborhood selection .based on the Bhattacharyya distance improves the embedding of sparsely sampled manifolds. Our experiments show a significant performance improvement compared to state-of-the-art landmark selection techniques on synthetic and medical data.
Holomorphic two-spheres in complex Grassmann manifold
Indian Academy of Sciences (India)
Home; Journals; Proceedings – Mathematical Sciences; Volume 118; Issue 3. Holomorphic Two-Spheres in Complex Grassmann Manifold (2, 4). Xiaowei Xu ... Author Affiliations. Xiaowei Xu1 Xiaoxiang Jiao1. School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, China ...
Sparse discriminant manifold projections for bearing fault diagnosis
Chen, Gang; Liu, Fenglin; Huang, Wei
2017-07-01
The monitored vibration signal of bearing is usually nonlinear and nonstationary, and may be corrupted by background noise. Thus, it is very difficult to accurately extract sensitive and reliable characteristics information from the vibration signal to diagnose bearing health conditions. This paper proposes a novel bearing fault diagnosis method based on sparse discriminant manifold projections (SDMP). The SDMP was developed based on sparsity preserving projections, and sparse manifold clustering and embedding. The SDMP can effectively extract the meaningful low-dimensional intrinsic features that hidden in a high-dimensional feature dataset. After dimensionality reduction with the SDMP, the least squares support vector machine (LS-SVM) is utilized to classify the different low-dimensional feature data for fault recognition. The effectiveness and superiorities of the proposed method are demonstrated through several comparative experiments with other three manifold learning methods. The experimental results validate that the SDMP is more effective than the other three manifold learning methods for implementation bearing fault diagnosis, and it is more robust when deal with noise interference signal.
Valve and Manifold considerations for Efficient Digital Hydraulic Machines
DEFF Research Database (Denmark)
Roemer, Daniel Beck; Nørgård, Christian; Bech, Michael Møller
2016-01-01
This paper seeks to shed light on the topic of design and sizing of switching valves and connecting manifolds found in large digital hydraulic motors, also known commercially as Digital Displacement Motors. These motors promise very high operation efficiencies with broad operation ranges, which s...
The topology of certain 3-Sasakian 7-manifolds
DEFF Research Database (Denmark)
A. Hepworth, Richard
2007-01-01
We calculate the integer cohomology ring and stable tangent bundle of a family of compact, 3-Sasakian 7-manifolds constructed by Boyer, Galicki, Mann, and Rees. Previously only the rational cohomology ring was known. The most important part of the cohomology ring is a torsion group that we descri...
Growth of fundamental group for Finsler manifolds with integral Ricci ...
Indian Academy of Sciences (India)
tant to reveal the relationship between the topology and geometry invariants for Finsler manifolds. As for the ... Ricci curvature bound was established and the integral Ricci curvature and topology was studied [13]. .... star-shaped at x ∈ T if for all y ∈ T there exists a minimal geodesic from x to y contained in T . For r > 0, let T ...
Rigid Body Energy Minimization on Manifolds for Molecular Docking.
Mirzaei, Hanieh; Beglov, Dmitri; Paschalidis, Ioannis Ch; Vajda, Sandor; Vakili, Pirooz; Kozakov, Dima
2012-11-13
Virtually all docking methods include some local continuous minimization of an energy/scoring function in order to remove steric clashes and obtain more reliable energy values. In this paper, we describe an efficient rigid-body optimization algorithm that, compared to the most widely used algorithms, converges approximately an order of magnitude faster to conformations with equal or slightly lower energy. The space of rigid body transformations is a nonlinear manifold, namely, a space which locally resembles a Euclidean space. We use a canonical parametrization of the manifold, called the exponential parametrization, to map the Euclidean tangent space of the manifold onto the manifold itself. Thus, we locally transform the rigid body optimization to an optimization over a Euclidean space where basic optimization algorithms are applicable. Compared to commonly used methods, this formulation substantially reduces the dimension of the search space. As a result, it requires far fewer costly function and gradient evaluations and leads to a more efficient algorithm. We have selected the LBFGS quasi-Newton method for local optimization since it uses only gradient information to obtain second order information about the energy function and avoids the far more costly direct Hessian evaluations. Two applications, one in protein-protein docking, and the other in protein-small molecular interactions, as part of macromolecular docking protocols are presented. The code is available to the community under open source license, and with minimal effort can be incorporated into any molecular modeling package.
Analytical and numerical manifolds in a symplectic 4-D map
Delis, N.; Contopoulos, G.
2016-05-01
We study analytically the orbits along the asymptotic manifolds from a complex unstable periodic orbit in a symplectic 4-D Froeschlé map. The orbits are given as convergent series. We compare the analytic results by truncating the series at various orders with the corresponding numerical results and we find agreement along a more extended length, as the order of truncation increases. The agreement is improved when the parameters approach those of the stability domain. Along the manifolds no terms with small divisors appear in the series. The same result is found if we use a parametrization method along the asymptotic curves. In the case of orbits starting close to the manifolds small divisors appear, but the orbits remain close to the manifolds for an extended period of time. If the parameters of the map are close to the stable domain the orbits recede and approach the origin several times and remain confined in a certain volume around the origin for a long time before escaping to large distances. For special sets of parameters we see resonance phenomena and the orbits take particular forms near every resonance.
An algorithmic approach to construct crystallizations of 3-manifolds ...
Indian Academy of Sciences (India)
dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely ..... from the construction of ˜r as in eq. (2.3) and Definition 2.2. D. DEFINITION 2.8. A crystallization ( , γ ) of a 3-manifold is called minimal with respect to the pair (〈S | R〉 ...
Gauge theory and the topology of four-manifolds
Friedman, Robert Marc
1998-01-01
The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying SU(2)-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space. One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the SU(2)-moduli spaces may one day be important for purposes beyond the algebraic invariants that ...
Curvature Properties of Lorentzian Manifolds with Large Isometry Groups
Energy Technology Data Exchange (ETDEWEB)
Batat, Wafaa [Ecole Normale Superieure de L' Enseignement Technique d' Oran, Departement de Mathematiques et Informatique (Algeria)], E-mail: wafa.batat@enset-oran.dz; Calvaruso, Giovanni, E-mail: giovanni.calvaruso@unile.it; Leo, Barbara De [University of Salento, Dipartimento di Matematica ' E. De Giorgi' (Italy)], E-mail: barbara.deleo@unile.it
2009-08-15
The curvature of Lorentzian manifolds (M{sup n},g), admitting a group of isometries of dimension at least 1/2n(n - 1) + 1, is completely described. Interesting behaviours are found, in particular as concerns local symmetry, local homogeneity and conformal flatness.
Convexity of spheres in a manifold without conjugate points
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
Abstract. For a non-compact, complete and simply connected manifold M without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres in M is a radial function, then the geodesic spheres are convex. We also show that if M is two or three dimensional and without ...
Deformations of coisotropic submanifolds in locally conformal symplectic manifolds
Czech Academy of Sciences Publication Activity Database
Le, Hong-Van; Oh, Y.-G.
2016-01-01
Roč. 20, č. 3 (2016), s. 553-596 ISSN 1093-6106 Institutional support: RVO:67985840 Keywords : locally conformal symplectic manifold * coisotropic submanifold * b-twisted differential * bulk deformation Subject RIV: BA - General Mathematics Impact factor: 0.895, year: 2016 http://intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0020/0003/a007/index.html
Stable Yang-Mills connections on special holonomy manifolds
Huang, Teng
2017-06-01
We prove that energy minimizing Yang-Mills connections on a compact G2-manifold has holonomy equal to G2 are G2-instantons, subject to an extra condition on the curvature. Furthermore, we show that energy minimizing connections on a compact Calabi-Yau 3-fold that has holonomy equal to SU(3) subject to a similar condition are holomorphic.
Two new variants of the manifold-mapping technique
D. Echeverria (David)
2006-01-01
htmlabstractManifold-mapping is an efficient surrogate-based optimization technique aimed at the acceleration of very time-consuming design problems. In this paper we present two new variants of the original algorithm that make it applicable to a broader range of optimization scenarios. The first
Quantum cohomology of flag manifolds and Toda lattices
International Nuclear Information System (INIS)
Givental, A.; Kim, B.
1995-01-01
We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice. (orig.)
Enhancing Low-Rank Subspace Clustering by Manifold Regularization.
Liu, Junmin; Chen, Yijun; Zhang, JiangShe; Xu, Zongben
2014-07-25
Recently, low-rank representation (LRR) method has achieved great success in subspace clustering (SC), which aims to cluster the data points that lie in a union of low-dimensional subspace. Given a set of data points, LRR seeks the lowest rank representation among the many possible linear combinations of the bases in a given dictionary or in terms of the data itself. However, LRR only considers the global Euclidean structure, while the local manifold structure, which is often important for many real applications, is ignored. In this paper, to exploit the local manifold structure of the data, a manifold regularization characterized by a Laplacian graph has been incorporated into LRR, leading to our proposed Laplacian regularized LRR (LapLRR). An efficient optimization procedure, which is based on alternating direction method of multipliers (ADMM), is developed for LapLRR. Experimental results on synthetic and real data sets are presented to demonstrate that the performance of LRR has been enhanced by using the manifold regularization.
Robust Semi-Supervised Manifold Learning Algorithm for Classification
Directory of Open Access Journals (Sweden)
Mingxia Chen
2018-01-01
Full Text Available In the recent years, manifold learning methods have been widely used in data classification to tackle the curse of dimensionality problem, since they can discover the potential intrinsic low-dimensional structures of the high-dimensional data. Given partially labeled data, the semi-supervised manifold learning algorithms are proposed to predict the labels of the unlabeled points, taking into account label information. However, these semi-supervised manifold learning algorithms are not robust against noisy points, especially when the labeled data contain noise. In this paper, we propose a framework for robust semi-supervised manifold learning (RSSML to address this problem. The noisy levels of the labeled points are firstly predicted, and then a regularization term is constructed to reduce the impact of labeled points containing noise. A new robust semi-supervised optimization model is proposed by adding the regularization term to the traditional semi-supervised optimization model. Numerical experiments are given to show the improvement and efficiency of RSSML on noisy data sets.
Manifold regularization for sparse unmixing of hyperspectral images.
Liu, Junmin; Zhang, Chunxia; Zhang, Jiangshe; Li, Huirong; Gao, Yuelin
2016-01-01
Recently, sparse unmixing has been successfully applied to spectral mixture analysis of remotely sensed hyperspectral images. Based on the assumption that the observed image signatures can be expressed in the form of linear combinations of a number of pure spectral signatures known in advance, unmixing of each mixed pixel in the scene is to find an optimal subset of signatures in a very large spectral library, which is cast into the framework of sparse regression. However, traditional sparse regression models, such as collaborative sparse regression , ignore the intrinsic geometric structure in the hyperspectral data. In this paper, we propose a novel model, called manifold regularized collaborative sparse regression , by introducing a manifold regularization to the collaborative sparse regression model. The manifold regularization utilizes a graph Laplacian to incorporate the locally geometrical structure of the hyperspectral data. An algorithm based on alternating direction method of multipliers has been developed for the manifold regularized collaborative sparse regression model. Experimental results on both the simulated and real hyperspectral data sets have demonstrated the effectiveness of our proposed model.
Morphological appearance manifolds for group-wise morphometric analysis.
Lian, Nai-Xiang; Davatzikos, Christos
2011-12-01
Computational anatomy quantifies anatomical shape based on diffeomorphic transformations of a template. However, different templates warping algorithms, regularization parameters, or templates, lead to different representations of the same exact anatomy, raising a uniqueness issue: variations of these parameters are confounding factors as they give rise to non-unique representations. Recently, it has been shown that learning the equivalence class derived from the multitude of representations of a given anatomy can lead to improved and more stable morphological descriptors. Herein, we follow that approach, by approximating this equivalence class of morphological descriptors by a (nonlinear) morphological appearance manifold fitting to the data via a locally linear model. Our approach parallels work in the computer vision field, in which variations lighting, pose and other parameters lead to image appearance manifolds representing the exact same figure in different ways. The proposed framework is then used for group-wise registration and statistical analysis of biomedical images, by employing a minimum variance criterion to perform manifold-constrained optimization, i.e. to traverse each individual's morphological appearance manifold until group variance is minimal. The hypothesis is that this process is likely to reduce aforementioned confounding effects and potentially lead to morphological representations reflecting purely biological variations, instead of variations introduced by modeling assumptions and parameter settings. Copyright © 2011 Elsevier B.V. All rights reserved.
On equations of motion on complex grassman manifold
International Nuclear Information System (INIS)
Berceanu, S.; Gheorghe, A.
1989-02-01
We investigate the equations of motion on the 'classical' phase space which corresponds to quantum state space in the case of the complex Grassmann manifold appearing in the Hartree-Fock problem. First and second degree polynomial Hamiltonians in bifermion operators are considered. The 'classical' motion corresponding to linear Hamiltonians is described by a Matrix Riccati equation.(authors)
The Geometry of Slow Manifolds near a Folded Node
Desroches, M.; Krauskopf, B.; Osinga, H.M.
2008-01-01
This paper is concerned with the geometry of slow manifolds of a dynamical system with one fast and two slow variables. Specifically, we study the dynamics near a folded-node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of
Invariant Distributionally Scrambled Manifolds for an Annihilation Operator
Directory of Open Access Journals (Sweden)
Xinxing Wu
2014-01-01
Full Text Available This note proves that the annihilation operator of a quantum harmonic oscillator admits an invariant distributionally ε-scrambled linear manifold for any 0<ε<2. This is a positive answer to Question 1 by Wu and Chen (2013.
Balanced metrics for vector bundles and polarised manifolds
DEFF Research Database (Denmark)
Garcia Fernandez, Mario; Ross, Julius
2012-01-01
We consider a notion of balanced metrics for triples (X, L, E) which depend on a parameter α, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of α, we prove that the limit of a convergent sequence of balanced metrics...
On some properties of the superposition operator on topological manifolds
Directory of Open Access Journals (Sweden)
Janusz Dronka
2010-01-01
Full Text Available In this paper the superposition operator in the space of vector-valued, bounded and continuous functions on a topological manifold is considered. The acting conditions and criteria of continuity and compactness are established. As an application, an existence result for the nonlinear Hammerstein integral equation is obtained.
Formation of a Chern-Simons cylindrical wormhole during evolution of manifolds
Sepehri, Alireza; Ghaffary, Tooraj; Naimi, Yaghoob; Ghaforyan, Hossein; Ebrahimzadeh, Majid
In this paper, the formation of cylindrical wormhole during evolution of manifolds is studied. It is shown that this type of wormholes may be produced at two stages and then disappeared very fast at the third stage. First, one N-dimensional is formed by joining point-like manifolds. Then, this manifold is torn and two child manifolds plus one Chern-Simons manifold appeared. Our universe is born on one of the child manifolds and connected to the other one by Chern-Simons manifold. At the third stage, this Chern-Simons manifold-which plays the role of cylindrical wormhole, dissolves into universes and gives its energy to them and causes inflation. Thus, the Chern-Simons cylindrical wormhole is unstable and dissolves in our four-dimensional universes and another universe very fast.
Person-Independent Head Pose Estimation Using Biased Manifold Embedding
Directory of Open Access Journals (Sweden)
Sethuraman Panchanathan
2008-02-01
Full Text Available Head pose estimation has been an integral problem in the study of face recognition systems and human-computer interfaces, as part of biometric applications. A fine estimate of the head pose angle is necessary and useful for several face analysis applications. To determine the head pose, face images with varying pose angles can be considered to be lying on a smooth low-dimensional manifold in high-dimensional image feature space. However, when there are face images of multiple individuals with varying pose angles, manifold learning techniques often do not give accurate results. In this work, we propose a framework for a supervised form of manifold learning called Biased Manifold Embedding to obtain improved performance in head pose angle estimation. This framework goes beyond pose estimation, and can be applied to all regression applications. This framework, although formulated for a regression scenario, unifies other supervised approaches to manifold learning that have been proposed so far. Detailed studies of the proposed method are carried out on the FacePix database, which contains 181 face images each of 30 individuals with pose angle variations at a granularity of 1Ã¢ÂˆÂ˜. Since biometric applications in the real world may not contain this level of granularity in training data, an analysis of the methodology is performed on sparsely sampled data to validate its effectiveness. We obtained up to 2Ã¢ÂˆÂ˜ average pose angle estimation error in the results from our experiments, which matched the best results obtained for head pose estimation using related approaches.
Quantum invariants of knots and 3-manifolds. 2. rev. ed.
International Nuclear Information System (INIS)
Turaev, Vladimir G.
2010-01-01
Due to the strong appeal and wide use of this monograph, it is now available in its second revised edition. The monograph gives a systematic treatment of 3-dimensional topological quantum field theories (TQFTs) based on the work of the author with N. Reshetikhin and O. Viro. This subject was inspired by the discovery of the Jones polynomial of knots and the Witten-Chern-Simons field theory. On the algebraic side, the study of 3-dimensional TQFTs has been influenced by the theory of braided categories and the theory of quantum groups. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFTs and 2-dimensional modular functors from so-called modular categories. This gives a vast class of knot invariants and 3-manifold invariants as well as a class of linear representations of the mapping class groups of surfaces. In Part II the technique of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFTs constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and skein modules of tangles in the 3-space. This fundamental contribution to topological quantum field theory is accessible to graduate students in mathematics and physics with knowledge of basic algebra and topology. It is an indispensable source for everyone who wishes to enter the forefront of this fascinating area at the borderline of mathematics and physics. From the contents: - Invariants of graphs in Euclidean 3-space and of closed 3-manifolds - Foundations of topological quantum field theory - Three-dimensional topological quantum field theory - Two-dimensional modular functors - 6j-symbols - Simplicial state sums on 3-manifolds - Shadows of manifolds and state sums on shadows - Constructions of modular categories. (orig.)
Action-angle variables and a KAM theorem for b-Poisson manifolds
Kiesenhofer, Anna; Miranda Galcerán, Eva; Scott, Geoffrey
2015-01-01
In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [14] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds. (C) 2015 Elsevier Masson SAS. All rights reserved.
30 CFR 250.444 - What are the choke manifold requirements?
2010-07-01
... 30 Mineral Resources 2 2010-07-01 2010-07-01 false What are the choke manifold requirements? 250... Blowout Preventer (bop) System Requirements § 250.444 What are the choke manifold requirements? (a) Your BOP system must include a choke manifold that is suitable for the anticipated surface pressures...
Birkhoff’s theorem in Lovelock gravity for general base manifolds
Ray, Sourya
2015-10-01
We extend the Birkhoff’s theorem in Lovelock gravity for arbitrary base manifolds using an elementary method. In particular, it is shown that any solution of the form of a warped product of a two-dimensional transverse space and an arbitrary base manifold must be static. Moreover, the field equations restrict the base manifold such that all the non-trivial intrinsic Lovelock tensors of the base manifold are constants, which can be chosen arbitrarily, and the metric in the transverse space is determined by a single function of a spacelike coordinate which satisfies an algebraic equation involving the constants characterizing the base manifold along with the coupling constants.
Out-of-sample extrapolation of learned manifolds.
Chin, Tat-Jun; Suter, David
2008-09-01
We investigate the problem of extrapolating the embedding of a manifold learned from finite samples to novel out-of-sample data. We concentrate on the manifold learning method called Maximum Variance Unfolding (MVU) for which the extrapolation problem is still largely unsolved. Taking the perspective of MVU learning being equivalent to Kernel PCA, our problem reduces to extending a kernel matrix generated from an unknown kernel function to novel points. Leveraging on previous developments, we propose a novel solution which involves approximating the kernel eigenfunction using Gaussian basis functions. We also show how the width of the Gaussian can be tuned to achieve extrapolation. Experimental results which demonstrate the effectiveness of the proposed approach are also included.
Slow manifold and Hannay angle in the spinning top
Energy Technology Data Exchange (ETDEWEB)
Berry, M V [H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL (United Kingdom); Shukla, P [Department of Physics, Indian Institute of Technology, Kharagpur (India)
2011-01-15
The spin of a top can be regarded as a fast variable, coupled to the motion of the axis which is slow. In pure precession, the rotation of the axis round a cone (without nutation), can be considered as the result of a reaction from the fast spin. The resulting restriction of the total state space of the top is an illustrative example, at graduate-student level, of the general dynamical concept of the slow manifold. For this case, the slow manifold can be calculated exactly, and expanded as a series of reaction forces (of magnetic type) in powers of slowness, corresponding to a modified precession frequency. The forces correspond to a series for the Hannay angle for the fast motion, describing the location of a point on the top.
Manifold: a Custom Analytics Platform to Visualize Research Impact
Directory of Open Access Journals (Sweden)
Steven Braun
2015-10-01
Full Text Available The use of research impact metrics and analytics has become an integral component to many aspects of institutional assessment. Many platforms currently exist to provide such analytics, both proprietary and open source; however, the functionality of these systems may not always overlap to serve uniquely specific needs. In this paper, I describe a novel web-based platform, named Manifold, that I built to serve custom research impact assessment needs in the University of Minnesota Medical School. Built on a standard LAMP architecture, Manifold automatically pulls publication data for faculty from Scopus through APIs, calculates impact metrics through automated analytics, and dynamically generates report-like profiles that visualize those metrics. Work on this project has resulted in many lessons learned about challenges to sustainability and scalability in developing a system of such magnitude.
Holomorphic Yukawa couplings for complete intersection Calabi-Yau manifolds
Energy Technology Data Exchange (ETDEWEB)
Blesneag, Stefan [Rudolf Peierls Centre for Theoretical Physics, Oxford University,1 Keble Road, Oxford, OX1 3NP (United Kingdom); Buchbinder, Evgeny I. [The University of Western Australia,35 Stirling Highway, Crawley WA 6009 (Australia); Lukas, Andre [Rudolf Peierls Centre for Theoretical Physics, Oxford University,1 Keble Road, Oxford, OX1 3NP (United Kingdom)
2017-01-27
We develop methods to compute holomorphic Yukawa couplings for heterotic compactifications on complete intersection Calabi-Yau manifolds, generalising results of an earlier paper for Calabi-Yau hypersurfaces. Our methods are based on constructing the required bundle-valued forms explicitly and evaluating the relevant integrals over the projective ambient space. We also show how our approach relates to an earlier, algebraic one to calculate the holomorphic Yukawa couplings. A vanishing theorem, which we prove, implies that certain Yukawa couplings allowed by low-energy symmetries are zero due to topological reasons. To illustrate our methods, we calculate Yukawa couplings for SU(5)-based standard models on a co-dimension two complete intersection manifold.
Laplacian manifold regularization method for fluorescence molecular tomography
He, Xuelei; Wang, Xiaodong; Yi, Huangjian; Chen, Yanrong; Zhang, Xu; Yu, Jingjing; He, Xiaowei
2017-04-01
Sparse regularization methods have been widely used in fluorescence molecular tomography (FMT) for stable three-dimensional reconstruction. Generally, ℓ1-regularization-based methods allow for utilizing the sparsity nature of the target distribution. However, in addition to sparsity, the spatial structure information should be exploited as well. A joint ℓ1 and Laplacian manifold regularization model is proposed to improve the reconstruction performance, and two algorithms (with and without Barzilai-Borwein strategy) are presented to solve the regularization model. Numerical studies and in vivo experiment demonstrate that the proposed Gradient projection-resolved Laplacian manifold regularization method for the joint model performed better than the comparative algorithm for ℓ1 minimization method in both spatial aggregation and location accuracy.
Multiview vector-valued manifold regularization for multilabel image classification.
Luo, Yong; Tao, Dacheng; Xu, Chang; Xu, Chao; Liu, Hong; Wen, Yonggang
2013-05-01
In computer vision, image datasets used for classification are naturally associated with multiple labels and comprised of multiple views, because each image may contain several objects (e.g., pedestrian, bicycle, and tree) and is properly characterized by multiple visual features (e.g., color, texture, and shape). Currently, available tools ignore either the label relationship or the view complementarily. Motivated by the success of the vector-valued function that constructs matrix-valued kernels to explore the multilabel structure in the output space, we introduce multiview vector-valued manifold regularization (MV(3)MR) to integrate multiple features. MV(3)MR exploits the complementary property of different features and discovers the intrinsic local geometry of the compact support shared by different features under the theme of manifold regularization. We conduct extensive experiments on two challenging, but popular, datasets, PASCAL VOC' 07 and MIR Flickr, and validate the effectiveness of the proposed MV(3)MR for image classification.
Gravity and supergravity as gauge theories on a group manifold
Directory of Open Access Journals (Sweden)
Yuval Ne'eman
1978-03-01
Full Text Available We construct generalizations of gravity, including supergravity, by writing the theory on the group manifold (Poincaré for gravity, the graded-Poincaré group for supergravity. The action involves forms over the group, restricted to a 4-dimensional submanifold. The equations of motion produce a Lorentz gauge in gravity and supergravity, and an additional anholonomic supersymmetric coordinate transformation which reduces to the “local supersymmetry” of supergravity.
Groups and manifolds lectures for physicists with examples in Mathematica
Fré, Pietro Giuseppe
2018-01-01
Groups and Manifolds is an introduction to the mathematics of symmetry with a variety of examples for physicists. It covers both classical symmetry as seen in crystallography as well as the mathematical concepts used in super-symmetric field theories. After a basic introduction of group theory, Lie algebras and a basic notion of differential geometry are discussed. Group-theoretical constructions are done using Mathematica.
Particle Spaces on Manifolds and Generalized Poincar\\'e Dualities
Kallel, Sadok
1998-01-01
It is quite an interesting phenomenon in Topology that configuration spaces on a manifold M are intrinsically related to certain mapping spaces from M. In this paper we interpret and greatly expand on this relationship. Building (mainly) on work of Segal, we introduce a new class of configuration spaces; the particle spaces, and these include the classical configuration spaces of distinct points, symmetric products, truncated products, divisor spaces, positive and negative particles of McDuff...
On symplectomorphisms of the symplectisation of a compact contact manifold
International Nuclear Information System (INIS)
Banyaga, A.
2004-03-01
Let (N, α) be a compact contact manifold and (N x R, d(e t α)) its symplectisation. We show that the group G which is the identity component in the group of symplectic diffeomorphisms Φ of (N x R, d(e t α)) that cover diffeomorphisms of Φ of N x S 1 is simple, by showing that G is isomorphic to the kernel of the Calabi homomorphism of the associated locally conformal symplectic structure. (author)
A note on generalized metrics on complex manifolds
International Nuclear Information System (INIS)
Rastogi, S.C.
1986-08-01
In 1981, Hojo introduced a generalized metric function Φ (P) , p(≠1) is a real number in a Finsler space and studied some beautiful consequences of such a metric function. The aim of this paper is to investigate the possibility of introducing a similar metric function on a complex manifold studied by Rund. It is interesting to note that such an introduction is unnatural for values of p other than 2, which corresponds to the metric function introduced by Rund. (author)
Homotopy classification of contact foliations on open contact manifolds
Indian Academy of Sciences (India)
64
Abstract. We give a homotopy classification of foliations on open contact manifolds whose leaves are contact ... The inverse images of FN under an f ∈ Trα(M, FN ), namely f−1FN , is a contact foliation on (M,α). ..... Let Γ be the graph of H; then the image of ϕ is contained in Γ. By Lemma 3.2 there exists a diffeomorphism Φ ...
Curved manifolds with conserved Runge-Lenz vectors
International Nuclear Information System (INIS)
Ngome, J.-P.
2009-01-01
van Holten's algorithm is used to construct Runge-Lenz-type conserved quantities, induced by Killing tensors, on curved manifolds. For the generalized Taub-Newman-Unti-Tamburino metric, the most general external potential such that the combined system admits a conserved Runge-Lenz-type vector is found. In the multicenter case, the subclass of two-center metric exhibits a conserved Runge-Lenz-type scalar.
One-loop effective potential on hyperbolic manifolds
International Nuclear Information System (INIS)
Cognola, G.; Kirsten, K.; Zerbini, S.
1993-01-01
The one-loop effective potential for a scalar field defined on an ultrastatic space-time whose spatial part is a compact hyperbolic manifold is studied using ζ-function regularization for the one-loop effective action. Other possible regularizations are discussed in detail. The renormalization group equations are derived, and their connection with the conformal anomaly is pointed out. The symmetry breaking and the topological mass generation are also discussed
Ultrasonic defect characterization using parametric-manifold mapping
Velichko, A.; Bai, L.; Drinkwater, B. W.
2017-06-01
The aim of ultrasonic non-destructive evaluation includes the detection and characterization of defects, and an understanding of the nature of defects is essential for the assessment of structural integrity in safety critical systems. In general, the defect characterization challenge involves an estimation of defect parameters from measured data. In this paper, we explore the extent to which defects can be characterized by their ultrasonic scattering behaviour. Given a number of ultrasonic measurements, we show that characterization information can be extracted by projecting the measurement onto a parametric manifold in principal component space. We show that this manifold represents the entirety of the characterization information available from far-field harmonic ultrasound. We seek to understand the nature of this information and hence provide definitive statements on the defect characterization performance that is, in principle, extractable from typical measurement scenarios. In experiments, the characterization problem of surface-breaking cracks and the more general problem of elliptical voids are studied, and a good agreement is achieved between the actual parameter values and the characterization results. The nature of the parametric manifold enables us to explain and quantify why some defects are relatively easy to characterize, whereas others are inherently challenging.
The α ' expansion on a compact manifold of exceptional holonomy
Becker, Katrin; Robbins, Daniel; Witten, Edward
2014-06-01
In the approximation corresponding to the classical Einstein equations, which is valid at large radius, string theory compactification on a compact manifold M of G 2 or Spin(7) holonomy gives a supersymmetric vacuum in three or two dimensions. Do α ' corrections to the Einstein equations disturb this statement? Explicitly analyzing the leading correction, we show that the metric of M can be adjusted to maintain supersymmetry. Beyond leading order, a general argument based on low energy effective field theory in spacetime implies that this is true exactly (not just to all finite orders in α '). A more elaborate field theory argument that includes the massive Kaluza-Klein modes matches the structure found in explicit calculations. In M-theory compactification on a manifold M of G 2 or Spin(7) holonomy, similar results hold to all orders in the inverse radius of M — but not exactly. The classical moduli space of G 2 metrics on a manifold M is known to be locally a Lagrangian submanifold of H 3( M,) ⊕ H 4( M,). We show that this remains valid to all orders in the α ' or inverse radius expansion.
Optimal reconfigurations of two-craft Coulomb formations along manifolds
Jones, Drew R.; Schaub, Hanspeter
2013-02-01
Coulomb formations refer to swarms of closely flying spacecraft, in which the net electric charge of each vehicle is controlled. Active charge control is central to this concept and enables a propulsion system with highly desirable characteristics, albeit with limited controllability. Numerous Coulomb formation equilibria have been derived, but to maintain and maneuver these configurations, some inertial thrust is required to supplement the nearly propellant-less charge control. In this work, invariant manifold theory is applied to two-craft Coulomb equilibria, which are admitted in a linearized two-body gravity model. The manifolds associated with these systems are analyzed for the first time, and are then utilized as part of a general procedure for formulating optimal reconfigurations. Specifically, uncontrolled flows along the manifolds are sought which provide near continuous transfers from one equilibrium to another. Control is then introduced to match continuity, while minimizing inertial thrusting. This methodology aims to exploit uncontrolled motions and charge control to realize the shape-changing ability of these formations, without large inertial control efforts. Some variations in formulating and parameterizing the optimal transfers are discussed, and analytical expressions are derived to aid in establishing control parameter limits, under certain assumptions. Numerical results are provided, as demonstrative examples of the optimization procedure, using relatively simple control approximations. Finally, Particle Swarm Optimization, a novel stochastic method, is used with considerable success to solve the numerically difficult parameter optimization problems.
Robust head pose estimation via supervised manifold learning.
Wang, Chao; Song, Xubo
2014-05-01
Head poses can be automatically estimated using manifold learning algorithms, with the assumption that with the pose being the only variable, the face images should lie in a smooth and low-dimensional manifold. However, this estimation approach is challenging due to other appearance variations related to identity, head location in image, background clutter, facial expression, and illumination. To address the problem, we propose to incorporate supervised information (pose angles of training samples) into the process of manifold learning. The process has three stages: neighborhood construction, graph weight computation and projection learning. For the first two stages, we redefine inter-point distance for neighborhood construction as well as graph weight by constraining them with the pose angle information. For Stage 3, we present a supervised neighborhood-based linear feature transformation algorithm to keep the data points with similar pose angles close together but the data points with dissimilar pose angles far apart. The experimental results show that our method has higher estimation accuracy than the other state-of-art algorithms and is robust to identity and illumination variations. Copyright © 2014 Elsevier Ltd. All rights reserved.
Hyperspherical Manifold for EEG Signals of Epileptic Seizures
Directory of Open Access Journals (Sweden)
Tahir Ahmad
2012-01-01
Full Text Available The mathematical modelling of EEG signals of epileptic seizures presents a challenge as seizure data is erratic, often with no visible trend. Limitations in existing models indicate a need for a generalized model that can be used to analyze seizures without the need for apriori information, whilst minimizing the loss of signal data due to smoothing. This paper utilizes measure theory to design a discrete probability measure that reformats EEG data without altering its geometric structure. An analysis of EEG data from three patients experiencing epileptic seizures is made using the developed measure, resulting in successful identification of increased potential difference in portions of the brain that correspond to physical symptoms demonstrated by the patients. A mapping then is devised to transport the measure data onto the surface of a high-dimensional manifold, enabling the analysis of seizures using directional statistics and manifold theory. The subset of seizure signals on the manifold is shown to be a topological space, verifying Ahmad's approach to use topological modelling.
GPU accelerated manifold correction method for spinning compact binaries
Ran, Chong-xi; Liu, Song; Zhong, Shuang-ying
2018-04-01
The graphics processing unit (GPU) acceleration of the manifold correction algorithm based on the compute unified device architecture (CUDA) technology is designed to simulate the dynamic evolution of the Post-Newtonian (PN) Hamiltonian formulation of spinning compact binaries. The feasibility and the efficiency of parallel computation on GPU have been confirmed by various numerical experiments. The numerical comparisons show that the accuracy on GPU execution of manifold corrections method has a good agreement with the execution of codes on merely central processing unit (CPU-based) method. The acceleration ability when the codes are implemented on GPU can increase enormously through the use of shared memory and register optimization techniques without additional hardware costs, implying that the speedup is nearly 13 times as compared with the codes executed on CPU for phase space scan (including 314 × 314 orbits). In addition, GPU-accelerated manifold correction method is used to numerically study how dynamics are affected by the spin-induced quadrupole-monopole interaction for black hole binary system.
Quantum fields on manifolds: PCT and gravitationally induced thermal states
International Nuclear Information System (INIS)
Sewell, G.L.
1982-01-01
We formulate an axiomatic scheme, designed to provide a framework for a general, rigorous theory of relativistic quantum fields on a class of manifolds, that includes Kruskal's extension of Schwarzchild space-time, as well as Minkowski space-time. The scheme is an adaptation of Wightman's to this class of manifolds. We infer from it that, given an arbitrary field (in general, interacting) on a manifold X, the restriction of the field to a certain open submanifold X/sup( + ), whose boundaries are event horizons, satisfies the Kubo--Martin--Schwinger (KMS) thermal equilibrium conditions. This amounts to a rigorous, model-independent proof of a generalized Hawking--Unruh effect. Further, in cases where the field enjoys a certain PCT symmetry, the conjugation governing the KMS condition is just the PCT operator. The key to these results is an analogue, that we prove, of the Bisognano--Wichmann theorem [J. Math. Phys. 17, (1976), Theorem 1]. We also construct an alternative scheme by replacing a regularity condition at an event horizon by the assumption that the field in X/sup( + ) is in a ground, rather then a thermal, state. We show that, in this case, the observables in X/sup( + ) are uncorrelated to those in its causal complement, X/sup( - ), and thus that the event horizons act as physical barriers. Finally, we argue that the choice between the two schemes must be dictated by the prevailing conditions governing the state of the field
Analyticity of the Free Energy of a Closed 3-Manifold
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Stavros Garoufalidis
2008-11-01
Full Text Available The free energy of a closed 3-manifold is a 2-parameter formal power series which encodes the perturbative Chern-Simons invariant (also known as the LMO invariant of a closed 3-manifold with gauge group U(N for arbitrary N. We prove that the free energy of an arbitrary closed 3-manifold is uniformly Gevrey-1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender-Gao-Richmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender-Gao-Richmond.
Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds
Buchbinder, Evgeny; Lukas, Andre; Ovrut, Burt; Ruehle, Fabian
2017-10-01
We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all ℙ1 curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a comparison with the genus zero Gromov-Witten invariants. We then use this construction to study instanton superpotentials on those manifolds and their quotients. We identify a non-toric quotient of a non-favorable CICY with a single genus zero curve in a certain homology class, so that a cancellation à la Beasley-Witten is not possible. In another example, we study a non-toric quotient of a favorable CICY and check that the superpotential still vanishes. From this and related examples, we conjecture that the Beasley-Witten cancellation result can be extended to toric and non-toric quotients of CICYs, but can be avoided if the CICY is non-favorable.
Compactifications of heterotic strings on non-Kaehler complex manifolds II
International Nuclear Information System (INIS)
Becker, Katrin; Becker, Melanie; Dasgupta, Keshav; Green, Paul S.; Sharpe, Eric
2004-01-01
We continue our study of heterotic compactifications on non-Kaehler complex manifolds with torsion. We give further evidence of the consistency of the six-dimensional manifold presented earlier and discuss the anomaly cancellation and possible supergravity description for a generic non-Kaehler complex manifold using the newly proposed superpotential. The manifolds studied in our earlier papers had zero Euler characteristics. We construct new examples of non-Kaehler complex manifolds with torsion in lower dimensions, that have nonzero Euler characteristics. Some of these examples are constructed from consistent backgrounds in F-theory and therefore are solutions to the string equations of motion. We discuss consistency conditions for compactifications of the heterotic string on smooth non-Kaehler manifolds and illustrate how some results well known for Calabi-Yau compactifications, including counting the number of generations, apply to the non-Kaehler case. We briefly address various issues regarding possible phenomenological applications
Dictionary Pair Learning on Grassmann Manifolds for Image Denoising.
Zeng, Xianhua; Bian, Wei; Liu, Wei; Shen, Jialie; Tao, Dacheng
2015-11-01
Image denoising is a fundamental problem in computer vision and image processing that holds considerable practical importance for real-world applications. The traditional patch-based and sparse coding-driven image denoising methods convert 2D image patches into 1D vectors for further processing. Thus, these methods inevitably break down the inherent 2D geometric structure of natural images. To overcome this limitation pertaining to the previous image denoising methods, we propose a 2D image denoising model, namely, the dictionary pair learning (DPL) model, and we design a corresponding algorithm called the DPL on the Grassmann-manifold (DPLG) algorithm. The DPLG algorithm first learns an initial dictionary pair (i.e., the left and right dictionaries) by employing a subspace partition technique on the Grassmann manifold, wherein the refined dictionary pair is obtained through a sub-dictionary pair merging. The DPLG obtains a sparse representation by encoding each image patch only with the selected sub-dictionary pair. The non-zero elements of the sparse representation are further smoothed by the graph Laplacian operator to remove the noise. Consequently, the DPLG algorithm not only preserves the inherent 2D geometric structure of natural images but also performs manifold smoothing in the 2D sparse coding space. We demonstrate that the DPLG algorithm also improves the structural SIMilarity values of the perceptual visual quality for denoised images using the experimental evaluations on the benchmark images and Berkeley segmentation data sets. Moreover, the DPLG also produces the competitive peak signal-to-noise ratio values from popular image denoising algorithms.
Toward Optimal Manifold Hashing via Discrete Locally Linear Embedding.
Rongrong Ji; Hong Liu; Liujuan Cao; Di Liu; Yongjian Wu; Feiyue Huang
2017-11-01
Binary code learning, also known as hashing, has received increasing attention in large-scale visual search. By transforming high-dimensional features to binary codes, the original Euclidean distance is approximated via Hamming distance. More recently, it is advocated that it is the manifold distance, rather than the Euclidean distance, that should be preserved in the Hamming space. However, it retains as an open problem to directly preserve the manifold structure by hashing. In particular, it first needs to build the local linear embedding in the original feature space, and then quantize such embedding to binary codes. Such a two-step coding is problematic and less optimized. Besides, the off-line learning is extremely time and memory consuming, which needs to calculate the similarity matrix of the original data. In this paper, we propose a novel hashing algorithm, termed discrete locality linear embedding hashing (DLLH), which well addresses the above challenges. The DLLH directly reconstructs the manifold structure in the Hamming space, which learns optimal hash codes to maintain the local linear relationship of data points. To learn discrete locally linear embeddingcodes, we further propose a discrete optimization algorithm with an iterative parameters updating scheme. Moreover, an anchor-based acceleration scheme, termed Anchor-DLLH, is further introduced, which approximates the large similarity matrix by the product of two low-rank matrices. Experimental results on three widely used benchmark data sets, i.e., CIFAR10, NUS-WIDE, and YouTube Face, have shown superior performance of the proposed DLLH over the state-of-the-art approaches.
Image reconstruction by domain-transform manifold learning
Zhu, Bo; Liu, Jeremiah Z.; Cauley, Stephen F.; Rosen, Bruce R.; Rosen, Matthew S.
2018-03-01
Image reconstruction is essential for imaging applications across the physical and life sciences, including optical and radar systems, magnetic resonance imaging, X-ray computed tomography, positron emission tomography, ultrasound imaging and radio astronomy. During image acquisition, the sensor encodes an intermediate representation of an object in the sensor domain, which is subsequently reconstructed into an image by an inversion of the encoding function. Image reconstruction is challenging because analytic knowledge of the exact inverse transform may not exist a priori, especially in the presence of sensor non-idealities and noise. Thus, the standard reconstruction approach involves approximating the inverse function with multiple ad hoc stages in a signal processing chain, the composition of which depends on the details of each acquisition strategy, and often requires expert parameter tuning to optimize reconstruction performance. Here we present a unified framework for image reconstruction—automated transform by manifold approximation (AUTOMAP)—which recasts image reconstruction as a data-driven supervised learning task that allows a mapping between the sensor and the image domain to emerge from an appropriate corpus of training data. We implement AUTOMAP with a deep neural network and exhibit its flexibility in learning reconstruction transforms for various magnetic resonance imaging acquisition strategies, using the same network architecture and hyperparameters. We further demonstrate that manifold learning during training results in sparse representations of domain transforms along low-dimensional data manifolds, and observe superior immunity to noise and a reduction in reconstruction artefacts compared with conventional handcrafted reconstruction methods. In addition to improving the reconstruction performance of existing acquisition methodologies, we anticipate that AUTOMAP and other learned reconstruction approaches will accelerate the development
Deep Learning on Sparse Manifolds for Faster Object Segmentation.
Nascimento, Jacinto C; Carneiro, Gustavo
2017-07-11
We propose a new combination of deep belief networks and sparse manifold learning strategies for the 2D segmentation of non-rigid visual objects. With this novel combination, we aim to reduce the training and inference complexities while maintaining the accuracy of machine learning based non-rigid segmentation methodologies. Typical non-rigid object segmentation methodologies divide the problem into a rigid detection followed by a non-rigid segmentation, where the low dimensionality of the rigid detection allows for a robust training (i.e., a training that does not require a vast amount of annotated images to estimate robust appearance and shape models) and a fast search process during inference. Therefore, it is desirable that the dimensionality of this rigid transformation space is as small as possible in order to enhance the advantages brought by the aforementioned division of the problem. In this paper, we propose the use of sparse manifolds to reduce the dimensionality of the rigid detection space. Furthermore, we propose the use of deep belief networks to allow for a training process that can produce robust appearance models without the need of large annotated training sets. We test our approach in the segmentation of the left ventricle of the heart from ultrasound images and lips from frontal face images. Our experiments show that the use of sparse manifolds and deep belief networks for the rigid detection stage leads to segmentation results that are as accurate as the current state of the art, but with lower search complexity and training processes that require a small amount of annotated training data.
Image reconstruction by domain-transform manifold learning.
Zhu, Bo; Liu, Jeremiah Z; Cauley, Stephen F; Rosen, Bruce R; Rosen, Matthew S
2018-03-21
Image reconstruction is essential for imaging applications across the physical and life sciences, including optical and radar systems, magnetic resonance imaging, X-ray computed tomography, positron emission tomography, ultrasound imaging and radio astronomy. During image acquisition, the sensor encodes an intermediate representation of an object in the sensor domain, which is subsequently reconstructed into an image by an inversion of the encoding function. Image reconstruction is challenging because analytic knowledge of the exact inverse transform may not exist a priori, especially in the presence of sensor non-idealities and noise. Thus, the standard reconstruction approach involves approximating the inverse function with multiple ad hoc stages in a signal processing chain, the composition of which depends on the details of each acquisition strategy, and often requires expert parameter tuning to optimize reconstruction performance. Here we present a unified framework for image reconstruction-automated transform by manifold approximation (AUTOMAP)-which recasts image reconstruction as a data-driven supervised learning task that allows a mapping between the sensor and the image domain to emerge from an appropriate corpus of training data. We implement AUTOMAP with a deep neural network and exhibit its flexibility in learning reconstruction transforms for various magnetic resonance imaging acquisition strategies, using the same network architecture and hyperparameters. We further demonstrate that manifold learning during training results in sparse representations of domain transforms along low-dimensional data manifolds, and observe superior immunity to noise and a reduction in reconstruction artefacts compared with conventional handcrafted reconstruction methods. In addition to improving the reconstruction performance of existing acquisition methodologies, we anticipate that AUTOMAP and other learned reconstruction approaches will accelerate the development of
On ruled surface in 3-dimensional almost contact metric manifold
Karacan, Murat Kemal; Yuksel, Nural; Ikiz, Hasibe
In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.
Parallel Transport Along Seifert Manifolds and Fractional Monodromy
Martynchuk, N.; Efstathiou, K.
2017-12-01
The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard (`integer') monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds.
Mücket-Treder's Two-Body Problem: Infinity Manifold
Mioc, Vasile
The two-body problem in the Mücket-Treder post-Newtonian classical field is revisited. Starting from the motion equations and first integrals in standard polar coordinates, we apply McGehee-type transformations of the first and second kind to suitably describe the escape/capture dynamics. To this end, the infinity manifold is defined, and the flow on it is depicted. The behaviour of orbits that neighbour infinity is wholly similar to the cases corresponding to some relativistic models.
Manifolds in random media: beyond the variational approximation
Goldschmidt, Yadin Y.
1994-01-01
In this paper we give a closed form expression for the 1/d corrections to the selfenergy characterizing the correlation function of a manifold in random media. This amounts to the first confection beyond the variational approximation. At this time we were able to evaluate these corrections in the high temperature "phase" of the notorious toy-model describing a classical particle subject to the influence of both a harmonic potential and a random potential. Although in this phase the correct solution is replica symmetric the calculation is non-trivial. The outcome is compared with previous analytical and numerical results. The corrections diverge at the "transition" temperature.
Computation of saddle-type slow manifolds using iterative methods
DEFF Research Database (Denmark)
Kristiansen, Kristian Uldall
2015-01-01
This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, which require mesh refinements to ensure uniform convergence...... with respect to , appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples, including a model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables, and the computation of homoclinic connections in the Fitz...
B-strings on non-Kählerian manifolds
Directory of Open Access Journals (Sweden)
Camillo Imbimbo
2016-11-01
Full Text Available We explain how to couple topological B-models whose targets are non-Kählerian manifolds to topological gravity and to thus define corresponding topological strings. We emphasize the need to take into account the coupling to the superghost field of topological gravity in order to obtain a consistent definition of the string model. We also review the importance of the superghost for correctly interpreting the holomorphic anomaly of the string amplitudes. We perform our analysis in the BV framework in order to make it completely gauge independent.
Tensor calculus for supergravity on a manifold with boundary
Energy Technology Data Exchange (ETDEWEB)
Belyaev, D.V. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Van Nieuwenhuizen, P. [New York State Univ., Stony Brook, NY (United States). C.N. Yang Institute for Theoretical Physics
2007-11-15
Using the simple setting of 3D N=1 supergravity, we show how the tensor calculus of supergravity can be extended to manifolds with boundary. We present an ex- tension of the standard F-density formula which yields supersymmetric bulk-plus-boundary actions. To construct additional separately supersymmetric boundary actions, we decompose bulk supergravity and bulk matter multiplets into co-dimension one submultiplets. As an illustration we obtain the supersymmetric extension of the York-Gibbons-Hawking extrinsic curvature boundary term. We emphasize that our construction does not require any boundary conditions on off-shell fields. This gives a significant improvement over the existing orbifold supergravity tensor calculus. (orig.)
The parametric manifold picture of space-time
International Nuclear Information System (INIS)
Perjes, Z.
1992-03-01
Parametric manifolds are reparametrization-invariant geometric structures describing space-time and internal degrees of freedom in a unified framework. Using the theory of parametric spinors, a decomposition of the space-time in General Relativity is developed with respect to the 3-space of trajectories of a time-like or space-like vector field. The parametric 3+1 decomposition surpasses the ADM formalism in generality since it is possible even in space-times which do not admit a space-like foliation. (author) 33 refs
Tensor calculus for supergravity on a manifold with boundary
International Nuclear Information System (INIS)
Belyaev, Dmitry V.; Nieuwenhuizen, Peter van
2008-01-01
Using the simple setting of 3D N = 1 supergravity, we show how the tensor calculus of supergravity can be extended to manifolds with boundary. We present an extension of the standard F-density formula which yields supersymmetric bulk-plus-boundary actions. To construct additional separately supersymmetric boundary actions, we decompose bulk supergravity and bulk matter multiplets into co-dimension one submultiplets. As an illustration we obtain the supersymmetric extension of the York-Gibbons-Hawking extrinsic curvature boundary term. We emphasize that our construction does not require any boundary conditions on off-shell fields. This gives a significant improvement over the existing orbifold supergravity tensor calculus
Optimality for totally real immersions and independent mappings of manifolds into C^N
Ho, Pak Tung; Jacobowitz, Howard; Landweber, Peter
2012-01-01
The optimal target dimensions are determined for totally real immersions and for independent mappings into complex affine spaces. Our arguments are similar to those given by Forster, but we use orientable manifolds as far as possible and so are able to obtain improved results for orientable manifolds of even dimension. This leads to new examples showing that the known immersion and submersion dimensions for holomorphic mappings from Stein manifolds to affine spaces are best possible.
Homotopy L-infinity spaces and Kuranishi manifolds, I: categorical structures
Tu, Junwu
2016-01-01
Motivated by the definition of homotopy $L_\\infty$ spaces, we develop a new theory of Kuranishi manifolds, closely related to Joyce's recent theory. We prove that Kuranishi manifolds form a $2$-category with invertible $2$-morphisms, and that certain fiber product property holds in this $2$-category. In a subsequent paper, we construct the virtual fundamental cycle of a compact oriented Kuranishi manifold, and prove some of its basic properties. Manifest from this new formulation is the fact ...
Manifold Based Low-rank Regularization for Image Restoration and Semi-supervised Learning
Lai, Rongjie; Li, Jia
2017-01-01
Low-rank structures play important role in recent advances of many problems in image science and data science. As a natural extension of low-rank structures for data with nonlinear structures, the concept of the low-dimensional manifold structure has been considered in many data processing problems. Inspired by this concept, we consider a manifold based low-rank regularization as a linear approximation of manifold dimension. This regularization is less restricted than the global low-rank regu...
New hyper-K{umlt a}hler manifolds by fixing monopoles
Energy Technology Data Exchange (ETDEWEB)
Houghton, C.J. [DAMTP, Silver Street, Cambridge, CB3 9EW (United Kingdom)
1997-07-01
The construction of new hyper-K{umlt a}hler manifolds by taking the infinite monopole mass limit of certain Bogomol{close_quote}nyi-Prasad-Sommerfield monopole moduli spaces is considered. The one-parameter family of hyper-K{umlt a}hler manifolds due to Dancer is shown to be an example of such manifolds. A new family of fixed monopole spaces is constructed. They are the moduli spaces of four SU{sub 4} monopoles, in the infinite mass limit of two of the monopoles. These manifolds are shown to be nonsingular when the fixed monopole positions are distinct. {copyright} {ital 1997} {ital The American Physical Society}
The Origin of Chern-Simons Modified Gravity from an 11 + 3-Dimensional Manifold
Directory of Open Access Journals (Sweden)
J. A. Helayël-Neto
2017-01-01
Full Text Available It is our aim to show that the Chern-Simons terms of modified gravity can be understood as generated by the addition of a 3-dimensional algebraic manifold to an initial 11-dimensional space-time manifold; this builds up an 11+3-dimensional space-time. In this system, firstly, some fields living in the bulk join the fields that live on the 11-dimensional manifold, so that the rank of the gauge fields exceeds the dimension of the algebra; consequently, there emerges an anomaly. To solve this problem, another 11-dimensional manifold is included in the 11+3-dimensional space-time, and it interacts with the initial manifold by exchanging Chern-Simon fields. This mechanism is able to remove the anomaly. Chern-Simons terms actually produce an extra manifold in the pair of 11-dimensional manifolds of the 11+3-space-time. Summing up the topology of both the 11-dimensional manifolds and the topology of the exchanged Chern-Simons manifold in the bulk, we conclude that the total topology shrinks to one, which is in agreement with the main idea of the Big Bang theory.
Heat shield manifold system for a midframe case of a gas turbine engine
Energy Technology Data Exchange (ETDEWEB)
Mayer, Clinton A.; Eng, Jesse; Schopf, Cheryl A.
2017-07-25
A heat shield manifold system for an inner casing between a compressor and turbine assembly is disclosed. The heat shield manifold system protects the outer case from high temperature compressor discharge air, thereby enabling the outer case extending between a compressor and a turbine assembly to be formed from less expensive materials than otherwise would be required. In addition, the heat shield manifold system may be configured such that compressor bleed air is passed from the compressor into the heat shield manifold system without passing through a conventional flange to flange joint that is susceptible to leakage.
Experimental Analysis of Exhaust Manifold with Ceramic Coating for Reduction of Heat Dissipation
Saravanan, J.; Valarmathi, T. N.; Nathc, Rajdeep; Kumar, Prasanth
2017-05-01
Exhaust manifold plays an important role in the exhaust system, the manifold delivers the waste toxic gases to a safe distance and it is used to reduce the sound pollution and air pollution. Exhaust manifold suffers with lot of thermal stress, due to this blow holes occurs in the surface of the exhaust manifold and also more noise is developed. The waste toxic gases from the multiple cylinders are collected into a single pipe by the exhaust manifold. The waste toxic gases can damage the material of the manifold. In this study, to prevent the damage zirconia powder has been coated in the inner surface and alumina (60%) combined with titania (40%) has been used for coating the outer surface of the exhaust manifold. After coating experiments have been performed using a multiple-cylinder four stroke stationary petrol engine. The test results of hardness, emission, corrosion and temperature of the coated and uncoated manifolds have been compared. The result shows that the performance is improved and also emission is reduced in the coated exhaust manifold.
Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold.
Palacios, Jonathan; Yeh, Harry; Wang, Wenping; Zhang, Yue; Laramee, Robert S; Sharma, Ritesh; Schultz, Thomas; Zhang, Eugene
2016-03-01
Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.
Computational studies of an intake manifold for restricted engine application
Prasetyo, Bagus Dwi; Ubaidillah, Maharani, Elliza Tri; Setyohandoko, Gabriel; Idris, Muhammad Idzdihar
2018-02-01
The Formula Society of Automotive Engineer (FSAE) student competition is an international contest for a vehicle that entirely designed and built by students from various universities. The engine design in the Formula SAE competition has to comply a tight regulation. Concerning the engine intake line, an air restrictor of circular cross-section less than 20 mm must be fitted between the throttle valve and the engine inlet. The throat is aimed to limit the engine air flow rate as it strongly influences the volumetric efficiency and then the maximum power. This article focuses on the design of the engine intake system of the Bengawan FSAE team vehicle to optimize the engine power output and its stability. The performance of engine intake system is studied through computational fluid dynamics (CFD). The objective of CFD is to know the pressure, velocity, and airflow of the air intake manifold for the best performance of the engine. The three-dimensional drawing of the intake manifold was made, and CFD simulation was conducted using ANSYS FLUENT. Two models were studied. The result shows that the different design produces a different value of the velocity of airflow and the kind of flow type.
Cyclic coverings, Calabi-Yau manifolds and complex multiplication
Rohde, Christian
2009-01-01
The main goal of this book is the construction of families of Calabi-Yau 3-manifolds with dense sets of complex multiplication fibers. The new families are determined by combining and generalizing two methods. Firstly, the method of E. Viehweg and K. Zuo, who have constructed a deformation of the Fermat quintic with a dense set of CM fibers by a tower of cyclic coverings. Using this method, new families of K3 surfaces with dense sets of CM fibers and involutions are obtained. Secondly, the construction method of the Borcea-Voisin mirror family, which in the case of the author's examples yields families of Calabi-Yau 3-manifolds with dense sets of CM fibers, is also utilized. Moreover fibers with complex multiplication of these new families are also determined. This book was written for young mathematicians, physicists and also for experts who are interested in complex multiplication and varieties with complex multiplication. The reader is introduced to generic Mumford-Tate groups and Shimura data, which are a...
Attracting manifolds for attitude estimation in flatland and otherlands
Akella, Maruthi R.; Seo, Dongeun; Zanetti, Renato
2006-12-01
Non-convex and non-affine parameterizations of uncertainty are intrinsic within every attitude estimation problem given the fact that minimal and/or nonsingular representations of the attitude matrix are invariably nonlinear functions of the unknown attitude variables. Of course, this fact remains true for rotation matrices both in the 2-D plane (flatland) and in higher dimensional spaces (otherlands). Therefore, estimation problems involving minimal nonsingular representations of unknown attitude matrices bring significant challenges to the adaptive estimation community. This paper develops a novel algorithm for attitude estimation. The proposed algorithm relies upon the design of an adaptive update law for the attitude estimate while preserving its inherent orthogonal structure. The underlying approach borrows from the classical Poisson differential equation in rigid-body rotational kinematics and endows certain manifold attractivity features within the adaptive estimation algorithm. Consequently, we are not only able to efficiently handle the non-affine and non-convex nature of the parameter uncertainty, but are also ensured of estimation algorithm stability and robustness under bounded measurement noise. In addition to a rigorous discussion on the overall methodology, the paper provides example simulations that help demonstrate the effectiveness of the attracting manifolds design.
Conformal thermal tensor network and universal entropy on topological manifolds
Chen, Lei; Wang, Hao-Xin; Wang, Lei; Li, Wei
2017-11-01
Thermal quantum critical systems, with partition functions expressed as conformal tensor networks, are revealed to exhibit universal entropy corrections on nonorientable manifolds. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy SK=lnk on the Klein bottle, where k relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy lng (g reflects noninteger ground-state degeneracy), SK has n o boundary dependence or surface energy terms accompanying it, and can be very conveniently extracted from thermal data. On the Möbius-strip manifold, we uncover an entropy SM=1/2 (lng +lnk ) in CFT, where 1/2 lng is associated with the only open edge of the Möbius strip and 1/2 lnk is associated with the nonorientable topology. As a useful application, we employ the universal entropy to accurately pinpoint the quantum phase transitions, even for those without local order parameters.
Data assimilation on the exponentially accurate slow manifold.
Cotter, Colin
2013-05-28
I describe an approach to data assimilation making use of an explicit map that defines a coordinate system on the slow manifold in the semi-geostrophic scaling in Lagrangian coordinates, and apply the approach to a simple toy system that has previously been proposed as a low-dimensional model for the semi-geostrophic scaling. The method can be extended to Lagrangian particle methods such as Hamiltonian particle-mesh and smooth-particle hydrodynamics applied to the rotating shallow-water equations, and many of the properties will remain for more general Eulerian methods. Making use of Hamiltonian normal-form theory, it has previously been shown that, if initial conditions for the system are chosen as image points of the map, then the fast components of the system have exponentially small magnitude for exponentially long times as ε→0, and this property is preserved if one uses a symplectic integrator for the numerical time stepping. The map may then be used to parametrize initial conditions near the slow manifold, allowing data assimilation to be performed without introducing any fast degrees of motion (more generally, the precise amount of fast motion can be selected).
Reconstructing spatial organizations of chromosomes through manifold learning.
Zhu, Guangxiang; Deng, Wenxuan; Hu, Hailin; Ma, Rui; Zhang, Sai; Yang, Jinglin; Peng, Jian; Kaplan, Tommy; Zeng, Jianyang
2018-02-02
Decoding the spatial organizations of chromosomes has crucial implications for studying eukaryotic gene regulation. Recently, chromosomal conformation capture based technologies, such as Hi-C, have been widely used to uncover the interaction frequencies of genomic loci in a high-throughput and genome-wide manner and provide new insights into the folding of three-dimensional (3D) genome structure. In this paper, we develop a novel manifold learning based framework, called GEM (Genomic organization reconstructor based on conformational Energy and Manifold learning), to reconstruct the three-dimensional organizations of chromosomes by integrating Hi-C data with biophysical feasibility. Unlike previous methods, which explicitly assume specific relationships between Hi-C interaction frequencies and spatial distances, our model directly embeds the neighboring affinities from Hi-C space into 3D Euclidean space. Extensive validations demonstrated that GEM not only greatly outperformed other state-of-art modeling methods but also provided a physically and physiologically valid 3D representations of the organizations of chromosomes. Furthermore, we for the first time apply the modeled chromatin structures to recover long-range genomic interactions missing from original Hi-C data. © The Author(s) 2018. Published by Oxford University Press on behalf of Nucleic Acids Research.
Manifold regularized multitask feature learning for multimodality disease classification.
Jie, Biao; Zhang, Daoqiang; Cheng, Bo; Shen, Dinggang
2015-02-01
Multimodality based methods have shown great advantages in classification of Alzheimer's disease (AD) and its prodromal stage, that is, mild cognitive impairment (MCI). Recently, multitask feature selection methods are typically used for joint selection of common features across multiple modalities. However, one disadvantage of existing multimodality based methods is that they ignore the useful data distribution information in each modality, which is essential for subsequent classification. Accordingly, in this paper we propose a manifold regularized multitask feature learning method to preserve both the intrinsic relatedness among multiple modalities of data and the data distribution information in each modality. Specifically, we denote the feature learning on each modality as a single task, and use group-sparsity regularizer to capture the intrinsic relatedness among multiple tasks (i.e., modalities) and jointly select the common features from multiple tasks. Furthermore, we introduce a new manifold-based Laplacian regularizer to preserve the data distribution information from each task. Finally, we use the multikernel support vector machine method to fuse multimodality data for eventual classification. Conversely, we also extend our method to the semisupervised setting, where only partial data are labeled. We evaluate our method using the baseline magnetic resonance imaging (MRI), fluorodeoxyglucose positron emission tomography (FDG-PET), and cerebrospinal fluid (CSF) data of subjects from AD neuroimaging initiative database. The experimental results demonstrate that our proposed method can not only achieve improved classification performance, but also help to discover the disease-related brain regions useful for disease diagnosis. © 2014 Wiley Periodicals, Inc.
Supersymmetric 3-branes on smooth ALE manifolds with flux
International Nuclear Information System (INIS)
Bertolini, M.; Campos, V.L.; Ferretti, G.; Fre, P.; Salomonson, P.; Trigiante, M.
2001-01-01
We construct a new family of classical BPS solutions of type IIB supergravity describing 3-branes transverse to a 6-dimensional space with topology R 2 xALE. They are characterized by a non-trivial flux of the supergravity 2-forms through the homology 2-cycles of a generic smooth ALE manifold. Our solutions have two Killing spinors and thus preserve N=2 supersymmetry. They are expressed in terms of a quasi harmonic function H (the 'warp factor'), whose properties we study in the case of the simplest ALE, namely the Eguchi-Hanson manifold. The equation for H is identified as an instance of the confluent Heun equation. We write explicit power series solutions and solve the recurrence relation for the coefficients, discussing also the relevant asymptotic expansions. While, as in all such N=2 solutions, supergravity breaks down near the brane, the smoothing out of the vacuum geometry has the effect that the warp factor is regular in a region near the cycle. We interpret the behavior of the warp factor as describing a three-brane charge 'smeared' over the cycle and consider the asymptotic form of the geometry in that region, showing that conformal invariance is broken even when the complex type IIB 3-form field strength is assumed to vanish. We conclude with a discussion of the basic features of the gauge theory dual
Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension
Energy Technology Data Exchange (ETDEWEB)
Medeiros, Paul de [Crawley, West Sussex (United Kingdom)
2016-02-01
We consider a class of smooth oriented Lorentzian manifolds in dimensions three and four which admit a nowhere vanishing conformal Killing vector and a closed two-form that is invariant under the Lie algebra of conformal Killing vectors. The invariant two-form is constrained in a particular way by the conformal geometry of the manifold. In three dimensions, the conformal Killing vector must be everywhere causal (or null if the invariant two-form vanishes identically). In four dimensions, the conformal Killing vector must be everywhere null and the invariant two-form vanishes identically if the geometry is everywhere of Petrov type N or O. To the conformal class of any such geometry, it is possible to assign a particular Lie superalgebra structure, called a conformal symmetry superalgebra. The even part of this superalgebra contains conformal Killing vectors and constant R-symmetries while the odd part contains (charged) twistor spinors. The largest possible dimension of a conformal symmetry superalgebra is realised only for geometries that are locally conformally flat. We determine precisely which non-trivial conformal classes of metrics admit a conformal symmetry superalgebra with the next largest possible dimension, and compute all the associated submaximal conformal symmetry superalgebras. In four dimensions, we also compute symmetry superalgebras for a class of Ricci-flat Lorentzian geometries not of Petrov type N or O which admit a null Killing vector.
Center manifold for nonintegrable nonlinear Schroedinger equations on the line
International Nuclear Information System (INIS)
Weder, R.
2000-01-01
In this paper we study the following nonlinear Schroedinger equation on the line, where f is real-valued, and it satisfies suitable conditions on regularity, on growth as a function of u and on decay as x → ± ∞. The generic potential, V, is real-valued and it is chosen so that the spectrum of H:= -d 2 /dx 2 +V consists of one simple negative eigenvalue and absolutely-continuous spectrum filling (0,∞). The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue of H, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all small solutions approach a particular periodic orbit in the center manifold as t→ ± ∞. In general, the periodic orbits are different for t→ ± ∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear bound state is asymptotic as t→ ± ∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different for t→ ± ∞. (orig.)
The construction of one unstable manifold for the dissipative Henon mapping
Valkering, T.P.
1984-01-01
The unstable manifold of a saddle point of the Henon mapping is constructed analytically via a contraction mapping, for a range of parameter values where the second fixed point is a stable node. One invariant piece of this manifold connects the saddle with the second fixed point. Rigorous error
21 CFR 870.4290 - Cardiopulmonary bypass adaptor, stopcock, manifold, or fitting.
2010-04-01
..., or fitting. 870.4290 Section 870.4290 Food and Drugs FOOD AND DRUG ADMINISTRATION, DEPARTMENT OF... Devices § 870.4290 Cardiopulmonary bypass adaptor, stopcock, manifold, or fitting. (a) Identification. A cardiopulmonary bypass adaptor, stopcock, manifold, or fitting is a device used in cardiovascular diagnostic...
Dirac-like operators on the Hilbert space of differential forms on manifolds with boundaries
Pérez-Pardo, Juan Manuel
The problem of self-adjoint extensions of Dirac-type operators in manifolds with boundaries is analyzed. The boundaries might be regular or non-regular. The latter situation includes point-like interactions, also called delta-like potentials, in manifolds of dimension higher than one. Self-adjoint boundary conditions for the case of dimension 2 are obtained explicitly.
Multi-view clustering via multi-manifold regularized non-negative matrix factorization.
Zong, Linlin; Zhang, Xianchao; Zhao, Long; Yu, Hong; Zhao, Qianli
2017-04-01
Non-negative matrix factorization based multi-view clustering algorithms have shown their competitiveness among different multi-view clustering algorithms. However, non-negative matrix factorization fails to preserve the locally geometrical structure of the data space. In this paper, we propose a multi-manifold regularized non-negative matrix factorization framework (MMNMF) which can preserve the locally geometrical structure of the manifolds for multi-view clustering. MMNMF incorporates consensus manifold and consensus coefficient matrix with multi-manifold regularization to preserve the locally geometrical structure of the multi-view data space. We use two methods to construct the consensus manifold and two methods to find the consensus coefficient matrix, which leads to four instances of the framework. Experimental results show that the proposed algorithms outperform existing non-negative matrix factorization based algorithms for multi-view clustering. Copyright © 2017 Elsevier Ltd. All rights reserved.
Manifolds, tensors and differential forms: some applications in physics
International Nuclear Information System (INIS)
Datta, S.
1989-01-01
The style of mathematics used in contemporary physics has evolved considerably during the last twentyfive years. Groups, topology and differential geometry have become an intergral part of the physicist's jargon in their attempt to express the laws of the nature in lucid and compact terms. The notes prepared are based on the lectures given by the author in the Mathematics Seminar of the Theoretical Physics Division in the latter half of 1985. These lecture notes contain an introduction to manifolds and differential forms in the most succinct manner that is possible. It is essentially an attempt to familiarise the reader with the requisite vocabulary in this area of mathematical physics without scaring them with excess of rigour. This hopefully will help in following the contemporary literature in physics. (author). 6 refs
RELATIVE CAMERA POSE ESTIMATION METHOD USING OPTIMIZATION ON THE MANIFOLD
Directory of Open Access Journals (Sweden)
C. Cheng
2017-05-01
Full Text Available To solve the problem of relative camera pose estimation, a method using optimization with respect to the manifold is proposed. Firstly from maximum-a-posteriori (MAP model to nonlinear least squares (NLS model, the general state estimation model using optimization is derived. Then the camera pose estimation model is applied to the general state estimation model, while the parameterization of rigid body transformation is represented by Lie group/algebra. The jacobian of point-pose model with respect to Lie group/algebra is derived in detail and thus the optimization model of rigid body transformation is established. Experimental results show that compared with the original algorithms, the approaches with optimization can obtain higher accuracy both in rotation and translation, while avoiding the singularity of Euler angle parameterization of rotation. Thus the proposed method can estimate relative camera pose with high accuracy and robustness.
Existence and equivalence of twisted products on a symplectic manifold
International Nuclear Information System (INIS)
Lichnerowicz, A.
1979-01-01
The twisted products play an important role in Quantum Mechanics. A distinction is introduced between Vey *sub(γ) products and strong Vey *sub(γ) products and it is proved that each *sub(γ) product is equivalent to a Vey *sub(γ) product. If b 3 (W) = 0, the symplectic manifold (W,F) admits strong Vey *sub(Gn) products. If b 2 (W) = 0, all *sub(γ) products are equivalent as well as the Vey Lie algebras. In the general case the formal Lie algebras are characterized which are generated by a *sub(γ) product and it proved that the existance of a *sub(γ)-product is equivalent to the existance of a formal Lie algebra infinitesimally equivalent to a Vey Lie algebra at the first order. (Auth.)
Spectral Quasi-Equilibrium Manifold for Chemical Kinetics.
Kooshkbaghi, Mahdi; Frouzakis, Christos E; Boulouchos, Konstantinos; Karlin, Iliya V
2016-05-26
The Spectral Quasi-Equilibrium Manifold (SQEM) method is a model reduction technique for chemical kinetics based on entropy maximization under constraints built by the slowest eigenvectors at equilibrium. The method is revisited here and discussed and validated through the Michaelis-Menten kinetic scheme, and the quality of the reduction is related to the temporal evolution and the gap between eigenvalues. SQEM is then applied to detailed reaction mechanisms for the homogeneous combustion of hydrogen, syngas, and methane mixtures with air in adiabatic constant pressure reactors. The system states computed using SQEM are compared with those obtained by direct integration of the detailed mechanism, and good agreement between the reduced and the detailed descriptions is demonstrated. The SQEM reduced model of hydrogen/air combustion is also compared with another similar technique, the Rate-Controlled Constrained-Equilibrium (RCCE). For the same number of representative variables, SQEM is found to provide a more accurate description.
Gauge theory of gravity and supergravity on a group manifold
International Nuclear Information System (INIS)
Ne'eman, Y.; Regge, T.
1977-12-01
The natural arena for the physics of gravity, supergravity and their enlargements appears to be the group manifold of the Poincare group P, the graded Poincare group GP of supersymmetry, and the corresponding enlargements. The dynamics of these theories correspond to geometrical algorithms in P and GP. Differential geometry on Lie groups is reviewed and results applied to P and GP. Curvature, gauge transformations and factorization are introduced. Also reviewed is the general coordinate transformation group and a hybrid gauge transformation, the anholonomized G.C.T. gauge. A study is made of the construction of an action, including the introduction of a set of special 2 forms, the ''pseudo curvatures.'' The possibilities of factorization in supersymmetry are analyzed. The version of supergravity is present which has now become a completely geometrical theory
Symplectic manifolds with no Kähler structure
Tralle, Aleksy
1997-01-01
This is a research monograph covering the majority of known results on the problem of constructing compact symplectic manifolds with no Kaehler structure with an emphasis on the use of rational homotopy theory. In recent years, some new and stimulating conjectures and problems have been formulated due to an influx of homotopical ideas. Examples include the Lupton-Oprea conjecture, the Benson-Gordon conjecture, both of which are in the spirit of some older and still unsolved problems (e.g. Thurston's conjecture and Sullivan's problem). Our explicit aim is to clarify the interrelations between certain aspects of symplectic geometry and homotopy theory in the framework of the problems mentioned above. We expect that the reader is aware of the basics of differential geometry and algebraic topology at graduate level.
Space Manifold Dynamics Novel Spaceways for Science and Exploration
Perozzi, Ettore
2010-01-01
This book presents an overview of the outcomes resulting from applying the dynamical systems approach to space mission design, a topic referred to as "Space Manifold Dynamics" (SMD). It is a natural follow-on to the international workshop "Novel Spaceways for Scientific and Exploration Missions," which was held in October 2007 at the Telespazio Fucino Space Centre (Italy) under the auspices of the Space OPS Academy. The benefits and drawbacks of using the Lagrangian points and the associated trajectories for present and future space missions are discussed. The related methods and algorithms are also described in detail. Each topic is presented in articles that were written as far as possible to be self consistent; the use of introductory sections and of extended explanations is included in order to address the different communities potentially interested in SMD: space science, the aerospace industry, manned and unmanned exploration, celestial mechanics, and flight dynamics.
Decision Manifold Approximation for Physics-Based Simulations
Wong, Jay Ming; Samareh, Jamshid A.
2016-01-01
With the recent surge of success in big-data driven deep learning problems, many of these frameworks focus on the notion of architecture design and utilizing massive databases. However, in some scenarios massive sets of data may be difficult, and in some cases infeasible, to acquire. In this paper we discuss a trajectory-based framework that quickly learns the underlying decision manifold of binary simulation classifications while judiciously selecting exploratory target states to minimize the number of required simulations. Furthermore, we draw particular attention to the simulation prediction application idealized to the case where failures in simulations can be predicted and avoided, providing machine intelligence to novice analysts. We demonstrate this framework in various forms of simulations and discuss its efficacy.
Generalized metric formulation of double field theory on group manifolds
International Nuclear Information System (INIS)
Blumenhagen, Ralph; Bosque, Pascal du; Hassler, Falk; Lüst, Dieter
2015-01-01
We rewrite the recently derived cubic action of Double Field Theory on group manifolds http://dx.doi.org/10.1007/JHEP02(2015)001 in terms of a generalized metric and extrapolate it to all orders in the fields. For the resulting action, we derive the field equations and state them in terms of a generalized curvature scalar and a generalized Ricci tensor. Compared to the generalized metric formulation of DFT derived from tori, all these quantities receive additional contributions related to the non-trivial background. It is shown that the action is invariant under its generalized diffeomorphisms and 2D-diffeomorphisms. Imposing additional constraints relating the background and fluctuations around it, the precise relation between the proposed generalized metric formulation of DFT WZW and of original DFT from tori is clarified. Furthermore, we show how to relate DFT WZW of the WZW background with the flux formulation of original DFT.
Low-rank matrix approximation with manifold regularization.
Zhang, Zhenyue; Zhao, Keke
2013-07-01
This paper proposes a new model of low-rank matrix factorization that incorporates manifold regularization to the matrix factorization. Superior to the graph-regularized nonnegative matrix factorization, this new regularization model has globally optimal and closed-form solutions. A direct algorithm (for data with small number of points) and an alternate iterative algorithm with inexact inner iteration (for large scale data) are proposed to solve the new model. A convergence analysis establishes the global convergence of the iterative algorithm. The efficiency and precision of the algorithm are demonstrated numerically through applications to six real-world datasets on clustering and classification. Performance comparison with existing algorithms shows the effectiveness of the proposed method for low-rank factorization in general.
Selecting protein families for environmental features based on manifold regularization.
Jiang, Xingpeng; Xu, Weiwei; Park, E K; Li, Guangrong
2014-06-01
Recently, statistics and machine learning have been developed to identify functional or taxonomic features of environmental features or physiological status. Important proteins (or other functional and taxonomic entities) to environmental features can be potentially used as biosensors. A major challenge is how the distribution of protein and gene functions embodies the adaption of microbial communities across environments and host habitats. In this paper, we propose a novel regularization method for linear regression to adapt the challenge. The approach is inspired by local linear embedding (LLE) and we call it a manifold-constrained regularization for linear regression (McRe). The novel regularization procedure also has potential to be used in solving other linear systems. We demonstrate the efficiency and the performance of the approach in both simulation and real data.
Topological strings on Grassmannian Calabi-Yau manifolds
International Nuclear Information System (INIS)
Haghighat, Babak; Klemm, Albrecht
2009-01-01
We present solutions for the higher genus topological string amplitudes on Calabi-Yau-manifolds, which are realized as complete intersections in Grassmannians. We solve the B-model by direct integration of the holomorphic anomaly equations using a finite basis of modular invariant generators, the gap condition at the conifold and other local boundary conditions for the amplitudes. Regularity of the latter at certain points in the moduli space suggests a CFT description. The A-model amplitudes are evaluated using a mirror conjecture for Calabi-Yau complete intersections in Grassmannians by Batyrev, Ciocan-Fontanine, Kim and Van Straten. The integrality of the BPS states gives strong evidence for the conjecture.
Generalized Metric Formulation of Double Field Theory on Group Manifolds
Blumenhagen, Ralph; Hassler, Falk; Lust, Dieter
2015-01-01
We rewrite the recently derived cubic action of Double Field Theory on group manifolds [arXiv:1410.6374] in terms of a generalized metric and extrapolate it to all orders in the fields. For the resulting action, we derive the field equations and state them in terms of a generalized curvature scalar and a generalized Ricci tensor. Compared to the generalized metric formulation of DFT derived from tori, all these quantities receive additional contributions related to the non-trivial background. It is shown that the action is invariant under its generalized diffeomorphisms and 2D-diffeomorphisms. Imposing additional constraints relating the background and fluctuations around it, the precise relation between the proposed generalized metric formulation of DFT${}_\\mathrm{WZW}$ and of original DFT from tori is clarified. Furthermore we show how to relate DFT${}_\\mathrm{WZW}$ of the WZW background with the flux formulation of original DFT.
Face recognition based on LDA in manifold subspace
Directory of Open Access Journals (Sweden)
Hung Phuoc Truong
2016-05-01
Full Text Available Although LDA has many successes in dimensionality reduction and data separation, it also has disadvantages, especially the small sample size problem in training data because the "within-class scatter" matrix may not be accurately estimated. Moreover, this algorithm can only operate correctly with labeled data in supervised learning. In practice, data collection is very huge and labeling data requires high-cost, thus the combination of a part of labeled data and unlabeled data for this algorithm in Manifold subspace is a novelty research. This paper reports a study that propose a semi-supervised method called DSLM, which aims at overcoming all these limitations. The proposed method ensures that the discriminative information of labeled data and the intrinsic geometric structure of data are mapped to new optimal subspace. Results are obtained from the experiments and compared to several related methods showing the effectiveness of our proposed method.
Analysis III analytic and differential functions, manifolds and Riemann surfaces
Godement, Roger
2015-01-01
Volume III sets out classical Cauchy theory. It is much more geared towards its innumerable applications than towards a more or less complete theory of analytic functions. Cauchy-type curvilinear integrals are then shown to generalize to any number of real variables (differential forms, Stokes-type formulas). The fundamentals of the theory of manifolds are then presented, mainly to provide the reader with a "canonical'' language and with some important theorems (change of variables in integration, differential equations). A final chapter shows how these theorems can be used to construct the compact Riemann surface of an algebraic function, a subject that is rarely addressed in the general literature though it only requires elementary techniques. Besides the Lebesgue integral, Volume IV will set out a piece of specialized mathematics towards which the entire content of the previous volumes will converge: Jacobi, Riemann, Dedekind series and infinite products, elliptic functions, classical theory of modular fun...
Multimodal manifold-regularized transfer learning for MCI conversion prediction.
Cheng, Bo; Liu, Mingxia; Suk, Heung-Il; Shen, Dinggang; Zhang, Daoqiang
2015-12-01
As the early stage of Alzheimer's disease (AD), mild cognitive impairment (MCI) has high chance to convert to AD. Effective prediction of such conversion from MCI to AD is of great importance for early diagnosis of AD and also for evaluating AD risk pre-symptomatically. Unlike most previous methods that used only the samples from a target domain to train a classifier, in this paper, we propose a novel multimodal manifold-regularized transfer learning (M2TL) method that jointly utilizes samples from another domain (e.g., AD vs. normal controls (NC)) as well as unlabeled samples to boost the performance of the MCI conversion prediction. Specifically, the proposed M2TL method includes two key components. The first one is a kernel-based maximum mean discrepancy criterion, which helps eliminate the potential negative effect induced by the distributional difference between the auxiliary domain (i.e., AD and NC) and the target domain (i.e., MCI converters (MCI-C) and MCI non-converters (MCI-NC)). The second one is a semi-supervised multimodal manifold-regularized least squares classification method, where the target-domain samples, the auxiliary-domain samples, and the unlabeled samples can be jointly used for training our classifier. Furthermore, with the integration of a group sparsity constraint into our objective function, the proposed M2TL has a capability of selecting the informative samples to build a robust classifier. Experimental results on the Alzheimer's Disease Neuroimaging Initiative (ADNI) database validate the effectiveness of the proposed method by significantly improving the classification accuracy of 80.1 % for MCI conversion prediction, and also outperforming the state-of-the-art methods.
Manifold regularized discriminative nonnegative matrix factorization with fast gradient descent.
Guan, Naiyang; Tao, Dacheng; Luo, Zhigang; Yuan, Bo
2011-07-01
Nonnegative matrix factorization (NMF) has become a popular data-representation method and has been widely used in image processing and pattern-recognition problems. This is because the learned bases can be interpreted as a natural parts-based representation of data and this interpretation is consistent with the psychological intuition of combining parts to form a whole. For practical classification tasks, however, NMF ignores both the local geometry of data and the discriminative information of different classes. In addition, existing research results show that the learned basis is unnecessarily parts-based because there is neither explicit nor implicit constraint to ensure the representation parts-based. In this paper, we introduce the manifold regularization and the margin maximization to NMF and obtain the manifold regularized discriminative NMF (MD-NMF) to overcome the aforementioned problems. The multiplicative update rule (MUR) can be applied to optimizing MD-NMF, but it converges slowly. In this paper, we propose a fast gradient descent (FGD) to optimize MD-NMF. FGD contains a Newton method that searches the optimal step length, and thus, FGD converges much faster than MUR. In addition, FGD includes MUR as a special case and can be applied to optimizing NMF and its variants. For a problem with 165 samples in R(1600), FGD converges in 28 s, while MUR requires 282 s. We also apply FGD in a variant of MD-NMF and experimental results confirm its efficiency. Experimental results on several face image datasets suggest the effectiveness of MD-NMF.
T4 fibrations over Calabi–Yau two-folds and non-Kähler manifolds in string theory
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Hai Lin
2016-08-01
Full Text Available We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial T4 fibrations over Calabi–Yau two-folds. These give rise to eight-dimensional non-Kähler Hermitian manifolds with SU(4 structure. The eight-manifold is also a circle fibration over a seven-dimensional G2 manifold with skew torsion. The eight-manifolds of this type appear as internal manifolds with SU(4 structure in type IIB string theory with F3 and F7 fluxes. These manifolds have generalized calibrated cycles in the presence of fluxes.
King, Nathan D.; Ruuth, Steven J.
2017-05-01
Maps from a source manifold M to a target manifold N appear in liquid crystals, color image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds M and N to the simpler problems of solving a PDE on M and projecting to the closest points on N. In our approach, an embedding PDE is formulated in the embedding space using closest point representations of M and N. This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes p-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and p-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance color images.
Learning an intrinsic-variable preserving manifold for dynamic visual tracking.
Qiao, Hong; Zhang, Peng; Zhang, Bo; Zheng, Suiwu
2010-06-01
Manifold learning is a hot topic in the field of computer science, particularly since nonlinear dimensionality reduction based on manifold learning was proposed in Science in 2000. The work has achieved great success. The main purpose of current manifold-learning approaches is to search for independent intrinsic variables underlying high dimensional inputs which lie on a low dimensional manifold. In this paper, a new manifold is built up in the training step of the process, on which the input training samples are set to be close to each other if the values of their intrinsic variables are close to each other. Then, the process of dimensionality reduction is transformed into a procedure of preserving the continuity of the intrinsic variables. By utilizing the new manifold, the dynamic tracking of a human who can move and rotate freely is achieved. From the theoretical point of view, it is the first approach to transfer the manifold-learning framework to dynamic tracking. From the application point of view, a new and low dimensional feature for visual tracking is obtained and successfully applied to the real-time tracking of a free-moving object from a dynamic vision system. Experimental results from a dynamic tracking system which is mounted on a dynamic robot validate the effectiveness of the new algorithm.
ON A FAIR MANIFOLD FARE RATING ON A LONG TRAFFIC LINE
Directory of Open Access Journals (Sweden)
Stanislav PALÚCH
2017-06-01
Full Text Available The paper studies the possibilities to design a fair manifold tariff on a long traffic line. If a single tariff is used on a long bus or railway line, passengers travelling long distances are favoured at the expense of those travelling short distances. The fairest approach to tariff is setting an individual tariff for every origin–destination relation of line stops that expresses real travel costs. However, sometimes the individual tariff is too complicated and is therefore replaced by double-, triple- or manifold tariff. This paper shows how to design a manifold tariff in order to minimize unfairness to passengers.
From Stein to Weinstein and back symplectic geometry of affine complex manifolds
Cieliebak, Kai
2013-01-01
A beautiful and comprehensive introduction to this important field. -Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a superb introduction to this area and also contains the authors' new results. -Tomasz Mrowka, MIT This book is devoted to the interplay between complex and symplectic geometry in affine complex manifolds. Affine co
Microwave transport in EBT distribution manifolds using Monte Carlo ray-tracing techniques
International Nuclear Information System (INIS)
Lillie, R.A.; White, T.L.; Gabriel, T.A.; Alsmiller, R.G. Jr.
1983-01-01
Ray tracing Monte Carlo calculations have been carried out using an existing Monte Carlo radiation transport code to obtain estimates of the microsave power exiting the torus coupling links in EPT microwave manifolds. The microwave power loss and polarization at surface reflections were accounted for by treating the microwaves as plane waves reflecting off plane surfaces. Agreement on the order of 10% was obtained between the measured and calculated output power distribution for an existing EBT-S toroidal manifold. A cost effective iterative procedure utilizing the Monte Carlo history data was implemented to predict design changes which could produce increased manifold efficiency and improved output power uniformity
Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds
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M. D. Siddiqi
2017-12-01
Full Text Available In the present paper, we study a new class of submanifolds of a generalized Quasi-Sasakian manifold, called skew semi-invariant submanifold. We obtain integrability conditions of the distributions on a skew semi-invariant submanifold and also find the condition for a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold to be mixed totally geodesic. Also it is shown that a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold will be anti-invariant if and only if $A_{\\xi}=0$; and the submanifold will be skew semi-invariant submanifold if $\
Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design
Gomez, G.; Koon, W. S.; Lo, Martin W.; Marsden, J. E.; Masdemont, J.; Ross, S. D.
2001-01-01
The invariant manifold structures of the collinear libration points for the spatial restricted three-body problem provide the framework for understanding complex dynamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold 'tubes' associated to libration point orbits are the phase space structures that provide a conduit for orbits between primary bodies for separate three-body systems. These invariant manifold tubes can be used to construct new spacecraft trajectories, such as 'Petit Grand Tour' of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. The current work extends the results to the spatial case.
Recent developments in the structural design and optimization of ITER neutral beam manifold
Chengzhi, CAO; Yudong, PAN; Zhiwei, XIA; Bo, LI; Tao, JIANG; Wei, LI
2018-02-01
This paper describes a new design of the neutral beam manifold based on a more optimized support system. A proposed alternative scheme has presented to replace the former complex manifold supports and internal pipe supports in the final design phase. Both the structural reliability and feasibility were confirmed with detailed analyses. Comparative analyses between two typical types of manifold support scheme were performed. All relevant results of mechanical analyses for typical operation scenarios and fault conditions are presented. Future optimization activities are described, which will give useful information for a refined setting of components in the next phase.
Energy Technology Data Exchange (ETDEWEB)
Diez, Rainer; Kornherr, Heinz; Pirntke, Frank; Schmidt, Juergen [Friedrich Boysen GmbH und Co. KG, Altensteig (Germany)
2010-05-15
In close interdisciplinary cooperation with BMW Group, Boysen has developed an air-gap-insulated exhaust manifold that encompasses both banks of the 4.4 l V8 spark-ignition twin turbo engine of the BMW X5 M and BMW X6 M. The manifold merges the exhaust gas flow from the cylinders of the left-hand and right-hand cylinder banks in opposing pairs, thus optimising gas exchange. Due to improvements in response, torque and power characteristics of the engine, the cylinder-bank comprehensive exhaust manifold helps achieve high fuel efficiency. (orig.)
Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis
Mohammed, Kitty; Narayanan, Hariharan
2017-01-01
We consider the problem of recovering a $d-$dimensional manifold $\\mathcal{M} \\subset \\mathbb{R}^n$ when provided with noiseless samples from $\\mathcal{M}$. There are many algorithms (e.g., Isomap) that are used in practice to fit manifolds and thus reduce the dimensionality of a given data set. Ideally, the estimate $\\mathcal{M}_\\mathrm{put}$ of $\\mathcal{M}$ should be an actual manifold of a certain smoothness; furthermore, $\\mathcal{M}_\\mathrm{put}$ should be arbitrarily close to $\\mathcal...
Fermat principles in general relativity and the existence of light rays on Lorentzian manifolds
International Nuclear Information System (INIS)
Fortunato, D.; Masiello, A.
1995-01-01
In this paper we review some results on the existence and multiplicity of null geodesics (light rays) joining a point with a timelike curve on a Lorentzian manifold. Moreover a Morse Theory for such geodesics is presented. A variational principle, which is a variant of the classical Fermat principle in optics, allows to characterize the null geodesics joining a point with a timelike curve as the critical points of a functional on an infinite dimensional manifold. Global variational methods are used to get the existence results and Morse Theory. Such results cover a class of Lorentzian manifolds including Schwarzschild, Reissner-Nordstroem and Kerr space-time. (author)
The parameterization method for invariant manifolds from rigorous results to effective computations
Haro, Àlex; Figueras, Jordi-Lluis; Luque, Alejandro; Mondelo, Josep Maria
2016-01-01
This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
Yang-Mills bar connections over compact K\\"ahler manifolds
Van Le, Hong
2008-01-01
In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact K\\"ahler manifolds as well as a short time existence of the negative Yang-Mills bar gradient flow. We also show a rigidity of holomorphic connections among a class of Yang-Mills bar connections over compact K\\"ahler manifolds of positive Ricci ...
de Hoop, Maarten V.; Ilmavirta, Joonas
2017-12-01
We study ray transforms on spherically symmetric manifolds with a piecewise C1, 1 metric. Assuming the Herglotz condition, the x-ray transform is injective on the space of L 2 functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a C1, 1 metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.
Nonlinear dimensionality reduction of data lying on the multicluster manifold.
Meng, Deyu; Leung, Yee; Fung, Tung; Xu, Zongben
2008-08-01
A new method, which is called decomposition-composition (D-C) method, is proposed for the nonlinear dimensionality reduction (NLDR) of data lying on the multicluster manifold. The main idea is first to decompose a given data set into clusters and independently calculate the low-dimensional embeddings of each cluster by the decomposition procedure. Based on the intercluster connections, the embeddings of all clusters are then composed into their proper positions and orientations by the composition procedure. Different from other NLDR methods for multicluster data, which consider associatively the intracluster and intercluster information, the D-C method capitalizes on the separate employment of the intracluster neighborhood structures and the intercluster topologies for effective dimensionality reduction. This, on one hand, isometrically preserves the rigid-body shapes of the clusters in the embedding process and, on the other hand, guarantees the proper locations and orientations of all clusters. The theoretical arguments are supported by a series of experiments performed on the synthetic and real-life data sets. In addition, the computational complexity of the proposed method is analyzed, and its efficiency is theoretically analyzed and experimentally demonstrated. Related strategies for automatic parameter selection are also examined.
Multimodal Medical Image Fusion by Adaptive Manifold Filter
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Peng Geng
2015-01-01
Full Text Available Medical image fusion plays an important role in diagnosis and treatment of diseases such as image-guided radiotherapy and surgery. The modified local contrast information is proposed to fuse multimodal medical images. Firstly, the adaptive manifold filter is introduced into filtering source images as the low-frequency part in the modified local contrast. Secondly, the modified spatial frequency of the source images is adopted as the high-frequency part in the modified local contrast. Finally, the pixel with larger modified local contrast is selected into the fused image. The presented scheme outperforms the guided filter method in spatial domain, the dual-tree complex wavelet transform-based method, nonsubsampled contourlet transform-based method, and four classic fusion methods in terms of visual quality. Furthermore, the mutual information values by the presented method are averagely 55%, 41%, and 62% higher than the three methods and those values of edge based similarity measure by the presented method are averagely 13%, 33%, and 14% higher than the three methods for the six pairs of source images.
Noise reduction in intracranial pressure signal using causal shape manifolds.
Rajagopal, Abhejit; Hamilton, Robert B; Scalzo, Fabien
2016-07-01
We present the Iterative/Causal Subspace Tracking framework (I/CST) for reducing noise in continuously monitored quasi-periodic biosignals. Signal reconstruction of the basic segments of the noisy signal (e.g. beats) is achieved by projection to a reduced space on which probabilistic tracking is performed. The attractiveness of the presented method lies in the fact that the subspace, or manifold, is learned by incorporating temporal, morphological, and signal elevation constraints, so that segment samples with similar shapes, and that are close in time and elevation, are also close in the subspace representation. Evaluation of the algorithm's effectiveness on the intracranial pressure (ICP) signal serves as a practical illustration of how it can operate in clinical conditions on routinely acquired biosignals. The reconstruction accuracy of the system is evaluated on an idealized 20-min ICP recording established from the average ICP of patients monitored for various ICP related conditions. The reconstruction accuracy of the ground truth signal is tested in presence of varying levels of additive white Gaussian noise (AWGN) and Poisson noise processes, and measures significant increases of 758% and 396% in the average signal-to-noise ratio (SNR).
Indoor localization using unsupervised manifold alignment with geometry perturbation
Majeed, Khaqan
2014-04-01
The main limitation of deploying/updating Received Signal Strength (RSS) based indoor localization is the construction of fingerprinted radio map, which is quite a hectic and time-consuming process especially when the indoor area is enormous and/or dynamic. Different approaches have been undertaken to reduce such deployment/update efforts, but the performance degrades when the fingerprinting load is reduced below a certain level. In this paper, we propose an indoor localization scheme that requires as low as 1% fingerprinting load. This scheme employs unsupervised manifold alignment that takes crowd sourced RSS readings and localization requests as source data set and the environment\\'s plan coordinates as destination data set. The 1% fingerprinting load is only used to perturb the local geometries in the destination data set. Our proposed algorithm was shown to achieve less than 5 m mean localization error with 1% fingerprinting load and a limited number of crowd sourced readings, when other learning based localization schemes pass the 10 m mean error with the same information.
Visual words assignment via information-theoretic manifold embedding.
Deng, Yue; Li, Yipeng; Qian, Yanjun; Ji, Xiangyang; Dai, Qionghai
2014-10-01
Codebook-based learning provides a flexible way to extract the contents of an image in a data-driven manner for visual recognition. One central task in such frameworks is codeword assignment, which allocates local image descriptors to the most similar codewords in the dictionary to generate histogram for categorization. Nevertheless, existing assignment approaches, e.g., nearest neighbors strategy (hard assignment) and Gaussian similarity (soft assignment), suffer from two problems: 1) too strong Euclidean assumption and 2) neglecting the label information of the local descriptors. To address the aforementioned two challenges, we propose a graph assignment method with maximal mutual information (GAMI) regularization. GAMI takes the power of manifold structure to better reveal the relationship of massive number of local features by nonlinear graph metric. Meanwhile, the mutual information of descriptor-label pairs is ultimately optimized in the embedding space for the sake of enhancing the discriminant property of the selected codewords. According to such objective, two optimization models, i.e., inexact-GAMI and exact-GAMI, are respectively proposed in this paper. The inexact model can be efficiently solved with a closed-from solution. The stricter exact-GAMI nonparametrically estimates the entropy of descriptor-label pairs in the embedding space and thus leads to a relatively complicated but still trackable optimization. The effectiveness of GAMI models are verified on both the public and our own datasets.
Descriptor Learning via Supervised Manifold Regularization for Multioutput Regression.
Zhen, Xiantong; Yu, Mengyang; Islam, Ali; Bhaduri, Mousumi; Chan, Ian; Li, Shuo
2017-09-01
Multioutput regression has recently shown great ability to solve challenging problems in both computer vision and medical image analysis. However, due to the huge image variability and ambiguity, it is fundamentally challenging to handle the highly complex input-target relationship of multioutput regression, especially with indiscriminate high-dimensional representations. In this paper, we propose a novel supervised descriptor learning (SDL) algorithm for multioutput regression, which can establish discriminative and compact feature representations to improve the multivariate estimation performance. The SDL is formulated as generalized low-rank approximations of matrices with a supervised manifold regularization. The SDL is able to simultaneously extract discriminative features closely related to multivariate targets and remove irrelevant and redundant information by transforming raw features into a new low-dimensional space aligned to targets. The achieved discriminative while compact descriptor largely reduces the variability and ambiguity for multioutput regression, which enables more accurate and efficient multivariate estimation. We conduct extensive evaluation of the proposed SDL on both synthetic data and real-world multioutput regression tasks for both computer vision and medical image analysis. Experimental results have shown that the proposed SDL can achieve high multivariate estimation accuracy on all tasks and largely outperforms the algorithms in the state of the arts. Our method establishes a novel SDL framework for multioutput regression, which can be widely used to boost the performance in different applications.
Discriminative semi-supervised feature selection via manifold regularization.
Xu, Zenglin; King, Irwin; Lyu, Michael Rung-Tsong; Jin, Rong
2010-07-01
Feature selection has attracted a huge amount of interest in both research and application communities of data mining. We consider the problem of semi-supervised feature selection, where we are given a small amount of labeled examples and a large amount of unlabeled examples. Since a small number of labeled samples are usually insufficient for identifying the relevant features, the critical problem arising from semi-supervised feature selection is how to take advantage of the information underneath the unlabeled data. To address this problem, we propose a novel discriminative semi-supervised feature selection method based on the idea of manifold regularization. The proposed approach selects features through maximizing the classification margin between different classes and simultaneously exploiting the geometry of the probability distribution that generates both labeled and unlabeled data. In comparison with previous semi-supervised feature selection algorithms, our proposed semi-supervised feature selection method is an embedded feature selection method and is able to find more discriminative features. We formulate the proposed feature selection method into a convex-concave optimization problem, where the saddle point corresponds to the optimal solution. To find the optimal solution, the level method, a fairly recent optimization method, is employed. We also present a theoretic proof of the convergence rate for the application of the level method to our problem. Empirical evaluation on several benchmark data sets demonstrates the effectiveness of the proposed semi-supervised feature selection method.
Towards representation of a perceptual color manifold using associative memory for color constancy.
Seow, Ming-Jung; Asari, Vijayan K
2009-01-01
In this paper, we propose the concept of a manifold of color perception through empirical observation that the center-surround properties of images in a perceptually similar environment define a manifold in the high dimensional space. Such a manifold representation can be learned using a novel recurrent neural network based learning algorithm. Unlike the conventional recurrent neural network model in which the memory is stored in an attractive fixed point at discrete locations in the state space, the dynamics of the proposed learning algorithm represent memory as a nonlinear line of attraction. The region of convergence around the nonlinear line is defined by the statistical characteristics of the training data. This learned manifold can then be used as a basis for color correction of the images having different color perception to the learned color perception. Experimental results show that the proposed recurrent neural network learning algorithm is capable of color balance the lighting variations in images captured in different environments successfully.
Supersymmetric Proof of the Hirzebruch-Riemann-Roch Theorem for Non-Kähler Manifolds
Directory of Open Access Journals (Sweden)
Andrei V. Smilga
2012-01-01
Full Text Available We present the proof of the HRR theorem for a generic complex compact manifold by evaluating the functional integral for the Witten index of the appropriate supersymmetric quantum mechanical system.
Yang-Mills instantons on cones and sine-cones over nearly Kähler manifolds
Gemmer, Karl-Philip; Lechtenfeld, Olaf; Nölle, Christoph; Popov, Alexander D.
2011-09-01
We present a unified eight-dimensional approach to instanton equations on several seven-dimensional manifolds associated to a six-dimensional homogeneous nearly Kähler manifold. The cone over the sine-cone on a nearly Kähler manifold has holonomy group Spin(7) and can befoliated by submanifolds with either holonomy group G 2, a nearly parallel G 2-structure or a cocalibrated G 2-structure. We show that there is a G 2-instanton on each of these seven-dimensional manifolds which gives rise to a Spin(7)-instanton in eight dimensions. The well-known octonionic instantons on {mathbb{R}^7} and {mathbb{R}^8} are contained in our construction as the special cases of an instanton on the cone and on the cone over the sine-cone, both over the six-sphere, respectively.
Estimating Turaev-Viro three-manifold invariants is universal for quantum computation
International Nuclear Information System (INIS)
Alagic, Gorjan; Reichardt, Ben W.; Jordan, Stephen P.; Koenig, Robert
2010-01-01
The Turaev-Viro invariants are scalar topological invariants of compact, orientable 3-manifolds. We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and (2+1)-dimensional topological quantum field theories. Its accuracy is shown to be nontrivial, as the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer. Thus approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a relation between the task of distinguishing nonhomeomorphic 3-manifolds and the power of a general quantum computer.
CERN Library
2014-01-01
Tuesday 25 March 2014 at 4 p.m. in the Library, bldg. 52-1-052 "Differential manifolds: a basic approach for experimental physicists" by Paul Baillon, World Scientific, 2013, ISBN 978-981-4449-56-4. Differential manifold is the framework of particle physics and astrophysics nowadays. It is important for all research physicists to be accustomed to it, and even experimental physicists should be able to manipulate equations and expressions in this framework. This book gives a comprehensive description of the basics of differential manifold with a full proof of elements. A large part of the book is devoted to the basic mathematical concepts, which are all necessary for the development of the differential manifold. This book is self-consistent; it starts from first principles. The mathematical framework is the set theory with its axioms and its formal logic. No special knowledge is needed. Coffee will be served from 3.30 p.m.
Soft-group-manifold structure of supergravity, and the proof of unitarity
Energy Technology Data Exchange (ETDEWEB)
Thierry-Mieg, J.; Ne' eman, Y.
1979-01-01
The Soft Group Manifold (SGM) method of gauging a non-internal group is reviewed. Applications to three different versions of Extended Supergravity are presented. Then the geometric identification of the ghost fields and BRS equations, another aspect of the Group Manifold method, is summarized. The results are applied to Supergravity, and a completion of the proof of Unitarity of Sterman, Townsend, and Van Nieuwenhuizen is provided. 3 figures.
A New Entropy Formula and Gradient Estimates for the Linear Heat Equation on Static Manifold
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Abimbola Abolarinwa
2014-08-01
Full Text Available In this paper we prove a new monotonicity formula for the heat equation via a generalized family of entropy functionals. This family of entropy formulas generalizes both Perelman’s entropy for evolving metric and Ni’s entropy on static manifold. We show that this entropy satisfies a pointwise differential inequality for heat kernel. The consequences of which are various gradient and Harnack estimates for all positive solutions to the heat equation on compact manifold.
Platonic polyhedra tune the three-sphere: II. Harmonic analysis on cubic spherical three-manifolds
International Nuclear Information System (INIS)
Kramer, Peter
2009-01-01
From the homotopy groups of two distinct cubic spherical three-manifolds, we construct the isomorphic groups of deck transformations acting on the three-sphere. These groups become the cyclic group of order eight and the quaternion group, respectively. By reduction of representations from the orthogonal group to the identity representation of these subgroups we provide two subgroup-periodic bases for the harmonic analysis on the three-manifolds, which have applications to cosmic topology.
International Nuclear Information System (INIS)
Kramer, Peter
2010-01-01
From the homotopy groups of three distinct octahedral spherical three-manifolds we construct the isomorphic groups H of deck transformations acting on the three-sphere. The H-invariant polynomials on the three-sphere constructed by representation theory span the bases for the harmonic analysis on three spherical manifolds. Analysis of the Cosmic Microwave Background in terms of these new bases can reveal a non-simple topology of the space part of space-time.
Alekseevsky, D. V.; Cortes, V.
1997-01-01
The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical K\\"ahler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo K\\"ahler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classification of homogeneous special pseudo K\\"ahler manifolds of semisimple groups with compact stabilizer.