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1
Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Mechanical Design 117, B (June 1, 1995): 87–92. http://dx.doi.org/10.1115/1.2836475.
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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.
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Park, F. C. "Optimal Robot Design and Differential Geometry." Journal of Vibration and Acoustics 117, B (June 1, 1995): 87–92. http://dx.doi.org/10.1115/1.2838681.
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In this article we survey some recent developments in optimal robot design, and collect some of the differential geometric approaches into a general mathematical framework for robot design. The geometric framework permits a set of coordinate-free definitions of robot performance that can be optimized for designing both open- and closed-chain robotic mechanisms. In particular, workspace volume is precisely defined by regarding the rigid body motions as a Riemannian manifold, and various features of actuators, as well as inertial characteristics of the robot, can be captured by the suitable selection of a Riemannian metric in configuration space. The integral functional of harmonic mapping theory also provides a simple and elegant global description of dexterity.
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Stavrinos, Panayiotis, and Christos Savvopoulos. "Dark Gravitational Field on Riemannian and Sasaki Spacetime." Universe 6, no. 9 (August 28, 2020): 138. http://dx.doi.org/10.3390/universe6090138.
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The aim of this paper is to provide the geometrical structure of a gravitational field that includes the addition of dark matter in the framework of a Riemannian and a Riemann–Sasaki spacetime. By means of the classical Riemannian geometric methods we arrive at modified geodesic equations, tidal forces, and Einstein and Raychaudhuri equations to account for extra dark gravity. We further examine an application of this approach in cosmology. Moreover, a possible extension of this model on the tangent bundle is studied in order to examine the behavior of dark matter in a unified geometric model of gravity with more degrees of freedom. Particular emphasis shall be laid on the problem of the geodesic motion under the influence of dark matter.
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Umair, H., H. Zainuddin, K. T. Chan, and Sh K. Said Husain. "The evolution of geometric Robertson–Schrödinger uncertainty principle for spin 1 system." Mathematical Modeling and Computing 9, no. 1 (2022): 36–49. http://dx.doi.org/10.23939/mmc2022.01.036.
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Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.
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Liu, Hong, Jie Li, Yongjian Wu, and Rongrong Ji. "Learning Neural Bag-of-Matrix-Summarization with Riemannian Network." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 8746–53. http://dx.doi.org/10.1609/aaai.v33i01.33018746.
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Symmetric positive defined (SPD) matrix has attracted increasing research focus in image/video analysis, which merits in capturing the Riemannian geometry in its structured 2D feature representation. However, computation in the vector space on SPD matrices cannot capture the geometric properties, which corrupts the classification performance. To this end, Riemannian based deep network has become a promising solution for SPD matrix classification, because of its excellence in performing non-linear learning over SPD matrix. Besides, Riemannian metric learning typically adopts a kNN classifier that cannot be extended to large-scale datasets, which limits its application in many time-efficient scenarios. In this paper, we propose a Bag-of-Matrix-Summarization (BoMS) method to be combined with Riemannian network, which handles the above issues towards highly efficient and scalable SPD feature representation. Our key innovation lies in the idea of summarizing data in a Riemannian geometric space instead of the vector space. First, the whole training set is compressed with a small number of matrix features to ensure high scalability. Second, given such a compressed set, a constant-length vector representation is extracted by efficiently measuring the distribution variations between the summarized data and the latent feature of the Riemannian network. Finally, the proposed BoMS descriptor is integrated into the Riemannian network, upon which the whole framework is end-to-end trained via matrix back-propagation. Experiments on four different classification tasks demonstrate the superior performance of the proposed method over the state-of-the-art methods.
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Newton, Nigel J. "Information geometric nonlinear filtering." Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, no. 02 (June 2015): 1550014. http://dx.doi.org/10.1142/s0219025715500149.
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This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's -1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.
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Torromé, Ricardo Gallego. "Maximal acceleration geometries and spacetime curvature bounds." International Journal of Geometric Methods in Modern Physics 17, no. 04 (March 2020): 2050060. http://dx.doi.org/10.1142/s0219887820500607.
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A geometric framework for metrics of maximal acceleration which is applicable to large proper accelerations is discussed, including a theory of connections associated with the geometry of maximal acceleration. In such a framework, it is shown that the uniform bound on the proper maximal acceleration implies a uniform bound for certain bilinear combinations of the Riemannian curvature components in the domain of the spacetime where curvature is finite.
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Campbell, Kristen M., Haocheng Dai, Zhe Su, Martin Bauer, P. Thomas Fletcher, and Sarang C. Joshi. "Integrated Construction of Multimodal Atlases with Structural Connectomes in the Space of Riemannian Metrics." Machine Learning for Biomedical Imaging 1, IPMI 2021 (June 16, 2022): 1–25. http://dx.doi.org/10.59275/j.melba.2022-a871.
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The structural network of the brain, or structural connectome, can be represented by fiber bundles generated by a variety of tractography methods. While such methods give qualitative insights into brain structure, there is controversy over whether they can provide quantitative information, especially at the population level. In order to enable population-level statistical analysis of the structural connectome, we propose representing a connectome as a Riemannian metric, which is a point on an infinite-dimensional manifold. We equip this manifold with the Ebin metric, a natural metric structure for this space, to get a Riemannian manifold along with its associated geometric properties. We then use this Riemannian framework to apply object-oriented statistical analysis to define an atlas as the Fréchet mean of a population of Riemannian metrics. This formulation ties into the existing framework for diffeomorphic construction of image atlases, allowing us to construct a multimodal atlas by simultaneously integrating complementary white matter structure details from DWMRI and cortical details from T1-weighted MRI. We illustrate our framework with 2D data examples of connectome registration and atlas formation. Finally, we build an example 3D multimodal atlas using T1 images and connectomes derived from diffusion tensors estimated from a subset of subjects from the Human Connectome Project.
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KALOGEROPOULOS, NIKOS. "TSALLIS ENTROPY COMPOSITION AND THE HEISENBERG GROUP." International Journal of Geometric Methods in Modern Physics 10, no. 07 (June 10, 2013): 1350032. http://dx.doi.org/10.1142/s0219887813500321.
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We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.
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Liu, Shuaiqi, Chuanqing Zhao, Yanling An, Pengfei Li, Jie Zhao, and Yudong Zhang. "Diffusion Tensor Imaging Denoising Based on Riemannian Geometric Framework and Sparse Bayesian Learning." Journal of Medical Imaging and Health Informatics 9, no. 9 (December 1, 2019): 1993–2003. http://dx.doi.org/10.1166/jmihi.2019.2832.
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Brucker, R., and M. Sorg. "Geometry and Topology of SO(4) Trivializable Gauge Fields." Zeitschrift für Naturforschung A 42, no. 6 (June 1, 1987): 521–37. http://dx.doi.org/10.1515/zna-1987-0601.
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The geometric and topological properties of SO(4)-trivializable SU(2) gauge fields are investigated in detail. The geometry of trivializable configurations is most adequately described by a conformally flat Riemannian structure, which yields a common geometric framework for both instantons and merons. In topologically nontrivial situations, the trivializable gauge fields exhibit a number of point defects (merons), which may be characterized by a quantized defect charge. The SO (2) reduction of the corresponding SU(2) bundle yields monopole configurations of the t'Hooft-Polyakov type.
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Sadik, Souhayl, and Arash Yavari. "Small-on-large geometric anelasticity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2195 (November 2016): 20160659. http://dx.doi.org/10.1098/rspa.2016.0659.
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In this paper, we are concerned with finding exact solutions for the stress fields of nonlinear solids with non-symmetric distributions of defects (or more generally finite eigenstrains) that are small perturbations of symmetric distributions of defects with known exact solutions. In the language of geometric mechanics, this corresponds to finding a deformation that is a result of a perturbation of the metric of the Riemannian material manifold. We present a general framework that can be used for a systematic analysis of this class of anelasticity problems. This geometric formulation can be thought of as a material analogue of the classical small-on-large theory in nonlinear elasticity. We use the present small-on-large anelasticity theory to find exact solutions for the stress fields of some non-symmetric distributions of screw dislocations in incompressible isotropic solids.
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Himpel, Benjamin. "Geometry of Music Perception." Mathematics 10, no. 24 (December 16, 2022): 4793. http://dx.doi.org/10.3390/math10244793.
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Prevalent neuroscientific theories are combined with acoustic observations from various studies to create a consistent geometric model for music perception in order to rationalize, explain and predict psycho-acoustic phenomena. The space of all chords is shown to be a Whitney stratified space. Each stratum is a Riemannian manifold which naturally yields a geodesic distance across strata. The resulting metric is compatible with voice-leading satisfying the triangle inequality. The geometric model allows for rigorous studies of psychoacoustic quantities such as roughness and harmonicity as height functions. In order to show how to use the geometric framework in psychoacoustic studies, concepts for the perception of chord resolutions are introduced and analyzed.
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Umair, H., H. Zainuddin, K. T. Chan, and Sh K. Said Husein. "THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM." Advances in Mathematics: Scientific Journal 10, no. 9 (September 30, 2021): 3241–51. http://dx.doi.org/10.37418/amsj.10.9.13.
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Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.
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Umair, H., H. Zainuddin, K. T. Chan, and Sh K. Said Husein. "THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`." Advances in Mathematics: Scientific Journal 10, no. 9 (September 30, 2021): 3253–62. http://dx.doi.org/10.37418/amsj.10.9.14.
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Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.
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van der Schaft, Arjan, and Bernhard Maschke. "Geometry of Thermodynamic Processes." Entropy 20, no. 12 (December 4, 2018): 925. http://dx.doi.org/10.3390/e20120925.
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Since the 1970s, contact geometry has been recognized as an appropriate framework for the geometric formulation of thermodynamic systems, and in particular their state properties. More recently it has been shown how the symplectization of contact manifolds provides a new vantage point; enabling, among other things, to switch easily between the energy and entropy representations of a thermodynamic system. In the present paper, this is continued towards the global geometric definition of a degenerate Riemannian metric on the homogeneous Lagrangian submanifold describing the state properties, which is overarching the locally-defined metrics of Weinhold and Ruppeiner. Next, a geometric formulation is given of non-equilibrium thermodynamic processes, in terms of Hamiltonian dynamics defined by Hamiltonian functions that are homogeneous of degree one in the co-extensive variables and zero on the homogeneous Lagrangian submanifold. The correspondence between objects in contact geometry and their homogeneous counterparts in symplectic geometry, is extended to the definition of port-thermodynamic systems and the formulation of interconnection ports. The resulting geometric framework is illustrated on a number of simple examples, already indicating its potential for analysis and control.
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Balan, Vladimir, and Jelena Stojanov. "Finslerian-type GAF extensions of the Riemannian framework in digital image processing." Filomat 29, no. 3 (2015): 535–43. http://dx.doi.org/10.2298/fil1503535b.
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Digital image processing was recently proved to be successfully approached by variational tools, which extend the Casseles-Kimmel-Sapiro weighted length problem. Such tools essentially lead to the socalled Geodesic Active Flow (GAF) process, which relies on the derived mean curvature flow PDE. This prolific process is valuable due to both the provided numeric mathematical insight, which requires specific nontrivial choices for implementing the related algorithms, and the variety of possible underlying specific geometric structures. A natural Finsler extension of Randers type was recently developed by the authors, which emphasizes the anisotropy given by the straightforward gradient, while considering a particular scaling of the Lagrangian. The present work develops to its full extent the GAF process to the Randers Finslerian framework: the evolution equations of the model are determined in detail, Matlab simulations illustrate the obtained theoretic results and conclusive remarks are drawn. Finally, open problems regarding the theoretic model and its applicative efficiency are stated.
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Gurzadyan, A. V., and A. A. Kocharyan. "Superspace and global stability in general relativity." International Journal of Modern Physics D 26, no. 05 (April 2017): 1741019. http://dx.doi.org/10.1142/s021827181741019x.
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A framework is developed enabling the global analysis of the stability of cosmological models using the local geometric characteristics of the infinite-dimensional superspace, i.e. using the generalized Jacobi equation reformulated for pseudo-Riemannian manifolds. We give a direct formalism for dynamical analysis in the superspace, the requisite equation pertinent for stability analysis of the universe by means of generalized covariant and Fermi derivative is derived. Then, the relevant definitions and formulae are retrieved for cosmological models with a scalar field.
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WANAS, M. I., and MONA M. KAMAL. "AN AP-STRUCTURE WITH FINSLERIAN FLAVOR II: TORSION, CURVATURE AND OTHER OBJECTS." Modern Physics Letters A 26, no. 27 (September 7, 2011): 2065–78. http://dx.doi.org/10.1142/s021773231103653x.
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An absolute parallelism (AP-) space having Finslerian properties is called FAP-space. This FAP-structure is wider than both conventional AP and Finsler structures. In the present work, more geometric objects as curvature and torsion tensors are derived in the context of this structure. Also second order tensors, usually needed for physical applications, are derived and studied. Furthermore, the anti-curvature and the W-tensor are defined for the FAP-structure. Relations between Riemannian, AP, Finsler and FAP structures are given. These relations facilitate comparison between results of applications carried out in the framework of these structures. We hope that the use of the FAP-structure, in applications may throw some light on some of the problems facing geometric field theories.
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Goel, Anmol, and Ponnurangam Kumaraguru. "Detecting Lexical Semantic Change across Corpora with Smooth Manifolds (Student Abstract)." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 18 (May 18, 2021): 15783–84. http://dx.doi.org/10.1609/aaai.v35i18.17888.
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Comparing two bodies of text and detecting words with significant lexical semantic shift between them is an important part of digital humanities. Traditional approaches have relied on aligning the different embeddings using the Orthogonal Procrustes problem in the Euclidean space. This study presents a geometric framework that leverages smooth Riemannian manifolds for corpus-specific orthogonal rotations and a corpus-independent scaling metric to project the different vector spaces into a shared latent space. This enables us to capture any affine relationship between the embedding spaces while utilising the rich geometry of smooth manifolds.
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Burnett, Christopher L., Darryl D. Holm, and David M. Meier. "Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2160 (December 8, 2013): 20130249. http://dx.doi.org/10.1098/rspa.2013.0249.
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We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler–Lagrange equations is obtained from a higher-order Hamilton–Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre–Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton–Pontryagin principle and preserves the geometric properties of the continuous-time solution.
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Iuras¸cu, C. C., and F. C. Park. "Geometric Algorithms for Kinematic Calibration of Robots Containing Closed Loops." Journal of Mechanical Design 125, no. 1 (March 1, 2003): 23–32. http://dx.doi.org/10.1115/1.1539512.
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We present a coordinate-invariant, differential geometric formulation of the kinematic calibration problem for a general class of mechanisms. The mechanisms considered may have multiple closed loops, be redundantly actuated, and include an arbitrary number of passive joints that may or may not be equipped with joint encoders. Some form of measurement information on the position and orientation of the tool frame may also be available. Our approach rests on viewing the joint configuration space of the mechanism as an embedded submanifold of an ambient manifold, and formulating error measures in terms of the Riemannian metric specified in the ambient manifold. Based on this geometric framework, we pose the kinematic calibration problem as one of determining a parametrized multidimensional surface that is a best fit (in the sense of the chosen metric) to a given set of measured points in both the ambient and task space manifolds. Several optimization algorithms that address the various possibilities with respect to available measurement data and choice of error measures are given. Experimental and simulation results are given for the Eclipse, a six degree-of-freedom redundantly actuated parallel mechanism. The geometric framework and algorithms presented in this article have the desirable feature of being invariant with respect to the local coordinate representation of the forward and inverse kinematics and of the loop closure equations, and also provide a high-level framework in which to classify existing approaches to kinematic calibration.
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Jawanpuria, Pratik, Arjun Balgovind, Anoop Kunchukuttan, and Bamdev Mishra. "Learning Multilingual Word Embeddings in Latent Metric Space: A Geometric Approach." Transactions of the Association for Computational Linguistics 7 (November 2019): 107–20. http://dx.doi.org/10.1162/tacl_a_00257.
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We propose a novel geometric approach for learning bilingual mappings given monolingual embeddings and a bilingual dictionary. Our approach decouples the source-to-target language transformation into (a) language-specific rotations on the original embeddings to align them in a common, latent space, and (b) a language-independent similarity metric in this common space to better model the similarity between the embeddings. Overall, we pose the bilingual mapping problem as a classification problem on smooth Riemannian manifolds. Empirically, our approach outperforms previous approaches on the bilingual lexicon induction and cross-lingual word similarity tasks. We next generalize our framework to represent multiple languages in a common latent space. Language-specific rotations for all the languages and a common similarity metric in the latent space are learned jointly from bilingual dictionaries for multiple language pairs. We illustrate the effectiveness of joint learning for multiple languages in an indirect word translation setting.
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Lv, Chenlei, and Junli Zhao. "3D Face Recognition based on Local Conformal Parameterization and Iso-Geodesic Stripes Analysis." Mathematical Problems in Engineering 2018 (October 25, 2018): 1–10. http://dx.doi.org/10.1155/2018/4707954.
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3D face recognition is an important topic in the field of pattern recognition and computer graphic. We propose a novel approach for 3D face recognition using local conformal parameterization and iso-geodesic stripes. In our framework, the 3D facial surface is considered as a Riemannian 2-manifold. The surface is mapped into the 2D circle parameter domain using local conformal parameterization. In the parameter domain, the geometric features are extracted from the iso-geodesic stripes. Combining the relative position measure, Chain 2D Weighted Walkthroughs (C2DWW), the 3D face matching results can be obtained. The geometric features from iso-geodesic stripes in parameter domain are robust in terms of head poses, facial expressions, and some occlusions. In the experiments, our method achieves a high recognition accuracy of 3D facial data from the Texas3D and Bosphorus3D face database.
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Yavari, Arash, and Alain Goriely. "Nonlinear elastic inclusions in isotropic solids." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2160 (December 8, 2013): 20130415. http://dx.doi.org/10.1098/rspa.2013.0415.
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We introduce a geometric framework to calculate the residual stress fields and deformations of nonlinear solids with inclusions and eigenstrains. Inclusions are regions in a body with different reference configurations from the body itself and can be described by distributed eigenstrains. Geometrically, the eigenstrains define a Riemannian 3-manifold in which the body is stress-free by construction. The problem of residual stress calculation is then reduced to finding a mapping from the Riemannian material manifold to the ambient Euclidean space. Using this construction, we find the residual stress fields of three model systems with spherical and cylindrical symmetries in both incompressible and compressible isotropic elastic solids. In particular, we consider a finite spherical ball with a spherical inclusion with uniform pure dilatational eigenstrain and we show that the stress in the inclusion is uniform and hydrostatic. We also show how singularities in the stress distribution emerge as a consequence of a mismatch between radial and circumferential eigenstrains at the centre of a sphere or the axis of a cylinder.
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Duruisseaux, Valentin, and Melvin Leok. "Time-adaptive Lagrangian variational integrators for accelerated optimization." Journal of Geometric Mechanics 15, no. 1 (2023): 224–55. http://dx.doi.org/10.3934/jgm.2023010.
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<abstract><p>A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> and <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in <sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup> to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.</p></abstract>
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Gonzalez-Caballero, Erick Gonzalez. "Applications of NeutroGeometry and AntiGeometry in Real World." International Journal of Neutrosophic Science 21, no. 1 (2023): 14–32. http://dx.doi.org/10.54216/ijns.210102.
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NeutroGeometries are those geometric structures where at least one definition, axiom, property, theorem, among others, is only partially satisfied. In AntiGeometries at least one of these concepts is never satisfied. Smarandache Geometry is a geometric structure where at least one axiom or theorem behaves differently in the same space, either partially true and partially false, or totally false but its negation done in many ways. This paper offers examples in images of nature, everyday objects, and celestial bodies where the existence of Smarandechean or NeutroGeometric structures in our universe is revealed. On the other hand, a practical study of surfaces with characteristics of NeutroGeometry is carried out, based on the properties or more specifically NeutroProperties of the famous quadrilaterals of Saccheri and Lambert on these surfaces. The article contributes to demonstrating the importance of building a theory such as NeutroGeometries or Smarandache Geometries because it would allow us to study geometric structures where the well-known Euclidean, Hyperbolic or Elliptic geometries are not enough to capture properties of elements that are part of the universe, but they have sense only within a NeutroGeometric framework. It also offers an axiomatic option to the Riemannian idea of Two-Dimensional Manifolds. In turn, we prove some properties of the NeutroGeometries and the materialization of the symmetric triad , , and .
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Calderhead, Ben, and Mark Girolami. "Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods." Interface Focus 1, no. 6 (August 24, 2011): 821–35. http://dx.doi.org/10.1098/rsfs.2011.0051.
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Mechanistic models based on systems of nonlinear differential equations can help provide a quantitative understanding of complex physical or biological phenomena. The use of such models to describe nonlinear interactions in molecular biology has a long history; however, it is only recently that advances in computing have allowed these models to be set within a statistical framework, further increasing their usefulness and binding modelling and experimental approaches more tightly together. A probabilistic approach to modelling allows us to quantify uncertainty in both the model parameters and the model predictions, as well as in the model hypotheses themselves. In this paper, the Bayesian approach to statistical inference is adopted and we examine the significant challenges that arise when performing inference over nonlinear ordinary differential equation models describing cell signalling pathways and enzymatic circadian control; in particular, we address the difficulties arising owing to strong nonlinear correlation structures, high dimensionality and non-identifiability of parameters. We demonstrate how recently introduced differential geometric Markov chain Monte Carlo methodology alleviates many of these issues by making proposals based on local sensitivity information, which ultimately allows us to perform effective statistical analysis. Along the way, we highlight the deep link between the sensitivity analysis of such dynamic system models and the underlying Riemannian geometry of the induced posterior probability distributions.
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Gay-Balmaz, François, and Hiroaki Yoshimura. "A free energy Lagrangian variational formulation of the Navier–Stokes–Fourier system." International Journal of Geometric Methods in Modern Physics 16, supp01 (January 29, 2019): 1940006. http://dx.doi.org/10.1142/s0219887819400061.
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We present a variational formulation for the Navier–Stokes–Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite-dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in [F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics, Part II: Continuum systems, J. Geom. Phys. 111 (2017) 194–212] as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier–Stokes–Fourier system on Riemannian manifolds.
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YANG, HYUN SEOK. "DARK ENERGY AND EMERGENT SPACETIME." International Journal of Modern Physics: Conference Series 01 (January 2011): 266–71. http://dx.doi.org/10.1142/s2010194511000389.
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A natural geometric framework of noncommutative spacetime is symplectic geometry rather than Riemannian geometry. The Darboux theorem in symplectic geometry then admits a novel form of the equivalence principle such that the electromagnetism in noncommutative spacetime can be regarded as a theory of gravity. Remarkably the emergent gravity reveals a noble picture about the origin of spacetime, dubbed as emergent spacetime, which is radically different from any previous physical theory all of which describe what happens in a given spacetime. In particular, the emergent gravity naturally explains the dynamical origin of flat spacetime, which is absent in Einstein gravity: A flat spacetime is not free gratis but a result of Planck energy condensation in a vacuum. This emergent spacetime picture, if it is correct anyway, turns out to be essential to resolve the cosmological constant problem, to understand the nature of dark energy and to explain why gravity is so weak compared to other forces.
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BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (August 2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.
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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explanation of the physical world, essentially depends on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, it still plays a fundamental role in non-Abelian gauge theories and in superstring theory, thanks to which a great variety of new mathematical structures has emerged. The scope of this presentation is to highlight the importance of these mathematical structures for theoretical physics. A very interesting hypothesis is that the global topological properties of the manifold's model of space–time play a major role in quantum field theory (QFT) and that, consequently, several physical quantum effects arise from the nonlocal changing metrical and topological structure of these manifold. Thus the unification of general relativity and quantum theory require some fundamental breakthrough in our understanding of the relationship between space–time and quantum process. In particular the superstring theories lead to the guess that the usual structure of space–time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections, and curvature in explaining the essential laws of nature. Kaluza–Klein theories and more remarkably superstring theory showed that space–time symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. This essentially means, in our view, that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain class of (presumably) nonsmooth manifolds. In the second part of this paper, we address the subject of topological quantum field theories (TQFTs), which constitute a remarkably important meeting ground for physicists and mathematicians. TQFTs can be used as a powerful tool to probe geometry and topology in low dimensions. Chern–Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of QFTs which can be exactly (nonperturbatively) and explicitly solved. Abelian Chern–Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link (i.e. the union of a finite number of disjoint knots). In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the well-known Jones polynomial. Powerful methods for complete analytical and nonperturbative computation of these knot and link invariants have been developed. From these invariants for unoriented and framed links in S3, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish–Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern–Simons field theory. Even perturbative analysis of Chern–Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. In Donaldson–Witten theory perturbative methods have proved their relations to Donaldson invariants. Nonperturbative methods have been applied after the work by Seiberg and Witten on N = 2 supersymmetric Yang–Mills theory. The outcome of this application is a totally unexpected relation between Donaldson invariants and a new set of topological invariants called Seiberg–Witten invariants. Not only in mathematics, Chern–Simons theories find important applications in three- and four-dimensional quantum gravity also. Work on TQFT suggests that a quantum gravity theory can be formulated in three-dimensional space–time. Attempts have been made in the last years to formulate a theory of quantum gravity in four-dimensional space–time using "spin networks" and "spin foams". More generally, the developments of TQFTs represent a sort of renaissance in the relation between geometry and physics. The most important (new) feature of present developments is that links are being established between quantum physics and topology. Maybe this link essentially rests on the fact that both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. One very interesting example is the super-symmetric quantum mechanics theory, which has a deep geometric meaning. In the Witten super-symmetric quantum mechanics theory, where the Hamiltonian is just the Hodge–Laplacian (whereas the quantum Hamiltonian corresponding to a classical particle moving on a Riemannian manifold is just the Laplace–Beltrami differential operator), differential forms are bosons or fermions depending on the parity of their degrees. Witten went to introduce a modified Hodge–Laplacian, depending on a real-valued function f. He was then able to derive the Morse theory (relating critical points of f to the Betti numbers of the manifold) by using the standard limiting procedures relating the quantum and classical theories. Super-symmetric QFTs essentially should be viewed as the differential geometry of certain infinite-dimensional manifolds, including the associated analysis (e.g. Hodge theory) and topology (e.g. Betti numbers). A further comment is that the QFTs of interest are inherently nonlinear, but the nonlinearities have a natural origin, e.g. coming from non-Abelian Lie groups. Moreover there is usually some scaling or coupling parameter in the theory which in the limit relates to the classical theory. Fundamental topological aspects of such a quantum theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit. This means that such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory. In the last decade much effort has been done to use these QFTs as a conceptual tool to suggest new mathematical results. In particular, they have led to spectacular progress in our understanding of geometry in low dimensions. It is most likely no accident that the usual QFTs can only be renormalized in (space–time) dimensions ≤4, and this is precisely the range in which difficult phenomena arise leading to deep and beautiful theories (e.g. the work of Thurston in three dimensions and Donaldson in four dimensions). It now seems clear that the way to investigate the subtleties of low-dimensional manifolds is to associate to them suitable infinite-dimensional manifolds (e.g. spaces of connections) and to study these by standard linear methods (homology, etc.). In other words we use QFT as a refined tool to study low-dimensional manifolds.
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Xu, Yilu, Hua Yin, Wenlong Yi, Xin Huang, Wenjuan Jian, Canhua Wang, and Ronghua Hu. "Supervised and Semisupervised Manifold Embedded Knowledge Transfer in Motor Imagery-Based BCI." Computational Intelligence and Neuroscience 2022 (October 17, 2022): 1–19. http://dx.doi.org/10.1155/2022/1603104.
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A long calibration procedure limits the use in practice for a motor imagery (MI)-based brain-computer interface (BCI) system. To tackle this problem, we consider supervised and semisupervised transfer learning. However, it is a challenge for them to cope with high intersession/subject variability in the MI electroencephalographic (EEG) signals. Based on the framework of unsupervised manifold embedded knowledge transfer (MEKT), we propose a supervised MEKT algorithm (sMEKT) and a semisupervised MEKT algorithm (ssMEKT), respectively. sMEKT only has limited labelled samples from a target subject and abundant labelled samples from multiple source subjects. Compared to sMEKT, ssMEKT adds comparably more unlabelled samples from the target subject. After performing Riemannian alignment (RA) and tangent space mapping (TSM), both sMEKT and ssMEKT execute domain adaptation to shorten the differences among subjects. During domain adaptation, to make use of the available samples, two algorithms preserve the source domain discriminability, and ssMEKT preserves the geometric structure embedded in the labelled and unlabelled target domains. Moreover, to obtain a subject-specific classifier, sMEKT minimizes the joint probability distribution shift between the labelled target and source domains, whereas ssMEKT performs the joint probability distribution shift minimization between the unlabelled target domain and all labelled domains. Experimental results on two publicly available MI datasets demonstrate that our algorithms outperform the six competing algorithms, where the sizes of labelled and unlabelled target domains are variable. Especially for the target subjects with 10 labelled samples and 270/190 unlabelled samples, ssMEKT shows 5.27% and 2.69% increase in average accuracy on the two abovementioned datasets compared to the previous best semisupervised transfer learning algorithm (RA-regularized common spatial patterns-weighted adaptation regularization, RA-RCSP-wAR), respectively. Therefore, our algorithms can effectively reduce the need of labelled samples for the target subject, which is of importance for the MI-based BCI application.
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Brozos-Vázquez, M., E. García-Río, P. Gilkey, and X. Valle-Regueiro. "Solutions to the affine quasi-Einstein equation for homogeneous surfaces." Advances in Geometry 20, no. 3 (July 28, 2020): 413–32. http://dx.doi.org/10.1515/advgeom-2020-0011.
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AbstractWe examine the space of solutions to the affine quasi–Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitons, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.
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Georgiadis, Athanasios G., and George Kyriazis. "Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators." Analysis and Geometry in Metric Spaces 8, no. 1 (January 1, 2020): 418–29. http://dx.doi.org/10.1515/agms-2020-0120.
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Abstract We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces are proved as well. Our result generalize the Euclidean case and are new for many settings of independent interest such as the ball, the interval and Riemannian manifolds.
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Finster, Felix, and Magdalena Lottner. "Banach manifold structure and infinite-dimensional analysis for causal fermion systems." Annals of Global Analysis and Geometry 60, no. 2 (May 31, 2021): 313–54. http://dx.doi.org/10.1007/s10455-021-09775-4.
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AbstractA mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.
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Wong, Ting-Kam Leonard, and Jiaowen Yang. "Pseudo-Riemannian geometry encodes information geometry in optimal transport." Information Geometry, July 30, 2021. http://dx.doi.org/10.1007/s41884-021-00053-7.
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AbstractOptimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate invariant properties of statistical inference. Their relations and applications in statistics and machine learning have started to gain more attention. In this paper we give a new differential-geometric relation between the two fields. Namely, the pseudo-Riemannian framework of Kim and McCann, which provides a geometric perspective on the fundamental Ma–Trudinger–Wang (MTW) condition in the regularity theory of optimal transport maps, encodes the dualistic structure of statistical manifold. This general relation is described using the framework of c-divergence under which divergences are defined by optimal transport maps. As a by-product, we obtain a new information-geometric interpretation of the MTW tensor on the graph of the transport map. This relation sheds light on old and new aspects of information geometry. The dually flat geometry of Bregman divergence corresponds to the quadratic cost and the pseudo-Euclidean space, and the logarithmic $$L^{(\alpha )}$$ L ( α ) -divergence introduced by Pal and the first author has constant sectional curvature in a sense to be made precise. In these cases we give a geometric interpretation of the information-geometric curvature in terms of the divergence between a primal-dual pair of geodesics.
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Jang, Cheongjae, Yung-Kyun Noh, and Frank Chongwoo Park. "A Riemannian geometric framework for manifold learning of non-Euclidean data." Advances in Data Analysis and Classification, November 27, 2020. http://dx.doi.org/10.1007/s11634-020-00426-3.
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Bellaard, Gijs, Daan L. J. Bon, Gautam Pai, Bart M. N. Smets, and Remco Duits. "Analysis of (sub-)Riemannian PDE-G-CNNs." Journal of Mathematical Imaging and Vision, April 16, 2023. http://dx.doi.org/10.1007/s10851-023-01147-w.
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AbstractGroup equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously (1) reduce network complexity, (2) increase classification performance, and (3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper, we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.
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Duruisseaux, Valentin, and Melvin Leok. "Accelerated Optimization on Riemannian Manifolds via Discrete Constrained Variational Integrators." Journal of Nonlinear Science 32, no. 4 (April 28, 2022). http://dx.doi.org/10.1007/s00332-022-09795-9.
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AbstractA variational formulation for accelerated optimization on normed vector spaces was recently introduced in Wibisono et al. (PNAS 113:E7351–E7358, 2016), and later generalized to the Riemannian manifold setting in Duruisseaux and Leok (SJMDS, 2022a). This variational framework was exploited on normed vector spaces in Duruisseaux et al. (SJSC 43:A2949–A2980, 2021) using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization, and it was observed that geometric discretizations which respect the time-rescaling invariance and symplecticity of the Lagrangian and Hamiltonian flows were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. As such, it is natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In this paper, we consider the case of Riemannian manifolds embedded in a Euclidean space that can be characterized as the level set of a submersion. We will explore how holonomic constraints can be incorporated in discrete variational integrators to constrain the numerical discretization of the Riemannian Hamiltonian system to the Riemannian manifold, and we will test the performance of the resulting algorithms by solving eigenvalue and Procrustes problems formulated as optimization problems on the unit sphere and Stiefel manifold.
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Hasan-Zadeh, Atefeh. "Application of Lie Algebras in Computer Animation." Asian Journal of Applied Sciences 8, no. 5 (October 30, 2020). http://dx.doi.org/10.24203/ajas.v8i5.6331.
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The mathematical theory behind the computer graphic enables one to develop the techniques for suitable creation of computer animation. This paper presents an application of Riemannian geometry in 3D animation via notions of in motion and deformation. By focusing on Lie algebras concepts, it provides a geometric framework for the implementation of computer animation.
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Huang, Qiao, and Jean-Claude Zambrini. "From Second-Order Differential Geometry to Stochastic Geometric Mechanics." Journal of Nonlinear Science 33, no. 4 (June 7, 2023). http://dx.doi.org/10.1007/s00332-023-09917-x.
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AbstractClassical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton–Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton–Jacobi–Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton’s equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler–Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton’s equations. A stochastic Noether’s theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schrödinger’s problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.
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You, Kisung, and Hae-Jeong Park. "Geometric learning of functional brain network on the correlation manifold." Scientific Reports 12, no. 1 (October 22, 2022). http://dx.doi.org/10.1038/s41598-022-21376-0.
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AbstractThe correlation matrix is a typical representation of node interactions in functional brain network analysis. The analysis of the correlation matrix to characterize brain networks observed in several neuroimaging modalities has been conducted predominantly in the Euclidean space by assuming that pairwise interactions are mutually independent. One way to take account of all interactions in the network as a whole is to analyze the correlation matrix under some geometric structure. Recent studies have focused on the space of correlation matrices as a strict subset of symmetric positive definite (SPD) matrices, which form a unique mathematical structure known as the Riemannian manifold. However, mathematical operations of the correlation matrix under the SPD geometry may not necessarily be coherent (i.e., the structure of the correlation matrix may not be preserved), necessitating a post-hoc normalization. The contribution of the current paper is twofold: (1) to devise a set of inferential methods on the correlation manifold and (2) to demonstrate its applicability in functional network analysis. We present several algorithms on the correlation manifold, including measures of central tendency, cluster analysis, hypothesis testing, and low-dimensional embedding. Simulation and real data analysis support the application of the proposed framework for brain network analysis.
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Spera, Mauro. "On some hydrodynamical aspects of quantum mechanics." Open Physics 8, no. 1 (January 1, 2010). http://dx.doi.org/10.2478/s11534-009-0070-4.
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AbstractIn this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the complex polynomial (i.e. Borel-Weil) realization of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schrödinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly determine the critical points of the pressure of this “Schrödinger fluid”, together with its vorticity, which turns out to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.
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Smets, Bart M. N., Jim Portegies, Etienne St-Onge, and Remco Duits. "Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations." Journal of Mathematical Imaging and Vision, September 18, 2020. http://dx.doi.org/10.1007/s10851-020-00991-4.
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Abstract Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.
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Ivancevic, Tijana. "Jet methods in time-dependent Lagrangian biomechanics." Open Physics 8, no. 5 (January 1, 2010). http://dx.doi.org/10.2478/s11534-009-0148-z.
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AbstractIn this paper we propose the time-dependent generalization of an ‘ordinary’ autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total mass-inertia matrix of the human body segments. This is the base manifold for standard autonomous biomechanics. To make its time-dependent generalization, we need to extend it with a real time axis. By this extension, using techniques from fibre bundles, we defined the biomechanical configuration bundle. On the biomechanical bundle we define vector-fields, differential forms and affine connections, as well as the associated jet manifolds. Using the formalism of jet manifolds of velocities and accelerations, we develop the time-dependent Lagrangian biomechanics. Its underlying geometric evolution is given by the Ricci flow equation.
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Morilla, Ian, Thibaut Léger, Assiya Marah, Isabelle Pic, Hatem Zaag, and Eric Ogier-Denis. "Singular manifolds of proteomic drivers to model the evolution of inflammatory bowel disease status." Scientific Reports 10, no. 1 (November 4, 2020). http://dx.doi.org/10.1038/s41598-020-76011-7.
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Abstract The conditions used to describe the presence of an immune disease are often represented by interaction graphs. These informative, but intricate structures are susceptible to perturbations at different levels. The mode in which that perturbation occurs is still of utmost importance in areas such as cell reprogramming and therapeutics models. In this sense, module identification can be useful to well characterise the global graph architecture. To help us with this identification, we perform topological overlap-related measures. Thanks to these measures, the location of highly disease-specific module regulators is possible. Such regulators can perturb other nodes, potentially causing the entire system to change behaviour or collapse. We provide a geometric framework explaining such situations in the context of inflammatory bowel diseases (IBD). IBD are severe chronic disorders of the gastrointestinal tract whose incidence is dramatically increasing worldwide. Our approach models different IBD status as Riemannian manifolds defined by the graph Laplacian of two high throughput proteome screenings. It also identifies module regulators as singularities within the manifolds (the so-called singular manifolds). Furthermore, it reinterprets the characteristic nonlinear dynamics of IBD as compensatory responses to perturbations on those singularities. Then, particular reconfigurations of the immune system could make the disease status move towards an innocuous target state.
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DEY, Santu, Pişcoran Laurian-ioan LAURİAN-IOAN, and Soumendu ROY. "Geometry of $*$-$k$-Ricci-Yamabe soliton and gradient $*$-$k$-Ricci-Yamabe soliton on Kenmotsu manifolds." Hacettepe Journal of Mathematics and Statistics, December 31, 2022, 1–16. http://dx.doi.org/10.15672/hujms.1074722.
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The goal of the current paper is to characterize $*$-$k$-Ricci-Yamabe soliton within the framework on Kenmotsu manifolds. Here, we have shown the nature of the soliton and find the scalar curvature when the manifold admitting $*$-$k$-Ricci-Yamabe soliton on Kenmotsu manifold. Next, we have evolved the characterization of the vector field when the manifold satisfies $*$-$k$-Ricci-Yamabe soliton. Also we have embellished some applications of vector field as torse-forming in terms of $*$-$k$-Ricci-Yamabe soliton on Kenmotsu manifold. Then, we have studied gradient $\ast$-$\eta$-Einstein soliton to yield the nature of Riemannian curvature tensor. We have developed an example of $*$-$k$-Ricci-Yamabe soliton on 5-dimensional Kenmotsu manifold to prove our findings.