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2,063 results on '"Tangent vector"'

1. Decomposition of Lorenz Trajectories Based on Space Curve Tangent Vector.

Author

Ma, Jingru, Hu, Lei, She, Hongke, Fan, Binghuai, and Da, Chaojiu

Subjects
*LORENZ equations, *DIFFERENTIAL equations, *NONLINEAR systems, *SPACE trajectories
Abstract

This article explores the evolution of Lorenz trajectories within attractors. Specifically, based on the characteristics of the tangents to trajectories, we derive quantitative standards for determining the spatial position of trajectory lines. The Lorenz trajectory is decomposed into four parts. This standard is objective and quantitative and is independent of the initial field of the Lorenz equation and the calculation scheme; importantly, it is designed based on the inherent dynamic characteristics of the Lorenz equation. Linear fitting of the trajectories in the left and right equilibrium point regions shows that the trajectories lie on planes, indicating the existence of linear features in the nonlinear system. This study identifies the fundamental causes of chaos in the Lorenz equation using the microscopic evolution and local characteristics of the trajectories, and indicating that the spatial position of the initial field is important for their predictability. We theoretically demonstrate that mutation is essentially self-regulation within chaotic systems. This scheme is designed according to the evolution characteristics of Lorenz trajectories, and thus has certain methodological limitations that mean it may not be applicable to other chaotic systems. However, it does depict the causes of chaos and elucidates the sensitivity of differential equations to initial values in terms of trajectory evolution. [ABSTRACT FROM AUTHOR]

Published
2024
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2. Decomposition of Lorenz Trajectories Based on Space Curve Tangent Vector

Author

Jingru Ma, Lei Hu, Hongke She, Binghuai Fan, and Chaojiu Da

Subjects
Lorenz equation, tangent vector, equilibrium state, decomposition of trajectories, data fitting, Meteorology. Climatology, QC851-999
Abstract

This article explores the evolution of Lorenz trajectories within attractors. Specifically, based on the characteristics of the tangents to trajectories, we derive quantitative standards for determining the spatial position of trajectory lines. The Lorenz trajectory is decomposed into four parts. This standard is objective and quantitative and is independent of the initial field of the Lorenz equation and the calculation scheme; importantly, it is designed based on the inherent dynamic characteristics of the Lorenz equation. Linear fitting of the trajectories in the left and right equilibrium point regions shows that the trajectories lie on planes, indicating the existence of linear features in the nonlinear system. This study identifies the fundamental causes of chaos in the Lorenz equation using the microscopic evolution and local characteristics of the trajectories, and indicating that the spatial position of the initial field is important for their predictability. We theoretically demonstrate that mutation is essentially self-regulation within chaotic systems. This scheme is designed according to the evolution characteristics of Lorenz trajectories, and thus has certain methodological limitations that mean it may not be applicable to other chaotic systems. However, it does depict the causes of chaos and elucidates the sensitivity of differential equations to initial values in terms of trajectory evolution.

Published
2024
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3. Some notes on directional curvature of a convex body in ℝn.

Author

Pereira, F. F.

Subjects
*CURVATURE, *DIRECTIONAL derivatives, *CONVEX bodies, *CONVEX sets
Abstract

Take a point ξ on the boundary of a convex body F in R n , near which the boundary is given by an implicit equation. We present some notes on the formula, proposed in Pereira [A directional curvature formula for convex bodies in R n . J Math Anal Appl. 2022;506(1):125656.], for calculating the curvature of F at ξ in the direction of its any tangent vector. Namely, we see that our formula is equivalent to the existing one for the curvature of a certain curve given by the intersection of n−1 implicit equations, but it is easier to apply. Furthermore, we show that when the directional curvature of F is positive, there is the directional derivative of the Minkowski functional of the polar set F o , and we propose a formula to calculate it. [ABSTRACT FROM AUTHOR]

Published
2022
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4. Some notes on directional curvature of a convex body in ℝn.

Author

Pereira, F. F.

Subjects
CURVATURE, DIRECTIONAL derivatives, CONVEX bodies, CONVEX sets
Abstract

Take a point ξ on the boundary of a convex body F in R n , near which the boundary is given by an implicit equation. We present some notes on the formula, proposed in Pereira [A directional curvature formula for convex bodies in R n . J Math Anal Appl. 2022;506(1):125656.], for calculating the curvature of F at ξ in the direction of its any tangent vector. Namely, we see that our formula is equivalent to the existing one for the curvature of a certain curve given by the intersection of n−1 implicit equations, but it is easier to apply. Furthermore, we show that when the directional curvature of F is positive, there is the directional derivative of the Minkowski functional of the polar set F o , and we propose a formula to calculate it. [ABSTRACT FROM AUTHOR]

Published
2022
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7. Symplectic Geometry

Author

Meyer, Kenneth R., Offin, Daniel C., Bell, J., Series editor, Constantin, P., Series editor, Antman, S.S., Editor-in-chief, Durrett, R., Series editor, Greengard, L., Editor-in-chief, Holmes, P.J., Editor-in-chief, Kohn, R., Series editor, Pego, R., Series editor, Ryzhik, L., Series editor, Singer, A., Series editor, Stevens, Angela, Series editor, Stuart, A., Series editor, Meyer, Kenneth R., and Offin, Daniel C.

Published
2017
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8. Geometric Aspects of Thermodynamics

Author

Scheck, Florian, Becker, Kurt H., Series editor, Di Meglio, Jean-Marc, Series editor, Hassani, Sadri, Series editor, Munro, Bill, Series editor, Needs, Richard, Series editor, Rhodes, William T., Series editor, Scott, Susan, Series editor, Stanley, H. Eugene, Series editor, Stutzmann, Martin, Series editor, Wipf, Andreas, Series editor, and Scheck, Florian

Published
2016
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9. Linearization of Manifolds

Author

Wedhorn, Torsten, Aigner, Martin, Series editor, Faßbender, Heike, Series editor, Gentz, Barbara, Series editor, Grieser, Daniel, Series editor, Gritzmann, Peter, Series editor, Kramer, Jürg, Series editor, Mehrmann, Volker, Series editor, Wüstholz, Gisbert, Series editor, and Wedhorn, Torsten

Published
2016
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13. Examples

Author

Lombardi, Henri, Quitté, Claude, Broué, Michel, Series editor, Fialowski, Alice, Series editor, Friedlander, Eric, Series editor, Greenlees, John, Series editor, Hiß, Gerhard, Series editor, Moerdijk, Ieke, Series editor, Reiten, Idun, Series editor, Schweigert, Christoph, Series editor, Teicher, Mina, Series editor, Verschoren, Alain, Series editor, Lombardi, Henri, and Quitté, Claude

Published
2015
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14. Moment Map of the Action of

Author

Morales, José Antonio Villa, Ventura, Enric, Series editor, Guillamon, Antoni, Series editor, Corbera, Montserrat, editor, Cors, Josep Maria, editor, Llibre, Jaume, editor, and Korobeinikov, Andrei, editor

Published
2015
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15. Path-Based Mathematical Morphology on Tensor Fields

Author

van de Gronde, Jasper J., Lysenko, Mikola, Roerdink, Jos B. T. M., Farin, Gerald, Series editor, Hege, Hans-Christian, Series editor, Hoffman, David, Series editor, Johnson, Christopher R., Series editor, Polthier, Konrad, Series editor, Rumpf, Martin, Series editor, Hotz, Ingrid, editor, and Schultz, Thomas, editor

Published
2015
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18. Barycentric Subspaces and Affine Spans in Manifolds

Author

Pennec, Xavier, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Nielsen, Frank, editor, and Barbaresco, Frédéric, editor

Published
2015
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19. Multivariate Differential and Integral Calculus

Author

Godement, Roger, Axler, Sheldon, Series editor, Capasso, Vincenzo, Series editor, Casacuberta, Carles, Series editor, MacIntyre, Angus, Series editor, Ribet, Kenneth, Series editor, Sabbah, Claude, Series editor, Süli, Endre, Series editor, Woyczynski, Wojbor A., Series editor, and Godement, Roger

Published
2015
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23. Tangent Linear Modeling and Adjoints

Author

Steven J. Fletcher

Subjects
Data assimilation, Advection, Mathematical analysis, Linear model, Tangent, Perturbation (astronomy), Tangent vector, Self-adjoint operator, Resolvent, Mathematics
Abstract

In this chapter we introduce three very important components of variational data assimilation: tangent linear modeling, perturbation forecast modeling, and the adjoint operator. We shall present a technique to linearize the semi-Lagrangian advection as well as the tangent linear model and the adjoint of parts of spectral modeling. Given these three different techniques, it is possible to determine different properties of the numerical model as well as observation impacts. We shall introduce the singular vectors which are used to identify properties of the model.

Published
2023

26. Parametrized Surfaces in

Author

Dineen, Seán, Chaplain, M.A.J., Series editor, Erdmann, K., Series editor, MacIntyre, Angus, Series editor, Süli, Endre, Series editor, Tehranchi, M R, Series editor, Toland, J.F., Series editor, and Dineen, Seán

Published
2014
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29. The Hairy Ball problem is PPAD-complete

Author

Alexandros Hollender and Paul Goldberg

Subjects
FOS: Computer and information sciences, Discrete mathematics, Computer Science::Computer Science and Game Theory, 000 Computer science, knowledge, general works, Hairy ball theorem, Toroid, General Computer Science, Computer Networks and Communications, Applied Mathematics, Computational Complexity (cs.CC), Computer Science::Computational Complexity, PPAD, Theoretical Computer Science, Computer Science - Computational Complexity, Computational Theory and Mathematics, Computer Science - Computer Science and Game Theory, Computer Science, Tangent vector, Ball (mathematics), Computational problem, Computer Science and Game Theory (cs.GT), Mathematics, Conceptual level
Abstract

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of END-OF-LINE, the canonical PPAD-complete problem. Our results on these new END-OF-LINE variants are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the IMBALANCE problem defined by Beame et al. in 1998., Journal version

Published
2021

30. Finite Type Conditions on Real Hypersurfaces with One Degenerate Eigenvalue

Author

Wanke Yin, Wei Chen, and Yingxiang Chen

Subjects
Pure mathematics, Mathematics::Complex Variables, General Mathematics, Degenerate energy levels, General Physics and Astronomy, Commutator (electric), Contact type, Field (mathematics), Type (model theory), law.invention, Hypersurface, law, Tangent vector, Eigenvalues and eigenvectors, Mathematics
Abstract

Let M be a smooth pseudoconvex hypersurface in ℂn+1 whose Levi form has at most one degenerate eigenvalue. For any tangent vector field L of type (1, 0), we prove the equality of the commutator type and the Levi form type associated to L. We also show that the regular contact type, the commutator type and the Levi form type of the real hypersurface are the same.

Published
2021

36. Parallel Transport with Pole Ladder: Application to Deformations of Time Series of Images

Author

Marco, Lorenzi, Pennec, Xavier, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Nielsen, Frank, editor, and Barbaresco, Frédéric, editor

Published
2013
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40. Vector Fields

Author

Bump, Daniel, Axler, Sheldon, Series editor, Ribet, Kenneth, Series editor, and Bump, Daniel

Published
2013
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48. Local theory of curves

Author

Abate, Marco, Tovena, Francesca, Quarteroni, A., editor, Ambrosio, L., editor, Biscari, P., editor, Ciliberto, C., editor, van der Geer, G., editor, Rinaldi, G., editor, Runggaldier, W. J., editor, Abate, Marco, Tovena, Francesca, and Gewurz, Daniele A., Translator

Published
2012
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49. Geodesics

Author

Abate, Marco, Tovena, Francesca, Quarteroni, A., editor, Ambrosio, L., editor, Biscari, P., editor, Ciliberto, C., editor, van der Geer, G., editor, Rinaldi, G., editor, Runggaldier, W. J., editor, Abate, Marco, Tovena, Francesca, and Gewurz, Daniele A., Translator

Published
2012
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50. Transformation Invariance in Pattern Recognition – Tangent Distance and Tangent Propagation

Author

Simard, Patrice Y., LeCun, Yann A., Denker, John S., Victorri, Bernard, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Montavon, Grégoire, editor, Orr, Geneviève B., editor, and Müller, Klaus-Robert, editor

Published
2012
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