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Duality and Riemannian cubics

Lyle Noakes
2006 Advances in Computational Mathematics  
To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday. Abstract Riemannian cubics are curves used for interpolation in Riemannian manifolds.  ...  It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation.  ...  Conclusion Duality is introduced for Riemannian cubics in Lie groups, especially SO(3) and SO(1, 2).  ... 
doi:10.1007/s10444-004-7621-4 fatcat:et7z4hxmxbfghnjp2u64b6sdg4

Sub-Riemannian cubics in SU(2) [article]

Michael Swaddle, Lyle Noakes
2018 arXiv   pre-print
Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration.  ...  Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space.  ...  In this situation critical points of S are called Riemannian cubics. We now consider the case where, M = G and˙ x is constrained to be in the distribution δ.  ... 
arXiv:1712.02930v2 fatcat:jzva3dr4enhqdiuzwlnc6bcmxu

Second order models for optimal transport and cubic splines on the Wasserstein space [article]

Jean-David Benamou , François-Xavier Vialard
2018 arXiv   pre-print
We explore two different numerical approaches, one based on multi-marginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport.  ...  On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation.  ...  Cubic splines on Riemannian manifolds In this section, we present Riemannian cubics, which are the extension of variational splines to a Riemannian manifold (M, g) where g is the Riemannian metric.  ... 
arXiv:1801.04144v2 fatcat:oxtw7eurnzfjzc4s7bzzn622ya

Page 3350 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews  
Then the cubic convergence is studied.  ...  P. (4-SHMPM; Southampton) Duality in mathematics and linear and integer programming. (English summary) J. Optim. Theory Appl. 90 (1996), no. 2, 257-278.  ... 

Parameters estimate of Riemannian Gaussian distribution in the manifold of covariance matrices

Paolo Zanini, Marco Congedo, Christian Jutten, Salem Said, Yannick Berthoumieu
2016 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM)  
In this paper we introduce a smooth estimate of these quantities using convex cubic spline, and we show that in this case the parameters estimate is coherent with theoretical results.  ...  We also present some simulations and a real EEG data analysis.  ...  Affine-invariance: for any invertible matrix B d R (BC 1 B T , BC 2 B T ) = d R (C 1 , C 2 ), (2) and self-duality d R (C −1 1 , C −1 2 ) = d R (C 1 , C 2 ). (3) Distance (1) is used to define the concept  ... 
doi:10.1109/sam.2016.7569687 dblp:conf/ieeesam/ZaniniCJSB16 fatcat:6b3s33ug4fdhrbbbrd5bgbuksy

On a constant curvature statistical manifold [article]

Shimpei Kobayashi, Yu Ohno
2022 arXiv   pre-print
and the curvatures satisfies R=R^*, where R^* is the curvature of the dual connection ∇^*.  ...  Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature -1 statistical structures and vice versa.  ...  Jun-ichi Inoguchi and Porf. Hitoshi Furuhata for comments on the manuscripts and letting us know several related references.  ... 
arXiv:2008.13394v4 fatcat:goex3ey57nht7fkfl4qxdvzab4

Invariant Higher-Order Variational Problems II

François Gay-Balmaz, Darryl D. Holm, David M. Meier, Tudor S. Ratiu, François-Xavier Vialard
2012 Journal of nonlinear science  
Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics on the group of transformations.  ...  This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with normal metrics. The prime examples of such object manifolds are the symmetric spaces.  ...  Examples: Riemannian cubic polynomials and generalizations. Riemannian cubics, as introduced in [GK85, NHP89] and [CS95] , generalize cubic polynomials in Euclidean space to Riemannian manifolds.  ... 
doi:10.1007/s00332-012-9137-2 fatcat:jbzbhrcez5hsthfcpn5voj4ble

Page 5900 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
The next chapter deals with the exact quantum theory and how it modifies the moduli space, which becomes a Riemannian surface.  ...  The dressing coset duality in this context ‘intersects’ with the Kiritsis-Obers duality.  ... 

Unified dark energy and dust dark matter dual to quadratic purely kinetic K-essence

Eduardo Guendelman, Emil Nissimov, Svetlana Pacheva
2016 European Physical Journal C: Particles and Fields  
by the square-root of the determinant of the Riemannian metric, as well as to another non-Riemannian volume-form in terms of an auxiliary maximal-rank antisymmetric tensor gauge field.  ...  Upon deriving the corresponding "Einstein-frame" effective action of the latter modified gravity-scalar-field theory we find explicit duality (in the sense of weak versus strong coupling) between the original  ...  S.P. and E.N. have received partial support from European COST actions MP-1210 and MP-1405, respectively, as well as from the Bulgarian National Science Fund Grant DFNI-T02/6.  ... 
doi:10.1140/epjc/s10052-016-3938-7 fatcat:onswnlsbuvgs7l4z7dbohk5sra

Canonical Barriers on Convex Cones

Roland Hildebrand
2014 Mathematics of Operations Research  
bounded from below by −(n − 2)g (Calabi 1972) duality realized by conormal map primal and dual ane spheres are isometric Outline Monge-Ampère equation statement and properties ane spheres Calabi  ...  Then the associated holomorphic cubic dierential U satises |U| 2 = 2e 6φ − 2e 4φ ∆φ = 2e 6φ (1 + K), where ∆ is the ordinary Laplacian and K the Gaussian curvature.  ... 
doi:10.1287/moor.2013.0640 fatcat:yw3ii7cqhvhrhkf7cvv7hl6nyy

Geometric Hamiltonian Formulation of a Variational Problem Depending on the Covariant Acceleration

Lígia Abrunheiro, Margarida Camarinha, Jesús Clemente-Gallardo
2013 Conference Papers in Mathematics  
We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials.  ...  This problem and the corresponding optimal control problem are described in the context of higher order tangent bundles using geometric tools.  ...  Riemannian cubic polynomials can be seen as a generalization of cubic polynomials to non-Euclidean spaces [3, 4] .  ... 
doi:10.1155/2013/243621 fatcat:dqqhngrpsrb6rioa27f2ekggeu

Metric-Independent Spacetime Volume-Forms and Dark Energy/Dark Matter Unification [article]

Eduardo Guendelman, Emil Nissimov, Svetlana Pacheva
2015 arXiv   pre-print
Also, an intrinsic relation between the above modified gravity + single scalar field model and a special quadratic purely kinetic "k-essence" model is established as a weak-versus-strong-coupling duality  ...  The method of non-Riemannian (metric-independent) spacetime volume-forms (alternative generally-covariant integration measure densities) is applied to construct a modified model of gravity coupled to a  ...  S.P. and E.N. have received partial support from European COST actions MP-1210 and MP-1405, respectively, as well from Bulgarian National Science Fund Grant DFNI-T02/6.  ... 
arXiv:1512.01395v1 fatcat:nitrfi365nfe3ppkujqg4joznq

Page 550 of Mathematical Reviews Vol. 6, Issue 11 [page]

1945 Mathematical Reviews  
Characterizations of the Euclidean sphere. 29 Grassmann and differential rings and their relations to the theory of multiple integrals. 194 Riemannian manifolds of four dimensions. 216 Curvatura integra  ...  Duality relations in topological spaces. Théoréme de dualité pour le polyédre infini. Choquet, G. (Cf. Bouligand, G.)  ... 

Page 1181 of Mathematical Reviews Vol. , Issue 89B [page]

1989 Mathematical Reviews  
They find that, under some conditions, it is necessary to include a cubic term to make the solution stable (classically).  ...  The one-soliton transformation is worked out explicitly and it is written as the product of a discrete trans- formation sandwiched between two duality transformations; this generalizes a known formula  ... 

Page 2845 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
At that time, the Erlanger program did not include Riemannian geometry. This situation was changed by Cartan’s idea of fibre bundles and connections.  ...  Cubic manifolds in lattices. (Russian. Russian summary) Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 2, 93-107. The category of cubic complexes and cubic maps is similar to the simplicial category.  ... 
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