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Duality and Riemannian cubics

2006
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Advances in Computational Mathematics
*

To Charles Micchelli, with warm greetings

doi:10.1007/s10444-004-7621-4
fatcat:et7z4hxmxbfghnjp2u64b6sdg4
*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). ...##
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Sub-Riemannian cubics in SU(2)
[article]

2018
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arXiv
*
pre-print

Sub-

arXiv:1712.02930v2
fatcat:jzva3dr4enhqdiuzwlnc6bcmxu
*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. ...##
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Second order models for optimal transport and cubic splines on the Wasserstein space
[article]

2018
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arXiv
*
pre-print

We explore two different numerical approaches, one based on multi-marginal optimal transport

arXiv:1801.04144v2
fatcat:oxtw7eurnzfjzc4s7bzzn622ya
*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. ...##
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Page 3350 of Mathematical Reviews Vol. , Issue 97E
[page]

1997
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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. ...##
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Parameters estimate of Riemannian Gaussian distribution in the manifold of covariance matrices

2016
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2016 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM)
*

In this paper we introduce a smooth estimate of these quantities using convex

doi:10.1109/sam.2016.7569687
dblp:conf/ieeesam/ZaniniCJSB16
fatcat:6b3s33ug4fdhrbbbrd5bgbuksy
*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 ...##
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On a constant curvature statistical manifold
[article]

2022
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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. ...

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Invariant Higher-Order Variational Problems II

2012
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Journal of nonlinear science
*

Finally, we apply second-order Lagrange--Poincar\'e reduction to the problem of

doi:10.1007/s00332-012-9137-2
fatcat:jbzbhrcez5hsthfcpn5voj4ble
*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. ...##
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Page 5900 of Mathematical Reviews Vol. , Issue 2000h
[page]

2000
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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*. ...##
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Unified dark energy and dust dark matter dual to quadratic purely kinetic K-essence

2016
*
European Physical Journal C: Particles and Fields
*

by the square-root of the determinant of the

doi:10.1140/epjc/s10052-016-3938-7
fatcat:onswnlsbuvgs7l4z7dbohk5sra
*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. ...##
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Canonical Barriers on Convex Cones

2014
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Mathematics of Operations Research
*

bounded from below by
−(n − 2)g (Calabi 1972)

doi:10.1287/moor.2013.0640
fatcat:yw3ii7cqhvhrhkf7cvv7hl6nyy
*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. ...##
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Geometric Hamiltonian Formulation of a Variational Problem Depending on the Covariant Acceleration

2013
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Conference Papers in Mathematics
*

We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of

doi:10.1155/2013/243621
fatcat:dqqhngrpsrb6rioa27f2ekggeu
*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] . ...##
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Metric-Independent Spacetime Volume-Forms and Dark Energy/Dark Matter Unification
[article]

2015
*
arXiv
*
pre-print

Also, an intrinsic relation between the above modified gravity + single scalar field model

arXiv:1512.01395v1
fatcat:nitrfi365nfe3ppkujqg4joznq
*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. ...##
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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.) ...##
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Page 1181 of Mathematical Reviews Vol. , Issue 89B
[page]

1989
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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 ...##
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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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