Papers by Augustin Batubenge
STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, 2018
Let G be a Lie group, \(\mathfrak {g}\) its Lie algebra, and \(\mathfrak {g^{*}}\) the dual of \(... more Let G be a Lie group, \(\mathfrak {g}\) its Lie algebra, and \(\mathfrak {g^{*}}\) the dual of \(\mathfrak {g}\). Let Φ be the symplectic action of G on a symplectic manifold (M, ω). If the momentum mapping \(\mu :M\rightarrow \mathfrak {g^{*}}\) is not Ad∗-equivariant, it is a fact that one can modify the coadjoint action of G on \(\mathfrak {g^{*}}\) in order to make the momentum mapping equivariant with respect to the new G-structure in \(\mathfrak {g^{*}}\), and the orbit of the coadjoint action is a symplectic manifold. With the help of a two cocycle \(\sum :\mathfrak {g}\times \mathfrak {g}\rightarrow \mathbf {R}\), \((\xi ,\eta )\mapsto \sum (\xi ,\eta )=d\hat {\sigma }_{\eta }(e)\cdot \xi \) associated with one cocycle \(\sigma :G\rightarrow \mathfrak {g^{*}};~~\sigma (g)=\mu (\phi _g(m))-Ad^*_g\mu (m)\), we show that a symplectic structure can be defined on the orbit of the affine action \(\Psi (g,\beta ):=Ad_{g}^{*}\beta +\sigma (g)\) of G on \(\mathfrak {g^{*}}\), the orbit of which is a symplectic manifold with the symplectic structure \(\omega _{\beta }(\xi _{\mathfrak {g^{*}}}(v),\eta _{\mathfrak {g^{*}}}(v))=-\beta ([\xi ,\eta ])+\sum (\eta ,\xi )\).
arXiv (Cornell University), Jan 6, 2020
Let (M, ω) be a symplectic and Riemannian manifold on which a Lie group acts in a hamiltonian way... more Let (M, ω) be a symplectic and Riemannian manifold on which a Lie group acts in a hamiltonian way. We determine conditions for which the reduced space inherits an induced Riemannian structure through a Riemannian submersion.
arXiv (Cornell University), Apr 6, 2020
Let G be an n-dimensional semisimple, compact and connected Lie group acting on both the Lie alge... more Let G be an n-dimensional semisimple, compact and connected Lie group acting on both the Lie algebra g of G and its dual g *. In this work it is shown that a nondegenerate Killing form of G induces an Ad *-equivariant isomorphism of g onto g * which, in turn, induces by passage to quotients a symplectic diffeomorphism between adjoint and coadjoint orbit spaces of G.
In this paper, the weakest topology underlying a Frölicher space provided with a Finsler metric i... more In this paper, the weakest topology underlying a Frölicher space provided with a Finsler metric is shown to coincide with the induced metric topology. M.S.C. 2010: 54A10, 54D20, 53D20 53B40, 54A10, 54F65, 57R55 and 55.XX.
In this paper, the weakest topology underlying a Frölicher space provided with a Finsler metric i... more In this paper, the weakest topology underlying a Frölicher space provided with a Finsler metric is shown to coincide with the induced metric topology. M.S.C. 2010: 54A10, 54D20, 53D20 53B40, 54A10, 54F65, 57R55 and 55.XX.
Quaestiones Mathematicae NISC Pty Ltd, Mar 31, 2009
This paper is an attempt to the symplectization of smooth unusual, but
standard imbedded subspac... more This paper is an attempt to the symplectization of smooth unusual, but
standard imbedded subspaces of Rn like a n-simplex, for the purpose of modelling
Hamiltonian and Lagrangian systems thereon. We show that these subspaces are
obtained by the process of smoothly gluing portions of Rn, all considered as Fr¨olicher
spaces. On working with some of them, we characterize smooth vector fields that
have a flow. Then, we show that both the modern and classical mechanical systems
can be written in a more larger category than that of Fr¨olicher spaces, which is the
category of differential spaces
Let G be a Lie group, \(\mathfrak {g}\) its Lie algebra, and \(\mathfrak {g^{*}}\) the dual of \(... more Let G be a Lie group, \(\mathfrak {g}\) its Lie algebra, and \(\mathfrak {g^{*}}\) the dual of \(\mathfrak {g}\). Let Φ be the symplectic action of G on a symplectic manifold (M, ω). If the momentum mapping \(\mu :M\rightarrow \mathfrak {g^{*}}\) is not Ad∗-equivariant, it is a fact that one can modify the coadjoint action of G on \(\mathfrak {g^{*}}\) in order to make the momentum mapping equivariant with respect to the new G-structure in \(\mathfrak {g^{*}}\), and the orbit of the coadjoint action is a symplectic manifold. With the help of a two cocycle \(\sum :\mathfrak {g}\times \mathfrak {g}\rightarrow \mathbf {R}\), \((\xi ,\eta )\mapsto \sum (\xi ,\eta )=d\hat {\sigma }_{\eta }(e)\cdot \xi \) associated with one cocycle \(\sigma :G\rightarrow \mathfrak {g^{*}};~~\sigma (g)=\mu (\phi _g(m))-Ad^*_g\mu (m)\), we show that a symplectic structure can be defined on the orbit of the affine action \(\Psi (g,\beta ):=Ad_{g}^{*}\beta +\sigma (g)\) of G on \(\mathfrak {g^{*}}\), the orb...
arXiv: Differential Geometry, 2020
In this paper we determine conditions of existence of an induced Riemannian structure on the symp... more In this paper we determine conditions of existence of an induced Riemannian structure on the symplectic quotient of a symplectic and Riemannian manifold following the action of a Lie group acting upon it in a hamiltonian way with equivariant momentum mapping.
Let G be an n-dimensional semisimple compact and connected Lie group acting on both the Lie algeb... more Let G be an n-dimensional semisimple compact and connected Lie group acting on both the Lie algebra g of G and its dual g*. We show that a nondegenerate Killing form of G induces an Ad*-equivariant isomorphism of g onto g* which, in turn, induces by passage to quotients a symplectomorphism between adjoint and coadjoint orbit spaces of G.
Demonstratio Mathematica, 2009
In this paper, we show that when the Frölicher smooth structure is induced on a subset or a quoti... more In this paper, we show that when the Frölicher smooth structure is induced on a subset or a quotient set, there are three natural topologies underlying the resulting object. We study these topologies and compare them in each case. It is known that the topology generated by strucure functions is the weakest one in which all functions and curves on the space are continuous. We show that on a subspace, it is rather the trace topology which has this property, while the three topologies are coincident on the quotient space. We construct a base for the Frölicher topology and using either a base or a subbase in the sense of A. Frölicher [
Demonstratio Mathematica, 2008
We define a class of Frölicher spaces locally diffeomorphic to Frölicher subspaces of the Euclide... more We define a class of Frölicher spaces locally diffeomorphic to Frölicher subspaces of the Euclidean space ℝ
Quaestiones Mathematicae, 2016
In this work we introduce a class of Sikorski differential spaces (M;D) called pre-Frolicher spac... more In this work we introduce a class of Sikorski differential spaces (M;D) called pre-Frolicher spaces, on which the process of yielding a Frolicher structure on the underlying set M is D preserving, their category we denote by preFrl. We investigate some algebraic properties on these spaces whose subsequent geometric properties are mostly similar to those of smooth manifolds, except for the invariance of dimension, and also that preFrl naturally induces a Cartesian closed subcategory of the category Frl in which there is no discrete object. Using this Cartesian property, it is shown that the Gelfand representation is a smooth map, that the tangent as well as cotangent bundles are made smooth spaces in an unusual but more natural way via smooth curves.
Quaestiones Mathematicae, 2015
Abstract This survey paper highlights a series of results in recent research on topology, geometr... more Abstract This survey paper highlights a series of results in recent research on topology, geometry and categorical properties of spaces provided with a new structure in the mathematical literature, called Frölicher spaces. Without any fear of contradiction, these smooth spaces are stated to generalize the theory of differentiable manifolds. More precisely, the present study will give the state of research on this topic which historically links a fundamental theorem of calculus in so-called Boman's theorem (and its generalization) to abstract spaces, whether they are normable or not.
Equivalence of star-products on symplectic manifolds, 2005
Besides properties related to the Hochshild cohomology of a symplectic manifold, model in analyti... more Besides properties related to the Hochshild cohomology of a symplectic manifold, model in analytical dynamics with applications in quantum theory, this paper also shows the equivalence of two star-products, more specifically, that every differential star-product of two functions u and v on a symplectic manifold is equivalent to one whose linear term is half of the Poisson bracket of these functions, i.e., 1/2{u,v}.
Demonstratio Mathematica, 2014
In this paper, the topologies underlying a product Frölicher space and a coproduct Frölicher spac... more In this paper, the topologies underlying a product Frölicher space and a coproduct Frölicher space are defined and compared. It is shown that the product topology, which is equal to the one induced by structure functions, is the weakest one in which all projections are continuous. On the other hand, it is proved that the three topologies arising from the coproduct structure are equal.
TURKISH JOURNAL OF MATHEMATICS, 2014
In this paper, we discuss some geometric properties of almost contact metric submersions involvin... more In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that this is obtained if the total space is an b-almost Kenmotsu manifold.
Quaestiones Mathematicae, 2005
ABSTRACT In this paper, we start by showing that one can obtain by way of diffeomorphisms some cl... more ABSTRACT In this paper, we start by showing that one can obtain by way of diffeomorphisms some classical smooth curves and surfaces. The notion of bundles is also introduced in the category of Frölicher spaces, and one may carry on investigating the analogue notion of ordinary G-bundles. We then review tangent and cotangent bundles of an arbitrary Frölicher space X and amend Cherenack's proof [4] that these notions coincide with the usual ones when X is a smooth manifold. Finally, we prove that the analogues of some results in [10] and [11] are true in the category of Frölicher Lie groups. But as for differential groups, the unicity of an integral curve of a vector field does pose problems. Nevertheless, one can show that the map c′ : R → TX, sending t ↦ c*t(1t), where c : R → X is a curve on a Frölicher space X, is smooth.Most of the results in our paper hold because the category of Frölicher spaces is Cartesian closed [5].
Quaestiones Mathematicae, 2009
This paper is an attempt to the symplectization of smooth unusual, but standard imbedded subspace... more This paper is an attempt to the symplectization of smooth unusual, but standard imbedded subspaces of IR^n like an-simplex, for the purpose of modelling Hamiltonian and Lagrangian systems thereon. We show that these subspaces are obtained by the process of smoothly gluing portions of IR^n, all considered as Froelicher spaces. On working with some of them, we characterize smooth vector fields that have a flow. Then we show that both the modern and classical mechanical systems can be written in a more larger category than that of Froelicher spaces, which is the category of differential spaces.
International Journal of Mathematical Education in Science and Technology, 2014
ABSTRACT The differential transform method (DTM) and the multi-step differential transform method... more ABSTRACT The differential transform method (DTM) and the multi-step differential transform method (MsDTM) are numerical methods that most undergraduate students are not familiar with. The methods provide solutions in terms of convergent series with easily computable components. The aim of this article is to introduce the DTM and MsDTM as efficient tools to solve linear and nonlinear differential equations, at undergraduate level. We choose a population growth problem and a mixing problem to illustrate the simplicity and accuracy of its variants by comparing the results with the Runge–Kutta method.
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Papers by Augustin Batubenge
standard imbedded subspaces of Rn like a n-simplex, for the purpose of modelling
Hamiltonian and Lagrangian systems thereon. We show that these subspaces are
obtained by the process of smoothly gluing portions of Rn, all considered as Fr¨olicher
spaces. On working with some of them, we characterize smooth vector fields that
have a flow. Then, we show that both the modern and classical mechanical systems
can be written in a more larger category than that of Fr¨olicher spaces, which is the
category of differential spaces
standard imbedded subspaces of Rn like a n-simplex, for the purpose of modelling
Hamiltonian and Lagrangian systems thereon. We show that these subspaces are
obtained by the process of smoothly gluing portions of Rn, all considered as Fr¨olicher
spaces. On working with some of them, we characterize smooth vector fields that
have a flow. Then, we show that both the modern and classical mechanical systems
can be written in a more larger category than that of Fr¨olicher spaces, which is the
category of differential spaces