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Numerical Algebraic Geometry and Differential Equations [PDF]

, 2014
In this paper we review applications of numerical algebraic geometry to differential equations. The techniques we address are direct solution, bootstrapping by filtering, and continuation and bifurcation. We review differential equations systems with multiple solutions and bifurcations.
Bei Hu, Wenrui Hao, Andrew J. Sommese
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Differential geometry on Grassmann algebras

Letters in Mathematical Physics, 1976
H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus ‘curvature’ is represented by a tensor of the third rank and ‘Killing's equations’ now involve this ‘curvature tensor ...
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Algebraic geometry of Abel differential equation

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2012
A solution $$y(x)$$ of an Abel differential equation $$(1) \ y^{\prime }=p(x)y^2 + q(x) y^3$$ is called “closed” on
Clara Shikhelman, Y. Yomdin, Y. Shelah, Sh. Giat   +3 more
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Differential algebra (contravariant analytic methods in differential geometry) [PDF]

Journal of Soviet Mathematics, 1980
The problems of developing the apparatus of differential-geometric investigations based on the calculus of differential operators on bundles of semiholonomic jets of Ehresmann are considered.
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Differential forms in algebraic geometry

2011
Before considering more general spaces we shall first discuss (1) the r-dimensional projective space Π r . In this space we shall consider a homogeneous coordinate system (Z0, Z1, ... , Z r ). Let U α be that part of Π r in which Z α ≠ 0. In U α we may then introduce non-homogeneous coordinates zαi = Zι/Zα (ι≠α).
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Noncommutative geometry with graded differential Lie algebras

Journal of Mathematical Physics, 1997
Starting with a Hilbert space endowed with a representation of a unitary Lie algebra and an action of a generalized Dirac operator, we develop a mathematical concept towards gauge field theories. This concept shares common features with the Connes–Lott prescription of noncommutative geometry, differs from that, however, by the implementation of unitary
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Current Algebras, the Sugawara Model, and Differential Geometry

Journal of Mathematical Physics, 1970
The Lie algebra defined by the currents in the Sugawara model is defined in a way that is natural from the point of view of Lie transformation theory and differential geometry. Previous remarks that the Sugawara model is associated with a field-theoretical dynamical system on a Lie group manifold are made more precise and presented in a differential ...
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Geometry of Differential Polynomial Functions, I: Algebraic Groups

American Journal of Mathematics, 1993
Introduction The present paper is a direct continuation of [B5]; we shall freely borrow terminology and notations from that paper which shall be referred to from now on as Part I. However understanding the Introduction of the present paper only requires familiarity with the Introduction and first Appendix of Part I.
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Differential Privacy Techniques for Cyber Physical Systems: A Survey

IEEE Communications Surveys and Tutorials, 2020
Jinjun Chen, Mubashir Husain Rehmani, Muneeb Ul Hassan   +2 more
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Programmable interactions and emergent geometry in an array of atom clouds

Nature, 2021
Avikar Periwal, Julian F Wienand, Eric S Cooper   +2 more
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