Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the Gener... more Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the General Evolution Research Group, and their 37-year partnership.
Complex dynamical systems theory and system dynamics diverged at some point in the recent past, a... more Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.
Progress in biophysics and molecular biology, Jan 14, 2015
Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathe... more Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathematics a cultural construct? In this short article we speculate on the place of mathematical reality from the perspective of the mystical cosmologies of the ancient traditions of meditation, psychedelics, and divination.
This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters... more This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will study the related topics of tensors and differential forms. A basic operation introduced in this chapter is the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.
The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ... more The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.
Manifolds have enough structure to allow differentiation of maps between them. To set the stage f... more Manifolds have enough structure to allow differentiation of maps between them. To set the stage for these concepts requires a development of differential calculus in linear spaces from a geometric point of view. The goal of this chapter is to provide this perspective.
We are now ready to study manifolds and the differential calculus of maps between manifolds. Mani... more We are now ready to study manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. This abstraction has proved useful because many sets that are smooth in some sense are not presented to us as subsets of Euclidean space. The abstraction strips away the containing space and makes constructions intrinsic to the manifold itself. This point of view is well worth the geometric insight it provides.
Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the Gener... more Mathematician Ralph Abraham recounts memories of Dave Loye, Riane Eisler, Ervin Laszlo, the General Evolution Research Group, and their 37-year partnership.
Complex dynamical systems theory and system dynamics diverged at some point in the recent past, a... more Complex dynamical systems theory and system dynamics diverged at some point in the recent past, and should reunite. This is a concise introduction to the basic concepts of complex dynamical systems, in the context of the new mathematical theories of chaos and bifurcation.
Progress in biophysics and molecular biology, Jan 14, 2015
Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathe... more Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathematics a cultural construct? In this short article we speculate on the place of mathematical reality from the perspective of the mystical cosmologies of the ancient traditions of meditation, psychedelics, and divination.
This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters... more This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will study the related topics of tensors and differential forms. A basic operation introduced in this chapter is the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.
The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ... more The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.
Manifolds have enough structure to allow differentiation of maps between them. To set the stage f... more Manifolds have enough structure to allow differentiation of maps between them. To set the stage for these concepts requires a development of differential calculus in linear spaces from a geometric point of view. The goal of this chapter is to provide this perspective.
We are now ready to study manifolds and the differential calculus of maps between manifolds. Mani... more We are now ready to study manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. This abstraction has proved useful because many sets that are smooth in some sense are not presented to us as subsets of Euclidean space. The abstraction strips away the containing space and makes constructions intrinsic to the manifold itself. This point of view is well worth the geometric insight it provides.
Uploads
Papers by Ralph Abraham