Search results for "Fiber bundle"

Tomita—Takesaki Theory in Partial O*-Algebras

2002

This chapter is devoted to the development of the Tomita-Takesaki theory in partial O*-algebras. In Section 5.1, we introduce and investigate the notion of cyclic generalized vectors for a partial O*-algebra, generalizing that of cyclic vectors, and its commutants. Section 5.2 introduces the notion of a cyclic and separating system (M, λ, λ c ), which consists of a partial O*-algebra M, a cyclic generalized vector λ for M and the commutant λ c of λ. A cyclic and separating system (M, λ, λ c ) determines the cyclic and separating system ((M w ′ )′, λ cc , (λ cc ) c ) of the von Neumann algebra (M w ′ )′, and this makes it possible to develop the Tornita-Takesaki theory. Then λ can be extende…

Families of Two-dimensional Vector Fields

1998

In this section we will consider individual vector fields. They can be considered as 0-parameter families. We assume these vector fields to be of class at least C 1. This will be sufficient to ensure the existence and uniqueness of the flow φ(t, x) (t is time, x ∈ S, the phase space) and the qualitative properties which we mention below.

Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline

2019

In this chapter we will state a result asserting that for elliptic semi-classical (pseudo-)differential operators the eigenvalues are distributed according to Weyl’s law “most of the time” in a probabilistic sense. The first three sections are devoted to the formulation of the results and in the last section we give an outline of the proof that will be carried out in Chaps. 16 and 17.

Bifurcations of Regular Limit Periodic Sets

1998

In this chapter, (X λ ) will be a smooth or analytic (in Section 3) family of vector fields on a phase space S, with parameter λ ∈ P, as in Chapter 1. Periodic orbits and elliptic singular points which are limits of sequences of limit cycles are called regular limit periodic sets. The reason for this terminology is that for such a limit periodic set Γ one can define local return maps on transversal segments, which are as smooth as the family itself. The limit cycles near Γ will be given by a smooth equation and the theory of bifurcations of limit cycles from Γ will reduce to the theory of unfoldings of differentiable functions. In fact, we will just need the Preparation Theorem and not the …

Invariants of equivariant algebraic vector bundles and inequalities for dominant weights

1998

Steady-state radiation heat transfer problem

1996

In Section 8.2, we shall see that the steady-state radiative heat transfer problem can be transformed to minimization of a smooth nonquadratic functional J over a convex and closed subset of a Banach space V. To this end we firstly shortly recall some basic definitions concerning differentiability of J, because these sometimes differ in the literature.

An Approach to a Version of the S(M, g)-pseudo-differential Calculus on Manifolds

2003

For appropriate triples (M,g,M), where M is an (in general non-compact) manifold, g is a metric on T* M, and M is a weight function on T* M, we develop a pseudo-differential calculus on.A4 which is based on the S(M,g))-calculus of L. Hormander [27] in local models. In order to do so, we generalize the concept of E. Schrohe [41] of so-called SG-compatible manifolds. In the final section we give an outlook onto topological properties of the algebras of pseudo-differential operators. We state the existence of “order reducing operators” and that the algebra of operators of order zero is a submultiplicative Ψ*-algebra in the sense of B. Gramsch [18] in \( \mathcal{L}\left( {{L^2}\left( M \right)…

*-Representations of Partial *-Algebras

2002

This chapter is devoted to *-representations of partial *-algebras. We introduce in Section 7.1 the notions of closed, fully closed, self-adjoint and integrable *-representations. In Section 7.2, the intertwining spaces of two *-representations of a partial *-algebra are defined and investigated, and using them we define the induced extensions of a *-representation. Section 7.3 deals with vector representations for a *-representation of a partial *-algebra, which are the appropriate generalization to a *-representation of the notion of generalized vectors described in Chapter 5. Regular and singular vector representations are defined and characterized by the properties of the commutant, and…

The Itô Integral

2014

The Ito integral allows us to integrate stochastic processes with respect to the increments of a Brownian motion or a somewhat more general stochastic process. We develop the Ito integral first for Brownian motion and then for generalized diffusion processes (so called Ito processes). In the third section, we derive the celebrated Ito formula. This is the chain rule for the Ito integral that enables us to do explicit calculations with the Ito integral. In the fourth section, we use the Ito formula to obtain a stochastic solution of the classical Dirichlet problem. This in turn is used in the fifth section in order to show that like symmetric simple random walk, Brownian motion is recurrent …

Some Afterthoughts on Hopfield Networks

1999

In the present paper we investigate four relatively independent issues, which complete our knowledge regarding the computational aspects of popular Hopfield nets. In Section 2 of the paper, the computational equivalence of convergent asymmetric and Hopfield nets is shown with respect to network size. In Section 3, the convergence time of Hopfield nets is analyzed in terms of bit representations. In Section 4, a polynomial time approximate algorithm for the minimum energy problem is shown. In Section 5, the Turing universality of analog Hopfield nets is studied. peerReviewed