Search results for "fiber"
showing 10 items of 2343 documents
General Theory: Algebraic Point of View
2009
It is convenient to divide our study of pip-spaces into two stages. In the first one, we consider only the algebraic aspects. That is, we explore the structure generated by a linear compatibility relation on a vector space V , as introduced in Section I.2, without any other ingredient. This will lead us to another equivalent formulation, in terms of particular coverings of V by families of subspaces. This first approach, purely algebraic, is the subject matter of the present chapter. Then, in a second stage, we introduce topologies on the so-called assaying subspaces \(\{V_r \}\). Indeed, as already mentioned in Section I.2, assuming the partial inner product to be nondegenerate implies tha…
A RAPID METHOD FOR CALCULATING THE TRANSVERSAL STRAIN IN THE MOIRE TECHNIQUE ALONG SECTIONS OF SYMMETRY
1970
An approximate rapid method is described for calculating in the moire technique the transversal strain directly from the longitudinal strain distribution across sections of symmetry. The method is based on evaluation of the load transmitted through the section and is corroborated by two examples.
Triangular irreducibility of congruences in quasivarieties
2014
Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. For any integer \({m \geqslant 3}\) and a quasivariety Q, the notion of an m-triangularily meet-irreducible Q-congruence in the algebras of Q is defined. In Section 2, some characterizations of finitely generated quasivarieties involving this notion are provided. Section 3 deals with quasivarieties with equationally definable m-triangular meets of relatively principal congruences. References to finitely based quasivarieties and varieties are discussed.
Some Special Foliations
2014
In this chapter we study two classes of ubiquitous foliations: Riccati foliations and turbulent foliations. A section will also be devoted to a very special foliation, which will play an important role in the minimal model theory.
Explanation of Notation
1991
For the presentation of all 288 polarization observables in the next section we have adopted the following scheme. Each observable is a function of the angle and the photon energy. With respect to the latter we have chosen five energies, namely 4.5 MeV, the maximum of the total cross section, 20 MeV, 60 MeV, 100 MeV, and 140 MeV. For each observable and for each of these energies we have studied the following topics: (i) The influence of meson exchange currents (MEC), isobar configurations (IC) and relativistic corrections (RC). Since the various potential models give qualitatively very similar results, we use in this case the r-space version of the Bonn model (OBEPR). (ii) The contribution…
Quasi-Modes and Spectral Instability in One Dimension
2019
In this section we describe the general WKB construction of approximate “asymptotic” solutions to the ordinary differential equation $$\displaystyle P(x,hD_x)u=\sum _{k=0}^m b_k(x)(hD_x)^ku=0, $$ on an interval α < x < β, where we assume that the coefficients bk ∈ C∞(]α, β[). Here h ∈ ]0, h0] is a small parameter and we wish to solve (above equation) up to any power of h. We look for u in the form $$\displaystyle u(x;h)=a(x;h)e^{i\phi (x)/h}, $$ where ϕ ∈ C∞(]α, β[) is independent of h. The exponential factor describes the oscillations of u, and when ϕ is complex valued it also describes the exponential growth or decay; a(x;h) is the amplitude and should be of the form $$\displaystyle a(x;h…
The “Maslov Anomaly” for the Harmonic Oscillator
2001
Specializing the discussion of the previous section to the harmonic oscillator we have for \(N = 1,\ \eta ^{a} = (p,x),\ a = 1,2,\ \eta ^{1} \equiv p,\ \eta ^{2} \equiv x\) $$\displaystyle{ H(p,x) = \frac{1} {2}\eta ^{a}\eta ^{a} = \frac{1} {2}{\bigl (p^{2} + x^{2}\bigr )}\;. }$$ (30.1) The only conserved quantity is J = H. In the action we need the combination $$\displaystyle{ \frac{1} {2}\eta ^{a}\omega _{ ab}\dot{\eta }^{b} -\mathcal{H}(\eta ) = \frac{1} {2}\eta ^{a}\left [\omega _{ ab} \frac{d} {dt} -{\bigl ( 1 + A(t)\bigr )}\mathrm{1l}_{ab}\right ]\eta ^{b} }$$ (30.2) and $$\displaystyle{ \tilde{M}_{\phantom{a}b}^{a} =\omega ^{ac}\partial _{ c}\partial _{b}(H + AJ\,) ={\bigl ( 1 + A(t)…
Berry Phase and Parametric Harmonic Oscillator
2001
Our concern in this section is once more with the time-dependent harmonic oscillator with Lagrangian $$\displaystyle{ L = \frac{1} {2}\dot{x}^{2} -\frac{1} {2}\omega ^{2}(t)x^{2}\;. }$$ To present a coherent picture of the whole problem, let us briefly review some of the results of Chap. 21. There we found the propagation function
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.