| 词汇 | example_english_foliation |
| 释义 | Examples of foliationThese examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. In particular, we obtain rigid foliations of all degrees. Bounds on distortion involve very technical and fastidious estimates on the differentiability of the invariant foliations (stable and unstable). In particular, the strong stable and strong unstable foliations are smooth. One can also work directly with vector fields instead of foliations. Moreover, these foliations have an invariant complex line. All foliations defined by the solutions of equation (3) also are singular. What makes foliations more useful than a mere partitioning is their added structure, which requires a smooth transition from one leaf to another. One way to understand the price dynamics for a pure exchange economy is to examine the geometry of the foliations. As illustrated, the mathematics of foliations provide a technical and conceptual tool to address and predict a variety of topics. By viewing this concern from the perspective of foliations, an approach becomes clear. Answers for these questions follow from the structure of foliations. Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups. The first and second cases correspond to both spacelike and timelike totally geodesible foliations. We classify them under a completeness hypothesis and deduce the dual classification of codimensionone geodesically complete timelike totally geodesic foliations. We deduce, in particular, that the weak stable and weak unstable foliations of admit transverse affine structures. The corresponding foliations are therefore also arbitrarily close. The main tool of our construction is the equisingular unfolding of foliations. As we have said, we restrict ourselves to smooth timelike totally geodesic foliations. We will see that they can be realized as geodesically complete timelike totally geodesic foliations. As equicontinuity is a delicate notion for foliations, admitting slightly different, and non-equivalent, definitions, we give the definition we consider. A homological characterization of foliations consisting of minimal surfaces. The latter are also called the weak unstable and weak stable foliations, respectively. There is an analogous definition for unstable foliations and center stable and center unstable. In this article, we classify the germs of holomorphic singular simple foliations of codimension 1 and dimension n 3, in the resonant case. The latter condition is motivated by the applications mentioned earlier to partially hyperbolic systems and to transversely projective foliations. Anastomosing shear planes producing shear lenses, wavy shear surfaces and foliations in basalt are commonly seen in the vicinity of the western thrusts. The examples in [2, 28] have closed periodic orbits and thus the two strong foliations are minimal. In particular, for such systems, it allows us to introduce a notion of duality between the induced affine structures on the stable and unstable foliations. The smoothness of the filled-in foliations is controlled by the bunching conditions. More generally, as described later, it is possible for the geometry of certain surfaces to force all foliations of certain dimensions to be singular. In the first case the foliations are given by the suspensions of pairs of commuting diffeomorphisms of the circle. We briefly recall a notion concerning the regularity of continuous foliations. The second step is to show that once the restriction of to the leaves of the foliations as above are smooth, then is smooth. We are starting with a product diffeomorphism, for which the unstable and stable u s foliations are jointly integrable. In applications occurring in rigidity of actions, there are situations when one has to consider other sets of foliations. The other goal of this article is to consider the relative position of incompressible tori with respect to the stable and unstable foliations. We also point out that the restriction to two foliations is not essential. The last corollary shows that it is easy to obtain examples of harmonically simple foliations. Several useful facts concerning finite energy foliations appear in [16, 18]. Therefore, we establish that the conjugacy is smooth when restricted to the stable and unstable foliations. Now we are left with classifying the parallelizable one-dimensional foliations. Magnetic foliations are parallel to the contact itself, and magnetic lineations plunge steeply outward (downdip) on the contact plane. Assume that the stable and unstable foliations of f are jointly integrable. The expansion growth of foliations is defined as an element of this extended growth set. We shall assume that the foliations both strong and weak are measurable in the following sense. As mentioned already in the introduction, one can also work directly with vector fields generating foliations. In other words, close vector fields generate close foliations. Here are two examples for linking numbers between measured foliations and divergence-free vector fields. In the next section, we will also want to use singular foliations. The indifference-sets-of-preferences illustration accurately suggests that one use of foliations is to conveniently catalogue information. By combining the continuity property with the structure of foliations once a more accurate census is taken leads to the following conclusion. In part, this is because important special cases of foliations are the level sets of smooth mappings. After providing a technical description of foliations (with references), the discussion emphasizes two general themes. I claimed that it is possible to use the structure of foliations to anticipate new kinds of properties for models from the social sciences. They are so common that it is fair to assert that after one learns what to look for, foliations can be found almost everywhere. Moreover, by invoking the structure of foliations, it now is not difficult to establish that any "reform" procedure also has serious faults. We can define transversely projective foliations in a similar vein, by simply replacing the affinities by homographies. First, a system may preserve several different foliations. We define the notion of the unstable and stable foliations stably carrying some unique nontrivial homologies. The results stated above (for both polynomial and projective foliations) are based on study of the monodromy group at infinity. Such foliations are untypical among all degree d + 1 foliations: they are specified by the fact that the infinite hyperplane is invariant. Regarded as real foliations, this implies that the corresponding leaves have conjugate holonomy groups. A well-known property of transversely parallelizable complete foliations is the following (see [14]). In both cases, there is a global circle transversal to the foliations. In this paper, we give the explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces. A structure theorem for such foliations is given. The crucial step in this technique is to ensure some analogue of the absolute continuity of the foliations of by stable and unstable fibers. Note that this definition is symmetric on the stable and unstable foliations. There are examples of partially hyperbolic diffeomorphisms with non-integrable center foliations (see [20]). Theorem 13 can sometimes be applied to singular foliations. Note that the proof is not restricted to considering just two foliations which are transversal. In particular, they are locally integrable and thus give rise to two orthogonal foliations. The answer about the existence of other dimensional foliations on an odd-dimensional sphere is more complicated and surprising. To illustrate this point with concerns from the social sciences, we first need to appreciate the more general geometry allowed by foliations. What facilitates this transfer is that foliations start with a common description of how changes among equivalence classes of information can occur. Indeed, foliations are used whenever indifference sets for preferences are drawn on a blackboard or the effects of different economic indicators are discussed. Singular foliations are where some partition sets-that is, some leafs-have lower dimensions. The basic problem is to determine the kinds of structures that define foliations. Geodesic flows of negative curved manifolds with smooth stable and unstable foliations. To prove mixing, it is enough to show that the stable and unstable foliations form a nonintegrable pair. Obviously, the stable and unstable foliations do not carry any non-trivial homology in this case. We show that it has the same qualitative properties as that of smooth codimension-one foliations on a compact manifold. As the definitions indicate, the entropy of foliations is positive if and only if the expansion growth of foliations has quasi-exponential growth type. Such foliations provide a detailed and intricate record of the tectonic evolution and structural relationships developed within the internal portions of orogenic belts. High-dimensional helicities and rigidity of linked foliations. In this paper we regard the problem of classifying such foliations. Furthermore, the isometry between two foliations is equivalent to the conjugacy between the dihedral group actions on the covering spaces. Looking only at the foliations, it is natural to take as a completion, then, the collection of all pairs of transverse linear foliations. The homoclinic points, foliations, and so on, all have a precise meaning in this expansion. We do not treat the case of foliations, bearing in mind that they are defined by involutive flows on covering manifolds. Approximating foliations is very delicate even with finite differentiability. In the later case, we can also consider the center-stable and center-unstable foliations. If we remove the hypothesis of completeness, the only thing we know is that the foliations are taut, which is a weaker property. In this work, singular orientable foliations which admit non-trivial recurrent leaves on 2-manifolds of finite or infinite genus are considered. Next, we describe a class of examples of harmonically non-simple foliations. In trying to construct examples of harmonically non-simple foliations, it is useful to have in mind the following easy fact. The measurability of coordinates is now taken for the weakly stable (or weakly unstable) foliations in addition to the previous assumption. Our approach was based on the construction of stable and unstable foliations for certain skew-products. In general, the stable and unstable foliations of a partially hyperbolic diffeomorphism do not have to be quasi-isometric. The existence and absolute continuity of the local invariant foliations is a local matter. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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