Embedding Theorem

Difference analogs of the embedding theorems. In the estimation of various properties of difference schemes such as stability, convergence, etc. we shall need yet inequalities corresponding to the simplest Sobolev embedding theorems. In this respect the appropriate results have been obtained with the following lemmas. 

The embedding theorem. Various a priori estimates for the equation Ay =

energy estimates imply a uniform estimate, that is, an estimate in the norm... 

It is worth noting here that on the square grid (h = h. = h) this condition is automatically fulfilled. A proper choice of (p guarantees the sixth order of accuracy of scheme (9) on any such grid. Convergence of scheme (9) with the fourth order in the space C can be established without concern of condition (11). An alternative way of covering this is to construct an a priori estimate for A z p and then apply the embedding theorem (see Section 4). 

The operators A and A are commuting and self-adjoint and, therefore, for the equation A z = i> the estimate A 2 < V is certainly true. By the embedding theorem from Section 4,... 

The Allan variance analyses of energy fluctuations can tell us about the nature of the configurational energy landscapes and the existence of (multidimensional) cooperativity among individual DOFs, enhanced for an intermediate time scale at the transition temperature. However, what can we learn or deduce from an (observed) scalar time series about the geometrical aspects of the underlying multidimensional state (or phase) space buried in the observations The so-called embedding theorems attributed to Whitney [75] and Takens [76] provide us with a clue to the answer of such a question (see also Section VI.A and Refs. 77-80). 

The embedding theorem holds irrespective of the choice of the delay time x, but, in practice, the observed time series are always contaminated by noise, computer round-off errors, or a finite resolution of observations, and they are sampled up to a certain finite time. This requires us to choose an optimal time delay x for an observable s. 

There does not exist delay embedding theorem for Hamiltonian systems. 

Embedding Theorem. Now suppose there exists an arbitrary d-dimensional smooth and compact manifold A in R. ... 

Then what will be the minimum number of dimensions m.( < k) required to embed a d-dimensional manifold A lying in Whitney s embedding theorem [75] states a condition that ensures producing the embedding of the /-dimensional manifold A in Uk onto a reduced state space Rm. Here, in order to capture its essence, let us consider an example of a one-dimensional manifold A, a twisted circle, that will be observed in Um (in 1,2, 3). As shown in Fig. 30, when the 1-manifold A is projected on to a one-dimensional Euclidean space R1, selfintersections (i.e., not one-to-one) occur at almost every point inevitably. For the... 

This illustrates Whitney s embedding theorem [75] to provide us with a sufficient condition to yield the minimum number of dimension m(> 2d), the so-called embedding dimension, required to embed a d-dimensional manifold. The mathematical description of the embedding theorem is as follows ... 

This embedding theorem holds irrespective of the choice of the delay time T if one could observe an infinitely long time series at infinitesimally small resolution. 

Sauer et al. [79] developed a new concept, the so-called prevalence, on infinitedimensional spaces, with a particular emphasis on function spaces, which corresponds to the concept of almost every (i.e., with the set of exceptions of Lebesgue measure zero) on finite-dimensional spaces. They generalized Whitney and Takens embedding theorems to ensure that they still hold for non-manifold A with noninteger fractal dimension with minor modifications. For a further detailed discussion, one should see Ref. 79. 

My treatment in the lecture follows more or less my book Abelian Varieties e.g., the 2 embedding theorems above are proven in Ch. 1 of this book. The group-theoretic aspects discussed in the lecture are in my book, 23, as well as in my paper and Igusa s book. The prime form Ee and its applications axe due to Riemann and are discussed at length with many more applications in Fay. 

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