Minimum embedding dimension

Although many methods have been developed so far to choose the time delay x and to determine the minimum embedding dimension, there exists no general method, especially for many-body systems where modes associated with hierarchical time and space scales are not necessarily decoupled from one another. The question of which geometrical information at a certain hierarchy on the state space can be reconstructed is not trivial at all for real finite time-series data with finite resolution. 

There are four parameters which must be fixed to use the MLE algorithm Embedding dimension, de, maximum scale, Sm, minimum scale, Sm and evolution time, O. Basically, de is the attractor dimension where the orbits were embedded, Sm is the estimate value of the length scale on which the local structure of the attractor is not longer being proved. Sm is the length scale in which noise is expected to appear. O is fixed for compute of divergence measurements which is the necessary time to renormalize the distances between trajectories (for more details see [50]). 

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 ... 

Leveling pads usually have section dimensions of roughly 6 X 12 in. The leveling pad shall have a minimum embedment depth of 1.5 ft measured from top of the pad. Reinforcement within the precast leveling pad is likely not required. 

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... 

The SWNT ropes used were the same as those used for bundle strength measurements discussed earlier. The matrix material used was Epicote 1006 epoxy resin, a room temperature curing system. 32 dog-bone SWNT/epoxy specimens with dimensions of 40 mm x 3.5 mm x 0.4-0.6mm were obtained. The gauge length of die specimens was about 15 mm, and the length of SWNT ropes embedded in the epoxy was about 20 mm. Details of the fabrication method can be foimd elsewhere. The volume fraction of SWNT ropes in the composite was controlled widiin die range of 0.1-0.9%. Specimens were cyclically tested by an Instron 8800 Microforce Tester under tension-tension at 5 Hz, using a sinusoidal wave function at R ratio (ratio of minimum to maximum cyclic stress) of 0.1. 

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