The homogenized equation has a unique solution in [H.sup.2.sub.0] ([OMEGA]), so every

subsequence of ([u.sub.n]) converges to the same limit u and this implies that the entire sequence ([u.sub.n]) converges to u.

(1) If y and z are limit points of ([x.sub.n]), then there exist

subsequences [mathematical expression not reproducible].

(b) [P'.sub.i] holds for some boundedly spaced

subsequence [mathematical expression not reproducible] of {[T.sup.n]}

Moreover, the DTW algorithm-based approach provides the advantage of searching all matched

subsequences automatically from a long sequence.

If {[[??].sub.n]} is a sequence in a bounded soft closed set S, then there exists a

subsequence [mathematical expression not reproducible] of {[[??].sub.n]} converging to some [??] [member of] S.

Section 3 introduces our approach for completing large missing

subsequences in low/uncorrelated multivariate time series.

Assume, up to a

subsequence, that [mathematical expression not reproducible].

(ii) {[x.sub.2n+1]} has a convergent

subsequence in V;

Setting [v.sup.[k]] = [u.sup.[k]] / [parallel] [u.sup.[k]] [parallel] and arguing as in the proof of Claim 1, we conclude that, possibly passing to a

subsequence,

We utilize LSTM network itself without additional model to complete

subsequence selection.

The traditionally LCS method measures the similarity by calculating the length of the longest common

subsequence. Given two strings [S.sub.1] and [S.sub.2] with length I and J respectively, [S.sub.1](i) and [S.sub.2](j) are the [i.sup.th] member of the string [S.sub.1] and the [j.sup.th] member of [S.sub.2] respectively, and L(i, j) represents the value of the LCS matrix at the location of (i, j).

A traversal sequence S' is said to be a

subsequence of sequence S if it is a subset of S.