By Leonid A Dickey

Dickey (mathematics, U. of Oklahoma) offers a close description of solitons, that have a variety of purposes in mechanics and physics. the hot version includes numerous additions and alterations together with dialogue of the Zakharov-Shabat matrix hierarchy with rational dependence on a spectral parameter, and its courting to isomonodromic deformations. total, the emphasis of the second one variation is on hierarchies of integrable equations instead of person equations. the second one variation additionally includes a new preface via the writer. Chapters contain Hamiltonian constitution of the GD hierarchies, Kupershmidt-Wilson Theorem and transformed KdV and GD, and tau services of matrix hierarchies.

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BJ ], we can express the squared Euclidean cost function as J(a1 , a1 , . . , aJ , b1 , b2 , . . , bJ ) = = 1 ||Y − ABT ||2F 2 1 ||Y − 2 J aj bTj ||2F . 50b) for j = 1, 2, . . , J subject to a ≥ 0 and b ≥ 0, respectively. 1 Large-scale NMF In many applications, especially in dimension reduction applications the data matrix Y ∈ RI×T can be very large (with millions of entries), but it can be approximately factorized using a rather smaller number of nonnegative components (J), that is, J << I and J << T .

Consider a mapping yt → φ(yt ) or Y → φ(Y) = [φ(y1 ), φ(y2 ), . . , φ(yT )], then the kernel NMF can be defined as φ(Y) ∼ = φ(Y) W BT . 37) which depends only on the kernel K = φT (Y)φ(Y). 12 Convolutive NMF The Convolutive NMF (CNMF) is a natural extension and generalization of the standard NMF. In the Convolutive NMF, we process a set of nonnegative matrices or patterns which are horizontally shifted (or time delayed) versions of the primary matrix X [126]. 38) p=0 where Y ∈ RI×T is a given input data matrix, Ap ∈ RI×J is a set of unknown nonnegative basis matrices, + + p→ 0→ X = X ∈ RJ×T is a matrix representing primary sources or patterns, X is a shifted by p columns version of + p→ X.

P − 1) knowing only the input data matrix Y ∈ RI×T . Each operator Sp = T1 (p = 1, 2, . . , P − 1) performs a horizontal shift of the columns in X by one spot. 13 Overlapping NMF In Convolutive NMF we perform horizontal shift of the columns of the matrix X. In some applications, such as in spectrogram decomposition, we need to perform different transformations by shifting vertically the rows of the matrix X. 39) → ←p Tp is the horizontal-shift matrix operator such that X = X T p and X = X T p .