By Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, Esteban G. Tabak
We regularly ponder our typical setting as being composed of very many interacting debris, present process person chaotic motions, of which merely very coarse averages are perceptible at scales common to us. besides the fact that, shall we in addition examine the realm as being made from person waves. this is often so not only as the contrast among waves and debris turns into relatively blurred on the atomic point, but additionally simply because even phenomena at a lot greater scales are larger defined when it comes to waves instead of of debris. it's infrequent in either fluids and solids to monitor strength being carried from one zone of area to a different by way of a given set of fabric debris; even more frequently, this move happens via chains of debris, neither of them relocating a lot, yet each one speaking with the following, and therefore developing those immaterial items we name waves.Waves happen at many spatial and temporal scales. lots of those waves have sufficiently small amplitude that they are often nearly defined via linear idea. in spite of the fact that, the joint influence of enormous units of waves is ruled through nonlinear interactions that are liable for large cascades of strength between very disparate scales. realizing those strength transfers is important so as to ensure the reaction of enormous structures, resembling the ambience and the sea, to exterior forcings and dissipation mechanisms which act on scales a long time aside. the sector of wave turbulence makes an attempt to appreciate the common habit of huge ensembles of waves, subjected to compelling and dissipation at contrary ends in their spectrum. It does so by way of learning person mechanisms for power move, similar to resonant triads and quartets, and trying to draw from them results that are meant to now not live to tell the tale averaging.This e-book offers the complaints of the AMS-IMS-SIAM Joint summer time examine convention on Dispersive Wave Turbulence held at Mt. Holyoke collage (MA). It drew jointly a bunch of researchers from many corners of the realm, within the context of a perceived renaissance of the sector, pushed through heated debate concerning the primary mechanism of strength move between huge units of waves, in addition to by means of novel applications-and previous ones revisited-to the certainty of the flora and fauna. those court cases mirror the spirit that permeated the convention, that of pleasant medical confrontation and actual ask yourself on the wealthy phenomenology of waves
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Additional info for Advances in Wave Interaction and Turbulence: Proceedings of an Ams-Ims-Siam Joint Summer Research Conference on Dispersive Wave Turbulence, Mount ... MA, June 11-15, 20
YT ] ∈ RI×T is defined as y = vec(Y) = yT1 , yT2 , . . , yTT T ∈ RIT . 68) The vec-operator applied on a matrix Y stacks its columns into a vector. The reshape is a reverse function to vectorization which converts a vector to a matrix. For example, reshape(y, I, T ) ∈ RI×T is defined as (using MATLAB notations and similar to the reshape MATLAB function): reshape(y, I, T ) = y(1 : I), y(I + 1 : 2I), . . , y((T − 1)I : IT ) ∈ RI×T . 69) Analogously, we define the vectorization of a tensor Y as a vectorization of the associated mode-1 unfolded matrix Y(1) .
5 Bilinear NMF model. The nonnegative data matrix Y ∈ RI×T is approximately represented by + a sum or linear combination of rank-one nonnegative matrices Y(j) = aj ◦ bj = aj bTj ∈ RI×T + . where the symbol ◦ denotes the outer product of two vectors. Thus, we can build an approximate representation of the nonnegative data matrix Y as a sum of rank-one nonnegative matrices aj bTj . , E = 0) then it is called the Nonnegative Rank Factorization (NRF) . Among the many possible series representations of data matrix Y by nonnegative rank-one matrices, the smallest integer J for which such a nonnegative rank-one series representation is attained is called the nonnegative rank of the nonnegative matrix Y and it is denoted by rank+ (Y).
AJ ] as follows: A ← ADA , where −1 −1 DA = diag(||a1 ||−1 p , ||a2 ||p , . . , ||aJ ||p ), p ∈ [0, ∞). , when the columns of A are normalized to unit 1 -norm. 1) helps to alleviate many numerical difficulties, like numerical instabilities or ill-conditioning, however, it makes searching for the global minimum more complicated. Moreover, to avoid rotational ambiguity of NMF, the rows of X should be sparse or zero-grounded. To achieve this we may apply some preprocessing, sparsification, or filtering of the input data.