Matrix memory

Figure 3.3 The errors produced by an iterative reconstruction algorithm have patterns which appear, at each iteration, completely random (A), but if successive patterns are memorised together, it is possible to observe regular structures appearing in the memory matrix (B).

The construction of the memory matrix is performed by taking as a starting point a totally blank matrix [m 0 = 0], and by applying the following operations ... 

The combination of a totally constrained algorithm with equations 3.7 of the memory matrix allows us to build, at each iteration, two very different matrices the structure matrix where the reconstruction appears, and the memory matrix where the parameters of the illegal values are gradually accumulated. 

If the distribution of these values were totally random, the memory matrix would tend to remain uniform, but in reality its behaviour is much more complex than that. At many points the illegal values do... 

Another important result is obtained by applying this method to pictures of many different kinds, because it has been noticed that the space distribution of the vortices is picture-dependent. The vortices pattern does not depend therefore on general characteristics of the algorithm, but on specific properties of the examined picture. It is as if a picture had a specific image in the memory space exactly as it has one in the real space. This brings us immediately to the following question Is it possible to use the information that appears in the memory matrix to improve the reconstruction in the structure matrix ... 

The hypotheses that were made about density modulation, therefore, are valid a memory matrix does allow us to obtain new information about the structure that we are reconstructing, and we can progressively move towards the point where a complete reconstruction becomes possible. 

This result is interesting because it focuses our attention on the individual features of the memory matrix. If only vortices are memorized, it is obvious that the algorithm performs better with pictures that have a high potential number of vortices, but if other features could be memorized, it would become possible to reduce substantially the unknowns even with grey pictures. We have therefore the problem of discovering if other features exist which allow us to... 

At this point we are left with the problem of discovering yet more memory matrices, and here we have plenty of suggestions. It is plausible, for example, that a memory of boundaries, or more generally a memory of discontinuities, could be built, but we can leave these developments to the future. We have seen that the memory matrix method can indeed perform reconstructions from incomplete information, and therefore we already have what we were looking for a model that may help us understand the logic of embryonic development. 

In the MRM model, the initial memory matrix is a tabula rasa, a white page that is gradually filled during the reconstruction process, while the reconstructed picture starts with a uniform image, and becomes progressively differentiated in the course of time. A reconstruction with the MRM model, in other words, is a set of two distinct reconstructions that are performed in parallel. The point is that this double reconstruction is necessary for reasons that are absolutely general. 

In the case of individual structures, a reconstruction matrix can receive new information only from its individual memory matrix, but in the case of multicellular structures there is also another option. Here it is possible to build a collective memory matrix, and this allows us to choose between two different reconstruction strategies. We can continue to adopt an individual approach, where each cell gets new information only from its individual memory matrix (Figure 7.6A), but we can also adopt an approach where,/row a certain point onwards, a cell can also receive new information from the collective memory of the system (Figure 7.6B). 

Figure 7.6 The reconstruction of multicellular structures can be performed by using only individual memory matrices (A), or by also exploiting a collective memory matrix (B).

The iterative algorithms that have been proposed for the reconstruction of structures from insufficient information differ from all other methods because they perform in parallel two distinct reconstructions one for the structure matrix, and one for the so-called memory matrix, i.e. for a matrix where any convenient feature can be stored. This is why these algorithms are collectively referred to as the Memory Reconstruction Method (MRM). 

The information that appears in the memory space cannot be transferred automatically to the structure space, and can be used only by employing specific conventions (the recognition of vortices in the memory matrix, for example, can be used only if a convention gives a meaning to the corresponding points of the structure matrix). This is another conclusion that leads to a universal principle, because it is necessarily valid for all systems. 

Tektronix 4051 Microcomputer w/16k memory Additional 16K memory Matrix function ROM pack Joystick input device... 

Integration of Eq. (11.4.1) then gives the relaxation matrix r in terms of the memory matrix K(f) as... 

Of great practical interest is the possibility of miniaturizing these devices and of integrating their thin film deposition and photolithographic processing technique with that of conventional silicon diodes. As an example. Figure 6.14 shows a part of a 256-bit memory matrix on a 3 X 3 mm Si chip (Neale, Nelson and Moore (1970) Neale and Aseltine (1973)). 

Note that the time evolution of experimental observables zl (t) is governed by real dynamics which is determined according to Eq. 83 by the real propagator exp fit. The situation with the time evolution of the stochastic force Fn t) and the memory matrix Knk t) is much more complicated. According to Eq. 85, their evolution is governed by projected dynamics the propagator of which is given by exp iQLQt. ... 

Note that the time evolution of the generahzed stochastic Langevin force F (T f t) and the memory matrix are governed by so-called projected dy-... 

The analytical form of the memory matrix (f t — t) given in Eq. 101 is much less clear. The memory matrix (t t — t) contains all the (unknown) information about intermolecular interactions, i.e., all consequences of entanglement effects. The fluctuating part of the intermolecular force F t) acting on the n segment of the tagged chain can be expressed by matrix density fluctuations around this segment ... 

On this basis the memory matrix can be expressed as an integral over the time-dependent autocorrelation function of matrix density fluctuations for projected dynamics ... 

Taking all approximations together, the memory matrix can be expressed... 

From an empirical point of view it is well known that the Rouse model describes satisfactorily the dynamical properties of polymer melts with Nfast decaying Rouse contribution from the total memory matrix given by Eq. 102 and rewrite Eq. 100 as... 

Renormalization in this context means an attempt to find a physically plausible ansatz for the unknown memory matrix given by Eq. 106. In principle one could postulate a power law with an exponent being a fitting parameter to experimental data after having derived expressions of observables on this basis. However, in the frame of the renormalized Rouse models (RRM) a somewhat less formal and less phenomenological approach is possible. 

The preaveraging approximation given in Eq. 107 is again used. The memory matrix then becomes a scalar relative to space rotations, and does not depend on the momentary configuration of the tagged chain ... 

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