Convergence of the iteration requires the norm of the objective vector 1g to be less than the convergence criterion, e. The initial estimates used, if not provided externally, are, in addition to Equation (7-28)  [c.122]


When network weights have been trained to appropriate values, the NSC is ready to start classifying. The data set to be classified is specified in the same manner as previously used for the training and validation sets. The classifier is applied to data through the use of a menu and generates a list including filenames, suggested class and the neuron outputs from the output layer (used for decision). The result is currently presented in a simple text editor, from which it can be saved and included in other documents.  [c.107]

It can be shown [3], that the projection of the measurement signals on the orthogonal basis made of eigen vectors of the square matrix M-M proceeds a source separation if either the matrix T is orthogonal ( difficult to realize as this matrix is unknown ), either the ratios between the different r, f cr. ( t being the norm of the /-th column of T) are very large. An eddy current probe satisfies this last condition a) the surface noise has high energy, b) the higher the used frequency, the higher the e.m.f and the more sensitive to surface noise. We can enhance this with a large measurement axial coil, mainly sensitive to the surface noise.  [c.364]

Additional requirements have to be met, such as norm DIN 25450 for manual testing. This norm describes the requirements of transmitters, receivers and other parts of the system.  [c.856]

By defining a norm defining the distance between two signals, one can easily spot its variations. This distance provides an image of the signal evolution. Signals are typically Lissajous ( orbits ), i.e. arrays 2 of successive complex-valued points  [c.1025]

One prefers to modify this norm to  [c.1025]

Scattered data are available on the electrocapillary effect when liquid metal phases other than pure mercury are involved. Frumkin and co-workers [138] have reported on the electrocapillary curves of amalgams with less noble metals than mercury, such as thallium or cadmium. The general effect is to shift the maximum to the right, and Koenig [116] has discussed the thermodynamic treatment for the adsorption of the metal solute at the interface. Liquid gallium give curves similar to those for mercury, again shifted to the right this and other systems such as those involving molten salts as the electrolyte are reviewed by Delahay [139]. Narayan and Hackerman [140] studied adsorption at the In-Hg-electrolyte interface. Electrocapillary behavior may also be studied at the interface between two immiscible electrolyte solutions (see Ref. 141).  [c.202]

R. Narayan and N. Hackerman, J. Electrochem. Soc., 118, 1426 (1971).  [c.222]

As the D matrix is a diagonal matrix with a complex number of norm exponent of Eq. (65) has to fulfill the following quantization mle  [c.69]

One of the main outcomes of the analysis so far is that the topological matrix D, presented in Eq. (38), is identical to an adiabatic-to-diabatic transformation matrix calculated at the end point of a closed contour. From Eq. (38), it is noticed that D does not depend on any particular point along the contour but on the contour itself. Since the integration is carried out over the non-adiabatic coupling matrix, x, and since D has to be a diagonal matrix with numbers of norm 1 for any contour in configuration space, these two facts impose severe restrictions on the non-adiabatic coupling terms.  [c.652]

Next, we refer to the requirements to be fulfilled by the matrix D, namely, that it is diagonal and that it has the diagonal numbers that are of norm 1. In order for that to happen, the veetor-funetion t(i) has to fulfill along a given (closed) path F the condition  [c.653]

Next, we determine the conditions for this matrix to become diagonal (with numbers of norm 1 in the diagonal), which will happen if and only if when p and q fulfill the following relations  [c.656]

Figure 1 presents the three y angles as a function of (p for various values of p and Ag. The two main features that are of interest for the present study are (1) following a full cycle, all three angles in all situations obtain the values either of tc or of zero. (2) In each case (viz., for each set of p and Ag), following a full cycle, two angles become zero and one becomes n. From Eq. (81) notice that the A matrix is diagonal at cp = 0 and ip = 2tc but in the case of ip = 0 the matrix A is the unit matrix and in the second case it has two (—1) terms and one (-1-1) in its diagonal. Again recalling Eq. (39), this implies that the D matrix is indeed diagonal and has in its diagonal numbers of norm 1. However, the most interesting fact is that D is not the unit matrix. In other words, the adiabatic-to-diabatic transformation matrix presented in Eq. (81) is not single valued in configuration space although the corresponding diabatic potential matrix is single valued, by definition [see Eqs. (E.l) and (E.2)]. The fact that D has two (—1) tenns and one (-fl) in its diagonal implies that the present x matrix produces topological effects, as was explained in the last two paragraphs of Section IV.A Two electronic eigenfunctions flip sign upon tracing a closed path and one electronic function remains with its original sign.  [c.730]

Forward Analysis In this type of analysis, we are interested in the propagation of initial perturbations Sxq along the flow of (1), i.e., in the growth of the perturbations 5x t xo) = (xo -h Sxq) — xq. The condition number K,(t) may be defined as the worst case error propagation factor (cf. textbook [4]), so that, in first order perturbation analysis and with a suitable norm j  [c.99]

Here 1 is the force constant of the harmonic spring, - denotes the Euclidian norm of a vector in the functions K —> K, f = 1,2, are assumed to be smooth but arbitrary otherwise, and qr , f = 1,2, are two fixed reference vectors.  [c.286]

For realizing (10), we need an adequate norm for measuring the error. It obviously ma,kes no sense to ise an Euclidian norm of z indiscriminately of  [c.403]

Here tp denotes the conjugate transpose of ip. Another conserved quantity is the norm of the vector ip, i.e., ip ip = const, due to the unitary propagation of the quantum part.  [c.413]

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via  [c.422]

An object, s, represented by a set of descriptors = Xj, Xj,. ..x ) is mapped into a 2D arrangement of neurons each containing as many weights, Wji, as the object has descriptors. The neuron that obtains the object, and is the winning neuron, has weights most similar to the descriptors of the object. A competitive learning algorithm will then adjust the weights of the neurons to make them even more similar to the descriptor values of the object (see Section 9.4). Objects having similar descriptors will be mapped into the same or closely adjacent neurons, thus leading to clusters of similar objects.  [c.193]

Such step-limiting is often helpful because the direction of correction provided by the Newton-Raphson procedure, that is, the relative magnitudes of the elements of the vector J G, is very frequently more reliable than the magnitude of the correction (Naphtali, 1964). In application, t is initially set to 1, and remains at this value as long as the Newton-Raphson correotions serve to decrease the norm (magnitude) of G, that is, for  [c.116]

One could think of the standard norm to define the distance between two signals Z, and Ziacquired during two successive inspections  [c.1025]

Narayana C, Luo FI, Orloff J and Ruoff A L 1998 Solid hydrogen at 342 GPa no evidenoe for an alkali metal Nature 393 46  [c.1964]

Another type of invariance, namely, with respect to unitary or gauge hans-formation of the wave functions (without change of norm) is a cornerstone of modem physical theories [66], Such transformations can be global (i.e., coordinate independent) or local (coordinate dependent). Some of the observational aspects arising from gauge hansfomiation have caused some conhoversy for example, what is the effect of a gauge transformation on an observable [232,233], The justification for gauge invariance goes back to an argument due to Dirac [134], reformulated more recently in [234], which is based on the observability of the moduli of overlaps between different wave function, which then leads to a definite phase difference between any two coordinate values, the same for all wave functions. Erom this, Dirac goes on to deduce the invariance of Abelian systems under an arbitrary local phase change, but the same argument holds hue also for the local gauge invariance of non-Abelian, multicomponent cases [70].  [c.109]

The first handling of the R-T effect in a A electronic state of a triatomic was carried out experimentally and theoretically by Merer and Travis [73]. It concerned the state of CCN. These authors derived the second-order perturbative formulas for the combined effect of the weak vibronic and spin-orbit couplings. The splitting of the bending potential curves due to the R-T interaction was assumed to involve a single term being of fourth order in the bending coordinate on the other hand, the mean adiabatic potential was assumed to be harmonic. Curiously enough, also in this case the pertnrbative formulas printed in the original reference were not correct (caused by a trivial error of a factor of 4 concerning the norm of the basis functions used, see [12]).  [c.510]

Thus B is a diagonal mati ix that contains in its diagonal (complex) numbers whose norm is 1 (this derivation holds as long as the adiabatic potentials are nondegenerate along the path T). From Eq. (31), we obtain that the B-matrix hansfomis the A-matrix from its initial value to its final value while tracing a closed contour  [c.647]

The control scheme tries to choose the stepsize t so that 111 11 = TOL in some adequate norm. In case of a tolerance exceeding error, i.e., for Herll > TOL, one reduces the stepsize according to  [c.403]

PICKABACK conserves total energy up to small fluctuations and the norm of the vector ip exactly. Its main drawback is the step-size restriction which is of the order of the inverse of the largest eigenvalue of the quantum operator if (q). Thus, if the evaluation of the operator V q) and the gradients VqV q) and VqUd q) are expensive due to long-range interactions, then the PICKABACK scheme can become inefficient, i.e., the permitted step-size might be much smaller than required by the pure classical dynamics. To overcome this problem, symmetric integration schemes are considered in [16, 17] and [13].  [c.415]

Sending the entire dataset through this network leads to a distribution of the I 20 reactions acro.ss the 2D arrangement of neurons. The question is now, doe.s this distribution make sense Remember we have used an unsupeiwised learning method and therefore have not said anything about the membership of a reaction in a certain reaction type. In order to analyze the mapping, these 120 reactions were classified intellectually by a chemist and the neurons were patterned aecord-ing to the assignment of a reaction to a certain type (this was done a posteriori, after training of the network we still have unsupervised learning ). Figure 3-20a shows the map thus patterned. It can be seen that reactions considered by a chemist to belong to one and the same reaction type arc to be found in contiguous parts of the Kolionen map. This becomes even dearei when we pattern the empty neurons, those neurons that did not obtain a reaction, on the basis of their k nearest neighbors a neuron obtains a pattern by a majority decision of its nearest neighbors (figure 3-2Db.  [c.195]

See pages that mention the term Nyrim : [c.475]    [c.254]    [c.501]    [c.655]    [c.746]    [c.2573]    [c.2636]    [c.2637]    [c.2637]    [c.2845]    [c.3073]    [c.3074]    [c.614]    [c.647]    [c.655]    [c.679]    [c.291]    [c.398]    [c.404]    [c.417]    [c.428]    [c.211]   
Plastics materials (1999) -- [ c.505 , c.527 ]