Summation operator

Practically all forms of neuron transfer functions include the summation operation, i.e., the sum of all inputs into the neuron (multiplied by their connection strength or weights) is calculated. In mathematical terms,... 

Another common name for the threshold value 0 is bias. The idea is that each neuron may have its own built-in bias term, independent of the input. One way of handling this pictorially and computationally is to add an extra unit to the input layer that always has a value of -1. Then the weight of the connections between this unit and the neurons in the next layer is the threshold or bias values for those neurons and the summation operation includes the bias term automatically. Then the summation formula becomes... 

After simplification by placing the summation operator with each term, one obtains... 

A summation operator of input signals, weighted by the respective synapses of the neuron. 

The application of the method of moments to the live polymer requires the application of the summation operator defined by Equation 12.24 on both sides of Equation 12.23, resulting in... 

A symbolic 0 appears on the left that does not actually represent the number 0, but a substance represented by a content formula in which all content numbers disappear. If we consider the substances to be numbered as discussed above, the expression can be written more compactly using the summation operator Y.-... 

The middle group represents the localization of the summation operator. Since this is also a 2-D operation, the localization form again looks similar. 

The following notation is used for the F elements R for reaction, S for separation, etc. The Boolean symbols c and u are used to denote sub-set and summation operations. The functionality of a module F is a sub-set of fimctions of the general set F, i.e. F /, c F (i = 7,r), where r is the number of modules (rank of the plant) that is defined by both the product and process envelopes. If N modules have the same functionality, Fu, then the functionality of the network of these modules is F = Fu- Thus the functionality of a network of three crystallisers (C/f) can be represented by the relation F=CR(jCRuCR. In terms of tasks, however, their number is 3. 

At this point, several physical conditions can be invoked to simplify the algebra. First, because gravity and friction are neglected in the present formulation, the Pij k factor can be moved across the summation operator since the pressure at any point within the well system is a constant. This constant is prescribed when the well is pressure-constrained but when the well is volume flow rate constrained, the unknown constant pressure level, whieh is different from well to well, must be found as part of the solution. 

Our wellbore model ignores friction and gravity for simplicity, so that pw can be moved across the summation operator for example, see Equations 15-12,13 and 15-29,30. Extend the finite difference model to include these effects. 

Incomputability generally refers to the question of whether or not a given function can be computed given a set of operators. So, for instance, given only the addition, subtraction, and summation operators, division cannot be computed. However, given... 

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