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Lemma 7. The relation between the area size of the ring sector Ti and that of T1 is Si = (2i 1) S1 . De nition 15.5.3. Let i the probability that an event will occur on the ring sector Ti . There are n ring sectors in the network. Lemma 8. Assuming a random uniform generation of events in the network area, the probability i of an event occurring on the ring sector Ti (1 i n) is i = (2i 1) n2



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The code at IL_000e starts out by loading two values onto the evaluation stack: the value of local variable 0, which was just initialized earlier to 1, and the constant 10 Then these two values are compared using the bles instruction This is a branch if lower or equal instruction that does both the comparing and the actual jumping, unlike IA-32 code, which requires two instructions, one for comparison and another for the actual branching The CLR compares the second value on the stack with the one currently at the top, so that lower or equal means that the branch will be taken if the value at local variable 0 is lower than or equal to 10.





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Let us now consider sector Ti . De nition 15.5.4. An area Ti handles an event generated in ring sector j if either the message was generated in the area Ti (i.e. j = i) or the message was propagated to Ti from the ring sector Ti+1 . De nition 15.5.5. Let hi be the number of the messages that are handled by the area Ti . We now de ne energy

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Since you happen to know that the local variable has just been loaded with the value 1, you know for certain that this branch is going to be taken at least on the first time this code is executed Finally, it is important to remember that in order for bles to evaluate the arguments passed to it, they must be popped out of the stack This is true for pretty much every instruction in IL that takes arguments through the evaluation stack those arguments are no longer going to be in the stack when the instruction completes Assuming that the branch is taken, execution proceeds at IL_0004, where the routine calls WriteLine, which is a part of the NET class library Write Line displays a line of text in the console window of console-mode applications.

De nition 15.5.6. Let ij a random variable which measures the energy that dissipates the sector Ti so as to handle the message j. For ij we have that = cR2 c(iR)2 with probability pi with probability 1 pi

The function is receiving a single parameter, which is the value of our local variable As you would expect, the parameter is passed using the evaluation stack One thing that s worth mentioning is that the code is passing an integer to this function, which prints text If you look at the line from where this call is made, you will see the following: void [mscorlib]System Console::WriteLine(int32) This is the prototype of the specific function being called Notice that the parameter it takes is an int32, not a string as you would expect Like many other functions in the class library, WriteLine is overloaded and has quite a few different versions that can take strings, integers, floats, and so on In this particular case, the version being called is the int32 version just as in C++, the automated selection of the correct overloaded version was done by the compiler.

where f is written in bold to distinguish it from the notation used for a single-argument function f . The values of f can be of the sigmoid or Gaussian type.

where cR2 is the energy dissipation for sending a message j from Ti to its adjacent ring sector Ti 1 and c is a constant. Thus, the expected energy dissipation in sector i for handling a message is E[

After calling WriteLine, the routine again loads two values onto the stack: the local variable and the constant 1 This is followed by an invocation of the add instruction, which adds two values from the evaluation stack and writes the result back into it So, the code is adding 1 to the local variable and saving the result back into it (in line IL_000d) This brings you back to IL_000e, which is where you left off before when you started looking at this loop Clearly, this is a very simple routine All it does is loop between IL_0004 and IL_0011 and print the current value of the counter It will stop once the counter value is greater than 10 (remember the conditional branch from lines IL_000e through IL_0011) Not very challenging, but it certainly demonstrates a little bit about how IL works..

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The McCulloch Pitts neuron uses the scalar product of the input vector and its weight vector [10]. The extension of this model to the geometric neuron requires the substitution of the scalar product with the Clifford or geometric product, i.e. wT x + wx + = w x + w x + (21.26)

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