Einstein's Diffusion and Probability-Wave Equations of Random ...
Einstein's Diffusion and Probability-Wave Equations of Random ...
Einstein's Diffusion and Probability-Wave Equations of Random ...
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
Consequently, there are countably many real numbers in the<br />
interval<br />
[,] ab<br />
, <strong>and</strong> the Integration Sum has countably many<br />
terms.<br />
While we do not sequence the real numbers in the interval,<br />
the summation takes place over countably many f ( xdx. )<br />
The Lower Integral is the Integration Sum where f ( x ) is<br />
replaced<br />
by its lowest value on each interval<br />
3.2<br />
∑<br />
x∈[ a, b]<br />
⎛<br />
⎜⎝<br />
dx dx<br />
2 2<br />
[ x − , x + ]<br />
⎞<br />
inf f ( t)<br />
dx<br />
⎠⎟<br />
x− ≤t≤ x+<br />
dx dx<br />
2 2<br />
The Upper Integral is the Integration Sum where f ( x ) is<br />
replaced by its largest value on each interval<br />
3.3<br />
∑<br />
x∈[ a, b]<br />
⎛<br />
⎜⎝<br />
⎞ sup f ( t)<br />
dx<br />
⎠⎟<br />
dx dx<br />
2 2<br />
x− ≤t≤ x+<br />
[ x − , x + ]<br />
dx dx<br />
2 2<br />
If the integral is a finite hyper-real, we have<br />
14
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