• The Detachment Operator •

Uploaded by amiruchka on 02.08.2010

Hello everyone,
today we will depict a semi-discrete approach in single variable calculus,
and show how it is related to the integral image algorithm.
Here's my name, and let's begin.
Let us recall that the formulation of this version of the discrete Green's theorem,
which is an extension of the integral image algorithm.
We won't get to the details of the theorem,
but we only remind that the parameter alpha_D is uniquely determined according to the corner's type.
Hence, a proper definition of alpha_D requires a tool for classification of corners in the plane,
and this is what we will try to do in the following slides.
Let us consider two corners in the plane, and let gamma be the curve that forms these corners.
A straight forward classification of the corners would include an inquiry of the curve's one-sided derivatives
at the point, where the curve has a corner.
But perhaps there's a simpler way?
Notice for example that whether the one-sided derivative of the function x that forms the curve equals 5 or 3 at the point,
it won't matter to the corner's type.
Hence, we can settle for less information, and inquire the SIGN of the one-sided derivatives.
But perhaps we can settle for less information than that?
Why calculate the tangent's slope if all we're interested in is its sign?
So now we will try to simplify the term of the sign of the derivative.
Let f be a function, and let us consider the term of "sign of the right derivative of f".
Let us detail the definition of the derivative.
Now since the sign function is discontinuous at zero,
then if we insert the sign function into the limit process,
Then these terms are not always equal.
Now, due to the multiplicative nature of the sign function,
and since h is always positive inside the limit process,
we can omit the denominator.
We call this term: "the function's detachment from right".
Note that it is not always equal to the sign of the function's derivative,
and as we will show, due to its simplicity, it is defined for a broader set of functions than the differentiable functions.
So now, instead of talking of the signs of the one-sided derivatives at the point,
a more natural way would be to settle with exactly the amount of information we're interested in,
and inquire the one-sided detachments.
Let us now analyze the definition of pointwise operators.
Given the term dy,
If we immediately take the limit, then the result is a non-informative operator,
since it is always zeroed if f is continuous.
the usual derivative is highly informative,
and the detachment is somewhere in between:
the information level it supplies is lower than "the tangent's slope",
but still it is capable of supplying an interesting information regarding the monotony behavior of the function.
Note that the computational cost of the detachment is preferable with respect to the computational cost of continuous operators.
A detailed discussion is held at the paper.
Sir Isaac Newton once said, that nature does nothing in vain, and affects not the pomp of superfluous information.
Indeed, the detachment operator is a simplified version of the derivative.
Here we see the geometric interpretation of the one-sided detachments.
An intuitive thumb-rule is: the detachment at a point from, say right, is say +1,
if and only if the point a one-sided minimum of the function from right.
The name "detachment" was chosen due to the fact that applying a one-sided version of this operator to a function
results with a step function that is torn, or "detached",
at the extrema points of the original function.
There are many differentiable functions,
and even more continuous functions.
Via the definition of the detachment, we can talk of:
signposted detachable,
and null disdetachable functions,
which may be even not continuous.
Here we see a function that demonstrates these pointwise properties.
Watch the red point change on the graph,
while the pink disk below indicates which of the pointwise properties hold at the point.
For example, at -5 the function is detachable and not continuous.
Make sure you understand it.
At -4, the function is signposted detachable, continuous and not differentiable.
And so on;
You can observe this demonstration at Wolfram Demonstrations project, just look for "Amir Finkelstein Wolfram".
Since the detachment and signposted detachment operators are not defined for null disdetachable functions,
it is required to define a different operator, that is defined whenever a function is left and right detachable.
This operator is named "tendency".
It is zeroed whenever a function is detachable,
because detachability occurs only at extrema points, where we would like the tendency to be zeroed,
just like the derivative.
In any other case the tendency is simply the detachment from right,
to "predict" the function's monotonic behavior.
In many cases, tendency agrees with the sign of the derivative,
however it is defined also for functions which are not even continuous.
Hence it can be thought of as an extension to the sign of the derivative.
Using it, we can build analogous theorems to those of single-variable calculus.
For example,
Let us recall Lagrange's mean value theorem.
And now we will see its semi-discrete analog..
Notice the similarity between the theorems!
We won't get to the details of the theorem in this demonstration,
but let us just remark that the theorem provides less information regarding the function,
and in return, it holds for a broader set of functions.
Let us compare between the theorems.
Both the mean value theorems require continuity in the closed interval.
The original version requires differentiability in the open interval,
while the semi-discrete analog requires tendability there,
that is, that the function is detachable from both sides in the interval.
This condition can be thought of as a weaker condition.
The statements of the theorem are similar,
they both resemble the definition of the pointwise operators from which they are derived.
While the original version guarantees the existence of one point for which the theorem's statement holds,
the semi-discrete version guarantees the existence of uncountably many points.
Let us review the structure of theorems in single variable calculus.
In this lecture we saw in what sense the discrete Green's theorem's formulation is derived from the detachment
(because classification of corners is most naturally performed using one-sided detachments of the curve).
We also gave an example to a theorem which is analog to Lagrange's theorem.
You can review other analogous theorems at the pre-print.
In the next lecture we will use the definition of the detachment to form a new semi-discrete integration method in the plane,
which in turn will enable to naturally extend the discrete Green's theorem to more general types of domains.
Thank you for attending this lecture.
Here's my mail address, feel free to mail me in case you have a question or if you wish to cooperate in research.
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Good-bye, stay in touch!