Uploaded by amiruchka on 22.05.2010

Transcript:

Hello everyone!

In this video we consider a simple, and to some extents an efficient approach to Calculus.

Let's take a look at the familiar derivative.

Now, perhaps we can simplify it? Let's take a look!

We've just simplified the derivative.

Let's look at it once again. Here's the derivative, and what we do is -

- we quantize the term: dy, via the sign function.

Sir Isaac Newton once said: ..

Now, in his era, computational efficiency was less important, since there were no computers.

However, nowadays we can implement his hint - to simplify the derivative.

This is the definition of the "Detachment": The limit of the sign of dy.

The idea of the definition is simple. Let us observe the term dy = f(x+h) - f(x).

Let's observe this graph, of computational efficiency Vs. information level.

The familiar derivative holds an information regarding the tangent's slope, hence the information level it provides is high.

However, its computational efficiency is not optimal, in a sense that will be introduced shortly.

The detachment provides a very little amount of information,

however it uses no superfluous information when it comes to classify the monontony behavior of a function.

In terms of computational cost, the detachment outperforms the derivative, due to the following two reasons:

First, it spares the division operator that is found in the definition of the derivative, and uses the "sign" function instead,

which requires from the computer significantly less computation time.

Second, the number of possible values of the limit at the definition of the derivative is uncountable,

due to the fact that the term dy/dx is continuous,

while the number of possible values of the limit at at the definition of the detachment is finite,

only three possible values, that is to say: {0, +1, -1}.

In fact, quantization of the derivative has been used by computer scientists in the past few decades,

and the detachment is really just a rigorous formulation of this approach.

Suppose that our goal is classification of the monotony of a function in an interval,

that is to say, we wish to determine whether it is ascending, descending or constant in an interval.

The derivative uses too much information to answer that question:

it calculates the tangent's slope, when all we are interested in is the monotony type.

Hence, metaphorically speaking, the derivative resembles the truck in this video: surely it will reach its destination,

but perhaps we can settle for a lighter, and hence faster vehicle?

Indeed, in the same manner that the scooter bypasses the truck because it is smaller and faster,

so is the detachment more efficient than the derivative due to its simplicity,

assuming that our aim is classification of monotony.

The geometrical interpretation of the derivative is depicted in this figure (the slope of the tangent).

The geometrical interpretation of the detachment is as follows.

Given a point, we divide the plane into four quadrants,

using two axis aligned lines that intersect at the given point.

Now, to answer the question: "what is the function's detachment from, say right, at the point", we simply have to ask:

is there a small enough right neighborhood of the given point,

for whom the function's values are fully contained in one of the quadrants,

or maybe they are all on the line which is parallel to the x-axis.

That is to say: we ask is there a right neighborhood of the given point

where all the function's values are either higher, lower or equal to the value of the function at the given point.

In this example, we can see that the detachment from left is -1 and the detachment from right is +1.

Why was the name "detachment" chosen?

Let's look at the operation of the detachment from right on a given function.

As we will see shortly, the result is a step function,

Obtained by "tearing" the given function, at its local extrema points,

where the detachment from right (and in this case, also the derivative), changes its sign.

Here we see a suggestion to the Calculus structure, where the detachable functions are embedded into it.

In blue, colored are detachable functions, that is to say,functions whose right detachment equals their left detachment;

In brown colored are signposted detachable functions, that is to say, functions

whose right detachment is the additive inverse of their left detachment;

And in pink, colored are the null disdetachable functions, that is, functions whose right and left detachments both exist,

and one of them equals zero while the other isn't (so that they are neither detachable nor signposted detachable).

Due to its simplicity, the detachment is defined also for functions which are not differentiable,

and even not continuous, such as f_1.

Thus, the detachment forms a simple tool for analyzing the monotony behavior

of functions for whom the derivative fails to be defined.

Please take a moment and make sure that you agree with this suggestion to the Calculus structure.

Let us consider the research potential of the detachment.

In elementary Calculus, analogous versions to the familiar theorems,

that rely on the definition of the detachment, can be formulated.

In advanced Calculus and discrete geometry, the discrete versions of Green's and Stokes' theorems (known as the antiderivative formula)

can be formulated to rather more general domains,

using definitions that rely on the semi-discrete approach that the detachment suggests.

For more information regarding this research direction, feel free to contact me by mail: amir.shachar@mail.huji.ac.il

In computer application, such as edges detection in images, the detachment has in fact already been used,

since as we said, computer scientists have been using the idea of quantization of the derivative for decades,

due to the efficiency that it suggests.

The detachment depicts interesting arithmetical properties, whose nature is multiplicative.

These arithmetical properties can be further explored.

In more advanced research areas of classical analysis, the idea of the definition of the detachment, that is,

the combination between continuous math and discrete math,

allows us to classify the discontinuity of a function at a point.

Thus, new definitions and theorems are available in these research areas.

Note that generally, given a function from one metric space to another, the derivative is undefined,

because we cannot know for sure that it is well defined to divide dy by dx,

since these terms are calculated in different metric spaces.

However, quantization of dy is always well defined.

Is a function differentiable from both sides almost everywhere if and only if it is detachable from both sides almost everywhere?

This question and others are available for research.

Thank you so much for attending this lecture.

Feel free to contact me if you wish to cooperate in research: amir.shachar@mail.huji.ac.il

You are welcome to take a look at my other mathematical lectures regarding the theory.

Thanks. Yours with love, -amir.

In this video we consider a simple, and to some extents an efficient approach to Calculus.

Let's take a look at the familiar derivative.

Now, perhaps we can simplify it? Let's take a look!

We've just simplified the derivative.

Let's look at it once again. Here's the derivative, and what we do is -

- we quantize the term: dy, via the sign function.

Sir Isaac Newton once said: ..

Now, in his era, computational efficiency was less important, since there were no computers.

However, nowadays we can implement his hint - to simplify the derivative.

This is the definition of the "Detachment": The limit of the sign of dy.

The idea of the definition is simple. Let us observe the term dy = f(x+h) - f(x).

Let's observe this graph, of computational efficiency Vs. information level.

The familiar derivative holds an information regarding the tangent's slope, hence the information level it provides is high.

However, its computational efficiency is not optimal, in a sense that will be introduced shortly.

The detachment provides a very little amount of information,

however it uses no superfluous information when it comes to classify the monontony behavior of a function.

In terms of computational cost, the detachment outperforms the derivative, due to the following two reasons:

First, it spares the division operator that is found in the definition of the derivative, and uses the "sign" function instead,

which requires from the computer significantly less computation time.

Second, the number of possible values of the limit at the definition of the derivative is uncountable,

due to the fact that the term dy/dx is continuous,

while the number of possible values of the limit at at the definition of the detachment is finite,

only three possible values, that is to say: {0, +1, -1}.

In fact, quantization of the derivative has been used by computer scientists in the past few decades,

and the detachment is really just a rigorous formulation of this approach.

Suppose that our goal is classification of the monotony of a function in an interval,

that is to say, we wish to determine whether it is ascending, descending or constant in an interval.

The derivative uses too much information to answer that question:

it calculates the tangent's slope, when all we are interested in is the monotony type.

Hence, metaphorically speaking, the derivative resembles the truck in this video: surely it will reach its destination,

but perhaps we can settle for a lighter, and hence faster vehicle?

Indeed, in the same manner that the scooter bypasses the truck because it is smaller and faster,

so is the detachment more efficient than the derivative due to its simplicity,

assuming that our aim is classification of monotony.

The geometrical interpretation of the derivative is depicted in this figure (the slope of the tangent).

The geometrical interpretation of the detachment is as follows.

Given a point, we divide the plane into four quadrants,

using two axis aligned lines that intersect at the given point.

Now, to answer the question: "what is the function's detachment from, say right, at the point", we simply have to ask:

is there a small enough right neighborhood of the given point,

for whom the function's values are fully contained in one of the quadrants,

or maybe they are all on the line which is parallel to the x-axis.

That is to say: we ask is there a right neighborhood of the given point

where all the function's values are either higher, lower or equal to the value of the function at the given point.

In this example, we can see that the detachment from left is -1 and the detachment from right is +1.

Why was the name "detachment" chosen?

Let's look at the operation of the detachment from right on a given function.

As we will see shortly, the result is a step function,

Obtained by "tearing" the given function, at its local extrema points,

where the detachment from right (and in this case, also the derivative), changes its sign.

Here we see a suggestion to the Calculus structure, where the detachable functions are embedded into it.

In blue, colored are detachable functions, that is to say,functions whose right detachment equals their left detachment;

In brown colored are signposted detachable functions, that is to say, functions

whose right detachment is the additive inverse of their left detachment;

And in pink, colored are the null disdetachable functions, that is, functions whose right and left detachments both exist,

and one of them equals zero while the other isn't (so that they are neither detachable nor signposted detachable).

Due to its simplicity, the detachment is defined also for functions which are not differentiable,

and even not continuous, such as f_1.

Thus, the detachment forms a simple tool for analyzing the monotony behavior

of functions for whom the derivative fails to be defined.

Please take a moment and make sure that you agree with this suggestion to the Calculus structure.

Let us consider the research potential of the detachment.

In elementary Calculus, analogous versions to the familiar theorems,

that rely on the definition of the detachment, can be formulated.

In advanced Calculus and discrete geometry, the discrete versions of Green's and Stokes' theorems (known as the antiderivative formula)

can be formulated to rather more general domains,

using definitions that rely on the semi-discrete approach that the detachment suggests.

For more information regarding this research direction, feel free to contact me by mail: amir.shachar@mail.huji.ac.il

In computer application, such as edges detection in images, the detachment has in fact already been used,

since as we said, computer scientists have been using the idea of quantization of the derivative for decades,

due to the efficiency that it suggests.

The detachment depicts interesting arithmetical properties, whose nature is multiplicative.

These arithmetical properties can be further explored.

In more advanced research areas of classical analysis, the idea of the definition of the detachment, that is,

the combination between continuous math and discrete math,

allows us to classify the discontinuity of a function at a point.

Thus, new definitions and theorems are available in these research areas.

Note that generally, given a function from one metric space to another, the derivative is undefined,

because we cannot know for sure that it is well defined to divide dy by dx,

since these terms are calculated in different metric spaces.

However, quantization of dy is always well defined.

Is a function differentiable from both sides almost everywhere if and only if it is detachable from both sides almost everywhere?

This question and others are available for research.

Thank you so much for attending this lecture.

Feel free to contact me if you wish to cooperate in research: amir.shachar@mail.huji.ac.il

You are welcome to take a look at my other mathematical lectures regarding the theory.

Thanks. Yours with love, -amir.