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Fourier Analysis is the Cell theory of True Calculus #43 Uni-text 8th



 
 
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  #1  
Old November 17th 13, 07:58 PM posted to sci.astro
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Default Fourier Analysis is the Cell theory of True Calculus #43 Uni-text 8th

Alright, this really belongs in Advanced Calculus (graduate school math) but here I am just going to mention some facts and details, for I never want to frighten students, in that math requires age to learn it. We cannot learn to read a clock at too young of an age but at a prime age, we pick it up almost instantly. Same thing for most of mathematics.

Now I must not excuse myself for being rather slow in realizing what the general formula other than the Power formula with its consequence of Chain rule, was for True Calculus. The general formula for all derivatives and integrals of functions is the Fourier Analysis method. All functions, I mean all, can be made into sine and cosine summations. In True Calculus all functions are continuous because of the gap or hole between finite numbers allows all functions to be continuous and when we divide by 0 we provide the function a patch over that division so as to force the function continuous.

So why was I slow in recognizing the Cell theory had a general method, more universal than the Power formula with it's consequence of the Chain Rule? Why was I slow to realize that the dy and the dx is naturally sine and cosine and that my new method of deriving the general formula of derivative and integral would be a Fourier transform? Why was I slow? I was slow because I always feel that pure geometry of lines and angles-- Power formula of y = x, is more basic and primal than is the trigonometry functions which build upon pure geometry. Somehow I refused to rise above pure geometry of the triangle and go to the levels higher of trigonometry to find that Universal Generalized Method of deriving the derivative and integral.

But now that I am here, and happy, let me just tell what the Fourier Analysis proof involves in proving that great theorem that all functions are summations of sine and cosine wavelets. For in the Fourier Analysis starts with Fourier Series and the series is built upon a *interval* and that interval is a uniform interval such as what Wikipedia describes as [x0, x0 + P]. In other words, Fourier Analysis is built upon the Cell theory of Calculus.

The Cell theory is built from the fact that finite and infinity require a borderline and that produces a **interval** of microinfinity, the holes and gaps between successive finite numbers. So when we fetch the macroinfinity border, we cause the existence of the microinfinity border and that creates uniform small intervals. That allows the existence of the Fourier Series of its interval.

Now out of curiosity, I wanted to see if any of my authors of Stewart, Strang, Ellis & Gulick, Fisher & Ziebur discuss the Fourier Analysis. I would have guessed the answer beforehand to be absolutely no.
But to my surprise, Stewart mentions the Fourier series on page 489 in his 2003, 5th ed. book. Strang goes really strong on the Fourier Analysis and let me quote his page 291 of his 1991 text Calculus:

--- quoting Strang page 291 ---
This is not for the sake of making up new problems. I believe these are the most important definite integrals in this chapter ( p and q are 0, 1, 2, ....). They may be the most important in all of mathematics, especially because the question has such a beautiful answer. The integrals are zero. On that fact rests the success of Fourier series, and the whole industry of signal processing.
--- end quote ---

Strang is skewed in his feelings about those integrals because Strang is in Old Math with its pollution of the phony limit concept and thus, nothing is really pretty or beautiful when surrounded by pollution.

What is beautiful is that the Fourier Analysis is all coming out of the Cell theory.

--
Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium
  #2  
Old November 18th 13, 12:10 AM posted to sci.astro
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Default generalized derivative and integral formula in True Calculus #44



Alright, in Old Math their only generalized formula was from the Power formula which gives rise to the Chain Rule. But in New Math, in True Calculus governed by the Cell theory and not the phony limit concept, we have a new means, a new method of fetching the generalized formula of both the derivative and integral starting from any given function. For what we apply is the Fourier Analysis, since the Fourier Analysis is a subset of the Cell theory..

Now in 10 Grid the smallest interval possible is 0.1 so that is the smallest cell and the cells in 10 Grid start with 0 to 0.1 then .1 to .2 then .2 to .3 etc etc.

So we plot the graph of the function in cell 1 then cell 2 then cell 3. Usually the plot will be a picketfence inside the cell of a right-triangle atop a rectangle. Now the derivative is the hypotenuse of the right triangle and the integral is the area under the hypotenuse.

So now, instead of applying the Power formula or the Chain Rule if possible, we apply instead a new method of Fourier Analysis. The derivative is after all dy/dx and so the sine function is the dy and the dx is the cosine function. So in the 1st cell we obtain a sine and a cosine for the Fourier transform and do the same in the 2nd and 3rd cell. Now the integral is a bit more difficult because we have to obtain the area under the hypotenuse first and then translate that area into a sine and cosine function.

Now, if we see the plot of the overall starting function is smooth overall, then we need not do many cells to know what the Fourier function for the derivative and integral are. If the plot of the starting function is altering in various regions, we have to section off the function and find the sine and cosine of the cells in those altering regions.

Now let me give some definitions of different starting functions. Let me define those smooth and patterned functions as Smooth Patterned functions which have the same shape all along its path. Such functions are the line functions and the power functions of y = x^2 or y = x^3 or y = 1/x or even sine and cosine.

Contrast that with a function that is not smooth patterned such as y = (3x^2-x-2)/(5x^2 + 4x +1) as shown in Stewart's text on page 139, where the function has regions of patterned by also has abrupt changes. Another function that has an abrupt change is the absolute value of a function such as y = |x| where it has a corner or kink.

What I want the reader to be aware of is that in the Fourier transform of building the derivative and integral, we need only a few cells to determine the overall general formula provided the function is smooth and patterned. If the function has abrupt changes in a region, then we have to work out more cells for the sine and cosine in those changed regions.

AP
  #3  
Old November 18th 13, 07:29 AM posted to sci.astro
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Default some functions written as a Fourier series #44.1 Uni-text 8th ed.:TRUE CALCULUS without the phony limit concept



Now I was hoping in Youtube, that someone would introduce a student new to the Fourier Synthesis, or whatever you want to call it-- analysis, transform, series. Show the student how to make a common simple function of y = x into a string of sine, cosine wavelets. Or take the function y = x^2 and make it a string of sine and cosine wavelets.

I found no such luck on any Google search or Youtube.

The best I could do was mathworld.wolfram.com/FourierSeries.html

one that website shows some common functions of the square wave and the triangle wave. Then lower on the page the Fourier Series of those functions are given as:

4/pi Summation1,3,5,... 1/n sin(n*pi*x/L) for the square wave

and

8/pi^2 Summation 1,3,5,... ((-1)^(n-1)/2)/n^2 sin (n*pi*x/L)

So, really, the Fourier theory is not adequate for College or University until graduate school. Perhaps a bit of information and facts about Fourier theory before they reach graduate school.

The points I want to stress is that the Fourier theory covers every function, which means every function (all functions are continuous in cell theory) can be written as a combination of sine and cosine. That is important because each cell allows us to find the derivative and integral of that cell, and so in True Calculus we thus have a means of finding the general formula of all derivatives and integrals of any given function. We no longer have just the Power formula and Chain Rule and various other restricted techniques, but we have a universal technique that covers every function of mathematics to fetch its general derivative and general integral.

AP
  #4  
Old November 18th 13, 07:05 PM posted to sci.astro
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Default common functions written as a Fourier series #44.2 Uni-text 8th ed.:TRUE CALCULUS without the phony limit concept

On Monday, November 18, 2013 12:29:46 AM UTC-6, wrote:
Now I was hoping in Youtube, that someone would introduce a student new to the Fourier Synthesis, or whatever you want to call it-- analysis, transform, series. Show the student how to make a common simple function of y = x into a string of sine, cosine wavelets. Or take the function y = x^2 and make it a string of sine and cosine wavelets.



I found no such luck on any Google search or Youtube.



The best I could do was mathworld.wolfram.com/FourierSeries.html



one that website shows some common functions of the square wave and the triangle wave. Then lower on the page the Fourier Series of those functions are given as:



4/pi Summation1,3,5,... 1/n sin(n*pi*x/L) for the square wave



and



8/pi^2 Summation 1,3,5,... ((-1)^(n-1)/2)/n^2 sin (n*pi*x/L)



So, really, the Fourier theory is not adequate for College or University until graduate school. Perhaps a bit of information and facts about Fourier theory before they reach graduate school.



The points I want to stress is that the Fourier theory covers every function, which means every function (all functions are continuous in cell theory) can be written as a combination of sine and cosine. That is important because each cell allows us to find the derivative and integral of that cell, and so in True Calculus we thus have a means of finding the general formula of all derivatives and integrals of any given function. We no longer have just the Power formula and Chain Rule and various other restricted techniques, but we have a universal technique that covers every function of mathematics to fetch its general derivative and general integral.



AP


common functions written as a Fourier series #44.2 Uni-text 8th ed.: TRUE CALCULUS without the phony limit concept

Now Jeffrey Chasnov on Youtube gives another take on the triangle function as an even function involving cosine, rather than MathWorld's odd function involving sine.

https://www.youtube.com/watch?v=edwG9x5v3Xo

And the triangle function here becomes f(x) = 1/2(a0) + Summation a_n cos(nx)

So that is far more easy than MathWorld's sine of triangle function.

Now let us see if the Cell theory can improve the Fourier theory, since the Cell theory is more basic.

AP


  #5  
Old November 18th 13, 08:43 PM posted to sci.astro
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Default Improvements of the Fourier theory #71 Math-Professor-text 8th ed.:TRUE CALCULUS

Improvements of the Fourier theory #71 Math-Professor-text 8th ed.: TRUE CALCULUS

This subtext is to teach college and university professors of mathematics what the real and true Fourier theory is all about.

Now many people reading my posts realize I scold math professors a lot. And the reason I do that is because they want to ignore their mistakes and ignore a person like me who wants to improve their math and make their math correct. So when you are a college or university professor of mathematics and teaching contradictory garbage like a dy/dx derivative where the limit makes dx go to 0 and division by 0; or when you teach that the integral is the summation of thin rectangles, for which the limit has forced out all the interior area of those rectangles so that the width of integral summations are 0 width, again, you as a college or university math professor is teaching contradictory garbage in your classroom.

When you teach a limit concept that is wholly irrelevant in finding the derivative or integral, then you are wasting the time and time of the life of students.

When you teach math that there is a infinity just beyond finite, yet you never in all your math career ever have the wit to find a **border between finite and where infinity starts** then you failed at logic and math and you failed in every piece of math that involves finite versus infinity.

So math professors have a lot of strikes against them. But I am willing to write this 10 page subtext of True Calculus just for the audience of college and university Math Professors. Even though they are short of wit of logic as per finite versus infinite and who have been "dolts of mathematics" in accepting dy/dx where dx =0 and accepting rectangles without interior area since the limit took away all the interior area.

You see, math professors would rather keep teaching garbage than correct their mistakes and teach the real true math, because they are embarrassed or simply the change is too much for them.

But I will not ignore the math professors and keep on scolding them until they do change for the better.

So I write this Fourier Improvement text. It is meant not for graduate students in college but meant for college math professors.

If you ignore me, I scold you, but if you jump in and participate, I praise you for your wit. That is my motto for writing this Fourier text. College professors of math can improve the Fourier theory, if they jump in and participate, but if they hide away and ignore these posts, well, that leaves me only to scold the entire community of college math professors.

Now this is only going to be 10 pages long and makes the 5th subtext of the whole text called True Calculus.

First off, I need to throw out the miserable terminology given to the Fourier theory. Such miserable and vague terms as that of Fourier series, Fourier analysis, Fourier transform, Fourier synthesis. Making up terms in science is not science itself but a clutter mess to hide the fact that you do not know much about the subject itself. So we dispense with this terminology nonsense and call it just simply Fourier theory.

Now the only reason I can write this subtext is because of the Cell theory which is far more general and universal over the Fourier theory. The Fourier theory is a subset of the Cell theory.

Now what gives rise to the Cell theory is the resolution of the border between finite and infinite. In the 10 Grid system where we pretend that 10 is the last and largest finite number and any number beyond is an infinite number causes a chain reaction in that the smallest nonzero finite number becomes 0.1. This number, 0.1 in 10 Grid is microinfinity whereas 10 is macroinfinity. What that does is tell us that the only numbers of mathematics in 10 Grid are those numbers of 0, .1, .2, .3 on up to 9.9 then finally 10. And any other number is an infinity number for which mathematics has no role for infinity numbers other than recognize that they take up Space between finite numbers.

Mathematics is only about Finite numbers because only finite numbers can be made precise and that is the definition of mathematics is equal to precision. Mathematics breaksdown when it involves itself into infinity numbers.

Now the true border of finite versus infinity is easily found, and in 2011 using the pseudosphere with associated sphere areas it is seen that the areas become exactly equal at Floor-pi*10^603. That number is huge and well beyond most every Physics numbers, except for perhaps magnetic monopoles. So the difference between a 10 Grid and a 10^603 Grid is a huge difference. In fact, our TV and computer screens work on just a simple 100 or 1000 Grid system where a circle looks like a perfect circle when it appears on our computer in the 100 Grid system.

So, the Fourier theory was discovered in the early 1800s and it had a large flaw, in that it went to infinity yet no-one bothered to fetch an infinity borderline. (An interesting side note is that Fourier is noted for the discovery of the Greenhouse gas effect that warms Earth.)

So what happens with the Fourier theory when you apply a finite to infinity border?

Well, a lot happens, and a lot changes.

What happens is that you can provide a Fourier Coordinate System that overlaps the Cartesian Coordinate System. In the Fourier Coordinate system every finite point is a sine and cosine component.

Now take a look at this website page

http://jwilson.coe.uga.edu/EMT668/EM...E/COSINE~1.HTM
y = cos x
jwilson.coe.uga.edu/EMT668/.../Dickerson/.../COSINE/COSINE~1.HTM

Laura Dickerson. To examine the graph of y = cos x, I will examine y = A cos (Bx +C) for different values of A, B, and C. ... Let's us first look at the graph y = cos x.


Now, take a look at the bottom of that web page where Laura graphs y = cos (1/3 x).

Can you see a trend there between cos x , cos 1/2 x
and then cos 1/3x. That the graph becomes more and more flat almost approaching a line, a straightline.

So, the question is in 10 Grid all in the 1st Quadrant only, what is the function y = cos (.1x)? Is it a straight line or where does it become a straightline so that x=0, y=1 and x=10, y=1? Do I need to have y = cos 0.01x in order to achieve a straightline in 10 Grid by using cosine function?

When you have a border between finite and infinite which causes a Grid System of finite points separated by holes and gaps of microinfinity, causes a major change in the Fourier theory, so that the Fourier theory becomes an alternative Coordinate System to the Cartesian Coordinate System.

--
Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium
  #6  
Old November 18th 13, 11:21 PM posted to sci.astro
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Default straightline of y = 1 becomes y = cos(10^-3x) in Fourier theory #71.1

On Monday, November 18, 2013 1:43:09 PM UTC-6, wrote:
Improvements of the Fourier theory #71 Math-Professor-text 8th ed.: TRUE CALCULUS



This subtext is to teach college and university professors of mathematics what the real and true Fourier theory is all about.



Now many people reading my posts realize I scold math professors a lot. And the reason I do that is because they want to ignore their mistakes and ignore a person like me who wants to improve their math and make their math correct. So when you are a college or university professor of mathematics and teaching contradictory garbage like a dy/dx derivative where the limit makes dx go to 0 and division by 0; or when you teach that the integral is the summation of thin rectangles, for which the limit has forced out all the interior area of those rectangles so that the width of integral summations are 0 width, again, you as a college or university math professor is teaching contradictory garbage in your classroom.



When you teach a limit concept that is wholly irrelevant in finding the derivative or integral, then you are wasting the time and time of the life of students.



When you teach math that there is a infinity just beyond finite, yet you never in all your math career ever have the wit to find a **border between finite and where infinity starts** then you failed at logic and math and you failed in every piece of math that involves finite versus infinity.



So math professors have a lot of strikes against them. But I am willing to write this 10 page subtext of True Calculus just for the audience of college and university Math Professors. Even though they are short of wit of logic as per finite versus infinite and who have been "dolts of mathematics" in accepting dy/dx where dx =0 and accepting rectangles without interior area since the limit took away all the interior area.



You see, math professors would rather keep teaching garbage than correct their mistakes and teach the real true math, because they are embarrassed or simply the change is too much for them.



But I will not ignore the math professors and keep on scolding them until they do change for the better.



So I write this Fourier Improvement text. It is meant not for graduate students in college but meant for college math professors.



If you ignore me, I scold you, but if you jump in and participate, I praise you for your wit. That is my motto for writing this Fourier text. College professors of math can improve the Fourier theory, if they jump in and participate, but if they hide away and ignore these posts, well, that leaves me only to scold the entire community of college math professors.



Now this is only going to be 10 pages long and makes the 5th subtext of the whole text called True Calculus.



First off, I need to throw out the miserable terminology given to the Fourier theory. Such miserable and vague terms as that of Fourier series, Fourier analysis, Fourier transform, Fourier synthesis. Making up terms in science is not science itself but a clutter mess to hide the fact that you do not know much about the subject itself. So we dispense with this terminology nonsense and call it just simply Fourier theory.



Now the only reason I can write this subtext is because of the Cell theory which is far more general and universal over the Fourier theory. The Fourier theory is a subset of the Cell theory.



Now what gives rise to the Cell theory is the resolution of the border between finite and infinite. In the 10 Grid system where we pretend that 10 is the last and largest finite number and any number beyond is an infinite number causes a chain reaction in that the smallest nonzero finite number becomes 0.1. This number, 0.1 in 10 Grid is microinfinity whereas 10 is macroinfinity. What that does is tell us that the only numbers of mathematics in 10 Grid are those numbers of 0, .1, .2, .3 on up to 9.9 then finally 10. And any other number is an infinity number for which mathematics has no role for infinity numbers other than recognize that they take up Space between finite numbers.



Mathematics is only about Finite numbers because only finite numbers can be made precise and that is the definition of mathematics is equal to precision. Mathematics breaksdown when it involves itself into infinity numbers.



Now the true border of finite versus infinity is easily found, and in 2011 using the pseudosphere with associated sphere areas it is seen that the areas become exactly equal at Floor-pi*10^603. That number is huge and well beyond most every Physics numbers, except for perhaps magnetic monopoles. So the difference between a 10 Grid and a 10^603 Grid is a huge difference. In fact, our TV and computer screens work on just a simple 100 or 1000 Grid system where a circle looks like a perfect circle when it appears on our computer in the 100 Grid system.



So, the Fourier theory was discovered in the early 1800s and it had a large flaw, in that it went to infinity yet no-one bothered to fetch an infinity borderline. (An interesting side note is that Fourier is noted for the discovery of the Greenhouse gas effect that warms Earth.)



So what happens with the Fourier theory when you apply a finite to infinity border?



Well, a lot happens, and a lot changes.



What happens is that you can provide a Fourier Coordinate System that overlaps the Cartesian Coordinate System. In the Fourier Coordinate system every finite point is a sine and cosine component.



Now take a look at this website page



http://jwilson.coe.uga.edu/EMT668/EM...E/COSINE~1.HTM

y = cos x

jwilson.coe.uga.edu/EMT668/.../Dickerson/.../COSINE/COSINE~1.HTM



Laura Dickerson. To examine the graph of y = cos x, I will examine y = A cos (Bx +C) for different values of A, B, and C. ... Let's us first look at the graph y = cos x.





Now, take a look at the bottom of that web page where Laura graphs y = cos (1/3 x).



Can you see a trend there between cos x , cos 1/2 x

and then cos 1/3x. That the graph becomes more and more flat almost approaching a line, a straightline.



So, the question is in 10 Grid all in the 1st Quadrant only, what is the function y = cos (.1x)? Is it a straight line or where does it become a straightline so that x=0, y=1 and x=10, y=1? Do I need to have y = cos 0.01x in order to achieve a straightline in 10 Grid by using cosine function?



When you have a border between finite and infinite which causes a Grid System of finite points separated by holes and gaps of microinfinity, causes a major change in the Fourier theory, so that the Fourier theory becomes an alternative Coordinate System to the Cartesian Coordinate System.





Now I did this only by eyeball so it must be checked and where I typed into a Google search of cos(.0001x) and up appears a graph where the cosine intersects the x-axis at about 20,000. But that is not our concern. Our concern is where does the cosine form a straightline that is equal to y = 1 in the 10 Grid? Which means the tip of the top of those crests form a straight enough line to be from 0 to 10.

Now at cos(.01x) I am not sure that the cosine forms that straight line so that we have x=0, y=1 and x=10, y =1 in 10 Grid. In 10 Grid, remember we truncate for the only numbers that exist are increments of 0.1.

So the question is, in 10 Grid what is the cosine Fourier equation for the line y = 1? And the answer appears to be that of y = cos(0.001x).

Now I must pause here, because it is likely possible to have a different sine and cosine combination to deliver a y = 1 function, a better one than the above of y = cos(0.001x).

So one may begin to see the entire game plan here. That I want to convert every function of the Cartesian Coordinate System into a equation of only sine and cosine.

What is the function y = x in Fourier theory?

Now most professors of math are very skeptical of any changes in what they learned or memorized learned of mathematics. So right away they raise red flags instead of a desire to "learn and learn something new and better." So right away they will complain why bother with converting functions to that of sine and cosine strings? And the answer is simple and easy. Because once a function is made into a pure sine and cosine string, its general formula for derivative and integral are immediately obtained. And why is that true? Because every function is built from numerous cells and the derivative and integral inside each cell is dependent on the sine and cosine of the right triangle in that cell. So if we know a function by pure sine and cosine string, then we know its general formula for derivative and integral. We no longer have to make up "tricks of the trade" to find the derivative and integral, we just simply write the function and it reveals its derivative and integral because it was a part of the function.

AP

  #7  
Old November 19th 13, 08:23 AM posted to sci.astro
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Default straightline of y = 0 becomes y = sin(0.1x) in Fourier theory #71.2

straightline of y = 0 becomes y = sin(0.1x) in Fourier theory #71.2 Math-Professor-text 8th ed.: TRUE CALCULUS

Alright, sorry about this for all readers concerned, for I made some rusty mistakes. I should have opened the calculator and done some calculations rather than just write up a page post coming purely out of my head.

So I made many pitiful mistakes in the prior two posts.

I need the sine function not the cosine function and I need to start at y = 0 and not y =1 to convert into a trigonometry function. That function would be y = sin(0.1x) which would be the x-axis straightline in 10 Grid

And to get to y=1, I make the adjustment of 2sin(0.1x)

Now I looked at the calculator for values of sin0.1 and they are .001 so that would mean for 10 Grid a value of 0. Now when x=10 we have 1/10 of 10 = 1 and the sin of 1 is .01 which would still be a 0 in 10 Grid.

So, I seem to have some good news and some bad news. The bad news is how rusty I am on trigonometry.

The good news is that in each Grid, the sine and cosine function have sufficient small numbers to cover the straightlines of y=0 or y=1 or y =5 etc etc.

Cosine was not the proper function to make for the straightline conversions to Fourier theory because at x=0 the cosine drops down to .99 and in 10 Grid where we truncate, would mean .9 which is not a continuation of 1 for y =0. So I had to drop to the sine function where the x value always delivered me a y value of 0.01 which in truncation is 0 in 10 Grid.

So the good news is that I need not go for infinite numbers inside a Grid to make straightlines in Fourier theory.

But there is a stickler on this. Since the function y = x, the identity function is longer in length than y =0 or y = 1, the question is whether the Fourier function can recover y = x without having to go to a infinity number?

So here is a question for the college and university math professors, for whom this text is written for, and please excuse the mistakes.

We have Grid systems. The smallest is 10 Grid which has only 100 finite numbers on x-axis and has 100x100 in the entire 1st quadrant Grid. Can I deliver every function (with truncation) as a Fourier function of only sine and cosine and how much must I use infinity numbers of the next higher Grids?

So that if I use 100 and 1000 Grid to plot y coordinate points in 10 Grid, can I deliver every function as a sine with cosine Fourier function?

And is there some unique pattern? That is to say, if the 100 Grid requires infinity numbers of 1000 and 10000 Grid in order to deliver all the functions in the 100 Grid?

What I am attempting to do is to make every function as a string of pure sine and cosines from this pyramid of Grids:

10
100
1000
10000
etc etc

If I can, then sometime in the distant future, in all the sciences, functions will be written for the most part as strings of sines with cosines, and rarely will anyone write a function like that of y = (x^2 +9x +8)/(x^2 -8x +9) but rather as strings of just sine and cosine.

Why go through that bother? Because all strings of sine and cosine are instantly differentiable and instantly integrable.

AP
  #8  
Old November 19th 13, 08:44 AM posted to sci.astro
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Default converting all functions of math into Fourier theory #71.3

converting all functions of math into Fourier theory #71.3 Math-Professor-text 8th ed.: TRUE CALCULUS



And to get to y=1, I make the adjustment of 2sin(0.1x)


Now the amplitude is too large in 2sin(0.1x) for that is y = 2 straightline in 10 Grid. So I need some more terms to drop the function down to where it is y = 1. And I run into another problem when I want y = 10 as sine and cosine strings, in that the 10sin(0.1x) at the value of x=10 becomes 10.1 for y value.

So I am rather sure that each Grid system such as the 10 Grid needs to employ y values of the 100 Grid in order to convert all the functions of 10 Grid into that of sine and cosine of Fourier theory.

AP
  #9  
Old November 19th 13, 08:06 PM posted to sci.astro
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Default Is the 10 Grid a Galois Algebra Field? #72 Math-Professor-text 8th

Is the 10 Grid a Galois Algebra Field #72 Math-Professor-text 8th ed.: TRUE CALCULUS

Alright, sorry about this rocky start of the professor's text of improving the Fourier theory so that it is all part and parcel of the Cell theory.

I have enough to start with the overall general conversion of all functions..

To get the straightline of y =0, our familiar x-axis line we have the Fourier function of y = sin(0.1x) for 10 Grid. But I could just as well made y = sin(0.01x) the y = 0. And for y=1 I could use the y = C + cos(0.1x) *where C is an increment of adding 0.1 to that of .9. Now as for the identity function y = x, the natural choice would be the tangent function of sin(x) / cos(x) for it is a identity in a small region around 0 and 1 and to exploit that to be from 0 to 10 for 10 Grid. But the tangent is not a Fourier of sine and cosine add or subtract. So we may have to exploit the sine function to be converted into the identity function in 10 Grid.

In 10 Grid the only numbers that exist are 0, .1, .2, on up to 9.8 then 9.9 then 10 so 100 numbers along the x-axis and y-axis and the grid is 100x100 = 10000 coordinate points in all. Now I used only 1st quadrant but I relax that requirement and use all 4 quadrants in this improvement of the Fourier theory.

The Grid systems use truncation since they are built from a borderline of finite with infinity numbers.

But it is alright to use 100 and 1000 Grid numbers for the 10 Grid. The subject of mathematics finds the final Grid to be 10^603, so as we use 100 and 1000 Grid numbers into the 10 Grid is just an increase in our precision.

Now I found out earlier in this book, that I need to use infinity numbers in each grid by using higher level grids, so that I could form low lying angles. By using numbers of 100 Grid and 1000 Grid for y-axis, I could get angles smaller than 45 degrees in the 10 Grid.

So I made a concession that I could use numbers of the 100 and 1000 Grid upon the 10 Grid for the y-axis. And it is not really much of a concession at all because 10^603 is the real true borderline of Infinity with finite. It is at 10^603 that we cannot use and borrow 10^604 or 10^605.

So here is a question for the professors of math. Does the Grid system with its microinfinity and macroinfinity, require the use and borrowing of 100 and 1000 Grid numbers within the 10 Grid form a pattern and is this use making the 10 Grid a Galois Algebra Field?

So in these first two pages of 10 pages of the Professor text, I ask two important questions.

(1) Can we convert all functions of mathematics, using 4 quadrants and using 10 Grid with borrowing from 100 and 1000 Grid, can we convert all functions to be Fourier functions?

(2) And if (1) is true, then is there a pattern such that all these Grid Systems form a Galois Algebra Field?

AP
  #10  
Old November 20th 13, 07:57 AM posted to sci.astro
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Default is the conversion process to Fourier theory a unique one? #73



Now in this professor of math text, I ask questions in each of the 10 pages, questions that I want the professors to solve.

In the first page I asked whether every function of the Grid System:

10 Grid
100 Grid
10^3 Grid
..
..
10^603 Grid

Where each grid has a pretend macroinfinity which gives a microinfinity, except the 10^603 Grid which is actually the true border between finite and infinity as the pseudosphere versus sphere surface area proves a crossover.

So we have truncation but we also are allowed borrowing of higher grids into lower grids, for example we borrow 100 and 1000 Grid numbers allowed to be used in 10 Grid, even though, strictly speaking above 10 is macroinfinity and smaller than 0.1 (except 0) are infinity numbers.

I have written over 70 pages explaining the Grid system and Cell theory for anyone wanting more information.

So the first question is: does the Fourier theory as embodied by the Bernoulli formula:

y = a_1 sin (pi*x/L) cos (pi*c*t/L) + a_2 sin (pi*x/L) cos (pi*c*t/L) + . .

capture every function that exists as a sine and cosine building blocks? Can every function graphed in the Grid systems be captured by a sine and cosine formula?

The question in the second page was whether if true about every function being represented as a Fourier function, whether that formed a Galois finite field?
Now let me make a side remark here because in Galois Finite Fields or Groups, there is a mystery requirement for it to work and is assumed that 1/0 = infinity and that 1/infinity = 0.

Now in True Calculus where we find the actual border between finite and infinity, we can dispense with that requirement, because in say 10 Grid infinity is any variety of number such as 11, or .001 which are just two infinity numbers within the 10 Grid.

Now this is the 3rd page of this professor of math text and the question I ask is whether that Bernoulli formula of sine and cosine is a unique Fourier representation of every function. I want to know if a unique process of converting every function to a sine and cosine in the Grid System exists. Unique in having to use two level higher grids, or one level higher grid to complete the conversion. So in this 3rd page, I am asking for whether a unique conversion process exists? My guess is yes, that a unique process exists with a unique formula. All because the Grids stop at 1*10^603.

--
Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

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