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highegg
science forum beginner

Joined: 11 Feb 2005
Posts: 5

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Proof of the fundamental theorem of Calculus (FTOC)

On 21 Nov 03 06:09:04 -0500 (EST), John Gabriel wrote:
 Quote: Your web page is interesting but I think there is a far easier way to prove not only the FTOC but every important calculus theorem - my average tangent theorem not only defines and proves the mean value theorem for the first time since Isaac Newton, but it proves both parts of the Ftoc in such simple mathematics so that even a high school student can understand it. It also proves finite differences, Taylor's Theorem, Quadrature/Cubature, Arc Lengths, Newton's Approximation Formula and much more. I invite you to check out: a href="http://www.geocities.com/john_gabriel">http://www.geocities.com/john_gabriel

Hi,
your ATT proof looks interesting, but I don't understand what simple
argument makes you think that you may exchange the summation and
limit, even when the summation contains increasing number of terms
(depending on n).

highegg
highegg
science forum beginner

Joined: 11 Feb 2005
Posts: 5

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Proof of the fundamental theorem of Calculus (FTOC)

On 21 Nov 03 06:09:04 -0500 (EST), John Gabriel wrote:
 Quote: Your web page is interesting but I think there is a far easier way to prove not only the FTOC but every important calculus theorem - my average tangent theorem not only defines and proves the mean value theorem for the first time since Isaac Newton, but it proves both parts of the Ftoc in such simple mathematics so that even a high school student can understand it. It also proves finite differences, Taylor's Theorem, Quadrature/Cubature, Arc Lengths, Newton's Approximation Formula and much more. I invite you to check out: a href="http://www.geocities.com/john_gabriel">http://www.geocities.com/john_gabriel

Hi John,

function f(x)=x^2 sin(1/x), f(0)=0,
which is differentiable everywhere (not continuously),
but the righthand side in your theorem doesn't converge to the secant

Jaroslav Hajek
Jason

Joined: 24 Mar 2005
Posts: 72

 Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Proof of the fundamental theorem of Calculus (FTOC) Jaroslav, You are incorrect! John states that it has to be continuous everywhere in the interval. Jason.
Lee
science forum beginner

Joined: 10 Jun 2005
Posts: 6

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: I need help in writing a recursion formula for a series.

Feb 15
Hello i need the same help with this formula

On 9 Feb 05 15:34:42 -0500 (EST), monica wrote:
 Quote: On 3 Feb 05 16:54:04 -0500 (EST), Amber wrote: I too need to know how to do this. On 17 Oct 04 00:23:07 -0400 (EDT), alethia wrote: a = 1 n / n 2 Please help, I do not understand how to write this formula into a recrusion formula. Thank you ***I too need help, did you ever find the formual for this? if so, can you show me... thanks!
Luis A. Afonso
science forum beginner

Joined: 24 Mar 2005
Posts: 1

 Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: I need help in writing a recursion formula for a series. u_n= 1 / (2^n) u_(n+1)= 1/ ( 2^(n+1)) u_(n+1)/ u_n = [1 / (2^n)] / [1/ ( 2^(n+1))] = [2^n] / [2^(n+1)] = 1/2 u_(n+1)= u_n / 2 u_0 = 1 / (2^0)=1 / 1 = 1
Antti Lange , lange@fkf.n
science forum beginner

Joined: 24 Mar 2005
Posts: 1

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: bordered block diagonal (bbd) matrix

Hi,
(3) of
Yours,
Antti
www.fkf.net

On Thu, 20 Jan 2005 10:19:47 -0600, Ben Gu wrote:
 Quote: Hi, i am doing some work on bordered block diagonal (bbd) matrix. Does anyone know any good books talking about this subject? Thanks a lot! Ben
maertens roger
science forum beginner

Joined: 24 Mar 2005
Posts: 1

 Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Pascals triangle Where can i find a summary of all the proven properties of the numbers of the pascals triangle. I need a property of the SUM OF TWO NUMBERS IN THE SAME COLUMN of the Pascals triangle. example: In the first column (1,1,1,1,...), the sum of two numbers equals 2. There is no number 2 in that column. So, in the first column, there are no solutions In the second column (1,2,3,4,....), the som of two numbers is another number included in that column So, in the second column, there are infinite solutions In the third column (1,3,6,10,15,21,..), the som of the two numbers 6 + 15 = 21 also is a member of that column. But it seems that there is only one. So, in the third column, there seems to be only one solution In the other colums, there seems to be no solutions. WHO CAN PROVE MATHEMATICALLY THERE ARE NO SULUTIONS FOR COLUMN FOUR AND FURHER ON? I need this prove to solve a practical problem. Thanks by Roger Maertens Roger Koeiendreef, 13 8310 Assebroek, Brugge Belgium 0032 (0)50 35 62 41 Who
Dave Rusin
science forum Guru

Joined: 25 Mar 2005
Posts: 487

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Pascals triangle

I have set follow-ups to sci.math since this isn't Numerical Analysis.

In article <9i6kvlu44aav@legacy>,
maertens roger <maertens.roger@skynet.be> wrote:
 Quote: Where can i find a summary of all the proven properties of the numbers of the pascals triangle.

Too many to summarize.

 Quote: I need a property of the SUM OF TWO NUMBERS IN THE SAME COLUMN of the Pascals triangle. example: In the first column (1,1,1,1,...), the sum of two numbers equals 2. There is no number 2 in that column. So, in the first column, there are no solutions In the second column (1,2,3,4,....), the som of two numbers is another number included in that column So, in the second column, there are infinite solutions In the third column (1,3,6,10,15,21,..), the som of the two numbers 6 + 15 = 21 also is a member of that column. But it seems that there is only one. So, in the third column, there seems to be only one solution

You didn't look very far, e.g. 55=45+10. More generally, you need the
solutions to (2n-1)^2 - 1 = (2m-1)^2 - (2k-1)^2 , which is to say you need
factorizations of x^2-1 subject to some parity constraints. There are
many solutions.

 Quote: In the other colums, there seems to be no solutions.

That may be; I didn't check. I note that in some contexts the products
n(n-1)(n-2)...(n-k+1) play a role similar to the powers n^k, so
the question you are asking is in some sense similar to Fermat's Last
Theorem, except that it lacks the homogeneity that make FLT pretty.
So if there is no obvious proof that all the solutions to
C(x,k) + C(y,k) = C(z,k) are the trivial ones, then any proof valid
for every k is likely to be very hard.

But of course this is pure speculation on my part; I didn't give it
more than a moment's consideration.

dave
bv

Joined: 16 May 2005
Posts: 59

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Efficient internal values calculation of Bulirsch-Stoer type integrator

guy wrote:
 Quote: I found it potentially good at some of my cases, anyhow the others include real second order kinetics such as in the system of two equations: dydt[0] = C1-C2*y[0]+C3*y[1]+C4*y[0]*y[1]; // C1,C2,C3,C4>0 dydt[1] = C5*y[0]-C3*y[1]-C4*y[0]*y[1]; // C5>0 Where I could not find a way of converting it into the form d\vec{y}/dt = A*\vec{y} + \vec{u}

It may not be obvious at a first glance, however DEs often need a bit of
"coaxing" into a right shape. In your case, either

- solve as linearized DEs: d/dt y = J*y, where J is system Jacobian
- solve as linear DEs with nonlinearities as input: d/dt y = Ay + u(y)

puts "expokit" squarely back in business.
Paul Fackler
science forum beginner

Joined: 24 Mar 2005
Posts: 2

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: matrix cookbook - new version

Kaare Brandt Petersen wrote:
 Quote: Dear Colleagues (Apollogies for multiple postings) A new and updated version of The Matrix Cookbook is available for download http://www.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf The Matrix Cookbook is a desktop reference on formulas regarding matrices such as the derivatives of determinants and traces, identities involving inverses, statistical moments and more. Apart from smaller additions and corrections, this version of The Matrix Cookbook has been updated with material on derivatives of complex matrices and matrix norm inequalities. This is largely due to the work of Michael Syskind Pedersen who has joined the project as author. Comments and corrections are most welcome. Best regards, Kaare -- Kaare Brandt Petersen * http://2302.dk

Hi Karre
A nice document. You may find some notes of mine of interest

http://www4.ncsu.edu/~pfackler/MatCalc.pdf

I take a somewhat different approach than you do in defining derivatives
but one that it makes it easy to solve fairly complicated matrix
derivatives.

Cheers
Paul
Gottfried Helms
science forum Guru

Joined: 24 Mar 2005
Posts: 301

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: matrix cookbook - new version

Am 24.02.05 08:38 schrieb Paul Fackler:
 Quote: Hi Karre A nice document. You may find some notes of mine of interest http://www4.ncsu.edu/~pfackler/MatCalc.pdf I take a somewhat different approach than you do in defining derivatives but one that it makes it easy to solve fairly complicated matrix derivatives. Hi Paul -

just took a short look into your text.
It reminds me to some operations, that I have done recently, which
look like a lengthy notation for a 3-D-matrix concept, which could
then be more concise, if elaborated.

For instance, computing the eigenvectors of a matrix having the
eigenvalues, (applying a vandermonde-like eigenvalue-matrix, which
I derived recently) seems to be expressible in 3-d-matrix-notation
much more concisely. The vec- and the kronecker-operator seem to
map the common 2-d-matrices into a 3-d-(notation-) space.

Have you seen a notation for 3-D-matrices anywhere?

Gottfried Helms
Guest

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: C++ lib for Complex Matrix Algebra

help involving complex libraries.
i have installed lapack++2.1.2 ,the latest distribution as available on
sourceforge.net.
but i dont know how to turn on complex types.
it says " Note: To switch on the support for complex-valued matrices,
you need to define the macro LA_COMPLEX_SUPPORT in your application "

the non-trivial question is about a suitable library for my
requirements.
*calculating pseudoinverses for complex matrices
*calculating eigenvalues and their corresponding LEFT eigenvectors
for a GENERALISED eigenvalue problem involving complex matrices.
i require the final code to be compiled in gcc on a linux platform.
it would be nice if you can tell me as to which of the following
packages would suit my requirements.
1. lapack++
2. MTL
3. PETSc and SLEPc

Mumit Khan wrote:
 Quote: In article <46bots\$s2@harbinger.cc.monash.edu.au>, Kai O'Yang wrote: pecora@zoltar.nrl.navy.mil (Lou Pecora) writes: There is a LaPack++, not sure of source. Check out: Netlib: http://netlib@att.com Math lib: http://math.jpl.nasa.gov, /start/html Just a note, you can't use LaPack++ 1.0 with gcc/g++ 2.7.0. The libg++ that comes with gcc 2.7.0 changed the format of complex numbers to templates and LaPack++ didn't like it at all. Have to use 2.6.3 for LaPack++ . (The authors know about it and suggested me to do so.) Kai A slight nit-pick: libg++-2.7.0 did not "change" the format of the complex numbers, rather it tries to adhere to the draft C++ standard and uses a templatized version of complex (eg., complex which is what most of the previous libraries used as complex). The changes to lapack++ to make it work with gcc-2.7.0 and libg++-2.7.0a is trivial: simple write a script that changes all the complex to complex and you're almost there. Can't remember if I had to also fix any of the new for-init-statement scope rule, but that's also trivial. I do remember that I didn't spend more than 20 minutes on it. Anybody who wants a copy of the shell script, please drop me a line; it's quite simple, but works. I actually have my code such that it works with cfront-based compilers that come with the older complex numbers as well as the newer ones such as libg++, and I do it by typedef'ing COMPLEX to be either complex (draft standard) or Complex (deprecated cfront-based and most other compilers probably). regards, mumit -- khan@xraylith.wisc.edu http://www.xraylith.wisc.edu/~khan/
Wendy E. McCaughrin
science forum beginner

Joined: 18 Jul 2005
Posts: 4

 Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: I need help in writing a recursion formula for a series. The OP did not indicate "^". For even n, the recurrence is: a(0) = 0, a(n) = a(n-2) + 1 for n>0. "Luis A. Afonso" wrote: : u_n= 1 / (2^n) : u_(n+1)= 1/ ( 2^(n+1)) : : u_(n+1)/ u_n = [1 / (2^n)] / [1/ ( 2^(n+1))] : : = [2^n] / [2^(n+1)] = 1/2 : u_(n+1)= u_n / 2 : u_0 = 1 / (2^0)=1 / 1 = 1
bv

Joined: 16 May 2005
Posts: 59

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: Re: Is there any: Simulation based parameter estimation method?

Neon wrote:
 Quote: How do you develop parameter estimation for complex system with some output observations, wich can only be simulated other than modeled with close form formula?

It's done routinely since few, if any, real world problems come with a
close form solution, except, homework problems stupefied for Matlab type
sw usage. It's a subject of active research, and is also (hopefully) at
the very core of modern cae/cad toolsets. Search for "inverse problems"
Jason Wells
science forum beginner

Joined: 24 Mar 2005
Posts: 1

Posted: Thu Mar 24, 2005 7:28 pm    Post subject: New Proof Of The Fundamental Theorem Of Calculus

John,
If indeed your Average Tangent Theorem is true, then it would be a
far more efficient way of proving both the Mean Value Theorem as well
as the FTOC (fundamental theorem of calculus). And yes it wouold
provide a direct link between the derivative and integral.

I can't see any flaws with it but some of your opponents, eg. Graeme
McRae and others don't seem to think so. Perhaps your most vehement
arguments were those on Cut-the-knot wher you refuted the
mathematician Vladimir. You have a very abrupt style of responding to
objections - if I may, you may try posing your responses as questions
rather than 'attacks' as has been interpreted by most.
I think McRae is not necessarily an opponent either because he plainly

I think you are on to something big here - at least as far as
understanding is concerned. Personally I have failed to see the direct
link even though I've always known there is one. An Archimedes may
have indeed known of differentiation should your theorem be proved.

Jason Wells.

On 21 Nov 03 06:09:04 -0500 (EST), John Gabriel wrote:
 Quote: Your web page is interesting but I think there is a far easier way to prove not only the FTOC but every important calculus theorem - my average tangent theorem not only defines and proves the mean value theorem for the first time since Isaac Newton, but it proves both parts of the Ftoc in such simple mathematics so that even a high school student can understand it. It also proves finite differences, Taylor's Theorem, Quadrature/Cubature, Arc Lengths, Newton's Approximation Formula and much more. I invite you to check out: a href="http://www.geocities.com/john_gabriel">http://www.geocities.com/john_gabriel

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