Over the weekend I was attempting to estimate the tension change in a bicycle spoke due to an applied load.  After various simplifications and approximations, the problem was reduced to the following.

Given a constraint, F(x,y) = 0, and functions G(x,y) and H(x,y), find dG/dH at a particular point, here (0,0). 

The constraint, F, was sufficiently complicated that solving for either variable was not feasible, so implicit differentiation seemed the best...

The following is the matter of so-called central limit theorems. We have the sum of random variables S:= ksi[1] + ksi[2] + .. ksi[n]. We know only that the number n is large, the variables are independent or weakly dependent, and each ksi[j] is small with respect to S in a certain sense.

By the  central limit theorems it implies that S is close to the normal distribution.

Here is the procedure which illustrates the Lindeberg-Levi theorem ( see

I've submitted an application to the Application Center: Great Expectations.  This is an interactive Maple document, suitable for instructional use in an undergraduate course in Probability.  The mathematical content is related to the Laws of Large Numbers and Central
Limit Theorem.  It requires no knowledge of Maple to use.

The iPad is a very exciting device and it has been gaining broad adoption from our academic and professional customers alike. It was a logical step for us to bring Maple technology to this platform.
 
The Maple Player for iPad is now available in the Apple App Store. It comes bundled with ready-made interactive Maple documents, covering topics like integration, differentiation, computing...

Russian Center of Maple.
02.X.2011
3.000.000 visits in 16 months.

http://webmath.exponenta.ru/

It would be good I hope to present symbolic-numeric CAE system for framed structures analysis.

It will be available soon as Preview version for enthusiasts

The main features are:

• One calculation act - all analytical dependencies.
• Fast designing process for structural systems in industry, consulting and design companies;
• Fastest parametric analysis of construction. New quality of designing in optimization tasks,...

   This is a promissory Maple package, which is rarely used (I found nothing  in MaplePrimes and in Application Center.). Let us see the ?padic package. It is well known that the field of rational numbers Q is not complete. For example, there does not exist a rational number k/n such that k^2/n^2=2. There are only two ways to complete Q ( http://en.wikipedia.org/wiki/Ostrowski's_theorem ) .  The first way is to create the field of real numbers R including Q. Every real number can be treated as a decimal fraction sum over [k in K] of a[k]*10^(k) with a[k] in {0,1,2,3,4,5,6,7,8,9}, finite or infinite. For example, the numbers 0.3+O(0.1), 0.33+O(0.01), 0.333+O(0.001), 0.3333+O(0.0001), ...  approximate the number  1/3.
   The second way is as follows (see http://en.wikipedia.org/wiki/P-adic_number  for more details). We choose a prime number p and consider the valuation v[p] of a rational number k/m=p^n*a/b <>0 where integers are supposed to be irreducible :v[p](k/m):=p^(-n) , v[p](0):=0. The completion of Q up to this valuation is the field of p-adic numbers Q[p] (also including Q).  Every p-adic number can be treated as a p-adic fraction sum over[k in K]of a[k]* p^(k) with a[k] in {0, 1, 2, 3, p-1}. For example, the numbers 2, 2+O(5),2+3*5+O(5^2),2+3*5+5^2+O(5^3) approximate the number 1/3 in Q[5]. These can be obtained with Maple as follows.
> with(padic);
> evalp(1/3, 5, 1);
                           2
> evalp(1/3, 5, 2);
                        2+O(5)
> evalp(1/3, 5, 3);
                          2+3*5+O(5^2)
> evalp(1/3, 5, 4);
                         2+3*5+5^2+O(5^3)
    The field Q[p] is a very strange object. For example, the set of integers is bounded in Q[p] because v[p](k) <= 1 for every integer k. Another striking statement: the sequence p^n tends to 0 in Q[p] as n approaches infinity. The functions expp(x), logp(x), sqrtp(x) and the others are defined in the usual way as the sums of power series (see ?padic,functions for more details). For example,
> Digitsp := 12;
> logp(2+3*5+5^2, 5);

               5+5^2+4*5^3+5^4+3*5^6+4*5^8+3*5^9+5^10+3*5^11+O(5^12)
> cosp(x, p, 2);

                            padic:-cosp(x, p, 2)
> eval(subs(x = 0, p = 5, padic:-cosp(x, p, 2)));

                             1
> eval(subs(x = 3*5, p = 5, padic:-cosp(x, p, 2)));

                             1                            
    The definition of the limit of a sequence in Q[p] is identical to the one in R (of course,  abs(x[n]-a)<epsilon should be replaced by v[p](x[n]-a)<epsilon for every rational epsilon) and the same with the derivative. But every continuous function is picewise-constant. There also exists a non-injective function on Q[p] having the  derivative 1 at every point of  Q[p] . It should also be noticed that the radius of convergence of the expp(x):=sum(x^n/n!,n=0..infinity) series equals p^(-1) if p >2 and 2^(-2) if p=2. Next, there exists a Haar measure d[p](x)=:dx on Q[p] such that d[p](Z)=1. The definite integral of a real-valued function f(x) over a subset D of Q[p] with respect to  dx is defined in certain cases. For example, the definite integral of 1 over
the ball B(0,p^n):={x in Q[p]: v[p](x)<=p^n} with respect to dx equals p^n, ie. the radius of B(0,p^n). It is clear that there does not exist any analog of the Newton-Leibniz formula in the p-adic case. Because of this reason every calculation of every definite p-adic integral is a hard problem.

        There are a lot of good and diffent books on p-adic analysis. In particular, see http://www.google.com/search?tbm=bks&tbo=1&q=p-adic&btnG= ,  http://books.google.com/books?id=H6sq_x2-DgoC&printsec=frontcover&dq=p-adic&hl=uk&ei=IgFuToupO8SL4gTE-tDOBA&sa=X&oi=book_result&ct=result&resnum=6&ved=
0CEYQ6AEwBQ#v=onepage&q&f=false
, and http://books.google.com/books?id=2gTwcJ55QyMC&printsec=frontcover&dq=p-adic&hl=ru&ei=UAxqTuabD5HGtAamhryxBA&sa=
X&oi=book_result&ct=result&resnum=4&ved=0CDkQ6AEwAw#v=onepage&q&f=false as a good introduction to the topic.
     Why  is it so important? Which are applications? There are indications that the space  we live in has not  the Archimedean property (see http://en.wikipedia.org/wiki/Archimedean_property) on a very small scale. To verify this hypothesis is  a dozen times more expensive than  the large hadron collider
 (see http://en.wikipedia.org/wiki/Large_Hadron_Collider ). However, the mathematicians already develop the necessary mathematical tools, in particular, p-adic analysis.  Concerning other applications, see the answer by Anatoly Kochubei in
 http://mathoverflow.net/questions/62866/recent-applications-of-mathematics.

Edit. The vanishing text and some typos.

3D Paper Physical Model

Td_Group_Adapted_Dou.zip 

Point Groups typical to cubic crystals are Tetrahedral (Td) in Zinc-blende and Octahedral (Oh = i x Td) in Diamond.  Symmetry operations give rise to the widespread application of Group Theory most notably to generate basis functions which transform according to irreducible representations.  Much work has been accomplished using Single Group basis, compatible...

I think I will continue posting some good questions from
Fridays Killer Questions 7city Learning:

Question:

There was a hit-and-run incident involving a taxi in a city in which 85% of the taxis
are green and the remaining 15% are blue. There was a witness to the crime who
says that the hit-and-run taxi was blue. Unfortunately this witness is only correct
80%...

Question:

Let say you have a revolver with six chambers. There are two bullets in the gun. The bullets are
located in two chambers next to each other. You now want to play Russian roulette. You spin
the barrel so that you don’t know where the bullets are and then pull the trigger. We assume
that you don’t kill yourself with this first attempt. Now assuming that you want to maximize
the chance of not killing yourself is it...