Non-calculator proof that ππ−π<1003\pi^\pi -\pi \lt \frac{100}{3}

I am looking for a few non-computational, non-calculator proof of the following inequality: ππ−π<1003 I can’t really seem to come up with a proof because of that killer ππ term. Answer The inequality xx−x<100/3 is true precisely when 0<x<3.14175798627357…, so intuitively a solution to this problem must in some fashion make use of the fact … Read more

Algorithm for isomorphic groups?

As I understand There can’t be a general algorithm to decide if two finite groups are isomorphic, Wikipedia. But are there efficient algorithms for all subgroups of $S_n$ for say $n=10$ or so? I accepted the answer since it really was the answer (I misread Wikipedia and became a bit confused) but I would appreciate … Read more

Avoiding Matrix Inversion

I have as input values the matrices A,B∈Rn×n, where B is invertible, vector →b∈Rn, and α∈R. Denoting the identity matrix by Id, I am computing the value of (αA+Id)(A+B−1)→b. However, as computing the value of B−1 is quite expensive, I wish to instead solve a system of linear equations. Letting →x:=(αA+Id)(A+B−1)→b, I tried →x=(αA+Id)B−1B(A+B−1)→b=(αAB−1+B−1)(BA+Id)→b. Then, … Read more

Calculation Mismatch for WolframAlpha vs RConsole [closed]

Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 3 years ago. Improve this question Getting different answers when the same input is entered on the mentioned consoles known as W and R respectively. Any … Read more

How to write a regular expression for every string w over {0,1} where w is a binary string with value of at least 40.

So far I have realized that minimum value is 101000 in binary, but we also have to accept strings like 110000 and 1000000. I came up with r = (0+1)^* 1 (01+10+11) (0+1)^3 but then realized the string could also be 10000000. I’m not sure how to take into consideration that any 1’s after the … Read more

Notation : What is the meaning of the (mod n) in factoring algorithms?

Pretty much every thing is in the title, really! I’m trying to come up with an efficient algorithm to factorize large integer as an homework for a parallel programming course. I’ve seen a few pages explaining the algorithms, but just can’t wrap my head around the notation enough to actually implement the thing. For exemple, … Read more

why do the Computability theory choose the natural number as the object of study?

I am wondering why the computable function is defined in the natural number set.Can people give me the answer or some resources that can solve my puzzle. Answer There’s a few points to make, here: First, the historical. The first real use of the idea of computable functions to prove a theorem was Goedel’s use … Read more

How to find the square root of a matrix

Could anyone help me to understand how the author was able to take the square root of the following matrix: $$ \rho = {1 \over Z}\left( {\matrix{ {e^{ – 2\beta B} } & 0 & 0 & 0 \cr 0 & {\cosh 2\beta \delta } & { – e^{i\theta } \sinh 2\beta \delta } & … Read more

unexpected high precision of the Simpson’s rule

I tried to use the Simpson’s rule to calculate the following integral π=4∫10dx1+x2. The Matlab code is as follows. I expected a scaling law error∝h4 Nlist = 2.^(2:6); elist = 0 * Nlist; for s1 = 1 : length(Nlist) N = Nlist(s1); h = 1/N; xlist = 0 : h/2 : 1; ylist = 1 … Read more