Zerotoepsilon

  Gödel’s Incompleteness Theorems (Stanford Encyclopedia of Philosophy)

 

  The idea of compactness comes from the fact that these are the nice set next to the finite sets. i.e. It shares many properties of finite sets in a topological space. The first thing common between finite and compact sets is, these are closed and bounded. Every sequence has a limit point. A continuous image of a compact set is compact and many more. Compactness has a very close relation to completeness . Let X be a topological space then the definition of compactness is defined in the following way-: X is said to be compact if every open covering of X has a finite sub-covering. Heine-Borel Theorem -: Heine-Borel make our work easy for Euclidean spaces by generalizing that, ‘ K is a compact subset of R if and only if it is closed and bounded. ’ But above theorem doesn’t work well for all metric spaces , for example -: Take N with discrete metric. What is missing here? Completeness? No! Boundedness? No! Closedness? No! Then what? The thing that is missing is totally bou...

 In 1827, Peter Lejeune-Dirichlet discovered this surprising result while working on conditions that ensured the convergence of the Fourier series. He was the first to notice that it is possible to rearrange the terms of certain series (now known as conditionally convergent series) so that the sum would change. Why is this result possible?  *ma049.qxd (washington.edu)

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 Primality testing is some sort of exciting stuff for a beginner in coding. There are various techniques for doing this. Following code is written by me that used the following fact that -: "N is a composite number if it has at least one non-trivial divisor d, such that d is less than or equal to sqrt(N)".  Proof of the fact is simple and can be proved using contradiction(Hint -; N is composite. What if all the non-trivial divisors are greater than sqrt(N)? ). Direct link -:  http://cpp.sh/7sng3 // To detect whether a number is prime or not. // Written by Sushant #include <iostream> #include<cmath> using namespace std; int main() { int p,q,i(0),d(0); cout<<"Enter the number -: \n"; cin>>p; // enter the prime q=int(sqrt(p)); // storing the integer part of square root                   for(i=2; i <= q;i++)                            {...

 Abel's partial summation formula is a very useful tool in analytic number theory to estimate the behavior of series. It converts the discrete sum into some integral.  Following link contain a nice proof with some example, which will make you comfortable with estimating various sums using this formula. Click here 👇👇👇 Abel Partial Summation Formula