Archive for the ‘Mathematics’ Category

I was trying to get a matrix for a projection of any point to a line given by the following equotations: After some time of research I’ve ended with the following result: Let then Of course the result isn’t new at all, it was just a big efford for me to do that at midnight […]

Today, I noticed an amazing and worth memorizing thing. I turns out that fibbonacci series shows up also in graph theory in one of the simplest graphs: Where’s Fibbonacci hiden here? Well, the graph can be represented as a folowing matrix: By multiplying the matrix by itself and using induction we get: 

Matrixes seem to have a lot of wonderful properties, for example they may help checking if given triangles represented each by three complex numbers (verticles) are simillar. So, two triangles represented by complex numbers and are similar if, and only if: Proof coming soon.

True 🙂 However I would put physicists a bit nearer mathematicans 😉

The polynomial argument is a very useful fact which can be used in order to prove, or justify a statement. It bases on simple fact that non-zero polynomial of k degree, can have no more than k solutions, futhermore the difference of two polynomials of k degree can also have max of k solutions if only […]

The proof for the limit points of the sequence haven’t given me a calm dream until today. Of course the limit points of that sequence (almost intuitively) Lemma. Lemma proof. from floor function definition web have the following inequality: Now we will prove that Let where So we have Let so of course Lets consider […]

Today, I decided to post a very eye-catching method for calculating an integral which shows very often in Statistical Mechanics in Physics. Let’s try to calc this First let’s mark it with a name , then What is just a double integral over the whole 2d surface. It can be also written in polar coordinate […]

The angle which arms contain chords or tangents of a circle is the half of the sum (if the verticle of the angle lies inside the circle or on it) or difference (if the verticle lies outside) of middle angles based on the curves, which set this angle. The proof can be illustrated by following […]

Problem: If you’ll eliminate every second person from the circle of people going clockwise, which one stays alive? (As first the second person dies) Aswer: if then the person with number stays alive. Equivalently it’s a one-bit-shift-left of number written binary I will not write here the whole solution, thus it can be found in […]

Prove that for every natural number there exists different natural numbers such as sum of any two is divisable by their diffrence. Proof: Because number is divisable by , we can write and of course . Adding those two together: Induction: 1. Sequence 1,2 works 2. Assume that sequence works, then the number is divisable […]

A few days ago I were stuck with one exercise which I can’t pushed off :] Meanwhile when I forgotten about it, friend of mine, Ecik, did it. It’s below with his solution. Prove the following: if Solution: I’m open on any suggestions or other solutions of yours. If you’d like to see Ecik’s blog, […]

Reading some of the Lev Kourliandtchik books, I’ve found a tricky observation how to write this gigantic expression in an extremely simple, with number of square roots. It’s easy to check that and using that equality we have cool, huh? 🙂

For , we do have \ Trigonometrics

Suppose that we have a grounded sphere made from a conductor and an electron far from the sphere. Let the radius of the sphere be and the beginning speed of the electron be orientated on the line in distance of from the center of the sphere. The smallest distance between electron and the center of […]

The solutions can be found here My solution for task number 9 is here: Solution for number 9

Lets calculate more compilated sum further, the last one equals to and now the first of the sums is equal to however the second one is easy, cause its a geometric sequence equal to under the law from previous post. Now we have , in the result of elementary algebraic transformation we recieve It’s easy […]

Asume that we want to calculate Of course, But the sum on the right side is equal to Therefore and so,

First identity comes from Lebesgue: and the second one from Euler:

Since my first met with imaginary unit have past about 3 years. I’ve been thinking about it many times. First step forward in understanding was made when I acquainted with Schrödinger equation, it gave me some kind of experience with complex numbers. Meantime I’ve heard a lot of voices like “What is it for if […]

Some of you may be not so surprised as me, but it truly was a discover for me. Terrence Tao who is a mentor in modern mathematics publishes his work on a WordPress blog! Here is the link