Carlos Beltrán, the resolver of problem 17 from Smale's list

Steven Smale, one of the most recognized mathematicians of the XXI century, left 18 problems known as Smale's list, which his creator hismself considered as the biggest problems of this century. And one of those problems was solved by a spanish mathematician called Carlos Beltrán.

Carlos Beltrán received the Jose Luis Rubio de Francia for resolving the 17th problem of the list. Despite his thesis director affirmed that Beltrán has enormous skills in a lot of math areas, his pupil stated that the problem was less harder than other problems from the list than aren't unresolved yet. But even that, the truth is that this problem never had a progress aside from Smale's contribution.

The problem number 17 consist of solving polynomial equations in polynomal time in the average case. But we're not talking about a second degree ecuation, we're talking about this kind of ecuation:

Normally, this kind of ecuation always has many ways to be solved, and has lots of solutions, but the problem is that solving this must be efficient (that's what ti means to solve the problem "in polynomal time"). Resolving this has caused quite a change in Maths, particulary in programming, because with this resolution, now someone who solves an ecuation on a math program knows if the program is going to take a long time or not, or even success on all the possible solutions. There are different ways on solving an ecuation, but with Beltran's method, a uniform probabilistic algorithm called Las Vegas algorirthm, will give always the correct results and will only gamble on the resources used for the computation. And it's no secret that on Maths, making sure the result is correct has to be checked with other ways.

"There are lots of ways to caculate things, but it's no easy to prove that the way you're doing if the right one" Carlos Beltrán.

How to solve a double integration

Hello everyone, welcome to my first post. Today I'm going to solve a double integration.

Find the volume of the solid bounded above by the plane z = 4 − x − y and below

by the rectangle R = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2}.

Solution

The volume under any surface z = f(x, y) and above a region R is given by

f(x,y) is, in our case, 4-x-y. And the limits on the integration are very simple: in the variable x, the limits are 0 and 1, and the y is from 0 to 2. We resolution is simple: You solve one of the integrals(in this problem, we'll solve the x first), considering the other variable a constant. With the first integral done, just solve the area left with the limits.

Repeat to solve the integral of y and the solution is done.

This double integral  are the easiest types to evaluate because all four limits of integration are constants. This happens when the region of integration is rectangular in shape. In non-rectangular regions of integration the limits are not all constant so we have to get used to dealing with non-constant limits.

The 3d plot has been done by Wolfram Alpha. It's a good site to see the solutions if you're stuck doing a math problem.

First post

Hello and welcome to my blog! Here, I'll write several posts about Calculus and other interesting things during these months. I come from the "Universidad Europea de Madrid" and this is my first year academy. I hope you enjoy this blog and don't be too nitpick, because this is the first time I'm doing one of those...

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