Tag: numerical-analysis

Fitting polynomials on points in the plane

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The aim of this blogpost is to prove that given $n$ points with distinct $x$ coordinates there is a unique polynomial function passing through all of them. Basically this polynomial is the (unique) polynomial of lowest degree that ‘fits’ given data. This theroem is known as the ‘Lagrange interpolation theorem’ in numerical analysis. From hence forth by a set of $n$-coordinate pairs we refer to a set indexed by $0,1,\cdots,n-1$, such that the $x$-coordinates of all tuples in the set are distinct. It is well know that given a set of $2$-coordinate pairs one can find a unique linear function(a...

The sum law for cardinals

vivaandaga4 Comments
The aim of this blog post is to prove that if $\kappa$ and $\gamma$ are two infinite cardinals then $\kappa$ + $\gamma =\max(\kappa,\gamma)$. It should be pretty clear that it suffices that we prove $\kappa+\kappa=\kappa$ for any infinite cardinal $\kappa$. We can proceed using Zorn’s lemma: For an infinite set $X$ with cardinality $\kappa$, consider the set of all pairs $(M,f)$, where $M$ is a subset of $X$ and $f$ is a bijection from $2\times M\to M$. Now we can partially order this set by extention on both coordinates. Further it is pretty clear that every chain has an upper...