Author Archives: topice

Existence of a sequence of polynomials using Rounge’s theorem

I want to use the Rounge’s theorem: if $Ksubsetmathbb{C}$ is compact and ${a_{j}}_{j=1}^{infty}subseteqmathbb{C}cup{infty}$ is such that foe every component of $mathbb{C}^{infty}smallsetminus K$ has at least one point $a_{j}$. Then, for all $f$ holomorphic in $K$, there exists $R_{n}$ rational function with poles in $a_{n}$ such that $R_{n}to f$ uniformly on $K$. For prove that there… Read More »

Can I change limits of integration from $[-P/2,P/2]$ to $[0,P]$?

For a function $f(x+2pi)=f(x)$, I have the usual FS coefficient $a_n=frac{1}{pi}int_{-pi}^{pi}f(t)cos(nt), dt$ and equivalent for $b_n$. Say I now have a function with period $P$, $g(x+P)=g(x)$ and use the change of variable $t=frac{2pi}{P}x$, $$ a_n=frac{2}{P}int_{-P/2}^{P/2}f(t)cosBig(frac{2pi n }{P}xBig), dx $$ Is it correct to change the limits from $[-P/2,P/2]$ to $[0,P]$, i.e. begin{align} a_n&=frac{2}{P}int_{-P/2}^{P/2}f(t)cosBig(frac{2pi n }{P}xBig),… Read More »

When a local langragian equation can be solved to port solution to be a non-local langragian to explain non-locality Euclidean physical action?

I see a continuous confusion between the definition of the word ‘non-local‘ between physicists because it is thought that non-local action is an exclusively physical concept and not, instead, purely mathematical. But the non-locality, in mathematics, does not equate to the definition of ‘non-local’ that is used in physics because while for physics it suggests… Read More »

Help with proving that a circle containing two complex numbers, $w=x+iy$ and $1/(bar{w})$ must intersect ${|z|=1}$ at right angles?

Question: Proving that a circle containing two complex numbers, $w=x+iy$ and $1/(bar{w})$ must intersect the circle, ${|z|=1}$ at right angles? I am currently in complex analysis and as a way to further our understanding of the topics he assigned us each different questions that would further our knowledge outside of what was taught in class.… Read More »

Relative Abelian Varieties

If $A$ is an abelian variety, we have an addition map $mu:Atimes Ato A$. Now, suppose we have a relative abelian variety $mathcal{A}to B$, i.e. the morphism is flat and proper and for any $bin B$, $mathcal{A}_b$ is an abelian variety. Can we define a morphism $mucolonmathcal{A}times_Bmathcal{A}tomathcal{A}$ such that it restricts to the addition map… Read More »