# All Questions

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Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ... 0answers 14 views ### Characteristics of a Character table and what it tells me. I am trying to solve the character table and some related questions. The questions are below, and what I have done is below that. Any help on any pieces I am sure will enlightening. For parts c and ... 0answers 27 views ### Proving “if” direction of continuous iff sequence x_n converging to x implies f(x_n) converges to f(x) Here is the theorem in mathjax: A real value function$f$is continuous at$x \in R$iff whenever a sequence of real numbers$x_{n}$converges to$x$then the sequence$f(x_{n})\rightarrow f(x)$. ... 2answers 13 views ### Is conditional expectation E(X|N) an a.e. equivalence class wrt N or underlying sigma algebra? Let$X$be a random variable defined on a measure space$(\Omega, F, P)$. Let$N$be a sub sigma algebra of$F$. Then conditional expectation$E(X|N)$is an a.e. equivalent class. Is the a.e. ... 4answers 42 views ### Why does the definiton of the Euler's number not violate the rule agaisnt division by zero? [duplicate] e= appears to be defined as the sum of the series 1/n! as n goes from zero to infinity. But this implies that the first term is 1/0! which appears to violate the rule against division by zero 0answers 16 views ### Weierstrass transform on a Riemmanin manifold As it has been written on this Wiki page, the Weierstrass transform can be defined on a Riemannian manifold. Even though, I couldn't find any references I guess this transform for a function$f: ...
I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...