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Apr
23
revised Let $A$ be an $n \times n$ matrix over $\mathbb{C}$ or $\mathbb{R}$. Does $\det(e^A) = e^{\mathrm{tr}(A)}$ always hold?
deleted 3 characters in body; edited title
Mar
29
revised If $f(z)$ maps the unit disk onto itself $k$ times, prove that f(z) must be a rational function and show that the degree of its denominator $\leq k$.
added 296 characters in body
Feb
3
revised If $f$ is analytic in $D$ and $|f(z)|<M$ everywhere on $|z|=1$, show for all $z:|z|<1$, $|f(z)| \leq M |\frac{z-a}{\bar a z - 1}|$
deleted 38 characters in body
Feb
1
revised Let $V$ be the space of complex polynomials on $[0,1]$. Is the differentiation operator self-adjoint?
added 87 characters in body
Jan
13
revised Let $f(x) = (x^n-1)/(x-1)$. Why does $f(1)=n$?
added 269 characters in body
Dec
20
revised What is the formula for a likelihood ratio $L$ that transforms martingale Geometric BM to Geometric BM with positive growth?
added 420 characters in body
Dec
13
revised Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense?
deleted 4 characters in body; edited title
Nov
19
revised Show $f_t- a x f_x + \frac{1}{2} b \sigma^2 f_{xx}=0$ has solutions of the form $f(x,t) = C(t) e^{-D(t)x^2}$
added 41 characters in body
Nov
4
revised Suppose $A$ is rank 1. If $A^2 = cA$ for some constant $c$, is it true that for a non-zero vector $x$, $Ax = cx$?
edited body
Apr
5
revised Show that a system of equations can be solved in terms of $x,y,z$ (Rudin, ex 9.19)
added 3 characters in body
Mar
25
revised Are $1/\sqrt{x}$ or $1/x$ Lebesgue integrable on $(0,1)$? If so, why?
deleted 37 characters in body
Mar
21
revised If $f,g: R^n \to R^3$, what is the derivative of the cross product $(f \times g)(\vec{a})$ where $\vec{a} \in R^n$?
added 14 characters in body
Mar
16
revised What is the limit of $f(a,b) = \frac{a^\beta}{a^2 + b^2}$ as $(a,b) \to (0,0)$?
deleted 9 characters in body; edited title
Mar
13
revised Does $\int_0^\infty f(x) dx = \lim_{n \to \infty} \int_0^n f(x) dx$ for $f \geq 0$ or $f$ not positive?
edited title
Mar
3
revised Prove that any bounded open set has an arbitrarily close closed subset
added 237 characters in body
Mar
2
revised Prove that if $E$ is measurable then $\forall \epsilon > 0$ $\exists F \subset E$ closed such that $m(E \setminus F) < \epsilon$.
added 8 characters in body
Feb
28
revised Prove $m_*( \bigcup_{i=1}^\infty I_i ) = \sum_{i=1}^\infty \ell ( I_i )$ if $I_i \cap I_j = \emptyset \forall i,j$.
added 11 characters in body
Feb
17
revised Show that every measurable set $A$ can be written $A=B \cup C$
added 12 characters in body
Feb
15
revised Prove $(\overline{A \cap B}) \subseteq \overline{A} \cap \overline{B}$.
added 42 characters in body
Feb
14
revised Show that $\,f(x) = \sum_{n=0}^\infty a_nx^n$, for $x \in [0,1]$, is of bounded variation
added 18 characters in body