Question about lifting of a function
Theorem 1Suppose that $p : X \to Y$ is a covering map. Suppose $\gamma_0, \gamma_1 : [0, 1] \to Y$ are continuous, $x_0 \in X$ and $p(x_0) = \gamma_0(0) = \gamma_1(0)$. Fie $\tilde{\gamma}_0$ and...
View ArticleIs the convolution between two CDF always well defined?
Given the integral convolution:$$(F_X * G_X)(x)=\int_{-\infty}^x F_X(t)G_X(x-t)dt $$and also that the CDF of random variables are bounded between the interval $(0,1)$ and the flipping of one of them in...
View ArticleFinding the value of this sum.
We want to find the value of this series $\sum_{n=1}^{\infty} \frac{n}{2^n(n+1)}$ According to Wolfram, tha value is $2 - ln(4)$What I did :$\frac{1}{1-x}$ = $\sum_{n=0}^{\infty} x^n$....
View ArticleThe set of algebraic numbers is countable.
I'm currently working through Rudin's PMA, and I'm on exercise 2.2 which tasks me to prove that the set of algebraic numbers is countable. There is a hint we are given to use, which is that the number...
View ArticleIntegrating slower growing functions to find a faster growing function
Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function of two variables. Now, consider the function $g(x) = \int f(x, y)\; dy$. It is known that $g$ has exponential growth, that is, for large values of...
View ArticleWhat is the measure of the set of values $S=\{0.x_1x_2... \in (0,1)$ (in...
I think the measure is zero but I'm not sure. I'm pretty sure the set is measurable because its construction does not require Axiom of Choice. I don't know where to begin.Edit: It was shown to be 1 by...
View ArticleFailure of Fubini's theorem for Riemann integral
This Wikipedia article says the following about Fubini's theorem:If a function is Lebesgue integrable on a rectangle $X\times Y$, then one can evaluate the double integral as an iterated...
View ArticleExtension of a $W^{1,p}_0(\Omega)$ function
Let $u \in W^{1,p}_0(\Omega)$ and let $\tilde{u}$ the function defined extending $u$ to zero on $\mathbb{R}^N \setminus \Omega$. I should prove that $\tilde{u}\in W^{1,p}(\mathbb{R}^N)$.The exercise do...
View ArticleProof of a Limit related to Gauss' Convergence test
So this is the question:if the series $\sum_{n=1}^{\infty} a_n$ is such that $$\frac{a_n}{a_{n+1}} = 1 + \frac pn + \alpha_n$$ and the series $\sum_{n=1}^{\infty} \alpha_n$ converges absolutely, then...
View ArticleHow to "draw" partial derivatives?
The idea of a total derivative is to find a linear function that is the tangent in a point, so to say. This is something I somehow can imagine.Do we have such an idea for partial derivatives to? That...
View ArticleWhat is the significance of a set having an accumulation point in a...
The identity theorem of complex analysis relates to the extrapolation of the equality of two complex functions between domains. In the statement, it is required that one domain have atleast one...
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An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$.We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle whose...
View Articlehow to prove that $\displaystyle \lim_{n \to \infty }...
In my exam there was this question :Prove convergence or divergence of :$\displaystyle \sum_{n=1}^ \infty \frac{(n!)^24^n}{(2n)!}$I noticed that :$\displaystyle \lim_{n \to \infty }...
View ArticleUniform Tightness of a Sequence of Probability Measures in $\mathbb...
Consider a sequence $(P_n)_{n\in\mathbb N}$ of probability measures on $(\mathbb R^{\mathbb N}, \mathcal B)$, where $\mathbb R^{\mathbb N}$ is the countable Cartesian product of $\mathbb R$, and...
View ArticleExample of proof of an infimum?
I'm doing an excercise about infimums and supremums and I've seen different examples of proving a =inf(S) $\iff \forall \epsilon >0\exists s\in S\colon a+\epsilon>s$ and to me they just seem to...
View ArticleBook recommendations for multi-variable analysis.
The only rigorous multi-variable analysis that I have read is from Rudin's PMA and it was only two chapters but I am sure that there are more theorems that calculus 3 course cover that was not in the...
View ArticleWhat kind of functions have radial Fourier series coefficients?
Suppose $f : \mathbb{T}^d \to \mathbb{R}$ has Fourier coefficients $\hat{f}(j) = (2\pi)^{-d}\int_{\mathbb{T}^d}f(\theta)e^{-ij^T\theta}$. If $\hat{f}(j) = \hat{f}(j')$ whenever $\|j\|^2 = \|j'\|^2$,...
View ArticleA question regarding inequality
let $0 < x < 1$ and $r,i_1,...,i_n > 0$.$$\frac{x^{i_1}+x^{i_2}+ \ldots +x^{i_n}}{n}\leq x^r$$for some x.Can we say that the inequality holds as x decreases?Note that it may not hold for all...
View Article$(f_n)$ sequence of differentiable functions on $[0,1]$ and converge...
Let $(f_n)$ sequence of differentiable functions on $[0,1]$ converging pointwise to $0$. Suppose$|f'_n(x)| \leqslant 2015 + \cos(x)$$\forall x \in [0,1]$ and $\forall n$. Show that $(f_n)$ converge...
View ArticleIf $f_n\to f$ uniformly on [a,b] and f is continuous on [a,b] then $f_n$ is...
Yesterday I wrote a test in calculus and had to answer the following question:Prove or contradict: if $f_n\to f$ uniformly on $[a,b]$ and f is continuous on [a,b] then $\exists n_0\in\mathbb N$ s.t...
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