Integration over a set
Let $\Omega=\{(x,y)\in \mathbb{R}^2:0<x<y<2x<2\}$Now I've got to integrate$\int_{\Omega}{ydxdy}$ over $\Omega$.Something is wrong with my integral because I get the false result.I've...
View ArticleRudin Ch 4 exercise 3: the zero set of a continuous function is closed
Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$) be the set of all $p \in X$ at which $f(p) = 0$. Prove that $Z(f)$ is closed.My attemptLet $p$ be a limit...
View ArticleProve that the function sequence $f_n(x)=n^2\left(\mathrm{e}^{\frac{1}{n...
Prove that the function sequence $f_n(x)=n^2\left(\mathrm{e}^{\frac{1}{n x}}-1\right) \sin \frac{1}{n x}(n=1,2, \cdots)$ convergent uniformly on $[a,+\infty)(a>0)$ .Proof1. For every $x...
View ArticleShow that if a continuous function is positive at a point it is positive on...
Question : Let $f: \Bbb R \rightarrow \Bbb R$ be continuous. Suppose $f(c) >0$. Show that there exists an $\alpha>0$ such that for all $x \in (c-\alpha, c+\alpha)$ we have $f(x)>0$.Attempt: By...
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Why does this series have a different sum when its terms are rearranged?
The problem is:Give an example of a convergent series such that, when the terms are rearranged, the series sums to a different value.A solution is:Although everything makes sense in this solution, I...
View ArticleIs this equality between supremum and infimums true?
$$\begin{multline}\sup_{(\pi, c, A)}\mathbb E \biggl[\int_0^\infty \!e^{-\beta t}\,u(c_t) \, \mathrm{d}t - \kappa A_0 - \kappa \int_0^\infty \!e^{-\beta t} \, \mathrm{d}A_t \biggr] \\=\sup_{(\pi,...
View ArticleContinuous $f:U \rightarrow \mathbb{R}$ on an open $U\subset\mathbb{R}^2$...
The title was too short to sumarize my question. I'm currently trying to do the following exercise:Let $f:U \rightarrow \mathbb{R}$ be continuous on an open $U \subset \mathbb{R}^2$, with partial...
View ArticlePlug-in estimator of expected value
Let $g$ be the statistical functional defined by $g(\mu) = \int x \,d\mu$. Then the plug-in estimator is defined as $\hat{g}=g(L_n)$ where$$L_n(\omega)=\frac 1 n \sum_{i=1}^n \delta_{X_i(\omega)}$$ is...
View ArticleShow that $\frac{1+z-\sqrt{z^2-6z+1}}{4}$ fits the Lagrangean framework
Let $S(z)$ be the OGF of bracketings. Show that the Lagrangean framework holds for $S(z)$.Remark: You can find the definition of Lagrangean framework below.From the Flajolet & Sedgewick book (p.81)...
View ArticleIntegration with spherically symmetric measure in $\mathbb R^d$
Let $\mu$ be a finite spherically symmetric measure over $\mathbb R^d$, so that $\mu(TB) = \mu(B)$ for all orthogonal transformations $T: \mathbb R^d \rightarrow \mathbb R^d$ and Borel set $B$. Let $g:...
View ArticleCentral limit theorem with changing bounds
Suppose $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of iid random variables with $X_n \sim \text{Beta}(1/2, 1)$. I would like to compute or get a good upper bound on$$ T = \lim_{d \to \infty}...
View ArticleLegendre Transform of MGF of B(1/2, 1)
If we have a Beta distribution with $\alpha = 1/2$ and $\beta =1$ then it has moment generating function given by$$M(t) = \frac {\sqrt {\pi}\text{ }\text{erfi}\left (\sqrt {t} \right)} {2\sqrt...
View ArticleAlways Closed Metric Space is Complete
I am trying to prove thatGiven a metric space $Y$, if for every metric space $X \supseteq Y$, $Y$ is $X$-closed, then $Y$ is complete.by contradiction or contraposition, such that I don't use the...
View Articlef a function is continuous differentiable and its derivative is monotonic, is...
f a function is continuous differentiable and its derivative is monotonic, is the derivative necessarily continuous? Please provide a proof, or if not, give a counterexample.I cannot provide a proof...
View ArticleIs $\frac{||f||_{L^p}^p}{||f||_{L^1}^2} \leq K ||f||_{L^r}^{s}$ possible?
Let $\Omega=(0,1) \subset \mathbb{R}$ (or, $\Omega$ is any open set in $\mathbb{R}^d$).Let's consider the usual $L^p(\Omega)$ space with the Lebesgue measure. I'm only assuming $f \geq0$ and $f \in...
View ArticleProof that continuous functions are bounded
So I'm following Calculus by Spivak and a particular part of a proof is troubling me. First off, the theorem that Spivak is trying to prove is the following: If f is continuous on $[a,b]$ then it is...
View ArticleExistence of degree $n$ best approximation polynomial implies Existence of...
There exist a degree $n+1$ polynomial of best approximation if there exist a degree $n$ polynomial of best approximationThis is the context of the question.If there exists a polynomial of best...
View ArticleIf $f_n(x)\rightrightarrows f(x),...
I want to prove following statement: Let $f_n:[0,1]\to\mathbb{R}$ be sequence of continuous functions uniformly convergent to $f\left(x\right)$. If...
View ArticleLebesgue measure generating set
In my cource of measure theory, Lebesgue measure were built starting with measure on semiring $S\subset 2^X$, after we defined outer measure for each $A\in2^X$ by $\inf\{ \sum{m[B] : A\subset \cup B ,...
View ArticleJensen's inequality with affine combination
From page 217 of 'Convex functions' by Arthur Wayne Roberts, Dale Varberg (exercise F) i want proof that: $f:\mathbb{R}\longrightarrow\mathbb{R}$ is affine iff$$...
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