Closed form expression for series $f_\alpha(x) = \sum_{n\geq 0} (-n)^{\alpha}...
Let $\alpha \in \mathbb N$, and consider the Taylor series$$f_\alpha(x) = \sum_{n >0} (-n)^{\alpha} x^{2n}$$which is convergent for $|x|<1$.Question: can we find a closed form to express...
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Local extreme points of $f(x)=\frac{5}{4} x^{\frac{4}{5}} - |x-2|$
Consider the real function $f(x)=\frac{5}{4} x^{\frac{4}{5}} - |x-2|$.$f$ is a continuous function and it is differentiable on $\mathbb{R}$ except at $\{0,2\}$.Indeed, $f$ is as follows:(i) If $x \geq...
View ArticleGeneral version of the fundamental lemma of calculus of variations
Let the measure $\mu$ on $\mathcal{B}(\mathbb{R}^n)$ be a general Borel measure.Let $f:\mathbb{R^n} \to \mathbb{R}$ be a locally integrable function and $\int f\varphi d\mu =0$ for any smooth function...
View ArticleNecessary and sufficient condition for $f(f(x))=x$
Based on the concept that $f$ is increasing then all the solutions of the equation $f(x)=f^{-1}(x)$ lie on the line $y=x$. I have two questions:Can I solve $f(f(x))=x$ by taking $f(x)=x$ if $f$ is...
View ArticleFind rate of convergence of two sequence [closed]
Consider the following two sequences $<a_n>$ and $<b_n>$ given by $a_n = \frac1{(n+1)} + \frac1{(n+2)} + \frac1{(n+3)} + ......\frac1{(2n)}$ and $b_n= \frac1n$. Which of the following...
View ArticleHow to prove that $\left(\frac{r_1}{r_2}\right)^n{\rm...
Let $\Omega\subset \mathbb{R}^n$ be a bounded connected domain. Introduce the notation$$\Omega_r=\left\{x\in\Omega|d(x,\partial\Omega)<\frac{1}{r}\right\},\quad r>0,$$where $d(x,\partial\Omega)$...
View Article$\mathbf{x}\perp f(\mathbf{x})$ and $\|f(\mathbf{x})\|=\|\mathbf{x}\|$ imply...
Let $f:\mathbf{R}^3\to \mathbf{R}^3$ be of class $C^1$ and $\mathbf{x}_0\in \mathbf{R}^3$. Suppose there exists an open neighbourhood $U$ of $\mathbf{x}_0$ such that for any $\mathbf{x}\in U$ we...
View ArticleUpper and lower integrals, piecewise function [duplicate]
Question: Let $f: [a,b] \to \mathbb{R}$ be a function such that$$f(x) = \begin{cases} x & x\in\Bbb Q \\ x+1 & x\not\in\Bbb Q\end{cases}.$$Compute the upper and lower integral of f.So I'm...
View Articleproof of polar coordinate change of variable in measure theory
I have to proveprove that $\sigma_{n-1}: 2^{S^{n-1}} \rightarrow[0,+\infty)$$$\sigma_{n-1}(E):=n m_n(\{r x: r \in[0,1], x \in E\})$$is a measure and then prove$$\int_{\mathbb{R}^n} f(x) d...
View ArticleMeasurability of $\|f(\cdot, x_{2})\|_{L^\infty(X_{1})}$ (proof of...
I am trying to prove Minkowski's inequality for integrals in the case $p = \infty$.Suppose $(X_{1}, \mu_{1})$ and $(X_{2}, \mu_{2})$ are two $\sigma$-finite measure spaces and $f(x_{1}, x_{2})$ is a...
View Article$f(x) = (\sum_{k=0}^n a_k x^k)^{\frac1n}$ is sublinear if $a_k \ge 0, x \ge...
$f(x) = (\sum_{k=0}^n a_k x^k)^{\frac1n}$ is sublinear if $a_k \ge 0, x \ge 0$, i.e. $f(a + b) \le f(a) + f(b)$ for $a, b \ge 0$.I have some solution. I know that there's a very short and cute solution...
View ArticleWrite squared norm of function in terms of squared norm of derivative.
Consider a continuously differentiable function $f:[0,1]\to\mathbb{R}$ which is integrable and has expectation zero with respect to a measure $\mu$ on the space $[0,1]$, that is $\int_{[0,1]} f(x) d\mu...
View ArticleDoes $f(\alpha x + (1-\alpha) y) - f(x) = f(y)-f(\alpha y + (1-\alpha) x)$...
If possible, I would like to find a convex set $X\subset\mathbb{R}^n$ and a function $f\colon X\to\mathbb{R}$ that satisfies $f(\alpha x + (1-\alpha) y) - f(x) = f(y)-f(\alpha y + (1-\alpha) x)$ for...
View ArticleSemidifferentiability at the extremum of an interval and continuous extension...
Let $f:[a,b] \to \mathbb{R}$ be differentiable on $(a,b)$ with continuous derivative $f'$.(i) Assuming that $f'$ can be continuously extended at $a$, is it true that $f$ is semidifferentiable at $a$...
View Articlepiecewise continuously differentiable path homotopy with fixed endpoints
Let $U$ be an open set in $\mathbb{C}$. Let $f_0, f_1:[a,b] \rightarrow U$ be (path) homotopic piecewise continuously differentiable functions with a homotopy $H$ and the additional property $f_0(a) =...
View ArticleElementary proof of the limit of $n^{n^{-p}}$
I'm going through George Bergman's exercises for Rudin's PMA Chapter 3 (Sequences and Series) and Bergman asks the following:Prove or disprove: For every positive real number $p$, one has $\lim_{n \to...
View ArticleProving that $f(x) = (2^x-1)^{1/x}+ (2^x+3^x-1)^{1/x}$ is increasing for $x...
I want to show that the function $f: [1, \infty) \rightarrow \mathbb{R}$ given by $$f(x) = (2^x-1)^{1/x}+ (2^x+3^x-1)^{1/x}$$ is increasing for $x \in [1, 2]$. By plotting it this seems to be true,...
View ArticleShow that $f$ is not differentiable but partially differentiable at $(0,0)$
Let be $f:\mathbb{R}^2\to\mathbb{R}$, where$$f(x,y):=\begin{cases}x,&x\neq y\\1,&x=y, x\neq 0,y\neq 0\\0,&x=y=0\end{cases}$$Show that $f$ is not differentiable but partially differentiable...
View ArticleStrong Law for dependent sequence of random variance
Let $\{X_i\}$ be a stationary sequence of random variables, $S_n = \sum_{i=1}^n X_i$, and $\text{Var}(S_n) \leq C$, where $C$ is a constant. Note $\{X_i\}$ need not be independent. I am wondering if...
View ArticleLet $a
Let $a<b$, can we have a form of $(a,b)$ that is closed in $S\subset\mathbb{R}$I know that $(a,b)$ itself is being closed in $S=(a,b)$.and $(a,b)$ is clopen in $S'=(a,b)\cup(b+1,b+2)$This is kind of...
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