Prove that $d(f,g) = \left(\int_0^1 (1+x)^3 |f(x) - g(x)|^3\;...
Prove that$$d(f,g) := \left(\int_0^1 (1+x)^3 |f(x) - g(x)|^3\; dx\right)^{1/3}$$is a metric on the set of all differentiable functions $f:[0,1]\to \mathbb{R}$ with $f$ continuous. The first two...
View ArticleHow to prove the Minkowski inequality (triangular) using Hölder and...
How to prove the Minkowski inequality (triangular) using Hölder and Cauchy-Schwarz for every space?$\|f+g\|_p\leq \|f\|_p+\|g\|_p$for$1\leq p \leq \infty$
View ArticleAny closed subset in a separable metric space is the union of a perfect set...
In problem 2.28 of Rudin's "Principles of Mathematical Analysis", the reader is asked to prove the following:Every closed subset of a separable metric space is the union of a perfect set and a set that...
View ArticleApproximating exponential function using piecewise constant function
I want to construct a piecewise function to approximate the function $f:[0,1]^d \to \mathbb{R}, ~f(x) = \exp(\|x\|^2)$. My approach is to partition the space $[0,1]^d$ into non-overlapping cubes...
View ArticleWhat does it mean to "solve forward/backward" a differential equation?
I know what it means to solve forward/backward difference equations. What I have not been able to wrap my head around is how that formally makes sense in continuous time. To make things concrete, here...
View ArticleHow to show the given function is differentiable...?
Consider the function $f:\mathbb{R^2}\to \mathbb{R}$ defined by$$\begin{equation}f(x,y)=\begin{cases}(x-y)^2\sin \frac{1}{x-y},&\text{ if } x\ne y\\0 ,&\text{ if }...
View ArticleFind the covariance matrix of the limiting distribution of the bivariate...
setupr.v.$$\begin{pmatrix}X_1\\Y_1\end{pmatrix},\ldots,\begin{pmatrix}X_n\\Y_n\end{pmatrix}\overset{i.i.d.}{\sim} N_2\left(\begin{pmatrix}\xi\\\eta\end{pmatrix},\begin{pmatrix}\sigma^2 &...
View ArticleHow to evaluate $\int_0^1 \frac{x^n}{x^2+x+1} \,dx$
How do I evaluate the integral$$I_n=\int_0^1 \frac{x^n}{x^2+x+1} \,dx$$I’ve tried so far integration by parts and partial fraction, each method has led to a dead end.So how I should evaluate this using...
View ArticleIf $f$ is differentiable at $a$, does it imply that $\lim\limits_{x,y\to...
Let $a \in \mathbb{R}$, $I \subseteq \mathbb{R}$ be a neighborhood of $a$, $f: I \rightarrow \mathbb{R}$ a function which is differentiable at $a$.Want (either/or) :A function $f$ for which there exist...
View Articleprove there exists neighbourhood around o such that $f'(x)>0 $
Assume $f(x)>0 $ for $x\in(0,\infty)$ and $f(0)=0$. Assume that $f(x)$ is continuously differentiable in $(0,\infty)$ and $\lim_{x\downarrow0}f'(x)$ exists. Can we prove that there exists...
View ArticleAbout the behaviour of an integral for $|x| > 1$ and $|x| < 1$
Let $f = \chi_{B(0,1)}$. Can anyone help me with the behavior of the following convolution $$f * |\cdot|^{-\alpha}(x) = \int_{B(0,1)}\frac{1}{|x-y|^{\alpha}}dy,$$for the cases $|x| > 1$ and $|x|...
View ArticleIf a series diverges, there exists a sequence convergent to zero such that...
Let $\sum y_n$ diverge. Then there exists a sequence $x_n$ such that $\lim_{n\rightarrow\infty}x_n=0$ and $\sum x_ny_n$ diverges.Is this true? How should I prove this?
View ArticleProof involving norm of an integral
I am totally stuck and have no idea whatsoever on how to prove the following inequality (by the way this is a problem from an undergraduate book in multivariable advanced calculus at Junior/Senior...
View ArticleCalculating the limit of $\lim_{n \to \infty}\prod_{i = 1}^{n -1}\bigl(1 -...
I want to calculate the limit of the sequence where $a_n = \bigl(1 - \frac{1}{n}\bigr) * \bigl(1 - \frac{2}{n}\bigr) * \dots * \bigl(1 - \frac{n-1}{n}\bigr)$ as $n \to \infty$. I want to use...
View ArticleProbability that Brownian Motion takes value in an $L^2$-Ball
Suppose $W:[0,1]\times \Omega \to \mathbb{R}$ is a standard Brownian motion on the unit interval. With $L^2[0,1]$ denoting the space of real-valued square-integrable functions with standard norm...
View ArticleIs there a formula for the derivative of Volterra's function? or at least...
I would really appreciate any description of Volterra's function's derivative, thank you!
View ArticleConvert $z$ = $1 - \cos(x) + i \sin(x)$ to polar form
I was given this question for homework.When it came time to solve $\theta$ I eventually got $\theta= \arctan$($\frac{\cos\frac{x}{2}}{\sin\frac{x}{2}}$)Where do I go from here since I can't see what to...
View ArticleFind the value of the integral of the real root of an equation
For $x\ge0$ let $f(x)$ represents the real root of the equation $$y^3+26xy=27$$ Find the value of $$\int_0^1 f^2(x) \ln(f(x)) dx$$Whenever I see such kind of questions, I instantly think of Lisant's...
View ArticleAsymptotic volume of n-sphere intersection with n-cube
I am looking for an lower bound on the following quantity in terms of $d, r_1, r_2$:$$\text{Vol}(C^d(r_1) \cap B^d(r_2))$$where $B^d(r) = \{(x_1, \dots, x_d) \mid x_1^2 + \dots + x_d^2 = r^2\}$ and...
View ArticleWhere is the mistake in the argument in favor of the (erroneous) claim "every...
A cut is a set $C$ such that:(a) $C\subseteq \mathbb Q $(b) $C \neq \emptyset $(c) $C \neq Q $(d) for all $a, c \in \mathbb Q $ , if $c\in C$ and $a\lt c$ , then $a\in C $(e) for all $c\in \mathbb Q$ ,...
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