Show that $\sum_{k>0}\frac{a_k}{f(b_k)} \geq \sum_{k>0}...
Given positive, by one bounded sequences $(a_k)_{k>0}, (b_k)_{k>0}$ and for some $x\in[0,1] \, f(y) = x(1-x) + y (1-2x)^2 $. for $\sum_{l>0} \lambda_l = 1$ I want to show that...
View ArticleDerivative of a piecewise constant function on a random domain
Imagine a domain $\Omega \subset \mathbb{R}^{n}$and a slightly perturbed domain $\Omega'(x;\xi) = x + \sum_{i=1}^{M} \xi_i \phi_i(x) $ for $x \in \Omega$, and one defines the following...
View ArticleODEs with parameter and continuous/smooth dependence on initial conditions...
For the purposes of differential geometry I would like to fully process the basics of ODEs, namely the existence/uniqueness theorem (EUT) and continuous/smooth dependence on initial conditions and...
View ArticleMinimum of $x+\frac{1}{x}+\frac{2-x}{1-2x}+\frac{1-2x}{2-x}$ where $x>2$
$$x+\frac{1}{x}+\frac{2-x}{1-2x}+\frac{1-2x}{2-x},\qquad (x>2)$$This is the function whose minimum value I want to find. I tried using the derivative but that is way too lengthy. The minimum value...
View ArticleHow to prove the following theorem by distribution function and series [closed]
Let $g$ be nonnegative and measurable function in $\Omega$ and $\mu_{g}$ be its distribution function, i.e.,$$\mu_{g}(t)=\left|\left\{ x\in \Omega:g(x)>t\right\}\right|,\; t>0.$$Let $\eta>1$...
View ArticleHow to apply Poincare's inequality to the following formula
We know that a generalized Poincare inequality holds: for $p>n$, there is a constant depending only on $p$ and $n$ so that\begin{equation}\sup _{s, y \in B_r}|u(x)-u(y)| \leq C...
View ArticleHow to prove a gradient is Lipschitz continuous?
I have read in textbooks the following phrase, "assuming the gradient is locally Lipschitz continuous...", but how does one prove that the gradient is locally Lipschitz continuous?If we have a...
View ArticleStability Index Forms an Open Set
Let the ODE$$x' = A(x),$$ where A is an linear vector field on $\mathbb{R}^d$, denote this set by $\mathcal{L}(\mathbb{R}^d)$. We say that $A$ is hiperbolic if A does not have pure imaginary...
View ArticleHow to adjust fraction parts to arrive to accurate rate
Let say the target rate is:$$0.2312076819339227309219662514224185$$How to adjust this fraction $984398920/4257639330$ that gives this $\mbox{rate}\...
View ArticleLargest inscribed $\ell_\infty$ ball
$\newcommand\eps\varepsilon\newcommand\reals{\mathbb R}$Let $f : \mathbb R^d \to \mathbb R^d$ be continuous and let $B_\infty(x_0,\eps)$ be the closed $\ell_\infty$ ball of radius $\eps$ centered at...
View ArticleRecursive definition depending on multiple previous values (Analysis I Tao)
I'm reading Analysis I from Tao, and he defines recursive definitions as:Proposition 2.1.16 (Recursive definitions). Suppose for each natural number $n$, we have some function $f_{n} : \mathbb{N}...
View ArticleImplicit function theorem and submanifolds in $\mathbb{R}^n$
I will use the following notations for tuples: $\mathbb R^{n+m} \ni (x, y) = (x_1, \dots, x_n, y_1, \dots, y_m)$. Furthermore, let $\mathbf f : \Omega \subset \mathbb{R}^{n + m} \rightarrow...
View ArticleFredholm alternative interpretation
Let $T \in \mathcal{K}(E)$, that is, $T$ is a compact operator in the Banach space $E$. Consider $N(T)=\{x \in E: Tx=0\}$ and $R(T)=\{y \in E:y=Tx ~(\exists x \in E)\}$.In Brézis book we have the...
View ArticleClik here to view.
Contraction Mapping Theorem. Prove $\{ y_{1},f(y_{1}),f(f(y_{1})),\ \ldots)...
Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$.(a) Show that $f$ is continuous on $R$ for all...
View ArticleMultivariate Taylor Expansion
I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and...
View ArticleIf $p$ is an equilibrium point of $f$ , then either $\varphi(t)\neq p$ for...
Let $p$ be an equilibrium point of a $C^1$ vector field $f: \mathbb{R}^n \to \mathbb{R}^n$ (that is, $f(p) = 0$). Let $\varphi: \mathbb{R} \to \mathbb{R}^n$ be a solution of the equation $\dot{x} =...
View ArticleWhy converge in measure does not imply converge almost everywhere?
I know that in general converge in measure does not imply converge almost everywhere and I know a counter example. But one of the proposition says that if $\{f_n\}$ converges to $f$ in measure, then...
View ArticleClik here to view.
Upper and lower bounds on sine integral?
Define the sine integral $\operatorname{Si}(x)$ by$$\def\Si{\operatorname{Si}}\Si(x) = \int^x_0 \frac{\sin t}{t} \, dt$$I want to establish upper and lower bounds on $\frac{1}{\pi} \Si(\pi x)$. The...
View ArticleIs an equivalence relation (= sign) needed for the real number system or is a...
My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further.Generally, the Real Number System is said to...
View ArticleBanach space divergent sequence.
Let $X$ be a real Banach space, and let $\{ x_n \} \subset X$ be a divergent sequence. Is it always true that there exist $\delta > 0$ and a strictly increasing sequence $\{ n_k \} \subset...
View Article