A Convergence Result Conjecture
When I was doing my own research, I am tempted to prove the following statement.Suppose $\{a_n\}$ and $\{b_n\}$ are sequences of real numbers such that $\frac{1}{n}\sum_{i = 1}^n a_ib_i \to G \neq 0$....
View ArticleUsing the Partition of Unity Method to build the Global PUM space from local...
I'm working to understand how the partition of unity method is used to build a global PUM space from local approximation spaces. Can someone please explain the mechanics of gluing the local spaces...
View ArticleDefinition of Entropy of probability density function
Definition of entropy of probability density function $f\ge 0$ is$$Ent(f)=\int_{\mathbb{R}} f(x)\log f(x)dx$$My question is how this definition defines an integral on a set such that f is zero.$0\log0$...
View ArticleInterchanging liminf and supremum under certain conditions?
Suppose a situation whereby I know for some bounded collection $a_{m,n}$ of real numbers the following:For any $n \geq 1$, $\displaystyle\liminf_{m\to\infty} a_{m,n} \leq B_{n}$ and we can choose the...
View Article"Leibniz's rule" for $t\in\mathbb{R}^n$
I am looking for a reference giving a measure-theoretic proof of a claim from the German Wikipedia. I have searched the references given on that site, as well as the English speaking Wikipedia and all...
View ArticleInterchange of supremum and lim sup under certain condition
Suppose a situation whereby I know for some bounded collection $a_{m,n}$of real numbers the following:For any n≥1, $\underset{m→∞}{lim sup}$$a_{m,n}$≤$B_n$and we can choose the $B_n$such that...
View ArticleShow that if $f$ is a measurable complex-valued function on...
I need to showIf $f$ is a measurable complex-valued function on $(X,\mathscr{A})$, then $|f|$ is also measurable.I tried it myself, but don't know if my work is correct or not? Could someone please...
View ArticleSmall question about Rudin's first example for the integral of differential...
This is a small question. Rudin's law for differential forms is given on page 254 as$$ \omega = \sum \alpha_{i_1 \cdots i_k}(x) \, dx_{i_1} \wedge \cdots \wedge dx_{i_k}$$according to the rule$$ \int...
View Articlelimit of two sequences tending to zero
Suppose there are two real-valued sequences $x_n$ and $y_n$ both tending to 0 as $n\rightarrow\infty$. Are there any examples such sequences for which\begin{equation*}\frac{x_n}{y_n}\rightarrow...
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What does it mean have multiple cluster points?
In my classes we have just talked about one cluster point but in an excercise says "has one and only one cluster point", does that mean a sequence can have more than one?
View ArticleSolution for mixed with two convex function
I am doing a mixed two convex function$F(1)$, $f(2)$ is convex function$F(3) = 0.7 f(1) + 0.3 f(2)$ , so $f(3)$ also will be convex functionIf i find the answer by optimizing $f(1)$ and $f(2)$, and i...
View ArticleWhat is $y$ in theorem 9.24 Rudin,s PMA
9.24 Theorem Suppose $\mathbf{f}$ is a $\mathscr{C}^{\prime}$-mapping of an open set $E \subset R^n$ into $R^n, \mathbf{f}^{\prime}(\mathbf{a})$ is invertible for some $\mathbf{a} \in E$, and...
View ArticleDoes absolute integrability imply square integrability or are these...
Given a function f(x) : $\mathbb{R} \to \mathbb{C}$f(x) is said to be absolute integrable iff $\int_{-\infty}^\infty |f(x)|dx$ is finite.f(x) is said to be square integrable iff $\int_{-\infty}^\infty...
View ArticleOpen, connected, bounded set with nice boundary is locally on one side of the...
Let $\Omega \subset \mathbb R^n$ be open, connected, and bounded. We assume that the boundary $\partial\Omega$ is locally the graph of a continuous (or differentiable) function, i.e., for all $x_0 \in...
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Why do we require intervals in Jensen's inequality?
In my notes I havewhere $I$ is the interval on which $X$ takes values. I am struggling to see why do we require for our random variable to be contained in an interval. I thought looking at the proof...
View ArticleWhy do we need the "Size $\delta$ Approximations" to Hausdorff measures?
The sources I've looked at define the Hausdorff measure in two stages. They first define, for dimension $s$ and some dimension-dependent constant $\alpha(s)$:$$ H_\delta^s(A) =\inf \left\lbrace...
View ArticleShow $\lim\limits_{n \to \infty}\frac{f\left(nx\right)}{n^2}=0$ a.e....
Show $\lim\limits_{n \to \infty}\frac{f\left(nx\right)}{n^2}=0$ a.e. $x\in\mathbb{R}$ if $f \in L^1\left([0,T]\right)$ and periodic in $\mathbb{R}$, where $T>0$ is the period.My idea is as follows:...
View ArticleTangent line of a convex function
Let $V : \mathbb{R}^n \rightarrow \mathbb{R}$ be (strictly) convex and continuously differentiable on $\mathbb{R}^n$. Show that for any $y \in \mathbb{R}^n$ and any $x \in \mathbb{R}^n$$$V(y) \geq V(x)...
View ArticleWhy $a+\delta w_0\in A$ given A is open?
Theorem 5.30 (Separation Theorem)Suppose that $X$ is a real or complex normed space and $A,B\subset X$ are non-empty, disjoint, convex sets.(a) If $A$ is open then there exists $f\in X^\prime$ and...
View ArticleShowing the identity log($z^n$) = n log (z) for a particular value of n and z
I was asked to show $\log(z^n) = n \log (z)$ where $z = 1 + i$ and $n = 5$.The worked solutions state that they are not equal for those values but I do not understand why given that we find the...
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