Proof of Limit of a Sequence Raised to a Positive Integer Power
Problem statement:If $\{a_n\}$ is a sequence of positive numbers such that $\lim\limits_{n \to \infty}a_n =L$, prove using $\epsilon, N$ definition\begin{align*}\lim\limits_{n \to \infty}a_n^p =...
View ArticleProve that $\lim _{t \rightarrow \infty}\|f\|=0$
I am reading Paper but I have a difficult in understand the following statementThey proved that $\frac{1}{2} \frac{d}{d t}\|f\|^2=\frac{1}{2} f^{\prime}\left[G+G^{\prime}\right] f+\Sigma \bar{u}_j...
View ArticleSequence with infinite number of zeros
My question reads:Let’s call a sequence $(x_n)$ zero-heavy if there exists $M\in\mathbb{N}$ such that for all $N\in\mathbb{N}$ there exists $n$ satisfying $N\leq\ n\leq\ N + M$ where $x_n = 0$.If a...
View ArticleSet f a lower semicontinuous real function. If the set of points such that...
Set $f$ lower semicontinuous real function.Supose the set of x such that $f(x)=c$ is dense.Then $f(x)<= c$ for all $x$ or $f(x) >= c$ for all $x$ ?Im inclined to belive that its the second...
View ArticleShow that $\partial ^\beta f(x)=\sum _{\alpha\in\mathbb{N}^n}c_{\alpha +\beta...
Firstly consider the multi-index notation.Let $\{c_\alpha\}_{\alpha\in\mathbb{N}^n}\subseteq\mathbb{R}$ and $x_0:=(x_{01},\cdots,x_{0n})\in\mathbb{R}$.Define $\rho :=\sup\big\{r\in [0,\infty...
View ArticlePoint of achieving minimum value of a function
In side the triangle lies with corner $(0,1),(0,1/2),(1/2,1/2)$ (lies in the XY Place) where does the function $$f(x,y)=\frac{15554x+5092y+12441}{35360x+11576y+28283}$$ achieves it's minimum value?
View ArticleProof of Great Picard Theorem
I need help please! So I'm reading 'Complex made simple' by David C. Ullrich. I made all the requirements for this proof but the author don't give the proof of this final theorem, instead it gives a...
View ArticleSufficient conditions for uniform convergence in probability
I have a sequence of continuous random variables $\{X_n\}$, with density $f_n(x \mid \theta)$ w.r.t. the Lebesgue measure. $X_n = o_{p_\theta}(1)$ for any $\theta \in \Theta$. For a fixed $\theta_0$ in...
View ArticleGive example of a countable collection of finite sets whose union is not finite
This is exercise 3.E from Elements of Real Analysis by Bartle. The exercise asks to give an example of a countable collection of finite sets whose union is not finite and I wanted to verify if my...
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 ArticleCalculate the upper sums Un and lower sums Ln,on a regular partition of the...
Calculate the upper sums $U_n$ and lower sums $L_n$, on a regular partition of the intervals, for the following integral:$$\int_1^2 \lfloor x\rfloor dx$$$$\Delta x=\frac{1}{n}$$And then I'm unsure as...
View ArticleCircular shift of a function.
Consider a function $f$ that maps real numbers to real numbers with domain $[-a,a]$. I would like to describe the circular shift of this function by an amount $\delta$ such that, if I shift the...
View ArticleIs this integral in its most simplified form?
The following integration $$F(x)= \int_{x}^{+\infty} \frac{t}{1+t^\alpha} dt$$ cannot be solved in general, however can be expressed when $\alpha=4$ as $$F(x)= 0.5 \text{tan}^{-1} (x^{-2}) $$it can...
View Articleunder what condition do we have $f(x)+g(y)=h(x+y)$?
Suppose we have functions $f(p_1,p_2,w_1)$ and $g(p_1,p_2,w_2)$. Under what condition do we have a function $h(p_1,p_2,w)=f(p_1,p_2,w_1)+g(p_1,p_2,w_2)$ where $w=w_1+w_2$?Or if we generalise this to...
View ArticleIs there a website that has all the special functions?
There are a lot of special functions, and I wonder if there is a website that collects all of them, similar to how the Encyclopedia of Triangle Centers collects information on triangle centers.Another...
View ArticleTopology of uniform convergence?
Stone starts with an arbitrary compact Hausdorff space X and considers the algebra C(X,R) of real-valued continuous functions on X, with the topology of uniform convergenceI am having a hard time...
View ArticleConstructing smooth paths between points in codomain
I have a differentiable non-bijective mapping f from $x \in R^T$ to $y = f(x) \in R^T$.Say I have some positions $x_s^*$ and $x_e^*$ with corresponding mappings $y_s*$ and $y_e^*$.Questions:I assume...
View ArticleLet a real measurable function $f$ map every open set to the whole real line....
Given $f:\mathbb{R}\rightarrow\mathbb{R}$ mapping each non-empty open set to the whole real line, is there always a set $A$ of measure zero such that the function $g:\mathbb{R}\rightarrow\mathbb{R}$...
View ArticleEquivalent definitions of affine function
For a function $f\colon\mathbb{R}^n\to\mathbb{R}$ the following two are equivalent\begin{align*}&\text{(i) there exist $a\in\mathbb{R}^n$, $b\in\mathbb{R}$ so that...
View ArticleExamining the function.
Given function: $$f(x)=|x+2|e^{-\frac{1}{|x|}}$$Find constants $a, b$ and $c$ such that $f(x)= ax +b+\frac{c}{x}+ \sigma(\frac{1}{x})$ when $x \rightarrow \infty$ and $x \rightarrow -\infty$?Examine...
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