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0
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1answer
24 views

Find the slope at $\theta = \pi/3$

We are given $r = 2\cos(\theta)$ Find the slope at $\theta = \pi/3$ This is a polar curve. Why would you find the slope at $dy/dx$ rather than $dr/d\theta$?
0
votes
1answer
20 views

A probably simple big $\mathcal{O}$ question

I have a probably simple big $\mathcal{O}$ question. Is the following statement correct? $$\mathcal{O}(x \log x)=\mathcal{O}(\sqrt x \log x)$$ why?
0
votes
0answers
14 views

Non-archimedean balls are/aren't abelian groups

I was reading a book and the author says the following, after defining $B_n(a)=\lbrace x\in\mathbb{Q}_p:|x-a|_p\le\tfrac{1}{p^n}\rbrace$ and $S_n(a)=\lbrace ...
2
votes
0answers
13 views

A $L^1$-bounded sequence from a $H^m$-bounded sequence

I am trying to show the following: for any $m > 0$ and $\alpha \in \mathbb{N}^n$, assume $(f_j)$ is a sequence of functions which is bounded in $H^m(\mathbb{R}^n).$ Assume moreover that all the ...
3
votes
0answers
41 views

Continuous functions and IVT

Suppose that $f$ is continuous on $[0,1]$ and $f(0) < 0$ and $f(1) > 1$. Show that there is some $x \in (0,1)$ such that $f(x) = x^{2014}$. I am guessing that here we should use the ...
0
votes
0answers
22 views

Sobolev space on closed surfaces

I was wondering if anybody here knows how the Sobolev space $H^2(\mathbb{S}^2)$ is defined? I.e. I want to integrate on this space with respect to the surface measure, but since this not the canonical ...
2
votes
1answer
31 views

The supermum of E

Let $f\ [0,1]\longrightarrow [0,1]$ be increasing function. let: $$E=\{x\in [0,1] \mid f(x)\geq x \} $$ Show that $E$ has a supermum $b$ and that $f(b)= b$. we have $x\leq 1$ since $f$ is ...
0
votes
1answer
23 views

the supremum of A

Let $A=\left\{ \left(\dfrac{m+n+1}{m+n}\right)^{m+n};\ m,n\in\mathbb{N}^* \right\}$. Calculate the supremum of A ($\sup A$). i tired To find the first few elements of A ...
0
votes
0answers
28 views

basic calculus/analysis question. why does the multivariable chain rule work?

Say $f$ is a function of $x(t)$ and $y(t)$ $$\frac{ df}{dt} = \frac{ \partial f}{\partial x} \frac{ dx}{dt} + \frac{ \partial f}{\partial y} \frac{ dy}{dt}$$ why is it so additively symmetric? (The ...
8
votes
6answers
774 views

How to know if a term is divisible by 10

I have some difficulties to solve this easy problem, could someone help me? Is $4^{1000}-6^{500}$ divisible by $10$?
0
votes
0answers
18 views

Integral inequality norm

I have read these one. It should be simple, but i can't find any answer: Let $D\subset \mathbb{R}^N$ measureable and bounded, and let $G:D\rightarrow\mathbb{C}^{N\times K}$ be measureable. Then, the ...
1
vote
0answers
22 views

Showing $\lim_{x\to\infty} [(\phi(t)-sin(wt))^2+(\phi'(t)-wcos(wt))^2]=0$

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^1 |b(t)| dx <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{x\to\infty} ...
0
votes
0answers
9 views

estimating function parameters from real world data

Sorry if this is to much of a newbie question. I have a function. it has two parameters. say f(x,y). The function models a real world process. I have real world sampled noisy output data for an ...
3
votes
1answer
32 views

Erwine Kryszeg's _Introductory Functional Analysis With Applications_: Section 2.3, Prob. 14

Here's problem 14 in the Problem Set immediately following Section 2.3 in the book, Introductory Functional Analysis With Applications by Erwine Kryszeg. Let $Y$ be a closed subspace of a normed ...
0
votes
2answers
10 views

Tip Top Landscaping Company-word problems

Max has only 9 months and has to take care of 1/4 of the yard, Alex has work for the company longer and has to takes care of 1/3 of the yard, Steven has been with the company the longest and must ...
2
votes
2answers
35 views

Which entire functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$?

Which holomorhpic functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$ on $\mathbb{C}$? So I've shown that $|f(z)| \leq |z|^k \implies f(z)$ is a polynomial of degree at most k. Therefore ...
1
vote
1answer
28 views

$I_{n}=\left[a+\dfrac{(k_n-1)}{2^n};\ a+\dfrac{k_{n}}{2^n}\right]$

Let $ \mathcal{P} \subset \mathbb{R}$,\ $\mathcal{P}\neq \emptyset $ and let $b$ be an upper bound of $\mathcal{P}$. Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
0
votes
2answers
58 views

Prove that if $|f(f(z)|>r$ then $f$ is constant

Let $r>0$. Prove that if $f$ is holomorphic on a whole complex plane and $|f(f(z))|>r$ for all $z\in\mathbb{C}$, then $f$ is constant. Can sb point me in the right direction?
0
votes
2answers
16 views

Does the function of a bounded sequence have a convergent subsequence?

Let {$x_n$} be a sequence in (s,t), and suppose f is continuous on [s,t]. Then does {f$(x_n)$} have a convergent subsequence? I know if {$x_n$} converges to some $x_0$ then {f$(x_n)$} converges to ...
2
votes
1answer
31 views

One measurable $\lim$ and one theorem

How we can prove following theorem? Let $f_n \ge 0 $ be measurable, $\lim f_n = f $ and $f_n \le f$ for each $n$. Show that $$\int f(x)dx=\lim_n \int f_n(x)dx $$ Any idea would be highly ...
0
votes
1answer
65 views

show $\exists\ m\in\mathbb{N} \text{ such that: } \quad a+\dfrac{m}{2^n}\geq b$

Let $ \mathcal{P} \subset \mathbb{R}$, $\ \mathcal{P}\neq \emptyset $ et let $b$ an upper bound of $\mathcal{P}$ Let $a \in \mathcal{P}$ and let $n\in \mathbb{N}^*$ Show that : ...
2
votes
1answer
39 views

Positiveness is a relative notion. [on hold]

Consider a ring $R=\mathbb{Z}[\sqrt7]=\{ a+b\sqrt7 ~| a,b\in \mathbb{Z}\}$ and let $p_1=\{ a+b\sqrt7 ~|a,b\in\mathbb{Z} ~~~ \text{and} ~~~ a+b\sqrt 7 ~~\text{is a positive real number} \}$ ...
0
votes
2answers
28 views

Spivak GENERAL limit law proof

Suppose $f(x) \le g(x)$ for all real $x$ Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$ Let limit for $f(x)$ be denoted by $L$ Let limit for $g(x)$ be denoted by $M$. ...
0
votes
1answer
50 views

Minimal surface and Weierstraß parametrization

If I have $f(z) = 1$ and $g(z) = \frac{1}{z}$ and I am looking for a minimal surface on $\mathbb{C} \backslash \{0\}$ using the Weierstraß-Enneper representation of minimal surfaces. Now I was ...
0
votes
1answer
10 views

Clarification of a transformation step (probably very simple).

A simple and quick question. Have been sitting over it for a while now but i can't get it right: Could someone just clarify how this transformation has been done? $$\frac{(v+1)^2 ...
0
votes
1answer
16 views

Find an example on a sequence of real-valued functions $(f_n(x))_{n\in\mathbb{N}}$ satisfying the given conditions

I am trying to find an example on a sequence of real-valued functions $(f_n(x))_{n\in\mathbb{N}}$ satisfying the following conditions: i) $f_n$'s are smooth and have compactly supported. ii) ...
-3
votes
2answers
64 views

Find this integral $\int_{0}^{1}f(x)dx$ [on hold]

let the function $$f(x)=\begin{cases} 1&x\in\{1,\dfrac{1}{2},\dfrac{1}{3}\cdots,\dfrac{1}{n},\cdots\}\\ 2&x\in other \end{cases}$$ Find this integral $$\int_{0}^{1}f(x)dx$$ where this ...
1
vote
2answers
28 views

Continuity of vector space operations in a normed space

Here's problem 4 immediately following section 2.3 in Erwine Kryszeg's book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar ...
0
votes
1answer
23 views

Equivalence of these two definitions of limit at a given point

Take a real $c$, a real $\delta > 0$, and a function $f: \mathbb{R} \to \mathbb{R}$. Then there are two definitions for $f$ to have a limit at $c$. Definition 1: If there is a real $l$ such that ...
1
vote
1answer
33 views

Prove that the subset $X$ of a normed vector space $(V,\|\cdot\|)$ is complete.

My subset $X$ has the Bolzano-Weierstrass property and I need to prove that $X$ is complete in the sense that every Cauchy sequence in $X$ converges to a point in $X$. I know that having the ...
1
vote
1answer
28 views

Proof under a suitable condition that $\lim_{x \to c}f(x) = 0$ implies $\lim_{x \to c}g(x) = 0$

I want to prove: Take a real $c$, a $\delta > 0$, and functions $f, g: \mathbb{R} \to \mathbb{R}$. If $$\lim_{x \to c}f(x) = 0$$ and $$|g(x)| \leq |f(x)|$$ for all $0 < |x-c| < \delta,$ ...
2
votes
0answers
23 views

Subtle Analysis Problem

Suppose you have a function $f \colon A \to \mathbf {R} $ and $ (a - \delta', a + \delta') \subseteq A$ for some $\delta' > 0$. Suppose also that $f$ is continuous at $a$. How do you prove that the ...
0
votes
0answers
26 views

An Introductory Question on Riemann Sums

I am a beginner at Riemann Integration and before I delve deeper into the subject, I would like it to be verified that I understand the working processes. Here is the question: Consider a function ...
2
votes
0answers
35 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribtution as $aX$ for some real $a$, what are the possible characteristic functions of $X$. Let $\varphi_X(t)$ be ...
2
votes
0answers
32 views

Is every convex function differentiable amost every where?

If a function $f: D \subset R^n \rightarrow R$ is convex, then for every $x,y \in D$ and $\alpha \in [0,1]$, $$ f(\alpha x + (1 - \alpha)y) \leq \alpha f(x) + (1 - \alpha)f(y). $$ I konw a convex ...
1
vote
0answers
27 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
0
votes
1answer
24 views

Uniformly Converging Sequence but Not Normly Converging

I want to find a sequence $(f_n) \subset \mathcal L^1(\mathbb R)$ of integrable functions which uniformly converges to $0$ but with $\int f_n \nrightarrow 0$. I came up with the following example: the ...
0
votes
1answer
27 views

Sequences, Bolzano Weierstrass theorem

According to Bolzano Weierstrass Theorem, Every bounded sequence has a convergent subsequence. And I saw the proof where if lets say we have this sequence bounded from [-M, M], you just kind of ...
0
votes
1answer
25 views

Prove that a subset is measurable is and only if the measurable of the set equal to the sum of that subset and its complement

Let $X$ be a set and $\mathscr{A}$ a collection of subsets of $X$ that form an algebra of sets. Suppose $l$ is a measure on $\mathscr{A}$ such that $l(X) < \infty$. Define $\mu^{*} $ as $$ ...
2
votes
1answer
35 views

Prove that the following set is dense in R

I need to show $ S = { m\cdot \sqrt{2}+ n\cdot \sqrt{3},where~m,~n~in~\mathbb{Z}} $ is dense in $\mathbb{R}$. I showed that S has an element in $(0,ε)$ for every $ε>0.$ How do I proceed to show ...
0
votes
0answers
44 views

Sum to infinity of the sum 1/n^2 [duplicate]

In my textbook it's mentioned that the sum $\lim\limits_{n\rightarrow\infty}(\sum\limits_{i=1}^n1/i^2=\pi^2/6)$. But how would you arrive at this result?
2
votes
1answer
26 views

Can anyone clarify the meaning of zero content?

I am having hard time understanding the definition of zero content. The following are the definitions of zero content in $\mathbb{R}$ and $\mathbb{R}^2.$ A set $Z \subset \mathbb{R}$ is said to have ...
1
vote
1answer
25 views

Introduction to Newtons method

I'm supposed to come up with two ways to introduce Newtons method for the approximation of zeros for highschool students. (That is the method using tangents and with the formula $ x_{n+1} = x_{n} - ...
0
votes
2answers
36 views

Is the map, $ f:(0,1)⊂ \mathbb{R}$ → $(1,∞)⊂ \mathbb{R}$ : $x ↦ 1/x $continuous?

I feel it is, but cannot prove why. Also is it bijective, and is its inverse continuous?
0
votes
0answers
22 views

Norm in $C(X,\Bbb{R})$

Let $X\subset\Bbb{R}$ a compact set and $f\in C(X,\Bbb{R})$. Define $$\|f\|_{\infty}=\sup A_f$$ with $A_f=\{|f(x)|\in \Bbb{R};x\in X\}$. Then $\|f\|_{\infty}=|f(x_0)|$, for some $x_0 \in X$, since ...
1
vote
3answers
69 views

Convergence and topology

Please what is the classical method to answer this question, does the sequence converge in the given topology ? 1) The sequence $\big(1+(-1)^n\big)_{n\in\mathbb N}$ in $(\mathbb{R},\tau)$ such that ...
0
votes
2answers
26 views

Proof that $[x+n] = [x] + n$ for all reals $x$ and all integers $n$

Let $$[x] := \sup \{n \in \mathbb{Z} \mid n \leq x \}$$ for all reals $x$. I want to prove that, if $x$ lies in $\mathbb{R}$ and $n$ lies in $\mathbb{Z}$, then $$[x+n] = [x] + n.$$ Nevertheless, by ...
8
votes
1answer
188 views

Show the sequence converges to M

Assume $f : [a,b] \to R$ is continuous and $f(x) \ge 0$ for all $x \in [a,b]$, and $M = \sup\{f(x) : x \in [a,b]\}$. Show that $$\lim_{n\to\infty}\left[\int_a^bf(x)^ndx\right]^{1/n}$$ converges to ...
1
vote
0answers
54 views

verify statement for |f(x)| is differentiable

I proved the following 'origin' statement If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $c$ and $f(c)=0$ , then the following are equivalent: (a) ...
2
votes
1answer
41 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...