Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1answer
65 views

Uniform continuity and limit

This is related to my other question. Consider the function $a(s)=\dfrac{1}{1+s^2}$. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous. How ...
2
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1answer
101 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
2
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3answers
112 views

Prove $\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$ using Dominated or Monotone Convergence

Is there a way to prove that $$\frac{\pi}{4} = \sum_{n = 0}^{\infty}\frac{(-1)^{n}}{2n + 1}$$ via the Dominated or Monotone Convergence Theorem?
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0answers
91 views

About an example

I want to find a function $f$ such that $|f(x)|\leq k |x|^{\alpha} $ where $k>0$ and $\alpha\in[0,1)$ for all $x\in \mathbb{R}$ $f'(0)=0$ Please thank you.
3
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1answer
97 views

Uniform continuity and translation invariance

Consider the function $a(s)=\dfrac{1}{1+s^2}$ and the space $X=\{f:\mathbb{R}\to \mathbb{R}$ such that $t\mapsto a(t)f(t)$ is bounded uniformly continuous$ \}$. I want to show that $X$ is ...
2
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1answer
331 views

Prove that the a modified Cantor Set is not Jordan-Measurable

Let $C_0 = [0,1]$ and if $C_n$ is given as a disjoint union of intervals, construct $C_{n+1}$ by removing from each interval $I$ an open interval of length $(n+2)^{-2}|I|$ in the middle of each ...
2
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1answer
36 views

Find $x$ for which the series converges

Find the $x$ s for which $$\sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)}$$ converges. How can I do this? My attempt is to write $\sum_{n=2}^{\infty} \frac{x^{n^2}}{n \log(n)}$ in the form $\sum ...
1
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2answers
79 views

f(x,y) jointly differentiable

What is the definition of "jointly continuously differentiable function"? I.e. when $f:\mathbb{R}^2\rightarrow \mathbb{R}:(x,y)\mapsto z$ is jointly continuously differentiable in $x,y$ ? Is it ...
1
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1answer
86 views

Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.

Let $F \subset \mathbb{R}$ be a closed set and define the distance from $x \in \mathbb{R}$ to $F$ by $d(x,F)= \inf_{y \in F} |x−y|.$ Prove that $$\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$$ for a.e. ...
7
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1answer
413 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
2
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1answer
35 views

Newtonian potential at (0, 0, – a)

I found this problem in the book Advanced Calculus, written by Friedman. "Newtonian potential at (0, 0, – a) due to a mass with constant densinty $\sigma$ on the hemisphere S: $x^2 + y^2 + z^2 = ...
0
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0answers
45 views

Calculate $a^2+b^2$

Factorize $f(x)=x^4+[3]x^3+[6]x^2+[5]x+[3] \in \mathbb{Z}_7[X]$.Let $g(x)$ the irreducible monic polynomial with the greatest degree.If $g([2])=[a] \text{ and } g([3])=[b]$,calculate $a^2+b^2$. I ...
0
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1answer
233 views

relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
0
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1answer
34 views

‎every ‎ring ‎automorphism‎ $\Phi$ ‎of ‎the ‎complex ‎algebra ‎‎

Why ‎every ‎ring ‎automorphism‎ $‎\Phi$ ‎of ‎the ‎complex ‎algebra $‎‎M‎‎‎_{‎n‎} ‎(‎\mathbb{C}‎)$ ‎of ‎all $‎n‎\times n$‎ ‎complex ‎matrices ‎has ‎the ‎form ‎‎ ‎$\Phi ‎(T)= ‎AT‎A‎^{-1} ‎‎$ ?
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3answers
329 views

Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
3
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1answer
99 views

a function with infinity L^p norm

Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$ or $$ ...
6
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2answers
150 views

Prove that $\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}$ is a strictly decreasing function.

This is part of an actuarial science problem. Unfortunately, the official solution of this problem takes the derivative of $$\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}\text{, } \quad x \geq 0\text{.}$$ and ...
7
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1answer
93 views

Show there exists a unique $f$ (in $\mathbb R^+$) such that $\frac{d}{dx}f(x)=f^{-1}(x)$

Question: Show there exists a unique bijection $f:\mathbb R^+\to\mathbb R^+$ such that $\frac{d}{dx}f(x)=f^{-1}(x)$, where the right-hand side is the functional inverse. I figured I would start by ...
1
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2answers
88 views

Why is the function integrable?

I was wonderind why the function $f(x)=x, x \in [0,1]$ is integrable in $[0,1]$,although $U(f,P) \neq L(f,P)$ $P$ a partition of $[0,1]$,let $P=\{ x_0=0,x_1=1\}$ $U(f,P)=(1-0) \sup f([0,1])=1$ ...
0
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1answer
32 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
1
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1answer
63 views

Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$

Let $(X,A,μ)$ be a finite measure space. Let $\{f_n\}^∞_{n=1} ⊂ L_2(μ)$ be a sequence of functions such that $∥f_n∥_2 ≤ 1.$ a.) Prove that if $f_n → 0$ in measure, then $f_n → 0$ in $L_1(μ).$ b.) If ...
1
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1answer
122 views

Can I easily deduce this stronger spectral theorem from this weaker one?

I've just read a nice proof that: For $T$ a self-adjoint bounded linear operator on a Hilbert space $E$, there exists a unique $C^*$-algebra isomorphism $C(\sigma (T)) \rightarrow A_T$, from the ...
2
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3answers
83 views

If $a$ is a limit point of $f^{-1}(b)$, then the linear mapping $f'(a)$ is not injective.

Let $U\subset\mathbb{R}^m$ be an open set and $f:U\to\mathbb{R}^n$ a differentiable function. Suppose that there exists $b\in\mathbb{R}^n$ and $a\in U$ such that $a$ is an accumulation point of ...
3
votes
2answers
109 views

Show $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$

As part of an analysis qual problem, I am having a hard time showing that $\lim_{n_\rightarrow \infty}\int_0^\infty ne^{-\frac{2n^2x^2}{x + 1}}dx = \infty$. Any suggestions? Thanks in advance. I ...
2
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1answer
113 views

Show that $e^{-a|x|}$ does not belong to Schwartz space

Let $f : \mathbb R \to \mathbb R$ and $a > 0$ given by $f(x) = e^{-a|x|}$. Show that $f$ is rapidly decreasing and belongs to $L_1(\mathbb R)$, but not to $\mathcal S(\mathbb R)$. I had shown that ...
1
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1answer
91 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
3
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1answer
42 views

Prove the inequality based on an infinite series

Define $$f(x)=\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n}.$$ It is easy to see that this series converges for $x\in(-1,1).$ Now we are asked to show that $(1-x)^2f(x)\geq x,$ for $x\in[0,1).$ I tried ...
1
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1answer
45 views

If $\| \psi \|_2=1$ can I say something about $\| \psi' \|_2$?

If I have a differentiable $L^2$ function $\psi:\mathbb R\rightarrow \mathbb C$ which is normalised $$ \int |\psi(x)|^2\;\text d x = 1 $$ can I say anything about the order of $$ \int ...
1
vote
1answer
97 views

example concerning Lusin's theorem

Is there any example satisfying the following: $f$ is a measurable function on $\mathbb{R}^n$ with lebesgue measure $\lambda$. For any subset $N\subseteq\mathbb{R}^N$ with $\lambda(N)=0$, ...
1
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1answer
48 views

A priori estimates for functions in $C_0^\infty (\overline{\Omega})$.

Let $u\in C_0^\infty (\overline{\Omega})$, where $\Omega\subset \mathbb{R}^N$ is a bounded domain. Fix some $a\in \Omega$ and choose $r>0$ such that $\overline{\Omega}\subset B(a,r)$. Define ...
0
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1answer
29 views

Continuity of function does not imply continuity of extension

Let $f$ be increasing on a dense subset $D$ of $\mathbb{R}$, and define $\tilde{f}$ on $x\in\mathbb{R}$ $\tilde{f}(x):=\inf_{x<t\in D}f(t)$. Show that the continuity of $f$ on $D$ does not imply ...
4
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1answer
91 views

Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
3
votes
1answer
48 views

$f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$

Is there exists $f:\mathbb{R^2}\setminus\{(0,0)\}\ \rightarrow \mathbb{R}$ of class $C^2$ for which $f_x(x,y)=\frac{y}{x^2+y^2}$ and $f_y(x,y)=\frac{-x}{x^2+y^2}$ for all ...
0
votes
2answers
73 views

Period of $\frac{\sin(Ny)}{\sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
4
votes
1answer
68 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
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2answers
67 views

Show that the function is Riemann integrable on $[a,b]$ and find $\int_a^b f$

Show that the function is Riemann integrable on $[a,b]$ and find $\int_a^b f$ $$ f(x) = \left\{ \begin{array}{c} 0, &a \le x < c \\ \frac{1}{2}, &x = c \\ 1, &c < x \le b ...
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0answers
48 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
0
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1answer
38 views

Integral over smooth, closed curve of vector field

Why doesn't vector field $v:\mathbb{R^3}\rightarrow\mathbb{R^3}$ given by $v(x,y,z)=(x,\cos y,e^z)$ does not meet $$\int_{\gamma} \langle {v\frac{\gamma'}{\|\gamma'\|}\rangle}\ d\sigma_1=0$$for every ...
0
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1answer
22 views

Surface measure of $A={\{(x,y,z)\in\mathbb{R^3}:z=f(x,y),x^2+y^2<1}\}$

Function $f:\mathbb{R^2}\rightarrow\mathbb{R},\ f\in C^{\infty}$ is Lipschitz continuous with constant $1$ and $$A={\{(x,y,z)\in\mathbb{R^3}:z=f(x,y),x^2+y^2<1}\}.$$ Why does it imply that ...
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0answers
94 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
0
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1answer
68 views

Sufficient condition on convex function such that $f(x) > -\infty$ for all $x$.

Let $f : \mathbb R^n \to [-\infty, \infty]$ convex and let $f(\overline x) > -\infty$ for $\overline x \in \mbox{int}(\mbox{dom}(f)$. Show that $f(x) > -\infty$ for all $x \in \mathbb R$. ...
0
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1answer
44 views

Suppose $f$ is continuously differentiable on $\mathbb R$, $f(0)=0$ and $f(x)f'(x)\leq2$ for all $x\in\mathbb R$. What is the maximum value of $f(4)$

I have assumed that the greatest value for $f(4)$ is obtained when $f(x)\cdot f'(x)=2$. Simply integrating yields the solution $f(x)=2\sqrt x$, so $f(4)=4$. This solution is technically not good ...
0
votes
1answer
34 views

What means $df(\tilde x) \in {\mathcal{L}(\mathbb{R}^n)}$

I'm trying to learn math on my own. The bad thing is, I can't google latex letters and they often have multiple meanings. For exmaple ${\mathcal{L}}$ could stand for lagrangian or something else. The ...
1
vote
0answers
48 views

On $\sigma$-algebra generated by sets

Given $\mathcal{S}$ a collection of subsets of $X$ and $A\subset X$. To show that $\sigma(\mathcal{S}\cap A)=\sigma(S)\cap A$, where for any collection of $\mathcal C$ of subsets of $X$, $\mathcal ...
1
vote
1answer
62 views

$\sigma$-algebra generated by a set

I want to show that if $X$ is an uncountable set then $\mathcal{S}=\{\{x\}:x\in X\}$ generates the $\sigma$-algebra $\mathcal{A}=\{A\subset X: A$ is countable or $X\setminus A$ is uncountable$\}$. I ...
4
votes
1answer
73 views

an argument that strengthen Lusin's theorem

Let $f$ be a measurable function on a subset $E$ of $\mathbb{R}^n$. Lusin's theorem states that for any $\epsilon>0$, there exists a measurable subset $F$ such that $F$ open in $E$, ...
0
votes
1answer
178 views

can anyone give me examples of open subspace of a metric space

Is there anyone who can give me an elegant example of non-empty subspace $A$ which is open in a metric vector space $H$? I know it cannot be found in $\mathbb R^n$..
2
votes
2answers
74 views

mean value property of derivatives in high dimensions

Let $E$ be a path-connected subset of $\mathbb{R}^n$ and $f$ a differentiable function on $E$. Prove or disprove: for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot ...
3
votes
0answers
72 views

Existence of increasing, smooth modulus of continuity

First, recall the definition: Given a function $f:M\to N$, where $M$ and $N$ are metric spaces, a modulus of continuity for $f$ is a function $\omega:[0,\infty)\to[0,\infty)$ such that ...
0
votes
1answer
59 views

Fractional Derivatives on a function with bounded Support

I have a question about functions that have bounded support in $\mathbb{R}$. In particular, suppose that I have a function $f$ with support $A\subset \mathbb{R}$ so that $A$ is compact. Without loss ...