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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2
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269 views

Prove if $\{f_n\}$ converges in measure to f and g, then f = g a.e.

In my class notes there is a really brief sketch for this proof. Basically by the triangle inequality $\forall \epsilon > 0$ and $\forall k \in \mathbb{N}$: $m(\{x:|f(x)-g(x)|>\epsilon\}) \le ...
0
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0answers
60 views

Points positioned on a surface with maximum distance

Given a spherical shell with area A. I want to arrange n points on this surface in such a way, that the distance between those n points is maximal. Do you know how to do this?(Can we say something ...
0
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1answer
104 views

Commuting Exponential Matrices

Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$. Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal. $A, B$ square matrices. ...
5
votes
1answer
76 views

Homework: Smooth mapping $f$ satisfying $f\circ f=f$

This is an exercise in Mathematical Analysis by Zorich, in the subsection 12.1. Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a smooth mapping satisfying condition $f\circ f=f$. $\quad$a) ...
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2answers
68 views

How can we describe different sorts of limits using ordinals, and what kind of “work” can they do in transfinite induction?

I recently encountered transfinite induction and ordinal numbers for the first time in a real analysis text (Bruckner, Bruckner, and Thomson) and I am still coming to grips with these tools and their ...
0
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1answer
147 views

Convergence to $\delta$ distribution

Show that $$v_{t}(x) = (4 \pi kt)^{- \frac{1}{2}} \exp \left( -\frac{a x ^2}{4kt} \right)$$ converges to $\delta_{0}$ in $D'(\mathbb{R})$ when $t \to 0^{+}$. Asumming that: $$\int_{\mathbb{R}} ...
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2answers
1k views

closure of a set is closed

I know that the closure of a set is closed and the proof can be found in Rudin. However this is done by using the fact that the complement of a closed set is open. I am using a different book now and ...
1
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1answer
197 views

Simple questions about Hausdorff measure

Let $s>0$ and $0 < \delta \leq \infty$. For a set $E \subset R^n$ define $$ H_{s}^{\delta} (E)=\inf \left\{ \sum_i r_{i}^{s}\right\},$$ where the infumum is taken over all coverings of $E$ by ...
3
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1answer
167 views

Question on a functional analysis exercise.

These days I am doing some independent study of functional analysis. While solving problems, I could not handle the following part of an exercise (exercise 13, chapter 1 of Rudin's Functional ...
2
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1answer
219 views

Cauchy-Lipschitz theorem

How to apply Cauchy-Lipshitz to say that the maximal solutions of $$x''+\alpha x'+a x +\sin x =L$$ are defined on $\mathbb{R}$? Thank you.
2
votes
1answer
129 views

If $Df(c)=0$ $\forall c\in V$ then $f$ is constant on $V$ [duplicate]

Theorem: Suppose that $V$ is open and connected in $\Bbb R^n$ $f:V\to \Bbb R^m$ is differentiable on $V$ If $Df(c)=0$ $\forall c\in V$ then $f$ is constant on $V$ I want to prove this theorem ...
2
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1answer
683 views

A sequence of measurable functions and the sup, lim sup of them

So I have stumbled upon the following theorem: Let $\left\{f_n\right\}$ be a sequence of measurable functions. For $x \in X$, put $$ g(x) = \sup \left\{ f_n (x) \mid n \in \mathbb{N} \right\} \\ ...
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0answers
72 views

In what Sobolev classes are the following functions

I need a little help. In what Sobolev classes are the following (give the answer for both $H^{s}$ and $H_{\mathrm{loc}}^{s}$) a. $\delta(x)$ b. $H(x)=\left\{\begin{matrix} 1, x\geq 0\\ 0, x<0 ...
2
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2answers
104 views

Prove $\varphi\in\mathcal{S}(\mathbb R^n)$ if and only if the following inequality holds..

I need some help for showing the following result: Let $\varphi\in C^\infty(\mathbb R^n)$. Then $\varphi\in \mathcal{S}(\mathbb R^n)$ if and only if for all $\alpha\in\mathbb N^n$ and $N\geq 0$ there ...
2
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3answers
78 views

How to show that the distance between these sets is positive?

Let $T_i=\{(1-t)x_i+ty_i;\;0\leq t\leq1\}$, where $x_i,y_i\in\mathbb{R}^n$; $i=1,2$. Could someone help me to prove that if $x_2= (1+\varepsilon)x_1$ and $y_2= (1+\varepsilon)y_1$ for some ...
4
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2answers
653 views

Is there an English translation of Jordan's “Cours D'analyse”

I am trying to find an English translation of Camille Jordan's work "Cours D'analyse". Only the French edition is on Amazon, so since this is a somewhat specialized topic, I thought perhaps someone in ...
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0answers
62 views

Bounded solutions of ODE

I have this ODE $(E)... x''+\alpha x'+ax+\sin x =L , t\geq 0$ we suppose that $a>0$and $\alpha \geq 0$ , and that there existe un constant $C>0$ such that $\frac{a}{4}x^2+\frac{y^2}{2}\leq ...
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2answers
70 views

The complement of the poles of a meromorphic function

Why does the complement of the set of poles of a meromorphic function form a connected subset of $\mathbb{C}$? Thanks for helping.
1
vote
1answer
49 views

How to show this equality?

I need some help for showing the following equality $$\displaystyle \left(\sum_{i=1}^nx_i\right)^m=\sum_{|\alpha|=m}\frac{m!}{\alpha!}x^\alpha$$ for all $x\in\mathbb R^n$. Here $\alpha\in\mathbb N^n$ ...
3
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0answers
76 views

Restriction on exponent; Weak Harnack inequality for strong solutions

The weak Harnack inequality for strong solutions goes as follows (Taking $Lu = a^{ij}(x)D_{ij}u + b^i(x)D_iu+c(x)u$ to be elliptic) Let $u\in W^{2,n}(\Omega)$ satisfy $Lu\leq f$ in $\Omega$ for ...
1
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2answers
176 views

Uniqueness of meromorphic continuation

Let $\Omega$ be a non-empty region of $\mathbb{C}$ and suppose $f$ is a holomorphic function on $\Omega$. How can one show that a meromorphic continuation of $f$ to all of $\mathbb{C}$ is unique, if ...
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2answers
327 views

Short way? Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$

Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$ Does there exist any short way? I have to calculate all partial dervatives. Is it?
5
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1answer
187 views

Can we derive that $A$ commutes with $B$ from this?

Based on some Physics backgrounds, I want to confirm the following thing. Let $[A,B]:=AB-BA$, where $A,B$ are matrices. Now the question is as follows: If for any real number $\lambda$, ...
2
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1answer
318 views

Inverse function theorem application

I have to solve this question with this solution way. But I made some mistakes while solving. I cannot see thesemistakes. And I cannot reach the wanted result properly. Please somebody helps me. ...
0
votes
1answer
36 views

Prove that there exist a function $g(s,t)$ continuously dif on $B_r(s_0,t_0)$ s.t. $x_0=g(s_0,t_0)$ and $x^2+s^2+t^2=1$ for $x=g(s,t)$

I have a question with its solution. But I dont understand some parts of the solution. I wrote why? near inapparent parts. Please can one explain there? Thank you. Question: If $x_0^2+ ...
1
vote
1answer
86 views

Estimate of $|(f*g)(x)-f(x)|$ where $g$ is approximation to the identity

Let $f: \mathbb R \rightarrow \mathbb R$ continuous with compact support $[0,1]$. Assume $|f(x)| \leq M$ for all $x \in [0,1]$. Let $\epsilon > 0$. Then let $0< \delta <1$ s.t. $$ \forall ...
2
votes
2answers
50 views

$f^{-1} $ is continuously differentiable.

Let $f(x,y)=(x^3+y^2, xy+y^4)$. I am trying to show that $f^{-1}$ is continuously differentiable at $(1,-1)$. Solution: $$Df(x,y)=\begin{pmatrix}3x^2 & 2y \\ y & x+4y^3\end{pmatrix} $$ ...
0
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2answers
85 views

What is basis of $\mathbb{R}$

I think it is just 1; but I am also under the impression that it is just any open interval on $\mathbb{R}$. Furthermore, I am trying to figure out how a compact interval $X = [0,1]$ inherite standard ...
4
votes
1answer
71 views

Proving that $\{ f_a \}_{a \in A}$ satisfying $\int \limits_{0}^{2\pi} |f_a(e^{i \phi})|^{1/2} d\phi \leq 1$ is a normal family in $\mathbb{D}$

This is another problem from a complex analysis qualifying exam from last year for the preparation course that I'm teaching right now. The question is the following. Let $F = \{ f_a \}_{a \in ...
5
votes
1answer
42 views

Integrability Question

A question that has popped up while studying for qualifying exams is the following: Prove that $\int_0^1 \int_0^1 \frac{1}{x^p + y^q} dx dy$ is integrable iff $p^{-1} + q^{-1} > 1$ I can handle a ...
0
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0answers
114 views

When does it make sense to say “the smallest measurable set containing x”?

We know for the Borel $\sigma$-algebra that each singleton set is measurable. I was working on the problem of proving that each infinite $\sigma$-algebra has uncountably many members. My solution went ...
1
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1answer
54 views

Induction Using Multi-Index

Does anyone know how to use induction in the context of multi-indices? I know the induction is done on the multi-index length, the main problem is how to split a multi-index of length $n+1$ into one ...
0
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1answer
1k views

Why is the set of all integers not open?

A point $p$ is an interior point of a set $E$ if there exists a neighborhood $N_r(p)$ such that $N_r(p)\subseteq E$, and a set is open if all of its points are interior points. Now my question is ...
1
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1answer
131 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
0
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1answer
65 views

If $u\in L^p$, is $u\in L^q$ for some $q>p$?

(Motivation is below) Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Let $p\in [1,\infty)$ and $u\in L^p(\Omega)$. Is there any $q>p$ such that $u\in L^q(\Omega)$? I already know that ...
3
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3answers
234 views

How to work out this integral?

Stimulated by the physics background of path integral, I want to know how to calculate this integral $\int_{-1}^{1}(1+x)^m(1-x)^ndx$? Where $m$, $n$ are positive integers. I'm struggling with it. ...
3
votes
1answer
88 views

The value of $ A \exp\left(\frac{-1}{2\pi} \int_{-\pi}^{\pi} \ln(1+A+2BC \cos x) dx \right)$

I'm looking for the value of: $$ A \exp\left(\frac{-1}{2\pi} \int_{-\pi}^{\pi} \ln(1+A+2BC \cos x) dx \right)$$ I know we could take $y=1+A+2BC \cos x$ but changing variable in this way makes the ...
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vote
1answer
79 views

construction of a smooth function using mollifiers

let $r>0$ and $B(x_0, r) \subset R^n$ . My problem is construct a function $u \in C^{\infty}_{0}(B(x_0, 2r))$ using mollification satisfying $$u = 1 \text{ on } \overline{B(x_0, r)} $$ and $$ ...
3
votes
2answers
399 views

Proving that a function has a removable singularity at infinity

I'm having trouble with the following exercise from Ahlfors' text (not homework) "If $f(z)$ is analytic in a neighborhood of $\infty$ and if $z^{-1} \Re f(z) \to0$ as $z \to \infty$, show that ...
7
votes
2answers
89 views

Are there any function which's derivate is the scaled one of the original?( $f'(x) = f(cx)$ )

which function could satisfy the following, for a certain $c\ne1$ $f'(x) = f(cx)$ ...beyond the trivial $f=0$ i've been thinking about it for a while. for a simpler case: $f'(x)= f(x+c)$ i've ...
2
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0answers
185 views

Outer measure defined in terms of open intervals vs. closed intervals

This should be a simple question (it's been a long summer break....). I am reviewing some measure theory and noticed in my notes from a class that we defined the exterior measure in terms of the ...
3
votes
1answer
565 views

Compactly supported continuous function is uniformly continuous

Let $f:\mathbb R \rightarrow \mathbb R$ be continuous and compactly supported. How can I prove that $f$ is uniformly continuous ? I was trying to prove it by contradiction but get stuck. My attempt ...
4
votes
3answers
124 views

Convergence of partial sums of real sequences

For all $i\in\mathbb{N}$, let $(a_{i,n})_{n\in\mathbb{N}}$ be a real sequence that tends to $0$ for $n\rightarrow\infty$. It holds also that $|a_{i,n}|\leq1$ for all $i,n\in\mathbb{N}$. Is it possible ...
2
votes
1answer
290 views

Taylor remainder of $f(x,y)=\sin x\cdot \cos y$

Given $f\colon \mathbb R^2\rightarrow\mathbb R,(x,y)\mapsto\sin x\cdot\cos y$ I want to show that there exists $M>0$ such that $$|f(x,y)-T_2(x,y)|\leq M(|x|+|y|)$$ for all $(x,y)\in\mathbb R^2$. ...
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1answer
41 views

Pushforward Filter

While reading some notes I came across the notion of a pushforward pre-filter. If $g:X\rightarrow Y$ is a continuous map of topological spaces and $F$ is a pre-filter on $X$, then then author claims ...
1
vote
1answer
356 views

LU decomposition that is not unique

How can I show that every matrix of the form $\begin{bmatrix}0& x\\0 & y\end{bmatrix}$ has an $LU$ factorization and that even if $L$ is unit lower triangular there is not a unique ...
0
votes
1answer
48 views

sequence question again - i think my book is wrong

So here it says: if a sequence $a(n)\rightarrow a\neq0$ then $a\cdot a(n)>\frac{a^2}{2}$ for every natural $n$ But that implies that if the limit is $2$ for example, then every term in that ...
2
votes
0answers
60 views

weak derivative and the value of a integral

Let $0 < r < R$ and $p>1$ and consider the function $$u(x) = \displaystyle\frac{\displaystyle\int_{|x|}^{R} t^{-1 }dt}{\displaystyle\int_{r}^{R} t^{-1 }dt},$$ if $r < |x|< R$ , and ...
1
vote
3answers
112 views

I should show that $\lim \limits_{(x,y)\to(0,0)}2x\log(x^2+y^2)=0$

I should show that $$\lim_{(x,y)\to(0,0)}2x\log(x^2+y^2)=0$$ which inequality should I use?
0
votes
2answers
227 views

Proving $\|f(x)-f(a)\|\le M\| x-a\|$

Theorem(1): (M.V.T for real valued functions) Let $V \subseteq \Bbb R^n$ be open. Suppose $ f: V \to \Bbb R$ is differentiable on $V$. If $x, a\in V$ then there is $c\in L(x;a)$ such that ...