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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1answer
148 views

Rudin: Problem Chp3.11 and need advice.

I am working on the following problems and I have a couple of questions. Suppose $a_n>0, s_n = \sum_{i = 1}^{n}$ and $\Sigma a_n$ diverges. RTP (a) $\Sigma \frac{a_n}{1+a_n}$ ...
0
votes
2answers
32 views

Integral related to harmonic functios

It suppose to be a easy task. But I couldn't solve it (I guess I can't learn much analysis). If $ u $ is harmonic, in the middle of my problem, I need to prove that the integral $ \int _ {\partial ...
1
vote
1answer
142 views

RObin problem (Laplace equation)

Let $\Delta u = 0 $, $ \frac{\partial u}{\partial v}(x) + \alpha u(x) = 0 $ be the Laplace equation with Robin conditions. How do I prove it has at most one solution. If I could prove that any two ...
2
votes
0answers
53 views

Energy Equality: how to derive the energy equalities?

Sorry! I'm wondering if there is some way of deriving the energy equality of a given equation. If possible, I would like to see a specific example, which is $ u_ {tt}- \Delta u + u^3 = 0 $ ($ u $ is ...
3
votes
2answers
288 views

derivative of exponential of matrix trace

What is the derivative of $\sum_{ij}e^{-d_{ij}^2(X)}=\sum_{ij}e^{-\operatorname{tr}(X^TC_{ij}X)}$, w.r.t $X$ where $C_{ij}$ is a constant matrix and $d_{ij}^2(X)$ denotes the squared Euclidean ...
3
votes
1answer
62 views

Can a sequence of a function with a single variable be thought about as a function with two variables?

Long title, but first off is it logically ok to think of $\{f_n(x)\}$ as $f(n,x)$ where $n$ is restricted to a natural number? Second, would this at all be useful? Thus far in my study of sequences ...
4
votes
2answers
444 views

Prove that if $s_n ≤ b$ for all but finitely many $n$, then $\lim s_n ≤ b$.

The question asks me to prove that if $s_n ≤ b$ for all but finitely many $n$, then $\lim s_n ≤ b$ where $(s_n)$ be a sequence that converges. . Here is how I did it but im not sure if its entirely ...
1
vote
2answers
65 views

How to evaluate $\lim_{n \to \infty} \sqrt[n]{\frac{1}{n^2}} $.

I was trying to find the radius of convergence of the power series $$\Sigma \frac{2^nz^n}{n^2}$$ and with the ratio test, found that the radius of convergence is $1 \over 2$. However, I am ...
7
votes
5answers
367 views

Is $\lim_{n \to \infty} \sqrt[n]{\frac{1}{n!}} = 0$? [duplicate]

When I was trying to solve a problem to find the radius of convergence of the power series $$\sum \frac{2^nz^n}{n!}$$ I fully understand that the ratio test works well in this one and the radius of ...
6
votes
1answer
339 views

Given $\Sigma a_n$ diverges show that $\Sigma \frac{a_n}{1+a_n}$ diverges.

Intuitively speaking, I first thought that if the series $\Sigma a_n$ is divergent then $$\lim_{n \to \infty} a_n \ne 0$$ therefore it was clear that $\Sigma \frac{a_n}{1+a_n} $ would be divergent, ...
1
vote
1answer
155 views

Given $\Sigma a_n \to \alpha$ show that $\Sigma \frac{\sqrt{a_n}}{n} \to \beta$ [duplicate]

I am trying to prove that if $\Sigma a_n$ is convergent, then $\Sigma {\sqrt {a_n} \over n}$ is also convergent. I tried to use the comparison test but $\sqrt{a_n} >a_n $ so I couldn't go that ...
4
votes
1answer
113 views

Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
3
votes
0answers
46 views

Differentiability of $\operatorname{dist}(x,\partial \Omega)$ function [duplicate]

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary and set $$\phi(x)=\operatorname{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|$$ for $x\in ...
1
vote
2answers
129 views

Integral defined as a limit using regular partitions

Definition. Given a function $f$ defined on $[a,b]$, let $$\xi_k \in [x_{k-1},x_k],\quad k=1,\ldots,n$$ where $$ x_k=a+k\frac{b-a}n, \quad k=0,\ldots,n \; .$$ One says that $f$ is integrable on ...
5
votes
1answer
79 views

Continuity of function given as a maximum

Let $f(x,y)$ is continuous in $[a,b]\times[c,d]$, and we define the function $g(y)$ as follows $$g(y):=\max_{x\in[a,b]}f(x,y),\quad\forall y\in[c,d].\tag{1}$$ The question is when we can conclude ...
5
votes
1answer
104 views

Proof that $\sum_{n=1}^{\infty} z^{1/n}$ doesn't converge

I believe I found a proof for the divergence of this sum for any value of $z$ besides 0. We can look on the telescopic series: $$\sum_{n=1}^{\infty}z^{1/(n+1)}-z^{1/n} = \lim_{N\rightarrow \infty} ...
3
votes
4answers
148 views

A smooth function instead of a piecewise function

I want to find a smooth function approximating f(x) as best as possible: \begin{equation*} f(x) = \begin{cases} x & \text{if } x \le a,\\ a & \text{if } x > a. \end{cases} \end{equation*} ...
4
votes
3answers
106 views

$\lim_{n\to\infty} a_n=a$ if and only if $\forall p\in \Bbb N$, $\lim_{n\to\infty} |a_{n+p}-a_n|=0$

I'm doing exercises. In the related book, there is a claim. Is this right? I'm not sure. For a sequence $\{a_n\}$, there exists a limit $a$ such that $\lim_{n\to\infty} a_n=a$ if and only if for ...
2
votes
2answers
554 views

Proving $C([0,1])$ Is Not Complete Under $L_1$ Without A Counter Example

I'd like to show that $C([0,1])$ (that is, the set of functions $\{f:[0,1]\rightarrow \mathbb{R} \, \textrm{ and } \, f \, \textrm{is continuous} \}$ is not a complete mertric space under the $L_1$ ...
1
vote
1answer
62 views

Fourier series representing a continuous function?

I am fairly sure the answer to my question is "No", so this is more of an affirmation/reference request question. Given a Fourier series $\sum\limits_{k \in \mathbb{Z}} a_k e^{kxi}$, we can view it ...
2
votes
2answers
830 views

Rudin Theorem 2.47 - Connected Sets in $\mathbb{R}$

I need help with the proof of the converse, as given by Rudin in Principles of Mathematical Analysis, to the following theorem: Theorem 2.47: A subset $E$ of the real line $\mathbb{R}^1$ is ...
1
vote
2answers
87 views

How prove $\mathbb Q$ is close in the following metric space?

assume $(d,\mathbb R)$ be a mertic space such that $$d:\mathbb R\times \mathbb R \to [0,\infty)$$$$d(x,y)= \begin{cases} 0, & \text{if x=y} \\ max\{|x|,|y|\}, & \text{if x$\neq$y} \\ ...
2
votes
0answers
51 views

Space of curves and differentiability

Let $A\subset\mathbb{R^2}$ be an open connected set, and $\Omega=\{c:[0,1]\rightarrow A| c\in C^1\textrm{ and } c(0)=c(1)\}$. Consider the $1$-form $\omega = F_1dx_1+F_2dx_2$ in $A$ such that ...
2
votes
1answer
199 views

left-invariant n-form and metric on a Lie group

These two questions are from my exam practice question sets , which are quite similar. I got some problem understanding and solving both of them . For (a) , I can only substite $dx\wedge dy\wedge ...
3
votes
2answers
287 views

Weak / Classical derivative

I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have $\int f' \varphi = -\int f\varphi'$ for all $\varphi\in ...
6
votes
1answer
108 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
0
votes
2answers
63 views

Derivative of $-e^y = 0$?

I stumbled upon this on wolfram alpha and still wonder why $-e^x$ equals $0$ (third step).
2
votes
1answer
116 views

Proving something is not differentiable

I am looking for confirmation so that I can be sure I understand what is being asked here. I need to show that the following function $f(x,y)$ is not differentiable at $(0,0)$ but that ...
3
votes
1answer
313 views

a sigma-algebra on a countable set is a topology

I am trying to prove the statement in the title, i.e. that a sigma-algebra $\Sigma$ on a countably infinite set $X$ is a topology on $X$. I feel like I have an intuition of why this is true, ...
2
votes
1answer
67 views

Is constructing a function that DNE a sufficient counterexample to show the function does not diverge to $\infty$?

Prove or disprove: If $f(x)\to 0$ as $x\to a^+$ and $g(x)\geq 1$ for all $x\in \mathbb{R}$, then $g(x)/f(x)\to\infty$ as $x\to a^+$. Counterexample: Let $f(x)=0$ and $g(x)=1$ for all ...
4
votes
1answer
289 views

Proof of weak maximum principle.

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
4
votes
1answer
179 views

Local integrability of the convolution of a function with a distribuition

Let $G_n$ be the following distribuitions for $n\geq3$ (for $n=2$ it is just a function) in $\mathbb{R^n}$ (the fundamental solutions of the Laplace equation in $\mathbb{R^n}$ ): ...
5
votes
1answer
823 views

Prove or disprove: if $f$ and $fg$ are continuous then $g$ is continuous.

Prove of provide a counterexample: Suppose that $f$ and $g$ are defined and finite valued on an open interval $I$ which contains $a$, that $f$ is continuous at $a$, and that $f(a)\neq 0$. Then $g$ is ...
1
vote
2answers
281 views

Application of the implicit function theorem

Assume that the equation $F(x,y,p)=0$ defines a regular submanifold $M$ of $R^3$. Consider the projection $\pi :M \rightarrow R^2$, given by $\pi (x,y,p)=(x,y)$. By the implicit function theorem, in ...
5
votes
3answers
150 views

How to calculate this integral

I have a function $u(x,y)$ defined on the domain $|x|<\infty, y>0$. I know that $$ \frac{\partial u(x,y)}{\partial y} = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{f(w)}{y^2 + (x-w)^2}dw$$ How ...
3
votes
1answer
291 views

A tricky analysis problem

Let $n,m \in \mathbb{N}, n>4,m\geq1$ We define \begin{equation} f_i(x)=\begin{cases} m_ix, & \text{if $x ...
4
votes
1answer
828 views

Can I exchange limit and differentiation for a sequence of smooth functions?

Let $(f_n)_{n\in \mathbb N}$ be a sequence of smooth functions converging to some $f$. Under what circumstances can I exchange limit and derivative?, i.e. $$\lim_{n\rightarrow \infty} \frac{\partial ...
1
vote
1answer
34 views

how can I get the continuity?

I'm reading 'Foruier analysis methods for PDE's'by R.Dancin.On page 43, at the end of thereom 2.2.3,to prove u belongs to $$C([0,T];\dot{B}_{p,r}^s)$$,the author used the density of ...
0
votes
1answer
74 views

showing $\inf\{|s_n| : n ∈ N\} > 0$ for a convergent sequence of real numbers

My book proves this as one of their examples which is Let $(s_n)$ be a convergent sequence of real numbers such that $s_n \ne 0$ for all $n ∈ N$ and $\lim s_n = s \neq 0$. Prove $\inf\{|s_n| : n ∈ N\} ...
4
votes
0answers
188 views

Stokes' Theorem problem

Let $M \subset \mathbf{R}^n$ be oriented compact smooth $k$-manifold and $\alpha$ be a $C^1$ diferential $(k-1)$-form defined in a neighborhood of M. Use Stokes' theorem to prove that \begin{align*} ...
5
votes
1answer
129 views

I want to study $\sqrt[n]{n}$ and its behavior.

As I was studying some limit problems, I came across $$\sqrt[n]{n}$$ and astoundingly found out that the graph of this has a maximum when $n = e$. I thought there is no way that this is not a ...
2
votes
0answers
100 views

First time dealing with limits with complex numbers in it.

I am solving the following problem. Investigate the behavior (convergence of divergence) of $\Sigma a_n$ if $$a_n = \frac{1}{1+z^n}, \quad \text{ for } z \in \Bbb C.$$ First of all, I am ...
1
vote
2answers
203 views

Definition of $C^k$ boundary

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition. I need this 'cause I'm not understanding what the Evan's book said. ...
2
votes
1answer
76 views

Continuity and Semi-continuity

$U=\{u=(x,y)\in R^2 : y\geq 0\}$ $J(u)= x^2+y^2$ for $x>0$ and $J(u)=0$ for $x\leq 0$ $U_1=\{(x,y): x>0, y\geq 0\}$ I have to show that $J$ is continuous on $U_1$ and semi-continuous on $U$. ...
3
votes
3answers
278 views

Introduction to Pseudodifferential operators

I'm interested in elementary introduction to pseduodifferential operators and its application to hyperbolic PDE's. I know measure theory, Fourier analysis and some elementary(linear) hyperbolic PDE's ...
2
votes
3answers
149 views

Without using L'Hospital's rule, I want to find a limit of the following.

Given a series with $a_n = \sqrt{n+1}-\sqrt n$ , determine whether it converges or diverges. The ratio test was inconclusive because the limit equaled 1. So I tried to use the root test. So the ...
8
votes
2answers
2k views

Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

I am stuck with the following problem. Prove that $$\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} a_n + \limsup_{n \to \infty} b_n$$ I was thinking of using the triangle inequality ...
1
vote
1answer
521 views

Finding the upper and lower limit of the following sequence.

$\{s_n\}$ is defined by $$s_1 = 0; s_{2m}=\frac{s_{2m-1}}{2}; s_{2m+1}= {1\over 2} + s_{2m}$$ The following is what I tried to do. The sequence is ...
4
votes
4answers
339 views

The limit $\lim\limits_{n\to\infty} (\sqrt{n^2-n}-n)$. Algebraic and intuitive thoughts.

I am working on the following problem. Find the limit of $$\lim_{n \to \infty} (\sqrt{n^2-n}-n)$$ Intuitively, I want to say it's $0$ because as $n \to \infty$, $\sqrt{n^2-n}$ behaves like $n$ ...
2
votes
1answer
121 views

How to show $C_p^k([-\ell, \ell])$ is not a Banach space?

I need to show the space $$C_p^k([-\ell, \ell])=\{f\in C^k(\mathbb R; \mathbb C); f(x+2\ell)=f(x), \forall x\in\mathbb R\},$$ is not a banach space with the norms ...