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
49 views

Measurability of multifunctions

If $(T,\mathcal{A})$ is a measurable space , $X$ a meatrizable separable space. $F$ a multifunction from $T$ to compacte subsets of $X$. We want to prove that $F$ is measurable if $\forall U\in X$ ...
1
vote
1answer
133 views

Glue Together smooth functions

Let's say that $f(x)$ is a $C^{1}$ function defined on a closed interval $I\subset \mathbb{R^{+}}$ and $g(x)\equiv c$ ($c$=constant) on an open interval $J\subset \mathbb{R^{+}}$ where ...
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0answers
88 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
1
vote
3answers
1k views

Limit superior and inferior

How can I find the limit superior/inferior of $a_n$, as $n \rightarrow \infty $? $$a_n=\frac{n^2+4n-5}{n^2+9}\sin^2\left(\frac{n\pi}{4}\right), n \in \mathbb N$$ I've tried Wolfram|Alpha, but it ...
7
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1answer
174 views

Stone-Čech compactification of completely regular space

Suppose $X$ is a completely regular space. Let $M$ be the set of nonzero algebra homomorphisms from $BC(X,\mathbb{R})$ to $\mathbb{R}$, equipped with the topology of pointwise convergence. Show that ...
2
votes
2answers
138 views

Absolute convergence of sum of convolutions

How do you prove that suppose $\sum a_n$ and $\sum b_n$ converge absolutely, then the series $\sum c_n$ also converges absolutely where $c_n = a_1b_{n-1} + a_2b_{n-2} + \cdots + a_{n-1}b_1$
5
votes
4answers
304 views

why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?

Why the following integral means the area of surface $f(x,y)=z$? $$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
2
votes
1answer
150 views

Finding a $C^1$ surface inside a convex open set (Rudin chapter 10 problem 29)

The problem is as follows (with $n>1)$: Let $E \subseteq \mathbb R^n$ be a convex open set, and let $F \subseteq \mathbb R^{n-1}$ be it's projection onto the first $n-1$ coordinates. It is clear ...
4
votes
3answers
93 views

Application of Weierstrass theorem

Let f be a continuously differentiable function on $[a.b]$. Show that there is a sequence of polynomials $\{P_n\}$ such that $P_n(x) \to f$ and $P'_n(x) \to f' (x)$ uniformly on $[a,b]$ My approach ...
5
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3answers
236 views

Continuity of $g(\theta) = \frac{1}{2\pi^2\theta^3}-\frac{\pi}{2}\cot(\pi\theta)\csc^2(\pi\theta)$ at $\theta=0$

I have the following function which I'm considering on $[0,1)$ $$g(\theta) = \frac{1}{2\pi^2\theta^3}-\frac{\pi}{2}\cot(\pi\theta)\csc^2(\pi\theta).$$ According to a graph in mathematica it is ...
3
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1answer
72 views

Evaluating difficult spectrum

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, ...
3
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0answers
53 views

Looking for an analysis book which uses linear maps notation for multivariable differentiation

I'm taking an analysis course and I find it quite hard to follow what the professor is saying. So far we've been following elementary real analysis by bruckner^2 and Thompson but for the topic on ...
3
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1answer
57 views

$\lim_{x\to\infty} x^{1+1/x}-x-\log x$ and $\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$

Evaluate $$\lim_{x\to\infty} x^{1+1/x}-x-\log x$$ and $$\lim_{x\to\infty}\frac{x^{1+1/x}-x}{\log x}$$ Would knowing one necessarily give the other?
4
votes
1answer
244 views

How to find $\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$

How to find$$\lim_{n\to\infty}n^2\left(\sin(2\pi en!)-\frac{2\pi}{n}\right)$$ I am very confused. I don't know how to show the limit exists, or what it is.
2
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1answer
35 views

Prove that $\lim_{t\to0}(\log t)(1-(2t)^{t/2})=0$

Please prove that $(\log t)(1-(2t)^{t/2})$ tends to 0 as t tends to 0. http://www.wolframalpha.com/input/?i=lim+t-%3E0+%281-%282t%29^%28t%2F2%29%29logt It seems the limit converges to 0 pretty ...
2
votes
1answer
96 views

Countable sum of continuous functionals.

Suppose $\{f_i\}_{i\in I}$ is a countable family of continuous functionals on a topological space $(X,\tau)$ such that $0\leq f_i \leq 1$. I want to show that $\sum_{i} \frac{f_i}{2^i}$ is continuous. ...
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2answers
130 views

A product of two sums of four squares

I am dealing with a problem and I hope you can help me. I have already proved this: Let us suppose that integers $m$ and $n$ can be written as sum of squares of two integers. Prove that m*n can also ...
2
votes
1answer
71 views

If $\mu$ is a complex measure, every set $E$ has $A \subset E$ so that $|\mu(A)| \ge \frac{1}{\pi}|\mu|(E).$

If $\mu$ is a complex measure on a $\sigma$-algebra $M$, show that every set $E \in M$ has a subset $A$ for which $$|\mu(A)| \ge \frac{1}{\pi}|\mu|(E).$$ The suggestion is as follows: Put ...
4
votes
1answer
74 views

Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$

Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the ...
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3answers
718 views

How to determine whether an integral is convergent

I missed up the last lecture and can't understand how to determine whether an integral with parameters is convergent or divergent? For example: For which values of the parameters $p,q \in ...
2
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1answer
70 views

If $f_k \to 0$ a.e. and $\sum_n n 2^n \mu\{|f_k| \in (2^{n-1}, 2^n]\} \leq 1$ for all $k$, then $\int f_k \to 0$.

(Stanford Real Analysis Qualifying Exam: Spring 2012) (Ideal time: 18 minutes) (a) Let $\mu$ denote Lebesgue measure on $[0,1]$. Let $f_k\colon [0,1] \to \mathbb{R}$ be Lebesgue measurable ...
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4answers
435 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
1
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2answers
85 views

A question about metric spaces

Assume that we have a metric space $(S,d)$ and points $a,b,c \in S$ which statisfy the following conditions: for all $x \in S$, $d(a,x) \leq d(a,b)$, for all $y\in S$, $d(b,y) \leq d(b,c)$. Does ...
1
vote
1answer
61 views

Simpler way to check whether or not a sequence is uniformly convergent

Let $f_n(x)=(1+x^n)^\frac{1}{n}$ on $[0, \infty)$. I want to check if this is uniformly convergent. It's pointwise limit is $$f(x)=\begin{cases} x \text{ if } |x|\geq1\\ 1 \text{ if } |x|<1\\ ...
6
votes
1answer
191 views

Series counterexample

Give an example of series $$\sum_{n=1}^{\infty} a_n \quad \text{ and } \quad \sum_{n=1}^{\infty} b_n$$ such that both converge, and the series $$\sum_{n=1}^{\infty} c_n$$ defined by $$c_n = ...
4
votes
1answer
169 views

Analysis on Improper Integrals

This question is from Munkres' Analysis on Manifolds, section 15 question 1. Let $f: \mathbb{R} \to \mathbb{R}$ be the function $f(x) = x$. Show that, given $\lambda \in \mathbb{R}$, there exists a ...
0
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2answers
50 views

surface integral help

I tried to solve this test question. I dont know if I have to use arc length any help would be appreciate please A water fountain sprays water so that when it falls, its height above the water ...
1
vote
1answer
178 views

Extending a $C^2$-function from a $C^{1,1}$-curve to some neighbourhood

Suppose I have a simple, compact $C^{1,1}$-curve $L$ in $\mathbb{R}^3$ and a $C^2$-function $f$ on it ($C^2$ meaning with two continuous arclength derivatives). Can it be extended to a $C^2$-function ...
2
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1answer
37 views

$f(x) = \inf_{y \in Y} c(x,y) - \inf_{\xi \in X} c(\xi,y) - f(\xi) \Rightarrow f$ is upper semicontinuous

Let $X, Y$ be metric spaces. Given $c: X \times Y \mapsto \mathbb{R}$ continuous, define $$ f(x) = \inf_{y \in Y} \left( c(x,y) - \inf_{\xi \in X} (c(\xi,y) - f(\xi)) \right).$$ Then is $f$ upper ...
0
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2answers
124 views

Is level set of sum of two continuous functions a closed set?

$f^i: R^{n}\to R^{n}$ is a continuous function for $i=1,2$. Let $$M=\{(x,y)\in R^{2n}~|~f^1(x)+f^2(y)=0\}$$ Is $M$ a closed set? If not, can you give a counter example.
0
votes
1answer
279 views

Weak Minimizer of a Functional

I showed that $u(x) = \frac{x^2}{2}$ is a potential minimizer for the functional $\int_0^2 \frac{n}{2}u'(x)^2-nu(x) \, dx$ in $C^2[0,2]$ with $u(0) = 0$ and $u(2)=2$ where $n$ is a positive constant ...
0
votes
1answer
60 views

Limit Involving Integral

I have been struggling with the following problem: $$\lim_{a\to\infty} \int_0^1 \frac {x^2e^x}{(2+ax)} dx .$$ My first reaction was to attempt to move the integral outside the limit, which would ...
1
vote
1answer
31 views

Interval of Uniform Convergence

If the sequence of functions $g_n$ converges to $g$ uniformly on the interval $[1/n, 1]$, where $n$ is a natural number, must it converge uniformly to $g$ on the interval $[0,1]$? I came across this ...
1
vote
1answer
55 views

Sequence of continued powers defined recursively

I have a series ${a_n}$ defined recursively by $a_1 = b$ and $a_{n+1} = b^{a_n}$, with $b \ge 1$ and $n \in \mathbb{Z} $. I am trying to show that ${a_n}$ is bounded above if $1 \le b < 3^{1/3}$, ...
2
votes
4answers
201 views

For $x > -1$ proof that $ \arctan x + \arctan\frac{1-x}{1+x} = \frac{\pi}{4} $

For $x > -1$ proof that $\arctan x + \arctan\dfrac{1-x}{1+x} = \dfrac{\pi}{4} $ I have no idea how to approach this, some kind of help would be greatly appreciated! edit: Thank you all!
1
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0answers
135 views

Rigorous hypothesis for Reynolds' transport theorem

I'm looking for rigorous hypothesis for the application of Reynolds' transport theorem : $$ \frac{\mathrm{d}}{\mathrm{d}t}\left[ \int_{\Omega(t)} \phi({\bf x},t) \mathrm{d}{\bf x} ...
4
votes
1answer
116 views

How much pure math should a physics/microelectronics person know [closed]

I do condensed matter physics modeling in my phd and I was struck up learning quite an amount of physics. But while having done lot of physics courses, I see that if I learn pure math I would ...
1
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1answer
94 views

Showing a function is discontinuous

I've used matlab to get some idea of how the following function behaves: $$g(\theta) = \frac{2}{\theta^3} - \frac{\pi\cos(\pi\theta)}{2\sin^3(\pi\theta)}.$$ It appears that it is discontinuous at ...
6
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1answer
494 views

Cauchy Integral Formula for Matrices

How do I evaluate the Cauchy Integral Formula $f(A)=\frac{1}{2\pi i}\int\limits_Cf(z)(zI-A)^{-1}dz$ for a matrix ...
7
votes
2answers
265 views

Prove the limit at infinity is 0

Suppose $f$ is continuous. For all $x>0$, the limit of $f(nx)$ when $n$ goes to infinity is $0$. Then please prove that the limit of $f(x)$ as $x$ goes to infinity is $0$. (I totally stuck at it) ...
1
vote
1answer
99 views

Correctness and help with Union and intersection proof of Open Sets

I need to prove the following: Let $A$ and $B$ be subsets of a metric space $(X, d)$ show that $A^o \cup B^o \subset (A \cup B)^o$. $(A \cap B)^o = A^o \cap B^o$ Here is my attempt: For 1) Let ...
0
votes
3answers
112 views

Uniform convergence of sequence of functions

Given that $$\gamma_n\rightarrow\gamma$$ uniformly, can we conclude that $$\int^b_a\|\gamma_n'\|\rightarrow\int^b_a\|\gamma'\|$$ uniformly? I know that we even do not have ...
2
votes
3answers
194 views

Some estimation a series by integral

Let $a \in (0,1)$. Does there exist a constant $C>0$ or function $C(a)>0$, which may be a function of $a$ but not $k$, such that $$ \sum_{n=1}^\infty a^n n^k \leq C(a) \int_0^\infty a^x x^k dx ...
2
votes
1answer
83 views

Is this function harmonic? [G-T] page 121

On page 121 of Gilbarg-Trudinger's book (Elliptic PDE of second order) they have the following Green's function in $\mathbb{R}^n (n\geq 3)$: \begin{equation} G(x, ...
2
votes
2answers
81 views

Radius of Convergence for$S=\sum^{\infty}_{0}\frac{2^n(x-2)^n}{(n+2)!}$

Given $$S=\sum^{\infty}_{0}\frac{2^n(x-2)^n}{(n+2)!}$$ After using root test, I got, $$-1\le\frac{2(x-2)}{(n+2)}\le 1$$ The n did not cancel. Now, How do I conclude about radius of convergence? ...
2
votes
1answer
156 views

Correctness of Converging sequence and Adherent Points

$x\in X$ is an adherent point of $A\subset X$ if for every $\epsilon>0$ there exists $y\in A$ s.t. $y\in B(x, \epsilon)$ $B(x, \epsilon)$ is the open ball centered at $x$ with radius $\epsilon$ ...
1
vote
1answer
336 views

Showing a sequence of functions converges uniformly on any bounded interval

Question: Let $\{f_n\}$ be a sequence of continuous functions on $\mathbb{R}$. Let $f_n \to f$ uniformly on $\mathbb{R}$. Let $g_n(x):=f_n(x+\frac{1}{n})$ for $n=1,2,3,....$ Then $g_n \to f$ ...
1
vote
1answer
58 views

how to prove this question about limit and derivative

Suppose $f:(a,b)\to\mathbb R$ that $ f $ satisfies: $$f\in C^1$$ $$\lim_{x\to a ^ +}f^2(x)=0$$ $$\lim_{x\to b ^ -}f^2(x)=e-1$$ if $\forall x \in(a,b) : 2f(x)f '(x)-f^2(x)\ge1 $, then how to ...
0
votes
1answer
127 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
17
votes
6answers
348 views

Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-…$

Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$ The number of signs increases by one in each "block". I have an idea. Group the series like ...