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
18 views

Countable and uncountable sets.

a) Show that $\left \{ n^{2}+m^{2}:n,m\in \mathbb{N} \right \}$ is countable. b) Show that $\left \{ x\in \mathbb{R}:x(x-2)<0 \right \}$ is uncountable. My answers: a) Is it possible to define ...
0
votes
1answer
9 views

Determine $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$, and is $u$ $\mathcal B(\mathbb{R})/\mathcal B(\mathbb{R})$-measurable?

Let $u:\mathbb{R}\to\mathbb{R}$ be given by $u(x)=\left \lfloor x \right \rfloor$. Determine the set $\left \{ u\geq a \right \}$ for all $a\in \mathbb{R}$. Show that $u$ is $\mathcal ...
1
vote
2answers
16 views

Compact operators, space of sequences

Let $\phi\in\ell^\infty$. For $p\in[1,\infty]$, define $M_\phi:\ell^p\to\ell^p$ by $$M_\phi(f)=\phi f.$$ Show that $\Vert M_\phi\Vert=\Vert\phi\Vert_\infty$, and $M_\phi$ is compact if and only if ...
1
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0answers
18 views

Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
2
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0answers
39 views

Real Analysis texts: Royden versus Stein & Shakarchi. Which is better? (and other suggestions welcome)

I am taking an introductory "graduate" analysis class and am comparing Analysis books that cover measure theory. I have had an "advanced calculus" class that covered the standard topics. I am having ...
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2answers
20 views

Use mathematical induction to prove Σ n,k=1 (1/k(k+1)) = (n/n+1) for all n in Natural numbers?

This is how far I can get: p(n): nΣk=1 (1/k(k+1)) = (n/n+1) p(1): 1Σk=1 (1/(1+1)) = (1/1+1) => 1/2 = 1/2 p(1) is true. Assume that p(k) is true. p(k) = kΣk=1, (1/k(k+1)) = k/k+1 Show ...
1
vote
1answer
26 views

Polar Coordinates in $\mathbb R^n$

After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the ...
0
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0answers
22 views

Basic analysis quetion [on hold]

If $f: \mathbb R→\mathbb R$ ($\mathbb R$ is real numbers) be a function and $f(x)=\inf\{|x-me|:m\in\mathbb Z\}$ ($\mathbb Z$ is integer numbers) then $\{f(n)\}_{n=1}^\infty$ have subsequences set ...
0
votes
2answers
25 views

Proof of the product rule for the divergence

How can I prove that $\nabla \cdot (fv) = \nabla f \cdot v + f\nabla \cdot v,$ where $v$ is a vector field and $f$ a scalar valued function? Many thanks for the help!
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0answers
6 views

Spaces of homogeneous type is seperable

Can anyone suggest me some reference for the following proof? $(X,d,\mu)$ is a space of homogenoeus type. Prove that it is seperable.
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votes
2answers
28 views

Fixed point in compact metric space

I guys! I try to solve the following small problem. However, I'm not able to prove the second part. In particular, I have some problems in using the compactness hypothesis on $X$ to find proper ...
0
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0answers
14 views

How can you find the integer part of y starting from this inequality?

How can you find the integer part of y starting from this equality? (I need a precise procedure, not only the number)
2
votes
1answer
45 views

Can the act of Taylor expansion be written as an exponential?

For example: $f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2!}f''(x_0)(x-x_0)^2+\dots=\exp\left((x-x_0)\frac{\mathrm{d}}{\mathrm{d}x}\right)f(x)\Big|_{x_0}$ I don't know how to write the $\Big|_{x_0}$ to the ...
1
vote
1answer
7 views

Help understanding the proof of $\mu$-completion of a sigma algebra $\mathfrak M$

The theorem (from Rudin) states: For a measure space $(X,\mathfrak M, \mu)$, let $\mathfrak M^*$ be the collection of all $E\subset X$ for which there exists sets $A$ and $B\in\mathfrak M$ such that ...
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0answers
20 views

Proving a subspace is a closed subspace of $C[0,1]$ with inner product?

Consider the inner product space of continuously differentiable functions $C^1 [0,1]$ with the inner product: $$<f,g> = \int^1_0 f(x) \overline{g(x)} dx + \int^1_0 f'(x) \overline{g'(x)} dx$$ ...
0
votes
1answer
20 views

Advanced Calculus Question. Prove (sn + tn) is a Cauchy sequence

Based on the definition of a Cauchy sequence, that if (sn) is a Cauchy sequence and (tn) is a Cauchy sequence, then (sn + tn) is a Cauchy sequence I try to work from the definition that |(Sn + tn) ...
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0answers
21 views

Regular surface/normal line

Let $\mathcal A$ be a regular surface in $\mathbb R^3$ and $P$ a point in $\mathbb R^3\setminus\mathcal A$. Suppose that $C$ is a point at minimum distance from $P$. Show that $P$ belongs to the ...
-1
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0answers
19 views

computational question concerning singular integral theory [on hold]

(I have posted this question yesterday, but it remained unanswered. I am changing the title of the question and posting it again hoping that other might pay attention) Let $m\in ...
-1
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0answers
32 views

Ideias for solve this problem in context PDE [on hold]

Im tryng solve this but Im not ideia how. Can someone help-me? Let $a_i$ for $ i=1,\cdots,n,$ be nonnegative $\cal{C}^1(\mathbb{R})$ functions such that $$\mid a\mid\leqslant\dfrac{1}{k}, ...
9
votes
3answers
121 views

Possible new definition of Gamma (Euler-Mascheroni Constant): $\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$

I think I've discovered a new definition for the Euler-Mascheroni Constant (Gamma) I can't find it online anywhere, has anyone seen it before? $$\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$$
1
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1answer
26 views

Continuous functions unbounded on set

For Each of the sets construct a continuous function that is unbounded on the set. $\Bbb N$ $(2,3)$ $\left\{\frac 1 n \mid n \in \Bbb N\right\}$ $[0, \sqrt 2]\cap \Bbb Q$ ...
2
votes
1answer
19 views

Basic analysis question: $\max_{1\leq i\leq n} a_i\geq n\epsilon\implies a_n\geq n\epsilon$

This is a follow up to something I asked earlier. (This question is self-contained so you don't need to click the link.) Thank you very much for your help! Question: $\{a_n\}$ is a sequence of ...
0
votes
1answer
13 views

Analysis: Proof checking and help on 2nd part (Integrals)

So I have the question, $f(x)= x$ if $0$ $\leq$ $x$ $\leq$ $1$ and $f(x)= x+2$ if $1<x$ $\leq$ $2$ (the same f(x) I just couldn't figure out how to do the big bracket) Part 1 is asking me to ...
0
votes
1answer
26 views

Show that $\int_{X}u\, \mathrm{d}\mu\leq 4$ and $\int_{X}u\, \mathrm{d}\mu=1$.

Let $(X,\mathcal{A},\mu)$ be a measureable space. Let $u\in \mathcal{M}_{\mathbb{R}}^{+}(\mathcal{A})$ and $\lbrace u_{j}\rbrace_{j\geq 1}$ be a sequence of functions in ...
0
votes
4answers
34 views

Verifying my proof that if $|S(x)| \leq 1$, then $\lim_{x \to 0} x\cdot S(1/x) = 0$

Question: Suppose that $S : R \to R$ is a function so that for all $x$, $−1 \leq S(x) \leq 1$. Prove from the limit definition that $$\lim_{x\to0} x \cdot S(1/x) = 0.$$ This is my ...
0
votes
0answers
24 views

Find the pointwise limit of {gn} on [0, ∞). Please help!!

Consider sequence of functions gn(x)=x\over 1+x^n over [0,\infty) (a) Find the pointwise limit of {gn} on [0, ∞). g(x)= x 0 \le x \lt x 1/2 x=1 00 x>1 Show gn(x) ...
3
votes
1answer
25 views

First order PDE with discontinuous coefficients

I want to consider the following equation $$u_t+\mathrm{sgn}(x)u_x=0,\,\,u(0,x)=u_0(x)$$ Now if $x>0$ or $x<0$ I can use the method of characteristics to obtain $u(t,x)=u_0(x-t)$ if $x>t$ and ...
1
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0answers
26 views

Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n $ in a textbook,provided that either $e$ or $n$ is large. We ...
2
votes
1answer
36 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
2
votes
1answer
53 views

How to show $\exp(tX)\exp(tY)=\exp(t(X+Y)+tR(t))$ with $\displaystyle \lim_{t\to 0} R(t)=0$?

Let $X\in GL(n, \mathbb R)$. The exponential of $X$ is the matrix given by $$\exp(X)=\sum_{n=0}^\infty \frac{X^n}{n!}.$$ I need some help for showing the following result: ...
0
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0answers
24 views

Determine integrals $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}$ and $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}$.

Consider the function $u:\mathbb{R}\to [0,\infty]$ given by $$ u(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(x) $$ I have determined that $\int_{\mathbb{R}}u\, \mathbb{d}\lambda=\pi^{2}/6$ where ...
1
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0answers
45 views

Ode with step function in the right-hand-side

I want to solve the following ODE: $$\dot{X}(t,x)=F(X(t,x))$$ where $F(x)=1$ if $x>0$ and $-1$ if $x<0$. How to treat this discontinuous right-hand-side?
0
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2answers
26 views

Limits of functions in metric spaces

My teacher said that in the definition of limit, the point in the domain, must be of accumulation, because otherwise the limit is not unique. Why? If the point is isolated, the function is continuous, ...
0
votes
1answer
61 views

Prove (or disprove) this $\sum_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergence? [on hold]

Let $a_{n}>0$ be a sequence, and $0<a\le 1$, such that $\sum\limits_{n=1}^{\infty}(a_{n})^a$ converges. Prove or disprove $\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergent. I ...
0
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0answers
54 views

Acceptance of Facts in Mathematics [on hold]

I have a simple question about acceptance of conventions/facts in mathematics. Tell me: Is it an accepted fact in the world of mathematics that something written like $\sin(x)$ would be considered ...
1
vote
2answers
26 views

Time derivative of operator

I have to compute, at least formally, the following derivative $$\partial_t \exp(it\Delta)f(x-ct)$$ where $\Delta$ is the Laplacian and $c$ is a constant. I know that $e^{it\Delta}$ is the Schrodinger ...
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0answers
14 views

Theorem about n=1 wave equation in Evans

In Evans, PDE edition 2 on p68 we have a Theorem that tells us some properties about the solution to the wave equation for $n=1$. It reads: Assume $g \in C^2(\mathbb{R})$, $h\in C^{1}(\mathbb{R})$, ...
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1answer
20 views

Let A be a bounded infinite subset of R^2, show that A has at least a limit point.

Let A be a bounded subset of R^2 with infinite points, show that A has at least one limit point. How can a prove that without using compactness?
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votes
1answer
15 views

Correction of Proof that if f:[0,1] is a continuous function and f(x)>2 with x being in [0,1) it is not necessary that f(1)>2

Here's how I wrote it up: To approach this consider an example of a continuous function which fails to satisfy f(1)>2 even though it satisfies f(x)>2 for x in [0,1). My counterexample is a negative ...
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0answers
21 views

Show that there is a θ, depending on α such that 0 < θ < 1 [on hold]

Here is the question that I am struggling, please help.
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0answers
30 views

Method of characteristic for second order pde

Can I use the method of characteristic to solve second order pdes? For instance I canconsider the equation $$u_t+u_x=u_{xx}$$
0
votes
1answer
30 views

Sequence of functions that converges a.e. but not in the $L^1$ norm

How can I construct a sequence of functions that converges a.e. but it does not in the $L^1$ norm?
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2answers
33 views

$f(x)=1$, for every $x \in [0,1]$ if $f:[0,1]\to\mathbb R$ is continuous and $f(p)=1$ for every $p\in [0,1]\cap\mathbb Q$.

How would you approach this if I have to use the fact that "every number is a sequence of rational numbers"? Currently, I am proving this by contradiction in the following way: Let f(p)=1 for all ...
1
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0answers
30 views

Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$

Given two sets $A,\; B$ and that $|A| = |B|$, show that $|2^A| = |2^B|$. Intuitively, I think this is true, but I am having trouble showing this formally. I know that there exists a bijection $f: A ...
0
votes
2answers
53 views

If $\sum a_n<\infty$ then $\exists (b_n)$ [duplicate]

If $\displaystyle\sum_{n=1}^\infty a_n<\infty$ and $a_n>0$, then $\exists (b_n)$ such that $b_n\geq1$, $b_n\to+\infty$ and $\displaystyle\sum_{n=1}^\infty a_nb_n<\infty$.
0
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0answers
14 views

P-a.s inequality meaning

If I have the following definition [ \begin{split} \mathcal{L}^0(\mathbb{R}^N)&=\mathcal{L}^0(\Omega,\mathcal{F},\mathbb{P},\mathbb{R}^N):=\{\textbf{X}=(X^1,\cdots,X^N)|X^n\in ...
0
votes
0answers
8 views

Proving properties of nth roots

First let me define some things. Let $x \gt 0$ and $n \ge 1$. Now $x^{\frac{1}{n}}:=\sup\ [ y\in \mathbb R : y \ge 0 \text{ and } y^n \le x]$ (a) If $x \gt 1$ then $x^{\frac{1}{k}}$ is a decreasing ...
0
votes
0answers
10 views

characterisation of continuity of a function in two variables in polarcoordinates

As the the titel of my question already indicates the question I have is about continuity. I know the "$\epsilon-\delta$ Definition" of continuity and think I have understood it. On the internet I ...
3
votes
2answers
74 views

Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$

Does the series $$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$ converges? My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n! $$\frac{(2n)!}{n!n!4^n} ...
0
votes
0answers
21 views

Integrating a particular function.

Could someone please show me how to integrate the following: $\int_{-\infty}^{\infty}p(x)L(p(x))dx$, where $L(x)$ is defined as $L(x)=\frac{x-1}{ln(x)}$ and $p(x)=(2\pi ...