Tagged Questions

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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1
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3answers
14 views

$C_n:=A_n\cap (A_1\cup…\cup A_{n-1})^c$ pairwise disjoint?

Let $\Omega$ be a set and $A_1,...\in Pot(\Omega)$. Why are the sets $C_n:=A_n\cap (A_1\cup...\cup A_{n-1})^c$ pairwise disjoint? I've tried to write it like $C_n\cap ...
3
votes
2answers
28 views

What can we say about the rate of growth of a function growing faster than all polynomials?

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the following: $$ \forall k \in \mathbb{N} \hspace{5pt} \lim_{t \rightarrow \infty} \frac{t^k}{f(t)} = 0.$$ Can we deduce a stronger growth ...
0
votes
4answers
38 views

How to show that $\sum_{n=1}^{\infty}(-1)^n \sin(a/n)$ converges but not absolutely.

How to show that, for any fixed constant $a\in(0,1)$, the series $$\sum_{n=1}^{\infty}(-1)^n \sin\frac{a}{n}$$ is convergent yet not absolutely convergent. My idea is to express sin(x) as series but ...
0
votes
0answers
6 views

Find a right inverse of a map with gauss brackets.

I am having a composition of two maps: $$ f:\mathbb{R}->\mathbb{R_0^+},f(x)=x^2 $$ $$ g:\mathbb{R_0^+}->\mathbb{\mathbb{N}},g(x)=\lfloor x\rfloor $$ $$h=g\circ f:\mathbb{R}->\mathbb{N_0}$$ ...
1
vote
1answer
14 views

Find relations on the real number: transitive and/or antisymmetric

$$I\ am\ searching\ for\ a\ relation\ on\ the\ real numbers\ (\mathbb R ),\ which\ sould\ be:$$ antisymmetric and transitive antisymmetric and NOT transitive NOT antisymmetric ,but ...
5
votes
1answer
29 views

A problem on the sum of the reciprocals of two derivatives

If $f(x)$ is continuous in the closed interval $[a,b]$ and differentiable in the open interval $a<x<b$, and if $f(a)=a$, $f(b)=b$, prove there exist points $x_1$ and $x_2$ with ...
0
votes
2answers
19 views

On converging and diverging sequences and their respective arithmetic mean

I'm working on a problem set which was given by our analysis lecturer (a) Let $(a_n)$ be a convergent sequence with limit $a$. show that the arithmetic mean $$s_n := \frac{1}{n} \sum_{k=1}^n ...
1
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0answers
12 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
2
votes
1answer
54 views

Finding the infinite sum of $e^{-n}$ using integrals

I am trying to understand this: $\displaystyle \sum_{n=1}^{\infty} e^{-n}$ using integrals, what I have though: $= \displaystyle \lim_{m\to\infty} \sum_{n=1}^{m} e^{-n}$ $= \displaystyle ...
0
votes
1answer
12 views

Absolute value of function in Sobolev Space

Assume $1<p<\infty$ and that $\Omega$ is bounded. Now I would like to prove that $u\in W^{1,p}$ implies that $|u|\in W^{1,p}$ but I have no idea on how do this. Could anyone help?
0
votes
1answer
38 views

Partitions of $[0,1]$

Trying to test my understanding of analysis, today I came up with two questions, that will probably look obvious (be patience, because I am self-thaught). Anyway, here they are: 1) Is possible to ...
3
votes
1answer
23 views

Existence of unique solution on $(-\delta,\delta)$ for $f(x)=1+x+\displaystyle\int^x_0\sin(tf(t))dt$

The following was a question previously given in a test at my university: Show that there exists some $\delta>0$ for which there is a unique continuous function ...
4
votes
1answer
43 views

How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) ...
2
votes
1answer
14 views

Prove that there is x, c < x < b, such that f(x) > f(c)

If $f : [a,b] \to R$ is differentiable at c, a < c < b and $f'(c) > 0$, prove that there is some x, c < x < b, such that $f(x) > f(c)$. I'm not totally sure where to begin with ...
0
votes
2answers
30 views

How integrate $ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $

I'm trying to resolve this integral $$ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b $$ I tried with polar coordinates: $$ x = r\cos{\theta} \\ ...
0
votes
0answers
25 views

Does an interval have a countinuous boundary

Does an interval on the real line, e.g. $(0,1)\subset \mathbb{R}$ have a $C^1$ boundary or does the boundary have to be connected, e.g. $B(0,1)$ where the boundary is the disc $D(0,1)$.
0
votes
0answers
8 views

contractive sequence

Consider the polynomial $p(x)=x^3-7x+2$ a.Find a contractive sequence $(x_n)$ that approximates the root $0<x_0<1$ of p. (I was thinking of writing $x=(x^3+2)/7$ b.Determine the minimal number ...
0
votes
1answer
23 views

Convergence of a sequence of linearly independent vectors in normed space

In an infinite dimensional normed vector space is it possible to find a sequence ${v_n}$ of linearly independent vector (so the sequence is a set of linearly independent vectors) each has norm 1 such ...
0
votes
1answer
38 views

Prove that B is vector subspace [on hold]

In $ℝ^∞$ we've got B={$\{x_n\}^{\infty}_{n=0}| \exists K\in~ (0,\infty) ∀n\in \mathbf{N}, | x_n|\leq K n^{-1}$}. Prove that B is vector subspace.
1
vote
1answer
32 views

Banach fixed-point theorem and Picard's theorem, exam revision

As the title suggests this is for exam revision so answers and not just hints are okay. I've got the current two problems I'm doing: $\mathbf{1}.$ Show that there exist a unique continuous function ...
2
votes
0answers
12 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
0
votes
1answer
24 views

Banach spaces, partial sums converge?

Let $(X,||.||)$ be a Banach space. Suppose the sequence $(x_n) \subset X$ is such that $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb{R}$. Prove that $S_n = x_1+x_2+...+x_n$ the "Partial sums" ...
3
votes
2answers
40 views

There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...
1
vote
1answer
30 views

Practical convergence in $C^{\infty}_c$

Let $C^{\infty}_c$ be the space of $C^{\infty}$ functions with compact support in $\mathbb{R}$ with the usual topology derived by the convergence in infinity norm in every $C^{k}_c$. I would like to ...
0
votes
0answers
12 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finit, with ...
0
votes
1answer
31 views

Proper definite of riemann integral (limit version)

I am sort of confused. Suppose we are given the series, $\displaystyle \lim_{n\to\infty}\sum_{k=1}^{n} \frac{k^{99}}{n^{100}}$ How can this be written as an integral, and what would the variable ...
0
votes
1answer
13 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = ...
2
votes
2answers
53 views

Finding the $nth$ partial sum for $e^{-n}$

Here is the question: $$\displaystyle \sum_{n=1}^{\infty} e^{-n}$$ Instead of using the formula of $\large\frac{1}{1-r}$ I want to try to get the partial sums. $S_1 = e^{-1}$ $S_2 = e^{-1} + ...
0
votes
0answers
5 views

Visualizing construction of a certain function

Consider a map $f$ defined on $\mathbb{R}^3$ with the following properties:\ 1) $f$ fixes the poles $(0,0,\pm1)$.\ 2) $f$ is symmetric in the plane $\{x=0\}$ and the plane $\{y=0\}$.\ 3) $F$ is ...
0
votes
1answer
22 views

Is the square root of the absolute functiom differentiable at x = 0?

I've been trying to solve the problem below for hours but so far I haven't managed to find a solution. Help would really be appreciated. Thanks a lot! Problem Show whether or not the function ...
1
vote
1answer
35 views

Is there a function $f \gt 0$ such that $\int f dx=0$?

If $$f \ge 0$$, then $$\int f dx \ge 0$$. There is a function such that $$f \gt 0$$ but $$\int f dx=0$$?
1
vote
1answer
34 views

A sequence of truncates of $f$

If $f$ is measurable and $A>0$ then the truncation $f_{A}$ defined by: $$f_{A}(x)=\begin{cases} f(x)&\text{if $\left | f(x) \right |\leq A$}\\ A&\text{if $ f(x)> A ...
0
votes
1answer
31 views

Integral of the log is less than the integral of the log of the average value

This is an interesting property that I came across while reading an old proof on this website. The poster didn't really explain it, so I thought I might ask. We suppose $u$ is a positive measure on ...
3
votes
1answer
35 views

Compactness and uniform equicontinuous family

Suppose $\mathcal{F} \subset C(A)$ be a family of continuous functions with domain $A$. If $\mathcal{F}$ is pointwise equicontinuous, is it true that $\cal F$ is uniformly equicontinuous? I ...
0
votes
0answers
8 views

Total variation of sum of measures

I'm working on proving that, given two signed measures $\nu_1$ and $\nu_2$ on $(X,M)$ that both omit either $\infty$ or $-\infty$, $|\nu_1 + \nu_2| \leq |\nu_1|+|\nu_2|$ using the definition $|\nu| = ...
0
votes
0answers
10 views

Subadditive sequence over n and convergence. [on hold]

For a subadditive sequence (a), how do I show that (a/n) converges?
2
votes
2answers
57 views

$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} \right)^n} = e^{-\frac{x^2}{6}} $

I am wondering about a limit that wolframalpha got me and that you can find here wolframalpha It says that $$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} ...
2
votes
0answers
13 views

Rearrangement of absolutely convergent series

I would be very grateful if someone would verify whether my proof below is correct. Many thanks. Theorem. $\,$ Let $(b_k)$ be a rearrangement of the complex sequence $(a_k)$. If $\sum_{k\geq 0}a_k = ...
2
votes
2answers
49 views

If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?

Let $x_0 \in \mathbb{R}$ and $f,g: \mathbb{R} \to \mathbb{R}$ such that $f$ is discontinuous at $x_0$ but $g$ is continuous at $x_0$, then $f+g$ at $x_0$ is... My approach: After some graphical ...
0
votes
1answer
22 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
1
vote
0answers
12 views

Determining the sign of a function containing ratio

Problem We want to know for which values of $x,y,z,w$ the function $\sigma$ is positive or negative: \begin{equation} \sigma = \frac{A}{A^2-B^2}, \end{equation} where \begin{eqnarray} A & = ...
2
votes
5answers
54 views

limit laws:$\lim_{n\to\infty}\max(a_n,b_n)=\max(\lim_{n\to\infty}a_n,\lim_{n\to\infty}b_n)$

Let $(a_n)^{∞}_{n=m}$ and $(b_n)^{∞}_{n=m}$ be convergent sequences of real numbers. Let $x$ and $y$ be the real numbers $x:=\lim_{n\to\infty}a_n$ and $y:=\lim_{n\to\infty}b_n$. Show that ...
0
votes
0answers
22 views

Does it really matter that we are using the Taylor polynomial and remainder?

Assuming that the quadrature rule $I_n$ integrates all polynomials of degree less than or equal to N exactly: $I_n(p)$=$I(p)$ for all p $\epsilon$ $P_N$. Using this it could be proved that for any ...
5
votes
2answers
104 views

Is $\mathbb{R}\setminus\mathbb{Q}$ a union of countable family of closed sets?

Can we represent set of irrational numbers as union of countable family of closed sets?
0
votes
2answers
17 views

Solution of $y'=xy^{1/3}, y(0)=0$ equal to $0$ in $[-c,c]$ and positive for $|x|>c$.

I'm looking for a continuous function $y(x)$ which satisfies the above and trying to make it depend on $c$ so that a solution exists for any $c>0$. I read it is possible, but I can't do it... Can ...
0
votes
0answers
17 views

Removable singularities for Dirichlet problems of Laplace equaions?

It is already known that if $u$ is harmonic in $\Omega\backslash\{x_0\}$ where $\Omega$ is a pre-compact domain in $\mathbb{R}^n$, $n\geq2$ and $u=o(|x-x_0|^{2-n})$ when $x\to x_0$, then the singular ...
3
votes
1answer
36 views

Verification of Proof that if f(x) is continuous and periodic then it is uniformly continuous on the reals.

Suppose f is defined on all reals. Then there is a positive p s.t. f(x+p)=f(x) for all x. This is my proof: Assume f is continuous on [0,p] then it is uniformly continuous on [-p,p]. Then for x,y ...
0
votes
1answer
26 views

Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ does.

Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
1
vote
5answers
76 views

Proof the formula

I have to prove the following formula: $$\sum_{k=0}^n \frac{(-1)^k}{k+1} \binom{n}{k} = \frac{1}{n+1}$$ I do have absolutely no clue ye about how to even start. I'm thinking about using binomial ...
-1
votes
2answers
33 views

equivalent statement to limsup [on hold]

How does one prove that, for a sequence $(x_n)$, we have the following equivalence $$\limsup_n x_n=L$$ iff a) and b) hold: \begin{align} a) \forall \varepsilon>0, \exists N: n>N: x_n\leq ...