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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3
votes
1answer
116 views

Help with Dedekind cuts

I am reviewing what Dedekind cuts are for my quiz tomorrow. I had posted a question before about Dedekind cuts and I thought that was the only problem but there were these two problems as well for ...
3
votes
1answer
68 views

Finite family of analytic functions linearly dependent if and only if Wronskian is 0

I know that given two analytic functions on some domain $D$ of the complex plane, then their Wronskian determinant being $0$ is equivalent to them being linearly dependent. I would like to generalise ...
2
votes
0answers
107 views

Boundedness for Reaction Diffusion BVP with Arbitrary Exponent $\alpha$

Let $U\subset\mathbb{R}^{n}$ be open, $U_{T}$ and $\Gamma_{T}$ be the parabolic cylinder and boundary of $U$ for arbitrary $0\leq t\leq T$, respectively, and suppose $u$ solves $$ ...
0
votes
1answer
101 views

Sobolev spaces doubt

Can somebody help me with this doubt? Let $\Omega$ an open set and $A$ be any finite subset of points of $\Omega.$ Is it true the following inequality? $\vert v(a) \vert \leq C \| v \|_p ...
4
votes
2answers
192 views

Tietze extension theorem for complex valued functions

Why is this theorem always only stated for real valued functions, and not for complex valued functions? Thanks.
4
votes
1answer
127 views

integration of $\int_0^\infty\frac{(\log(x))^3}{x^3-1}\,\mathrm{d}x$

I have this integral to solve with complex analysis: $$\int_0^\infty\frac{(\log(x))^3}{x^3-1}\,\mathrm{d}x$$ My result is $$\frac{\pi^3}{54}$$ But I don't know if it is ok..and so I ask you if it ...
6
votes
2answers
403 views

$f$ is integrable, but $f$ has no indefinite integral

Let $$f(x)=\cases{0,& $x\ne0$\cr 1, &$x=0.$}$$ Then $f$ is clearly integrable, but has no antiderivative (primitive), at least on the entire domain of $f$, since any antiderivative function ...
3
votes
2answers
827 views

Does Riemann integrability on closed interval implies uniform boundedness?

Does Riemann integrability on closed interval implies uniform boundedness? My thought process points to yes, because if f is Riemann integrable then it is bounded pointwise on [a,b]. I could be ...
3
votes
1answer
139 views

Show that $(1+ \frac{1}{n})^n$ and $(1- \frac{1}{n})^{-n}$ have the same limit

Let $x$ be positive and $$ a_n = \left( 1 + \frac{x}{n} \right)^n \qquad b_n = \left( 1 - \frac{x}{n} \right)^{-n}. $$ Show that a) The sequence $(a_n)$ and $(b_n)$ have the same limit $\xi =: ...
2
votes
3answers
453 views

Prove integral is greater than $0$

$f(x)$ is Riemann integrable on $I=[a,b]$ and $f(x)>0$ for all $x \in I$, prove $\int_a^b f(x) dx >0$ . Need help on this question, please help me
1
vote
0answers
91 views

Analysis and real analysis

Let $X$ and $Y$ be compact metric spaces. Let $X\times Y = \{(x; y) \,:\, x \in X;\, y \in Y \} $be the cartesian product. Show that any $f \in C(X \times Y )$ can be uniformly approximated by ...
8
votes
1answer
234 views

Does $f\colon \Omega \to \mathbb R$ differentiable imply $f$ locally Lipschitz?

Let $f\colon \Omega \subseteq \mathbb R^n \to \mathbb R$ be a differentiable function. Is it true that $f$ is locally Lipschitz, i.e. Lipschitz on every compact $K \subset \Omega$? If $f$ were ...
5
votes
2answers
202 views

Taimanov's extension theorem [collecting applications]

In a topology course we proved the following theorem: Let $X$ be any space, $D \subseteq X$ dense, $Y$ a compact $T_3$ space and $f: D \to Y$ be any continuous map, s.t. for all disjoint closed ...
1
vote
0answers
34 views

Boundaries- regularity and local parametrization

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^3$ with $C^2$ boundary.Let $x_0 \in \partial \Omega$. We choose a $X_1,x_2,x_3$- coordinate system such that the $x_1,x_2$-plane is ...
2
votes
1answer
189 views

Analyticity of $\frac{Log(z+4)}{z^2+i}$

This problem is from Churchill and Brown. How do I prove that $f(z)=\frac{Log(z+4)}{z^2+i}$ is analytics everywhere except $\pm\frac{(1-i)}{\sqrt{2}}$ and on the portion $x \le -4$ of the real axis. ...
2
votes
1answer
191 views

Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$?

Is there a Bolzano-Weierstrass theorem for the function space $C^k(\bar\Omega)$? More precisely, I want to prove that THEOREM. A sequence $\{f_n\}$ is convergent in $C^k(\bar\Omega)$ (or some more ...
3
votes
2answers
68 views

Showing Solution to Some Random PDE Tends to Zero Uniformly

Another qual problem that is causing me some difficulty... Consider the PDE $$ \left\{\begin{array}{rl} u_{xxt}+u_{xx}-u^{3}=0&\text{in}\;[0,1]\times(0,\infty)\\ ...
3
votes
2answers
125 views

Cauchy Problem for Heat Equation with Holder Continuous Data

This exercise comes from a past PDE qual problem. Assume $u(x,t)$ solves $$ \left\{\begin{array}{rl} u_{t}-\Delta u=0&\text{in}\mathbb{R}^{n}\times(0,\infty)\\ ...
1
vote
4answers
93 views

Simple analytic proof.

If asked to prove that $$e^x>1+x: x>0$$ Can I argue that $$\lim_{x\rightarrow0}\frac{e^x-1}{x}=1$$ and this limit is approached from right side. However, am not confident how I justify it ...
4
votes
1answer
80 views

Strong convergence of multiplication operator

I am looking for a necessary and sufficient condition for a sequence of multiplication operators $T^{(k)}$ to converge to zero strongly. (i.e. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
1
vote
2answers
114 views

Radius of convergence is 1

Let's assume that $\displaystyle\sum_{n=0}^{\infty}a_{n}$ is convergent conditionally. Then prove that the radius of convergence of $\displaystyle\sum_{n=0}^{\infty}a_{n}x^{n}$ is equal to $1$ Please ...
2
votes
1answer
317 views

Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$

Let $(a_n)_n$ be a convergent sequence of integers , what can we say about $(a_n)_n$? (I don't understand what is meant by this question)
3
votes
3answers
265 views

Can a non-zero vector have zero image under every linear functional?

Let $X$ be an infinite-dimensional vector space, and let $x_0$ be an element of $X$ such that $f(x_0)=0$ for every linear functional $f$ defined on $X$. Then can we prove that $x_0$ is the zero vector ...
11
votes
3answers
151 views

Is the function differentiable

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for $x_0 \in \mathbb{R}$ $$ \lim_{\mathbb{Q} \ni h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$$ exists. Is this function ...
3
votes
2answers
357 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
1
vote
3answers
649 views

How to find the norm of this bounded linear functional?

Let $C^\prime[a,b]$ denote the normed space of all continuously differentiable real (or complex) valued functions defined on the closed, bounded interval $[a,b]$ in $\mathbf{R}$ with the norm defined ...
6
votes
5answers
214 views

$(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$.

When I solved a problem, I could solve it if I assumed that $(1+\frac{1}{n\log n})^n-1=O(\frac{1}{n})$ I tried to prove it, but I failed. Actually, I don't convince if it is true. Is it correct? If ...
2
votes
2answers
104 views

Find $\delta$ such that $0<|x-3|<\delta \Rightarrow |\frac{1}{x} - \frac{1}{3}| < 10^{-4}$

I was working on a calculus problem: Find $\delta$ such that $0<|x-3|<\delta \Rightarrow |\frac{1}{x} - \frac{1}{3}| < 10^{-4}$ I did some algebra on the consequent obtaining: $$ ...
0
votes
0answers
40 views

Difficult Series to analyse

I have the following series, $$ \sum_{t=N}^\infty F(t,N),$$ $$ F(t,N) = Max \, \{ \, \, \,1 - A(t) \, \,e^{-\frac{ N \,b}{ t} }\, , \, \, \, \, 0\, \, \, \} $$ where $A(t)$ cannot grow faster ...
4
votes
1answer
99 views

Improper Riemann integral of bounded function is proper integral

Let $f:[a,b) \rightarrow \mathbb R$ be Riemann integrable on each compact subinterval of $[a,b)$ and bounded on $[a,b)$. Let $g:[a,b] \rightarrow \mathbb R$ be arbitrary extension $f$ ( i.e. ...
2
votes
4answers
688 views

Solving a quadratic equation with precision when using floating point variables

I know how to solve a basic quadratic equation with the formula $t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ but I learned that if $b\approx\sqrt{b^2-4ac}$ floating point precision may give slightly ...
0
votes
1answer
27 views

What is $| g(t,x) |$ for multidimensional $g$?

I'm reading a book on ODE, and find $|\cdot|$ is confusing. It says: Consider a function $g:\Omega \rightarrow \mathbb{R}^n$. For every compact $K\subset \Omega$, there exist constants $C$ and $L$ ...
0
votes
0answers
56 views

variational problem

I have: $\Omega \subset R$, be open and bounded, assume that $q \in L^{\infty}(\Omega)$ satisfies $q\geq 0$ a.e in $\Omega$, and let $f :\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such ...
2
votes
1answer
53 views

Morphology of binary images

During the lecture we talked about analysis of pictures and got some exrecises. Other students say that this is very easy but I don't get a good answer. Here the facts: Suppose $A$ is a bounded ...
2
votes
2answers
31 views

Does $ \sum_{n=0}^{\infty} a_n |\ln a_n| < +\infty$?

Assume $\sum_{n=0}^{\infty} a_n < +\infty$ and $a_n > 0$. dose $$ \sum_{n=0}^{\infty} a_n |\ln a_n| < +\infty$$ hold?
5
votes
2answers
89 views

Prove that $\displaystyle\prod_{q\in \mathbb{Q}^{\times}}|q|=1$

Prove that $\displaystyle\prod_{q\in \mathbb{Q}^{\times}}|q|=1$. I don't have a lot of experience working with infinite products, but I read a couple of theorems that say that absolute convergence of ...
0
votes
0answers
35 views

Boundedness in dimension 1 and 2.

Consider $\psi$ in $\mathbb{R}^m$ such that $$\lim_{|x|\rightarrow\infty}\frac{|\psi(x)|}{|x|^{1-m/2}}\leq C,$$ where $C$ is a positive constant. Why in dimensions 1 and 2 is sufficies to assume that ...
0
votes
1answer
70 views

Question about boundness of derivatives.

My doubt is in the paper: Further qualitative properties for elliptic equations in unbounded domains, by Berestycki, Caffarelli and Niremberg (page: 93) My question is simples. For any direction ...
2
votes
1answer
221 views

Rational Function Theorem related to Integration?

I know how to use this algorithm when I am integrating rational functions, but my textbook has omitted the actual proof for why it works. If someone could please help me with this question:
1
vote
2answers
125 views

$\sin x$ does not satisfy this quadratic equation

Prove that $\sin x$ is not a rational function using the fact that it is not of the form $p(x)/q(x)$ where $p$ and $q$ are polynomials. Then, by using the above proof, prove that $\sin x$ does not ...
1
vote
1answer
79 views

Find a majorizing function

Please, could somebody help me find a function $f(x)$ such that $| \frac{1}{n+n^2 \sin(xn^{-2})}| \le f(x)$ for each $n \in (0, \infty)$. $f(x)$ has to be $\ge 0$ for every $x \in (0, \infty)$ and ...
1
vote
1answer
55 views

Analysis convergence

This is a past exam question which im not sure how to solve.. ${For \ n=1,2,3,... consider \ f_n: [0,2] \rightarrow \mathbb{R} \ given \ by}$ $f_n(x)=$ $\left\{\begin{array}{l l}nx & \quad ...
0
votes
2answers
63 views

Trigonometric proof query

I am having trouble proving the following identity (where $m,n \in \mathbb{R}$ are arbitrary): $$\sin(mx)\sin(nx) = \frac{1}{2}[\cos(m -n )x - \cos(m + n)x] \quad (1)$$ By expanding the RHS, I can ...
0
votes
1answer
202 views

Analysis.. Norm on C([a,b])

Let $w:[a,b]\rightarrow \mathbb{R}$ with $ w(x)\geq c>0 $ for some $c \in \mathbb{R}$ and all $x \in [a,b]$. Prove that $$\lVert f\rVert_w \ = \ \displaystyle\int^b_a \lvert f(t)\rvert w(t)\ ...
0
votes
1answer
82 views

Analysis.. Convergence of sequence

I really struggle with understanding convergence and have the following questions.. Determine whether the following sequences converge and if so, give the limit: $x_n = ...
0
votes
2answers
176 views

Product of limsup

Let $f(x)$ be positive and increasing and $g(x)$ satisfy $\limsup_x g(x)=1$. I want to show $\limsup_x f(x) g(x)=\infty$ Is that true and how do i show it? I'm thinking that since $f(x)$ is ...
2
votes
2answers
180 views

If $f_n(x)=x^n$ converges to $f$, why is $f$ not continuous?

I was reading my Analysis course notes and had some trouble. I hope you can help me. Let $C(X)=\{ f | f:X \longrightarrow \mathbb{R} \text{ is a continuous function}\}$. It was already stated and ...
2
votes
1answer
42 views

Anharmonic series, find a permutation of its indices so that its sum is 0

Could you help me solve this problem? Find such a permutation of indices of anharmonic series ($\sum _{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$ ) so that after this permutation its sum equals $0$.
4
votes
1answer
122 views

Polynomial expression of $\frac{\sin x}{x} $

Could you explain to me why $$\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi ^2}\right)\left(1-\frac{x^2}{(2 \pi) ^2}\right)\left(1-\frac{x^2}{(3 \pi )^2}\right)\cdots$$ I've read in this article ...
2
votes
1answer
71 views

Is multiplying by a measurable function $V$ always self-adjoint?

There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator: $$-\Delta + V(x)$$ Is (essentially) self-adjoint on ...