Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
0
votes
0answers
22 views
Which numbers of [0,1) have a unique base g expansion?
Good evening,
i know that is question is rather standard, but unfornunately I have not much knowledge of number theory.
Take $2 \leq g\in \mathbb{N}$. I know that every $x \in [0,1)$ can be ...
3
votes
0answers
31 views
The derivative of a family of flows
If one has a family of flows, can one describe the derivative in the "family" direction?
Specifically, let $M$ be a smooth manifold and let
$X_{s,t}$
be a 2-parameter family of fields on $M$. That ...
3
votes
3answers
72 views
Regarding $\lim_{n \to \infty} n^{\frac{1}{n}}$
Suppose $\lim_{n \to \infty} n^{\frac{1}{n}} = l \in \mathbb{R}$. The function $f(x) = x^n$ is continuous, then
$$l^n=\left (\lim_{n \to \infty} n^{\frac{1}{n}} \right)^n=\lim_{n \to \infty} \left ( ...
9
votes
0answers
63 views
$\int_{\mathbb{R}}|f(t)|^2dt=\int_{\mathbb{R}}|f'(t)|^2dt$ implies $f(t)=\mathbb{x}_{i}|f(t)|$
Let $f \in C^{1}(\mathbb{R},\mathbb{R}^m)$ be such that $f$ and $f'$ are integrable and
$$\{t:f(t)=0\} \subset \{t:f'(t)=0\}$$
$$ |\{t:f(t)=0\}|=n\in \mathbb{N}$$
Prove that if ...
6
votes
3answers
85 views
Question about a proof that $\mathbb{Q}$ is dense in $\mathbb{R}$
This is from Ross's elementary analysis book. The statement is if $a,b \in \mathbb{R}$ such that $a<b$ then there exists a rational $r \in \mathbb{Q}$ such that $a<r<b$.
I don't understand ...
1
vote
1answer
46 views
Wealth indicator function for bidder agent logic
I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others).
Given:
Total number of competitors $n$
Amount of all ...
1
vote
1answer
34 views
question about Morse theory in Hilbert space
This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of :
The qth singular relative homology groupe
The qth critical group
Please;
...
6
votes
2answers
110 views
If $\left| f'(x) \right| \leq A |f(x)|^\beta $ then f is a constant function
Problem Let $f(x)$ be a differentiable function on $[a,b]$ satisfying $f(a)=0$. If there exist $A \ge 0$ and $\beta \ge 1$ such that the inequality
$$\left| f'(x) \right| \leq A \left| f(x) ...
1
vote
2answers
25 views
Showing that an absolute integrable monotone decreasing function $f: [1,\infty[ \rightarrow \mathbb{R}$ is in $L^p([1,\infty[)$
For an exercise in my analysis course, I have to show that: if $f: [1,\infty[ \rightarrow \mathbb{R}$ is monotone decreasing and $f \in L^1([1,\infty[)$, then $f \in L^p([1,\infty[)$ for every $p > ...
0
votes
1answer
38 views
Question about eigenvalues
I have this :
i dont understand why they write $\lambda=m^2 , m\in \mathbb{N}\cup\lbrace0\rbrace$ ,
it's right that $\lambda=m^2$ is the eigenvalues of $(P_0)$ ,but $0$ is not an eigenvalue !.
...
2
votes
1answer
39 views
Does $f(x)=x^{2}\sin\left(\frac{1}{x^2}\right)$ satisfy the relation $f(x)+f(y)−2f\left(\frac{x+y}{2}\right)=O\left(\left|x−y\right|^2\right)$?
Does $f(x)=x^{2}\sin\left(\frac{1}{x^2}\right)$, $x\in(0,1)$ satisfy the relation $f(x)+f(y)−2f\left(\frac{x+y}{2}\right)=O\left(\left|x−y\right|^2\right)$?
1
vote
0answers
30 views
Asymptotic growth over an interval
Given a function $f(x)$, we can define the new function
$$
A_f(t) = \limsup\limits_{x\to\infty}\ (f(x+t) - f(x))
$$
Is there a place that this transformation has been studied?
Also, given a positive ...
1
vote
1answer
39 views
Does $f(x)=x^{2}\sin(\frac{1}{x^{2}})$ satisfy the relation $f(x)+f(y)-2f(\frac{x+y}{2})=O(|x-y|^{2})$?
Does $f(x)=x^{2}\sin(\frac{1}{x^{2}})$ satisfy the relation $f(x)+f(y)-2f(\frac{x+y}{2})=O(|x-y|^{2})$?
I can't check it. Who will hint it? Please.
2
votes
0answers
49 views
Symmetry between differentiation and integration [duplicate]
I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!!
as experience told me, ...
0
votes
1answer
26 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
7
votes
3answers
121 views
Does $f, f' \in L^1([0, \infty))$ imply that $\lim_{x \to \infty} xf(x) = 0$?
Does $\int_0^\infty |f(x)| \, dx$ and $\int_0^\infty |f'(x)| \, dx$ being finite imply that $\lim_{x \to \infty} xf(x) = 0$?
(Context: I am working through an analytic number theory textbook. In a ...
0
votes
1answer
32 views
Work to provide explanation on the definition of the area of a Jordan-measurable set
The problem is as follows:
Given this theorem:
Let $D$ be bounded & Jordan-measurable set
Let $f$ be a bounded function on $D$
And $f$ is continuous except for a set of zero ...
3
votes
1answer
49 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
1
vote
1answer
65 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
0
votes
1answer
44 views
Continuous function on a closed set
Let $f: F \to \mathbb R$ be defined in a closed set $F \subset \mathbb R$. Show that $f$ is continuous if and only if for all $c \in \mathbb R$, the sets $E[f \le c]=\{x \in F; f(x) \le c\}$ and $E[f ...
4
votes
3answers
80 views
What is the domain of $x^x$ when $ x<0$
I know that $x^x$ for all $x>0$
but what is negative values for that function which give a real number
for example $$f(-1)=(-1)^{-1}=-1\in R$$
I try to put sequence for that but i faild
is ...
1
vote
1answer
29 views
Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)
Got a quick question from a past exam paper.
If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
1
vote
0answers
48 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
0
votes
0answers
19 views
Step functions are dense in the intergrable functions
I am trying to show that the set of step functions $X=\{\mbox{step functions} \ I\rightarrow\mathbb{C}\}$ are dense in the set of $Y=\{\mbox{intergrable functions}\ I\rightarrow\mathbb{C}\}$ with ...
0
votes
0answers
30 views
If a function $f:J\to\mathbb{R}$ satisfies the Zygmund condition, is it $C^1$?
A function $f\colon J\rightarrow \mathbb{R}$ on an open interval $J$ satisfies Zygmund condition if, for all
$x,y\in J$, $$f(x)+f(y)-2f\left(\frac{x+y}{2}\right)=o(|x-y|).$$ It is clear, if $f\in ...
1
vote
2answers
41 views
Can a function be uniformly continuous on an open interval?
I am learning analysis and all the uniformly continuous functions I have seen are over a closed interval. So, can a uniformly continuous function be defined on an open interval?
2
votes
2answers
58 views
Proof on showing $\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq (b-a) f(\frac{a+b}{2})$ for class $C^2$ function $f$
The task is as follows:
Given:
(a) function $f \in C^2$
(b) $f \geq 0$ and (c) $f'' \leq 0$ on $[a,b]$
Goal:
Show
$$\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq ...
-1
votes
0answers
50 views
Mathematical Metric spaces
How can we show that any finite measure on a separable complete metric space is tight?
By tight, given $\epsilon > 0$, showing that there exists finitely many points $x_{1},\ldots,x_{n}$
3
votes
1answer
50 views
Question regarding $\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right)$
I wanted to find out whether the following limit exists, and find the value if it does.
$$\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right).$$
Attempt
After many ...
4
votes
1answer
52 views
Closed form of an integral
Is there a closed form of $$\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx \quad \text{?}$$
I just know that ...
2
votes
2answers
41 views
Is the following version of the fundamental lemma of the calculus of variations valid?
Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary $\partial U$. Consider a function $f$ in $L^2(u)$. Suppose that for every $h$ in the Sobolev space$ H^2_0(U)$ it holds that
$$\int_U f ...
2
votes
2answers
32 views
Finding a strong enough solution to a specific PDE problem.
Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression
$$\Delta u = f.$$
$$\text{+"convenient boundary conditions"}$$
In my specific case $f\in H^2_0$. Under ...
-1
votes
0answers
17 views
Help with Toeplitz operators applications. [closed]
I am trying to find a physics problem which solution involves Toeplitz operators.
2
votes
4answers
64 views
The value of $\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}$
I want to find the value of
$$\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}.$$
Since $x \rightarrow + \infty$, I only consider the value of the function for $x \ge 0$, i.e.
$$\lim_{x\to +\infty} ...
0
votes
1answer
48 views
Show $C\geq \mathrm{max}\left \{ A,B \right \}$.
Let $\sum_{n=0}^{\infty}a_{n}x^{n}$ and $\sum_{n=0}^{\infty}b_{n}x^{n}$ be the power series with the convergent of radius respectively $A>0$ and $B>0$.
Define $c_{n}=\mathrm{min}\left \{ ...
0
votes
0answers
43 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
0
votes
1answer
41 views
Find for all value of constant $a>0$; the interval of convergence of the power series $\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$
Find for all value of constant $a>0$; the interval of convergence of the power series
$\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$.
What I have tried is; if we let $b_{n}=\frac{1}{1+a^{n}}x^{n}$ so ...
0
votes
2answers
26 views
How do I show that the degree $n$ Taylor polynomials of $f$ about two points are equal?
Question
Suppose that $f(x)$ is a polynomial of degree $d$, and that $n \ge d$. Let $x_0 \neq x_1$. Show that the degree $n$ Taylor polynomials of $f$ about $x_0$ and $x_1$ are equal.
Attempt
Let the ...
1
vote
2answers
40 views
corollary to the completeness axiom
The corollary states "Every nonempty subset $S$ of $\mathbb{R}$ that is bounded below has a greatest lower bound inf S.
The part I don't get in the proof is from where they came up with the set $-S$ ...
1
vote
0answers
59 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
5
votes
1answer
84 views
Continuous function differentiable on $[0,1]\setminus\mathbb{Q}$, but nondifferentiable on all of $\mathbb{Q}\cap[0,1]$?
I'm trying to work out an example of a continuous function which is differentiable at all irrationals but nondifferentiable at all rationals in $[0,1]$.
Since $\mathbb{Q}$ is countable, list it as ...
6
votes
3answers
112 views
If $\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$ does it imply that $\lim_{x \to \infty}f'(x)$ = $\lim_{x \to -\infty}f'(x)=0$?
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and
$\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$,
there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$.
Can we ...
1
vote
2answers
45 views
Prove the convergence of the sequence.
Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$
$$x_{n+1} = \sqrt{a + x_n}$$
1
vote
0answers
28 views
What's the need of $^{S}_{T}$ in $f^{S}_{T}:S\rightarrow T$?
I'm reading Lang's Undergraduate Analysis:
In the chapter about mappings, he says that we should denote the set of arrival and the set of departure with the following notation:
...
0
votes
1answer
17 views
Is there extension of function from a curve on the whole space preserving smoothness?
Assume that $\alpha: (a,b) \rightarrow \mathbb R^3$ and $f: (a,b) \rightarrow \mathbb R$ are given smooth functions. Let $t_0 \in (a,b)$.
Do there exist a $\delta>0$ and a smooth function $V: ...
1
vote
2answers
235 views
A less known definition of the definite integral of a continuous function
The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110.
(link to full book) (screenshots: page ...
0
votes
0answers
30 views
question about Bernoulli number
we know that we can generate the Bernoulli number using this equation
$(1+B)^n=B^{[n]}$
where $B_n$ is Bernoulli number
but how we can prove it ?
is there any help
thanks for all
2
votes
1answer
67 views
Prove that $(1+1/x)^x$ is concave for $x>0$
From the graph it looks like $(1+1/x)^x$ is concave for $x>0$. But in this post, I can only prove that it is concave for $x\ge 1$. It is of interest to see a proof for $x>0$.
5
votes
1answer
60 views
Chain rule proof
Let $a \in E \subset R^n, E \mbox{ open}, f: E \to R^m, f(E) \subset U
\subset R^m, U \mbox{ open}, g: U \to R^l, F:= g \circ f.$ If $f$ is
differentiable in $a$ and $g$ differentiable in ...
2
votes
1answer
28 views
Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
The question is as follows:
Given:
(1) function $f: U \subset \mathbb R^n ==> \mathbb R$
(2) $U$ is open and convex set
(3) $f \in C^1$ in $U$
Goal: Show that $f$ is ...
