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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2
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1answer
19 views

Does $\log$ minimize this functional for its Abel equation?

Suppose that we have the functional equations ("Abel equation", it is called) for a function $F: [1, \infty) \rightarrow \mathbb{R}$ given by $$F(1) = 0$$ $$F(ex) = F(x) + 1$$ where $e$ is the ...
0
votes
1answer
23 views

Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions. My attempt: I dont know how to approximate $f(x)$ to ...
1
vote
1answer
18 views

Strong convexity and strong smoothness duality

A function $f$ is said to be strongly convex with respect to a norm $\|\cdot\|$ at a point $y$ if $f(x) \geq f(y) + \nabla f(y)^T(x-y) + \frac{1}{2}\|x-y\|^2.$ It is said to be strongly smooth with ...
1
vote
0answers
25 views

Laplacian of composition

Let $U \subset \mathbb{R}^n$ be open and $u \in C^2(U)$ with $\Delta u(y)=0$ for all $y \in U.$ Let $\phi: V \rightarrow U$ be in $C^2(U)$, too with $V \subset \mathbb{R}^n$ open. Now, I want to ...
1
vote
1answer
28 views

Real Analysis: Determine $|J|$ where $J=J_1 \cup J_2$

Let $J_1= \{-2,-1,0,1,2\}$ and $J_2=\{-2,-\frac{2}{3}, \frac{2}{3}, 2\}$ be partitions of $[2,2]$. Let $J = J_1 \cup J_2$. Determine $|J| = (max_i (x_i - x_{i-1})$. Would the answer be $1$ as the ...
0
votes
0answers
32 views

Idea of the proof of Lusternik-Schnirelmann

I have this theorem of Lusternik-Schnirelmann from Chang's book: " Let $M$ be a smooth Banach-Finsler manifold. Suppose that $f\in C^1(M,\mathbb{R})$ is a function bounded from below, satisfying ...
6
votes
1answer
50 views

the elements of Cantor's discontinuum

Let $(A_n)_{n \in \mathbb{N}}$ the sequence of subsets of $\mathbb{R}$, given by $A_0 := \bigcup_{k \in \mathbb{Z}}[2k, 2k + 1]$ und $A_n := \frac{1}{3}A_{n-1}$ for $n ≥ 1$. Also, we define $$ A := ...
0
votes
2answers
31 views

Periodic function

Considering the function $F(x)=x-E(x)$ such as $E(x)$ is the integer part of $x$ So here just with observation we can see that : $f(x+1)=(x+1)-E(x+1)=f(x)$. But here mathematics is not based just ...
2
votes
2answers
37 views

$\inf_{x\in[a,b]}f(x)=\inf_{x\in[a,b]\cap\mathbb{Q}}f(x)$ for a continuos function $f:[a,b]\to\mathbb{R}$

Let $f:[a,b]\to\mathbb{R}$ be continuos. I'm sure it's not hard, but I'm unsure what exactly we need to do to prove $$\inf_{x\in[a,b]}f(x)=\inf_{x\in[a,b]\cap\mathbb{Q}}f(x)$$
0
votes
0answers
19 views

Let $ f:R_{\ge 0} \rightarrow R_{\ge 0}$ for every $\lambda \in [0,1]$ and $x,y \ge 0$

Let $ f:R_{\ge 0} \rightarrow R_{\ge 0}$ for every $\lambda \in [0,1]$ and $x,y \ge 0$ such that $f(\lambda x +(1-\lambda) y) \ge \lambda f(x) +(1-\lambda)f(y)$.Know that $f(0)=0$ so that $f(x) > ...
1
vote
1answer
17 views

Proving an inequality involving the summation of Riemann zeta function.

If $0\lt a \le 1, s\gt 1,$ define $\zeta(s,a)=\sum (n+a)^{-s}$. Show that this series converges absolutely for $s \gt 1$ and prove that $$\sum_{h=1}^k \zeta(s,\frac{h}{k})=k^s\zeta (s)\ \ \ ...
1
vote
1answer
16 views

Proving convergence for series containing ln and factorial

I am trying to show whether or not the following series ${a_n}$ converges. Based on the hint, I have tried using Bertrand's test, but I am having a hard time simplifying the absolute value of the ...
-5
votes
1answer
108 views

Translate this proof from German to English

I need your help to translate some exercises from German to English. I will attach like images. Thanks :) Satz 3. Es sei $(X,d)$ ein ultrametrischer Raum. $X$ ist genau dann transvollständig, wenn ...
0
votes
2answers
66 views

The necessity of the axiom of induction

$\underline{First\ question}$ Let $P(n)$ be a proposition about $n$. In standard mathematical induction, we require: (1)$P(0)$ holds. (2)If $P(n)$ holds, $P(n+1)$holds. Here we use "the axiom of ...
0
votes
1answer
40 views

What are all the indeterminate forms for which L'Hopitals rule is valid?

What are all the indeterminate forms for which L'Hopitals rule is valid? I know the basic ones are $\frac{\infty}{\infty}$ and $\frac{0}{0}$. Are there any other ones? Thanks,
2
votes
1answer
24 views

Minimizer of a quadratic form

Suppose I have a quadratic form of the form: $$q(x)=\frac{1}{2} x^T Q x$$ Now I want to find the minimum step length w.r.t the steepest descent. So I know the descent direction is $\nabla q(x)$. So ...
-2
votes
0answers
62 views

Any mathematical analysis text that covers Rudin but is written like Munkres? [closed]

Is there any book on mathematical analysis that covers (all) the (same) topics (as) in Principles of Mathematical Analysis by Walter Rudin, 3rd edition, but which is written like Topology by James R. ...
0
votes
0answers
23 views

convergence, contracting sequence

I need a little help on the following question: let $x(0)$ be an element of reals and define a sequence $\{x(n)\}$ by: $x(1) =f(x(0)), x(2)=f(x(1)),...., x(n+1)= f(x(n))...$ show that if $m>0, ...
1
vote
1answer
19 views

A continuously differentiable bijection implies its inverse is Lipschitz continuous

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable bijection. Does this imply that $f^{-1}$ is Lipschitz continuous? (of course, not globally, take for instance $f(x)=x^3$) If ...
7
votes
0answers
56 views
+100

Why are diffeomorphism-invariant PDE not elliptic?

In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ...
-1
votes
0answers
40 views

On some counterexamples about continuity, intermediate value propriety, Riemann integrability, and antiderivatives

At school, I have been studying the relationship between continuity, monotonicity, and Riemann integrability. In doing so, I tried to make up some examples and counterexamples, but there are some ...
3
votes
2answers
55 views

Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
3
votes
3answers
69 views

Show that: $\int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1}$

How do you show that: $$ \int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1} $$ Without using Gamma function?
2
votes
2answers
27 views

problem of numeric sequence

Give an example of a sequence $(A_n)$ which is not convergent but the sequence $(B_n)$ defined by $\displaystyle B_n = \frac{A_1+A_2+\cdots+A_n}{n}$ is convergent.
1
vote
0answers
16 views

Difference of linear transformation of convex function

I'm trying to show that for constants $a,b > 0$, and a convex, continuously differentiable function $f$ with $f(0) = 0$ that $x_1 > x_2 > 0$ implies $f(-a-b x_1) - f(-b x_1) \geq f(-a-b x_2) ...
1
vote
1answer
63 views

Inequality with differential equations solutions

I would love some help working through this problem: Let $f_1,f_2,f : [0,\infty) \to \mathbb{R}$ be three bounded, continuous and absolutely Riemann integrable functions so that $|f_1(x)|, |f_2(x)| ...
2
votes
2answers
80 views

How to do this integral.

I need to do this: $$\int_0^\infty e^{ikt}\sin(pt)dt$$ I already have the solution, I'm just clueless on how to actually calculate it. I've tried several changes,integration by parts and extending ...
1
vote
0answers
17 views

Must the definition of the limit of a complex function be an inequality?

Looking at a few books, the definition of the limit of a complex function is of the form: DEFINITION. If $f :S\rightarrow \mathbb{C}$ is an arbitrary complex function and $z_0$ is a limit point of ...
0
votes
0answers
15 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
0
votes
1answer
20 views

What is a Bi-Analytic function

I want to know what the definition of a Bi-analytic function is. I have tried looking it up online, but all I am able to find are research papers/books on the theory of bi-analytic functions. Can ...
2
votes
0answers
40 views

Gamma Function Representation

I have some questions relating to the gamma function. My question is simply to evaluate the integral $$\int_0^\infty {t^{z-1}\sin t}\,dt$$ whenever $-1<\text{Re}(z)<1$. If we take $z$ ...
3
votes
2answers
105 views

Does the proof of Bolzano-Weierstrass theorem require axiom of choice?

When selecting the terms of subsequence from each bisections, I thought axiom of choice might be required. But I'm not so sure whether or not, so please tell me. [edited] I'm sorry for the lack of ...
0
votes
1answer
13 views

Maximal Magnitude of Fourier Transform

Assume you are given a length-$n$ vector $x\in\mathbb{C}$ with elements $x_0$ through $x_{n-1}$. Define the Fourier transform of $x$ as $$ X(e^{j\theta}) = \sum_{k=0}^{n-1} x_k e^{-jk\theta}. $$ I'm ...
0
votes
1answer
53 views

multiplying two metrics

Let $(X,d)$ and $(X,d^\prime)$ are metric spaces ,is $d×d^\prime$ metric on $X$ ?I try to prove triangle inequality , I write two triangle inequalities for $d $ and $ d^\prime$ but it not true.
1
vote
1answer
44 views

Calculation of integral including exponentials

So I have this integral \begin{equation} \int_1^{\infty}...\int_1^{\infty} \frac{ e^{min\{x_1,x_2,...,x_n,c\}}}{e^{x_1+x_2+...+x_n}} dx_1dx_2...dx_n \end{equation} where $c>1$. I thought that what ...
1
vote
2answers
38 views

Measure of intersection of set and its translation

I came across an old qualifying exam question: Let $A\subset [0,1)$ be a Lebesgue measurable subset of unit intreval such that $0<\mu(A)<1$. For every $x\in [0,1)$ let $A+x=\{x+y$ mod 1$:y\in ...
1
vote
1answer
93 views

Analysis Constructing a Sequence

I'm looking for a sequence of functions that is continuous and absolutely integrable, but pointwise divergent for every $z$ $\in [0,1]$. In other words, $ \int_0^1 |f_n(z)| dz \rightarrow 0$ as $n ...
3
votes
1answer
76 views

Show that $\limsup$ of $\sin n$ is 1

I want to prove that the $\limsup\limits_{n\rightarrow \infty} \sin n=1$. I know that $1$ is an upper bound for $\sin n$ but I cannot find a subsequence of $\sin n$ that converges to $1$. Can somebody ...
2
votes
1answer
21 views

Why do the dimensions not line up when I calculate this (directional) derivative using the chain rule?

an arbitrary, differentiable function $f : \mathbb{R}^n \to \mathbb{R}$ and the function $\gamma: \mathbb{R} \to \mathbb{R}^n$ defined as $\gamma(t) = u + tv$, where $u, v$ are fixed vectors in ...
0
votes
3answers
25 views

Uniform convergence and maximum of an absolute difference

I am trying to prove that: Consider the sequence $a_n = \sup_{x\in S}|f_n(x) - f(x)|$. Then $f_n$ converges to $f$ uniformly if and only if $a_n$ tends to $0$. But I can't prove that if $f_n$ ...
1
vote
3answers
81 views

Can anyone explain why this series converges?

Can anyone explain why this series converges? $1+\frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6}...$ The answer is given, but i do not understand it: $$**|S_{6n}-S_{3n}|= ...
0
votes
1answer
44 views

Prove that one series converges and that the other diverges .. [closed]

Let $a_n >0$, for $n \geq1$ the series $\sum_{n=1}^{\infty}a_n$ diverges. Let $S_n=a_1+a_1+a_3...+a_n > 1$ for $n \geq 1 $ prove that : $$\sum_{n=1}^{\infty}{a_{n+1}\over S_n \ln S_n}$$ diverges ...
1
vote
2answers
65 views

Do the following series converge if $a_n>0$ and $ \sum_{n=1}^{\infty}a_n$ diverges?

Do the following series converge if $a_n>0$ and $\sum_{n=1}^{\infty}a_n$ diverges ? a.) $\sum_{n=1}^{\infty}{a_n \over 1+ a_n}$ b.)$\sum_{n=1}^{\infty}{a_n\over 1+ a_n ^2}$ ...
6
votes
0answers
89 views

Why are the fundamental theorems of calculus usually associated to the Riemann Integral?

I am writing a "textbook" on Analysis, and I've reached the time I must talk about integrals. I prefer to approach directly the Lebesgue Integral theory. This question is not about the status of this ...
0
votes
1answer
53 views

How am I suppose to find the area between the given curves (2 seperate, based on same, areas)

Could do I find the area inside the curve which bounds the area, curve is $$1.\ \ \ \ \ (y-\arcsin x)^2=x-x^2$$ $$2.\ \ \ \ x=\frac {1-t^2}{(1+t^2)^2}, y= \frac {2at}{(1+t^2)^2}$$ I was traditionally ...
2
votes
2answers
32 views

Verify that the set $\Omega = \lbrace (u,v) \in \mathbb{R}^2 \mid |u| + |v| \leq 1 \rbrace$ is Jordan measurable

Motivation: I am currently in a rather uncomfortable spot in my Analysis studies. In class we introduced the Jordan measure in a very vague way, meaning no proofs, no examples. (Because next Semester ...
0
votes
3answers
58 views

How is it possible that this series converges?

The divergence of: $$\sum_{n=1}^{\infty}{1 \over \sqrt{n(n+1)}}$$ is proved by using the negative form of Cauchy partial sum theorem: $$\begin{eqnarray*}|S_{2n}-S_{n}|&=&{1\over ...
-1
votes
1answer
39 views

Is the distance function continuous? [closed]

Is the distance function continuous? I know that distance function is continuous, give an example of distance function that is not continuous.
1
vote
1answer
54 views

Example 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be open?

Let $N$ be the following subset of $\mathbb{R}^2$: $$N \colon= \{ \ (x,y) \in \mathbb{R}^2 \ \colon \ \vert x \vert < \frac{1}{y^2+1} \ \}.$$ Then intuitively it is apparent that $N$ is open. ...
3
votes
1answer
53 views

A possible inequality related to binomial theorem (or, convex/concave functions)

Let $x, \ y, \ p$ be any real numbers with $x>0$, $y>0$, and $p>1$. The question is about (most probably) an elementary inequality: Is it always true that $x^p+y^p\leq (x+y)^p$ ? Note ...