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
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1answer
625 views

Proof of the classical div-curl-lemma

let $1 = \frac{1}{p} + \frac{1}{q}$ as usual. Let $f \in L^p, g \in L^q$ be vector fields from $\mathbb R^n$ to itself. Assume $div f = 0$ and there exists a function $G$ s.t. $\nabla G = g$. Then $f ...
2
votes
2answers
276 views

Question about $\liminf$ and $\limsup$ of a sequence

Suppose that a sequence $\{x_k\}$ has a clusterpoint (or more..) $c\in\mathbb{R}$. what conclusion, if any, can be drawn about either $\liminf x_k$ or $\limsup x_k$ ? I don't know the conclusion... ...
0
votes
1answer
122 views

How to prove that every nondegenerate critical point of $f$ has a neighbourhood which does not contain any further critical points?

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}, f \in C^2$. I have to prove that for every nondegenerate critical point of $f$, there exists a neighbourhood which does not contain any further critical ...
1
vote
5answers
142 views

two examples in analysis

I want to ask for two examples in the following cases: 1) Given a bounded sequence $\{a_n\}$, $$\lim_{n\to \infty}{(a_{n+1}-a_n)}=0$$ but $\{a_n\}$ diverges. 2) A function defined on real-line ...
2
votes
3answers
2k views

small o(1) notation

It's probably a vey silly question, but I'm confused. Does o(1) simply mean $\lim_{n \to \infty} \frac{f(n)}{\epsilon}=0$ for some $n>N$?
1
vote
3answers
103 views

Sequence of subspaces of $\ [ 0,2{\pi} ]$ with length that goes to $\ 0 $

Can you give me an example of a sequence of subspaces of $\ [0,2{\pi}] $ that the legth of them tends to $\ 0 $ but for every $\ x \in [0,2{\pi}]$ there are infinitely many of them such as x lives in ...
8
votes
2answers
455 views

What is the root linear coefficient theorem?

MathWorld gives the root linear coefficient theorem as The sum of the reciprocals of roots of an equation equals the negative coefficient of the linear term in the Maclaurin series. The theorem ...
5
votes
1answer
427 views

Applying Mean Value Theorem to formula

As I understand it, the mean value theorem is where $${f}'(c)=\frac{f(b)-f(a)}{b-a}$$ if $f$ is continuous on the open interval (a, b) and differentiable on the closed interval [a,b]. A problem ...
8
votes
2answers
2k views

question on second mean value theorem for integration

I am wondering two different forms of the second mean value theorem for integration. For the one in wikipedia, I also wonder where I can find a proof. The form I read from another reference is that: ...
4
votes
1answer
438 views

Testing convergence of $\sum\limits_{n=2}^{\infty} \frac{\cos{\log{n}}}{n \cdot \log{n}}$

Does the series: $$\sum\limits_{n=2}^{\infty} \frac{\cos(\log{n})}{n \cdot \log{n}}$$ converge or diverge? I know that $|\cos(\log{n})| \leq 1$, but i really cannot apply it here. Any ideas on how ...
0
votes
1answer
419 views

Ky Fan Norm Question

How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix) Thanks.
8
votes
1answer
184 views

Continuity of a function that maps a point to the closest point on a compact convex set

Let $K$ be a nonempty compact convex subset of $\mathbb R^n$ and let $f$ be the function that maps $x \in \mathbb R^n$ to the unique closest point $y \in K$ with respect to the $\ell_2$ norm. I want ...
40
votes
5answers
2k views

Why are gauge integrals not more popular?

A recent answer reminded me of the gauge integral, which you can read about here. It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue ...
4
votes
2answers
709 views

On distributions over $\mathbb R$ whose derivatives vanishes

Let $I \subset \mathbb R$ be open, $u \in \mathcal D'(I)$ be a distribution whose distributional derivatives vanishes (i.e. is zero for all test functions, which we may assume to be complex valued ...
2
votes
3answers
167 views

How can I simplify my $\Delta f$?

Given $f: \mathbb{R}^n \backslash \{0\} \rightarrow \mathbb{R}$, a twice differentiable, rotationally symmetric function, that is to say $\exists \varphi: \mathbb{R}_{>0} \rightarrow \mathbb{R}$ ...
9
votes
2answers
1k views

Countability of local maxima on continuous real-valued functions

I am working through a bank of previous exams and couldn't figure a problem out to my satisfaction. Let $f(x) : \mathbb{R} \to \mathbb{R}\,$ be a continuous function. Show that $f$ can ...
2
votes
2answers
117 views

$\bar{z}$ and rotation by quaternion multiplication

Does there exist a quaternion $q$ on the unit sphere such that, given the vanilla complex plane $\mathbb{C}$, $q\mathbb{C}q^{-1} = \bar{\mathbb{C}}$? Motivation: ordinarily, the plane is rotated by ...
1
vote
1answer
130 views

Does the integral of squared Shah function exist?

Let $$f(x)=\sum_{s=-\infty}^{\infty}e^{-2\pi ixsk}$$ $k$ integer. Does this integral exist? $$\int_{0}^{1}(f(x))^{2}dx$$
8
votes
1answer
425 views

Largest $\sigma$-algebra on which an outer measure is countably additive

If $m$ is an outer measure on a set $X$, a subset $E$ of $X$ is called $m$-measurable iff $$ m(A) = m(A \cap E) + m(A \cap E^c) $$ for all subsets $A$ of $X$. The collection $M$ of all ...
6
votes
2answers
2k views

Compactness in the weak* topology

Let $X$ be a Banach space, and let $X^*$ denote its continuous dual space. Under the weak* topology, do compactness and sequential compactness coincide? That is, is a subset of $X^*$ weakly* ...
10
votes
3answers
561 views

How does Lambert's W behave near ∞?

How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of $$\frac{W(x)}{\ln(x)}$$ near $\infty$ (but along the positive real line, if that ...
1
vote
2answers
170 views

Analysis, determing pointwise and uniform convergence

Let I(x) be a function such that I(x)=0 if x$\leq$0, and I(x)=1 if x$>0$. Let $c_{n}$ be a sequence of numbers such that $\sum$ c$_n$ converges absolutely. Let $x_n$ be some sequence of distinct ...
28
votes
3answers
753 views

On calculating $\int_0^1\ln(1-x^2)\;{\mathrm dx}$ — where is the mistake?

I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...
1
vote
1answer
177 views

Harmonic polynomial

I am working on a project regarding the number of zeros of a harmonic polynomial and am stuck with the proof of this: The zero set of the harmonic polynomial h(z)= $z^n$ - $\bar{z}^n$ consists of n ...
1
vote
2answers
209 views

dividing samples in equal slabs

I have sample data like in above format along X, Y axis. Now what i would like to do is to devide it in "n" number of slabs having fixed values. Now how do i achive this in mathematics(statistics). ...
15
votes
6answers
2k views

Please show $\int_0^\infty x^{2n} e^{-x^2}\mathrm dx=\frac{(2n)!}{2^{2n}n!}\frac{\sqrt{\pi}}{2}$ without gamma function?

Prove: $$\int_0^\infty x^{2n} e^{-x^2}\mathrm dx=\frac{(2n)!}{2^{2n}n!}\frac{\sqrt{\pi}}{2}$$ Thanks!
5
votes
1answer
302 views

The average of a bounded, decreasing-difference sequence

(not sure if "decreasing-difference" is the right way to put it - please edit away if you know the proper technical term) From my analysis homework: Does there exist a bounded sequence ...
4
votes
2answers
269 views

How can I compute this limit

I want to prove that $\lim_{h\rightarrow\infty}\left(\int_{0}^{\infty}\left(\cos ...
3
votes
2answers
102 views

A sufficient condition for $U \subseteq \mathbb{R}^2$ such that $f(x,y) = f(x)$

I have another short question. Let $U \subseteq \mathbb{R}^2$ be open and $f: U \rightarrow \mathbb{R}$ be continuously differentiable. Also, $\partial_y f(x,y) = 0$ for all $(x,y) \in U$. I want to ...
1
vote
2answers
113 views

For which $p, q$ exists $C > 0$ with $||f||_p \leq C ||f||_q$ for all $f \in C([0,1])$?

I want to find out for which $p, q$ exists $C > 0$ with $||f||_p \leq C ||f||_q$ for all $f \in C([0,1]), p \in \mathbb{R}_{\geq 1} \cup \{\infty\}$. I first let $p,q \geq 1$ and I looked at the ...
8
votes
1answer
421 views

Monotonic behavior of a function

I have the following problem related to a statistics question: Prove that the function defined for $x\ge 1, y\ge 1$, ...
1
vote
3answers
198 views

Does my function exist?

Is it possible to define a bijection $f:\mathbb{R}\setminus\mathbb{R}^{-}\rightarrow[0,1)$ such that $f$ is continuously differentiable on its entire domain?
2
votes
2answers
442 views

directional derivative in a manifold

Let us assume that directional derivative of a function $f$ exists at a point $p$ (i.e.,$ D_v(f)$) for all vectors $v \in \mathbb{R}^{n}$. Does it imply that the function is differentiable?
2
votes
2answers
339 views

Prove a property holds on a metric space if it holds on a dense subset

For the past two weeks, I've tried to prove two different results that hold the same structure: Suppose a property that holds for a dense subset of a metric space. Prove that it holds for the entire ...
1
vote
1answer
97 views

Help with the proof of a Gaussian type inequality and some numerical results

I am trying to figure out if I made a mistake in the following proof - in particular I have been trying to verify the inequality (*) below. I have attached my proof but I have been getting some error ...
5
votes
3answers
1k views

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard ...
2
votes
3answers
990 views

Practice problem from Mean Value Theorem in Real Analysis

Can someone give an insight on the following problem? I'm not sure how to start the problem. It's a practice problem for "mean value theorem" and "Taylor's Theorem" so I'm assuming they might be ...
1
vote
3answers
141 views

Show that $ \displaystyle x^n+x^{\frac{1}{n}}+n = 0$ has no real zeros for $n\geqslant 2$

I'm unable to prove the following two statements: $ \displaystyle x^n+x^{\frac{1}{n}}+n = 0$ has no real zeros for all $n\geqslant 2.$ $ \displaystyle x^n+x^{\frac{1}{n}}-n = 0$ has exactly one ...
1
vote
2answers
176 views

subharmonic functions

Let $U\subseteq \mathbb{C}$ be an open set and let $u(z)=-\log(\mathrm{dist}(z,\partial U))$. I need to show that $u$ is subharmonic on $U$? $\partial U$ it means the boundary of $U$.
0
votes
1answer
817 views

Accounting Question: Computing margin of Safety Ratio

Smith Company produces desk lamps. The information for June indicated that the selling price was $\$25$ per unit, variable costs were $\$15$ per unit, fixed costs totaled $\$6,000$, and the margin of ...
1
vote
3answers
735 views

$\sin(2\pi nx)$ does not converge for $x \in (0,1/2)$

How to show that $\sin(2 \pi nx)$ does not converge as n goes to infinity? $x \in (0,1/2)$
4
votes
1answer
392 views

A measurable function on an atom is almost everywhere constant

Let $f \in m(\Omega,\mathcal{F})$, i.e. $f \mapsto [-\infty,\infty]$ and let $A \in \mathcal{F}$ be an atom. Prove that $f$ is almost everywhere constant on A: there exists $k \in [-\infty,\infty]$ ...
7
votes
1answer
121 views

Has this Extension to a Series been Studied Before?

We know from Calculus what a series is, and you might have seen infinite products as well. But the Elementary Symmetric Polynomials give an entire spectrum of operators between a sum and product over ...
4
votes
3answers
517 views

How to find a Newton-like approximation for that function?

I want to find the complex fixpoint $t=b^t $ for real bases $b> \eta = \exp(\exp(-1))$. added remark: I'm aware that there is a solution using branches of the Lambert-W-function, but I've no ...
2
votes
2answers
155 views

A problem on summing real numbers all taken to the same exponent

Let there be given a set of $n$ real numbers, $\{r_i\} \subset (0,1)$; is it possible to find some conditions satisfied by a real number $m \in \mathbb{R}$ to ensure that: ...
2
votes
0answers
304 views

Uniformly continuous functions

Given a uniformly continuous function $f(x)$ on the real numbers $\Bbb R$, then by the definition of uniform continuity this means: for any $\epsilon>0$ there exists $\delta >0$ such that ...
4
votes
2answers
783 views

Uniform continuity

I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is bounded. Thank you.
2
votes
2answers
348 views

arc length of $\dfrac{e^x - e^{-x}}{2}$

how can I please calculate an arc length of $\dfrac{e^x-e^{-x}}{2}$. I tried to substitute $\dfrac{e^x-e^{-x}}{2}=\sinh x$, which leads to $\int\sqrt{1+\cosh^2x}dx$, which unfortunately I can't solve. ...
1
vote
1answer
258 views

Integration by parts

Hi I want to integrate this integral and ask if my work is correct or not. $$\int^\infty_0 dx x^{\alpha-1} e^{-x} (a+bx)^{-\alpha}$$ I want to integrate it by parts, so I have $$(a+bx)^{-\alpha} ...
4
votes
1answer
295 views

Archimedean property of $\mathbb{R}$

Theorem: If $x,y \in \mathbb{R}$ and $x > 0$, $\exists$ a positive integer $n$ such that $nx > y$ I read the proof by Rudin and understood it. I think it is very elegant and uses the LUB ...