Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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16
votes
2answers
678 views

doubly periodic functions as tessellations (other than parallelograms)

I think of a snapshot of a single period of a doubly periodic function as one parallelogram-shaped tile in a tessellation, could a function have a period that repeats like honeycomb or some other not ...
2
votes
3answers
323 views

Improper Integrals

I don't get how we're supposed to use analysis to calculate things like: a) $$ \int_0^1 \log x \mathrm dx $$ b) $$\int_2^\infty \frac{\log x}{x} \mathrm dx $$ c) $$\int_0^\infty \frac{1}{1+x^2} ...
9
votes
1answer
851 views

Nice way of thinking about the Laplace operator… but what's the proof?

Here's a nice fact: roughly speaking, the Laplace operator gives you the difference between the value of a function at a point and the average value at "neighboring" points. More precisely, in ...
0
votes
1answer
248 views

Continuity of Functions & Uniform Convergence

I am trying to solve the following problems. Do you have any hints? Let $g_n(x)$ be functions defined on interval $I = [a,b]$ and suppose $g(x) =\lim_{n \to \infty} g_n(x)$ is defined for every $x ...
31
votes
5answers
2k views

Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several ...
4
votes
1answer
416 views

Dual of $\ell_{\infty}$ is not $\ell_1$

As the title indicates I'm trying to show that $\ell_{\infty}^{*}$ is not $\ell_1$. I've shown that for p, q conjugate and finite we do indeed have $\ell_{p}^{*} = \ell_q$, with the correspondence ...
7
votes
1answer
206 views

Rectifiable functions

A continuous function $\alpha: [a,b] \to \mathbb{R}^k$ is called a curve. For each partition $P = \{t_0<t_1<....<t_n=b\}$, define $l(\alpha, P) = \sum_{i=1}^n \left|\alpha(t_i) - ...
13
votes
1answer
691 views

Classifying the compact subsets of $L^p$

Some of my favorite theorems in analysis are those which classify the (pre-)compact subsets of a particular space. For example: The Heine-Borel Theorem classifies the compact subsets of ...
17
votes
2answers
2k views

Why Doesn't Cantor's Diagonal Argument Also Apply to Natural Numbers?

In my understanding of Cantor's diagonal argument, we start by representing each of a set of real numbers as an infinite bit string. My question is, why can't we begin by representing each natural ...
1
vote
3answers
702 views

Show the function is differentiable

$F(x) = \int_{x-1}^{x+1}f(t)dt$ for x an element of the reals. Show that $F$ is differentiable on Reals, and compute $F^\prime$. I am unsure about how to showing $F$ is differentiable. I know that I ...
0
votes
2answers
57 views

a numerical concluding about (a/(a+b)) and (c/(c+b))

Let $a,b,c$ be three integers greater than $0$, and assume there is a real number $t$ such that $$ \frac{a}{a+b}=\frac{\left\lfloor t\right\rfloor}{\left\lfloor t\right\rfloor+1}. $$ Is there a way to ...
1
vote
2answers
188 views

Name of Formula $x^3+y^3=z^3+1$

I encountered the formula $$x^3+y^3=z^3+1$$ with the condition, that $$x<y<z$$ and wonder, whether it has got a specific name or whether it can be easily transformed into another well-known ...
3
votes
1answer
193 views

Analysis Convergence/Divergence

Prove there exists a function $f$ such that $$\int_1^{\infty}f(x)\,dx\text{ converges, but }\int_1^{\infty}|f(x)|\,dx\text{ diverges.}$$ Similarly, prove that there exists a function $g$ ...
9
votes
2answers
233 views

For which $p \in \mathbb{R}_{>0}$ does the integral $\int_{[0,1]^n} \frac{\mathrm dx}{(x_1^p+2x_2^p + … + nx_n^p)^{1/3}}$ converge?

I want to find out for which $p \in \mathbb{R}_{>0}$ the integral $$\int_{[0,1]^n} \frac{\mathrm d x}{(x_1^p+2x_2^p + ... + nx_n^p)^{1/3}}$$ converges. To be honest, I have no idea or whatsoever ...
0
votes
1answer
104 views

An approximation of rational function with polynomials

To compute some asymptotic expression I need to approximate $$\frac{(x-1)^{r+u}\left((x-1)^{p-r+1}+x^{p-r+1}\right)\left(x^{p-u+1}+(x-1)^{p+u+1}\right)}{\left(x^{2p+2}+(x-1)^{2p+2}\right)}$$ by some ...
3
votes
2answers
116 views

Sequence of functions (convergence)

Let be $f(x)=\frac{2x}{1+x} $ function and $ x_0 > 0 $. With the help of this, form the $x_{n+1}=f(x_n)$ sequence. Is $x_n$ convergent and if yes what is the limit? Thank you very much in advance! ...
16
votes
3answers
524 views

How can I sum the infinite series $\frac{1}{5} - \frac{1\cdot4}{5\cdot10} + \frac{1\cdot4\cdot7}{5\cdot10\cdot15} - \cdots\qquad$

How can I find the sum of the infinite series $$\frac{1}{5} - \frac{1\cdot4}{5\cdot10} + \frac{1\cdot4\cdot7}{5\cdot10\cdot15} - \cdots\qquad ?$$ My attempt at a solution - I saw that I could ...
1
vote
1answer
544 views

Power Series and Sequences and Series of Functions

Observe that $e^{-x^2} = \sum_{n = 0}^{\infty} \frac{(-1)^n}{n!} x^{2n}$ for $x$ an element of the reals. Express $F(x) = \int_{0}^x e^{-t^2}\,dt$ as a power series For part 1, I don't understand ...
0
votes
1answer
53 views

How can I calculate max value, if I know just number of records, minimal value and average value of all records?

How can I calculate max value or all possible and relevant maximum values, if I know number of records, minimal value and average value of all records? For example: Number of records (persons): 92 ...
4
votes
1answer
263 views

How I can calculate this partial derivative of $f(a,b)=\int_0^\infty e^{-ax^3-bx^2}\mathrm dx$?

My question is: How to prove that the function: $$f(a,b)=\int_0^\infty e^{-ax^3-bx^2}\mathrm dx$$ is a solution of the differential equation: $$3ab\frac{{{\partial ^2}f}}{{\partial {b^2}}} - ...
1
vote
1answer
75 views

Lipschitz contradiction

Assume that $\phi:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is a smooth vector field, and assume that we can find vectors $y_k,x_k$ ($k$ positive integer) such that $(\phi(x_k)-\phi(y_k),x_k-y_k)\geq k ...
1
vote
1answer
83 views

an estimate for derivative

let $F$ a closed convex subset of $\mathbb{R}^n$, let $x,y\in F$ and assume that for any $s\in[0,1]$ we have $f(s):=\mid sx+(1-s)y-z\mid\geq \mid y-z\mid$ why is it true that ...
0
votes
1answer
75 views

Concatenation of differentiable paths/piecewise differentiable paths

I am going through J.W. Anderson's book $Hyperbolic$ $Geometry$, and I come along to the following statement made: "We note here that the concatenation of piecewise differentiable paths is again ...
5
votes
1answer
467 views

Mean Value theorem and Newton's Method

I am trying to prove that: given $x_0, x_1, x_2 \ldots$ the sequence of approximations to $\pi$, use the mean value theorem to show that $|\pi-x_{j+1}| = |\tan c_j||\pi - x_j|$, where $c_j$ is some ...
1
vote
1answer
393 views

On the tightness of Chernoff bounds for sum of Poisson trials

For the sum $X$ of independent 0-1 random variables $X_i$ ($0 \le i \le n-1$) with $Pr(X_i)=p_i$, namely $X=\sum_{i=0}^{n-1}{X_i}$ the following Chernoff bound holds, $$ Pr(X \ge (1+\delta)\mu) \le ...
0
votes
1answer
637 views

Differentiation of expectation of a nonlinear function

I encountered the following problem and need some help. Let $X$ be a continuous random variable. (You can assume $X$ to be very nice: it has a smooth density function with bounded support, bounded ...
4
votes
1answer
818 views

Stuck at the proof of the Riemann-Lebesgue lemma

I'm currently trying to prove the Riemann-Lebesgue lemma using lower Darboux-sums and an approximation of any integrable function $f: [0,1] \to \mathbb{R}$ defined as $$t(x) := \begin{cases} m_i & ...
3
votes
2answers
133 views

Inequality problem

Prove that if $$|x-x_0| < \min\left(\frac{\epsilon}{2(|y_0| + 1)}, 1\right)$$ and $$|y-y_0| < \frac{\epsilon}{2(|x_0| + 1)},$$ then $|xy - x_0y_0| < \epsilon.$ I am doing some problems in ...
6
votes
1answer
258 views

Algorithmic Analysis Simplified under Big O

Hi I am revising for my exams and I have the following inhomogeneous first order recurrence relation defined as follows: f(0) = 2 f(n) = 6f(n-1) - 5, n > 0 I ...
7
votes
0answers
266 views

How can we prove a simple case of the High Indices Theorem?

Let $(a_n)$ be a sequence of real numbers such that $$f(x) = \sum_{n=1}^{\infty} a_n x^{2^n}$$ converges for $|x| < 1$ and $f(x)$ converges to $a$ as $x \to 1^{-}$. Then I have to prove that $\sum ...
1
vote
2answers
204 views

How to solve combinations of differentials and integrals?

In the first chapter of Nearing's book "Mathematical tools for physics" (available online) I encountered an interesting combination of differentials and integrals - which I don't fully understand: ...
3
votes
2answers
233 views

Question about a puzzle on injecting a subset of $\mathbb{R}$ into $\mathbb{Q}$

I was just browsing through the Puzzle section on Noam Elkies website. The puzzle can be found here. The solution to the puzzle proves that any well-ordered subset of $\mathbb{R}$ is countable. In ...
3
votes
3answers
580 views

Proof of a binomial theorem based inequality?

Let $k \in N, x \gt 0$. Show that there exists some $n_2 \in \mathbb{N}$ so that $\forall n \geq n_2: (1+x)^n \gt n^k$. Hint: binomial theorem. My thought on this is first to make the substitution ...
5
votes
1answer
184 views

Abelian theorem regarding Riesz summability

This is my first time to post something here. If there is anything wrong, please inform me... Anyway, here is my question: Let $k$ be a nonnegative integer. We say a sequence $(a_n)$ is $(R, ...
23
votes
2answers
913 views

Number of local maxima of a function

Let $z_j$ ($j=1,\dots, k$) be $k$ points on the complex plane none of which lies on the real line. Is it always true that the function $$ F(x)=\sum_{j=1}^k \frac{1}{|x-z_j|^2} $$ has at most $k$ ...
2
votes
3answers
657 views

Continuous function with local maxima everywhere but no global maxima

Can there be such a function: $f \colon \mathbb R \to \mathbb R$ is continuous and non-constant. It has a local maxima everywhere, i.e., for all $x \in \mathbb R$ there is some $\delta_x>0$ such ...
1
vote
2answers
409 views

Differentiation of Taylor Series

Let $g(x) = e^{-1/x^2}$ for $x$ not equal to zero, and $g(0) = 0$. a) Please Show that $g^{(n)}(0) = 0$, for all $n = 0,1,2,3,4, \ldots$ Can someone please elaborate on the comments below for this ...
4
votes
1answer
118 views

Root estimation

What is the estimation for the positive root of the following equation $$ ax^k = (x+1)^{k-1} $$ where $a > 0$ (specifically $0 < a \leq 1$). Could you point out some reference related to the ...
4
votes
2answers
190 views

Same symbol “$\partial$” - different things ( the boundary $\partial A$ / partial derivative $\frac{\partial f}{\partial x}$)?

Are there any deep reasons, why we use the same symbol, $\partial$, when describing two (apparently fundamental) different mathematical objects, namely the boundary of a set (in topology), as well as ...
0
votes
1answer
140 views

Can $\sin (\theta n)$ be bounded from below by a decreasing exponential for all naturals $n$ [Update]

To put it more rigorously: Does there exist constants $0 < a < 1$ and $0 < b < 1$ such that for all real $\theta$ which are non-rational multiples of $\pi$ and all natural numbers $n$ it ...
2
votes
1answer
416 views

Cauchy-Schwarz for Multiple Integrals

Is there a generalization of the Cauchy-Schwarz Inequality for multiple integrals?
2
votes
1answer
242 views

About real analytic functions

Suppose a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is analytic at 0, i.e., it is given by its Taylor series on a neighborhood $U$ of 0. Then is it necessarily true that $f$ is analytic at every ...
6
votes
1answer
275 views

Generalization of the series for $\frac{\pi^2}{6}$? Is there a more elementary proof?

In the same vein as: $ \frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots $ Starting with: $ \displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = ...
3
votes
3answers
140 views

compose two functions to make them differentiable

Are there two continuous functions ( $f:R^n\to R$, $g:R\to R$ ) which are not differentiable at point $0$, but if you compose them, $h=g(f(x))$ is differentiable at $0$? ($f(0)=0$)
2
votes
1answer
139 views

Ring germs of $C^{\infty}$ functions on the real line

Is the ring $\mathcal{O}$ of germs of $C^{\infty}$ functions defined on the neighborhoods of $0\in\mathbb{R}$ the localization of the ring of $C^{\infty}$ functions on $\mathbb{R}$ at the maximal ...
1
vote
2answers
331 views

Power Series Expansion Problem Analysis

Please show that for all $x,y\in\mathbb{R}$, $$e^{x+y} - e^xe^y = \lim_{k\to\infty} \sum_{n=1}^k \sum_{j = 0}^n\left(\frac{x^{k+j}}{(k+j)!}\frac{y^{n-j}}{(n-j)!} + ...
3
votes
1answer
650 views

What are the rules for transformation (translation, dilation, etc.) of integral on n-dimensional sphere?

We know that in $\mathbb{R}^n$, we have transformation rules such as: $\int_{\mathbb{R}^n}f(x-h) dx=\int_{\mathbb{R}^n}f(x) dx$ $\delta^n \int_{\mathbb{R}^n}f(\delta x) dx=\int_{\mathbb{R}^n}f(x) ...
0
votes
1answer
185 views

Analysis of convergence 2

Let $f: \mathbb{R} \to \mathbb{R}$. Prove that if $f$ is bounded such that for all $x, y \in \mathbb{R}$ $x\neq y$ implies $|f(x) - f(y)| \lt |x -y|$ and for all $x \in \mathbb{R}$, $f$ is ...
0
votes
2answers
275 views

Derivatives of infimum

Let $\zeta: \mathbb{R^m} \times \mathbb{R^n} \mapsto \mathbb{R}$ be a smooth function and define $\phi(x) = \inf_{y \in C} \zeta(x,y)$ where $C \subset \mathbb{R^n}$ is compact. Suppose that for every ...
0
votes
1answer
139 views

Arc length of level sets

I have a function $z = B \sin x \ \sin y+\cos x \ \cos y$. Where $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. I need to find the length of the curve that describes a level set for any value of $B$. ...