**54**

votes

**2**answers

15k views

### Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...

**32**

votes

**7**answers

4k views

### Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to
$ f(f(x)) = \sin x.$ It is displayed, below, identified by the symbol $g$ because I am not entirely sure ...

**9**

votes

**3**answers

834 views

### Limit cycles as closed geodesics(in negatively curved space)

The classical Van der Pol equation is the following vector field on $\mathbb{R}^{2}$:
\begin{equation}\cases{\dot{x}=y-(x^{3}-x)\\ \dot{y}=-x}\end{equation}
This equation defines a foliation on ...

**74**

votes

**9**answers

16k views

### How to solve $f(f(x)) = \cos(x)$?

I found the following interesting equation on some web page I cannot remember:
$f(f(x))=\cos(x)$
Out of curiosity I tried to solve it, but realized that I do not have a clue how to approach such an ...

**6**

votes

**5**answers

6k views

### Beginners text on calculus of variations

I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options:
http://tinyurl.com/36koaq4
I work on Machine Learning, and that where ...

**8**

votes

**1**answer

504 views

### Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$.
It seems that the solution of linear ODE
$$\dot x= Ax$$
can not be written in a closed form using $\int$, $A$, $x(0)$ ...

**57**

votes

**15**answers

8k views

### What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?

Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...

**17**

votes

**21**answers

24k views

### Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?

**18**

votes

**8**answers

8k views

### Does the exponential function have a square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM)
Is there a "half-exponential" function h(x) such that h(h(x))=ex? Is it unique? Is it analytic?
Related question: Is there an invertible ...

**10**

votes

**0**answers

522 views

### Multiple Integral (American Mathematical Monthly problem 11621 and its generalization)

AMM problem 11621 asks to calculate the integral
$$I_2=\int\limits_{-\infty}^{\infty}ds_1\int\limits_{-\infty}^{s_1}ds_2
\int\limits_{-\infty}^{s_2}ds_3\int\limits_{-\infty}^{s_3}ds_4
...

**15**

votes

**2**answers

760 views

### Asymptotic approximation of $x^\alpha$ by entire functions

Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with ...

**3**

votes

**0**answers

100 views

### Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...

**119**

votes

**16**answers

17k views

### How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...

**72**

votes

**4**answers

12k views

### Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.)
Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...

**27**

votes

**2**answers

4k views

### Is there a category structure one can place on measure spaces so that category-theoretic products exist?

The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...

**92**

votes

**5**answers

7k views

### Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$?

Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the ...

**30**

votes

**1**answer

2k views

### Geometric interpretation of the half-derivative?

For $f(x)=x$, the half-derivative of $f$ is
$$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$
Is there some geometric interpretation of (Q1) this specific derivative, and, ...

**14**

votes

**5**answers

10k views

### Criteria to determine whether a real-coefficient polynomial has real root?

Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not?
I ...

**14**

votes

**2**answers

2k views

### Ideals of the ring of smooth functions

The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...

**11**

votes

**5**answers

988 views

### Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same.
Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...

**15**

votes

**11**answers

2k views

### Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...

**10**

votes

**2**answers

3k views

### Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb ...

**6**

votes

**1**answer

2k views

### Existence/Uniqueness of solutions to quasi-Lipschitz ODEs

Would the Picard–Lindelöf theorem still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be almost Lipschitz in y?
If not, are there any moduli ...

**18**

votes

**1**answer

462 views

### Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...

**11**

votes

**1**answer

1k views

### Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$
runs over the integers?
The existence of the limes inferior follows from Dirichlet's approximation theorem,
but the ...

**14**

votes

**3**answers

1k views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...

**10**

votes

**1**answer

309 views

### On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := ...

**9**

votes

**1**answer

926 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**4**

votes

**1**answer

686 views

### Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...

**1**

vote

**2**answers

546 views

### Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left ...

**10**

votes

**1**answer

450 views

### A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ ...

**2**

votes

**2**answers

979 views

### How to compute \prod_{n=1}^{\infty} (1-p^{-n})

We know it converges for any prime p. I just want to know how to compute its exact value:
$$\prod_{n=1}^{\infty} (1-p^{-n})$$

**1**

vote

**1**answer

304 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...

**1**

vote

**1**answer

439 views

### New differintegral formula: how is it related to other differintegral formulas?

Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x)= \lim_{t\to s} ...

**0**

votes

**1**answer

80 views

### Examples of Bivariate Analytic, Globally Continuous Functions, whose Set of Zeros are Boundaries of Polygons

Are there any known examples of analytic, globally continuous functions $f(x,y): (x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f(x,y)\lt 0\Leftrightarrow ...

**98**

votes

**26**answers

31k views

### What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...

**85**

votes

**17**answers

19k views

### Why is differentiating mechanics and integration art?

It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by ...

**61**

votes

**21**answers

8k views

### Probabilistic Proofs of Analytic Facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...

**44**

votes

**22**answers

13k views

### Interesting Calculus Questions/Exercises

I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...

**47**

votes

**9**answers

8k views

### Why do functions in complex analysis behave so well? (as opposed to functions in real analysis)

Complex analytic functions show rigid behavior while real-valued smooth functions are flexible. Why is this the case?

**74**

votes

**8**answers

6k views

### Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$?
(I think it is.) If so, how would one prove this? (To confirm: This is the power
series for $e^x$, except with the ...

**29**

votes

**11**answers

27k views

### Why is the gradient normal?

This is a somewhat long discussion so bear with me. There is a theorem that I have always been curious about from an intuitive standpoint. This is an issue that has been glossed over in most ...

**58**

votes

**1**answer

4k views

### A hard integral identity on MATH.SE

The following identity on MATH.SE
$$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$
seems to be ...

**39**

votes

**16**answers

6k views

### How helpful is non-standard analysis?

So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...

**31**

votes

**15**answers

10k views

### What is the Implicit Function Theorem good for?

What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are:
the result that the solution space of a non-degenerate system of equations ...

**29**

votes

**6**answers

3k views

### Taylor's theorem and the symmetric group

Anytime I see an $n!$ in some formula, my instinct is to look for the symmetric group on $n$ letters coming in somewhere. I have never done this seriously with the $n!$ in Taylor's theorem.
Question: ...

**29**

votes

**4**answers

2k views

### Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
...

**12**

votes

**4**answers

8k views

### Visualization of Riemann–Stieltjes Integrals

The Riemann–Stieltjes integral $\int_a^b f(x)\,dg(x)$ is a generalization of the Riemann integral. It is e.g. heavily used as a starting point for stochastic integration. The approximating ...

**23**

votes

**2**answers

7k views

### Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?

**33**

votes

**4**answers

2k views

### Which functions of one variable are derivatives ?

This is motivated by this recent MO question.
Is there a complete characterization of those functions $f:(a,b)\rightarrow\mathbb R$ that are pointwise derivative of some everywhere differentiable ...