**15**

votes

**3**answers

3k views

### Why is $ \frac{\pi^2}{12}=\ln(2)$ not true?

This question may sound ridiculous at first sight, but let me please show you all how I arrived at the aforementioned 'identity'.
Let us begin with (one of the many) equalities established by Euler:
...

**62**

votes

**9**answers

13k 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 ...

**5**

votes

**1**answer

140 views

### Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...

**0**

votes

**1**answer

110 views

### Complete solution set of a Convolutional Equation?

Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..
Setup: In what ...

**2**

votes

**1**answer

184 views

### All solutions to a set of integral equations

I would like a better understanding of the set of pairs $(f_1,f_2)$ of functions $[0,1] \times [0,1] \to [0,1]$ which satisfy the following conditions:
For all $y \in [0,1]$, $f_1(x,y) \geq ...

**2**

votes

**1**answer

261 views

### Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds.
(Cf. discussion on p. 45.)
Definition
Let $E$ and $F$ be two Banach spaces together with a plain subset ...

**12**

votes

**1**answer

592 views

### A generalization of intermediate value theorem on R^k

Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$.
Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...

**2**

votes

**1**answer

369 views

### A proof from Lang's undergraduate analysis

This is from P.580 of Serge Lang's undergraduate analysis (2nd edition).
$\textbf {Proposition 2.3.}$ Let $A$ be an admissible set in $\mathbb R^n$ and assume that its closure $\bar{A}$ is contained ...

**-9**

votes

**1**answer

149 views

### Do the mathematicians really know the exact values of what usually called “indeterminate forms”? [closed]

First of all I would point out that exact value of a function and the limit of the function in that point do not necessarily coincide. For instance, it is often assumed that $0^0=1$ even though the ...

**1**

vote

**0**answers

261 views

### Is evaluating limits with dual numbers sound?

Let $D$ be the ring $\mathbb{C}[\epsilon]/\langle \epsilon^2\rangle$. Define the functions $dual : \mathbb{C} \to D$ and $stdPart : D \to \mathbb{C}$ by $dual(x) := x+0\cdot \epsilon$ and ...

**6**

votes

**1**answer

129 views

### Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$.
By this I mean that there exists a polynomial $P$
with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!)
such that $P(\phi) = 0$
Let ...

**3**

votes

**1**answer

406 views

### The Weyl algebra modules which are also rings

Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...

**3**

votes

**1**answer

257 views

### First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...

**7**

votes

**1**answer

239 views

### A conjecture about the measure estimates of a trigonometric polynomial

Formulation of the Conjecture
Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( ...

**8**

votes

**1**answer

358 views

### Basic examples of induction on scales arguments

An important ingredient in recent progress on Euclidean harmonic analysis has been that of "inductions on scales". A few examples are the papers of Wolff, Tao, and Bourgain and Guth.
Here is a ...

**9**

votes

**2**answers

333 views

### On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation
...

**9**

votes

**18**answers

17k 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?

**1**

vote

**1**answer

109 views

### Another type of derivative, and the associated primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that:
$
...

**1**

vote

**2**answers

64 views

### Finding conditions to guarantee existence of solutions to IVP [closed]

Consider the following IVP:
$x'=f(t,x)$ and $x(0)=x_0$, where $x\in \mathbb{R}^n$ and $t\in \mathbb{R}$.
Suppose that for all $(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$.
In order for the ...

**2**

votes

**1**answer

115 views

### Relationship between Laplacian and Hessian on compact Lie groups

If $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is smooth and compactly supported, one has
$$\int |\Delta f(\mathbf{x})|^2\,d\mathbf{x} = \int \| Hf(\mathbf{x}) \|_F^2\,d\mathbf{x}\,,$$
where $\Delta$ ...

**2**

votes

**2**answers

204 views

### Is this infinite series related to some well-known special functions?

Please allow me to resort once again to the expertise of the MathOverflow community :
During research I encoutered the following infinite series :
$$\sum_{n=-\infty}^{+\infty} ...

**1**

vote

**0**answers

19 views

### Taking the potential of a super additive measure

In recent research of my coauthors and me, it has become necessary to consider the Riesz potential of a superadditive measure.
Recall that the $s$-dimensional Riesz potential of a finite Borel ...

**1**

vote

**1**answer

124 views

### Generalization of the triple tangent identity

It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$
I came across the following generalization of this equality:
$$\sqrt{1-k^2} {\rm ...

**4**

votes

**0**answers

260 views

### Transforming a multivariable integral to make it separable

In the following I will omit requirements of smoothness, extent of domain, finiteness, etc, both to simplify the exposition and because I don't know exactly what the requirements are. Please imagine ...

**1**

vote

**0**answers

91 views

### How to find a invariant surface of a diffeomorphism

Recently, I read a paper about discrete Schrödinger operator. There is a map related to trace map from $C^3$ to $C^3$ as follows:
$$T(x,y,z)=(y,z,yz-x).$$
We can calculated that $T$ has the folliwng ...

**12**

votes

**1**answer

327 views

### Lower-Hölder embeddings of the sphere

My question is very simple:
Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that
$$
|f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r,
$$
for ...

**0**

votes

**0**answers

26 views

### Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions
Let
$L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric),
$a,b$ be arbitrary $n$-dimensional points,
$c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...

**4**

votes

**1**answer

110 views

### PDEs on torus $\mathbb T$

(Hope this question is o.k. for MO)
I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular ...

**17**

votes

**6**answers

3k views

### Square roots of the Laplace operator

In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...

**28**

votes

**19**answers

6k views

### Interesting applications (in pure mathematics) of first-year calculus

What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...

**5**

votes

**2**answers

194 views

### Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin ...

**4**

votes

**1**answer

356 views

### Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...

**0**

votes

**0**answers

114 views

### Notion of solution of pde

Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...

**3**

votes

**1**answer

255 views

### Does this function have any exponential growth?

Has anyone seen any function of the following type?
$$
g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0.
$$
The question is whether for some constant ...

**0**

votes

**0**answers

74 views

### Asymptotic analysis of a sum of complex summands using integral

I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, ...

**-2**

votes

**2**answers

48 views

### Systems of ODEs that fulfill a matrix relationship at steady state [closed]

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...

**2**

votes

**2**answers

125 views

### Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...

**20**

votes

**9**answers

2k views

### When does the zeta function take on integer values?

Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...

**0**

votes

**1**answer

111 views

### Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define
$$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...

**28**

votes

**7**answers

3k views

### formal power series convergence

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 ...

**0**

votes

**0**answers

101 views

### A question about the duality principle

Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by ...

**1**

vote

**0**answers

52 views

### How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...

**8**

votes

**1**answer

645 views

### How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial?

Suppose that $\sum^\infty_{n=1} u_n = s,$ where the series converges conditionally, and $s'>s$. How to prove the existence of such a permutation $\sigma,$ such that
1) $u_n\geq 0 \rightarrow ...

**1**

vote

**1**answer

38 views

### Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:
One factor would be the condition number of the approximation operator. The other factor ...

**1**

vote

**0**answers

97 views

### Is there a unique solution? [closed]

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...

**3**

votes

**2**answers

1k views

### Arctangents and the golden ratio

Why is the golden ratio lurking in $(d/dx)\arctan\left( x + \frac{1}{x} \right)$
$$
= \frac{\left(\frac{1+\sqrt{5}}{2}\right)}{x^2 + \left(\frac{1+\sqrt{5}}{2}\right)^2} + ...

**0**

votes

**0**answers

18 views

### Composition of Lossless Systems from Delay and Mixing regarding junction admittance

Given $m_1, \dots m_N \in \mathbb{N}$ and matrix $\mathbf{A} \in \mathbb{C}^{N\times N}$.
Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{-m_1}, \dots, z^{-m_N})$ with $z\in\mathbb{C}$. ...

**0**

votes

**0**answers

143 views

### Hammerstein integral equation with inverse of the solution

In signal processing theory I found this integral equation that I recognized to be of Hammerstein type:
$$u(t)-\int_{0}^{1}d\phi cos(\omega t+\phi)\frac{1}{u(\phi)}=0$$
Unfortunately the solution ...

**6**

votes

**1**answer

138 views

### Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which
$$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$
and
$$x_1+\ldots+x_k\leq 1.$$
I am ...

**0**

votes

**0**answers

64 views

### How to solve a couple of ODEs

Let $\phi_+ (\phi_-)$ be a strictly increasing (decreasing) function defined on $R_+$ such that $\phi_+(\phi_-)\in\mathcal{C}^0(R_+)\cap\mathcal{C}^1(R_+^{\ast})$ and $\phi_+(0)=0(\phi_-(0)=0)$. ...