24
votes

Accepted

### Writing a function on $\mathbb{R}$ as a sum of two injections

The answer is yes. Every function on the reals is the sum of two injective functions, and this can be done in a highly effective manner, constructing the two functions $g,h$ from $f$ without any need ...

16
votes

Accepted

### Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Consider the Dirichlet series
$$F(s)=\sum_{n=1}^\infty\frac{f(n)}{n^s},$$
where $f(n)$ is the completely multiplicative function which satisfies $f(2)=-1$ and $f(p)=(-1)^{(p-1)/2}$ for odd primes $p$. ...

13
votes

### Writing a function on $\mathbb{R}$ as a sum of two injections

It works at least for (locally) absolutely continuous functions.
Such a function is the integral of a locally $L^1$ function.
This weak derivative can be written as a sum of a positive and negative ...

11
votes

### How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$

Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th ...

8
votes

Accepted

### Are “most” bounded derivatives not Riemann integrable?

In 1977 Clifford E. Weil showed that $A$ is a first Baire category set (i.e. a meager set) in $X$ (sup norm) -- see The space of bounded derivatives. So the situation, at least with respect to one ...

6
votes

### Fourier coefficients of the logarithm of a given function

What you stated is correct. The crude statement is
$|d_n|\leq C(\mu)e^{-\mu n}$, for every $\mu<\mu_0$, where
$$\mu_0=\min\{\lambda,\theta_0\},\; \theta_0=\min\{|\theta|:
f(x+i\theta)=0\}.$$
Proof. ...

5
votes

Accepted

### Uniqueness of the $J$ invariant

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...

5
votes

Accepted

### Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

No, this is usually not possible. There's a previous MO question discussing this, but to add to the material there:
A time-one map $\phi_1$ commutes with the 1-parameter family of diffeomorphisms $\...

4
votes

### What is the minimum and the maximum perimeter of a triangle with area $x$ that can be inscribed in a circle?

The extrema are attained for symmetric (isosceles) triangles. Trigonometry gives for the pairs (area, perimeter) as a function of the half tip angle $\psi$ the relations
$$\left( 4 \,\sin \psi\,\cos^3\...

3
votes

Accepted

### Can a solution to this parameterized ODE converge to zero?

The answer is yes and the proof splits in some steps. In what follows $\sigma$ is just positive, decreasing and in $L^1(0, \infty)$.
If $y$ is a bounded solution of the ODE (forgetting the initial ...

3
votes

### Can I accurately approximate solutions for m for any k being an integer : $\sum_{n=1}^{k+1} \frac{k ! m^{(k-n+1)}}{(k-n+1) !}-\frac{k !}{2} e^m = 0$

$\newcommand\Ga\Gamma$Your conjecture is correct. Indeed, using repeated integration by parts (see e.g. formula (2.6) in this paper or in its arXiv version), we have
$$\Ga(k+1,m)=k!e^{-m}\sum_{i=0}^k\...

3
votes

Accepted

### Can we integrate arbitrary rational functions of Jacobian elliptic functions?

The answer is positive. Rational function of Jacobi elliptic function is elliptic, and for every elliptic function $f$,
$f(z)dz$ is an Abelian differential, and integral of it is an Abelian integral. ...

2
votes

### Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems? (time-varying case)

Any orientation-preserving diffeomorphism of $\mathbb{R}^d$ is isotopic to the identity. Differentiating the isotopy gives you a flow whose time-one is the diffeomorphism. See also this MO answer.

2
votes

### When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?

Weak$^*$-convergence of a net means pointwise convergence on singletons, and therefore (uniform) convergence finite sets. Uniform convergence on compact subsets means exactly that. Since finite sets ...

2
votes

Accepted

### Matrices and vectors of intervals

$\newcommand\R{\mathbb R}$Any operation you can define on intervals on the real line, you can define (entry-wise) on any arrays of such intervals.
For any function $f\colon\R^n\to\R$, you can define ...

2
votes

Accepted

### Computation of tangent functional

$\newcommand{\om}{\omega}\newcommand{\Om}{\Omega} \newcommand{\de}{\delta}\renewcommand{\th}{\theta}$For each real $t>0$ and some $\om_t\in\Om$,
\begin{equation*}
\|x+ty\|
=|(x+ty)(\om_t)|=|...

2
votes

### Computation of tangent functional

The idea is that, for $|x(\omega)|\sim 1$ and $t$ small, $$|x(\omega)+ty(\omega)|\sim|x(\omega)^2+tx(\omega)y(\omega)|\sim 1+tx(\omega)y(\omega)$$
Indeed, if $|x(\omega)|<1-\varepsilon$, then for ...

2
votes

### If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point?

The answer is negative. Take a bounded nowhere differentiable continuous map $f : \mathbb{R} \to \mathbb{R}$ and consider the map $g(x) = f(x) \cdot x^2$, which is differentiable at $x = 0$, but not ...

2
votes

### Eigenvalues of the modified Mathieu equation with normalizable solution

Solutions of this equation, normalized at $x\to\pm\infty$ are called Mathieu functions of the third kind. See, for example, D. Naylor, On a simplified asymptotic formula for the Mathieu function of ...

1
vote

Accepted

### If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?

This is more suited for math.SE but I'll still post the answer here, although the post will likely be closed soon.
You can simply apply Lax-Milgram in the Hilbert space
$$
H=\Big\{f\in H^1(0,1): \...

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