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 ...
- 219k
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 $p$. ...
- 95.1k
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 ...
- 7,981
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 ...
- 95.1k
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 ...
- 3,097
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. ...
- 87.2k
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 ...
- 35.7k
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 $\...
- 5,221
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
$$\left( 4 \,\sin \psi\,\cos^3\...
- 1,371
3
votes
Accepted
Can a solution to this parameterized ODE converge to zero?
The answer is yes and the proof splits in some steps. In what follows $\sigma$ is just positive, decreasing and in $L^1(0, \infty)$.
If $y$ is a bounded solution of the ODE (forgetting the initial ...
- 4,765
3
votes
Can I accurately approximate solutions for m for any k being an integer : $\sum_{n=1}^{k+1} \frac{k ! m^{(k-n+1)}}{(k-n+1) !}-\frac{k !}{2} e^m = 0$
$\newcommand\Ga\Gamma$Your conjecture is correct. Indeed, using repeated integration by parts (see e.g. formula (2.6) in this paper or in its arXiv version), we have
$$\Ga(k+1,m)=k!e^{-m}\sum_{i=0}^k\...
- 110k
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. ...
- 87.2k
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.
- 19k
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 ...
- 41
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 ...
- 110k
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)|=|...
- 110k
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 ...
- 354
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 ...
- 46.7k
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 ...
- 87.2k
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): \...
- 4,826
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