17
votes
Accepted
$f'=e^{f^{-1}}$, again
There is no analytic local solution at $0$ to $f'=e^{f^{-1}}$, $f(0)=0$, that is, the formal power series solution is diverging. Together with the solution given in comments by fedja, this means the ...
- 60.5k
10
votes
Continuum hypothesis and cardinality of infinite tree paths
If the nodes add a fixed amount of new nodes at each level, then the number of infinite paths seems to be countable.
It does not seem so for me. Even if the number of nodes $k_n$ on level $n$ ...
- 109k
9
votes
Accepted
Oscillation of monotone real-analytic function
An elementary example is given by the formula
$$f(x)=\sum_{k=1}^\infty k!\,\Phi(x-k!)$$
for real $x>1$, where $\Phi$ is the standard normal cumulative distribution function. Then the function $f$ ...
- 128k
8
votes
Accepted
Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
UPD. Bound simplified.
Here is a constructive bound for the number of solutions to $\phi(x)=m$.
Let $\varphi(a) = m$. If $p^k\mid a$ for some $k\geq 1$, then $p^{k-1}(p-1)\mid m$, and thus $k\leq 1+\...
- 34.3k
7
votes
Accepted
Does the rate of decay of an entire function dictate the global growth rate?
This is a typical walk-to-the-library problem. I used Boas, but probably other standard books would have worked too.
Boas proves the following results: (1) if $f$ is of order $1$, then $\limsup m(r)M(...
- 24.2k
7
votes
Accepted
Continuum hypothesis and cardinality of infinite tree paths
The set of all branches is a closed set of reals. Cantor proved that closed sets are either countable or of size continuum.
- 18.6k
6
votes
Accepted
Functions with at most linear growth at infinity: is the constant itself continuous?
The answer is no to both your hopes: it can happen that neither $M_{f_n}\to M_f$ nor $\sup_n M_{f_n}<+\infty$ hold, although $M_f<\infty$.
As a counter-example take
$$
f_n(x)=\max(0,n(x-n)).
$$
(...
- 5,380
5
votes
Accepted
Cardinality of growth rates
Let me rather define $f=o(g)$ as $\forall \varepsilon > 0, \exists x_0, \forall x \geq x_0, |f(x)| \leq \varepsilon |g(x)|$.
Let $f_0,f_1$ be such that $f_0 = o(f_1)$. Let us assume that we have ...
- 7,239
4
votes
Accepted
Milnor-Wolf theorem for topological groups
I found the following paper:
Yves Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bulletin de la Société Mathématique de France (1973) Volume: 101, page 333-379
The ...
- 4,728
4
votes
Accepted
Rate of convergence of Fejer kernel to the Dirac delta function
For any $\delta,x$ such that $0<\delta\leq |x|<\pi$ one has $|F_N(x)|\leq[2\pi(N+1)\sin^2(\delta/2)]^{-1}$, so the error in the delta-function approximation is of order $1/N$.
Two explicit ...
- 188k
3
votes
Accepted
How large must algebras with a given congruence lattice be?
What do we know about the growth rate of $C(n)$?
We know the exact value of $C(n)$ if, in its definition,
we restrict to the class of distributive lattices.
Otherwise we only have partial results. Let ...
- 14.6k
3
votes
Accepted
Largest asymptotic growth for $2f(x)-f(2x)$
Let us discretise the problem by setting $a_n=2^{-n}f(2^n)$, $b_n=2^{-n-1}\Delta_f(2^n)$. Then your relation becomes,
$$b_n=a_n-a_{n+1}.$$
since $a_n,b_n$ are non-negative, we conclude that
$$\sum_{n=...
- 91.8k
3
votes
Accepted
Geometry in Hilbert spaces / spheres in high dimensions
Set $c_n:=\frac{1}{\alpha_n}$. Clearly, for any $N>0$ one must have $\DeclareMathOperator{\spa}{span}$ $\DeclareMathOperator{\conv}{conv}$ $\newcommand{\bsB}{\boldsymbol{B}}$
$$
\bsB_1\cap\spa\{...
- 34.7k
3
votes
Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
We have
$$\frac n{\varphi(n)}=\prod_{p\mid n}\bigl(1-p^{-1}\bigr)^{-1}
\le2\prod_{\substack{p\mid n\\p\ge3}}\frac32
=2\prod_{\substack{p\mid n\\p\ge3}}3^{\log_3(3/2)}
\le2\prod_{\substack{p\mid n\\p\...
- 47.1k
3
votes
Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
Here is a simpler bound, based on the comment of R. van Dobben de Bruyn.
Let a solution of the equation be broken into two parts, c and d, where c is the n-smooth part of the solution, and is coprime ...
- 13k
2
votes
Proving a sum to be sublinear in growth
$\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\...
- 128k
2
votes
Accepted
When does this recurrence stop?
I always like to answer these questions by approximating the difference equation with a differential equation (for which the result is always easier). In this case, the d.e. is $\dot x=-x^{1/k}$, $x(0)...
- 23.2k
2
votes
Does the rate of decay of an entire function dictate the global growth rate?
The answer to your question is no. More generally, for functions of exponential type $b$ (which means $|f(z)|\leq Me^{b|z|}$), define the indicator:
$$h(\theta)=\limsup_{r\to\infty}\frac{\log|f(re^{i\...
- 91.8k
2
votes
Poisson Furstenberg Boundary of topological groups, reference request
In their paper "Existence of positive harmonic functions on groups and on covering manifolds", Bougerol & Elie give an overview of the connection between the three properties growth, amenability, ...
- 527
2
votes
Poisson Furstenberg Boundary of topological groups, reference request
Edit: I found the introduction of this article of A. Furman (mainly quoting results of Furstenberg) useful for one approach to non-amenability, although it maybe isn't quite what you are looking for:
...
- 4,728
1
vote
Accepted
Inferring polynomial rate of convergence from polynomial bound
The answer is no (assuming that $\|x\|:=x$ for real $x\ge0$). Indeed, as noted by mathworker21, your condition on the existence of $p$ is always satisfied, by choosing, e.g., $p(t)=c+t$ with $c:=\...
- 128k
1
vote
Accepted
Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?
When $M$ has positive curvature, the limit should always be 0. Indeed if $M$ has non-negative Ricci curvature, Bishop-Gromov tells you that $\frac{Vol S(p, r)}{n\omega_nr^{n-1}}\leq 1$, so the volumes ...
- 320
1
vote
Does the rate of decay of an entire function dictate the global growth rate?
I don't have the answer to your question but on top of my head :
consider an entire function $f$ that goes to zero on $\mathbb R^+$,
the function $z\mapsto f(1/z)$ is holomorphic on $\mathbb C^*$.
...
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