Questions tagged [growth-rate]

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5
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3answers
543 views

Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?

Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
0
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1answer
126 views

Natural candidates for sub-half-exponential which limit to half-exponential function from below

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However sub-half-exponentials (functions whose composition grows ...
1
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0answers
89 views

Natural candidates for super-half-exponential which limit to half-exponential function from above

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However super-half-exponentials (functions whose composition grows ...
2
votes
0answers
76 views

Are all intermediate growth branch groups just-infinite?

Are all branch groups of intermediate growth just-infinite? I can't seem to find an answer to this one way or another; the question is motivated by the fact all examples of intermediate growth branch ...
0
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2answers
73 views

Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?

In $1969$, Margulis proved, for suitable constant $h>0$and $r$ is a positive constant that : $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ with ($(S(p,r)$ is geodesic spheres), exists at ...
2
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0answers
98 views

Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation \begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \...
3
votes
1answer
112 views

Cardinality of growth rates

$Let f=o(g)$ stand for $\lim_{x\to\infty} f(x)/g(x)=0$. It is simple to prove the following fact. Proposition. Let $f_0,f_1:\mathbb{R}\to(0,\infty)$ be such that $f_0=o(f_1)$. Then there is a family ...
1
vote
1answer
116 views

Proving a sum to be sublinear in growth

Suppose that $\alpha \in (0,1)$. The goal is to prove that the following sum is of $o(T)$ (or, if possible, give a more accurate growth rate, e.g. $O(T^{1-\alpha})$ or something like that): $$ \sum_{t=...
0
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1answer
143 views

When does this recurrence stop?

Denote $n_i=n_{i-1}-\sqrt[k]{n_{i-1}}$. If $n_0=n$ then what is the minimum $i$ at which $n_i<2$ holds? Is there a standard technique to solve such problems? Any references?
0
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0answers
54 views

Polynomial growth of random walks: critical values?

Consider a sequence of i.i.d. random variables $(X_n)_{n\geq 0}$, and set $S_N = \sum_{n=1}^N X_n$. For which $p> 0$ do we have that \begin{equation} \lim \inf \frac{|S_N|}{N^{1/p}} > 0 \text{ ...
1
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0answers
153 views

Should the power series solution to $y' = y, y(0) = 1$ be obvious? [closed]

My Understanding: I would derive the Poisson random variable as follows: I consider an experiment which consists of a continuum of trials on an interval $[0,t)$. The result of the experiment takes the ...
0
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0answers
48 views

Questions on the behaviour of functions of exponential type 1

I am interested in understanding the properties of entire functions of exponential type 1. I have few questions about their growth. How many sectors can a function of exponential type have, in which ...
5
votes
3answers
315 views

Does the rate of decay of an entire function dictate the global growth rate?

If $f$ is entire and we assume a decay rate on $\mathbb{R}^+$, does this dictate a growth rate on $\mathbb{C}$? For example, is there a function that decays as $e^{-x^2}$ on $\mathbb{R}^+$ and is of ...
1
vote
0answers
38 views

Necessary additive and multiplicative properties to characterize a mildly growing function

Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...
0
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2answers
362 views

Poisson Furstenberg Boundary of topological groups, reference request

I'm trying to understand the relations between between the following group properties, in the case of (say, compactly generated locally compact) topological groups: Group growth. Amenability. Poisson ...
4
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1answer
145 views

Milnor-Wolf theorem for topological groups

The Milnor-Wolf theorem states that any finitely generated solvable group has either polynomial or exponential growth. Is there an analogous result for locally compact compactly generated groups? ...
30
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1answer
2k views

$f'=e^{f^{-1}}$, again

This question is a spin-off of this one, in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is ...
13
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2answers
693 views

When are growth series rational?

For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by $ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...
3
votes
2answers
179 views

The minimal growth rate of the countable family of sequences

Let us consider a countable set of sequences of positive numbers $\{(x_n^{(1)}),(x_n^{(2)}),\dots\}$ for which we have $(\forall k\in\mathbb{N}) \ \lim_n x_n^{(k)} = +\infty$, and $(\forall k\in\...
0
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0answers
127 views

Entropy, convergence and invariant measures

Could you give conditions that a sequence of shift invariant measures $\eta_{n}$ has to satisfy in order to happen this convergence in terms of entropies $h(\eta_{n})$: $h(\eta_{n}) \rightarrow \log2$...
5
votes
1answer
362 views

The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$

I was reading this question on functions "in the middle" of linear and exponential growth, and it caused me to think of this function: $$ G(x) = \left(1 + \frac{x}{f(x)}\right)^{f(x)} $$ for some ...
10
votes
1answer
189 views

Free subgroups in algebras of polynomial growth

What is known about free non-abelian subgroups in finitely generated associative algebras of polynomial growth (e.g., over finite fields, to avoid finite-dimensional free subgroups)? For example, are ...
4
votes
1answer
293 views

Functions in “gaps” in Hardy hierarchy

Recall the definition of Hardy's hierarchy: $H_0(n)=n+1\\ H_{\alpha+1}(n)=H_\alpha(n+1)\\ H_\alpha(n)=H_{\alpha[n]}(n)$, where the last rule applies if $\alpha$ is limit ordinal, and $\alpha[n]$ ...
5
votes
1answer
668 views

Milnor-Wolf result on growth of solvable groups

The Milnor-Wolf theorem says that any solvable group has either polynomial or exponential growth. I wonder about the existence of alternative proofs of this fact. I have an impression that the ...
1
vote
0answers
208 views

Growing rate of f(n) = $10 \rightarrow 10 \rightarrow … \rightarrow 10 \rightarrow 10$ with n 10's [closed]

I would like to have an estimate for a really fast growing function and a number. Define the following function f(n) = $10 \rightarrow 10 \rightarrow ... \rightarrow 10 \rightarrow 10$ with n 10's. ...
2
votes
0answers
174 views

geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent. see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French) http://link.springer.com/...
6
votes
1answer
363 views

growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?

I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...
5
votes
2answers
652 views

Do all finitely generated nilpotent semigroups have polynomial growth?

The notion of nilpotency passes nicely from groups to semigroups. Define $q_1(x,y)=xy$ and $$q_{i+1}(x,y,z_1,\cdots,z_i)=q_i(x,y,z_1,\cdots,z_{i-1})z_iq_i(y,x,z_1,\cdots,z_{i-1})$$ inductively for all ...