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Questions tagged [growth-rate]

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Rate of convergence of Fejer kernel to the Dirac delta function

This seems like something one might find in a book so I would be grateful for any references you think may be helpful. I am interested in the rate at which of a function integrated against the $N$th ...
JMcB's user avatar
  • 131
3 votes
0 answers
148 views

Growth of the constants from the Stone-Weierstrass Theorem

The Stone Weierstrass theorem for $C([0,1])$ claims that for any continuous function $f:[0,1]\to\mathbb{R}$ and each $n\in\mathbb{N}$, there is a polynomial $p_{n,f}(x)=\sum_ia_{f,n,i}x^i$ such that $\...
Saúl RM's user avatar
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4 votes
1 answer
151 views

How large must algebras with a given congruence lattice be?

This is a follow-up to a recent question of mine: For $n\in\mathbb{N}$ let $C(n)$ be the smallest $k$ such that every bounded lattice with cardinality $\le n$ which is isomorphic to the congruence ...
Noah Schweber's user avatar
0 votes
0 answers
117 views

A weaker condition on Fourier coefficients for boundedness of the function

Let $f : \mathbb{S}_1 \rightarrow \mathbb{C}$ be a square-integrable fnction and let $(\widehat{f}_k)_{k \in \mathbb{Z}}$ be its Fourier-coefficients. It is very well known, that the condition $\sum\...
Tardis's user avatar
  • 1,253
1 vote
0 answers
49 views

When are ellipsoids in a convex hull of a sequence with prescribed growth rate?

I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like ...
Tardis's user avatar
  • 1,253
2 votes
1 answer
204 views

Largest asymptotic growth for $2f(x)-f(2x)$

I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$ is positive for $x$ large enough and has the greatest asymptotic growth. ...
cs89's user avatar
  • 981
2 votes
1 answer
192 views

Geometry in Hilbert spaces / spheres in high dimensions

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are ...
Tardis's user avatar
  • 1,253
3 votes
1 answer
420 views

Functions with at most linear growth at infinity: is the constant itself continuous?

I am considering the family $\mathcal{F}$ of functions $f \colon \mathbb{R} \to \mathbb{R}$ which have at most linear growth at infinity, that is there exists a constant $M_f$ such that \begin{...
A. Pesare's user avatar
  • 192
7 votes
0 answers
173 views

Upper bounds for a sequence of integers

Given $\alpha\geq0$ we consider the sequence $$ C_k=k^\alpha\sum_{j=0}^{k-1}C_jC_{k-1-j} $$ with $C_0=1$. I'm interested in upper bounds (in terms of $\alpha$) for such a sequence. I know that when $\...
guacho's user avatar
  • 833
-3 votes
2 answers
238 views

Continuum hypothesis and cardinality of infinite tree paths [closed]

Suppose a tree with nodes located at levels $1,2,3...$. At each level the nodes branch into several nodes or do not branch. Does the cardinality of the set of all infinite paths in this tree depend on ...
Anixx's user avatar
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1 vote
0 answers
71 views

Approximation Rates for Multivariate Taylor Series

Let $k,n,m$ be positive integers and suppose that $f$ is $C^{k}(\mathbb{R}^n,\mathbb{R}^m)$ functions. For any given $\epsilon>0$ and $x_0\in \mathbb{R}^n$, are there known sharp approximation ...
ABIM's user avatar
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-1 votes
1 answer
87 views

Inferring polynomial rate of convergence from polynomial bound

Let $x_n$ be a non-negative valued sequence and suppose that the following hold: $\lim\limits_{n\to\infty} x_n =0$ There exists some polynomial function $p$ of degree at-least $1$ such that: $$ \|x_n\...
ABIM's user avatar
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1 vote
0 answers
287 views

Growth rate of exponential sum of $S_j$

Let $X_j$, $1 \leq j \leq n$ be chosen i.i.d. uniform over $[0,2\pi)$. Denote $S_j \triangleq X_1 +X_2+\cdots+X_j$ and suppose that $c_j$, $1 \leq j \leq n$ are some constants such that $|c_j|=1$. I'...
Mini's user avatar
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5 votes
3 answers
808 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. ...
zeraoulia rafik's user avatar
-1 votes
1 answer
230 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 ...
VS.'s user avatar
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1 vote
0 answers
116 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 ...
VS.'s user avatar
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3 votes
0 answers
109 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 ...
Thomas Meyer's user avatar
0 votes
2 answers
102 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 ...
zeraoulia rafik's user avatar
2 votes
0 answers
123 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) = \...
Joker123's user avatar
  • 153
3 votes
1 answer
137 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 ...
Federico's user avatar
  • 423
1 vote
1 answer
150 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=...
M.R.Karimi's user avatar
0 votes
1 answer
158 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?
Turbo's user avatar
  • 13.8k
0 votes
0 answers
60 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{ ...
Goulifet's user avatar
  • 2,226
1 vote
0 answers
155 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 ...
jaslibra's user avatar
  • 111
0 votes
0 answers
55 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 ...
tst's user avatar
  • 483
5 votes
3 answers
387 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 ...
tst's user avatar
  • 483
1 vote
0 answers
40 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 ...
Turbo's user avatar
  • 13.8k
1 vote
2 answers
748 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 ...
Snoop Catt's user avatar
4 votes
1 answer
169 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? ...
Snoop Catt's user avatar
31 votes
1 answer
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 ...
Fan Zheng's user avatar
  • 5,139
14 votes
2 answers
1k 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 ...
Tom Ultramelonman's user avatar
3 votes
2 answers
196 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\...
Marko Erceg's user avatar
0 votes
0 answers
161 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 $h(\eta_{n})$: $h(\eta_{n}) \rightarrow \log2$...
Bruno Brogni Uggioni's user avatar
5 votes
1 answer
482 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 ...
usul's user avatar
  • 4,489
10 votes
1 answer
201 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 ...
Volodymyr Nekrashevych's user avatar
4 votes
1 answer
352 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 $\alpha[n]$ ...
Wojowu's user avatar
  • 27.7k
6 votes
1 answer
971 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 ...
Kate Juschenko's user avatar
2 votes
0 answers
218 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. ...
Peter's user avatar
  • 1,203
2 votes
0 answers
196 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/...
alphanzo's user avatar
  • 113
6 votes
1 answer
467 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 ...
Jiang's user avatar
  • 1,528
5 votes
2 answers
733 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 ...
Victor's user avatar
  • 1,427