Questions tagged [growth-rate]
The growth-rate tag has no usage guidance.
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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 ...
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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 ...
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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 $\...
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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 ...
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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\...
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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 ...
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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.
...
- 88.9k
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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 ...
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$f'=e^{f^{-1}}$, 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 ...
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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{...
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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 $\...
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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 ...
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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 ...
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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\...
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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'...
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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. ...
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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]$ ...
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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 ...
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108
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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 ...
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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 ...
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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) = \...
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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 ...
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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=...
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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?
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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 ...
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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{ ...
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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 ...
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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 ...
- 88.9k
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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 ...
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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 ...
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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?
...
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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 ...
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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\...
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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 $h(\eta_{n})$:
$h(\eta_{n}) \rightarrow \log2$...
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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 ...
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1
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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 ...
CommunityBot
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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 ...
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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.
...
- 1,203
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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/...
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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 ...
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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 ...
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