text:
\section{Generalized Cauchy Process}
\label{sec:genCauchy}

The stationary Gaussian process defined by the following generalized Cauchy (GC) covariance parametrized by two indices
\begin{align}
  C_{\alpha,\beta}(t) & = \Bigl<U_{\alpha,\beta}(s)U_{\alpha,\beta}(t+s)\Bigr>
                    = \bigl(1+ |t|^\alpha\bigr)^{-\beta}, &
                  & t \in \mathbb{R}, \ 0 < \alpha \leq 1, \ \beta > 0,
\label{eq:genCauchy_0010}
\end{align}
is known as generalized Cauchy process.
\cite{LimLi06}
When $\alpha = 2$, $\beta = 1$ one gets the usual Cauchy process.
This process was first introduced by Gneiting and Schlather.\cite{GneitingSchlather04}
It has a nice and useful property which allows separate characterization of fractal dimension and LRD by two different parameters.

It is well-known that a stationary process cannot be self-similar.
$U_{\alpha,\beta}(t)$ satisfies a weaker self-similar property known as local self-similarity.
\cite{Kent97,Adler81}
A Gaussian stationary process is locally self-similar of order $\kappa$ if its covariance $C(t)$ satisfies for
$t \to 0$,
\begin{align}
  C(t) & = 1 - \beta|t|^\kappa \Bigl[1+ O\bigl(|t|^\nu\bigr)\Bigr], &
       & \nu > 0.
\label{eq:genCauchy_0020}
\end{align}
A more intuitive alternative definition is the following.
A Gaussian process $U(t)$ is said to be locally self-similar of index $\kappa$ if
\begin{align}
  U_{\alpha,\beta}(s) - U_{\alpha,\beta}(rt) & = r^\kappa \Bigl[U_{\alpha,\beta}(s) - U_{\alpha,\beta}(t)\Bigr], &
                                      & \text{as} \ |t - s| \to 0,
\label{eq:genCauchy_0030}
\end{align}
where the equality is in the sense of finite joint distributions.
The above two definitions and also the locally asymptotically self-similarity defined by
(\ref{eq:multifracBM_0010})
are all equivalent.\cite{LimLi06}
It is straight forward to show that the tangent process at a point $t_\circ$ is FBM indexed by $\alpha$.
In other words, GC process behaves locally like a FBM.
The fractal dimension of the graph of a locally self-similar process of order $\alpha$ is
$5/2 - \alpha$.

GC process is LRD for $0 < \alpha\beta \leq 1$ and is SRD if $\alpha\beta > 1$.
The large time lag behaviour of the covariance
(\ref{eq:genCauchy_0010}) is given by the hyperbolically decaying covariance
$C(t) \sim |t|^{-\alpha\beta}$, $t \to \infty$ which is characteristic of LRD.
If the covariance is re-expressed as
$\bigl(1 + |t|^\alpha\bigr)^{-\zeta/\alpha}$
then the parameters $\alpha$ and $\zeta$, respectively, provide separate characterization of fractal dimension and LRD.

It is interesting to point out that the covariance of GC process has the same functional form as the characteristic function of generalized Linnik distribution
\cite{LimTeo10}
and spectral density of the generalized Whittle-Mat\'{e}rn process.\cite{LimTeo09b}
There are also laws in physics which have this same analytic form.
One example is the Havriliak-Negami relaxation law in the non-Debye relaxation theory.
\cite{LimLi06}
Thus, all of these quantities should have the same analytic and asymptotic properties,
and results obtained in any one of them are of relevance to the other.

Applications of GC process can be found in geostatistics, telecommunication
and climate modelling.
\cite{LimLi06,LiLim08,Schlather10}
Extension to GC field\cite{LimTeo09c}
is particularly useful for geological modeling.
Generalization of GC process to multifractional GC process so far has not been carried out.
However, it is expected such a generalization would be similar to MRBM indexed by two variable parameters.

config: arxiv


text:
\begin{document}

\maketitle

\begin{abstract}
We determine statistical and computational limits for estimation of a rank-one matrix (the spike) corrupted by an additive gaussian noise matrix, in  a sparse limit, where the underlying hidden vector (that constructs the rank-one matrix) has a number of non-zero components that scales sub-linearly with the total dimension of the vector, and the signal-to-noise ratio tends to infinity at an appropriate speed. We prove explicit low-dimensional variational formulas for the asymptotic mutual information between the spike and the observed noisy matrix and analyze the approximate message passing algorithm in the sparse regime. For Bernoulli and Bernoulli-Rademacher distributed vectors, and
when the sparsity and signal strength satisfy an appropriate scaling relation, we find all-or-nothing phase transitions for the asymptotic minimum and algorithmic mean-square errors. These jump from their maximum possible value to zero, at well defined signal-to-noise thresholds whose asymptotic values we determine exactly. In the asymptotic regime the statistical-to-algorithmic gap diverges indicating that sparse recovery is hard for approximate message passing.
\end{abstract}

\input{sections/intro}

\input{sections/InfoTheor_XX}

\input{sections/secAMP_XX}

\input{sections/broaderimpact}
\section*{Acknowledgments}
J.B.\ acknowledges discussions with Galen Reeves during his visit of Duke University. C.R.\ acknowledges support from NSF CCF \#1849883 and N.M. from Swiss National Foundation for Science grant number 200021E 17554.

\bibliographystyle{unsrt_abbvr}
\bibliography{refs}

 \appendix

 \input{sections/APP_Wigner_generalized}

 \input{sections/APP_interp_XX}

 \input{sections/APP_concentratrion_f_XX}

 \input{sections/APP_conc_overlap_XX}

 \input{sections/APP_remarkable}

 \input{sections/APP_ITthresh}

\input{sections/APP_AMP_trans}

 \input{sections/APP_nishi}

 \input{sections/APP_properties_gaussian}

 \input{sections/APP_MMSE}

\input{sections/APP_AMP}

\input{sections/APP_AMP_regime}

\end{document}


TITLE: Complex logarithm of a product
QUESTION [0 upvotes]: In Apostol Calculus 1, the complex logarithm is defined as follows:
$$\text{Log}(z) = \log(|z|) + i \; \text{arg}(z)$$
We assume $\text{arg}(z) \in (-\pi, \pi]$. However, in the next exercise, we need to show that:
$$\text{Log}(z_1z_2) = \text{Log}(z_1) + \text{Log}(z_2) + 2n\pi i, n \in Z$$
I do not see how he gets to this outcome using the definition:
$$\text{Log}(z_1z_2) =  \log(|z_1||z_2|) + i \; \text{arg}(z_1z_2) = \log(|z_1|) +\log(|z_1|) + i \; \big(\text{arg}(z_1) + \text{arg}(z_2)\big)$$
or $\text{Log}(z_1z_2) = \text{Log}(z_1) + \text{Log}(z_2)$. Where is the summand $2n\pi i$ supposed to appear from?
This detail necessary to show the restrictions on the following fact:
$$(z_1z_2)^w = z_1^wz_2^w$$
where this summand plays an important role to show the domain of the complex numbers $z_1$ and $z_2$.

REPLY [1 votes]: The point is $\arg(z_1z_2)\neq\arg(z_1)+\arg(z_2)$, since the argument function is valued in a finite interval: $[0,2\pi)$, usually.
Thus if $z_1=z_2=-i$, $z_1z_2=-1$ and
$$\arg(z_1z_2)=\pi\neq 3\pi=\arg(z_1)+\arg(z_2)$$
What is true is that the argument of the product of two complex numbers is either the sum of the arguments or the sum plus $2\pi$. Hence, generalizing to any number of numbers, the formula holds up to adding an entire number of full angles, i.e. $2n\pi$.


TITLE: Let $E= A_1 \times A_2 \times \dots A_d$ and $F = B_1 \times B_2 \times \dots B_d$. Is there a way to express $E \setminus F$ as an union somehow?
QUESTION [0 upvotes]: Let $E= A_1 \times A_2 \times \dots A_d$ and $F = B_1 \times B_2 \times \dots B_d$. Is there a way to express the set $E \setminus F$ as an union somehow?

For say $d=2$ case one has $(A_1 \times A_2) \setminus (B_1 \times B_2) = (A_1 \times (A_2 \setminus B_2)) \cup ((A_1 \setminus B_1) \times B_2)$, but I have trouble expressing this with more elements. What should I consider here?

REPLY [1 votes]: How about:$$A_1\times\cdots\times A_i\times\cdots\times A_d-B_1\times\cdots\times B_i\times\cdots\times B_d=\bigcup_{i=1}^dA_1\times\cdots\times (A_i\setminus B_i)\times\cdots\times A_d$$


TITLE: Covering radius and Binary Even weight codes
QUESTION [3 upvotes]: Stumbled up on the statement: "if we puncture a  binary even weight code the covering radius decreases  by one", yet I'm having some difficulty proving it.
It's mainly because I don't see where the even weight of the codewords comes into play. I was able to prove that puncturing a binary code would not necessarily change the covering radius with an example, but can't prove said statement. Am I missing an obvious step?

REPLY [1 votes]: Let $\mathcal{C}$ be your code (of length $n$), and let $\rho$ be its covering radius.
To each vector $x\in\Bbb{F}_2^n$ let $d_{\mathcal{C}}(x)$ be the minimum Hamming distance from a codeword of $\mathcal{C}$ to $x$. It follows that $$\rho=\max\{d_{\mathcal{C}}(x)\mid x\in\Bbb{F}_2^n\}.$$
Let $\mathcal{C}'$ be the punctured code. Similarly, for all $x\in\Bbb{F}_2^n$ let $x'$
be the corresponding punctured vector. Consider a vector $y\in\Bbb{F}_2^{n-1}$.
Justify the following:

There are two vectors, $x$ and $\overline{x}$, differing at the punctured position, such that $x'=y=\overline{x}'$.
The weights of $x$ and $\overline{x}$ have opposite parities.
Because all the words of $\mathcal{C}$ have an even weight, the distances $d_{\mathcal{C}}(x)$ and $d_{\mathcal{C}}(\overline{x})$ have opposite parities also.
Therefore either $d_{\mathcal{C}}(x)$ or $d_{\mathcal{C}}(\overline{x})$ is $\le\rho-1$.
Therefore $d_{\mathcal{C'}}(y)\le\rho-1$. In other words, the covering radius of $\mathcal{C}'$ is at most $\rho-1$.
But if $d_{\mathcal{C}}(x)=\rho$, then $d_{\mathcal{C'}}(x')\ge\rho-1$, so the covering radius of $\mathcal{C'}$ is exactly $\rho-1$.


text:
\begin{document}

\title{A Simple Lower Bound on the Noncoherent Capacity of Highly Underspread Fading Channels}

\author{Steven~Herbert,~\IEEEmembership{Student~Member,~IEEE,}
        Ian~Wassell,
        Tian-Hong~Loh,~\IEEEmembership{Member,~IEEE,}
\thanks{This work is supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) and National Physical Laboratory (NPL) under an EPSRC-NPL Industrial CASE studentship programme on the subject of Intra-Vehicular Wireless Sensor Networks. The work of T. H. Loh was supported by the 2009-2012 Physical Program and 2012-2015 Electromagnetic Metrology Program of the National Measurement Office, an Executive Agency of the U.K. Department for Business, Innovation and Skills, under Projects 113860 and EMT13020, respectively.}
\thanks{Steven Herbert is with the Computer Laboratory, University of Cambridge, CB3 0FD, UK, and the National Physical Laboratory, TW11 0LW, UK. Contact: sjh227@cam.ac.uk}
\thanks{Ian Wassell is with the Computer Laboratory, University of Cambridge, CB3 0FD, UK. Contact: ijw24@cam.ac.uk}
\thanks{Tian Hong Loh is with the National Physical Laboratory, TW11 0LW, UK. Contact: tian.loh@npl.co.uk}
}

\maketitle

\begin{abstract}

\input{abst}
\end{abstract}

\begin{IEEEkeywords}
Underspread channels, noncoherent capacity, vehicle communications, Kalman filters, hidden markov processes.
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\section{Introduction}
\label{int}
\input{intro}

\input{chcap}

\appendices
\section{Proof of Lemmas}
\input{appall}

\section*{Acknowledgment}

The authors would like to thank Tobi Koch and Jossy Sayir for generously lending their expertise in a series of introductory discussions. Also to Edward Mottram for his help with the Lemma's, and to Ramji Venkataramanan for proof reading and internally reviewing the paper.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi


TITLE: How to treat small number within square root
QUESTION [3 upvotes]: guys.I am reading a math book. It has a equation shown as follows,
$\sqrt{(1+\Delta^2)}$
And then,since $\Delta$ is very small, it can be written as,
$\sqrt{(1+\Delta^2)} = (1+\frac12\Delta^2)$
What is the theory behind this equation?

REPLY [0 votes]: What @Mohamed_Fadel noted is NOT true.
The correct taylor expansion for $\sqrt {1+x^2}$ at the point $x_0=0$ is:
$$\sqrt {1+x^2} = 1 + \frac {1}{2} x^2 - \frac {1}{8} x^4 + \frac {1}{16} x^6...$$
That you get by just going through the standard taylor expansion.
For getting $\sqrt{(1+\Delta^2)} = (1+\frac12\Delta^2)$, all you need to do is neglect every order following the quadratic.
@Mohamed_Fadel: You cannot treat $x^2$ like a single variable $x$ when expanding, because inner derivatives will arise when doing the Taylor expansion.


TITLE: Epidemic threshold
QUESTION [5 upvotes]: Need some help / ideas to proceed. Stuck for a while on this.
In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{\max}(A)$ where $\lambda_{\max}(A)$ is the largest eigenvalue of the adjacency matrix $A$, thus implying that the outbreak is basically dependent on the spectral radius of the underlying contact graph.
Say, now, there's a new kind of virus that spreads by a different mechanism (i.e., no longer purely based on the existence of a link between two nodes). For instance, say each time it infects a specific type of people (nodes) (e.g., male/female, young/old, Asian/European/American/African etc.) and then it mutates and infect different kind of people (nodes) next time instance.
For this, I come up with a matrix, say $X$, that records the probability of the virus infecting each node (i.e., $x_{ij}$ = probability of node $j$ being infected if $i$ is the infectant). It is a non-symmetric square hollow matrix with positive values for all off-diagonal elements in the matrix.
With this infection probability matrix $X$ in mind, I'm struggling on how to derive the epidemic threshold. Is it still $1/\lambda_{\max}(X)$ by analogy of the conventional epidemic theory? Any idea, suggestions, hints or pointers will be appreciated.
Secondly, if I have two of such matrices $X_1$ and $X_2$ essentially describing the behaviour of two viruses, how do I compare them using their corresponding probability matrix, $X$?
Thank you in advance.

REPLY [2 votes]: The problem is addressed in

Peng, Chengbin, Xiaogang Jin, and Meixia Shi. "Epidemic threshold and immunization on generalized networks." Physica A: Statistical Mechanics and its Applications 389.3 (2010): 549-560. Link

An epidemic will become extinct if and only if the spectral radius of matrix $X$ is smaller than 1.


TITLE: adding general "linear" quaternion polynomials
QUESTION [1 upvotes]: The starting point is: What is the most general form of a "linear quaternion polynomial"?
By linear here, I do not mean a linear map, but general polynomials limited to at most first degree in the variable. This is complicated by quaternion multiplication not commuting, so there may also be a more appropriate term for these "polynomials", but hopefully it is clear from context below what I mean.
As a first stab, I guessed that every such polynomial could be put in the form:
$$ f(q) = aqb + cq + qd + e $$
with quaternion constants $a,b,c,d,e$.
But I'm really struggling to prove that adding two functions of this form results in another function of this form. So maybe that is not the most general form?
More explicitly, let
$$
\begin{aligned}
f(q) &= f_0qf_1 + f_2q + qf_3 + f_4 \\
g(q) &= g_0qg_1 + g_2q + qg_3 + g_4 \\
h(q) &= f(q) + g(q) = h_0qh_1 + h_2q + qh_3 + h_4 \\
\end{aligned}
$$
How can I solve for the constants in $h(q)$ given the constants in $f(q),g(q)$?
Or if this is provably impossible, what is the most general form of a linear quaternion polynomial?

EDIT:
Hand wavy, but maybe I should expect that the most general form would allow
$$
\begin{aligned}
 (q_0, q_1, q_2, q_3) \rightarrow &( \\
&a_{00} q_0 + a_{01} q_1 + a_{02} q_2 + a_{03} q_3 + a_{04}, \\
&a_{10} q_0 + a_{11} q_1 + a_{12} q_2 + a_{13} q_3 + a_{14}, \\
&a_{20} q_0 + a_{21} q_1 + a_{22} q_2 + a_{23} q_3 + a_{24}, \\
&a_{30} q_0 + a_{31} q_1 + a_{32} q_2 + a_{33} q_3 + a_{34} \\
&) \\
\end{aligned}
$$
Which suggests that what I wrote is not even close to the most general form for a quaternion linear polynomial.
If we label the quaternion basis as $h_0 = 1, h_1 =i, h_2 = j, h_3 = k$, then maybe the general form is:
$$ f(q) = a_0 + \sum_{i,j} c_{ij}\ h_i\ q\ h_j $$
where $c_{i,j}$ are real value constants and $a_0$ is a quaternion constant.
That seems excessive. Is it really not possible to reduce this further?
In this form it is at least easy to add two linear polynomials, so could be a good stepping stone if there is a more reduced version.

REPLY [2 votes]: Let $Q$ be a quaternion algebra (I assume that you want to work with $Q=\mathbb{H}$, the usual Hamilton quaternions, but this works for general quaternion algebras). Then we have a linear map (sometimes called the "sandwich map")
$$ Q\otimes Q\to L(Q)$$
where I use $L(Q)$ for the linear maps $Q\to Q$, given by
$$a\otimes b\mapsto (q\mapsto aqb).$$
So the image of $a\otimes b$ is one of your "linear polynomials". You also use the image of $1\otimes d$ and $c\otimes 1$ when you have "one-sided" products. In the end the linear part of your initial guess is exactly the image of the tensors of the form $a\otimes b+c\otimes 1+1\otimes d$.
Now the result is the following: this map $Q\otimes Q\to L(Q)$ is an isomorphism (of vector spaces).
First, this means that your "linear polynomials" are in fact all affine maps $Q\to Q$ (since you allow arbitrary sums), since all elements of $Q\otimes Q$ are sums of pure tensors $a\otimes b$.
But since not all tensors can be written as $a\otimes b+c\otimes 1+1\otimes d$, your initial guess is not enough, as you suspected. So actually your question can be reduced to: how many pure tensors are necessary to write a general element of $Q\otimes Q$? I'll let you think about that.


TITLE: How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$ and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$?
QUESTION [2 upvotes]: How to compute taylor series for $f(x)=\frac{1}{1-x}$ about $a=3$  and $f(x)=\sin{x}$ about $a=\frac{\pi}{4}$ ?
For the first one, using substitution, let $t=x-3$, then $x=t+3$. Then  $f(x)=\frac{1}{1-(t+3)}=\sum^{\infty}_{n=0}(t+3)^n=\sum^{\infty}_{n=0}(x)^n$. Apparently my solution wrong as it returns to the original series.
For the second one,  let $t=x-\frac{\pi}{4}$. Then $\sin({t+\frac{\pi}{4}})=\sum^{\infty}_{n=0}\frac{(-1)^n}{(2n+1)!}(t+\frac{\pi}{4})^{2n+1}$. Then I fail to countinue.
Could someone suggest a solution?

REPLY [1 votes]: Hint. One may write
$$
f(x):=\frac{1}{1-x}=-\frac12\frac{1}{1+\frac{(x-3)}2}=-\frac12\sum_{n=0}^\infty(-1)^n\frac{(x-3)^n}{2^n}, \quad |x-3|<2,
$$ and one may write, for any real number $x$,
$$
\begin{align}
f(x)&:=\sin x
\\\\&=\frac{\sqrt{2}}2 \left(\sin\left(x-\frac{\pi}4\right)+ \cos\left(x-\frac{\pi}4\right)\right)
\\\\&=\frac{\sqrt{2}}2\left(\sum^{\infty}_{n=0}\frac{(-1)^n}{(2n+1)!}\left(x-\frac{\pi}{4}\right)^{2n+1}+\sum^{\infty}_{n=0}\frac{(-1)^n}{(2n)!}\left(x-\frac{\pi}{4}\right)^{2n}\right).
\end{align}
$$


text:
\begin{document}
\maketitle
\date{}
\begin{abstract}
We explicit and clarify better the contraction method that Bacry and Levy-Leblond\cite{jmll} used to link all the kinematical Lie groups. Firstly, we use the kinematical parameters: the speed $c$ of light, the radius $r$ of the universe  and the period $\tau$ of the universe constrained by $r=c\tau$. Secondly, we use the dynamical parameters that are mass $m$, energy $E_0$ and compliance $C$. The kinematical and the dynamical parameters are related by the three relations $c^2=\frac{E_0}{m}$, $\tau^2=mC$ and $r^2=CE_0$. For each kinematical Lie algebra, we express the associated physical quantities in function of these dynamical parameters.
\end{abstract}
\section{Introduction}
Contraction is a method permitting to construct a new group (algebra) from an old one. Contraction of Lie groups and Lie algebras started with E.Inonu and E.P.Wigner\cite{inonuwigner} in $1953$. Inonu and Wigner were trying to connect Galilean relativity and special relativity. Eight years later, in $1961$, E.Saletan\cite{saletan} provided a mathematical foundation for the Inonu-Wigner method. Since then, various papers have been produced and the method of contraction has been applied to various Lie groups and Lie algebras(\cite{dooley},\cite{fialowski}, \cite{gromov}, \cite{guo}, \cite{jmll2}, \cite{mcrae},\cite{ancilla1},\cite{sanjuan}).\\In addition this method has been used by H.Bacry J.M.Levy-Leblond \cite{jmll} to link the de Sitter Lie algebras to all other kinematical Lie algebras through three kinds of contractions: speed-space contractions, speed-time contractions and space-time contractions. These naming being related to the fact that Bacry and Levy-Leblond have first of all scaled the speed-space generators, the speed-time translation generators ,the space-time translation generators by a parameter $\epsilon$ to obtain in the limit $\epsilon\rightarrow 0$ the speed-space contractions, speed-time contractions and space-time contractions respectively. Within the corresponding eleven Lie groups, four of them, the Galilei group $G$ governing the Newtonian physics (Galilean relativity), the Poincare group $P$ governing the Einstein physics (special relativity), the Newton-Hooke groups $NH_{\pm}$ describing a galilean relativity in presence of a cosmological constant and the de Sitter Lie groups $dS_{\pm}$ governing the de Sitter relativity of a space-time in expansion or oscillating, are well known in physics litterature. Within the remaining five ones, the Para-Poincare groups $P_{\pm}$ and the Static $S$ are still unknown in physics but the Para-Galilei group $G_+$ and the Carroll group $C$ are gaining more interest in nowadays. The Para-Galilei group as governing a light spring\cite{ancilla} while the Carroll group as associated to tachyon dynamics\cite{escamilla} , to Carrollian electromagnetim\cite{duval} versus Galilean electromagnetism\cite{Lebellac} or to dynamics of Carroll particles\cite{joakimgomis}.\\The aim of this paper is to explicit the three kinds of contraction methods and clarify them better. First, we recall the Inonu-Wigner contraction method (section $2$). We recall also the twelve kinematical Lie algebras found by A.Ngendakumana and al.\cite{ancilla1} following the Bacry and Levy-Leblond method (section $3$). We recover these twelve kinematical Lie algebras by using the kinematical parameters which are the universe radius $r$ related to the cosomlogical constant by $r^2=\frac{3}{\Lambda}$, the universe period $\tau$ and the speed of light $c$. This clarifies the speed-space, speed-time and the space-time contractions of Bacry and Levy-Leblond (section $4$). In so doing, the kinematical Lie algebras are found by keeping one parameter finite and by letting the other two tend to infinity while keeping their ratio finite. With section $5$ we parameterize the de Sitter Lie algebras by the dynamical parameters mass $m$, compliance $C$ (inverse of stifness or Hooke constant) and energy $E_0$ related to the kinematical parameters by $c^2=\frac{E_0}{m}$, $\tau^2=mC$ and $r^2=CE_0$. Each of the three kinds of contraction is then obtained by letting one of the dynamical parameter tend to infinity. The three Bacry and Levy-Leblond contractions, i.e. the speed-space contraction, the speed-time contraction and the space-time contraction correspond to an infinite energy $E_0$, an infinite mass $m$ and an infinite compliance $C$ respectively. They are the Newtonian limit, the static limit and the flat limit of Dyson\cite{dyson}. Finally with section $6$ , the Kirillov form permits us to establish for each kinematical Lie algebra, a Poisson Lie algebra and the motion equations that clarify better the links and differences between the twelve kinematical Lie algebras.
\section{Inonu-Wigner Contractions of Lie algebras}
We start with a Lie algebra $(\cal{G},\varphi)$ generated by $X_i$ and defined by $\varphi(X_i,X_j)=X_kC^k_{ij}$ where the constants $C^k_{ij}$ are called the structure constants of $\cal{G}$. The mapping $\varphi:\cal{G}\times \cal{G}\rightarrow \cal{G}$ is \textbf{bilinear skewsymmetric } and satisfy the Jacobi identity
\begin{eqnarray}
\varphi(X_i,\varphi(X_j,X_k))+c.p.=0
\end{eqnarray}
meaning that a Lie algebra is non associative. \\We then define a mapping $\psi_{\epsilon}:{\cal{G}}\rightarrow {\cal{G}}$ which is singular for a certain value $\epsilon_0$ of $\epsilon$. If the mapping $\varphi^{\prime}:\cal{G}\times \cal{G}\rightarrow \cal{G}$ is defined by
\begin{eqnarray}
\varphi^{\prime}(X,Y)=\lim_{\epsilon \rightarrow \epsilon_0}\psi^{-1}_{\epsilon}(\psi_{\epsilon}(X),\psi_{\epsilon}(Y))
\end{eqnarray}
then $(\cal{G},\varphi^{\prime})$ is a new Lie algebra called the \textbf{contraction of the Lie algebra} $(\cal{G},\varphi)$.\\
The pioneer contraction method is that of E.Inonu and E.P.Wigner \cite{inonuwigner} which starts with a Lie algebra  $\cal{G}=\cal{H}+\cal{P}$ where $\cal{H}$ is generated $X_a$, $\cal{P}$ is generated by $X_{\alpha}$; the structure of $\cal{G}$ being a priori given by
\begin{eqnarray*}
\varphi(X_a,X_b)=X_cC^c_{ab}+X_{\gamma}C^{\gamma}_{ab}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\~\varphi(X_a,X_{\alpha})=X_cC^c_{a\alpha}+X_{\gamma}C^{\gamma}_{a\alpha}~,~\varphi(X_{\alpha},X_{\beta})=X_cC^c_{\alpha\beta}+X_{\gamma}C^{\gamma}_{\alpha\beta}
\end{eqnarray*}
The Inonu-Wigner method uses the parameterized change of basis $\psi_{\epsilon}: (X_a,X_{\alpha})\rightarrow (Y_a,Y_{\alpha})$ defined by  $Y_a=X_a~,~Y_{\alpha}=\epsilon X_{\alpha}$. The structure of the Lie algebra $\cal{G}$ becomes
\begin{eqnarray}\label{mother1a}
\varphi(Y_a,Y_b)=Y_cC^c_{ab}+\epsilon^{-1}Y_{\gamma}C^{\gamma}_{ab}
\end{eqnarray}
\begin{eqnarray}\label{mother1b}
\varphi(Y_a,Y_{\alpha})=\epsilon Y_cC^c_{a\alpha}+Y_{\gamma}C^{\gamma}_{a\alpha}~,~\varphi(Y_{\alpha},Y_{\beta})=\epsilon^2Y_cC^c_{\alpha\beta}+\epsilon Y_{\gamma}C^{\gamma}_{\alpha\beta}
\end{eqnarray}
In the limit $\epsilon\rightarrow 0$ , the term $\epsilon^{-1}Y_{\gamma}C^{\gamma}_{ab}$  diverges. A limit will exist if only if the structure constants $C^{\gamma}_{ab}$ vanish. Hence to get a Inonu-Wigner contraction, $\cal{H}$ must be a subalgebra of $\cal{G}$.  The structure of the contracted Lie algebra is then
\begin{eqnarray}\label{contracted}
\varphi^{\prime}(Y_a,Y_b)=Y_cC^c_{ab}~,~\varphi^{\prime}(Y_a,Y_{\alpha})=Y_{\beta}C^{\beta}_{a\alpha}~,~\varphi^{\prime}(Y_{\alpha},Y_{\beta})=0
\end{eqnarray}
The Lie algebra $(\cal{G},\varphi^{\prime})$ defined by (\ref{contracted}) is a Inonu-Wigner contraction of the mother Lie algebra $(\cal{G},\varphi)$ with respect to the Lie subalgebra $\cal{H}$. It is a semi-direct sum of $(\cal{H},\varphi^{\prime})$ and the abelian Lie algebra $(\cal{P},\varphi^{\prime})$.
\section{Possible kinematical Lie algebras a la Levy-Leblond}
According to H.Bacry and J.M.Levy-Leblond \cite{jmll}, a kinematical group is a space-time transformation group keeping laws of physics invariant. Due to the assumptions of space isotropy, space-time homogeneity and the existence of inertial transformations, a kinematical group is a ten dimensional Lie group whose the Lie algebra is generated by three rotation generators $J_i$ (isotropy of space), three space translation generators $P_i$ (homogeneity of space), a time translation generator $H$ (homogeneity of time) and three inertial transformation generators $K_i$. Following Bacry and J.M.Levy-Leblond \cite{jmll}, A.Ngendakumana and coauthors \cite{ancilla1}  have shown that under some mathematical physics assumptions there are only twelve mathematical possible kinematical Lie algebras. Their Lie algebraic structures have in common the Lie brackets defining the adjoint representation of the rotation generators
\begin{eqnarray*}
[J_i,J_j]=J_k\epsilon^k_{ij},~[J_i,K_j]=K_k\epsilon^k_{ij},~[J_i,P_j]=P_k\epsilon^k_{ij},~[J_i,H]=0
\end{eqnarray*}
The remaining Lie brackets are given by the Table I \cite{ancilla1} below where $c$ is a speed, $r$ is a length and $\tau$ is a time.\\
The ParaPoincare Lie algebra ${\cal{P}}_+$ which is isomorphic to the Euclidean Lie algebra ${\cal{E}}(4)$, the "translations" generated by $K_i$ and $H$ form an abelian Lie algebra. This Lie algebra has been removed in the kinematical ones by Bacry and Levy-Leblond \cite{jmll} with the argument that the inertial transformations are compact. However they are noncompact, only space translations are compact.\\Using the Inonu-Wigner contraction method \cite{inonuwigner}, Bacry and Levy-Leblond \cite{jmll} have established that these Lie algebras are approximations of the de Sitter Lie algebras. Their links are summarized by the contractions scheme (see Figure $1$ on page $1610$ of \cite{jmll}). We will refer to these nomenclatura in the next two sections.
\newpage
\begin{table}[htbp]
\center{}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Lie symbol&Lie algebra Name& $[K_i,H]$&$[K_i,K_j]$&$[K_i,P_j]$&$[P_i,P_j]$& $[P_i,H]$\\
\hline
 $dS_{\pm}$& $de~Sitter$ &$P_i$ & $-\frac{1}{c^2}J_k\epsilon^k_{ij}$ & $\frac{1}{c^2}H\delta_{ij}$ & $\pm \frac{1}{r^2}J_k\epsilon^k_{ij}$& $\pm \frac{1}{\tau^2} K_i$ \\
 \hline
  $P$& $Poincare$ & $P_i$ & $-\frac{1}{c^2}J_k\epsilon^k_{ij}$ & $\frac{1}{c^2}H\delta_{ij}$ & $0$& $0$ \\
  $NH_{\pm}$& $Newton-Hooke$ & $P_i$ & $0$ & $0$ & $0$& $\pm \frac{1}{\tau^2} K_i$ \\
  $P_{\pm}$& $Para~Poincare$ & $0$ & $0$ & $\frac{1}{c^2}H\delta_{ij}$ & $\pm \frac{1}{r^2}J_k\epsilon^k_{ij}$& $\pm \frac{1}{\tau^2} K_i$ \\
  \hline
$G$& $Galilei$ & $P_i$ & $0$ & $0$ & $0$& $0$ \\
$G_{\pm}$& $Para-Galilei$ & $0$ & $0$ & $0$ & $0$& $\pm \frac{1}{\tau^2} K_i$ \\
$C$& $Carroll$ & $0$ & $$ & $\frac{1}{c^2}H\delta_{ij}$ & $0$& $0$ \\
\hline
 $S$& $Static$ & $0$ & $0$ & $0$ & $0$& $0$ \\
 \hline
\end{tabular}
\caption{The kinematical Lie algebras in term of $c$,$r$ and $\tau$}
\end{table}
\pagebreak
\section{Kinematical contractions of the de Sitter Lie algebras}
We propose to recover the Bacry and Levy-Leblond contractions scheme by using the kinematical parameters $r$, $\tau$ and $c$ which are the radius of universe, the period of the universe and speed of light, respectively. We first introduce the de Sitter Lie algebras $dS_{\pm}$ as isomorphic to the pseudo-orthogonal Lie algebras $O_{\pm}(5)$, i.e. that $dS_+(3)$ [$dS_-(3)$] is isomorhic to $O(1,4)$ [$O(2,3)$] Lie algebra. The aim of this section is to clarify better and in another way the naming speed-space contractions, speed-time contractions and space-time contractions of Levy-Leblond \cite{jmll}.
\subsection{The Lie algebras ${\cal{O}}_{\pm}(5)$}
Let $V$ be a five dimensional manifold equipped with the metric
\begin{eqnarray}\label{metric}
ds^2=\delta_{ij}dx^idx^j-(dx^4)^2\pm(dx^5)^2\equiv \eta_{ab}dx^adx^b,~~i,j=1,2,3
\end{eqnarray}
where the dimension of the $x^a$ is that of length. The matrix elements $\eta_{ab}$ form the matrix
\begin{eqnarray}
\eta_{\pm}=\left(
\begin{array}{ccc}
I_{3\times 3}&0&0\\0& -1&0\\0&0&\pm 1
\end{array}
\right)
\end{eqnarray}
The isometry group of the metric (\ref{metric}) is the group of real square matrices $g$ with order $5$ satisfying
\begin{eqnarray}
g^t\eta_{\pm}g=\eta_{\pm}
\end{eqnarray}
The Lie group $SO_0(4,1)$ is the connected component of $G_{+}$ while $SO_0(3,2)$ is the connected component of $G_-$.\\
The Lie algebra ${\cal{O}}_{\pm}(5)$ is the set of the real square matrices $X$ of order $5$ satisfying
\begin{eqnarray}
^tX\eta_{\pm}+\eta_{\pm}X=0
\end{eqnarray}
It is a linear combination
\begin{eqnarray}
X=J_k\theta^k+A_k\alpha^k+B_k\beta^k+\gamma \Gamma,~~~k=1,2,3
\end{eqnarray}
where the dimensionless matrices
\begin{eqnarray*}
J_k=\left(
\begin{array}{ccc}
\epsilon_{kj}^i&0&0\\0&0&0\\0&0&0
\end{array}
\right)~,~A_k=\left(
\begin{array}{ccc}
0&e_k&0\\ ^te_k&0&0\\0&0&0
\end{array}
\right)~,~B_k=\left(
\begin{array}{ccc}
0&0&e_k\\0&0&0\\ \mp ^te_k&0&0
\end{array}
\right)~,~\Gamma=\left(
\begin{array}{ccc}
0&0&0\\0&0&1\\0&\pm 1&0
\end{array}
\right)
\end{eqnarray*}
form a basis of ${\cal{O}}_{\pm}(5)$ whose the Lie structure is defined by the Lie brackets
\begin{eqnarray}\label{rotations}
[J_i,J_j]=J_k\epsilon^k_{ij},~[J_i,A_j]=A_k\epsilon^k_{ij},~[J_i,B_j]=B_k\epsilon^k_{ij},~[J_i,\Gamma]=0
\end{eqnarray}
\begin{eqnarray}\label{boosts}
[A_i,A_j]=- J_k\epsilon^k_{ij},~[A_i,B_j]=\Gamma \delta_{ij},~[A_i,\Gamma]=B_i
\end{eqnarray}
\begin{eqnarray}\label{translations}
[B_i,B_j]=\pm J_k\epsilon^k_{ij},~[B_i,\Gamma]=\pm A_i
\end{eqnarray}
Those generators can be realized as the differential operators
\begin{eqnarray}\label{vectorfields}
J_k=\epsilon^i_{kj}x^j\frac{\partial}{\partial x^i}~,~A_k=x^4\frac{\partial}{\partial x^k}+x^k\frac{\partial}{\partial x^4}~,~B_k=\mp x^5\frac{\partial}{\partial x^k}+x^k\frac{\partial}{\partial x^5}~,~\Gamma=x^4\frac{\partial}{\partial x^5}\pm x^5\frac{\partial}{\partial x^4}
\end{eqnarray}
\subsection{The de Sitter Lie algebras}
Let us set $K_i=\frac{1}{c}A_i$, $P_i=\frac{1}{r}B_i$ and $H=\frac{1}{\tau}\Gamma$ where $\sigma=\frac{1}{c}$ is a slowness, $\kappa=\frac{1}{r}$ is a curvature and $\omega=\frac{1}{\tau}$ is a frequency. It then follows from (\ref{vectorfields}) that
\begin{eqnarray}\label{vectorfields1}
K_i=t\frac{\partial}{\partial x^i}+\frac{x^i}{c^2}\frac{\partial}{\partial t}~,~P_i=\mp \lambda\frac{\partial}{\partial x^i}+\frac{x^i}{r^2}\frac{\partial}{\partial \lambda}~,~H=\frac{c}{r\tau}t\frac{\partial}{\partial \lambda}\pm \frac{r}{c\tau}\lambda\frac{\partial}{\partial t}
\end{eqnarray}
where $t=\frac{x^4}{c}$ is a time coordinate while $\lambda=\frac{x^5}{r}$ is a dimensionless coordinate. Hence parameters associated to $K_i$ ,$P_i$ and $H$ have velocity, length and time as respective dimension.\\
The Lie brackets (\ref{rotations}), (\ref{boosts}) and (\ref{translations}) become then
\begin{eqnarray}\label{dsrotations1}
[J_i,J_j]=J_k\epsilon^k_{ij},~[J_i,K_j]=K_k\epsilon^k_{ij},~[J_i,P_j]=P_k\epsilon^k_{ij},~[J_i,H]=0
\end{eqnarray}
\begin{eqnarray}\label{dsboosts1}
[K_i,K_j]=-\frac{1}{c^2} J_k\epsilon^k_{ij},~[K_i,P_j]=\frac{\tau}{c r}H\delta_{ij},~[K_i,H]=\frac{r}{c \tau}P_i
\end{eqnarray}
\begin{eqnarray}\label{dstranslations1}
[P_i,P_j]=\pm\frac{1}{r^2} J_k\epsilon^k_{ij},~[P_i,H]=\pm \frac{c}{r \tau}K_i
\end{eqnarray}
Let us now study the limits of the de Sitter Lie algebras as the constants tend to infinity. Normally the three constants are constrained by $r=c\tau$. However, we momentally ignore it. We will use it at the end of the section to show that we have recovered the results of table $1$. It is first of all evident that (\ref{dsrotations1}) does not change. We are then interested in the behavior of (\ref{dsboosts1}) and (\ref{dstranslations1}) only.
\subsection{The Newton-Hooke, Poincare and Para-Poincare Lie algebras}
In this section we look for the limits of the de Sitter Lie algebras as two of the constants tend to infinity while their ratio if kept finite.\\
 The limits of (\ref{dsboosts1}) and (\ref{dstranslations1}) as the speed $c$ and the radius $r$ tend to infinity while their ratio $\frac{r}{c}$ is kept finite are
\begin{eqnarray}\label{dsboosts2}
[K_i,K_j]=0,~[K_i,P_j]=0,~[K_i,H]=\frac{r}{c\tau}P_i
\end{eqnarray}
\begin{eqnarray}\label{dstranslations2}
[P_i,P_j]=0,~[P_i,H]=\pm \frac{c}{r\tau}K_i
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts2}) and (\ref{dstranslations2}) define the Newton-Hooke Lie algebra ${\cal{NH}}_{\pm}(3)$.\\
 The limits of (\ref{dsboosts1}) and (\ref{dstranslations1}) as the period $\tau$ and the radius $r$ tend to infinity while their ratio $\frac{r}{\tau}$ is kept fnite are
\begin{eqnarray}\label{dsboosts3}
[K_i,K_j]=-\frac{1}{c^2}J_k\epsilon^k_{ij},~[K_i,P_j]=\frac{\tau}{rc}H\delta_{ij},~[K_i,H]=\frac{r}{c\tau}P_i
\end{eqnarray}
\begin{eqnarray}\label{dstranslations3}
[P_i,P_j]=0,~[P_i,H]=0
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts3}) and (\ref{dstranslations3}) define the Poincare Lie algebra ${\cal{P}}(3)$.\\
 The limits of (\ref{dsboosts1}) and (\ref{dstranslations1}) as the speed $c$ and the period $\tau$ tend to infinity while their ratio $\frac{c}{\tau}$ is kept finite are
\begin{eqnarray}\label{dsboosts4}
[K_i,K_j]=0,~[K_i,P_j]=\frac{\tau}{rc}H\delta_{ij},~[K_i,H]=0
\end{eqnarray}
\begin{eqnarray}\label{dstranslations4}
[P_i,P_j]=\pm\frac{1}{r^2}J_k\epsilon^k_{ij},~[P_i,H]=\pm\frac{c}{\tau r}K_i
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts4}) and (\ref{dstranslations4}) define  the Para-Poincare Lie algebra ${\cal{P}}_{\pm}(3)$.\\
As in \cite{jmll}, the Newton-Hooke Lie algebras, the Poincare Lie algebra and the Para-Poincare Lie algebras are speed-space, space-time and speed-time contractions of the de Sitter Lie algebras respectively.
\subsection{The Galilei, Para-Galilei and Carroll Lie algebras}
The limit of the Lie brackets (\ref{dsboosts2}) and (\ref{dstranslations2}) as the radius $r$ and the period $\tau$ tend to infinity while $\frac{r}{\tau}$ is kept finite and the limit (\ref{dsboosts3}) and (\ref{dstranslations3}) as the radius $r$ and the speed $c$ tend to infinity while $\frac{c}{r}$ is kept finite are the same, i.e.
\begin{eqnarray}\label{dsboosts5}
[K_i,K_j]=0,~[K_i,P_j]=0,~[K_i,H]=\frac{r}{c\tau} P_i
\end{eqnarray}
\begin{eqnarray}\label{dstranslations5}
[P_i,P_j]=0,~[P_i,H]=0
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts5}) and (\ref{dstranslations5}) define the Galilei Lie algebra $\cal{G}$.\\ The limit of the Lie brackets (\ref{dsboosts2}) and (\ref{dstranslations2}) as the speed $c$ and the period $\tau$ tend to infinity while $\frac{c}{\tau}$ is kept finite and the limit (\ref{dsboosts4}) and (\ref{dstranslations4}) as the radius $r$ and the speed $c$ tend to infinity while $\frac{c}{r}$ is kept finite are the same, i.e.
\begin{eqnarray}\label{dsboosts6}
[K_i,K_j]=0,~[K_i,P_j]=0,~[K_i,H]=0
\end{eqnarray}
\begin{eqnarray}\label{dstranslations6}
[P_i,P_j]=0,~[P_i,H]=\pm\frac{c}{r\tau} K_i
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts6}) and (\ref{dstranslations6}) define the Para-Galilei Lie algbra $\cal{G}_{\pm}$.\\
The limit of the Lie brackets (\ref{dsboosts3}) and (\ref{dstranslations3}) as the speed $c$ and the period $\tau$ tend to infinity while $\frac{c}{\tau}$ is kept finite and the limit (\ref{dsboosts4}) and (\ref{dstranslations4}) as the radius $r$ and the period $\tau$ tend to infinity while $\frac{r}{\tau}$ is kept finite are the same, i.e.
\begin{eqnarray}\label{dsboosts7}
[K_i,K_j]=0,~[K_i,P_j]=\frac{\tau}{rc} H\delta_{ij},~[K_i,H]=0
\end{eqnarray}
\begin{eqnarray}\label{dstranslations7}
[P_i,P_j]=0,~[P_i,H]=0
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts7}) and (\ref{dstranslations7}) define the Carroll Lie algebra ${\cal{C}}(3)$.\\
Hence the Galilei, the Para-Galilei, the Carroll Lie algebras are contractions of the Newton-Hooke or Poincare Lie algebras, the Newton-Hooke or the Para-Poincare Lie algebras, the Poincare or the Para-Poincare Lie algebras respectively.
\subsection{The Static Lie algebra}
The limit of the Lie brackets (\ref{dsboosts5}) and (\ref{dstranslations5}) as the speed $c$ and the period $\tau$ tend to $\infty$ while $\frac{c}{\tau}$ is kept finite ,the limit (\ref{dsboosts6}) and (\ref{dstranslations6}) as the radius $r$ and the period $\tau$ tend to $\infty$ while $\frac{r}{\tau}$ is kept finite and the limit of the Lie brackets (\ref{dsboosts7}) and (\ref{dstranslations7}) as the speed $c$ and the radius $r$ tend infinity while $\frac{c}{r}$ is kept finite are the same; i.e
\begin{eqnarray}\label{dsboosts8}
[K_i,K_j]=0,~[K_i,P_j]=0,~[K_i,H]=0
\end{eqnarray}
\begin{eqnarray}\label{dstranslations8}
[P_i,P_j]=0,~[P_i,H]=0
\end{eqnarray}
The Lie brackets (\ref{dsrotations1}), (\ref{dsboosts8}) and (\ref{dstranslations8}) define the Static Lie algebra $\cal{S}$.\\When the constraint $r=c\tau$ is taken in account, the Lie brackets in the table I are recovered.\\These approximations through kinematical parameters are summarized in the following cube (see figure $1$).  The horizontal arrows represent the contractions as {\color{green}$c,\tau \rightarrow \infty$}(speed-time contractions), the vertical arrows represent the contractions as {\color{red}$c,r \rightarrow \infty$}(speed-space contractions) and the oblique arrows represent the contractions as {\color{blue}$r,\tau \rightarrow \infty$}(space-time).
\newpage
\begin{figure}[h!]
\begin{center}
\definecolor{qqqqff}{rgb}{0,0,1}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm,scale=1.1]

\draw[color=red] (1.5,1) node {S};
\draw[color=red] (1.5,4.84) node {C};
\draw[color=red] (5.44,1) node {G};
\draw[color=qqqqff] (5.44,4.84) node {P};
\draw[color=red] (3.26,2.24) node {$G_{\pm}$};
\draw[color=qqqqff] (3.26,6.76) node {$P_{\pm}$};
\draw[color=red] (7.46,2.24) node {$NH_{\pm}$};
\draw[color=qqqqff] (7.46,6.76) node {$dS_{\pm}$};
{\color{green}\draw[ ->,line width=1.3pt] (5.3,1) -- (1.6,1);}
{\color{green}\draw[ ->,line width=1.3pt] (7.1,2.24) -- (3.5,2.24);}
{\color{green}\draw[ ->,line width=1.3pt] (5.3,4.84) -- (1.6,4.84);}
{\color{green}\draw[ ->,line width=1.3pt] (7.1,6.84) -- (3.5,6.84);}
{\color{red}\draw[ ->,line width=1.3pt] (1.5,4.6) -- (1.5,1.1);}
{\color{red}\draw[ ->,line width=1.3pt] (5.3,4.6) -- (5.3,1.1);}
{\color{red}\draw[ ->,line width=1.3pt] (3.3,6.5) -- (3.3,2.5);}
{\color{red}\draw[ ->,line width=1.3pt] (7.46,6.5) -- (7.46,2.5);}
{\color{blue}\draw[ ->,line width=1.3pt] (3.1,2.2) -- (1.6,1);}
{\color{blue}\draw[ ->,line width=1.3pt] (7.1,2.2) -- (5.6,1);}
{\color{blue}\draw[ ->,line width=1.3pt] (3.1,6.7) -- (1.6,4.9);}
{\color{blue}\draw[ ->,line width=1.3pt] (7.2,6.7) -- (5.6,4.9);}

\end{tikzpicture}
\end{center}
\caption{Contractions in terms of $c$, $r$ and $\tau$}
\end{figure}
\pagebreak
\section{Dynamical contractions of the de Sitter Lie algebras}
We are going to characterize the kinematical Lie algebras by the compliance, the mass and the energy in place of the curvature, the speed of light and the period $\tau$. For this we define the impulse generators $Q_i=\frac{1}{m}K_i$ where $m$ is a mass, the compliance $C=\frac{\tau^2}{m}$ and the energy $E_0=mc^2$.
 \subsection{Three parameters Lie algebras} If we inject these ingredients in (\ref{dsrotations1}), (\ref{dsboosts1}) and (\ref{dstranslations1}) we obtain that the de Sitter Lie algebras $dS_{\pm}$ is defined in the basis $(J_i,Q_i,P_i,H)$ by the Lie brackets
\begin{eqnarray}\label{dsrotations9}
[J_i,J_j]=J_k\epsilon^k_{ij},~[J_i,Q_j]=Q_k\epsilon^k_{ij},~[J_i,P_j]=P_k\epsilon^k_{ij},~[J_i,H]=0
\end{eqnarray}
\begin{eqnarray}\label{dsboosts9}
[Q_i,Q_j]=-\frac{1}{mE_0} J_k\epsilon^k_{ij},~[Q_i,P_j]=\frac{1}{E_0}H\delta_{ij},~[Q_i,H]=\frac{1}{m}P_i
\end{eqnarray}
\begin{eqnarray}\label{dstranslations9}
[P_i,P_j]=\pm\frac{1}{CE_0} J_k\epsilon^k_{ij},~[P_i,H]=\pm \frac{1}{C}Q_i
\end{eqnarray}
The de Sitter Lie algebras $dS_{\pm}$ are then characterized by the three parameters $m$, $C$ and $E_0$.
\subsection{Two parameters Lie algebras}
When the energy $E_0$ tends to infinity, the compliance $C$ tends to infinity and the mass tends to infinity, the Lie brackets (\ref{dsrotations9}), (\ref{dsboosts9}) and (\ref{dstranslations9}) tend to the Lie brackets defining the Newton-Hooke Lie algebras ($NH_{\pm}$), defining the Poincare Lie algebra $P$ and defining the Para-Poincare Lie algebras $P_{\pm}$, respectively. The Newton-Hooke Lie algebras are then characterized by a mass $m$ and a compliance $C$ related by the frequency $\omega=\frac{1}{\sqrt{mC}}$ (time $\tau=\sqrt{mC}$), the Poincare Lie algebra is characterized by a mass and an energy related by the speed $c=\sqrt{\frac{E_0}{m}}$ (slowness $s=\sqrt{\frac{m}{E_0}}$) and the Para-Poincare Lie algeberas are characterized by an energy and a compliance related by the curvature $\kappa=\frac{1}{\sqrt{CE_0}}$ (the radius $r=\sqrt{CE_0}$).
\subsection{One parameter Lie algebras}
The Lie brackets defining the Galilei Lie algebra $G$ are obtained from those defining the Poincare Lie as the energy tends to infinity or from those defining the Newton-Hooke Lie algebras as the compliance tends to infinity; the Lie brackets defining the Carroll Lie algebra $C$ are obtained from those defining the Poincare Lie algebra as the mass tends to infinity or from those defining the Para-Poincare Lie algebras as the compliance tends to infinity; the Lie brackets defining the Para-Galilei Lie algebras $G_{\pm}$ are obtained from those defining the Para-Poincare Lie algebras as the energy tends to infinity or from those defining the Newton-Hooke Lie algebras as the mass tend to infinity. The Galilei Lie algebra is then characterized by a mass, the Carroll Lie algebra is characterized by an energy while the Para-Galilei Lie algebras are charcterized by a compliance.
\subsection{Zero parameters Lie algebra}
The zero parameters Lie algebra is the static Lie algebra which is obtained from the Galilei Lie algebra as the mass tends to infinity, from the Carroll Lie algebra as the energy tends to infinity or from the Para-Galilei Lie algebra as the compliance tends to infinity.\\All the Lie brackets defining these Lie algebras have in common the Lie brackets
\begin{eqnarray*}
[J_i,J_j]=J_k\epsilon^k_{ij},~[J_i,K_j]=K_k\epsilon^k_{ij},~[J_i,P_j]=P_k\epsilon^k_{ij},~[J_i,H]=0
\end{eqnarray*}
the others are summarized in the table II while the limiting process is given by the figure $2$ where the horizontal arrows represent the contractions as {\color{green}$m \rightarrow \infty$}(static limit), the vertical arrows represent the contractions as {\color{red}$E_0 \rightarrow \infty$}(Newtonian limit) and the oblique arrows represent the contractions as {\color{blue}$C \rightarrow \infty$}(flat limit).
\newpage
\begin{table}[htbp]
\center{}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Lie symbol&Lie algebra Name& $[Q_i,H]$&$[Q_i,Q_j]$&$[Q_i,P_j]$&$[P_i,P_j]$& $[P_i,H]$\\
\hline
 $dS_{\pm}$& $de~Sitter$ &$\frac{1}{m}P_i$ & $-\frac{1}{mE_0}J_k\epsilon^k_{ij}$ & $\frac{1}{E_0}H\delta_{ij}$ & $\pm \frac{1}{CE_0}J_k\epsilon^k_{ij}$& $\pm \frac{1}{C} Q_i$ \\
 \hline
  $P$& $Poincare$ & $\frac{1}{m}P_i$ & $-\frac{1}{mE_0}J_k\epsilon^k_{ij}$ & $\frac{1}{E_0}H\delta_{ij}$ & $0$& $0$ \\
  $NH_{\pm}$& $Newton-Hooke$ & $\frac{1}{m}P_i$ & $0$ & $0$ & $0$& $\pm \frac{1}{C} Q_i$ \\
  $P_{\pm}$& $Para~Poincare$ & $0$ & $0$ & $\frac{1}{E_0}H\delta_{ij}$ & $\pm \frac{1}{CE_0}J_k\epsilon^k_{ij}$& $\pm \frac{1}{C} Q_i$ \\
  \hline
$G$& $Galilei$ & $\frac{1}{m}P_i$ & $0$ & $0$ & $0$& $0$ \\
$G_{\pm}$& $Para-Galilei$ & $0$ & $0$ & $0$ & $0$& $\pm \frac{1}{C} Q_i$ \\
$C$& $Carroll$ & $0$ & $$ & $\frac{1}{E_0}H\delta_{ij}$ & $0$& $0$ \\
\hline
 $S$& $Static$ & $0$ & $0$ & $0$ & $0$& $0$ \\
 \hline
\end{tabular}
\caption{The kinematical Lie algebras in term of the mass, the compliance and the energy}
\end{table}
\begin{figure}[h!]
\begin{center}
\definecolor{qqqqff}{rgb}{0,0,1}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm,scale=1.1]

\draw[color=red] (1.5,1) node {S};
\draw[color=red] (1.5,4.84) node {C};
\draw[color=red] (5.44,1) node {G};
\draw[color=qqqqff] (5.44,4.84) node {P};
\draw[color=red] (3.26,2.24) node {$G_{\pm}$};
\draw[color=qqqqff] (3.26,6.76) node {$P_{\pm}$};
\draw[color=red] (7.46,2.24) node {$NH_{\pm}$};
\draw[color=qqqqff] (7.46,6.76) node {$dS_{\pm}$};
{\color{green}\draw[ ->,line width=1.3pt] (5.3,1) -- (1.6,1);}
{\color{green}\draw[ ->,line width=1.3pt] (7.1,2.24) -- (3.5,2.24);}
{\color{green}\draw[ ->,line width=1.3pt] (5.3,4.84) -- (1.6,4.84);}
{\color{green}\draw[ ->,line width=1.3pt] (7.1,6.84) -- (3.5,6.84);}
{\color{red}\draw[ ->,line width=1.3pt] (1.5,4.6) -- (1.5,1.1);}
{\color{red}\draw[ ->,line width=1.3pt] (5.3,4.6) -- (5.3,1.1);}
{\color{red}\draw[ ->,line width=1.3pt] (3.3,6.5) -- (3.3,2.5);}
{\color{red}\draw[ ->,line width=1.3pt] (7.46,6.5) -- (7.46,2.5);}
{\color{blue}\draw[ ->,line width=1.3pt] (3.1,2.2) -- (1.6,1);}
{\color{blue}\draw[ ->,line width=1.3pt] (7.1,2.2) -- (5.6,1);}
{\color{blue}\draw[ ->,line width=1.3pt] (3.1,6.7) -- (1.6,4.9);}
{\color{blue}\draw[ ->,line width=1.3pt] (7.2,6.7) -- (5.6,4.9);}

\end{tikzpicture}
\end{center}
\caption{Contractions in terms of mass, energy and compliance}
\end{figure}
\pagebreak
It follows from table III below that
 \begin{itemize}
 \item the de Sitter, Poincare, Newton-Hooke and the Galilei Lie algebras are finite mass Lie algebras while the Para-Poincare, the Carroll, the Para-Galilei and the static Lie algebras are their respective limits as mass becomes infinite;
 \item the de Sitter, Para-Poincare, Newton-Hooke and the Para-Galilei Lie algebras are finite compliance Lie algebras while the Poincare, the Carroll, the Galilei and the Static Lie algebras are their respective limits as compliance becomes infinite;
 \item the de Sitter, Poincare, Para-Poincare and the Carroll Lie algebras are finite energy Lie algebras while the Newton-Hooke, the Galilei, the Para-Galilei and the Static Lie algebras are their respective limits as energy becomes infinite.
 \end{itemize}
If we use the letter $\cal{K}$ for kinematical, the de Sitter, Poincare, Para-Poincare, Newton-Hooke, Galilei, Para-Galilei, and Carroll  Lie algebras can be labeled as ${\cal{K}}_{(m,E_0,C)}$, ${\cal{K}}_{(m,E_0)}$, ${\cal{K}}_{(E_0,C)}$, ${\cal{K}}_{(m,C)}$, ${\cal{K}}_{m}$, ${\cal{K}}_{C}$ and ${\cal{K}}_{E_0}$, respectively.
\newpage
\begin{table}[htbp]
\center{}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Lie symbol&Lie algebra Name& $m<\infty$&$m=\infty$&$C<\infty$&$C=\infty$& $E_0<\infty$&$E_0=\infty$\\
\hline
 $dS_{\pm}$& $de~Sitter$ &$yes$ & $no$ & $yes$ & $no$& $yes$&$no$ \\
 \hline
  $P$& $Poincare$ & $yes$ & $no$ & $no$ & $yes$& $yes$&$no$ \\
  $NH_{\pm}$& $Newton-Hooke$ & $yes$ & $no$ & $yes$ & $no$& $no$&$yes$ \\
  $P_{\pm}$& $Para~Poincare$ & $no$ & $yes$ & $yes$ & $no$& $yes$&$no$ \\
  \hline
$G$& $Galilei$ & $yes$ & $no$ & $no$ & $yes$& $no$&$yes$ \\
$G_{\pm}$& $Para-Galilei$ & $no$ & $yes$ & $yes$ & $no$& $no$&$yes$ \\
$C$& $Carroll$ & $no$ & $yes$ & $no$ & $yes$& $yes$&$no$ \\
\hline
 $S$& $Static$ & $no$ & $yes$ & $no$ & $yes$& $no$&$yes$ \\
 \hline
\end{tabular}
\caption{The kinematical Lie algebras according the values of the dynamical parameters}
\end{table}
\pagebreak
 \section{Kinematical Poisson Lie algebras and motion equations}
 In this section, motion equations and physical quantities associated to each kinematical Lie algebra are precized.
 \subsection{Poisson brackets and motion equations}
 We Know that the Poisson bracket of two functions defined on the dual ${\cal{G}}^*$ of any Lie algebra $\cal{G}$ is defined by
 \begin{eqnarray}
 \{f,g\}=K_{ij}(a)\frac{\partial f}{\partial a_i}\frac{\partial g}{\partial a_j}
 \end{eqnarray}
where $a_i$ are the coordinates on ${\cal{G}}^*$ and $K_{ij}(a)=-a_kC^k_{ij}$ are the matrix elements of the Kirillov form.
 From the Lie brackets defining the kinematical Lie algebras in function of $m$,$E_0$ and $C$ follows that the dual of the kinematical Lie algebras are Poisson-Lie algebras defined by
  \begin{eqnarray}\label{poissonrotations}
\{\tau_i,\tau_j\}=-\frac{1}{E_0}\tau_k\epsilon^k_{ij},~\{\tau_i,q_j\}=-\frac{1}{E_0}q_k\epsilon^k_{ij},~\{\tau_i,p_j\}=-\frac{1}{E_0}p_k\epsilon^k_{ij},~\{\tau_i,E\}=0
\end{eqnarray}
 where $\tau_k=\frac{j_k}{E_0}$ is a time pseudo-vector. The other Poisson brackets are given by the table IV
 \newpage
 \begin{table}[htbp]
\center{{\it \textbf{Table $4$}: The kinematical Poisson Lie algebras}}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Lie symbol&Lie algebra Name& $\{E,q_i\}$&$\{q_i,q_j\}$&$\{p_i,q_j\}$&$\{p_i,p_j\}$& $\{E,p_i\}$\\
\hline
 $dS_{\pm}$& $de~Sitter$ &$\frac{1}{m}p_i$ & $\frac{1}{m}\tau_k\epsilon^k_{ij}$ & $\frac{E}{E_0}\delta_{ij}$ & $\mp \frac{1}{C}\tau_k\epsilon^k_{ij}$& $\pm \frac{1}{C} q_i$\\
  $NH_{\pm}$& $Newton-Hooke$ & $\frac{1}{m}p_i$ & $0$ & $0$ & $0$& $\pm \frac{1}{C} q_i$ \\
 \hline
  $P$& $Poincare$ & $\frac{1}{m}p_i$ & $\frac{1}{m}\tau_k\epsilon^k_{ij}$ & $\frac{E}{E_0}\delta_{ij}$ & $0$& $0$ \\
  $G$& $Galilei$ & $\frac{1}{m}p_i$ & $0$ & $0$ & $0$& $0$ \\
  \hline
  $P_{\pm}$& $Para~Poincare$ & $0$ & $0$ & $\frac{E}{E_0}\delta_{ij}$ & $\mp \frac{1}{C}\tau_k\epsilon^k_{ij}$& $\pm \frac{1}{C} q_i$ \\
  $G_{\pm}$& $Para-Galilei$ & $0$ & $0$ & $0$ & $0$& $\pm \frac{1}{C} q_i$ \\
\hline
$C$& $Carroll$ & $0$ & $$ & $\frac{E}{E_0}\delta_{ij}$ & $0$& $0$ \\
 $S$& $Static$ & $0$ & $0$ & $0$ & $0$& $0$ \\
 \hline
\end{tabular}
\caption{The kinematical Poisson Lie algebras}
\end{table}
 \begin{table}[htbp]
\center{}
\begin{tabular}{|c|c|}
\hline
Lie~algebras&Motion~equations\\
\hline
$dS_{\pm}$ and $NH_{\pm}$&$\frac{dq_i}{dt}=\frac{p_i}{m}~,~\frac{dp_i}{dt}=\pm \frac{q_i}{C}$\\
\hline
$P$ and $G$&$\frac{dq_i}{dt}=\frac{p_i}{m}~,~\frac{dp_i}{dt}=0$\\
\hline
$P_{\pm}$ and $G_{\pm}$&$\frac{dq_i}{dt}=0~,~\frac{dp_i}{dt}=\pm \frac{q_i}{C}$\\
\hline
$C$ and $S$&$\frac{dq_i}{dt}=0~,~\frac{dp_i}{dt}=0$\\
\hline
\end{tabular}
\caption{Motion Equations}
\end{table}
\pagebreak
It follows from (\ref{poissonrotations}) that the linear momenta $p_i$ and the positions $q_i$ behave as components of vectors while the energy behave as a scalar under time pseudo-vector which is is constant in time. The three brackets are all zero for an infinte energy $E_0$ (the bottom face of the cube above).  The motion equations for the $p_i$ and the $q_i$ are given by the table V. They are common for any pair of groups related by the limit $E_0 \rightarrow \infty$.\\For finite energy $E_0$ (the top face in the cube above) the Poisson bracket $\{p_i,q_j\}=\frac{E}{E_0}\delta_{ij}$ is different from zero (a kind of incertitude relation). The Poisson brackets $\{p_i,p_j\}=\mp \frac{\tau_k}{C}\epsilon^k_{ij}$ do not vanish for finite compliance and $\{q_i,q_j\}=\frac{\tau_k}{m}\epsilon^k_{ij}$  do not vanish for finite mass.  Following \cite{ancilla} we can say that the magnetic field $B_k$ and the exotic magnetic field $B^*_k$ are related to the time pseudo-vector $\tau_k$ by $eB_k=-\frac{\tau_k}{C}$ and $e^*B^*_k=-\frac{\tau_k}{m}$. Then $\frac{e^*}{C}B_k^*=\frac{e}{m}B_k$.
\subsection{Physical quantities}
 To each function $f$ on the dual Lie algebra is associated the vector field
\begin{eqnarray}
X_f=K_{ij}(a)\frac{\partial f}{\partial a_i}\frac{\partial }{\partial a_j}
\end{eqnarray}
such that $\{f,g\}=X_f(g)$.\\
If the general element the dual of a kinematical Lie algebra is $j_kJ^{k*}+q_kQ^{k*}+p_kP^{k*}+EH^*$, we verify that the de Sitter Lie algebra $dS_{\pm}$ is represented by the vector fields
\begin{eqnarray}
X_{j_i}=-j_k\epsilon^k_{ij}\frac{\partial}{\partial j_j}-q_k\epsilon^k_{ij}\frac{\partial}{\partial q_j}-p_k\epsilon^k_{ij}\frac{\partial}{\partial p_j}
\end{eqnarray}
\begin{eqnarray}
X_{q_i}=q_k\epsilon^k_{ij}\frac{\partial}{\partial j_j}+\frac{j_k}{mE_0}\epsilon^k_{ij}\frac{\partial}{\partial q_j}-\frac{E}{E_0}\frac{\partial}{\partial p_i}-\frac{p_i}{m}\frac{\partial}{\partial E}
\end{eqnarray}
\begin{eqnarray}
X_{p_i}=p_k\epsilon^k_{ij}\frac{\partial}{\partial j_j}\mp\frac{j_k}{CE_0}\epsilon^k_{ij}\frac{\partial}{\partial p_j}+\frac{E}{E_0}\frac{\partial}{\partial q_i}\mp\frac{q_i}{C}\frac{\partial}{\partial E}
\end{eqnarray}
\begin{eqnarray}
X_E=\frac{p_i}{m}\frac{\partial}{\partial q_i}\pm\frac{q_i}{C}\frac{\partial}{\partial p_i}
\end{eqnarray}
Those realizing the other kinematical Lie algebras are obtained by the contraction process described in section $5$. On the contact manifold $V\times R$ equipped with the symplectic form $\sigma=dp_i\wedge dq_i-dE\wedge dt$ where $V=dS_{\pm}^*/so(3)^*$, the corresponding physical quantities $\mu(a_k)$, solutions of the differential $d\mu(a_k)=-i_{X_{a_k}}(\sigma),~k=1,2,...,10$, are given by the table VI where $j_{0i}~,q_{0i},~p_{0i}$ and $U$ are constants of integration.
\newpage
\begin{table}[htbp]
\center{}
\begin{tabular}{|c|c|c|c|c|}
\hline
~&angular~momentum&position&linear momentum&energy\\
\hline
$dS_{\pm}$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$\frac{j_k}{mE_0}\epsilon^k_{ij}p^j+\frac{E}{E_0}q_i+\frac{p_i}{m}t+q_{0i}$&$\pm \frac{j_k}{CE_0}\epsilon^k_{ij}q^j+\frac{E}{E_0}p_i\pm\frac{q_i}{C}t+p_{0i}$&$\frac{\vec{p}^2}{2m}\mp \frac{\vec{q}^2}{2C}+U$\\
\hline
$NH_{\pm}$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$\frac{p_i}{m}t+q_{0i}$&$\pm \frac{q_i}{C}t+p_{0i}$&$\frac{\vec{p}^2}{2m}\mp \frac{\vec{q}^2}{2C}+U$\\
\hline
$P$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$\frac{j_k}{mE_0}\epsilon^k_{ij}p^j+\frac{E}{E_0}q_i+\frac{p_i}{m}t+q_{0i}$&$\frac{E}{E_0}p_i+p_{0i}$&$\frac{\vec{p}^2}{2m}+U$\\
\hline
$G$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$\frac{p_i}{m}t+q_{0i}$&$p_{0i}$&$\frac{\vec{p}^2}{2m}+U$\\
\hline
$P_{\pm}$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$\frac{E}{E_0}q_i+q_{0i}$&$\pm \frac{j_k}{CE_0}\epsilon^k_{ij}q^j+\frac{E}{E_0}p_i\pm \frac{q_i}{C}t+p_{0i}$&$\mp \frac{\vec{q}^2}{2C}+U$\\
\hline
$G_{\pm}$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$q_{0i}$&$\pm \frac{q_i}{C}+p_{0i}$&$\mp \frac{\vec{q}^2}{2C}+U$\\
\hline
$C$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$\frac{E}{E_0}q_i+q_{0i}$&$\frac{E}{E_0}p_i+p_{0i}$&$U$\\
\hline
$S$&$q_k\epsilon^k_{ij}p^j+j_{0i}$&$q_{0i}$&$p_{0i}$&$U$\\
\hline
\end{tabular}
\caption{The physical quantities associtated to kinematical Lie algebras}
\end{table}
\pagebreak
\section{Conclusion}
We can conlude by saying that the de Sitter groups (Newton-Hooke groups) are associated to a massive spring in a phase space where positions as well momenta do not commute, where the incertitude relation $\{p_i,q_j\}=\frac{E}{E_0}\delta_{ij}$ is satisfied (in phase space where positions as well momenta do commute and there is no incertitude relation ); the Poincare group (Galilei group ) is associated to a massive system in a phase where positions do not commute and momenta commute and the incertitude relation $\{p_i,q_j\}=\frac{E}{E_0}\delta_{ij}$ is satisfied (in phase space where positions as well momenta do commute and the incertitude relation is not satisfied); the Para-Poincare groups (Para-Galilei group) is associated to a light spring\cite{ancilla} with compliance $C$ in a phase space where positions do commute , momenta do not commute and the incertitude relation $\{p_i,q_j\}=\frac{E}{E_0}\delta_{ij}$ is satisfied in phase space where positions as well momenta do commute and there is no incertitude relation ); the Carroll group (Static group) is associated to a non moving system where positions as well momenta commute but the incertitude relation $\{p_i,q_j\}=\frac{E}{E_0}\delta_{ij}$ is satisfied ( there is no incertitude relation).


text:
\begin{document}

\maketitle

\bibliographystyle{amsplain}

\begin{abstract}In this paper, asymptotic behavior of the uncertainty product of Gauss-Weierstrass wavelet is investigated. It is shown that the uncertainty product is bounded from above, a feature that distinguishes Gauss-Weierstrass wavelet among other wavelet families.
\end{abstract}

\begin{bfseries}Key Words and Phrases:\end{bfseries} uncertainty product, time--frequency localization, Gauss--Weierstrass wavelet\\
\begin{bfseries}2010 MSC:\end{bfseries} 42C40

\section{Introduction}

The uncertainty product of spherical functions, a notion introduced by Narcowich and Ward in~\cite{NW96}, is a measure for the trade-off between the spatial and frequency localization. A short history of the notion and a discussion about various nomenclatures can be found in \cite[Introduction]{IIN16USW}. Similarly as in physics (Heisenberg's uncertainty principle), the uncertainty product is bounded from below. It can be interpreted as one of indicators of 'goodness' of spherical wavelets (for theory of wavelet transforms see~\cite{IIN15WT}), because well-localized wavelets yield a well-localized wavelet transform, i.e., such that 'fuzziness' of the transform reflects mostly the 'fuzziness' of the analyzed signal. In the paper~\cite{IIN16USW} I proved that the uncertainty product of a wide class of wavelets is in limit $\rho\to0$, where $\rho$ denotes the scale parameter, bounded from above by $\mathcal O(\rho^{-\alpha})$ for some $\alpha>0$. Consequently, its finite boundedness seems to be something exceptional and wavelets having this property are quite rare.

In the case of Poisson wavelets~$g_\rho^m$, $m\in\mathbb N$, over the $n$-dimensional sphere~$\mathcal S^n$ \cite{HI07,IIN15PW} -- a wavelet family which is the most popular one for applications because of their explicit representation as well as existence of discrete frames \cite{IH10,IIN16WF} -- the uncertainty product $U(g_\rho^m)$ is indeed bounded in limit $\rho\to0$ \cite{IIN16UPW}. Moreover, $\lim_{\rho\to0}U(g_\rho^m)$ approaches the optimal value for a certain sequence of $(n,m)$.

The present paper is devoted to another very popular wavelet family, namely Gauss-Weierstrass wavelet. It is shown that its uncertainty product is bounded in limit $\rho\to0$. I was not able to compute the value of the limit with analytical tools, but in view of \cite[Theorem~3.4]{IIN16USW}, sole boundedness of the uncertainty product is a significant advantage of the wavelet family.

The paper is organized as follows. After an introduction of the necessary notions and statements in Section~\ref{sec:sphere}, the main result of the paper, Theorem~\ref{thm:uncertainty_GW} is proven in Section~\ref{sec:uncertaintyGW}.

\section{Preliminaries}\label{sec:sphere}

Let $\mathcal S$ denote the $2$-dimensional unit sphere in the Euclidean space~$\mathbb{R}^3$ and $d\sigma$ be the Lebesgue measure of~$\mathcal S$. Integrable zonal functions over the sphere (i.e., those depending only on~$\vartheta=\left<\hat e,x\right>$, where~$\hat e$ is the north-pole of the sphere $\hat e=(1,0,0)$) have the Legendre expansion
$$
f(\cos\vartheta)=\sum_{l=0}^\infty\widehat f(l)\,P_l(\cos\vartheta)
$$
with the Legendre coefficients
$$
\widehat f(l)=a_l\int_{-1}^1 f(t)\,P_l(t)\left(1-t^2\right)dt,
$$
where~$a_l$ is a constant that depends on~$l$, and~$P_l$, $l\in\mathbb N_0$, denote the Legendre polynomials.

The variances in space and momentum domain of a $\mathcal C^2(\mathcal S)$--function~$f$ with $\int_{\mathcal S}x\,|f(x)|^2\,d\sigma(x)\ne0$ are given by \cite{nLF03}
$$
\text{var}_S(f)=\left(\frac{\int_{\mathcal S}|f(x)|^2\,d\sigma(x)}{\int_{\mathcal S}x\,|f(x)|^2\,d\sigma(x)}\right)^2-1
$$
and
$$
\text{var}_M(f)=-\frac{\int_{\mathcal S}\Delta^\ast f(x)\cdot \bar f(x)\,d\sigma(x)}{\int_{\mathcal S}|f(x)|^2\,d\sigma(x)},
$$
where $\Delta^\ast$ is the Laplace--Beltrami operator on~$\mathcal S$. The quantity
\begin{equation}\label{eq:def_Uf}
U(f)=\sqrt{\text{var}_S(f)}\cdot\sqrt{\text{var}_M(f)}
\end{equation}
is called the uncertainty product (constant) of~$f$.

The uncertainty product of zonal functions may be computed from their Legendre coefficients \cite{nLF03,IIN16MR} and according to the spherical uncertainty principle it is bounded from below by~$1$, see \cite{NW96,RV97} and \cite[formula (4.37)]{GG04a}, \cite[formula (12)]{GG04b}.

\begin{lem}\label{lem:varS_varM} Let a zonal square integrable and continuously differentiable function over $\mathcal S^2$ be given by its Legendre expansion
$$
f(\cos\vartheta)=\sum_{l=0}^\infty\frac{2l+1}{4\pi}\,c_l(f)P_l(\cos\vartheta),
$$
where $\{\sqrt{2l+1}\,c_l\}\in l^2(\mathbb N_0)$. Its variances in space and momentum domain are equal to
\begin{align}
\text{var}_S(f)&=\left(\frac{\sum_{l=0}^\infty(2l+1)\,|c_l|^2}{\sum_{l=0}^\infty(l+1)\,(\overline{c_l}\,c_{l+1}+c_l\,\overline{c_{l+1}})}\right)^2-1,\label{eq:varS}\\
\text{var}_M(f)&=\frac{\sum_{l=1}^\infty l(l+1)(2l+1)\,|c_l|^2}{\sum_{l=0}^\infty(2l+1)\,|c_l|^2},\label{eq:varM}
\end{align}
whenever the series are convergent.
\end{lem}

\begin{thm}For $f\in\mathcal L^2(\mathcal S^2)\cap\mathcal C^1(\mathcal S^2)$, $U(f)\geq1$.
\end{thm}

As a consequence of~\cite[Theorem3.4]{IIN16USW} we have the following upper estimation.

\begin{thm}Let $\{\Psi_\rho\}$ be a zonal wavelet family with
$$
\widehat\Psi_\rho(l)=\frac{2l+1}{4\pi}\left[\rho^a q_\nu(l)\right]^c\,e^{-\rho^a q_\nu(l)},
$$
where $a>0$, $c>0$, and $q_{\nu}(l)=a_\nu l^\nu+a_{\nu-1}l^{\nu-1}+\dots+a_1 l+a_0$ is a polynomial of degree~$\nu$, positive and monotonously increasing for $l\geq1$. The
uncertainty product of~$\Psi_\rho$ for $\rho\to0$ behaves like
$$
U(\Psi_\rho)\leq\mathcal O\left(\rho^{\frac{-a}{2\nu}}\right).
$$
\end{thm}

\section{The uncertainty product of Gauss-Weierstrass wavelet}\label{sec:uncertaintyGW}

Gauss--Weierstrass wavelet (with respect to the measure~$\alpha(\rho)=\frac{1}{\rho}$) is given by
$$
\Psi_\rho^G(x)=\sum_{l=0}^\infty\frac{2l+1}{4\pi}\sqrt{2\rho l(l+1)}\,e^{-\rho l(l+1)}\,P_l(\cos\vartheta),\qquad x=(\vartheta,\varphi),\,\rho\in(0,\infty),
$$
see \cite[Subsec.~10.2.3]{FGS-book}.

\begin{thm}\label{thm:uncertainty_GW} The uncertainty constant of Gauss--Weierstrass wavelet is in limit $\rho\to0$ bounded by
\begin{equation}\label{eq:UPsi_upperbound}
 U(\Psi_\rho^G)\leq\sqrt{2\left(1+\frac{6}{e}+\frac{16}{e^2}\right)}+o(1).
\end{equation}
\end{thm}

In order to prove theorem, we need the following lemmas, which proofs are postponed to the end of the section.

\begin{lem}\label{lem:estimations}The following estimations are valid for $\rho\to0$:
\begin{align}
\tfrac{1}{8\rho^2}\!-\!\tfrac{3\sqrt 3e^{-3/2}}{8\rho\sqrt\rho}\!+\!\mathcal O(\tfrac{1}{\rho})
   &\leq\sum_{l=1}^\infty\! l(l\!+\!\tfrac{1}{2})(l\!+\!1)\,e^{\!-\!2\rho l(l\!+\!1)}
\leq\tfrac{1}{8\rho^2}\!+\!\tfrac{3\sqrt 3e^{-3/2}}{8\rho\sqrt\rho}\!+\!\mathcal O(\tfrac{1}{\rho}),\label{eq:est_v1}\\
\tfrac{1}{8\rho^2}\!-\!\tfrac{\sqrt{2\pi}\!+\!6\sqrt 3e^{\!-\!3/2}}{16\rho\sqrt\rho}\!+\!\mathcal O(\tfrac{1}{\rho})&\leq\sum_{l=1}^\infty(l\!+\!1)^2\sqrt{l(l\!+\!2)}\,e^{\!-\!2\rho(l\!+\!1)^2}
   \leq\tfrac{1}{8\rho^2}\!+\!\tfrac{3\sqrt 3e^{\!-\!3/2}}{8\rho\sqrt\rho}\!+\!\mathcal O(\tfrac{1}{\rho}),\label{eq:est_v23}\\
\tfrac{1}{8\rho^3}\!-\!\tfrac{25\sqrt 5e^{5/2}}{32\rho^2\sqrt\rho}\!+\!\mathcal O(\tfrac{1}{\rho\sqrt\rho})
   &\leq\sum_{l=1}^\infty l^2(l\!+\!\tfrac{1}{2})(l\!+\!1)^2\,e^{\!-\!2\rho l(l\!+\!1)}
   \leq\tfrac{1}{8\rho^3}\!+\!\tfrac{25\sqrt 5e^{-5/2}}{32\rho^2\sqrt\rho}\!+\!\mathcal O(\tfrac{1}{\rho\sqrt\rho}).\label{eq:est_v4}
\end{align}
\end{lem}

\begin{lem}\label{lem:estimations_varS}The series
$$
\sum_{l=1}^\infty\left[l(l+\tfrac{1}{2})(l+1)\,e^{-2\rho l(l+1)}-(l+1)^2\sqrt{l(l+2)}\,e^{-2\rho(l+1)^2}\right]
$$
is in limit $\rho\to0$ bounded from above by
$$
\left(\frac{1}{8}+\frac{3}{4e}+\frac{2}{e^2}\right)\frac{1}{\rho}+\mathcal O\left(\frac{1}{\sqrt\rho}\right).
$$
\end{lem}

\begin{bfseries}Proof of Theorem~\ref{thm:uncertainty_GW}. \end{bfseries}Substitute
$$
c_l=\sqrt{2\rho l(l+1)}\,e^{-\rho l(l+1)}
$$
to~\eqref{eq:varS} and~\eqref{eq:varM} to obtain
\begin{align*}
\text{var}_S(\Psi_\rho^G)&=\left(\frac{4\rho\sum_{l=1}^\infty l(l+\frac{1}{2})(l+1)\,e^{-2\rho l(l+1)}}
   {4\rho\sum_{l=1}^\infty(l+1)^2\sqrt{l(l+2)}\,e^{-2\rho(l+1)^2}}\right)^2-1,\\
\text{var}_M(\Psi_\rho^G)&=\frac{4\rho\sum_{l=1}^\infty l^2(l+\frac{1}{2})(l+1)^2\,e^{-2\rho l(l+1)}}
   {4\rho\sum_{l=1}^\infty l(l+\frac{1}{2})(l+1)\,e^{-2\rho l(l+1)}}.
\end{align*}
In order to estimate~$\text{var}_S(\Psi_\rho^G)$, we write it as
$$
\text{var}_S(\Psi_\rho^G)=\frac{(A+B)(A-B)}{B^2},
$$
where
\begin{align*}
A&:=\sum_{l=1}^\infty l(l+\tfrac{1}{2})(l+1)\,e^{-2\rho l(l+1)}\qquad\text{and}\\
B&:=\sum_{l=1}^\infty(l+1)^2\sqrt{l(l+2)}\,e^{-2\rho(l+1)^2}.
\end{align*}
Using estimations~\eqref{eq:est_v1} and~\eqref{eq:est_v23}, and the fact that
$$
\frac{a+\mathcal O(\rho)}{\alpha+\mathcal O(\rho)}=\frac{a}{\alpha}+o(1),\qquad\rho\to0,
$$
we conclude that
$$
\frac{A+B}{B}=\frac{\frac{1}{4\rho^2}+\mathcal O(\tfrac{1}{\rho\sqrt\rho})}{\frac{1}{8\rho^2}+\mathcal O(\tfrac{1}{\rho\sqrt\rho})}=2+o(1).
$$
On the other hand, by Lemma~\ref{lem:estimations_varS} and~\eqref{eq:est_v23},
$$
\frac{A-B}{B}\leq\frac{\left(\frac{1}{8}+\frac{3}{4e}+\frac{2}{e^2}\right)\frac{1}{\rho}+\mathcal O(\frac{1}{\sqrt\rho})}{\frac{1}{8\rho^2}+\mathcal O(\frac{1}{\rho\sqrt\rho})}
   =\left(1+\frac{6}{e}+\frac{16}{e^2}\right)\rho+o(\rho).
$$
Consequently,
$$
\text{var}_S(\Psi_\rho^G)\leq2\left(1+\frac{6}{e}+\frac{16}{e^2}\right)\rho+o(\rho).
$$
for $\rho\to0$. Similarly,
$$
\text{var}_M(\Psi_\rho^G)=\frac{1}{\rho}+o\left(\frac{1}{\rho}\right),\qquad\rho\to0,
$$
and the assertion~\eqref{eq:UPsi_upperbound} follows by definition~\eqref{eq:def_Uf}.
\hfill$\Box$

In the proofs of Lemmas~\ref{lem:estimations} and~\ref{lem:estimations_varS} the following estimation of the rest term in the trapezoidal quadrature rule will be used.

\begin{lem}\label{lem:quadrature}
Let~$v$ be a continuous function over~$[0,\infty)$, monotonously increasing for $t\in(0,t_0)$, monotonously decreasing for $t\in(t_0,\infty)$ and such that
\begin{equation}\label{eq:0_limit_condition}
f(0)=\lim_{t\to\infty}f(t)=0.
\end{equation}
Then,
\begin{equation}\label{eq:quadrature}
\int_0^\infty v(t)\,dt-v(t_0)\leq\sum_{l=1}^\infty v(l)\leq\int_0^\infty v(t)\,dt+v(t_0).
\end{equation}
\end{lem}

\begin{bfseries}Proof. \end{bfseries}Denote by~$[t_0]$ the greatest integer less than or equal to~$t_0$. By the monotonicity of~$v$,
$$
\sum_{l=0}^{[t_0]-1}v(l)\leq\int_0^{[t_0]}v(t)\,dt\leq\sum_{l=1}^{[t_0]}v(t)
$$
and
$$
\sum_{l=[t_0]+2}^\infty v(l)\leq\int_{[t_0]+1}^\infty v(t)\,dt\leq\sum_{l=[t_0]+1}^\infty v(t).
$$
These estimations, together with
$$
\int_{[t_0]}^{[t_0]+1}v(t)\,dt\leq v(t_0)\cdot1
$$
yield the lower estimation in~\eqref{eq:quadrature}. Further, we have
$$
\min\{v([t_0]),v([t_0]+1)\}\leq\int_{[t_0]}^{[t_0]+1}v(t)\,dt
$$
and
$$
\max\{v([t_0]),v([t_0]+1)\}\leq v(t_0).
$$
Thus,
$$
v([t_0])+v([t_0]+1)\leq\int_{[t_0]}^{[t_0]+1}v(t)\,dt+v(t_0),
$$
and the upper estimation in~\eqref{eq:quadrature} follows.\hfill$\Box$

\begin{bfseries}Proof of Lemma~\ref{lem:estimations}. \end{bfseries} For the proof of estimation~\eqref{eq:est_v1} consider the function
$$
v_1(t)=t(t+\tfrac{1}{2})(t+1)e^{-2\rho t(t+1)}.
$$
It satisfies condition~\eqref{eq:0_limit_condition}. Further, its derivative equals
$$
v_1^\prime(t)=\left[3t^2+3t+\frac{1}{2}-\rho t(t+1)(2t+1)^2\right]e^{-2\rho t(t+1)}.
$$
The expression in brackets disappears for $t=t_1$ such that
\begin{equation}\label{eq:rho_t1}
\rho=\frac{6t_1^2+6t_1+1}{2t_1(t_1+1)(2t_1+1)^2},
\end{equation}
and it is nonnegative for $t<t_1$ and it is negative for $t\geq t_1$. Thus, the function~$v_1$ is monotonously increasing for $t\in[0,t_1]$ and monotonously decreasing for $t\in[t_1,\infty)$. Moreover,
$$
\frac{6t_1^2+6t_1+1}{2t_1(t_1+1)(2t_1+1)^2}=\frac{3}{4t_1^2}+\mathcal O\left(\frac{1}{t_1^3}\right)\qquad\text{for }t_1\to\infty.
$$
Multiply both sides of
$$
\rho=\frac{3}{4t_1^2}+\mathcal O\left(\frac{1}{t_1^3}\right)
$$
by~$t_1^2$ to see that
$$
t_1=\frac{\sqrt 3}{2\sqrt\rho}+\mathcal O(1)
$$
satisfies equation~\eqref{eq:rho_t1} for $\rho\to0$. Consequently,
\begin{align}
\max_{t\in\mathbb R_+}&v_1(t)=v_1(t_1)=\left(\frac{\sqrt 3}{2\sqrt\rho}+\mathcal O(1)\right)^3e^{-2\rho\left(\frac{\sqrt 3}{2\sqrt\rho}+\mathcal O(1)\right)^2}\notag\\
&=\left[\frac{3\sqrt 3}{8\rho\sqrt\rho}+\mathcal O\left(\frac{1}{\rho}\right)\right]e^{-\frac{3}{2}+\mathcal O(\sqrt\rho)}\qquad\text{for }\rho\to0.\label{eq:max_v1}
\end{align}
Since
$$
e^{-\frac{3}{2}+\mathcal O(\sqrt\rho)}=\sum_{j=0}^\infty\frac{\left(-\frac{3}{2}+\mathcal O(\sqrt\rho)\right)^j}{j!}=e^{-\frac{3}{2}}+\mathcal O(\sqrt\rho),
$$
we obtain from~\eqref{eq:max_v1},
$$
\max_{t\in\mathbb R_+}v_1(t)=\frac{3\sqrt 3}{8\rho\sqrt\rho}\,e^{-\frac{3}{2}}+\mathcal O\left(\frac{1}{\rho}\right)\qquad\text{for }\rho\to0.
$$
On the other hand, it can be checked by derivation that
$$
\int v_1(t)\,dt=-\frac{e^{-2\rho t(t+1)}(2\rho t^2+2\rho t+1)}{8\rho^2}+C,
$$
and, consequently,
$$
\int_0^\infty v_1(t)\,dt=\frac{1}{8\rho^2}.
$$
Estimation~\eqref{eq:est_v1} follows by Lemma~\ref{lem:quadrature}.

In order to obtain~\eqref{eq:est_v23} note that
$$
v_2(t)\leq(t+1)^2\sqrt{t(t+2)}e^{-2\rho(t+1)^2}\leq v_3(t)
$$
for
\begin{align*}
v_2(t)&:=t(t+1)^2\,e^{-2\rho(t+1)^2},\\
v_3(t)&:=(t+1)^3\,e^{-2\rho(t+1)^2}.
\end{align*}
Similarly as in the previous case, we obtain
\begin{align*}
v_2^\prime(t)&=(t+1)\left[3t+1-4\rho t(t+1)^2\right]e^{-2\rho(t+1)^2},\\
v_3^\prime(t)&=(t+1)^2\left[3-4\rho(t+1)^2\right]e^{-2\rho(t+1)^2},
\end{align*}
and
$$
\max_{t\in\mathbb R_+}v_2(t)=v_2(t_2),\qquad\max_{t\in\mathbb R_+}v_3(t)=v_3(t_3)
$$
with~$t_2$ such that
$$
\rho=\frac{3t_2+1}{4t_2(t_2+1)^2}=\frac{3}{4t_2^2}+\mathcal O\left(\frac{1}{t_2^3}\right)\qquad\text{for }t_2\to\infty,
$$
i.e.,
$$
t_2=\frac{\sqrt 3}{2\sqrt\rho}+\mathcal O(1)\qquad\text{for }\rho\to0,
$$
and
$$
t_3=\frac{\sqrt 3}{2\sqrt\rho}-1.
$$
Consequently,
\begin{align*}
\max_{t\in\mathbb R_+}v_2(t)&=\frac{3\sqrt 3}{8\rho\sqrt\rho}\,e^{-\frac{3}{2}}+\mathcal O\left(\frac{1}{\rho}\right)\qquad\text{for }\rho\to0,\\
\max_{t\in\mathbb R_+}v_3(t)&=\frac{3\sqrt 3}{8\rho\sqrt\rho}\,e^{-\frac{3}{2}}.
\end{align*}
Further, it follows from
\begin{align*}
\int v_2(t)\,dt&=-\frac{2[1+2\rho t(t+1)]e^{-2\rho(t+1)^2}+\sqrt{2\pi\rho}\,\text{erf}\left[\sqrt{2\rho}\,(t+1)\right]}{16\rho^2}+C\qquad\text{and}\\
\int v_3(t)\,dt&=-\frac{[1+2\rho(t+1)^2]\,e^{-2\rho(t+1)^2}}{8\rho^2}+C.
\end{align*}
that
\begin{align*}
\int_0^\infty v_2(t)\,dt&=\frac{2e^{-2\rho}-\sqrt{2\pi\rho}\,\text{erfc}(\sqrt{2\rho})}{16\rho^2}\qquad\text{and}\\
\int_0^\infty v_3(t)\,dt&=\frac{(1+2\rho)\,e^{-2\rho}}{8\rho^2}.
\end{align*}
Estimation~\eqref{eq:est_v23} is an implication of
$$
\int_0^\infty\!\!\!\!v_2(t)\,dt-\max_{t\in\mathbb R_+}v_2(t)\leq\sum_{l=1}^\infty(l+1)^2\sqrt{l(l+2)}\,e^{-2\rho(l+1)^2}
\leq\int_0^\infty\!\!\!\!v_3(t)\,dt+\max_{t\in\mathbb R_+}v_3(t)
$$
and
$$
\text{erfc}(\sqrt{2\rho})=1-2\,\sqrt\frac{2}{\pi}\sqrt\rho+\mathcal O(\rho\sqrt\rho)\qquad\text{for }\rho\to0.
$$

Finally, the derivative
$$
v_4^\prime(t)=t(t+1)\left[5t^2+5t+1-\rho t(t+1)(2t+1)^2\right]e^{-2\rho t(t+1)}
$$
of
$$
v_4(t):=t^2(t+\tfrac{1}{2})(t+1)^2\,e^{-2\rho t(t+1)}
$$
disappears for~$t=t_4$ such that
$$
\rho=\frac{5t_4^2+5t_4+1}{t_4(t_4+1)(2t_4+1)^2}=\frac{5}{4t_4^2}+\mathcal O\left(\frac{1}{t_4^3}\right)\qquad\text{for }t_4\to\infty,
$$
i.e.,
$$
t_4=\frac{\sqrt 5}{2\sqrt\rho}+\mathcal O(1))\qquad\text{for }\rho\to0.
$$
It is nonnegative for $t<t_4$ and negative for $t>t_4$. Thus,
$$
\max_{t\in\mathbb R_+}v_4(t)=v_4(t_4)=\frac{25\sqrt 5}{32\rho^2\sqrt\rho}\,e^{-\frac{5}{2}}+\mathcal O\left(\frac{1}{\rho\sqrt\rho}\right)\qquad\text{for }\rho\to0.
$$
In order to obtain~\eqref{eq:est_v4} note that
$$
\int v_4(t)\,dt=-\frac{1+2\rho t(t+1)+2\rho^2t^2(t+1)^2}{8\rho^3}\,e^{-2\rho t(t+1)}+C
$$
yields
$$
\int_0^\infty v_4(t)\,dt=\frac{1}{8\rho^3}.
$$
\hfill$\Box$

\begin{bfseries}Proof of Lemma~\ref{lem:estimations_varS}. \end{bfseries}We shall express the difference of the series as
\begin{align*}
\sum_{l=1}^\infty l(l+\tfrac{1}{2})&(l+1)\,e^{-2\rho l(l+1)}-\sum_{l=1}^\infty(l+1)^2\sqrt{l(l+2)}e^{-2\rho(l+1)^2}\\
&=\sum_{l=1}^\infty\left[v_1(l)+v_2(l)+v_3(l)\right]
\end{align*}
for
\begin{align*}
v_1(t)&:=\left[t(t+\tfrac{1}{2})(t+1)-(t+1)^3\right]e^{-2\rho t(t+1)},\\
v_2(t)&:=(t+1)^3\left[e^{-2\rho t(t+1)}-e^{-2\rho(t+1)^2}\right],\\
v_3(t)&:=\left[(t+1)^3-(t+1)^2\sqrt{t(t+2)}\right]e^{-2\rho(t+1)^2},
\end{align*}
and for each of the functions~$v_1$, $v_2$, $v_3$ apply the trapezoidal quadrature rule from Lemma~\ref{lem:quadrature} (note, however, that $v_1$ is a negative function, monotonous\-ly decreasing to a minimum, and then monotonously increasing with limit equal to~$0$ in infinity).

It follows from
$$
\int v_1(t)\,dt
=\frac{e^{-2\rho t(t+1)}\left[2\sqrt\rho\,(6t+7)-e^{\tfrac{1}{2}\rho(2t+1)^2}\sqrt{2\pi}\,(3+\rho)\,\text{erf}\left(\frac{\sqrt\rho\,(2t+1)}{\sqrt 2}\right)\right]}{32\rho\sqrt\rho}+C
$$
that
\begin{align*}
\int_0^\infty v_1(t)\,dt&=-\frac{14\sqrt\rho+e^{\,\rho/2}\sqrt{2\pi}\,(3+\rho)\,\text{erfc}\left(\frac{\sqrt\rho}{\sqrt 2}\right)}{32\rho^{\tfrac{3}{2}}}\\
&=-\frac{3\sqrt\frac{\pi}{2}}{16\rho\sqrt\rho}-\frac{1}{4\rho}+\mathcal O\left(\frac{1}{\sqrt\rho}\right)\qquad\text{for }\rho\to0.
\end{align*}
On the other hand, the derivative
$$
v_1^\prime(t)=-\frac{1}{2}\left[6t+5-2\rho(t+1)(2t+1)(3t+2)\right]e^{-2\rho t(t+1)}
$$
disappears for~$t=t_1$ such that
$$
\rho=\frac{6t_1+5}{2(t_1+1)(2t_1+1)(3t_1+2)}=\frac{1}{2t_1^2}+\mathcal O\left(\frac{1}{t_1^3}\right)\qquad\text{for }t_1\to\infty,
$$
i.e.,
$$
t_1=\frac{1}{\sqrt{2\rho}}+\mathcal O(1)\qquad\text{for }\rho\to0.
$$
Thus,
$$
\min_{t\in\mathbb R_+}v_1(t)=v_1(t_1)=-\frac{3}{4e\rho}+\mathcal O\left(\frac{1}{\sqrt\rho}\right).
$$
Further,
\begin{align*}
\int&v_2(t)\,dt=\frac{1}{32\rho^2}\biggl[4\left[1+2\rho(t+1)^2\right]e^{-2\rho(t+1)^2}\\
&-2\left[2+\rho(4t^2+10t+7)\right]e^{-2\rho t(t+1)}+(3+\rho)\sqrt{2\pi\rho}\,e^{\frac{\rho}{2}}\,\text{erf}\left(\sqrt\frac{\rho}{2}\,(2t+1)\right)\biggr]
\end{align*}
and
\begin{align*}
\int_0^\infty v_2(t)\,dt&=\frac{-4(1+2\rho)e^{-2\rho}+2(2+7\rho)+(3+\rho)\sqrt{2\pi\rho}\,e^{\frac{\rho}{2}}\,\text{erfc}\left(\sqrt\frac{\rho}{2}\right)}{32\rho^2}\\
&=\frac{3\sqrt\frac{\pi}{2}}{16\rho\sqrt\rho}+\frac{1}{4\rho}+\mathcal O\left(\frac{1}{\sqrt\rho}\right).
\end{align*}
The derivative
$$
v_2^\prime(t)=(t+1)^2\left[\left[3-2\rho(t+1)(2t+1)\right]e^{2\rho(t+1)}-3+4\rho(t+1)^2\right]e^{-2\rho(t+1)^2}
$$
disappears when
$$
\left[3-2R(2t+1)\right]e^{2R}-3+4R(t+1)=0,
$$
where $R=\rho(t+1)$, i.e., for
$$
t=\frac{(3-2R)\,e^{2R}-3+4R}{4R\,(e^{2R}-1)}=\frac{1}{R}-\frac{3}{4}+\mathcal O(R)\qquad\text{for }R\to0.
$$
The last equation is satisfied for
$$
t=t_2=\frac{1}{\sqrt\rho}-\frac{7}{8}+\mathcal O(\sqrt\rho)\qquad\text{for }\rho\to0.
$$
Consequently,
$$
\max_{t\in\mathbb R_+}v_2(t)=v_2(t_2)=\frac{2}{e^2\rho}+\mathcal O\left(\frac{1}{\sqrt\rho}\right)\qquad\text{for }\rho\to0.
$$

In order to estimate~$v_3$ note that
$$
(t+1)^3-(t+1)^2\sqrt{t(t+2)}=\frac{t+1}{2}+\mathcal O\left(\frac{1}{t}\right)\qquad\text{for }t\to\infty.
$$
Hence, there exists a constant~$c$ such that
$$
\int_0^\infty v_3(t)\,dt=\underbrace{\int_0^1 v_3(t)\,dt}_{=:I_1}+\underbrace{\int_1^\infty\frac{t+1}{2}\,e^{-2\rho(t+1)^2}dt}_{=:I_2}+R
$$
with
$$
|R|\leq c\int_1^\infty\frac{e^{-2\rho(t+1)^2}}{t}\,dt=:cI_3.
$$
Since
$$
I_1=\int_0^1 v_3(t)\,dt\leq1
$$
independently of~$\rho$,
$$
I_2=\int_1^\infty\frac{t+1}{2}\,e^{-2\rho(t+1)^2}\,dt=\frac{e^{-8\rho}}{8\rho}=\frac{1}{8\rho}+\mathcal O(1)\qquad\text{for }\rho\to0,
$$
and
$$
I_3=\int_1^\infty\frac{e^{-2\rho(t+1)^2}}{t}\,dt\leq\int_0^\infty e^{-\rho t^2}dt=\frac{\sqrt\pi}{2\sqrt\rho},
$$
we have
$$
\left|\int_0^\infty v_3(t)\,dt-\frac{1}{8\rho}\right|\leq\mathcal O\left(\frac{1}{\sqrt\rho}\right)\qquad\text{for }\rho\to0.
$$
Further, the derivative
$$
v_3^\prime(t)=(t+1)\left(t+1-\sqrt{t(t+2)}\right)\left[2-\frac{t+1}{\sqrt{t(t+2)}}-4(t+1)^2\rho\right]e^{-2\rho(t+1)^2}
$$
changes its sign from positive to negative in $t=t_3$ such that
$$
\rho=\frac{2\sqrt{t_3(t_3+2)}-t_3-1}{4(t_3+1)^2\sqrt{t_3(t_3+2)}}.
$$
Since
$$
\frac{2\sqrt{t_3(t_3+2)}-t_3-1}{4(t_3+1)^2\sqrt{t_3(t_3+2)}}=\frac{1}{4t_3^2}+\mathcal O\left(\frac{1}{t_3^3}\right)\qquad\text{for }t_3\to\infty,
$$
we have
$$
t_3=\frac{1}{2\sqrt\rho}+\mathcal O(1)\qquad\text{for }\rho\to0
$$
and
$$
\max_{t\in\mathbb R_+}v_3(t)=v_3(t_3)=\frac{1}{4\sqrt{e\rho}}+\mathcal O(\sqrt\rho)\qquad\text{for }\rho\to0.
$$

The desired estimation follows from
\begin{equation*}\begin{split}
&\left|\sum_{l=1}\left[v_1(l)+v_2(l)+v_3(l)\right]-\int_0^\infty\left[v_1(t)+v_2(t)+v_3(t)\right]dt\right|\\
   &\qquad\leq-\min_{t\in[0,\infty)}v_1(t)+\max_{t\in[0,\infty)}v_2(t)+\max_{t\in[0,\infty)}v_3(t).
\end{split}\end{equation*}
\hfill$\Box$

\begin{bfseries}Remark. \end{bfseries}With more sophisticated methods we probably would be able to find a better bound for the error made by replacing $\sum_{l=1}^\infty\left[v_1(l)+v_2(l)\right]$ by $\int_0^\infty\left[v_1(t)+v_2(t)\right]dt$. However, numerical experiments have shown that it would be of order~$\frac{1}{\rho}$, anyway. Consequently, we are not able to compute the exact value of~$U(\Phi_\rho^G)$, as it was the case for Gauss-Weierstrass kernel in~\cite{LFP03}. On the other hand, sole boundedness of the uncertainty product distinguishes Gauss--Weierstrass wavelet from other ones, see~\cite[Theorem~3.4]{IIN16USW}.


TITLE: Show that $n(2n + 1)(7n + 1)$ is divisible by 6 for all integers $n$ and find all integers $n$ such that $n(2n + 1)(7n + 1)$ is divisible by 12.
QUESTION [2 upvotes]: a) Show that $n(2n + 1)(7n + 1)$ is divisible by 6 for all integers $n$.
b) Find all integers $n$ such that $n(2n + 1)(7n + 1)$ is divisible by 12.
I tried expanding but it doesn't really help much.  Thanks in advance!

REPLY [6 votes]: $$n(2n+1)(7n+1)\equiv n(2n+1)(n+1)\equiv 0\pmod{2},$$
because exactly one of $n$, $n+1$ is even.
$$n(2n+1)(7n+1)\equiv n(-n+1)(n+1)\pmod{3}$$
$$\equiv n(-1)(n-1)(n+1)\equiv 0\pmod{3},$$
because exactly one of $n$, $n-1$, $n+1$ is divisible by $3$.
Therefore $2\cdot 3\mid n(2n+1)(7n+1)$ for all $n\in\mathbb Z$.
The condition $2^2\cdot 3\mid n(2n+1)(7n+1)$ is therefore equivalent to $2^2\mid n(2n+1)(7n+1)$.
Since $2n+1$ is odd for all $n\in\mathbb Z$, this is equivalent to $2^2\mid n(7n+1)$ (by common sense, or you can see Generalized Euclid's lemma. We have $\gcd\left(2^2,2n+1\right)=1$ for all $n\in\mathbb Z$).
$$n(7n+1)\equiv n(-n+1)\equiv -n(n-1)\pmod{2^2}$$
is equivalent to $0$ if and only if either $2^2\mid n$ or $2^2\mid n-1$ (because $n$, $n-1$ are coprime).
The answer is: all $n\in\mathbb Z$ such that $n\equiv \{0,1\}\pmod{4}$.


TITLE: Why does Finset not have a natural numbers object?
QUESTION [0 upvotes]: Let $C$ be an elementary topos with final object $1$. A natural numbers object is an object $\mathbb{N}$ equipped with morphisms $q:1\longrightarrow \mathbb{N}$ and $s:\mathbb{N}\longrightarrow\mathbb{N}$ such that for any object $A$ and morphisms $z:A\longrightarrow A$ and $f:A\longrightarrow A$ there exists an unique morphism $\phi:\mathbb{N}\longrightarrow A$ such that the obvious diagram is commutative.
For proving that the category Finset doesn't have a NNO I was thinking to use an especific object $A=\{a_{0},...,a_{n}\}$ and morphisms $q:1\longrightarrow A$ such that $q(0):=a_{0}$ and $f:A\longrightarrow A$ such that $f(a_{i})=a_{i+1}$ for all $0\leq i\leq n-1$ and $f(a_{n})= a_{0}$. Then there exists the object $\mathbb{N}$ with $|\mathbb{N}|=m$ with the morphisms $s$ and $z$.
Then I was thinking that we will conclude that $\mathbb{N}\cong \mathbb{Z}_{n}$, because we will have $\phi(z(0)))=a_{0}$, $\phi(s(z(0)))=f(a_{0})=a_{1}$ and $\phi(s^{n}(z(0)))=a_{n}$, but from here I don't know how to get a contradiction.
Thank you very much.

REPLY [1 votes]: Your solution basically works aside from the last paragraph. As Niven pointed out in their answer the trick to make this work is to consider a family of sets like the one you constructed and show that this forces the cardinality of the hypothetical natural numbers object to be infinite.
Alternatively, and this is the main reason I'm writing this answer, you could follow the route suggested by Mark Saving and prove that in general $\mathbb{N}\cong 1 \amalg \mathbb{N}\newcommand\NN{\mathbb{N}}$, which would immediately give a contradiction.
Proof:
Note on terminology. I'm going to call triples $(A,z:1\to A, s:A\to A)$ counting systems. I don't know that this is standard terminology, but I need a name for these things, and this is what I settled on.
Suppose $(\NN,z:1\to \NN, s:\NN\to\NN)$ is a natural numbers object. To show $\NN\cong 1\amalg \NN$,
it suffices to show that $1\amalg \mathbb{N}$ is also a natural numbers object. We define $z' : 1\to 1\amalg \NN$ to be the inclusion $\iota_1$ in the coproduct. We define $s' : 1\amalg \NN\to 1\amalg \NN$ to be $\iota_\NN\circ z + \iota_\NN \circ s$.
Now if $(A,z_A, s_A)$ is another counting system, suppose $f'=f_0+f : 1\amalg \NN\to A$ is a map compatible with the counting systems, i.e., $f'\circ z' = z_A$, $f'\circ s' = s_A\circ f'$. The first equality says $f_0 = z_A$, so $f_0$ is uniquely determined by these equations, and the second equality says that
$$f'(\iota_\NN z+\iota_\NN s) = fz + fs = s_A f' = s_Af_0 + s_Af.$$
This gives us that
$fz=s_Af_0=s_Az_A$ and $fs=s_Af$.
So $f$ is a map of counting systems from $(\NN,z,s)$ to $(A,s_Az_A,s_A)$, and is therefore unique.
Therefore $f'$, if it exists, is unique. On the other hand, we can also use these equations to construct $f_0$ and $f$ to produce a valid map $f'$.


TITLE: How could the BOE (barrel of oil equivalent) be measured?
QUESTION [0 upvotes]: The barrel of oil equivalent was given as an example of gigajoule scale energy.
"The barrel of oil equivalent (BOE) is a unit of energy based on the approximate energy released by burning one barrel (42 U.S. gallons or 158.9873 litres) of crude oil."
https://en.wikipedia.org/wiki/Barrel_of_oil_equivalent
Now any idea how's that actually measured?

REPLY [2 votes]: The obvious method is to burn a barrel of crude oil and measure how much energy is released.
The only slightly less obvious method is to burn a small amount of oil and measure how much energy is released, and then mathematically figure how much energy a whole barrel would release, as @CuriousOne points out.
The latter method is superior in both the economical and ecological senses.