TITLE: Expectation of the norm of a random vector
QUESTION [9 upvotes]: Suppose $X$ is a random vector denoted as $(X_1,\cdots,X_n)$, where $X_1,\cdots,X_n$ are iid random variables with sub-Gaussian distributions. For all $i$, suppose $E[X_i^2]=1$ for simplicity and $\|X_i\|_{\psi_2}=K$ where$\|\cdot\|_{\psi_2}$ is the sub-Gaussian norm.
Let $Y=\|X\|$ be the 2-norm of $X$. A known fact is $E[Y]-\sqrt n$ can be const bounded. On the other hand, one may ask when $n\rightarrow \infty$, will $E[Y]-\sqrt n \rightarrow 0$?
For example, consider the standard normal distribution. Then $Y^2$ is $\chi^2(n)$ distribution, and $$E[Y]=\sqrt 2 \frac {\Gamma\left(\frac {n+1} 2\right)} {\Gamma\left(\frac {n} 2\right)}$$ so $E[Y]-\sqrt n\rightarrow 0$ when $n\rightarrow \infty$. So for arbitary random variable $X_i$ which satisfy the above condition, does $E[Y]-\sqrt n\rightarrow 0$ always hold?

REPLY [13 votes]: $\newcommand{\si}{\sigma}$
Let us prove a stronger estimate of $EY$, and let us do that under less restrictive conditions. Namely, let us prove that
\begin{equation*}
 EY-\sqrt n=O(1/\sqrt n) \tag{1}
\end{equation*}
assuming only that $EX_1^4<\infty$ (instead of the $X_i$'s being sub-Gaussian).
Substituting
$$U:=\frac{Y^2}n=\frac1n\,\sum_1^n X_i^2$$
for $u$ in the inequalities
$$\frac{1+u-(u-1)^2}2\le\sqrt u\le\frac{1+u}2$$
for $u\ge0$, taking the expectations, and using that $EU=1$ and $E(U-1)^2=\operatorname{Var}\,U=\sigma^2/n$, where $\si^2:=\operatorname{Var}(X_1^2)<\infty$,
we have
$$1-\frac{\sigma^2}{2n}\le\frac{EY}{\sqrt n}\le1,$$
so that (1) indeed follows.


text:
In the following, various simulation results are presented to illustrate the influence of the algorithm parameters, i.e., matrix $T_d$, singular values $\sigma_k$, $k = 1,\ldots,r$, of $W$ and map parameters $\alpha_1,\alpha_2$.  For the sake of visualization, we focus on examples with $n=2$.
\begin{figure}[t]
	\centering
	\setlength\figureheight{4.5cm}
	\setlength\figurewidth{4.5cm}
	{\small \input{Fig/TW_examp.tikz}}
	\caption[]{Generated areas of the exploration sequence $\{w_k\}_{k=0}^{m-1}$ in \eqref{eq:input_coord} for $T(W) \in \mathbb{R}^{2n \times 2n }$ as given in \eqref{eq:TW_pq} for $n = 2$. Hence, $T(W)_{1,3}= T(W)_{2,4} = -1,\  T(W)_{3,1}= T(W)_{4,2} = 1$ and the rest $0$.

	The filled green area ($\areapos$) have a area surface value of $1$, the striped orange area ($\areaneg$) of  $-1$, and the rest of $0$.

	 \correct{The coordinates of each subplot are given by $x_{p,i}$ and $y_{q,i}$ defined in \eqref{eq:area_points} for $p,q=1,\ldots,2n$.}}
	\label{fig:TW_examp}
\end{figure}

\begin{figure*}[t]
	\centering
	\setlength\figureheight{2.8cm}
	\setlength\figurewidth{3.2cm}
	{\small \input{Fig/2D_Lie_1-1_1-1_1_1.tikz}}
	{\small \input{Fig/2D_Lie_1-5_0-2_1-5_0-2.tikz}}
	\caption[]{\corrected{An illustration of the algorithm's behavior (setup as in Simulation 1) with $T_d$ as in \eqref{eq:cases_H1} and singular values $[\sigma_1\ \sigma_2\ \sigma_3\ \sigma_4]=[1\ 1\ 1\ 1]$ (top) and $[\sigma_1\ \sigma_2\ \sigma_3\ \sigma_4]=[1.5\ 0.2\ 1.5\ 0.2]$ (bottom) of the exploration matrix $W$. In (a)+(e) the trajectories $x_k$ ({\color{orange90}$\thinmin$}), the first and last $m$ steps ({\color{red}$\boldmin$}), the exact gradient descent algorithm ({\color{blue}$\thinmins\,\thinmins$}), initial state $x_0$ ({\color{green60}$\bullet$}), and optimizer $x^*$ ({\color{green60}$\blacklozenge$}) are depicted. The plots (b)+(f) and (c)+(g) show the exploration sequence $\{u_k\}_{k=0}^{m-1}$ and $\{v_k\}_{k=0}^{m-1}$, respectively, where ({\color{green60}$\bullet$}) is the first and ({\color{orange90}$\cross$}) the second component. In (d)+(h) $T(W)$ in form of the areas generated by $\{w_k\}_{k=0}^{m-1}$ is visualized, where the filled green areas ($\areapos$) have an area surface value of $1$, the striped orange areas ($\areaneg$) of  $-1$, and the rest of $0$.}}
	\label{fig:sim1}
\end{figure*}

\begin{figure*}[t]
	\centering
	\setlength\figureheight{2.8cm}
	\setlength\figurewidth{3.2cm}
	{\small \input{Fig/2D_Lie_2_2.tikz}}
	{\small \input{Fig/2D_Lie_0-2_0-2.tikz}}
	\caption[]{\corrected{An illustration of the algorithm's behavior (setup as in Simulation 2) with $T_d$ as in \eqref{eq:cases_H2} and singular values $[\sigma_1\ \sigma_2]=[2\ 2]$ (top) and $[\sigma_1\ \sigma_2]=[0.2\ 0.2]$ (bottom) of the exploration matrix $W$. In (a)+(e) the trajectories $x_k$ ({\color{orange90}$\thinmin$}), the first and last $m$ steps ({\color{red}$\boldmin$}), the exact gradient descent algorithm ({\color{blue}$\thinmins\,\thinmins$}), initial state $x_0$ ({\color{green60}$\bullet$}), and optimizer $x^*$ ({\color{green60}$\blacklozenge$}) are depicted. The plots (b)+(f) and (c)+(g) show the exploration sequence $\{u_k\}_{k=0}^{m-1}$ and $\{v_k\}_{k=0}^{m-1}$, respectively, where ({\color{green60}$\bullet$}) is the first and ({\color{orange90}$\cross$}) the second component. In (d)+(h) $T(W)$ in form of the areas generated by $\{w_k\}_{k=0}^{m-1}$ is visualized, where filled green area ($\areapos$) have an area surface value of $1$, the striped orange area ($\areaneg$) of  $-1$, and the rest of $0$.}}
	\label{fig:sim2}
\end{figure*}

An extensive bench-marking study, including the best choice of parameters for certain classes of the objective, is beyond the scope of this paper and carried out in ongoing and future work. Hence, we keep $f,g,$ (sinusoidal) and $h$ fixed and

provide only a limited number of simulation examples to get some qualitative insight in the degrees of freedom and how they influence the algorithms behavior.

In the figures below, we show trajectories and the exploration sequences.

In addition, we provide a geometric interpretation and a visualization of the matrix $T(W)$
which is of interest by its own and which is explained next.

The values of the components of $T(W)$ in \eqref{eq:cond_equ_gen} for $[\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}]$, i.e.,

\begin{align}
	T(W)_{pq} = \sum_{i=0}^{m-1}\sum_{j=0}^{i-1} \frac{1}{2} e_p^\top w_i w_i^\top e_q +   e_p^\top w_i w_j^\top e_q ,
	\label{eq:TW_pq}
\end{align}
with \correct{$p,q=1,\ldots,2n$}, where the index $pq$ specifies the element of $T(W)$ in the $p$-th row and $q$-th column,  can be interpreted as the projected areas spanned by the exploration sequences $\{e_p^\top w_\ell\}_{\ell=0}^{m-1},\ \{e_q^\top w_\ell\}_{\ell=0}^{m-1}$.

The net area $A_{pq}$ of an $n$-sided polygon with corner points $(x_{p,i},y_{q,i}) \in \mathbb{R}^2$, $i=0...n-1$ and \correct{$p,q=1,\ldots,2n$}, known as \textit{Shoelace} or \textit{Gauss} area formula \cite{spivak2018calculus} is obtained as a  special case of Green's Theorem and is given by

\begin{align}
	\correct{A_{pq} = \frac{1}{2}\sum_{i=0}^{n-1} \left(x_{p,i+1}  y_{q,i} - x_{p,i} y_{q,i+1}\right)}
	\label{eq:area}
\end{align}

with \correct{$x_{p,0} = x_{p,n} = 0$ and $y_{q,0}=y_{q,n} = 0$}. In \Cref{thm:gen_area_skew} (\Cref{sec:pre_lem}) we show that \eqref{eq:area} is equivalent to \eqref{eq:TW_pq} for

\begin{align}
    \correct{x_{p,i} = \sum_{k=0}^{i-1}e_p^\top w_{k},\ y_{q,i} = \sum_{k=0}^{i-1}e_q^\top w_{k}.}
    \label{eq:area_points}
\end{align}

This geometric interpretation is not surprising in the light of
\Cref{rem:brockett_int2} and the area generating rule appearing in the study of nonholonomic systems \cite{bloch2004nonholonomic}.

In particular, \eqref{eq:area} represents a (double) iterated summation
over the exploration sequence.
In the continuous-time setting, this would correspond to double iterated integrals (or in general $k$-fold iterated integrals, which are called the signature of a path) and which play a fundamental role in nonholonomic control systems.

Note that for $[\alpha_1\ \alpha_2] \neq [\nicefrac{1}{2}\ \nicefrac{1}{2}]$, \eqref{eq:area} with \eqref{eq:area_points} does not hold.
However, we believe it is related to some kind of weighted area.

To illustrate this geometric interpretation, the exploration sequence in \eqref{eq:input_coord} with $[\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}]$ generates $T(W)$ as depicted in \Cref{fig:TW_examp}, where the singular values of $W$ are $\sigma_1 = \sigma_{2} = \sigma_3 =  \sigma_4 = \sqrt{2}$. Obviously and as presented in the sequel, the singular values of $W$ influence the shape of the areas and give rise to various interpretations of the algorithms behavior.

\subsubsection*{Simulation 1}
\label{sec:sim1}

In the first simulation setup we consider the objective $J(x) = \|x-[1\ 2]^\top\|^2_2$ and setting $f(J(x)) = \sin(J(x)),\ g(J(x))=\cos(J(x))$, $[\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}]$, $T_d$ as in \eqref{eq:cases_H1}, $h = 0.05$, and $x_0 = [0\ 1]^\top$. The simulation results for two different choices of singular value pairs are depicted in \Cref{fig:sim1}. On the one hand $\sigma_1=\sigma_2=\sigma_3=\sigma_4=1$ yields $m = 8$ and on the other hand $\sigma_1=\sigma_3=1.5,\ \sigma_2 = \sigma_4= 0.2$ results in $\corrected{m = 21}$. In the latter case the areas with net area value $\pm 1$ have an elongated elliptical shape, which yield a small amplitude of $\{v_k\}_{k=0}^{m-1}$ and therefore a small steady-state amplitude since $\sin(J(x^*))=0$ and $\cos(J(x^*))=1$ (cf. \eqref{eq:TM}). Concluding, the amplitude of $\{e_i^\top w_\ell\}_{\ell=0}^{m-1}$ is proportional to $\sigma_i$ with $i=1,\ldots,2n$ for this choice of $T_d$.

\subsubsection*{Simulation 2.}
\label{sec:sim2}

In the second simulation study, we consider the same setup as in Simulation 1, but choose $T_d$ as in \eqref{eq:cases_H2} with $Q$ as specified in the proof of \Cref{thm:min_steps}. The simulation results for two different choices of singular values pairs are depicted in \Cref{fig:sim2}. On one hand $\sigma_1 = \sigma_2=2$ results in $m = 4$ and on the other hand $\sigma_1 =  \sigma_2 = 0.2$ in $m = 154$. The singular values can be interpreted as a kind of energy measurement of the exploration sequence, and therefore in the latter case, $m$ is much larger, but reveals a smoother behavior. Moreover, $\sigma_i$ influences the amplitude of $\{e_{2i-1}^\top w_\ell\}_{\ell=0}^{m-1}$ and $\{e_{2i}^\top w_\ell\}_{\ell=0}^{m-1}$ for $i=1,\ldots,n$ for the given choice of $T_d$, compared to the previous simulation example. However, due to space limitations we omit the plots for $\sigma_1 \neq \sigma_2$.

Interestingly, as observed in \Cref{fig:sim1} and \Cref{fig:sim2} (d)+(h), the areas of $T(W)_{ij}$ for $(i,j) \in \{(1,2),(2,1)\}$ and $(i,j) \in \{(3,4),(4,3)\}$ have the same shape as the last $m$ steps of $x_k$.

\subsubsection*{Simulation 3.}
\label{sec:sim3}

Consider the same setup as in Simulation 1, but choosing $[\alpha_1\ \alpha_2] = [1\ 0]$ and $T_d$ as described in \eqref{eq:Td_E}. Choosing $\sigma_1 = \sigma_2 = \sigma_3 = \sigma_4 = 1$ results in the behavior depicted in \Cref{fig:sim3} (a)-(c), which is similar to the behavior from \Cref{fig:sim1} (a)-(d), where $m=8$, too. On the one hand, the number of evaluations of the objective $J$ is reduced by half when compared to Simulation 1. On the other hand, in this parameter setup, the choice of singular values is restricted to $\sigma_1 = \sigma_2$ and $\sigma_3 = \sigma_4$, hence, a behavior as in \Cref{fig:sim1} (e)-(h) can not be achieved. However, reducing $\sigma_3 = \sigma_4$ to $0.4$ yields a scaling in the coordinate directions as illustrated in \Cref{fig:sim3} (d)-(f).

\begin{figure*}[t]
	\centering
	\setlength\figureheight{2.8cm}
	\setlength\figurewidth{3.2cm}
	{\small \input{Fig/2D_normal_1_1.tikz}}
	{\small \input{Fig/2D_normal_1_0-4.tikz}}
	\caption[]{\corrected{An illustration of the algorithm's behavior (setup as in Simulation 3) with $T_d$ as in \eqref{eq:Td_E} with singular values $[\sigma_1\ \sigma_2\ \sigma_3\ \sigma_4]=[1\ 1\ 1\ 1]$ (top) and $[\sigma_1\ \sigma_2\ \sigma_3\ \sigma_4]=[1\ 1\ 0.4\ 0.4]$ (bottom) of the exploration matrix $W$. In (a)+(d) the trajectories $x_k$ ({\color{orange90}$\thinmin$}), the first and last $m$ steps ({\color{red}$\boldmin$}), the exact gradient descent algorithm ({\color{blue}$\thinmins\,\thinmins$}), initial state $x_0$ ({\color{green60}$\bullet$}), and optimizer $x^*$ ({\color{green60}$\blacklozenge$}) are depicted. The plots (b)+(e) and (c)+(f) show the exploration sequence $\{u_k\}_{k=0}^{m-1}$ and $\{v_k\}_{k=0}^{m-1}$, respectively, where ({\color{green60}$\bullet$}) is the first and ({\color{orange90}$\cross$}) the second component.}}
	\label{fig:sim3}
\end{figure*}

\subsubsection*{Simulation 4.}
\label{sec:sim4}

\corrected{
In this simulation scenario we consider the cost function $J(x) = \|x-[1\ 2]^\top + 0.5\sin(10\pi x)\|^2$, i.e., an objective with many local minima. We define $f(J(x))=\sin(J(x)),\ g(J(x)) = \cos(J(x)),\ [\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}],\ h=0.05$, and $x_0 = [1\ 2]^\top$. The simulation results with $T_d$ as in Simulation 2 and $\sigma_1= \sigma_2 =  1$ are depicted in \Cref{fig:sim4}. As motivated in the introduction, the proposed algorithm is able to overcome local minima and converges into a neighborhood of the global minimum. Due to the definition of the maps $M_k^{\sqrt{h}}$, the gradient of $J$ is gained by a procedure similar to numerical integration. It has been observed in simulations that this procedure is numerically more stable than numerical differentiation (finite differences).
In summary, this integrating behavior has the effect to even and flatten out local minima and noise respectively (cf. \cite{wildhagen2018characterizing}).}

\subsubsection*{Simulation 5.}
\label{sec:sim5}

\corrected{One of the key advantages of derivative-free optimization is demonstrated, namely to deal with non-smooth objectives. To this end, in its ability to a simulation experiment, the cost function is set to $J(x) = \|x\|$. We use the same algorithm parameters as in Simulation 2, i.e., $f(J(x))=\sin(J(x)),\ g(J(x)) = \cos(J(x)),\ [\alpha_1\ \alpha_2] = [\nicefrac{1}{2}\ \nicefrac{1}{2}],\ h=0.05$, and $x_0 = [1\ 2]^\top$ with $T_d$ as in \eqref{eq:cases_H2}, $Q$ specified in \cref{thm:min_steps}, and singular values $\sigma_1 = \sigma_2 =  0.4$. The algorithm's behavior is depicted in \cref{fig:sim5}. Since no first-order information (derivatives) needs to be competed, the algorithm is, as expected, converging to the local minima.}

\begin{rem}
	It is worth to mention that a very promising generating function class is

	\begin{align}
	f(z) = \sqrt{z}	\sin(\text{ln}(z)\mu),\ g(z) = \sqrt{z}	\cos(\text{ln}(z)\mu)
	\end{align}

	with $\mu \in \mathbb{R}_{>0}$, which belong to the setting of \eqref{eq:cases_H1} and \eqref{eq:cases_E1} (with some adaptions). Specifically, the Lie bracket between the generating functions results in $[f,g](z) = -\mu$, i.e., it holds $x_{m+k} = x_k - h\mu\nabla J(x_k) + \mathcal{O}(h^{3/2})$. In this view, $\mu$ can be chosen large and $h$ small, hence, a large enough gradient step is executed while the oscillations can be kept small.

\end{rem}

config: arxiv


TITLE: Testing divisibility by 7 of number in base 8
QUESTION [0 upvotes]: Is the number 3776543210123456773, in base 8, divisible by 7?
I started by expanding the number:
3 * $8^0$ + 7 * $8^1$ + 7 * $8^2$ + ...
3 * $(7 + 1)^0$ + 7 * $(7 + 1)^1$ + 7 * $(7 + 1)^2$ + ...
Then, by using Newton's Binomial Theorem, I reasoned that for each binomial, only the first term wouldn't be divisible by 7 since it would be equal to 1. This led me to the conclusion that each digit of the initial number would be multiplied by one once. Therefore, if the sum of all digits was divisible by 7, the number would also be divisible by 7.
However, I converted this number to base 10 and the remainder of the division by 7 was equal to 2, instead of 0, which was the result I found using the reasoning above.
I can't seem to find where I made a mistake. :c

REPLY [3 votes]: Your reasoning is correct. But the sum of the digits of the digits of your number is $76$. And the remainder of the division of $76$ by $7$ is $6$, of course. So, the remainder of the division of your number by $7$ is $6$ too.
And, in base $10$, your number is $72\,011\,638\,983\,515\,643$, whose remainder, when divided by $7$, is $6$ too.


TITLE: In how many ways can we distribute 10 identical ice creams to 4 people so one of them gets a maximum of 4 ice cream?
QUESTION [0 upvotes]: I know this formula should be used but I do not know how to apply the maximum part $\binom{n+r−1}{r−1}$
Edit:
Possible answers are: $\binom{8}{5}$,
$\binom{13}{10}$-$\binom{8}{5}$,$\binom{10}{4}$-$\binom{7}{4}$,$\binom{13}{4}$-$\binom{5}{4}$ or $\binom{10}{4}$-$\binom{6}{3}$
Edit2: since there are identical ice creams, some people may get 0 while one may even get 10. But one of them can get MAXIMUM 4.

REPLY [1 votes]: I believe using principle of inclusion and exclusion (PIE) is the easiest.
if nobody gets more than 4 ice creams:
$$
\begin{aligned}
\sum_{i=0}^{\left\lfloor\frac{10}{5}\right\rfloor}{\binom{4}{i}\binom{13-5i}{3}\left(
-1\right)^{i}}&=\binom{4}{0}\binom{13}{3}-\binom{4}{1}\binom{8}{3}+\binom{4}{2}\binom{3}{3}\\
&=68
\end{aligned}
$$
if only one particular person cannot get more than 4 ice creams:
Say that person A cannot have more than 4. We want to count the number of distribution subtracted by number of distribution in which A get more than 4.
$$
\binom{13}{3}-\binom{8}{3}=230
$$
here, $\binom{8}{3}$ means we distribute 5 candies to 4 people then add 5 candies to person A. this way he always get more than 4.
You may want to read about this PIE, a very powerful tools in combinatorics.

REPLY [0 votes]: Since there are $4$ people and $10$ ice creams, someone must get at least $3$, so the maximum anyone gets is either $3$ or $4$.  The possible patterns when the maximum is $3$ are $$3,3,3,1\\3,3,2,2$$  The possible patterns when the maximum is $4$ are $$4,4,2,0\\4,4,1,1\\4,3,3,0\\4,3,2,1\\4,2,2,2$$
Now you just have to figure out how many ways each of these patterns can arise and add them up.
Usually, there are too many cases fo this to be a practical procedure, but I think it's the best way for this problem.


TITLE: Smoothness and my typing speed
QUESTION [1 upvotes]: I practice typing on a site called keybr.com. I get statistics of my typing speed in graph format as well.

Here, I guess the straight green line is a line fitted to my typing speed data. There is some function called 'Smoothness'. From Wikipedia, I see that smoothing filters noise and modifies some data. I see that as I change the value of smoothness, the line changes its slope and my typing speed apparently decreases when smoothness is high.
Can anyone give an intuitive explanation of why does this happen?
What is the correct/optimum amount of smoothness so that I get the truest trend of my typing speed over time? According to the graph, does my typing speed actually increase or decrease with time?

REPLY [1 votes]: The original plot is very noisy. If you remove the green best-fit line, then the connection between typing speed and time is not very interpretable.
The idea behind smoothing is to fit the data to a (potentially non-linear) curve that best represents your typing speed over time. However, this requires adjusting the data and what you see is just a smooth approximation of the data. In general, there is no optimal amount of smoothing, it is usually subjective (to make your data more interpretable)!
In your case, adding too much smoothness resulted in a curve that didn't resemble the initial trend very well. There is a pretty clear upward trend with 0 smoothness (as shown by the green linear regression line). In my opinion, I think the straight green line tells you what you need to know about how your typing speed has changed over time. Ultimately, it is all about whether you value interpretability or accuracy.


TITLE: A question about relative homology group
QUESTION [0 upvotes]: I want to find two space pairs $(X,A)$ and $(Y,B)$ such that $X$ is homotopic to $Y$ and $A$ is homotopic to $B$, but their relative homology group is not isomorphic, that is
$$
H_*(X,A)\neq H_*(Y,B)
$$

REPLY [1 votes]: Hint use reduced homology ie. take $A=\{x_1\}$, $B=\{x_2\}$ in distinct connected components of a space space $X=Y$, such that the two connected components $C_1, C_2$ have different homologies.


text:
\begin{document}
\date{}
\maketitle
\begin{abstract}
We study some stronger forms of sensitivity, namely, $\mathcal{F}$-sensitivity and weakly$\mathcal{F}$-sensitivity for non-autonomous discrete dynamical systems. We obtain a condition under which these two forms of sensitivity are equivalent. We also justify the difference between $\mathcal{F}$-sensitivity and some other stronger forms of sensitivity through examples. We explore the relation between the $\mathcal{F}$-sensitivity of the non-autonomous system $(X,f_{1,\infty})$ and autonomous system  $(X,f)$, where $f_n$ is a sequence of continuous functions converging uniformly to $f$. We also study the $\mathcal{F}$-sensitivity of a non-autonomous system $(X,f_{1,\infty})$, generated by a finite family of maps $\mathbb{F}=\{f_1, f_2, \ldots, f_k\}$ and give an example showing that such non-autonomous systems can be $\mathcal{F}$-sensitive, even when none of the maps in the family $\mathbb{F}$ is $\mathcal{F}$-sensitive.
\end{abstract}
\keywords{Non-autonomous dynamical system, Furstenberg Families,  $\mathcal{F}$-sensitive, weakly $\mathcal{F}$-sensitive}
\\ \msc{Primary 54H20; Secondary 37B55}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnotetext{\hspace*{-5mm}
\renewcommand{\arraystretch}{1}
\begin{tabular}{@{}r@{}p{15cm}@{}}
$^\dag$& the corresponding author. Email address: radhika.vasisht92@gmail.com (R. Vasisht)\\
$^1$&Department of Mathematics, University of Delhi, Delhi-110007, India\\
\end{tabular}}

\section{Introduction}
Beginning with the contributions of Poincar\'e and Lyapunov, theory of dynamical systems has seen significant developments in recent years. This theory has gained considerable interest and has been found to have useful connections with many different areas of mathematics. An autonomous discrete dynamical system is a dynamical system which has no external input and always evolves according to the same unchanging law. Most of the natural systems in this world are subjected to time-dependent external forces and their modelling leads to a mathematical theory of what is called non-autonomous discrete dynamical system.  The theory of non-autonomous dynamical systems helps characterizing the behaviour of various natural phenomenons which cannot be modelled by autonomous systems. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamic systems \cite{MR2960260, MR3206430, MR3344142, MR3485444}. We first introduce some notations. Consider the following non-autonomous discrete dynamical system (N.D.S) $(X,f_{1,\infty})$:
\begin{equation} x_{n+1}=f_n(x_n) , n \geq 1,\nonumber \end{equation}
where $(X,d)$ is a compact metric space and $f_n: X \rightarrow X$ is a continuous map. For convenience, we denote $({f_n})_{n =1}^\infty $ by $f_{1,\infty}$. Naturally, a difference equation of the form $x_{n+1}=f_n(x_n)$ can be thought of as the discrete analogue of a non-autonomous differential equation $\frac{dx}{dt}=f(x,t)$.
\\ Chaos is a universal dynamical behavior of nonlinear dynamical systems and one of the central topics of research in nonlinear science. It is well known that sensitivity characterizes the unpredictability of chaotic phenomena, and is one of the essential conditions in various definitions of chaos. Therefore, the study on sensitivity has attracted a lot of attention from many researchers. For continuous self maps of compact metric spaces, Moothathu \cite{MR2351026} gave an insight of some stronger forms of sensitivity based on the largeness of subsets of $\mathbb{N}$. Since then several other stronger forms of sensitivity have been studied by different researchers for both autonomous and non-autonomous systems\cite{MR2922208, MR3528201}.
\\ In \cite{MR2600324}, the authors introduced and studied the concept of sensitivity via Furstenberg families for autonomous systems. A lot of work has been done in this direction since then \cite{MR3339062, MR3543466}. Recently, some researchers have introduced and studied various versions of sensitivity via Furstenberg families for non-autonomous discrete systems \cite{MR3623040, MR3584037}. Motivated by all the research done for sensitivity via Furstenberg families, we study $\mathcal{F}$-sensitivity and weakly $\mathcal{F}$-sensitivity for non-autonomous discrete dynamical systems.
\\In section 2, we recall some concepts about Furstenberg families and also give preliminaries required for other sections.
In section 3, we study the relations between $\mathcal{F}$-sensitivity and some other stronger forms of sensitivity . We obtain a condition under which $\mathcal{F}$-sensitivity and weakly $\mathcal{F}$-sensitivity for the system $(X,f_{1,\infty})$ are equivalent. We give some characterizations of $\mathcal{F}$-sensitivity. In section 4, we investigate $\mathcal{F}$-sensitivity and weakly $\mathcal{F}$-sensitivity for different non-autonomous systems. We explore the relation between the $\mathcal{F}$-sensitivity of the non-autonomous system $(X,f_{1,\infty})$ and autonomous system  $(X,f)$, where $f_n$ is a sequence of continuous functions converging uniformly to $f$. We also study the $\mathcal{F}$-sensitivity of a non-autonomous system $(X,f_{1,\infty})$, generated by a finite family of maps $\mathbb{F}=\{f_1, f_2, \ldots, f_k\}$ and give an example showing that such non-autonomous systems can be $\mathcal{F}$-sensitive, even when none of the maps in the family $\mathbb{F}$ is $\mathcal{F}$-sensitive.

\section{Preliminaries}

Let $(X,f_{1,\infty})$ be a non-autonomous dynamical system. For any two open sets U and V of $X$, let$N_{f_{1,\infty}}(U,V)= \{n\in \mathbb{N} : f_1^n(U)\cap V\ne \emptyset \}$. In particular, $N_{f_{1,\infty}}(x,V)= \{n\in \mathbb{N} : f_1^n(x)\in V\}$, where $f_1^i(x) = f_{i}\circ\cdots\circ f_1(x)$.

\begin{defn}
A system $(X,f_{1,\infty})$ is said to $\textit{feeble open}$ if for any non-empty open set $U$ in $X$, $int(f_{n}(U))$ is non-empty for each $n\in\mathbb{N}$.
\end{defn}

\begin{defn}
The system $(X,f_{1,\infty})$ is said to have $\textit {sensitive dependence on initial}$ $ \textit{conditions}$ if there exists a constant $ \delta_0>0$ such that for any $x_0 \in X$ and any neighbourhood U of $x_0$, there exists $y_0\in X\cap U$ and a positive integer n such that $d(x_n,y_n)>\delta_0$, where $\{x_i\}_{i=0}^\infty$ and  $\{y_i\}_{i=0}^\infty$ are the orbits of the system $(X,f_{1,\infty})$ starting from $x_0$ and $y_0$ respectively, the constant $\delta_0>0$ is called a sensitivity constant of the system $(X,f_{1,\infty})$.
Here  $({x_i})_{i=0}^\infty = \{x\in X$ such that $f_1^i(x) , i\geq 1\}$.
Thus, system $(X,f_{1,\infty})$ has sensitive dependence on initial conditions or is $\textit{sensitive}$ in $X$ if there exists a constant $\delta>0$ such that for any non-empty open set V of $X$, $N_{f_{1,\infty}}(V,\delta)$ is non-empty, where $N_{f_{1,\infty}}(V,\delta) = \{n\in \mathbb{N}$ such that, there exist   $x,y \in V$ with $ d(f_1^n(x),f_1^n(y))> \delta \}$.
\end{defn}

\begin{defn}
The system $(X,f_{1,\infty})$ is called $\textit{cofinitely sensitive}$ in $X$ if there exists a constant $\delta>0$ such that for any non-empty open set V of $X$,there exists $N\geq1$ such that $[N,\infty)\cap\mathbb{N} \subset N_{f_{1,\infty}}(V,\delta)$; $\delta$ is called a constant of cofinite sensitivity.
\end{defn}

\begin{defn}
The system $(X,f_{1,\infty})$ is said to have $\textit{syndetic sensitivity}$ in $X$ if there exists a constant $\delta>0$ such that for any non-empty open set V of $X$, $N_{f_{1,\infty}}(V,\delta)$ is syndetic; $\delta$ is called a constant of syndetic sensitivity.
\end{defn}
Now, we recall some concepts related to Furstenberg families.
Let $\mathcal{P}$ be the collection of all subsets of $\mathbb{Z^+}$. A collection $\mathcal{F}\subseteq \mathcal{P}$ is called a $\mathit{Furstenberg\hspace{0.15cm} family}$, if it is hereditary upwards, that is, $F_1 \subset F_2$ and $F_1\in \mathcal{F}$ implies $F_2\in \mathcal{F}$. A family $\mathcal{F}$ is proper if it is a proper subset of $\mathcal{P}$, that is, it is neither whole $\mathcal{P}$ nor empty. Throughout this paper, all Furstenberg families are proper.
\\For a Furstenberg family $\mathcal{F}$, the dual family\begin{align*}
\mathit{k} \mathcal{F} = \{F\in \mathcal{P}:F\cap F'\neq  \emptyset, for\hspace{0.15cm} all\hspace{0.15cm} F'\in \mathcal{F}\}=\{F\in \mathcal{P}: \mathbb{Z^+}\setminus F\notin \mathcal{F}\}
\end{align*}
Clearly, if $\mathcal{F}$ is a Furstenberg family, then so is $\mathit{k}\mathcal{F}$. One can note that $\mathit{k}(\mathit{k}\mathcal{F})=\mathcal{F}$. Note that the family $\mathcal{B}$ of all infinite subsets of $\mathbb{Z^+}$ is a Furstenberg family and $\mathit{k}\mathcal{B}$ is the family of all cofinite subsets of $\mathbb{Z^+}$.
\\ For Furstenberg families $\mathcal{F}_1$ and $\mathcal{F}_2$, let $\mathcal{F}_1.\mathcal{F}_2=\{F_1 \cap F_2: F_1 \in \mathcal{F}_1, F_2 \in \mathcal{F}_2\}$. A Furstenberg family $\mathcal{F}$ is said to be filterdual if $\mathcal{F}$ is proper and $\mathit{k}\mathcal{F}.\mathit{k}\mathcal{F}\subseteq \mathit{k}\mathcal{F}$. A Furstenberg family $\mathcal{F}$ is said to be translation invariant if for any $F\in \mathcal{F}$ and any $i\in \mathbb{Z^+}, \hspace{0.15cm} F+i \in \mathcal{F}$ and $F-i\in \mathcal{F}$.
\\Let $A\subseteq X$ and $x\in X$. If $N_{f_{1,\infty}}(x,A)\in \mathcal{F}$, then $x$ is called an $\mathcal{F}$-attaching point of $A$. The set of all $\mathcal{F}$-attaching points of $A$ is called the $\mathcal{F}$-attaching set of $A$ and is denoted by $\mathcal{F}(A)$. Clearly, \begin{align*}
\mathcal{F}(A)= \bigcup_{F\in\mathcal{F}}\bigcap_{n\in\mathcal{F}}f_1^n(A).
\end{align*}
Let $(X,f_{1,\infty})$ be a N.D.S. A Furstenberg family $\mathcal{F}$ is compatible with the system $(X,f_{1,\infty})$ if the $\mathcal{F}$-attaching set of $U$ is a $G_{\delta}$ set of $X$, for each open set $U$ of $X$.
\\Throughout this paper, we will use the following relations on $X$. For a given $\delta>0$:
\begin{align*}
 V_{\delta}=\{(x,y): d(x,y)< \delta\},
\end{align*}
\begin{align*}
\overline{V_{\delta}} = \{(x,y): d(x,y)\leq \delta \}
\end{align*}

\begin{defn}
The set of $\mathcal{F}-\textit{asymptotic pairs}$, $Asym_{\delta}(\mathcal{F})=\{(x,y): N_{f_{1,\infty}}((x,y),\overline{V_{\delta}}\in \mathit{k}\mathcal{F})\}= \mathit{k}\mathcal{F}(\overline{V_{\delta}})$ and $Asym_{\delta}(\mathcal{F})(x)=\{y: (x,y)\in Asym_{\delta}(\mathcal{F})\}$.
\\Clearly, $Asym(\mathcal{F})=\bigcap_{\delta>0}Asym_{\delta}(\mathcal{F})$ and $Asym(\mathcal{F})(x)=\bigcap_{\delta>0}Asym_{\delta}(\mathcal{F})(x)$.
\end{defn}

\begin{defn}
The system $(X,f_{1,\infty})$ is $\textit{weakly}\hspace{0.15cm} \mathcal{F}-\textit{sensitive}$ if there exists a positive $\delta>0$, such that for every open subset $U$ of $X$, there exists $x,y \in U$ such that the pair $(x,y)$ is not $\mathcal{F}$-$\delta$-asymptotic, that is, $\{n\in \mathbb{N}: d(f_1^n(x),f_1^n(y))>\delta\}\in \mathcal{F}$ or we can say $N_{f_{1,\infty}}((x,y),X\times X\setminus \overline{V_{\delta}})\in \mathcal{F}$,  where $\delta$ is called a weakly $\mathcal{F}$-sensitive constant.
\end{defn}

\begin{defn}
The system $(X,f_{1,\infty})$ is $\mathcal{F}- \textit{sensitive}$ if there exists a positive $\delta>0$, such that for every $x\in X$ and every open neighbourhood $U$ of $x$, there exists $y\in U$ such that the pair $(x,y)$ is not $\mathcal{F}$-$\delta$-asymptotic, that is, the set $N_{f_{1,\infty}}(U,\delta)\in \mathcal{F}$,  where $\delta>0$ is an $\mathcal{F}$-sensitive constant.
\end{defn}

Let $X$ be a topological space and $\pazocal{K}$($X)$ denote the hyperspace of all non-empty compact subsets of $X$ endowed with the \textit{Vietoris Topology}. A basis of open sets for Vietoris topology is given by following sets:
\bigskip

\noindent $< U_1, U_2, \ldots , U_k >$ = $\{K \in$ $\pazocal{K}$($X)$: $K \subset \bigcup_{i=1}^{k} U_{i}$ and $K\cap U_{i}$  $\ne \emptyset$, for each $i$ $\in \{1, 2, \ldots ,k\}$\},

\bigskip\noindent where $U_1, U_2, \ldots ,U_k$ are non-empty open subsets of $X$.

Given metric space $(X, d)$, a point $x \in X$ and $A \in \pazocal{K}(X)$, let $d(x, A)$ = $\inf \{d(x, a): a \in A \}$. For every $\epsilon > 0$, let  open $d$-ball in $X$ about $A$ and radius $\epsilon$ be given by  $B_{d}(A, \epsilon) = \{x \in X: d(x, A) < \epsilon\} = \bigcup_{a \in A} B_d(a, \epsilon)$, where $B_d(a, \epsilon)$ denotes the open ball in $X$ centred at $a$ and of radius $\epsilon$. The Hausdorff metric on $\pazocal{K}$($X)$  induced by $d$, denoted by $d_H$, is defined  as follows:

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $d_H$($A, B) = \inf \{ \epsilon > 0: A \subseteq B_{d}(B, \epsilon) \  \text{and} \  B \subseteq B_{d}(A, \epsilon)\}$,

\bigskip\noindent where $A$, $B \in$ $\pazocal{K}$($X)$. We shall recall that the topology induced by the Hausdorff metric coincides with the Vietoris topology if and only if the space $X$ is compact. Also, for a compact metric space $X$ and  $A, B \in$ $\pazocal{K}$($X)$, we get that $d_H$($A, B) < \epsilon$ if and only if $A \subseteq B_d(B, \epsilon)$ and $B \subseteq B_d(A, \epsilon)$.
\\Let $\pazocal{F}(X)$ denote the set of all finite subsets of $X$. Under Vietoris topology, $\pazocal{F}(X)$ is dense in $\pazocal{K}(X)$ \cite{MR2665229, MR1269778}.
 Given a continuous function $f: X \to X$, it induces a continuous function $\overline{f}$: $\pazocal{K}$($X) \to$ $\pazocal{K}$($X)$ defined by $\overline{f}(K) = f(K)$, for every $K \in$ $\pazocal{K}$($X)$, where $f(K)$ = $\{f(k) : k\in K\}$. Note that continuity of $f$ implies continuity of $\overline{f}$.

\bigskip Let $(X, f_{1,\infty})$ be a non-autonomous discrete dynamical system and $\overline{f}_n$ be the function on $\pazocal{K}$($X)$, induced by $f_n$ on $X$, for every $n\in\mathbb{N}$. Then the sequence $\overline{f}_{1,\infty}$ = ($\overline{f}_1, \overline{f}_2$, $\ldots ,\overline{f}_n, \ldots )$ induces a non-autonomous discrete dynamical system ($\pazocal{K}$($X), \overline{f}_{1,\infty})$, where $\overline{f}_1^n = \overline{f}_n \circ \ldots \circ \overline{f}_2\circ \overline{f}_1$.  Note that $\overline{f}_1^n = \overline{f^n_1}$.
\\Let $(X,d_X)$ and $(Y,d_Y)$ be compact metric spaces. For non-autonomous discrete dynamical systems $(X,f_{1,\infty})$ and $(Y,g_{1,\infty})$, put $(f_{1,\infty}\times g_{1,\infty})= (h_{1,\infty}) = (h_1,h_2,\ldots,h_n,\ldots)$, where $h_n=f_n\times g_n$ , for each $n\in \mathbb{N}$. Thus, $(X\times Y, f_{1,\infty}\times g_{1,\infty})$ is a non-autonomous dynamical system, where $(X\times Y)$ is a compact metric space endowed with the product metric $d_{X\times Y}((x,y),(x',y'))= d_X(x,x')+d_Y(y,y')$. Here, $h_{1}^{n}=h_n\circ h_{n-1}\circ \cdots \circ h_2\circ h_1 = (f_n\times g_n)\circ (f_{n-1}\times g_{n-1})\circ \cdots \circ (f_2\times g_2)\circ (f_1\times g_1)$ \cite{MR3584171}.
\\Let $(X,d)$ be a compact metric space and let $C(X)$ denote the collection of continuous self-maps on $X$. For any $f,g\in C(X)$, the \textit{Supremum metric} is defined by \\$D(f,g)=\underset{x\in X}{sup}$ $d(f(x),g(x))$. It is easy to observe that a sequence $(f_n)$ in $C(X)$ converges to $f$ in $(C(X),D)$ if and only if $f_n$ converges to $f$ uniformly on $X$ and hence the topology generated by the \textit{Supremum metric} is called the topology of uniform convergence.
\\ The $k$-th iterate is considered as $(X, f_{1,\infty}^{[k]})$, where $f_{1,\infty}^{[k]} = (f_{k(n-1)+ 1})_{n=1}^{\infty})$ and $f_{k(n-1)+ 1} = f_{kn-k+1+k-1}\circ f_{kn-1}\circ\cdots\circ f_{kn-k+1}$.
\\We recall the following results:
\begin{lem}[Corollary 2.2, \cite{MR3019971}]
Assume that the non-autonomous system $(X,f_{1,\infty})$ converges uniformly to a map $f$. Then for any $\epsilon>0$ and any $k\in \mathbb{N}$, there exists $\eta(\epsilon)>0$ and $N(k)\in \mathbb{N}$ such that for any pair $x,y \in X$ with $d(x,y)<\eta(\epsilon)$ and any $n\geq N(k)$, $d(f_n^k(x),f_n^k(y))<\epsilon/2$.
\end{lem}

\begin{lem}[Corollary 1, \cite{MR2}]
Let $(X,f_{1,\infty})$ be a N.D.S generated by a family $f_{1,\infty}$ and let $f$ be any continuous self map on $X$. If the family $f_{1,\infty}$ commutes with $f$ then for any $x\in X$ and any $k\in\mathbb{N}$, $d(f_{1}^{n+k}(x),f^{k}(f_{1}^{n}(x))) \leq \sum_{i=1}^k D(f_{i+1},f)$.
\end{lem}

\section{Relations of $\mathcal{F}$-sensitivity with some other stronger forms of sensitivity.}
In this section, we present our results about $\mathcal{F}$-sensitivity and weakly$\mathcal{F}$-sensitivity of the non-autonomous systems and discuss a condition under which these two are equivalent. We also give an example to show that $\mathcal{F}$-sensitivity need not imply syndetic sensitivity and cofinite sensitivity.
\begin{thm}
Let $(X,f_{1,\infty})$ be a non-autonomous system and $\mathcal{F}$ be a filterdual. Then $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive if and only if it is $\mathcal{F}$-sensitive.
\end{thm}

\begin{proof}
First suppose that $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive, with constant of $\mathcal{F}$-sensitivity $\delta>0$. Let $U$ be any open set in $X$. For any $x\in U$, since $U$ is an open neighbourhood of $x$, by $\mathcal{F}$-sensitivity of $(X,f_{1,\infty})$, there exists a $y\in U$ such that $(x,y)$ is not $\mathcal{F}-\delta$-asymptotic. Therefore, $U$ contains a pair $(x,y)$ which is not $\mathcal{F}-\delta$-asymptotic and hence $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive.
\\Conversely, suppose that $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive, with constant of weakly$\mathcal{F}$-sensitivity $\delta>0$. If $(X,f_{1,\infty})$ is not $\mathcal{F}$-sensitive, then for each $\delta>0$, there exist $x\in X$ and an open neighbourhood $U$ of $x$ such that $\{n\in\mathbb{N}: d(f_1^n(x),f_1^n(y))>\delta\} \notin \mathcal{F}$, for every $y\in X$. Therefore, $N_{f_{1,\infty}}((x,y),X\times X \setminus \overline{V_{\delta}})\notin \mathcal{F}$, for every $y\in X$ and hence
$\mathbb{N}\setminus N_{f_{1,\infty}}((x,y),\overline{V_{\delta}})\notin \mathcal{F}$ which implies $
N_{f_{1,\infty}}((x,y),\overline{V_{\delta}})\in \mathit{k}\mathcal{F}$.
Since $\mathcal{F}$ is filterdual, using triangle inequality, we get that $N_{f_{1,\infty}}((a,b),\overline{V_{2\delta}})\in \mathit{k}\mathcal{F}$. Therefore, $\mathbb{N}\setminus N_{f,\infty}((a,b),X\times X\setminus\overline{V_{2\delta}})\in \mathit{k}\mathcal{F}$ and hence $N_{f_{1,\infty}}((x,y),X\times X\setminus \overline{V_{2\delta}})\notin \mathcal{F}$ which contradicts that $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive.
\end{proof}

\begin{rmk}
If $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive, then $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive, even when $\mathcal{F}$ is not filterdual.
\end{rmk}

\begin{lem}
Let $(X,f_{1,\infty})$ be a N.D.S. A Furstenberg family $\mathcal{F}$ is compatible with system $(X,f_{1,\infty})$ if and only if the $\mathit{k}\mathcal{F}$-attaching set of V is a $F_{\sigma}$ set of $X$ for each closed subset $V$ of $X$.
\end{lem}

\begin{proof}
Suppose $V$ is a closed subset of $X$, then $x\in\mathit{k}\mathcal{F}$
\begin{align*} & \iff N_{f_{1,\infty}}(x,V)\in \mathit{k}\mathcal{F} \\
& \iff \mathbb{N} \setminus N_{f_{1,\infty}}(x,X\setminus V) \in \mathit{k}\mathcal{F} \\
& \iff N_{f_{1,\infty}}(x,X\setminus V) \notin \mathcal{F} \\
& \iff x \notin \mathcal{F}(X\setminus V). \end{align*} Therefore, we have that $\mathit{k}\mathcal{F}(V)=
X\setminus \mathcal{F}(X\setminus V)$. Hence, $\mathit{k}\mathcal{F}(V)$ is an $F_{\sigma}$-set if and only if $\mathcal{F}(X\setminus V)$ is a $G_{\delta}$-set which is true by definition.

\end{proof}

The following result gives characterizations for the $\mathcal{F}$-sensitivity of non-autonomous system $(X,f_{1,\infty})$.
\begin{thm}
Let $(X,f_{1,\infty})$ be a N.D.S. If the Furstenberg family $\mathcal{F}$ is a filterdual and is compatible with the system $(X\times X, f_{1,\infty}\times f_{1,\infty})$, then the following are equivalent.
\begin{enumerate}
\item $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive.
\item There exists a positive $\delta$ such that for every $x\in X$, $Asym_{\delta}(\mathcal{F})(x)$ is a first category subset of $X$.
\item There exists a positive $\delta$ such that $Asym_{\delta}(\mathcal{F})$ is a first category subset of $X\times X$.
\item There exists a positive $\delta$ such that $\mathcal{F}(X\times X \setminus \overline{V_{\delta}})$ is dense in $X\times X$.
\item $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive.
\end{enumerate}
\end{thm}

\begin{proof}
(1)$\implies$(2) We have that $\mathcal{F}$ is compatible with the system $(X\times X, f_{1,\infty}\times f_{1,\infty})$, so by Lemma 3.1, $Asym_{\delta}(\mathcal{F})$ is an $F_{\sigma}$ set of $X\times X$.

Suppose $Asym_{\delta}(\mathcal{F})=\bigcup_{i=1}^{\infty}A_n$, where every $A_i$ is a closed subset of $X\times X$, then $Asym_{\delta}(\mathcal{F})(x)=\bigcup_{i=1}^{\infty}A_n(x)$.
Suppose that for each $\delta>0$, there exists $x\in X$ such that $Asym_{\delta}(\mathcal{F})(x)$ is not of first category, then by Baire's Category Theorem there exists an open subset $U$ of $X$ such that $U\subset A_i(x)$, for some $i$. Hence, for each $y\in U$, $N_{f_{1,\infty}}((x,y),\overline{V_{\delta}})\in \mathit{k}\mathcal{F}$ and since $\mathcal{F}$ is filterdual, using triangle inequality we get that $N_{f_{1,\infty}}((a,b),\overline{V_{2\delta}})\in \mathit{k}\mathcal{F}$, for all $a,b \in U$. Then, $\mathbb{N}\setminus  N_{f_{1,\infty}}((a,b),X\times X\setminus  \overline{V_{2\delta}})\in \mathit{k}\mathcal{F}$. Therefore $N_{f_{1,\infty}}((a,b),X\times X\setminus \overline{V_{2\delta}})\notin \mathcal{F}$, which contradicts that $(X,f_{1,\infty})$ is weakly $\mathcal{F}$-sensitive.\\
(2)$\implies$(3) We know that $Asym_{\delta}(\mathcal{F})=\mathit{k}\mathcal{F}(\overline{V_{\delta}})$, therefore by Lemma 3.1, we get that $Asym_{\delta}({\mathcal{F}})$ is an $F_{\sigma}$ set. Suppose that $Asym_{\delta}(\mathcal{F})=\bigcup_{i=1}^{\infty}A_i$, where each $A_i$ is a closed subset of $X\times X$. Then, $Asym_{\delta}(\mathcal{F})(x)=\bigcup_{i=1}^{\infty}A_i(x)$. If $Asym_{\delta}(\mathcal{F})$ is not first category, then by Baire's Category Theorem, some $A_i$ has non-empty interior. If $U\times V \subset A_i$ and $x\in U$, then $V\subset A_i$. Hence, $Asym_{\delta}(\mathcal{F})(x)$ is not first category.\\
(3)$\implies$(1) If $(X,f_{1,\infty})$ is not weakly $\mathcal{F}$-sensitive, then for every $\delta>0$, there exists an open set $U$ in $X$ such that $N_{f_{1,\infty}}((x,y),X\times X \setminus \overline{V_{\delta}})\notin \mathcal{F}$, for each $(x,y)\in U\times U$, that is, $\mathbb{N}\setminus N_{f_{1,\infty}}((x,y),\overline{V_{\delta}})\notin \mathcal{F}$. Therefore, $N((x,y),\overline{V_{\delta}})\in \mathit{k}\mathcal{F}$ which implies $(x,y)\in Asym_{\delta}(\mathcal{F})$. Hence, $U\times U\subset Asym_{\delta}(\mathcal{F})$. Thus, $Asym_{\delta}(\mathcal{F})$ is not of first category.\\
\\Thus, we have proved that (1), (2) and (3) are equivalent.
\\(2)$\implies$(4) We first observe that $\mathcal{F}(X\times X\setminus \overline{V_{\delta}})=X\times X\setminus\mathit{k}\mathcal{F}(\overline{V_{\delta}})=X\times X\setminus Asym_{\delta}(\mathcal{F})$. Since, complement of a first category set is dense in a Baire's Space, we get the required result.
\\(4)$\implies$(1) Let $U$ be any open set in $X$. Since $\mathcal{F}(X\times X\setminus \overline{V_{\delta}})$ is dense in $X\times X$, therefore we have $U\times U\bigcap  \mathcal{F}(X\times X\setminus \overline{V_{\delta}})$ is non-empty which implies $U\times U\bigcap X\times X\setminus  Asym_{\delta}(\mathcal{F})$ is non-empty. Thus, there exists $(x,y)\in U\times U$ such that $(x,y)\notin Asym_{\delta}(\mathcal{F})$. Hence, $(X,f_{1,\infty})$is weakly $\mathcal{F}$-sensitive.
\\(1)$\iff$(5) follows directly from Theorem 3.1.
\end{proof}

The following example justifies that an $\mathcal{F}$-sensitive non-autonomous system need not be syndetic sensitive and cofinite sensitive.
\begin{exm} \end{exm}
Let $\mathbb{Z}$ denote the set of all integers, $\cal{A}$ = $\{0,1\}$ and $\Sigma_2$=$\cal{A}^\mathbb{Z}$ = $\{( \ldots, x_{-2}, x_{-1}$, \fbox{$x_0$}, $x_1, \ldots): x_i \in \{0,1\}, \ \text{for every} \  i  \in \mathbb{Z}\}$ with product metric  \[\rho(x,y) = \sum_{j=-\infty}^{\infty}\frac{|x_j-y_j|}{2^{|j|}} \]
for any pair $ x= ( \ldots, x_{-2}, x_{-1}$, \fbox{$x_0$}, $ x_1, \ldots)$;  $y = ( \ldots, y_{-2}, y_{-1}$, \fbox{$y_0$}, $ y_1, \ldots) \in \Sigma_2$. The space $(\Sigma_2, \rho)$ is called the \emph{the two sided symbolic space}. Define $\sigma: \Sigma_2 \to \Sigma_2$ by $\sigma(x) = ( \ldots, x_{-2}, x_{-1}, x_0, \fbox{$x_1$}, x_2, \ldots)$, where $x = ( \ldots, x_{-2}, x_{-1}, $ \fbox{$x_0$}, $ x_1, x_2 \ldots) \in \Sigma_2$, then $\sigma$ is a homeomorphism and is called the \emph{shift map} on $\Sigma_2$.
\\Consider the non-autonomous system $(\Sigma_2, f_{1,\infty})$, where $f_{1, \infty}$  given by
\begin{align} f_{1,\infty} :=  \{ I, I, \sigma, \sigma^{-1}, I, I, I, I, \sigma^{2}, \sigma^{-2}, \ldots,  \underbrace{I, I, \ldots, I}_{\text{$(2)^{r}$-times}},  \sigma^{r}, \sigma^{-r}, \ldots \}  \nonumber\end{align}
Let $\mathcal{F}$ be the Furstenberg family of all infinite subsets of $\mathbb{Z^+}$. For any $x\in \Sigma_2$, and any open neighbourhood $U$ of $x$, we will get a $y\in U$ such that $(x,y)$ is not an $\mathcal{F}$-asymptotic pair for $(\Sigma_2, f_{1,\infty})$.
Therefore, $(\Sigma_2, f_{1,\infty})$ is $\mathcal{F}$-sensitive.
\\Note that in the above example, the non-autonomous system $(\Sigma_2, f_{1,\infty})$ is not cofinite sensitive because for any open set $U\in \Sigma_2$, the compliment of the set $N_{f_{1,\infty}}(U)$ is not finite.\\ Also, this system is not syndetically sensitive because the set $N_{f_{1,\infty}}(U)$ has unbounded gaps, for any open set $U\in \Sigma_2$.
\\Thus, $\mathcal{F}$-sensitivity need not imply cofinite sensitivity and syndetic sensitivity.

\section{$\mathcal{F}$-sensitivity for different non-autonomous dynamical systems}
In this section,  we investigate $\mathcal{F}$-sensitivity and weakly $\mathcal{F}$-sensitivity for different non-autonomous systems. In the following theorem, we discuss the relation between the $\mathcal{F}$-sensitivity of a non-autonomous system $(X,f_{1,\infty})$ and its hyperspace $(\pazocal{K}(X),\overline{f_{1,\infty}})$.

\begin{thm}
Let $(X,f_{1,\infty})$ be a non-autonomous dynamical system and let $\mathit{k}\mathcal{F}$ be filterdual. Then, $(X,f_{1,\infty})$ is $\mathcal{F}$- sensitive if and only $(\pazocal{K}(X),\overline{f_{1,\infty}})$ is $\mathcal{F}$-sensitive.
\end{thm}

\begin{proof}
Let $(\pazocal{K}$($X),\overline{f}_{1,\infty}$) be $\mathcal{F}$-sensitive with constant of $\mathcal{F}$-sensitivity $\delta>0$. For any $\epsilon>0$ and $x\in X$ let $U=B_d(x,\epsilon)$. Since $B_{d_H}(\{x\},\epsilon)$ is an $\epsilon$- neighbourhood of $\{x\}$ in $\pazocal{K}$($X)$ and $\overline{f_{1}^{\infty}}$ is $\mathcal{F}$-sensitive, therefore $N_{\overline{f_{1,\infty}}}[{B_{d_H}(\{x\},\epsilon),\delta)}]\in \mathcal{F}$ which implies there exist $A\in B_{d_H}(\{x\},\epsilon)$ and $n\geq 0$ such that $d_{H}(\overline{f_{1}^{n}}(\{x\}),\overline{f_{1}^{n}}(A))>\delta$.
\\Hence, we get $y\in A \subset U$ such that $d(f_{1}^{n}(x),f_{1}^{n}(y))>\delta$ which implies $N_{\overline{f_{1,\infty}}}$ $[\cup{B_{d_H}(\{x\},\epsilon),\delta)};x\in U] \subset N_{f_{1,\infty}}(U,\delta)$. As $N_{\overline{f_{1,\infty}}}[\cup{B_{d_H}(\{x\},\epsilon),\delta)};x\in U]\in \mathcal{F}$, therefore $N_{f_{1,\infty}}(U,\delta)\in \mathcal{F}$. Hence, $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive.

Conversely, assume that $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive with constant of $\mathcal{F}$-sensitivity $\delta>0$. Since $\pazocal{F}(X)$ is dense in $\pazocal{K}(X)$, it suffices to prove the result for $(\pazocal{F}(X),\overline{f|_{\pazocal{F}(X)}})$.

Let $A=\{x_1, x_2,\ldots, x_k\}\in \pazocal{F}(X)$ and $\epsilon>0$ be arbitrarily chosen. Then $\mathcal{F}$-sensitivity of $(X,f_{1,\infty})$ implies that the set $\{n\in \mathbb{N}$: diam$f_{1}^{n}(B_d(x_i,\epsilon/2))>\delta\}\in \mathcal{F}$, holds for each $i, 1\leq i\leq k$. Also we have $\mathit{k}\mathcal{F}$ is a filterdual, therefore \begin{align}
\Sigma = \bigcap_{i=1}^k\{n\in\mathbb{N}: diam f_{1}^{n}(B_d(x_i,\epsilon/2))>\delta\}\in \mathcal{F}. \nonumber
\end{align}
For any $n \in \Sigma$, there exists $(y_1,y_2,\ldots, y_k)\in B_d(x_1,\epsilon/2)\times \ldots \times B_d(x_k,\epsilon/2)$ such that $d(f_1^n(x_i),f_1^n(y_i))>\delta/2$ for each $i, 1\leq n\leq k$. Take $B=\{z_1,z_2,\ldots, z_k\}$ with \begin{enumerate}
\item $z_i=y_i, d(f_1^n(x_1),f_1^n(x_i))\leq \delta/2$,
\item $z_i=x_i$, otherwise.
\end{enumerate}

Then, \begin{align}
d_H(\overline{f_1^n}(A), \overline{f_1^n}(B))\geq min \{d(f_1^n(x_1),f_1^n(z_i)): 1\leq n\leq k\}>\delta/2.
\nonumber \end{align}
Note that $B\in \pazocal{F}(X)$ and $d_H(A,B)<\epsilon$. Therefore, \begin{align}
\Sigma \subset \{n\in \mathbb{N}: diam \overline{f_1^n}|_{\pazocal{F}(X)}(B_{d_{H}}(A,\epsilon)\cap \pazocal{F}(X))>\delta/2\}\in \mathcal{F}.
\nonumber \end{align}
Hence, $(\pazocal{F}(X), \overline{f_{1,\infty}}|_{\pazocal{F}(X)})$ is $\mathcal{F}$-sensitive. Therefore, $(\pazocal{K}(X),\overline{f_{1,\infty}})$ is $\mathcal{F}$-sensitive.
\end{proof}

\begin{rmk}
Note that the sufficiency part of the above theorem doesn't require $\mathcal{F}$ to be a filterdual. Therefore, $(\pazocal{K}(X),\overline{f_{1,\infty}})$ is $\mathcal{F}$-sensitive $\implies$ $(X,f_{1,\infty})$ is $\mathcal{F}$- sensitive in general.
\end{rmk}

\begin{thm}
Let $(X,f_{1,\infty})$ be a N.D.S generated by a family $f_{1,\infty}$ of feeble open maps commuting with $f$ and $\mathcal{F}$ be a filterdual and translation invariant Furstenberg family. If $\sum_{i=1}^\infty D(f_i,f) < \infty $, then $(X,f)$ is $\mathcal{F}$-sensitive if and only if $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive.
\end{thm}

\begin{proof}
Let $(X,f)$ be $\mathcal{F}$-sensitive with constant of $\mathcal{F}$-sensitivity $\delta > 0$. Let $\epsilon > 0$ be given and $U=B(x,\epsilon)$ be a non-empty open set in $X$. As $\sum_{i=1}^{\infty} D(f_n,f) < \infty$, by Lemma 2.2 we get that, for any $\epsilon > 0$, there exists $ n \in \mathbb{N} $ such that $d(f_{1}^{n+k}$ $(x),f^k(f_{1}^{n}$ $(x)))$ $ < \epsilon$, for every $x \in X$ and every $k\in \mathbb{N}$. Choose $m \in \mathbb{N}$ such that $1/m < \delta/4$. Then, there exists $n_0$ such that $d(f_{1}^{n_{0}+k}(x),f^k(f_{1}^{n_0}(x)))< 1/m$, for every $x\in X$ and every $k \in \mathbb{N}$. As $f_n's$ are feeble open, $U'=$int$f_{1}^{n_0}(U)$ is non-empty open and by $\mathcal{F}$-sensitivity of $(X,f)$, we get $N_{f}(U',\delta)\in \mathcal{F}$. Let $k \in N_{f}(U',\delta)$ then there exist $v_1$ and $v_2 \in f_{1}^{n_0}(U)$ such that $d(f^k(v_1),f^k(v_2))> \delta$. As $ v_1, v_2 \in f_{1}^{n_0}(U)$, there exist $v_{1}'$, $v_{2}'$ $\in U$ such that $v_1 = f_{1}^{n_0}(v_1'), v_2 = f_{1}^{n_0}(v_2')$ and \begin{align}d(f^k(f_{1}^{n_0}(v_1'),f^k(f_{1}^{n_0}(v_2'))) > \delta.\nonumber\end{align} Also $d(f_{1}^{n_{0}+k}(v_i'),f^k(f_{1}^{n_{0}}(v_i')))< 1/m$, for $i=1,2$.

Therefore, by triangle inequality, \begin{align}d(f_{1}^{n_{0}+k}(v_1'),f_{1}^{n_{0}+k}(v_2'))> (\delta-2/m) > \delta/2.\nonumber\end{align} Thus, we obtain $v_1'$ and $v_2' \in U$ such that  \begin{align}d(f_{1}^{n_{0}+k}(v_1'),f_{1}^{n_{0}+k}(v_2'))> \delta/2\nonumber\end{align} which implies $n_{0}+k \in N_{f_{1,\infty}}(U,\delta/2)$ and hence $N_f(U',\delta) + n_{0} \subseteq N_{f_{1,\infty}}(U,\delta/2)$.
Since $N_f(U',\delta)\in \mathcal{F}$ and $\mathcal{F}$ is translation invariant therefore $N_{f_{1,\infty}}(U,\delta/2)\in \mathcal{F}$ and hence $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive.

Conversely, let $(X,f_{1,\infty})$ be $\mathcal{F}$-sensitive with constant $\delta>0$, $U$ be a non-empty open set in $X$ and $m \in \mathbb{N}$ be such that $1/m<\delta/4$. Then there exists $n_{0}\in\mathbb{N}$ such that $d(f_{1}^{n_{0}+k}(x),f^k(f_{1}^{n_{0}}(x)))<1/m$, for every $x\in X$ and every $k\in \mathbb{N}$. Since $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive, therefore for any $n \in \mathbb{N}$, the set \{$k:diam(f_{1}^{k}(U))>\delta$\} is infinite. Thus, for open set $(f_{1}^{n_0})^{-1}(U) $, the set $N_{f_{1,\infty}}((f_{1}^{n_0})^{-1}(U),\delta)\in \mathcal{F}$. So there exists $k \in \mathbb{N}$ such that $k+n_0 \in N_{f_{1,\infty}}((f_{1}^{n_0})^{-1}(U),\delta)$ implying there exist $v_1$ , $v_2$ $\in (f_{1}^{n_0})^{-1}(U)$ such that \begin{align}d(f_{1}^{n_{0}+k}(v_1),f_{1}^{n_{0}+k}(v_2))>\delta.\nonumber\end{align} Since $v_1$, $v_2$ $\in ((f_{1}^{n_0})^{-1}(U))$ therefore there exist $v_{1}'$ and $v_{2}'$ $\in U$ such that $v_{1}' = f_{1}^{n_0}(v_1)$ and $v_{2}'=f_{1}^{n_0}(v_2)$ and hence \begin{align}d((f_{n_{0}+k}\circ \ldots \circ f_{n_{0+1}})(v_{1}'),(f_{n_{0}+k}\circ \ldots \circ f_{n_{0+1}})(v_{2}')) > \delta.\nonumber\end{align} Also $d(f_{1}^{n_{0}+k}(v_i),f^k(f_{1}^{n_{0}}(v_i)))<1/m$ or $d(f_{1}^{n_{0}+k}(v_i),f^k(v_i'))<1/m$, for $i=1,2$.

Therefore, by triangle inequality \begin{align}d(f^k(v_{1}'),f^k(v_{2}'))>(\delta-2/m)>\delta/2.\nonumber\end{align} Hence, we obtain $v_{1}'$ and $v_{2}' \in U$ such that $d(f^k(v_{1}'),f^k(v_{2}'))>\delta/2$ which implies $k \in N_{f}(U,\delta/2)$ so $N_{f_{1,\infty}}((f_{1}^{n_0})^{-1}(U),\delta)- n_{0} \subseteq N_{f}(U,\delta/2)$ and $N_{f_{1,\infty}}((f_{1}^{n_0})^{-1}(U),\delta)\in \mathcal{F}$ implies $N_{f}(U,\delta/2)\in \mathcal{F}$.
Therefore, $(X,f)$ is $\mathcal{F}$-sensitive.
\end{proof}

\begin{thm}
Let $(X,f_{1,\infty})$ be a N.D.S. which converges uniformly to a map $f$. If $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive, then for any $k\geq 1$, $(X,f_{1,\infty}^{[k]})$ is also $\mathcal{F}$-sensitive.
\end{thm}

\begin{proof}
For any fixed $k\geq 2$, as $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive with constant of $\mathcal{F}$-sensitivity $\delta>0$, therefore, for any $x\in X$ and any $\epsilon>0$, there exists $y\in B_d(x,\epsilon)$ such that $(x,y)\notin Asym_{\epsilon}(\mathcal{F})$. Hence, there exists $n_{x}(\epsilon)\in \mathbb{N}$ such that $d(f_{1}^{n_x(\epsilon)}(x),f_1^{n_x(\epsilon)}(y))>\delta$. Now, by Lemma 2.1, we have that for any given $\delta>0$, there exist $\eta>0$ and $N_0\geq 3k$ such that for any pair $(x,y)\in X$ with $d(x,y)<\eta$, we have \begin{align*}
& d(f_n^i(x),f_n^i(y))<\delta,\hspace{0.2cm}for\hspace{0.2cm} all\hspace{0.2cm} n\geq N_0\hspace{0.2cm} and\hspace{0.2cm} 0\leq i\leq k \\ \iff & d(f_n^i(x),f_n^i(y))\geq \delta \implies d(x,y)\geq \eta,\hspace{0.2cm} for \hspace{0.2cm} all\hspace{0.2cm} n\geq N_0 \hspace{0.2cm}and\hspace{0.2cm} 0\leq i\leq k.
\end{align*}
Now, for any $1\leq i\leq 2N_0$, $f_1^i$ is uniformly continuous, therefore for given $\delta>0$, there exists $\epsilon'>0$ such that \begin{align}
d(x_1,x_2)< \epsilon' \implies d(f_1^i(x),f_1^i(y))<\delta,\hspace{0.2cm} for\hspace{0.2cm} any\hspace{0.2cm} x_1, x_2 \in X \hspace{0.2cm}and\hspace{0.2cm} 1\leq i\leq 2N_0.
\end{align}
Hence, for any $x\in X$ and any $\epsilon< \epsilon'$, we have $n_x(\epsilon)> 2N_0\geq 6k$. By division algorithm, there exists $i_0\in \{0,1,2,\ldots, n-1\}$ such that $n_x(\epsilon)=kl+i_0$, for some $l\in \mathbb{N}$ which implies $n_x(\epsilon)-i_0=kl$. Combining this with the fact that for any $\epsilon< \epsilon'$, $d(f_1^{n_x(\epsilon)}(x),f_1^{n_x(\epsilon)}(y))$ = $d(f_{n_x(\epsilon)-i_0+1}^{i_0}(f_1^{n_x(\epsilon)-i_0}(x)),f_{n_x(\epsilon)-i_0+1}^{i_0}(f_1^{n_x(\epsilon)-i_0}(y)))>\delta$. Also, since $n_x(\epsilon)>2N_0\geq 6k$ and $0\leq i_0< k$, therefore $n_x(\epsilon)-i_0+1\geq N_0$.\\ Using (1), we get that for any $\epsilon< \epsilon'$, $d(f_1^{n_x(\epsilon)-i_0}(x),f_1^{n_x(\epsilon)-i_0}(y))\geq\eta.$\\ Now since $k$ divides $n_x(\epsilon)-i_0$, therefore we have $d(f_{k(l-1)+1}^k\circ \ldots \circ f_1^k(x),f_{k(l-1)+1}^k\circ \ldots \circ f_1^k(y))\geq$ $d(f_1^{n_x(\epsilon)-i_0}(x),f_1^{n_x(\epsilon)-i_0}(y))$, and hence we get $d(f_{k(l-1)+1}^k\circ \ldots \circ f_1^k(x),f_{k(l-1)+1}^k\circ \ldots \circ f_1^k(y))\geq\eta$. Thus, $(x,y)$ is not a $\mathcal{F}$-$\delta_0$-asymptotic pair for $(X,f_{1,\infty}^{[k]})$, $0< \delta_{0} < \eta$ implying $(X,f_{1,\infty}^{[k]})$ is $\mathcal{F}$- sensitive, with $\mathcal{F}$-sensitivity constant $0< \delta_{0} < \eta$.
\end{proof}

\begin{rmk}
Converse of the above result is also true. If $(X,f_{1,\infty}^{[k]})$ is $\mathcal{F}$-sensitive, for any $k\in \mathbb{N}$, then in particular for $n=1$, we get that $f_{1,\infty}^{[1]}=(f_{(n-1)+1}^1)_{n=1}^{\infty}$ which implies $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive.
\end{rmk}

\begin{rmk}
Based on similar arguments, one can prove the above result for $(X,f_{1,\infty})$ being weakly $\mathcal{F}$-sensitive.
\end{rmk}

In \cite{MR3661658}, the authors have investigated various dynamical properties of a non-autonomous system generated by a finite family of maps. We explore $\mathcal{F}$-sensitivity for such non-autonomous systems.
\begin{thm}
Let $(X,f_{1,\infty})$ be a non-autonomous system generated by a finite family of maps, $\mathbb{F}=\{f_1, f_2, \ldots, f_k\}$. If the autonomous system $(X,f_k\circ f_{k-1} \circ \ldots \circ f_1)$ is $\mathcal{F}$-sensitive, then $(X,f_{1,\infty})$ is also $\mathcal{F}$-sensitive.
\end{thm}

\begin{proof}
Let $x\in X$ and $U$ be any open neighbourhood of $x$. Suppose $(X,f_k, f_{k-1}\circ f_1)$ is $\mathcal{F}$-sensitive with $\mathcal{F}$-sensitivity constant $\delta>0$. By $\mathcal{F}$-sensitivity of $(X,f_k\circ f_{k-1}\circ \ldots \circ f_1)$, we get a $y\in U$ such that the set $\{n\in \mathbb{N}:d(({f_1^k})^n(x),({f_1^k})^n)(y))>\delta\} \in \mathcal{F}$. Therefore, the set $\{m\in \mathbb{N}:d(f_1^m(x),f_1^m(y))>\delta\}\in \mathcal{F}$. Hence, $(X,f_{1,\infty})$ is $\mathcal{F}$-sensitive.
\end{proof}

\begin{rmk}
A similar result, as Theorem 3.6, holds true for weakly $\mathcal{F}$-sensitivity.
\end{rmk}

In the following example, we show that the non-autonomous system generated by the family $\mathbb{F}$ can be $\mathcal{F}$-sensitive, even when none of the maps in the family $\mathbb{F}$ is $\mathcal{F}$-sensitive.

\begin{exm} \end{exm}
Let $I$ be the interval $[0,1]$, $\mathcal{F}$ be the Furstenberg family of all infinite subsets of $\mathbb{Z^+}$ and $f_1, f_2$ on $I$ be defined by:
\[ f_1(x) = \begin{cases}
4x,  & \text{for} \ x \in \left[0, \frac{1}{4}\right] \\

5/4-x, & \text{for} \ x \in \left[\frac{1}{4},1\right].
\end{cases} \]

\[ f_2(x) = \begin{cases}
1/4-x,  & \text{for} \ x \in \left[0, \frac{1}{4}\right] \\
4x-1, & \text{for} \ x \in \left[\frac{1}{4},\frac{1}{2}\right]\\
2-2x, & \text{for} \ x \in \left[\frac{1}{2},1\right].
\end{cases} \]

Let $\mathbb{F}=\{f_1,f_2\}$ and $(X,f_{1,\infty})$ be non-autonomous system generated by $\mathbb{F}$. Note that [1/4,1] is an invariant set for $f_1$ and [0,1/4] is an invariant set for $f_2$. Therefore, neither $f_1$ nor $f_2$ is $\mathcal{F}$-sensitive. However, their composition given by

\[ f_2\circ f_1(x) = \begin{cases}
1/4-4x,  & \text{for} \ x \in \left[0, \frac{1}{16}\right] \\
16x-1, & \text{for} \ x \in \left[\frac{1}{16},\frac{1}{8}\right]\\
2-8x, & \text{for} \ x \in \left[\frac{1}{8},\frac{1}{4}\right]\\
2x-1/2, & \text{for} \ x \in \left[\frac{1}{4},\frac{3}{4}\right]\\
4-4x, & \text{for} \ x \in \left[\frac{3}{4},1\right].
\end{cases} \]
is $\mathcal{F}$-sensitive and hence by Theorem 3.6, the non-autonomous system $(X,f_{1\infty})$ is $\mathcal{F}$-sensitive.

\section*{Acknowledgement}
The first author is funded by GOVERNMENT OF INDIA, MINISTRY OF SCIENCE and TECHNOLOGY No: DST/INSPIRE Fellowship/[IF160750].


TITLE: Polynomial GCD in the presence of floating-point errors
QUESTION [7 upvotes]: The crucial requirement for using root isolation methods based on Vincent's theorem is that the input polynomial does not have multiple zeros. One way to remove the multiple zeros is to use polynomial GCD. However, when it is implemented with the usual floating-point numbers (IEEE doubles), most probably the various rounding errors will cause the multiple roots to disintegrate into clusters of roots.
Does anyone know about good (and simple) approximate polynomial GCD algorithm, which does not require exact arithmetic (i.e. is amenable to implementation in 64-bits floating points numbers)?
The closest thing I found is the "Computing multiple roots of inexact polynomials" by Zhonggang Zeng but unfortunately it is way over my tiny brain.

REPLY [5 votes]: One of the reasons why the methods you mentioned do not work for multiple roots is that, in this situation, the root-finding problem is numerically ill-posed. That is, an arbitrarily small perturbation of the input will change the structure of your solution (just as you recognized, multiple roots will split into clusters). The same holds for the computation of the square-free part. Trying to remedy this instability by an inexact calculation is, in general, doomed, unless you have additional information (e.g., the total number of real and complex roots, or your actual goal is to isolate a multiple root of which you know a priori that it is unique). In such situations, you should check whether it is indeed necessary to make the polynomial squarefree, or if you can infer all the required information by other means.
Also, it might be easier to look for the complex roots using methods like Aberth-Ehrlich (as implemented in MPSolve), Durand-Kerner-Weierstraß, or homotopy continuation methods (e.g., as in PHCpack or Bertini). In some sense, if the precision of the input is not high enough, they work just as if the polynomial would have a cluster of roots instead of a multiple root, and you can simply stop after a sufficient precision or "resolution" is reached. Then postprocess or combine the clustered approximations into multiple roots. Of course, that's a heuristic, but an approximate GCD is less than trivial to implement in a certified manner, too.
By the way, the heart of the Aberth-Ehrlich method is maybe a five-liner, and it performs impressively well in practice. I'd definitely give that one a try, even if you are just looking for real roots.
But back to your actual question: As far as I know, approximate GCDs are still not sufficiently well understood, and in particular I know of no ready-made implementations using machine precision (probably because all of them will fail for some instances). YMMV, but if you want to try something in machine precision, I would take the following steps:

Use the Extended Euclidean Algorithm for the GCD computation, nothing fancier (in particular, no asymptotically fast algorithm like the Half-GCD unless you have a very good reason to believe that it should be as stable).
The critical steps will be divisions with remainder where the leading coefficient of the remainder vanishes. If it doesn't due to numerical instabilities, the output will be garbage. Thus:
Along with the floating-point version of the polynomial GCD (possibly in interval arithmetic), compute the GCD modulo some prime(s) (I assume your input consists of polynomials in $\mathbb{Q}[x]$). Not only that this will, with high probability, give you the correct degree of the GCD; unless you have chosen an "unlucky" prime, the sequence of the degrees and coefficients occuring throughout the EEA should be identical modulo the prime. So you can use these results as a sanity check for any rounding to or away from zero.

But AFAICS, if you ever run into a situation where you do not have enough precision to sufficiently bound a non-zero leading coefficient (as certified by the modular computation) away from zero, you will have trouble and need to rely on guesswork (e.g., heuristics for root distances or magic epsilons in your code).


text:
\section{Conclusion}
\label{sec:conclusion}

While the AMP algorithm has been shown to be a very desirable
approach for signal recovery and statistical inference problems
in terms of both computational efficiency and accuracy, it is also
very sensitive to problems which deviate from its fundamental assumptions.
In this work, we propose the \swamp~algorithm which matches AMP's accuracy
while remaining robust to such variations, all without unduly increasing
computation or memory requirements. We also demonstrate how \swamp~can be
used to solve practical problems for which AMP and GAMP cannot be
applied, namely, group testing and 1-bit CS with troublesome projections. In
all cases, \swamp~provides superior accuracy as compared to $\ell_1$-
minimization, as well as convergence properties superior to AMP and GAMP, and all
with less computational and memory burden than BP or r-BP.

Exact analysis of the asymptotic state evolution of \swamp, as well as a
thorough analytical proof of its convergence,
remains a challenging open problem for future work.


TITLE: If $S = \{ (a_1, a_2): a_1, a_2 \in R \}$, then is $R$ necessarily the field over which the vector space $S$ is defined?
QUESTION [0 upvotes]: I am reading through an introductory linear algebra textbook. My understanding is that a vector space, defined as a set, is defined over a given field.
If $S = \{ (a_1, a_2): a_1, a_2 \in R \}$, then is $R$ necessarily the field over which the vector space $S$ is defined? If not, why? Please give a counterexample.
Thank you.

REPLY [3 votes]: Your $S$ is just a set, unless you define the vector space operations and the scalar multiplication. There is an obvious definition for this set which will make it a two dimensional $S$ vector space, namely by defining addition as
$$((a_1, a_2), (b_1, b_2)) \mapsto (a_1+a_2, b_1+b_2)$$
and scalar multiplication by
$$(\lambda,(a_1, a_2))\mapsto (\lambda a_1, \lambda a_2)$$
with $\lambda, a_i , b_i\in R$.
But if, e.g., $R=  \mathbb{R}$ you can make the set $S$ a one dimensional complex vector space if you define the operations appropriately. Addition would be the same, but multiplication would be defined as
$$(z, (a_1, a_2) )\mapsto (z_1a_1-z_2a_2, z_1a_2 + z_2a_1)$$
for $z\in \mathbb{C}$ and $a_i\in \mathbb{R}$. (This is just the complex product of $z_1+iz_2$ with $a_1+ia_2$ written down component wise). With this definition of scalar multiplication the set $S$ is then a one dimensional complex vector space.