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Every think you should be considered for the role?” or even, “Why did you apply for this position?”
They sound very much different, yet the answers will be the same, and you will need to prepare in the same manner.
Either and many more like them are common interview questions, and here you can learn more about what employers wish to know; and how you can fly through any questions and answers part of your interview.
What is the Interviewer Asking?
The purpose of this question is to see how well candidates sell themselves during the job interview.
The answer to these interview questions are similar to others, and they do require you to look far beyond your first position in the company; and what you have to offer further on.
Any recruitment will be an investment in the company, so they wish to know the following from the answer to the question:
- Are you a good fit for the company?
- Are you suitable for long-term work in the company?
- How much do you know about the company, and have you done sufficient research?
- Do you possess excellent influencing and communication skills?
How Do You Think You Will Contribute to Our Team?
Even without thinking of your answer to this common interview question can backfire if you are not careful.
There are two particular areas where this can happen.
Keep the answer to you: You need to highlight what makes you the best option for the position than other candidates. It would help if you sold yourself, but not at the expense of making, other candidates appear bad. Never speak negatively of others.
Describing your strengths: Job interview questions are designed to see the best of you. It would help if you sold your strongest attributes and strengths. However, if you list the answers to things, you are good at; it can negatively portray you.
To come up with the best answers to the “What can you bring to the team?” question, you need to practice your answers. The reason being, it may not be a natural way you speak, as you are talking purely about yourself to others.
You can use the information you had in your cover letter, yet be sure you do not repeat that as the interviewer will more than likely check as you deliver your answer.
Here are the areas to focus your practice ready for your job interview.
Why are you unique?
Up until you hear this interview question, you will be trying to set yourself apart from every other candidate. The way you answer here does the same thing, so explaining with cliché terms do you no good such as “I am hardworking.”
Many other candidates will say the same, so you can easily blend in. Refer to the job description to see if anything stands out you may think others cannot deliver.
How Can You Apply Your Skills?
After the first part of the question and what it is, that makes you stand out; you need to make this as the key section of the “What can you bring to the company,” question.
You have unique skills, yet the employer wants to see how you can use these for their benefit. Any person can list skills as part of their job search, yet understanding where they can use them is very different.
If you possess excellent problem-solving skills, you can expand on this and explain which particular part of the company or team, these skills can benefit.
If you are not applying for your first position, it may be easier to see how your skills helped you in other jobs if they relate to your new application.
Show Your Value
When trying to show your value, if you have any past professional or academic experience, you should mention this and relate to your position.
The best career advice will always be to make sure your answers using your sills relate to the job in question and the job description.
You may have references from past supervisors or talking about how previous team members saw you as a valuable asset.
As we said, you will need to practice the answers to this and other common interview questions.
Sample Answers to What Can You Bring to the Table
I have always been hardworking, and throughout my employment history, such energy has manifested itself in my confident attitude and ability to keep busy.
The job was more than just a chore list for me. I endeavour to help with special projects and daily tasks to be sure the assignments are completed.
Naturally, proper time management skills drove my energy, so I focus on the most important tasks before looking for new busy work.
If I got this job, my energy and all that comes with it would lead to thinking outside the box and being innovative and efficient.
Conclusion
You can find many interview tips, yet these are only guidelines for what you need to do. You can scour every page on a site right down to the policy terms and conditions, yet nothing serves you better than your research and practice.
No matter what you find, no information can help you because you are unique, and you do have unique benefits that others cannot deliver. | 407,554 |
TITLE: Does uniform convergence in $L^1$ imply uniform convergence in $L^2$?
QUESTION [2 upvotes]: I had to keep the title reasonably long, so below are the full details. This is a question that came to my mind while working on a distantly related problem; since I do not know whether the answer is negative or affirmative, I have no idea whether I should look for counterexamples or, on the contrary, I should try to come up with a proof.
Let $X$ be a space endowed with a finite measure $m$ (also, there exist non-empty subsets of measure $0$, to prevent some trivial counterexamples). Let $T$ be some compact Hausdorff topological space. Let $f_k : T \times X \to \mathbb R$ be a family of functions indexed by $k \in \mathbb N$ such that:
$f_k (t, \cdot) \in L^2 (X) \subseteq L^1(X)$ for all $k \ge 0$;
the map $T \ni t \mapsto f_k (t, \cdot) \in L^1 (X)$ is continuous with respect to the $L^2$ norm;
$f_k (t, \cdot) \to 0$ in $L^1(X)$, uniformly with respect to $t \in T$;
$f_k (t, \cdot) \to 0$ in $L^2(X)$ for every $t \in T$.
Is the convergence in $L^2$ also uniform with respect to $t \in T$, like the convergence in $L^1$?
REPLY [1 votes]: The answer is negative.
Consider the space $X = \left( [0, 1], \mathcal{B}\left( [0, 1] \right), \lambda \vert_{[0, 1]} \right)$, where $\mathcal{B}\left( [0, 1 ] \right)$ denotes the Borel $\sigma$-algebra and $\lambda \vert_{[0, 1]}$ denotes the restriction of the Lebesgue measure. Consider also the space $T = [0, 1]$ endowed with its usual topology. Given an integer $k \geq 1$, define the map $f_{k} \colon T \times X \rightarrow \mathbb{R}$ by $$f_{k} \colon (t, x) \mapsto \begin{cases} \max\left\lbrace k \left( 1 -\lvert k t -1 \rvert \right), 0 \right\rbrace & \text{if } x \in \left[ \frac{1}{k^{2}}, \frac{2}{k^{2}} \right]\\ 0 & \text{otherwise} \end{cases} \, \text{.}$$
The desired conditions are satisfied:
For every integer $k \geq 1$ and every $t \in T$, we have $f_{k}(t, .) \in L^{2}(X)$ since $f_{k}(t, .)$ is bounded.
For every integer $k \geq 1$ and every $t_{0} \in T$, the map $f_{k}(t, .)$ tends to $f_{k}\left( t_{0}, . \right)$ in $L^{\infty}(X)$ as $t \rightarrow t_{0}$. Therefore, for every $k \geq 1$, the map $t \in T \mapsto f_{k}(t, .) \in L^{2}(X)$ is continuous.
For every integer $k \geq 1$ and every $t \in T$, we have $$\left\lVert f_{k}(t, .) \right\rVert_{L^{1}(X)} = \int_{\frac{1}{k^{2}}}^{\frac{2}{k^{2}}} \max\left\lbrace k \left( 1 -\lvert k t -1 \rvert \right), 0 \right\rbrace \, dx \leq \int_{\frac{1}{k^{2}}}^{\frac{2}{k^{2}}} k \, dx = \frac{1}{k} \, \text{.}$$ Therefore, we have $$\lim_{k \rightarrow +\infty} \sup_{t \in T} \left\lVert f_{k}(t, .) \right\rVert_{L^{1}(X)} = 0 \, \text{.}$$
We have $f_{k}(0, .) = 0$ for all $k \geq 1$, and hence $f_{k}(0, .)$ tends to $0$ in $L^{2}(X)$ as $k \rightarrow +\infty$. For every $t \in ]0, 1]$, we have $f_{k}(t, .) = 0$ for all integers $k \geq \frac{2}{t}$, and hence $f_{k}(t, .)$ tends to $0$ in $L^{2}(X)$ as $k \rightarrow +\infty$.
For every integer $k \geq 1$, we have $$\left\lVert f_{k}\left( \frac{1}{k}, . \right) \right\rVert_{L^{2}(X)}^{2} = \int_{\frac{1}{k^{2}}}^{\frac{2}{k^{2}}} k^{2} \, dx = 1 \, \text{.}$$ | 114,978 |
Analysis of On-Site Conditions
Analyzing the operating conditions on a site
Haul Road Maintenance Audit
Tire while maintaining tire performance, it is sometimes useful to reconsider the profile of the haul roads (slopes, cambers, curve radii, etc.).
Improvement in productivity leads to an increase in the rate of operation. While preserving the general safety of the site, this increase must also take account of the limits linked to the machines and their tires.
This is why Michelin technicians offer to carry out complete site audits, during which different parameters on the vehicles are analyzed;ering of loads)
Depending on the density of the materials transported, completely filling a loader bucket or a dumper truck may result in the machines working overloaded.
Similarly, uneven loading of a dumper truck results in overloading on one side, concentrated on certain tires.
Weighing the trucks and loaders regularly (or routinely when working with high density materials) means that loading can be optimized and controlled.
By 3D modelling of the loading of a rigid dump truck (by passing the truck under a lazer camera), we can assess any improvement to be made in the centring of loads.
Michelin technicians can carry out very precise loading checks, or recommend companies who can perform these studies.
Measuring the Carbon Footprint of Your MICHELIN® Tires
Very often, requests for propsals for major Civil Engineering or construction projects include an environmental section in which the carbon footprint is one of the major aspects.
The tires on your machines are part of this carbon footprint; Michelin has a tool, which by measuring parameters such as the nominal load of machines, the profile of the haul roads (rises, descents, curves, etc.) and the traffic cycles (acceleration, braking, etc.), can calculate the carbon footprint of your tires.
Services
- Our Service Offer
- Analysis of On-Site Conditions
- MICHELIN Tyre Pressure Monitoring System (TPMS): MEMS Evolution3
- Business Model TK
- Recovery of Scrap Tire Products at the End of Their Service Life | 416,883 |
B. That will be in conjunction with the opening for Kathleen Scott, whose exhibit is called Nocturne. Click here for more information. | 56,596 |
Product Info
The Hozelock AC Plus water timer is hugely popular and is perfect for automatic garden watering systems. Using the simple dial and four buttons, you can set the water timer to turn on multiple times per day (every 6, 8, 12 or 24 hours), or once every 2, 3 or 7 days. Watering duration can be set from 1 minute up to 180 minutes.. Audible beep confirms selection.
- Watering cycles from once per week to four times per day.
- Maximum watering cycle is 180 minutes.
Download: Hozelock AC Plus Electronic Water Timer 2700 - Instruction Manual
Programming Info
1. Set the current time
Turn dial to the nearest hour on the Red scale (24 hour clock) then press the time button. An audible beep confirms correct setting.
2. Set the watering start time
Turn dial to the nearest hour to your desired start time on the Red scale (24 hour clock) then press the S button. An audible beep confirms correct setting.
3. Set the watering frequency
This can be set in hours (every 6, 8, 12, or 24 hours) or it can be set in days (every 2, 3 or 7 days). Turn dial to desired setting on the Green scale and then press the Green button. An audible beep confirms correct setting.
4. Set the watering duration
Turn dial to number of watering minutes required on the Blue scale and then press the Blue button. An audible beep confirms correct setting.
5. Turn dial to AUTO to start the selected programme | 206,893 |
TITLE: Direction of t (Vector Space)
QUESTION [1 upvotes]: Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x = e^{-t}\cos t, y = e^{-t} \sin t, z = e^{-t}; (1, 0, 1). $$
The solution states I form $r(t)$ then get the $r'(t)$. After, it says: The point $(1, 0, 1)$ correspond to $t = 0$. Then they solve $r'(0)$ and thus, the tangent line is parallel to the vector $r'(0)$.
I'm pretty confused by the whole process.
REPLY [2 votes]: The curve you mention is given by $$r(t) = (e^{-t}\cos t, e^{-t}\sin t, e^{-t}).$$ (You may have $\textbf{r}(t)$ instead) More precisely, the image of this function is the curve under consideration, so every point on the curve is of the form $r(t)$ for some $t$. When they say the point $(1, 0, 1)$ corresponds to $t = 0$, they mean that the point $(1, 0, 1)$ is on the curve and $r(0) = (1, 0, 1)$.
How did they determine this? By solving the equation $r(t) = (1, 0, 1)$ which reduces to simultaneously solving the three equations \begin{align} e^{-t}\cos t &= 1\\ \\e^{-t}\sin t &= 0\\ \\e^{-t} &= 1.\end{align} The third equation is the easiest to solve as the other two involve products. The equation $e^{-t} = 1$ has unique solution $t = 0$. However, we need to solve these equations simultaneously. That is, we need to find the value(s) of $t$ such that all three equations hold, not just one of them. So we know that $e^t = 1$ holds if $t = 0$. As $e^0\cos 0 = 1\times 1 = 1$ and $e^0\sin 0 = 1\times 0 = 0$, $t = 0$ also satisfies the first and second equations as well. We can then conclude that the point $(1, 0, 1)$ is on the curve, and it corresponds to $t = 0$; that is, $r(0) = (1, 0, 1)$.
If you really want to make sure you have understood this process, I encourage you to try the following for yourself. I have hidden the answers below the questions. You can see them by placing your mouse over the box but you should have a go at them yourself first.
Is the point $(e, 0, e)$ on the curve? If so, which $t$ does it correspond to?
Looking at the three equations as above, but for the point $(e, 0, e)$, the third equation becomes $e^{-t} = e$ which has unique solution $t = -1$. However, $e^{-(-1)}\cos(-1) = e\cos(-1) \neq e$ as $\cos(-1) \neq 1$. Therefore there is no $t$ such that $r(t) = (e, 0, e)$, so the point $(e, 0, e)$ is not on the curve.
Is the point $(\frac{1}{\sqrt{2}}\sqrt[4]{e^{\pi}}, -\frac{1}{\sqrt{2}}\sqrt[4]{e^{\pi}}, \sqrt[4]{e^{\pi}})$ on the curve? If so, which $t$ does it correspond to?
Again, looking at the three equations, the third becomes $e^{-t} = \sqrt[4]{e^{\pi}} = e^{\pi/4}$ so $t = -\frac{\pi}{4}$ is the unique solution. Now we need to check the other two equations. As $e^{-(-\pi/4)}\cos\left(-\frac{\pi}{4}\right) = \sqrt[4]{e^{\pi}}\frac{1}{\sqrt{2}}$, $t = -\frac{\pi}{4}$ satisfies the first equation, and as $e^{-(-\pi/4)}\sin\left(-\frac{\pi}{4}\right) = \sqrt[4]{e^{\pi}}\frac{1}{\sqrt{2}}$, $t = -\frac{\pi}{4}$ also satisfies the second equation. Therefore the point $(\frac{1}{\sqrt{2}}\sqrt[4]{e^{\pi}}, -\frac{1}{\sqrt{2}}\sqrt[4]{e^{\pi}}, \sqrt[4]{e^{\pi}})$ is on the curve and corresponds to $t = -\frac{\pi}{4}$; that is, $r\left(-\frac{\pi}{4}\right) = (\frac{1}{\sqrt{2}}\sqrt[4]{e^{\pi}}, -\frac{1}{\sqrt{2}}\sqrt[4]{e^{\pi}}, \sqrt[4]{e^{\pi}})$.
Does the curve intersect the $xy$-plane?
The $xy$-plane is given by the equation $z = 0$, so if the curve intersects the $xy$-plane, it does so at a point of the form $(a, b, 0)$. That is, if the curve does intersect the $xy$-plane, there is some $t$ corresponding to that point such that $r(t) = (a, b, 0)$. But then the third equation reads $e^{-t} = 0$ which has no solutions. Therefore the curve does not intersect the $xy$-plane. | 38,433 |
Former Central Intelligence Agency Director, David Petraeus, in Sept. 2011. Mark Wilson/Getty Images hide caption | 219,940 |
TITLE: Prob. 14, Sec. 2.10 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: application to a system of equations?
QUESTION [2 upvotes]: Let $M$ be a non-empty subset of a normed space $X$, and let $M^a$ denote the subspace of the dual space $X'$ that consists of all those bounded linear functionals that vanish at each point of set $M$.
Now here's Prob. 14, Sec. 2.10 in Introductory Functional Analysis With Applications by Erwin Kreyszig:
If $M$ is an $m$-dimensional subspace of an $n$-dimensional normed space $X$, show that $M^a$ is an ($n-m$)-dimensional subspace of $X'$. Formulate this as a theorem about solutions of a system of linear equations.
My effort:
Since $X$ is finite-dimensional, every linear functional on $X$ is bounded; so we can write $X' = X^*$.
Let $\{e_1, \ldots, e_m \}$ be a basis for $M$; extend it to a basis $\{ e_1, \ldots, e_m, \ldots, e_n \}$ for $X$. Then each element of $X$ has a unique representation as a linear combination of the $e_j$s.
Suppose that $x \in X$ has the unique representation $$x = \sum_{j=1}^n \xi_j e_j.$$
Then, for any $f \in X'$, we have $$f(x) = \sum_{j=1}^n \xi_j f(e_j).$$
Now, for each $j= 1, \ldots, n$, let $f_j \in X'$ be deined as
$$f_j(x) = \xi_j.$$
Then we can write
$$f(x) = \sum_{j=1}^n f(e_j) f_j(x).$$
So $$f = \sum_{j=1}^n \alpha_j f_j, \ \mbox{ where } \ \alpha_j \colon= f(e_j) \ \mbox{ for each } \ j= 1, \ldots, n.$$
It can also be shown that the set $\{ f_1, \ldots, f_n \}$ is linearly independent and therefore a basis for $X^* = X'$.
Now suppose that $f \in M^a$. Then $$ f(e_j) = 0 \ \mbox{ for each } \ j= 1, \ldots, m.$$ So $$f(x) = \sum_{j=m+1}^n \xi_j f(e_j) = \sum_{j=m+1}^n \alpha_j f_j(x).$$
Thus each $f \in M^a$ can be written as
$$f = \sum_{j=m+1}^n \alpha_j f_j.$$
Moreover, the set $\{f_{m+1}, \ldots, f_n \}$, being a subset of a linearly independent set, is also linearly independent and hence forms a basis for $M^a$. So $M^a$ has dimention $n-m$.
Is the above proof correct?
Now it is my feeling that this result yields the following result:
Let $m < n$. Then any system of $m$ independent homogeneous simultaneous linear equations in $n$ unknowns (with real or complex numbers as co-efficients) has $n-m$ linearly independent solutions.
Is my conclusion correct?
But I'm not exactly sure how to relate the above formulation to this conclusion.
REPLY [1 votes]: Your proof is correct. Well done!
As for the other result, here's one way to think about it:
Take $e_j$ to be the standard basis of $\Bbb R^n$, and take $f_j$ to be the corresponding dual basis. The system of equations
$$
a_{11} x_1 + \cdots + a_{1n} x_n = 0\\
a_{21} x_1 + \cdots + a_{2n} x_n = 0\\
\vdots \\
a_{m1} x_1 + \cdots + a_{mn} x_n = 0
$$
can be rewritten as
$$
(a_{11}f_1 + \cdots + a_{1n}f_n) x = 0\\
(a_{21}f_1 + \cdots + a_{2n}f_n) x = 0\\
\vdots\\
(a_{m1}f_1 + \cdots + a_{mn}f_n) x = 0
$$
That is, the solution set of the system of equation is the common zero set of the linearly independent set of functionals $\sum_{j=1}^n a_{ij}f_j$ where $i = 1,\dots,m$. That is, if $M$ denotes the solution set to the homogeneous system of equations, we have
$$
M^a = \text{span}\left\{\sum_{j=1}^n a_{1j}f_j,\dots,\sum_{j=1}^n a_{mj}f_j\right\}
$$
By your result, we may conclude $\dim(M) = n-m$, as desired. | 25,670 |
TITLE: Proving that a right angle triangle is always formed when two vectors from either side of a semicircle join at a point on the circles circumferance
QUESTION [1 upvotes]: The goal is to prove that $\vec{r}_1$ is perpendicular to $\vec{r}_2$.
To begin, I started with the definition that if two vectors are perpendicular to each other, then their scalar product will be zero. For the case of this question,
$$
\vec{r}_1 \cdot \vec{r}_2 = 0
$$
Expanding this would give
$$
\vec{r}_1 \cdot \vec{r}_2 = r_{1x}r_{2x} + r_{1y}r_{2y}
$$
From the diagram, we know that $R + r_x = r_{1x}$, and $R - r_x = r_{2x}$, so
\begin{align}
\vec{r}_1 \cdot \vec{r}_2 &= (R + r_x)(R - r_x) + r_{1y}r_{2y}\\
&= R^2 - r_x^2 + r_{1y}r_{2y}
\end{align}
Since $r_y = r_{1y} = r_{2y}$,
$$
\vec{r}_1 \cdot \vec{r}_2 = R^2 - r_x^2 + r_y^2
$$
Then substituting in $r_x^2 = r^2 - r_y^2$
\begin{align}
\vec{r}_1 \cdot \vec{r}_2 &= R^2 - (r^2 - r_y^2) + r_y^2\\
&= R^2 - r^2 + r_y^2 + r_y^2\\
&= R^2 - r^2 + 2r_y^2
\end{align}
Since $\|\vec{r}\| = r = R$
\begin{align}
\vec{r}_1 \cdot \vec{r}_2 &= R^2 - R^2 + 2r_y^2\\
&= 2r_y^2
\end{align}
But this cannot be so. The scalar product must be equal to $0$. How is it that its equal to $2r_y^2$? Where did I go wrong in my math?
REPLY [0 votes]: What I am going to show is that if the dot product $\vec{r_1}.\vec{r_2}=0$, then the triangle is a right triangle.
Let us denote by $\vec{R}=\vec{\Omega O}$ where $\Omega$ is the origin of axes and $O$ the center of the circle.
We have:
$$\vec{r_1}+\vec{r_2}=2 \vec{R}$$
Take the scalar product of both sides with themselves:
$$(\vec{r_1}+\vec{r_2})^2=4 \vec{R}^2$$
$$\vec{r_1}^2+\vec{r_2}^2 + 2 \vec{r_1}.\vec{r_2}=4 \vec{R}^2$$
As we have assumed $\vec{r_1}.\vec{r_2}=0$ we see that
$$r_1^2+r_2^2=(2 R)^2$$
which means (reciprocal of Pythagoras' theorem) that we have a right triangle. | 176,249 |
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Accommodation Mt Buller welcomes Forest Park Riding And Equitation School, the ideal Stromlo ACT Holiday outlet for all the family to enjoy. We have made sure that we have thought of everything you would expect from your vacation with many features, click on the email or phone now button to get in touch with the owner.Is Forest Park Riding And Equitation School in Stromlo your property on Accommodation Mt Buller, please claim your listing and subscribe to update your contact details and receive your phone calls and emails from your clients. | 79,123 |
Crises in the workplace can happen anywhere, and at any time. Human Resources plays a vital role and needs to ensure all departments are ready and have a plan in place to deal with any major event safely and successfully.
This whitepaper walks you through the key systems involved in handling crises effectively, including employee information tracking, succession planning, and supporting crisis management planning.
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TITLE: Find the solution to $x=\log_{10}(\log_{10}(\log_{10}(x)))$
QUESTION [1 upvotes]: This is one practice in my real analysis class on proof writing. Is my proof completely solid? Is there anything to be improved on on the logic? Thanks in advance
Problem: Find the solution to $x=log_{10}(log_{10}(log_{10}(x)))$
My attempt:
+Let $f:(0,+\infty] \rightarrow R$ be defined by:
$$f(x)=x-log_{10}(x)$$
$$\Rightarrow f'(x)=1-\frac{1}{xln(10) }$$
We have:$\left\{\begin{matrix}
f'(x)=0\Leftrightarrow x=\frac{1}{ln10}\\
f'(100)>0\\
f'(0.1)<0
\end{matrix}\right.$
and $\lim_{x\rightarrow 0^+}{f(x)=+\infty}$
$\Rightarrow $ f(x) attains its local and global minimum at $x=\frac{1}{ln10}$
$\Rightarrow $$f(x)\geq f(\frac{1}{ln10})>0$ ∀ $x\in(0,+\infty]$ $(1)$
$$$$
+Let $f_1(x)=f(x)$
and $(f_n)_{n\in N}:A_n\subset (0,+\infty] \rightarrow R$ be defined as:
$$f_n(x)=f_{n-1}(log_{10}(x))$$
By (1), $∀n\in N,f_n(x)>0$ $∀ $ $x\in A_n$
Let $(g_n)_{n\in N}:A_n\rightarrow R$ be defined as:
$g_n=x-log_{10}(log_{10}(log_{10}(...(x))))$ (n
times)
$\Rightarrow ∀n\in N,g_n(x)=\sum_{i=1}^n{f_n(x)}>0$ $∀ $ $x\in(0,+\infty]$
$\Rightarrow g_3(x)=x-log_{10}(log_{10}(log_{10}(x)))$ >0 $∀ $ $x\in(0,+\infty]$
Hence the equation $x=log_{10}(log_{10}(log_{10}(x)))$ has no solution
REPLY [0 votes]: Hints: if $x$ is a solution, then
one must have $x>0$;
Also, $f(x)>0$ and $f(f(x))>0$ where $f(y)=\log_{10}(y)$ in increasing.
$x>f(x)$. | 206,084 |
As keen as Mustardé
A Scottish surgeon was recruited to go to Ghana by a charlatan. He went just the same, and ended up founding a vital new hospital, writes Anna Burnside
In 1991, at a conference in Argentina, an American doctor asked if he would join a Rotary-funded medical team bound for Ghana. Of course he said yes. “I didn’t want to sit and do nothing,” says Mustardé, now 89 and back home in Ayr. “I don’t play golf. I’ve got no eye for a ball.” He shrugs, as if it is inconceivable that a septuagenarian might have any qualms about heading off to a Third World country for three weeks. “And it was meant to be no surgery, only advice.”
It transpired that he had been recruited by a plausible con artist who had never performed surgery of any kind and was later struck off the medical register. However, when Mustardé arrived in Ghana, where… | 254,197 |
: Use the Support tab on your company's platform to get in touch with our friendly customer support team for help using the platform features.
For any other questions simply fill out the form to the right and our Small Business team will be in touch. | 171,234 |
TITLE: Model of spread of a rumor
QUESTION [5 upvotes]: From Stewart 7e pg 614 # 9
"One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard th eremor and the fraction who have not hear the rumor.
a) Write a differential equation that is satisfied by y.
b) Solve the differential equation
c) A small town has 1000 inhabitants. At 8 am 80 people have heard a rumor. By noon half the town has heard it. At what will 90 percent of the population have heard the rumor?
"
The wording of this is very ambiguous to me and I can't really make sense of it.
They mention a product, so I know that something is being multiplied and that y is a fraction which belongs to the population who have seen it so I think that "have not" heard is a constant, and that y is a fraction taht represents who have. I tried to set this up and it is the wrong answer. I am not sure what they want from that, the English usage is too ambiguous to make sense of it. The complete lack of punctuation is what really does it.
REPLY [5 votes]: A start: Let $y=y(t)$ be the fraction who have heard by time $t$. Then the fraction who have not is $1-y$. The rate of change of $y$, we are told, is proportional to the product $y(1-y)$. Our differential equation is therefore
$$\frac{dy}{dt}=ky(1-y).$$
This is a special case of the logistic equation, which you know how to solve.
It is convenient to let $t=0$ at $8\colon00$. So $y(0)=\frac{80}{1000}$. We are told that $y(4)=\frac{1}{2}$. These two items are enough to tell us everything about the equation, including the constant $k$. Some algebraic manipulation will be needed.
Now that you have the equation for $y(t)$ in terms of $t$, you can find the $t$ such that $y(t)=0.9$. Note that this $t$ is the time elapsed since $8\colon00$ AM. You will need to give the answer in clock terms.
REPLY [1 votes]: Let $y$ be the fraction of the population that has heard the rumor. The fraction that has not heard the rumor is then $1-y$. Our model would then be $y'=ay(1-y)$ where $a$ is the proportionality constant. Intuitively, the rumor spreads any time somebody who has heard the rumor meets somebody who has not. For a given meeting, the chance that exactly one has heard the rumor is $2y(1-y)$ and the $2$ can be absorbed in the definition ofFor c), your solution from b) has two unknowns: the constant of integration and $a$. You are given two data points, so should be able to determine both constants, then find the time when $y=0.90$ for the final answer. | 70,354 |
TITLE: Creating a function $f$ to measure "distance" in $\mathbb{R}_{> 0}$
QUESTION [0 upvotes]: I have two numbers $a, b \in \mathbb{R}_{ > 0}$.
Does there exist a function $f: \mathbb{R}_{> 0} \times \mathbb{R}_{> 0} \to \mathbb{R}$ such that:
if $a = b$, $f(a, b) = 0$;
$f(a, b) \in [0, 1]$ for all $a, b \in \mathbb{R}_{> 0}$;
$f$ is monotonically increasing with respect to $|a - b|$ (i.e., as $|a - b|$ increases, $f$ is increasing)?
I was considering using the logistic function (i.e., $$f(a, b) = \dfrac{\exp(|a-b|)}{1+\exp(|a-b|)}$$)
but if $a = b$, then $f(a, b) = 1/2$. Perhaps the most difficult part is trying to bound this in $[0, 1]$, for which I only know of the probit and logit transformations.
This is not a homework problem. Triangle inequality is a plus, but is not necessary.
REPLY [2 votes]: I think the usual thing to do here would be to use $$f(a,b) = \frac{\lvert a - b\rvert}{1 + \lvert a - b \rvert}.$$ This defines a bounded metric on $\mathbb R$ which generates the same topology as the standard metric $d(a,b) = \lvert a - b\rvert$. It is also clearly monotonically increasing in $\lvert a - b\rvert$ since the map $$g(x) = \frac{x}{1+x}, \,\,\,\,\, x \ge 0$$ has derivative $$g'(x) = \frac{1}{1+x} - \frac{x}{(1+x)^2} = \frac{1}{(1+x)^2} > 0$$ for all $x \ge 0$. | 151,854 |
Updated: Thursday the 24th of May, 2012 Fasteners: Flange
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SUPERTURN - Leicester
TFC Ltd - Heathfield | 325,372 |
\begin{document}
\maketitle
\begin{abstract}
A family $\mathcal A$ of subsets of an $n$-element set is called an
\textit{eventown} (resp. \textit{oddtown}) if all its sets have even
(resp. odd) size and
all pairwise intersections have even size. Using tools from linear
algebra, it was shown by Berlekamp and Graver that the maximum size
of an eventown is $2^{\left\lfloor n/2\right\rfloor}$. On the other
hand (somewhat surprisingly), it was proven by Berlekamp, that
oddtowns have size at most $n$. Over the last four decades, many
extensions of this even/oddtown problem have been studied.
In this paper we present new results on two such extensions. First,
extending a result of Vu, we show that a $k$-wise eventown (i.e.,
intersections of $k$ sets are even) has for $k \geq 3$ a unique
extremal configuration and obtain a stability result for this problem.
Next we improve some known bounds for the defect version of an
$\ell$-oddtown problem. In this problem we consider sets of size $\not\equiv 0 \pmod \ell$ where $\ell$ is a prime number $\ell$ (not necessarily $2$) and allow a few pairwise intersections to also have size $\not\equiv 0 \pmod \ell$.
\end{abstract}
\section{Introduction}
Let $\mathcal A=\{A_1,\ldots, A_m\}$ be a family of subsets of $[n]:=\{1,2,\ldots, n\}$. We say that $\mathcal A$ is an \textit{eventown} (resp. \textit{oddtown}) if all its sets have even (resp. odd) size and
\begin{align*}
|A_i\cap A_j| \;\text{ is even}&\text{ for }1\le i< j\le m
\end{align*}
Answering a question of Erd\H{o}s, Berlekamp~\cite{B69} and Graver~\cite{G75} showed independently that the maximum size of an eventown is $2^{\left\lfloor n/2\right\rfloor}$. Somewhat surprisingly, the answer changes drastically when one considers oddtowns. Indeed, Berlekamp~\cite{B69} proved that oddtowns have size at most $n$, which is easily seen to be best possible. The proofs of these two results relied on a technique known as the \textit{linear algebra bound} method, which has been widely used to tackle problems in Extremal Combinatorics ever since.
Over the last decades, many extensions of this even/oddtown problem have been studied. A natural extension is to consider the problem modulo $\ell \ge 2$. We say that $\mathcal A$ is a $\ell$-\textit{eventown} (resp. $\ell$-\textit{oddtown}) if all its sets have size $\equiv 0 \; (\bmod\; \ell)$ (resp. $\not \equiv 0 \; (\bmod\; \ell)$) and
\begin{align}
\label{eq:leventown}
|A_i\cap A_j| \equiv 0\; (\bmod\; \ell)\, \text{ for }1\le i< j\le m.
\end{align}
The problem of estimating the maximum possible size of an $\ell$-oddtown is nowadays fairly well understood. One can modify Berlekamp's proof for oddtowns slightly to show that if $\ell$ is a prime number then an $\ell$-oddtown has size at most $n$. With a bit of effort one can prove that the same still holds when $\ell$ is a prime power and that a weaker bound of $m\le c(\ell)n$ holds in general, where $c(\ell)$ is a constant depending on $\ell$. It remains an open problem whether one can take $c(\ell) = 1$ when $\ell$ is a composite number. For further details and related problems see the excellent monograph~\cite{BF92} of Babai and Frankl.
For $\ell$-eventowns a bit less is known. A natural lower bound construction for the maximum size of an $\ell$-eventown is $2^{\lfloor n/\ell\rfloor}$. This arises from considering $\lfloor n/\ell\rfloor$ disjoint subsets $B_1,\ldots, B_{\lfloor n/\ell\rfloor}$ of $[n]$ of size $\ell$ and taking $\mathcal A=\left\{\bigcup_{i\in S} B_i: \;S\subseteq \left[\lfloor n/\ell\rfloor\right]\right\}$. It turns out surprisingly that for large $\ell$ there are significantly larger $\ell$-eventowns. Indeed, Frankl and Odlyzko~\cite{FO83} found a nice construction of $\ell$-eventowns of size at least $(c\ell)^{\lfloor n/(4\ell)\rfloor}$, where $c>0$ is an absolute constant. Their construction relies on a clever use of Hadamard matrices. In addition, they showed that any $\ell$-eventown has size at most $2^{O\left(\log \ell/\ell\right)n}$ as $n\rightarrow \infty$. These two results combined certify that the maximum possible size of an $\ell$-eventown is of order $2^{\Theta(\log \ell/\ell)n}$ as $n\rightarrow \infty$.
Our results will focus on two other extensions of the even/oddtown problem that have been considered in the past. The first one extends property (\ref{eq:leventown}) to multiple intersections. The second one is a defect version of the $\ell$-oddtown problem, obtained by relaxing condition (\ref{eq:leventown}). We shall discuss these two extensions as well as our results in the next two subsections.
\subsection{Multiple intersections}
We say that $\mathcal A=\{A_1,\ldots, A_m\}$ is a \textit{$k$-wise $\ell$-eventown} if
\begin{align}
\label{eq:kwiseleventown}
\left|\bigcap_{i\in S}A_{i}\right|\equiv 0\; (\bmod\; \ell)\text{ for every non-empty }S\subseteq [m] \text{ of size } |S| = k.
\end{align}
For simplicity, we refer to a $k$-wise $2$-eventown simply as a $k$-wise eventown. We remark that a $2$-wise eventown is not the same as an eventown, since in the former we do not require that the sets themselves have even size.
The problem of maximizing the size of $k$-wise eventowns is nowadays well understood. For $k=1$, a $k$-wise eventown $\mathcal A$ is just a family of even-sized sets. Thus, $|\mathcal A| \le \sum_{i = 0}^{\lfloor n/2\rfloor}\binom{n}{2i}=2^{n-1}$, a bound which is attained by taking $\mathcal A$ to be the family of all subsets of $[n]$ of even size. The case $k=2$ was first considered in the papers of Berlekamp~\cite{B69} and Graver~\cite{G75} who showed that the maximum size of a $2$-wise eventown is $n+1$ if $n\le 5$, $2^{\lfloor n/2\rfloor}$ if $n\ge 6$ is even and $2^{\lfloor n/2\rfloor}+1$ if $n\ge 7$ is odd. Later, Vu~\cite{V97} addressed the general case:
\begin{thm}[Vu \cite{V97}]
\label{thm:vukwiseeventown}
There is a constant $c>0$ such that for any $k\ge 2$ the maximum size of a $k$-wise eventown in a universe of size $n\ge c\log_2 k$ is $2^{\lfloor n/2 \rfloor}$ if $n$ is even and $2^{\lfloor n/2\rfloor}+k-1$ if $n$ is odd.
\end{thm}
In Extremal Combinatorics, given an extremal result like Theorem~\ref{thm:vukwiseeventown}, it is common to ask what possible extremal configurations exist. In many problems, one can classify all the extremal configurations or at least describe some structural properties of these. When there is a unique extremal configuration, it is often the case that a \textit{stability} result holds. This means that one can give a precise structural description not just of the extremal configuration but also of nearly extremal configurations.
Given Theorem~\ref{thm:vukwiseeventown}, it is therefore natural to investigate what $k$-wise eventowns of maximum possible size look like, and whether a stability version of Theorem~\ref{thm:vukwiseeventown} exists. The next construction provides $k$-wise eventowns with the sizes indicated in Theorem~\ref{thm:vukwiseeventown}, for any $k\ge 2$ and $n\ge 2\lceil\log_{2}(k-1)\rceil$.
\begin{cons}
\label{cons:maxkwiseeventown}
(i) Let $B_1,\ldots, B_{\lfloor n/2\rfloor}$ be $\lfloor n/2\rfloor$ disjoint subsets of $[n]$ of size $2$. The family $\mathcal A=\left\{\bigcup_{i\in S} B_i: \;S\subseteq \left[\lfloor n/2\rfloor\right]\right\}$ is a $k$-wise eventown of size $2^{\lfloor n/2\rfloor}$ for every $k\in \NN$.
(ii) If $n$ is odd, let $B_1,\ldots, B_{\lfloor n/2\rfloor}$ and $\mathcal A$ be as in (i). Let $i\in[n]$ be the unique element not covered by the sets $B_1,\ldots, B_{\lfloor n/2\rfloor}$ and let $C_1,\ldots, C_{k-1}$ be any $k-1$ distinct sets in $\mathcal A$ (for this we need that $n\ge 2\lceil\log_{2}(k-1)\rceil$). If we add to $\mathcal A$ the $k-1$ sets $C_1\cup\{i\},\ldots, C_{k-1}\cup\{i\}$ then the resulting family is a $k$-wise eventown of size $2^{\lfloor n/2\rfloor}+k-1$.
\end{cons}
For $k=2$, the families considered in Construction~\ref{cons:maxkwiseeventown} are by no means the only examples of $2$-wise eventowns of maximum size. For example, for $n$ even, one can show that for any $2$-wise eventown $\mathcal A$ with even-sized sets, there exists a $2$-wise eventown $\mathcal B$ containing $\mathcal A$ of size $2^{\lfloor n/2\rfloor}$ (see, e.g., Ex. 1.1.10 of Babai-Frankl~\cite{BF92}). This allows one to produce many highly non-isomorphic $2$-wise eventowns of maximum possible size, by starting with very different looking small $2$-wise eventowns $\mathcal A$ with even-sized sets and then extending them to $2$-wise eventowns of maximum possible size. Given this phenomena, it is natural to ask what happens for $k\ge 3$. We prove that in this case the extremal construction of a $k$-wise eventown is unique. Moreover, a stability result holds.
\begin{thm}
\label{thm:kwiseeventown}
Let $\mathcal A$ be a $k$-wise eventown on $[n]$ for some $k\ge 3$. If $|\mathcal A|> \frac{3}{4}2^{\lfloor n/2\rfloor}+(k-1)n$ and $n\ge 2\lceil \log_{2}(k-1)\rceil + 4$ then $\mathcal A$ is a subfamily of a family in Construction~\ref{cons:maxkwiseeventown}.
\end{thm}
In order to establish Theorem~\ref{thm:kwiseeventown} it will be convenient for us to consider a strengthening of (\ref{eq:kwiseleventown}). We say that $\mathcal A$ is a \textit{strong $k$-wise $\ell$-eventown} if it is a $k'$-wise $\ell$-eventown for every $k'\in \{1,2,\ldots, k\}$. The problem of estimating the maximum size of a strong $k$-wise eventown is a simple one. For $k=1$, a strong $k$-wise eventown is the same as a $k$-wise eventown and so, as mentioned earlier, its maximum possible size is $2^{n-1}$. For $k\ge 2$, a strong $k$-wise eventown is also an eventown and thus has size at most $2^{\lfloor n/2\rfloor}$. Construction~\ref{cons:maxkwiseeventown} (i) certifies that strong $k$-wise eventowns of this size exist for every $k$. As was the case with $2$-wise eventowns, there are many highly non-isomorphic strong $2$-wise eventowns of size $2^{\lfloor n/2\rfloor}$. However, as our next result shows, for $k\ge 3$ the families in Construction~\ref{cons:maxkwiseeventown} (i) are the only strong $k$-wise eventowns of size $2^{\lfloor n/2\rfloor}$ and, furthermore, a stability result holds.
\begin{thm}
\label{thm:strongkwiseeventown}
If $\mathcal A$ is a $k$-wise eventown in $[n]$ for every $k\in \NN$, then there exist disjoint even-sized subsets $B_1,\ldots, B_s$ of $[n]$ such that $\mathcal A\subseteq \{\bigcup_{i\in S}B_i:S\subseteq [s]\}$. Furthermore, for $k\ge 2$, if $\mathcal A$ is a strong $k$-wise eventown in $[n]$ but not a $(k+1)$-wise eventown then $|\mathcal A|\le 2^{\left\lfloor n/2\right\rfloor-\left(2^{k}-k-2\right)}$.
\end{thm}
We remark that strong $k$-wise eventowns which are not $(k+1)$-wise eventowns only exist for $n\ge 2^{k+1}-1$. Moreover, the upper bound in Theorem~\ref{thm:strongkwiseeventown} is best possible as there exist strong $k$-wise eventowns of size $2^{\lfloor n/2\rfloor -(2^k-k-2)}$ which are not $(k+1)$-wise eventowns for any $n\ge 2^{k+1}-1$. We discuss this in Section~\ref{section:kwiseeventowns} after proving Theorem~\ref{thm:strongkwiseeventown}.
Far less is known about the maximum possible size of (strong) $k$-wise $\ell$-eventowns when $\ell > 2$. We address this problem in Section~\ref{section:concludingremarks}.
\subsection{Defect version for $\ell$-oddtowns}
We say that $\mathcal A=\{A_1,\ldots, A_m\}$ is a \textit{$d$-defect $\ell$-oddtown} if for every $i\in [m]$ we have $|A_i|\not \equiv 0\; (\bmod\; \ell)$ and there are at most $d$ indices $j\in [m]\setminus \{i\}$ such that $|A_i\cap A_j|\not \equiv 0\;(\bmod \;\ell)$. Note that a $0$-defect $\ell$-oddtown is the same as an $\ell$-oddtown. For simplicity, we refer to a $d$-defect $2$-oddtown simply as a $d$-defect oddtown. Vu~\cite{V99} considered the problem of maximizing the size of a $d$-defect oddtown, solving it almost completely. His results imply the following:
\begin{thm}[Vu~\cite{V99}]
\label{thm:vu}The maximum size of a $d$-defect oddtown in $[n]$ is $(d+1)(n-2\lceil \log_{2}(d+1)\rceil)$,
for any $d\ge 0$ and $n\ge d/8$.
\end{thm}
For $\ell>2$, Vu observed that the maximum size of a $d$-defect $\ell$-oddtown is at most $(d+1)n$ if $\ell$ is a prime number and at least $(d+1)(n-\ell\lceil \log_{2}(d+1)\rceil)$ for every $\ell$. Our next result improves Vu's upper bound of $(d+1)n$ on the maximum size of a $d$-defect $\ell$-odtown, when $\ell >2$ is a prime number.
\begin{thm}
\label{thm:ddefectloddtown}
Let $\ell$ be a prime number and suppose $\mathcal A$ is a $d$-defect $\ell$-oddtown in the universe $[n]$. There is a constant $C>0$ such that if $n\ge Cd\log d$ then $|\mathcal A|\le (d+1)\left(n-2\left(\lceil \log_{2}(d+2)\rceil-1\right)\right)$.
\end{thm}
For $d=1$ we can show that this upper bound is essentially best possible:
\begin{thm}
\label{thm:1defectloddtown}
Let $\ell$ be a prime number. If $\mathcal A$ is a $1$-defect $\ell$-oddtown in $[n]$ then $|\mathcal A|\le \max\{n,2n-4\}$. Moreover, there exist $1$-defect $\ell$-oddtowns of size $2n-4$ for infinitely many values of $n$.
\end{thm}
It turns out that Vu's lower bound of $(d+1)(n-\ell\lceil \log_{2}(d+1)\rceil)$ can also be improved for some values of $d$ and $\ell$. We discuss this briefly in the last section of the paper.
\medskip
\noindent
\textbf{Organization of the paper:} In Section~\ref{section:auxiliaryresults} we introduce some auxiliary lemmas which we need in the proofs of our results. In Section~\ref{section:kwiseeventowns} we present the proofs of Theorems~\ref{eq:kwiseleventown} and \ref{thm:strongkwiseeventown}. In Section~\ref{section:ddefectloddtowns} we prove Theorems~\ref{thm:ddefectloddtown} and \ref{thm:1defectloddtown}. Finally, in Section~\ref{section:concludingremarks} we discuss further extensions of the problems considered as well as related open problems.
\section{Auxiliary results}
\label{section:auxiliaryresults}
The following lemma (see, e.g. Ex. 1.1.8 of \cite{BF92}) will be useful for us in the proof of Theorem~\ref{thm:kwiseeventown}.
\begin{lemma}[Skew Oddtown Theorem]
\label{lemma:skewoddtown}
Suppose $R_1,\ldots, R_m$ and $B_1,\ldots, B_m$ are subsets of $[n]$ such that the following conditions hold:
\begin{enumerate}
\item[$(a)$] $|R_i\cap B_i|\not\equiv 0 \pmod 2$ for every $i\in [m]$;
\item[$(b)$] $|R_i\cap B_j|\equiv 0 \pmod 2$ for $1\le i< j\le m$.
\end{enumerate}
Then $m\le n$.
\end{lemma}
For any graph $G$ we denote by $\chi(G)$ and $\Delta(G)$ the chromatic number and maximum degree of $G$, respectively. Recall that for any graph $G$ one has $\chi(G)\le \Delta(G)+1$ (see, e.g., \cite{D10}). In the proof of Theorem~\ref{thm:ddefectloddtown} we will be interested in the cases in which equality holds. For that matter we make use of Brooks' Theorem~\cite{B41}.
\begin{thm}[Brooks' Theorem]
\label{thm:Brooks}
For any graph $G$, we have $\chi(G)\le \Delta(G)$ unless $G$ contains a copy of $K_{\Delta(G)+1}$ or $\Delta(G)=2$ and $G$ contains a cycle of odd length.
\end{thm}
The next auxiliary lemmas use basic linear algebra. All the vector spaces considered will be over the field $\mathbb{F}_{\ell}$ where $\ell$ is a prime number and the dot product considered will always refer to the standard inner product such that $(x_1,\ldots,x_n)\cdot (y_1,\ldots, y_n) = \sum_{i=}^{n}x_iy_i$ for $(x_1,\ldots, x_n),(y_1,\ldots,y_n)\in \mathbb F_{\ell}^{n}$. We will say that a subspace $U$ of $\mathbb F_{\ell}^{n}$ is \textit{non-degenerate} if the dot product in $U$ is a non-degenerate bilinear form, meaning that for any non-zero vector $u\in U$ there exists $v\in U$ such that $u\cdot v \neq 0$. The next well-known lemma follows from Proposition 1.2 of Chapter XV of \cite{L02}.
\begin{lemma}
\label{lemma:linearalgebra}
Let $V$ be a non-degenerate subspace of $\mathbb{F}_{\ell}^{n}$ and $U$ a subspace of $V$. Denote by $U^{\perp}$ the orthogonal complement of $U$ in $V$ with respect to the dot product. Then:
\begin{enumerate}
\item[$(a)$] $\dim U+\dim U^{\perp}=\dim V$.
\item[$(b)$] If $U$ is non-degenerate then $U^{\perp}$ is also non-degenerate.
\end{enumerate}
\end{lemma}
Note that any $d$ linearly independent vectors in $\mathbb F_{\ell}^{n}$ span a subspace of size $\ell^{d}$. Therefore, given $t$ distinct vectors $v_1,\ldots, v_t$ in $\mathbb F_{\ell}^{n}$ one can always find $\lceil\log_{\ell}t\rceil$ of them which are linearly independent (e.g. take a basis of the subspace spanned by $v_1,\ldots, v_t$ consisting of vectors from this set). This is best possible in general but it can be improved under certain conditions on these vectors. A good example of this, is the following theorem of Odlyzko \cite{O81} which will be useful for us.
\begin{thm}
\label{thm:(0,1)-vectors}
Let $\ell$ be a prime number and $n$ a natural number. Given $t$ distinct $\{0,1\}$-vectors in $\mathbb F_{\ell}^{n}$ one can find at least $\lceil\log_{2}t\rceil$ of them which are linearly independent.
\end{thm}
In the proof of Theorem~\ref{thm:ddefectloddtown} we will make use of the following lemma of this type.
\begin{lemma}
\label{lemma:dimension}
Suppose $b_1,\ldots,b_{t}$ are distinct $\{0,1\}$-vectors in a non-degenerate subspace $W$ of $\mathbb{F}_{\ell}^{n}$ such that $(b_1\cdot b_1)(b_i\cdot b_j)=(b_1\cdot b_i)(b_1\cdot b_j)\neq 0$ for every $i,j\in [t]$. Then $\dim W\ge 2\lceil \log_{2}(t+1)\rceil-1$.
\end{lemma}
\begin{proof}
For each $i\in [t]$ define $c_i:=(b_1\cdot b_1)b_{i}-(b_1\cdot b_i)b_{1}$. Let $B$ and $C$ be the linear subspaces generated by $b_1,\ldots, b_{t}$ and $c_1,\ldots, c_{t}$, respectively, and let $C^{\perp}$ denote the orthogonal complement of $C$ in $W$ with respect to the dot product. Note that
\[c_{i}\cdot b_{j}=(b_1\cdot b_1)(b_i\cdot b_j)-(b_1\cdot b_i)(b_1\cdot b_j)=0\]
for every $i,j\in [t]$ and so it follows that $C\subseteq B\subseteq C^{\perp}$. Moreover, we know that $b_1\notin C$ since $b_1\cdot b_1\neq 0$ and so $\dim C\le \dim B -1$. In addition, by the definition of the vectors $c_1,\ldots, c_{t}$ it follows that $B=C+\text{span}(b_1)$ and so $\dim C\ge \dim B-1$. We conclude then that $\dim C=\dim B-1$.
By (a) of Lemma~\ref{lemma:linearalgebra} we have $\dim C+\dim C^{\perp}=\dim W$ and so we get:
\[\dim W \ge \dim B+\dim C=2\dim B-1\]
Finally, since $b_1,\ldots, b_{t}$ and the $0$-vector are $t+1$ distinct $\{0,1\}$-vectors (because $b_i\cdot b_i\neq 0$) it follows from Theorem~\ref{thm:(0,1)-vectors} that $\dim B\ge \lceil\log_{2}(t+1)\rceil$.
\end{proof}
\noindent
\textbf{Remark:} For $\ell = 2$, since all the vectors in $\mathbb{F}_{2}^{n}$ are $\{0,1\}$-vectors, one can apply Theorem~\ref{thm:(0,1)-vectors} to the vectors in $C$ to get the stronger bound $\dim W\ge 2\lceil\log_{2}t\rceil+1$. We believe that one should be able to get the same bound for any prime $\ell$.
\section{$k$-wise eventowns}
\label{section:kwiseeventowns}
In this section we present the proofs of Theorems~\ref{thm:kwiseeventown} and \ref{thm:strongkwiseeventown}. The main ingredient in the proof of Theorem~\ref{thm:kwiseeventown} is the structure of large strong $k$-wise eventowns obtained from Theorem~\ref{thm:strongkwiseeventown}. Therefore, we start with the proof of the latter and later use it to deduce the proof of the former.
\subsection{Proof of Theorem~\ref{thm:strongkwiseeventown}}
\label{subsection:strongkwiseeventown}
In the next lemma, we prove the first half of the statement in Theorem~\ref{thm:strongkwiseeventown}, characterizing the families which are $k$-wise eventowns for every $k\in \NN$.
\begin{lemma}
\label{lemma:blockfamily}
If $\mathcal A$ is a $k$-wise eventown for every $k\in \NN$, then there exist disjoint even-sized subsets $B_1,\ldots, B_s$ of $[n]$ such that $\mathcal A\subseteq \{\bigcup_{i\in S}B_i:S\subseteq [s]\}$.
\end{lemma}
\begin{proof}
Suppose $\mathcal A=\{A_1,\ldots, A_m\}$ is a $k$-wise eventown for every $k\in \NN$. Define for each $i\in [m]$ the sets $A^{0}_{i}:=A_i$ and $A^{1}_{i}:=[n]\setminus A_i$. Set $\mathcal T =\{0,1\}^{m}\setminus\{(1,1,\ldots,1)\}$ and given a tuple $t=(t_i)_{i\in [m]}\in \mathcal T$ let $B_t:=\bigcap_{i\in [m]}A^{t_i}_i$. To prove Lemma~\ref{lemma:blockfamily} it suffices to show that the sets $\{B_t:t\in \mathcal T\}$ satisfy:
\begin{enumerate}
\item[(a)] for every $i\in [m]$ there exists a set $T_i\subseteq \mathcal T$ such that $A_i=\cup_{t\in T_i}B_t$
\item[(b)] for any $t,t'\in \mathcal T$, if $t\neq t'$ then $B_t\cap B_{t'}=\emptyset$.
\item[(c)] $|B_t|$ is even for every $t\in \mathcal T$.
\end{enumerate}
We start by showing that (a) holds. Given $i\in [m]$ let $T_i=\{t\in \mathcal T: t_i=0\}$. Note that for any $t\in T_i$ we have $B_t=\bigcap_{j\in [m]}A^{t_j}_j\subseteq A_i$ since the term $A_i^{t_i}=A_i$ appears in this intersection. Thus, it follows that $\bigcup_{t\in T_i}B_t\subseteq A_i$. Now, note that for each $a\in A_i$ there exists $t\in T_i$ such that $a\in B_t$. Indeed, just consider $t_j=0$ if $a\in A_j$ and $t_j=1$ otherwise. Thus, it follows also that $A_i\subseteq \bigcup_{t\in T_i}B_t$.
Next, we show that (b) holds. Suppose $t\neq t'$ and let $i\in [m]$ be such that $t_i\neq t'_i$. Then $B_t\subseteq A_i^{t_i}$ and $B_{t'}\subseteq A_i^{t'_i}$. Since $t_i\neq t'_i$ it follows that $A_i^{t_i}\cap A_i^{t'_i}=\emptyset$ and so $B_t\cap B_{t'}=\emptyset$.
Finally, we show that $(c)$ holds. Given $t\in \mathcal T$ we have:
\begin{align*}|B_t|=&\left|\left(\bigcap_{i\in [m], t_i=0}A_i\right)\cap\left(\bigcap_{i\in [m],t_i=1}[n]\setminus A_i\right)\right|\\
=&\left|\left(\bigcap_{i\in [m], t_i=0}A_i\right)\setminus\left(\bigcup_{i\in [m],t_i=1}A_i\right)\right|\\
=& \left|\left(\bigcap_{i\in [m], t_i=0}A_i\right)\right|-\left|\left(\bigcap_{i\in [m], t_i=0}A_i\right)\cap \left(\bigcup_{i\in [m],t_i=1}A_i\right)\right|.
\end{align*}
The first term is the intersection of a positive number of sets in $\mathcal A$ (since $t\neq (1,1,\ldots,1)$) and thus has even size since $\mathcal A$ is a $k$-wise eventown for every $k\in \NN$. Moreover, the second term can be written, by the inclusion-exclusion principle, as a sum of signed intersection sizes of sets in $\mathcal A$. Thus, the second term is also even, implying that $|B_t|$ is even.
\end{proof}
\medskip
For the second half of the statement of Theorem~\ref{thm:strongkwiseeventown} we will use basic linear algebra techniques. Given a set $A\subseteq [n]$ let $v_A\in\mathbb{F}_{2}^{n}$ denote its $\{0,1\}$-characteristic vector. We consider the following two correspondences between families $\mathcal A\subseteq 2^{[n]}$ and linear subspaces $V\subseteq\mathbb{F}_{2}^{n}$:
\[\mathcal A\mapsto V_{\mathcal A}:=\text{span}\{v_{A}:A\in\mathcal A\}\;\;\text{and}\;\;V\mapsto \mathcal A_V:=\{A\subseteq [n]: v_{A}\in V\}\]
Given $\mathcal A\subseteq 2^{[n]}$, we define $\overline{\mathcal A}:=\mathcal A_{V_{\mathcal A}}$ which we call the \textit{linear closure of $\mathcal A$}. Note that $\mathcal A\subseteq \overline{\mathcal A}$, but equality does not necessarily hold. As the next lemma shows, an important property of linear closure is that it preserves the property of being a strong $k$-wise eventown.
\begin{lemma}
\label{lemma:linearclosure}
If $\mathcal A$ is a strong $k$-wise eventown then $\overline{\mathcal A}$ is also a strong $k$-wise eventown.
\end{lemma}
\begin{proof}
Given a set $B\subseteq [n]$ define the function $f_{B}:[n]\rightarrow \mathbb F_{2}$ such that
\[f_{B}(i)=\left\{\begin{matrix}
1 & \text{ if } i\in B\\
0 & \text{ if } i\notin B
\end{matrix}\right.\]
and note that:
\begin{enumerate}
\item[(i)] for any $B\subseteq [n]$ we have $|B|= \sum_{i\in [n]}f_B(i)\pmod 2$;
\item[(ii)] for any $t$ sets $B_1,\ldots,B_t\subseteq [n]$ we have $f_{\cap_{i\in [t]}B_i}=\prod_{i\in [t]}f_{B_i}$;
\item[(iii)] if $A_1,\ldots, A_{t}, B\subseteq [n]$ are such that $v_{B}=\sum_{i\in [t]}v_{A_i}$ then $f_{B}=\sum_{i\in [t]}f_{A_{i}}$.
\end{enumerate}
Now, let $B_1,\cdots,B_k$ be any $k$ not necessarily distinct sets in $\overline{\mathcal A}$. We want to show that $\bigcap_{j\in [k]}B_j$ has even size. Since $\overline{\mathcal A}$ is the span of the vectors $\{v_A\}_{A\in \mathcal A}$, we know that for each $j\in [k]$ there are sets $A_1^{j},\ldots, A_{t_{j}}^{j}\in \mathcal A$ such that $v_{B_j}=\sum_{i\in [t_j]}v_{A_{i}^{j}}$. Thus, by properties (i), (ii) and (iii) it follows that
\begin{align*}
\left|\bigcap_{j\in[k]} B_j\right|&= \sum_{i\in [n]}f_{\cap_{j\in[k]} B_j}(i)\\
&= \sum_{i\in [n]}\prod_{j\in [k]}f_{B_j}(i)\\
&= \sum_{i\in [n]}\prod_{j\in [k]}\sum_{h\in [t_j]}f_{A_h^{j}}(i)\\
&=\sum_{i\in [n]}\sum_{(h_1,\ldots, h_k)}f_{\cap_{j\in [k]}A_{h_j}^{j}}(i)\\
&= \sum_{(h_1,\ldots,h_k)}\sum_{i\in [n]}f_{\cap_{j\in [k]}A_{h_j}^{j}}(i)\\
&= \sum_{(h_1,\ldots,h_k)}\left|\cap_{j\in [k]}A_{h_j}^{j}\right|\pmod 2
\end{align*}
where the sums indexed with $(h_1,\ldots,h_k)$ run over all tuples in $[t_1]\times\ldots\times[t_k]$.
Since $\mathcal A$ is a strong $k$-wise eventown we conclude that all the terms in the last sum are even. Thus, for any $k$ not necessarily distinct sets $B_1,\ldots, B_k\in\overline{\mathcal A}$ the set $\bigcap_{j\in[k]} B_j$ has even size, i.e., $\overline{\mathcal A}$ is a strong $k$-wise eventown.
\end{proof}
With Lemma~\ref{lemma:linearclosure} we are ready to present the proof of the second half of the statement of Theorem~\ref{thm:strongkwiseeventown}.
\begin{lemma}
\label{lemma:upperbound}
For $k\ge 2$, if $\mathcal A\subseteq 2^{[n]}$ is a strong $k$-wise eventown but not a $(k+1)$-wise eventown then:
\[|\mathcal A|\le 2^{\left\lfloor n/2\right\rfloor-\left(2^{k}-k-2\right)}\]
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lemma:upperbound}]
Suppose $\mathcal A\subseteq 2^{[n]}$ is a strong $k$-wise eventown which is not a $(k+1)$-wise eventown and let $A_1,\ldots, A_{k+1}\in \mathcal A$ be such that $|A_1\cap\ldots\cap A_{k+1}|$ is odd. For each $S\subseteq [k+1]$ define the set $A_{S}:=\bigcap_{i\in S}A_i$, let $\mathcal S=\{S\subseteq [k]: 2\le |S|\le k-1\}$ and define $\mathcal B:=\{A_{S}\}_{S\in\mathcal S}$. We claim that the family $\mathcal C=\mathcal A\cup\mathcal B$ is an eventown. Indeed, this holds since:
\begin{enumerate}
\item[1)] all sets in $\mathcal A$ and pairwise intersections between sets in $\mathcal A$ have even size since $\mathcal A$ is a strong $k$-wise eventown and $k\ge 2$;
\item[2)] all sets in $\mathcal B$ have even size since they are the intersection of at most $k-1$ sets in $\mathcal A$;
\item[3)] for any $A\in \mathcal A$ and $S\in\mathcal S$ the set $A\cap A_{S}=A\cap \left(\bigcap_{i\in S}A_i\right)$ is the intersection of at most $k$ sets in $\mathcal A$, and thus has even size;
\item[4)] for any $S_1,S_2\in\mathcal S$ the set $A_{S_1}\cap A_{S_2}=\bigcap_{i\in S_1\cup S_2}A_i$ is the intersection of at most $k$ sets in $\mathcal A$, and thus has even size.
\end{enumerate}
We claim now that $\dim V_{\mathcal C}=\dim V_{\mathcal A} + \dim V_{\mathcal B}$ and that $\dim V_{\mathcal B} = |\mathcal S|=2^{k}-k-2$. If this is the case then:
\[\left|\overline{\mathcal C}\right|=2^{\dim V_{\mathcal C}}=2^{\dim V_{\mathcal A}}\cdot 2^{\dim V_{\mathcal B}}\ge |\mathcal A|\cdot 2^{2^{k}-k-2}\]
and since $\overline{\mathcal C}$ is an eventown by Lemma~\ref{lemma:linearclosure}, we conclude that
\[|\mathcal A|\le \left|\overline{\mathcal C}\right|\cdot 2^{-\left(2^{k}-k-2\right)}\le 2^{\left\lfloor n/2\right\rfloor-\left(2^{k}-k-2\right)}\]
as desired. Thus, it remains to prove the claim. For that, it suffices to prove that if there is a linear relation
\begin{align}
\label{eq:lindep2}
\sum_{A\in \mathcal A}\alpha_{A}v_{A}+\sum_{S\in \mathcal S}\beta_{S}v_{A_S}=0
\end{align}
then $\beta_{S}=0$ for any $S\in \mathcal S$. Define for each $S\in \mathcal S$ the set $S^{c}:=[k+1]\setminus S$ and note that for any $A\in \mathcal A$ and $S,T\in \mathcal S$ we have:
\begin{enumerate}
\item[(i)] $v_{A}\cdot v_{A_{T^{c}}}=|A\cap\left(\bigcap_{i\in T^{c}}A_i\right)|= 0\pmod 2$ because the latter is the intersection of at most $k$ sets in $\mathcal A$, since $|T^{c}|=k+1-|T|\le k-1$;
\item[(ii)] if $S\cup T^{c}\neq [k+1]$ then $v_{A_S}\cdot v_{A_{T^{c}}}=|\bigcap_{i\in S\cup T^{c}}A_i|= 0\pmod 2$ because the latter is the intersection of at most $k$ sets in $\mathcal A$;
\item[(iii)] $v_{A_T}\cdot v_{A_{T^{c}}}=|\bigcap_{i\in[k+1]}A_i|= 1\pmod 2$.
\end{enumerate}
Consider now a linear relation as in equation~(\ref{eq:lindep2}) and suppose that there is some set $S\in \mathcal S$ such that $\beta_{S}\neq 0$. Let $T\in \mathcal S$ be such a set of maximum possible size and note that for any $S\in \mathcal S\setminus \{T\}$ with $\beta_{S} \neq 0$ we have $T\not\subseteq S$, or equivalently $
S\cup T^{c}\neq [k+1]$. Therefore, it follows from (i), (ii) and (iii) that
\[0 = \left(\sum_{A\in \mathcal A}\alpha_{A}v_{A}+\sum_{S\in \mathcal S}\beta_{S}v_{A_S}\right) \cdot v_{A_{T^{c}}}=\sum_{A\in \mathcal A}\alpha_{A}\left(v_{A}\cdot v_{A_{T^{c}}}\right)+\sum_{S\in \mathcal S}\beta_{S}\left(v_{A_S}\cdot v_{A_{T^{c}}}\right) =\beta_{T}\]
contradicting the choice of $T$. This proves the claim.
\end{proof}
Note that Lemma~\ref{lemma:upperbound} implies that there is no strong $k$-wise eventown in $[n]$ that is not a $(k+1)$-wise eventown if $\lfloor n/2\rfloor < 2^{k}-k-2$. In fact, one actually needs that $n\ge 2^{k+1}-1$ for such families to exist. The reason for this is quite simple. If $\mathcal A$ is not a $(k+1)$-wise eventown then there exist sets $A_1,\ldots, A_{k+1}\in \mathcal A$ for which $|A_1\cap \ldots, A_{k+1}|$ is odd. Since the intersection of the sets in any proper non-empty subfamily of $\{A_1,\ldots, A_{k+1}\}$ has even size then one can use the principle of inclusion-exclusion to show that in fact $|A'_1\cap \ldots \cap A'_{k+1}|$ is odd for any choice of $A'_i \in \{A_i, [n]\setminus A_i\}$ for $i\in [k+1]$, with the exception of the choice $A'_i = [n]\setminus A_i$ for every $i\in [k+1]$ (when $n$ is odd). This implies that there are at least $2^{k+1}-1$ disjoint non-empty sets in $[n]$, implying that $n\ge 2^{k+1}-1$.
We show next that for any $n\ge 2^{k+1}-1$ there are strong $k$-wise eventowns $\mathcal A$ in $[n]$ of size $|\mathcal A|=2^{\lfloor n/2\rfloor-(2^{k}-k-2)}$ which are not $(k+1)$-wise eventowns. We start by constructing a strong $k$-wise eventown consisting of $2^{k+2}$ subsets of $[2^{k+1}]$ which is not a $(k+1)$-wise eventown.
For convenience, let us denote by $2^{[k+1]}$ the family of all subsets of the set $[k+1] = \{1,\ldots, k+1\}$ and let $f:2^{[k+1]}\rightarrow [2^{k+1}]$ be any bijection. Let $B_0=[2^{k+1}]$ and for each $i\in [k+1]$ define $B_i=\{f(S):i\in S\subseteq [k+1]\}$. Note that for any set $I\subseteq \{0,1,\ldots,k+1\}$ we have:
\[\left|\bigcap_{i\in I}B_i\right|=\left|\bigcap_{i\in I\setminus \{0\}}B_i\right|=\left|\left\{f(S):\left(I\setminus \{0\}\right)\subseteq S\subseteq [k+1]\right\}\right|=2^{k+1-\left|I\setminus \{0\}\right|}\]
and so the family $\mathcal B=\{B_0,B_1,\ldots, B_{k+1}\}$ is a strong $k$-wise eventown but not a $(k+1)$-wise eventown. Hence, by Lemma~\ref{lemma:linearclosure} it follows that $\overline{\mathcal B}$, the linear closure of $\mathcal B$, is also a strong $k$-wise eventown but not a $(k+1)$-wise eventown.
We claim now that the vectors $v_{B_0},\ldots, v_{B_{k+1}}$ are linearly independent. Indeed, this follows from the next observations:
\begin{itemize}
\item $v_{\{f(\emptyset)\}}\cdot v_{B_0} = 1$ and $v_{\{f(\emptyset)\}}\cdot v_{B_i} = 0$ for $i\in [k+1]$ since $f(\emptyset)\not\in B_i$.
\item for $i,j\in[k+1]$ we have $v_{\{f(\{i\})\}}\cdot v_{B_j} = \left\{\begin{matrix}
1 & \text{if }i=j\\
0 & \text{if }i\neq j
\end{matrix}\right.$.
\end{itemize}
Therefore, $\overline{\mathcal B}$ is a strong $k$-wise eventown in $[2^{k+1}]$ of size $|\overline{\mathcal B}| = 2^{\dim V_{\mathcal B}} = 2^{k+2}$ which is not a $(k+1)$-wise eventown.
Now, if $n\ge 2^{k+1}$ let $\mathcal C$ be a strong $k$-wise eventown in $[n]\setminus [2^{k+1}]$ of size $2^{\lfloor (n-2^{k+1})/2\rfloor}$ as in Construction~\ref{cons:maxkwiseeventown} (i). Since $\overline{\mathcal B}$ and $\mathcal C$ are both strong $k$-wise eventowns and $\overline{\mathcal B}$ is not a $(k+1)$-wise eventown, a moment's thought reveals that the family
\[\mathcal A = \{B\cup C: B\in \overline{\mathcal B}, C\in \mathcal C\}\]
is a strong $k$-wise eventown in $[n]$ of size
\[|\mathcal A|= 2^{k+2}\cdot 2^{\lfloor (n-2^{k+1})/2\rfloor} = 2^{\lfloor n/2\rfloor-(2^{k}-k-2)}\]
which is not a $(k+1)$-wise eventown.
When $n=2^{k+1}-1$, note that if we choose the bijection $f$ above such that $f(\emptyset) = 2^{k+1}$ then the sets $B_1,\ldots, B_{k+1}$ are subsets of $[2^{k+1}-1] = [n]$. Therefore, in a similar way as above, we can conclude that the linear closure of $\{B_1,\ldots, B_{k+1}\}$ will be a strong $k$-wise eventown in $[n]$ of size $2^{k+1} = 2^{\lfloor n/2\rfloor - (2^{k}-k-2)}$ which is not a $(k+1)$-wise eventown.
\subsection{Proof of Theorem~\ref{thm:kwiseeventown}}
We will use Theorem~\ref{thm:strongkwiseeventown} in order to prove Theorem~\ref{thm:kwiseeventown}. The reason why we can do this is because, as the next lemma shows, any $k$-wise eventown contains a large strong $k$-wise eventown.
\begin{lemma}
\label{lemma:strongsubfamilies}
If $\mathcal A$ is a $k$-wise eventown on $[n]$ then it contains a subfamily $\mathcal A'$ of size $|\mathcal A'|\ge |\mathcal A|-(k-1)n$ which is a strong $k$-wise eventown.
\end{lemma}
\begin{proof}
Set $\mathcal A_0 := \mathcal A$ and, for $i\ge 0$, as long as $\mathcal A_{i}$ is not a strong $k$-wise eventown let $A^{i}_1,\ldots,A^{i}_{k_{i}}$ be a maximal collection of less than $k$ distinct sets in $\mathcal A_{i}$ such that $|A^{i}_1\cap\ldots\cap A^{i}_{k_{i}}|\not\equiv 0\pmod 2$ and set $\mathcal A_{i+1}:=\mathcal A_{i}\setminus\{A^{i}_1,\ldots, A^{i}_{k_i}\}$. After a finite number of iterations of this procedure, say $s$ iterations, we obtain a (possibly empty) subfamily $\mathcal A'$ of $\mathcal A$ which is a strong $k$-wise eventown. Since at each step $i<s$ the family $\mathcal A_{i+1}$ is obtained from $\mathcal A_i$ by removing $k_i \le k-1$ sets, we have:
\[|\mathcal A'|\ge |\mathcal A|-(k-1)s.\]
Thus, it suffices then to show that $s\le n$.
For each $i\in [s]$ define the sets $R_i:=A_1^{i}\cap\ldots A_{k_i}^{i}$ and $B_i:=A_1^{i}$ and note that:
\begin{enumerate}
\item[(a)] $|R_i\cap B_i|=|A_1^{i}\cap\ldots A_{k_i}^{i}|\not\equiv 0 \pmod 2$ for every $i\in [s]$;
\item[(b)] $|R_i\cap B_j|=|A_1^{i}\cap\ldots A_{k_i}^{i}\cap A_1^{j}|\equiv 0 \pmod 2$ for $i< j$ since otherwise $A_1^{i},\ldots, A_{k_i}^{i}, A_1^{j}$ would be a collection of $k_i+1$ distinct sets in $\mathcal A_i$ whose intersection has odd size, contradicting the maximality in the choice of the sets $A_1^{i},\ldots, A_{k_i}^{i}$ (note that $k_i + 1 < k$ since $\mathcal A$ is a $k$-wise eventown).
\end{enumerate}
Thus, by Lemma~\ref{lemma:skewoddtown} it follows that $s\le n$, as desired.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:kwiseeventown}]
By Lemma~\ref{lemma:strongsubfamilies}, the family $\mathcal A$ contains a subfamily $\mathcal A'$ of size
\begin{align}
\label{eq:sizeofsubfamily}
|\mathcal A'|\ge |\mathcal A|-(k-1)n> \frac{3}{4}2^{\lfloor n/2\rfloor}
\end{align}
which is a strong $k$-wise eventown. Thus, since $2^{\lfloor n/2\rfloor -(2^{k}-k-2)}\le \frac{1}{8}2^{\lfloor n/2\rfloor}\le\frac{3}{4}2^{\lfloor n/2\rfloor}$, it follows from Theorem~\ref{thm:strongkwiseeventown} that there are non-empty disjoint subsets $B_1,\ldots, B_s$ of $[n]$ of even size such that $\mathcal A'\subseteq \mathcal B:=\{\bigcup_{i\in S}B_i : S\subseteq [s]\}$. Note that $|\mathcal A'|\le |\mathcal B|\le 2^{s}$ and so it follows from (\ref{eq:sizeofsubfamily}) that $s\ge\lfloor n/2\rfloor$. Furthermore, since the sets $B_1,\ldots, B_s$ are non-empty disjoint subsets of $[n]$ of even size, we must have $s=\lfloor n/2\rfloor$ and $|B_i|=2$ for every $i\in [s]$.
We claim now that for any $A\in \mathcal A$ and $i\in [s]$, if $A\cap B_i\neq \emptyset$ then $B_i\subseteq A$. Suppose that this is not the case and let $A^{*}\in \mathcal A$ and $i\in [s]$ be such that $|A^{*}\cap B_i|=1$. Let $\mathcal A''=\{A\in \mathcal A': B_i\subseteq A\}$ and note that $\mathcal A'\setminus \mathcal A'' \subseteq \{\bigcup_{j\in S}B_j: S\subseteq [s]\setminus \{i\}\}$. Therefore $|\mathcal A'\setminus \mathcal A''|\le 2^{s-1}$ and so by (\ref{eq:sizeofsubfamily}):
\[|\mathcal A''|\ge |\mathcal A'|-2^{s-1}=|\mathcal A'|-\frac{1}{2}2^{s}> \frac{1}{4}2^{s} = \frac{1}{2}2^{s-1}.\]
Thus, if we define $\mathcal S=\left\{S\subseteq [s]\setminus \{i\}:B_i\cup\left(\bigcup_{j\in S}B_j\right)\in \mathcal A''\right\}$, we see that $\mathcal S\subseteq 2^{[s]\setminus\{i\}}$ and that $|\mathcal S|=|\mathcal A''|>\frac{1}{2}\left|2^{[s]\setminus\{i\}}\right|$. Hence, there must exist two distinct disjoint sets $S_{1},S_{2}\subseteq [s]\setminus \{i\}$ such that $A_1:=B_i\cup\left(\bigcup_{j\in S_{1}}B_j\right)\in \mathcal A''$ and $A_2:=B_i\cup\left(\bigcup_{j\in S_{2}}B_j\right)\in \mathcal A''$. Since $S_1$ and $S_2$ are disjoint, this implies that $A_1\cap A_2=B_i$. Finally, let $A_3,\ldots, A_{k-1}$ be $k-3$ distinct sets in $\mathcal A''\setminus \{A_1,A_2\}$ and note that
\[|A^{*}\cap A_1\cap A_2\cap\ldots\cap A_{k-1}|=|A^{*}\cap B_i|=1\]
contradicting the fact that $\mathcal A$ is a $k$-wise eventown. Thus, we conclude that for any $A\in \mathcal A$ and $i\in [s]$ if $A\cap B_i\neq \emptyset$ then $B_i\subseteq A$.
If $n$ is even then $\bigcup_{j\in [s]}B_j=[n]$ and so it follows that $\mathcal A\subseteq \mathcal B$, the latter being a family in Construction~\ref{cons:maxkwiseeventown}. If $n$ is odd, then $\bigcup_{j\in [s]}B_j=[n]\setminus \{i\}$ for some $i\in [n]$, and thus $\mathcal A\subseteq \mathcal B\cup\{C\cup \{i\}:C\in \mathcal B\}$. Since the intersection of any number of sets of the form $\{C\cup\{i\}\}_{C\in \mathcal B}$ has odd size, and since $\mathcal A$ is a $k$-wise eventown, we conclude that there are at most $k-1$ sets $C\in \mathcal B$ such that $C\cup\{i\}\in\mathcal A$. Thus, we conclude that $\mathcal A$ is a subfamily of a family in Construction~\ref{cons:maxkwiseeventown}.
\end{proof}
\noindent
\textbf{Remark:} In the proof of Theorem~\ref{thm:kwiseeventown} we implicitly use the fact that $|\mathcal A''|\ge k-1$ when we consider $k-3$ distinct sets $A_3,\ldots, A_{k-1}$ from $\mathcal A''\setminus \{A_1,A_2\}$. This follows from the fact that $|\mathcal A''|\ge \frac{1}{4}2^{\lfloor n/2\rfloor}$ and the condition $n\ge 2\lceil \log_{2}(k-1)\rceil+4$ in the theorem statement.
\section{$d$-defect $\ell$-oddtowns}
\label{section:ddefectloddtowns}
\subsection{Proof of Theorem~\ref{thm:ddefectloddtown}}
Given a family of sets $\mathcal A=\{A_1,\ldots, A_{m}\}$ we define its $\ell$-auxiliary graph $G_{\ell}(\mathcal A)$ to be the simple graph with vertex set $\mathcal A$ where $A_iA_j$ is an edge if and only if $|A_i\cap A_j|\not\equiv 0\pmod \ell$. We will often abuse notation slightly and refer to the properties of $G_{\ell}(\mathcal A)$ as being properties of $\mathcal A$. In particular, we use $\Delta(\mathcal A)$, $\chi(\mathcal A)$ and $\alpha(\mathcal A)$ to denote the maximum degree, chromatic number and independence number of $G_{\ell}(\mathcal A)$, respectively.
Let $\mathcal A$ be a $d$-defect $\ell$-oddtown in $[n]$, where $\ell$ is a prime number. Note that $\Delta(\mathcal A)\le d$ and so, in particular, $\alpha(\mathcal A)\ge |\mathcal A|/(d+1)$.
Moreover, observe crucially that an independent set in $G_{\ell}(\mathcal A)$ corresponds to an $\ell$-oddtown inside $\mathcal A$, which as we discussed in the introduction has size at most $n$. Hence, we conclude that $|\mathcal A|\le (d+1)n$. We now wish to improve this simple upper bound to $(d+1)(n-t)$ where $t=2\left(\lceil\log_{2}(d+2)\rceil-1\right)$. We consider the following two cases:
\begin{enumerate}
\item[(a)] $G_{\ell}(\mathcal A)$ contains at most $t$ copies of $K_{d+1}$
\item[(b)] $G_{\ell}(\mathcal A)$ contains more than $t$ copies of $K_{d+1}$
\end{enumerate}
and show that in any case we have $|\mathcal A|\le (d+1)(n-t)$.
We consider case (a) first. Let $\mathcal A'$ be a family obtained from $\mathcal A$ by removing one set from each copy of $K_{d+1}$ in $G_{\ell}(\mathcal A)$. We claim that $\alpha(\mathcal A')\ge |\mathcal A'|/\left(d+\frac{1}{2}\right)$. Indeed, note that the graph $G_{\ell}(\mathcal A')$ does not contain a copy of $K_{d+1}$. Therefore, if $d\neq 2$, it follows from Brooks' Theorem (Theorem~\ref{thm:Brooks}) that $\chi(\mathcal A')\le d$, which implies that $\alpha(\mathcal A')\ge |\mathcal A'|/d\ge |\mathcal A'|/\left(d+\frac{1}{2}\right)$. If $d=2$ then, since $\Delta(\mathcal A')\le 2$, the graph $G_{\ell}(\mathcal A')$ is a disjoint union of cycles of length at least $4$ (recall that $G_{\ell}(\mathcal A')$ is $K_3$-free) and paths. A path of length $\ell$ has an independent set of size at least $\ell/2$ and a cycle of length $\ell \ge 4$ has an independent set of size at least $2\ell/5$. Thus, for $d=2$, it follows that $\alpha(\mathcal A')\ge 2|\mathcal A'|/5=|\mathcal A'|/\left(d+\frac{1}{2}\right)$. Since an independent set in $G_{\ell}(\mathcal A')$ corresponds to an $\ell$-oddtown inside $\mathcal A'$ and since an $\ell$-oddtown in $[n]$ has at most $n$ sets, we conclude that $|\mathcal{A'}|/\left(d+\frac{1}{2}\right)\le n$ and hence:
\[|\mathcal{A}|\le t+|\mathcal A'|\le t+ \left(d+\frac{1}{2}\right)n\le (d+1)(n-t)\]
provided $n\ge Cd\log d$ for some constant $C>0$.
\medskip
We consider now case (b). Let $C_1,\ldots, C_{r}$ denote the connected components of the graph $G_{\ell}(\mathcal A)$. For each $A\in \mathcal A$, let $v_{A}$ denote its characteristic vector in $\mathbb{F}_{\ell}^{n}$ and consider the $n\times |\mathcal A|$ matrix $M$ whose column vectors are the vectors $\{v_{A}\}_{A\in \mathcal A}$, ordered according to the connected components $C_1,\ldots, C_r$. Note that the matrix $\mathcal M=M^{T}M$ is a square matrix of dimension $|\mathcal A|$ and that the entry corresponding to two sets $A,B\in \mathcal A$ in $\mathcal M$ is precisely $v_{A}\cdot v_{B}=|A\cap B|\pmod \ell$. Moreover, since the rows and columns of $\mathcal M$ are ordered according to the connected components of $G_{\ell}(\mathcal A)$ and since $|A\cap B|=0\pmod \ell$ for $A,B\in \mathcal A$ in different connected components, it follows that $\mathcal M$ is a block diagonal matrix, with each block $\mathcal M_i$ corresponding to a connected component $C_i$. Thus, we have:
\begin{align}
\label{eq: sum of ranks}
\sum_{i=1}^{r} \text{rank}(\mathcal M_i)=\text{rank}(\mathcal M)\le \text{rank}(M)\le n.
\end{align}
Note that if $\mathcal I=\{A_1,\ldots, A_{|\mathcal I|}\}$ is an independent set in $C_i$ then $v_{A_{j}}\cdot v_{A_{j'}}=|A_{j}\cap A_{j'}|\neq 0\pmod \ell$ if and only if $j=j'$, implying that the submatrix of $\mathcal M_i$ whose rows and columns correspond to the sets in $\mathcal I$ has full rank $|\mathcal I|$. Thus, since $\Delta(\mathcal A)\le d$ it follows that for each $i\in [r]$:
\begin{align}
\label{eq: bound on rank}
\text{rank}(\mathcal M_i)\ge \alpha(C_i)\ge |C_i|/(d+1).
\end{align}
We claim now that there is at least one component $C_i$ which is a copy of $K_{d+1}$ such that $\text{rank}(\mathcal M_i)=1$, or else $|\mathcal A|<(d+1)(n-t)$. Indeed, since we are looking at case (b), we know that more than $t$ components of $G_{\ell}(\mathcal A)$ are copies of $K_{d+1}$. Moreover, if all the corresponding blocks have rank at least $2$ then there are more than $t$ values of $i\in [r]$ for which inequality (\ref{eq: bound on rank}) can be improved to $\text{rank}(\mathcal M_i)\ge 1+|C_i|/(d+1)$. Thus, in that case it follows from (\ref{eq: sum of ranks}) that:
\[n\ge \sum_{i=1}^{r}\text{rank}(\mathcal M_i)> t+\sum_{i=1}^{r}|C_i|/(d+1)=t+|\mathcal A|/(d+1) \Rightarrow |\mathcal A|<(d+1)(n-t)\]
Thus, we may assume that there is one connected component $C_{i^{*}}$ of $G_{\ell}(\mathcal A)$ which is a copy of $K_{d+1}$ and whose corresponding block matrix $\mathcal M_{i^{*}}$ in $\mathcal M$ has rank $1$. Note that this implies that any two rows/columns in $\mathcal M_{i^{*}}$ are multiples of one another. Let $B_1,\ldots, B_{d+1}$ be the sets in $\mathcal A$ corresponding to such a connected component. Note that since $b_i\cdot b_i = |B_i| \neq 0 \pmod \ell$ for any $i\in [d+1]$ and since the rows of $\mathcal M_{i^{*}}$ are multiples of one another, it follows that $b_i\cdot b_j\neq 0 \pmod \ell$ for any $i,j\in [d+1]$ and that $(b_1\cdot b_1)(b_i\cdot b_j) = (b_1\cdot b_i)(b_1\cdot b_j)$.
Now, let $\mathcal A'$ denote the family $\mathcal A\setminus \{B_1,\ldots, B_{d+1}\}$ and let $A_1,\ldots, A_{s}$ be sets corresponding to an independent set of maximum size in $G_{\ell}(\mathcal A')$. Since $\Delta(\mathcal A')\le d$ it follows that
\begin{align}
\label{eq:bound}
s=\alpha(\mathcal A')\ge \frac{|\mathcal A'|}{d+1} =\frac{|\mathcal A|}{d+1}-1.
\end{align}
Let $a_1,\ldots, a_{s}$ and $b_1,\ldots,b_{d+1}$ be the characteristic vectors in $\mathbb{F}_{\ell}^{n}$ of $A_1,\ldots, A_{s}$ and $B_1,\ldots, B_{d+1}$, respectively. Because of the choices of these sets, it follows that:
\begin{enumerate}
\item[(i)] For every $i,j\in [s]$: $a_i\cdot a_j\neq 0$ if and only if $i=j$.
\item[(ii)] For every $i,j\in [d+1]$: $(b_1\cdot b_1)(b_{i}\cdot b_j)=(b_1\cdot b_i)(b_1\cdot b_j)\neq 0$.
\item[(iii)] For every $i\in [s]$ and $j\in [d+1]$: $a_i\cdot b_j=0$.
\end{enumerate}
Denoting by $U$ the space generated by $a_1,\ldots, a_{s}$, it follows from (i) that $U$ is a non-degenerate subspace of $\mathbb{F}_{\ell}^{n}$ (see Section~\ref{section:auxiliaryresults} for the definition) and that $\dim U=s$. Furthermore, by Lemma~\ref{lemma:linearalgebra} we know that $U^{\perp}$ is non-degenerate and that
\begin{align}
\label{eq:dimension}
s+\dim U^{\perp} = \dim U + \dim U^{\perp}=n
\end{align}
Since the vectors $b_1,\ldots, b_{d+1}$ are distinct $\{0,1\}$-vectors satisfying (ii) and are in $U^{\perp}$ by (iii), we obtain by Lemma~\ref{lemma:dimension} that
\begin{align}
\label{eq:lowerbounddimension}
\dim U^{\perp}\ge 2\lceil\log_{2}(d+2)\rceil-1 = t+1.
\end{align}
Finally, putting (\ref{eq:bound}), (\ref{eq:dimension}) and (\ref{eq:lowerbounddimension}) together, we conclude that
\[\left(\frac{|\mathcal A|}{d+1}-1\right)+(t+1)\le n\; \Leftrightarrow\; |\mathcal A|\le (d+1)(n-t)\]
as claimed. This finishes the proof of Theorem~\ref{thm:ddefectloddtown}.
\subsection{Proof of Theorem~\ref{thm:1defectloddtown}}
We start by giving constructions of $1$-defect $\ell$-oddtowns of size $2n-4$ for infinitely many values of $n$, when $\ell$ is a prime number. Our constructions rely on the use of Hadamard matrices. A Hadamard matrix of order $n$ is an $n \times n$ matrix whose entries are either $+1$ or $-1$ and whose rows are mutually orthogonal. A necessary condition for a Hadamard matrix of order $n > 2$ to exist is that $n$ is divisible by $4$. The most important open question in the theory of Hadamard matrices, known as the Hadamard conjecture, is whether this condition is also sufficient. For more on Hadamard matrices see e.g. \cite{H07}.
Suppose a Hadamard matrix $H$ of order $n-1$ exists. We may assume the last column has every entry equal to $1$, by multiplying some rows by $-1$ if necessary. For $j\in [n-2]$ define sets $A_j, B_j\subseteq [n-1]$ by taking $i \in A_j$ if and only if $H_{i,j} = 1$ and setting $B_j=[n-1]\setminus A_j$. The fact that $H$ is a Hadamard matrix of order $n-1$ with the last column being the all-$1$ vector ensures that for any $j$:
\[|A_{j}|=|B_{j}|=\frac{n-1}{2}\;\;\text{and}\;\;|A_{j}\cap B_{j}|=0\]
and for $j_{1}\neq j_{2}$:
\[|A_{j_1}\cap A_{j_2}|=|A_{j_1}\cap B_{j_2}|=|B_{j_1}\cap B_{j_2}|=\frac{n-1}{4}\]
Thus, one can easily check that
\[\mathcal A=\{A_{1}\cup\{n\},B_1\cup\{n\},\ldots, A_{n-2}\cup \{n\},B_{n-2}\cup\{n\}\}\]
is a $1$-defect $\ell$-oddtown in $[n]$ of size $2n-4$, provided $n\equiv 5 \pmod 8$ if $\ell=2$ or $\ell\mid n+3$ if $\ell>2$. Thus, a $1$-defect $\ell$-oddtown in $[n]$ of order $2n-4$ exists provided a Hadamard matrix of order $n-1$ exists and these divisibility conditions on $n$ are satisfied. We claim now that there are infinitely many values of $n$ for which this holds.
For $\ell=2$, this is ensured by a construction of Paley~\cite{P33} of Hadamard matrices of order $q+1$ for any odd prime power $q$. For $\ell>2$, this is ensured by a result of Wallis~\cite{W76} which states that for any $q\in\mathbb{N}$ there is $s_0\in \mathbb{N}$ such that a Hadamard matrix of order $2^{s}q$ exists for any $s\ge s_0$ (just take $n$ to be of the form $2^{s}q+1$, where $q=\ell-1$ and $s$ is any sufficiently large multiple of $\ell-1$). We conclude that for any prime $\ell$ there are $1$-defect $\ell$-oddtowns in $[n]$ of size $2n-4$ for infinitely many values of $n$.
\medskip
Now we prove that any $1$-defect $\ell$-oddtown in $[n]$ has size at most $\max\{n,2n-4\}$ if $\ell$ is a prime number. Suppose $\mathcal A$ is a $1$-defect $\ell$-oddtown in $[n]$. If all pairwise intersections of sets in $\mathcal A$ have size $= 0\pmod \ell$ then $\mathcal A$ is an $\ell$-oddtown and so, as discussed in the introduction, we have $|\mathcal A|\le n$. Otherwise, we can label the sets in $\mathcal A$ as $A_1,B_1,\ldots, A_{t},B_{t},A_{t+1},\ldots,A_{s}$ ($1\le t\le s$) such that the pairs $(A_i,B_i)$ have pairwise intersection of size $\neq 0\pmod \ell$ and all other pairwise intersections have size $= 0 \pmod \ell$. Let $a_1,\ldots,a_{s}$ and $b_1,\ldots, b_{t}$ be the characteristic vectors in $\mathbb F_{\ell}^{n}$ corresponding to the sets $A_1,\ldots, A_{s}$ and $B_1,\ldots, B_{t}$, respectively. Note crucially that for $i\neq j$ we have $a_i\cdot a_j = |A_i\cap A_j| = 0 \pmod \ell$, $a_i\cdot b_j = |A_i\cap B_j| = 0\pmod \ell$, $b_i\cdot b_j = |B_i\cap B_j| = 0\pmod \ell$, $a_i\cdot a_i = |A_i| \neq 0 \pmod \ell$, $a_i\cdot b_i = |A_i\cap B_i| \neq 0 \pmod \ell$ and $b_i\cdot b_i = |B_i| \neq 0 \pmod \ell$.
We consider now two separate cases:
\begin{enumerate}
\item[1)] $(a_1\cdot a_1)(b_1\cdot b_1)=(a_1\cdot b_1)^2$
\item[2)] $(a_1\cdot a_1)(b_1\cdot b_1)\neq (a_1\cdot b_1)^2$.
\end{enumerate}
In each case, we will either show that $|\mathcal A|\le n$ or we will find $s+2$ linearly independent vectors in $\mathbb{F}_{\ell}^{n}$. This then implies that if $|\mathcal A| >n$ then $s+2 \le n$ and so:
\[|\mathcal A|=s+t\le 2s\le 2(n-2)=2n-4.\]
We consider case 1 first. Let $v=(a_1\cdot a_1)b_1-(a_1\cdot b_1)a_1$ and note that $v\cdot a_i=0$ for any $i\in [s]$. Indeed, we have $v\cdot a_1 = (a_1\cdot a_1)(b_1\cdot a_1) - (a_1\cdot b_1)(a_1\cdot a_1) = 0$ and for $i>1$ we have $a_1 \cdot a_i = 0 $ and $b_1\cdot a_i = 0$, implying that $v\cdot a_i = 0$. Moreover, since $a_1$ and $b_1$ are distinct $\{0,1\}$-vectors one has $v\neq 0$, and
\[v\cdot v=(a_1\cdot a_1)\left[(a_1\cdot a_1)(b_1\cdot b_1)-(a_1\cdot b_1)^2\right]=0.\]
Since $v\neq 0$, we can find a vector $v_{1}\in \mathbb{F}_{\ell}^{n}$ so that $v\cdot v_1\neq 0$. Define $v_{2}:=v-v_{1}$ and note that $v\cdot v_2=-v\cdot v_1$ since $v\cdot v=0$. We claim now that the vectors $a_1,\ldots, a_{s},v_{1},v_{2}$ are linearly independent. Indeed, if
\[\sum_{i=1}^{s}\alpha_{i}a_i+\beta_{1} v_{1}+\beta_{2} v_{2}=0\]
is a linear combination of these vectors, then doing the dot product with $v$ allows us to conclude that
\[0=\beta_{1}(v\cdot v_{1})+\beta_{2}(v\cdot v_2)=(\beta_1-\beta_2)(v\cdot v_1)\]
and therefore, since $v\cdot v_1 \neq 0$, we must have $\beta_1=\beta_2$. Then, since $\beta_{1}v_{1}+\beta_{2}v_{2}=\beta_1 v$, doing the dot product with $a_{i}$ for $i\in [s]$ we can deduce that $\alpha_{i}=0$. Finally, since $v\neq 0$ we can conclude then that $\beta_{1}=\beta_{2}=0$, and so the vectors $a_1,\ldots, a_s,v_1,v_2$ are linearly independent as claimed.
We now consider case 2 and assume for the moment that $t\ge 2$. We claim that the vectors $a_1,\ldots, a_{s},b_1,b_2$ are linearly independent. Indeed, if
\[\sum_{i=1}^{s}\alpha_{i}a_i+\beta_{1} b_{1}+\beta_{2} b_{2}=0\]
is a linear combination of these vectors, then doing the dot product of the above with $a_{i}\in [s]\setminus \{1,2\}$ allows us to conclude that $\alpha_{i}=0$ and so
\[\alpha_{1}a_1+\alpha_2a_2+\beta_{1}b_{1}+\beta_{2}b_2=0.\]
Now, doing the dot product of the latter with $a_1$ and $b_1$ we see that:
\[\begin{bmatrix}
(a_1\cdot a_1) & (a_1\cdot b_1)\\
(a_1\cdot b_1) & (b_1\cdot b_1)
\end{bmatrix}
\begin{bmatrix}
\alpha_1\\
\beta_1
\end{bmatrix}=\begin{bmatrix}
0\\
0
\end{bmatrix}\]
Since the determinant of this matrix is non-zero (because we are in case 2), we conclude that $\alpha_1=\beta_1=0$. Then, since $a_2$ and $b_2$ are distinct $\{0,1\}$-vectors we conclude that $\alpha_2=\beta_2=0$ and so the vectors $a_1,\ldots, a_s,b_1,b_2$ are linearly independent as claimed. Finally, if $t=1$ then one can show, similarly to the above, that the $s+1$ vectors $a_1,\ldots,a_{s},b_1$ are linearly independent, implying that $|\mathcal A|=s+1\le n$. This finishes the proof of Theorem~\ref{thm:1defectloddtown}.
\section{Further remarks and open problems}
\label{section:concludingremarks}
Theorems \ref{thm:vukwiseeventown}, \ref{thm:kwiseeventown} and \ref{thm:strongkwiseeventown} establish the maximum size of (strong) $k$-wise $\ell$-eventowns and characterize
their structure for $\ell = 2$. Far less is known for (strong) $k$-wise $\ell$-eventowns with $\ell > 2$. A natural analogue of Construction~\ref{cons:maxkwiseeventown} for $\ell>2$ arises from the next construction:
\begin{cons}
\label{cons:strongkwiseleventown}
Let $B_1,\ldots, B_{\lfloor n/\ell\rfloor}$ be $\lfloor n/\ell\rfloor$ disjoint subsets of $[n]$ of size $\ell$. Then the family $\mathcal A=\left\{\bigcup_{i\in S} B_i: \;S\subseteq \left[\lfloor n/\ell\rfloor\right]\right\}$ is a strong $k$-wise $\ell$-eventown of size $2^{\lfloor n/\ell\rfloor}$ for every $k\in \NN$.
\end{cons}
Construction~\ref{cons:strongkwiseleventown} provides a strong $k$-wise $2$-eventown of maximum possible size for any $k\ge 2$, and, in light of Theorem~\ref{thm:strongkwiseeventown}, this is the unique such family for $k\ge 3$, up to the choice of the sets $B_1,\ldots, B_{\lfloor n/2\rfloor}$. Surprisingly, for $\ell > 2$, Construction~\ref{cons:strongkwiseleventown} is far from best possible. As mentioned in the introduction, Frankl and Odlyzko~\cite{FO83} constructed a strong $2$-wise $\ell$-eventown of size $2^{\Omega(\log \ell/\ell)n}$, as $n\rightarrow \infty$, which is significantly larger than the families in Construction~\ref{cons:strongkwiseleventown} for large $\ell$.
Interestingly, this phenomena does not hold only for $k=2$. Indeed, Frankl and Odlyzko's construction can be used to construct a strong $3$-wise $\ell$-eventown of size $2^{\Omega(\log \ell/\ell)n}$, as $n\rightarrow \infty$. This follows from the next simple lemma which shows how to create a large strong $k$-wise $\ell$-eventown from a large strong $(k-1)$-wise $\ell$-eventown if $k$ is odd. We leave its proof as an exercise to the interested reader.
\begin{lemma}
\label{lemma:stepup}
Suppose $\mathcal A=\{A_1,\ldots, A_m\}$ is a strong $(k-1)$-wise $\ell$-eventown on the universe $[n]$. For each $i\in [m]$ define the sets $A^{*}_{i}=(([n]\setminus A_i)+n)\subseteq [2n]\setminus [n]$ and $B_i=A_i\cup A^{*}_{i}$. If $\ell\mid n$ and $k$ is odd then $\mathcal B=\{B_1,\ldots, B_m\}$ is a strong $k$-wise $\ell$-eventown on the universe $[2n]$ of size $|\mathcal B|=|\mathcal A|$.
\end{lemma}
We can also show that for any fixed $k\in \NN$ there are strong $k$-wise $\ell$-eventowns of size $2^{\Omega(\log \ell/\ell)n}$ as $n\rightarrow \infty$ when $\ell$ is a power of $2$:
\begin{lemma}
\label{lemma:beatingblockfamilies}
For any $k\in \NN$ and $\ell$ a power of $2$, there are strong $k$-wise $\ell$-eventowns in the universe $[n]$ of size $\left(2^{k+1}\ell\right)^{\lfloor n/(2^k\ell)\rfloor}$.
\end{lemma}
We give a brief sketch on how to construct such families. We start by recursively defining for $r\ge 0$ a family $\mathcal A^{r}$ with $2^{r+1}$ subsets $A_1^{r},\ldots, A_{2^{r+1}}^{r}$ of $[2^{r}]$ with the property that
\begin{align}
\label{eq:property}
2^{r-|S|}\; \text{ divides }\; \left|\bigcap_{i\in S}A_i^{r}\right| \;\text{ for any set }\; S\subseteq [2^{r+1}]\; \text{ of size } \;|S|\le r.
\end{align}
For $r=0$ we define $A_{1}^{0} = \emptyset$ and $A_{1}^{1}=\{1\}$. For $r> 0$, define for $1\le i\le 2^{r}$ the sets $A_{i}^{r}:= A_i^{r-1}\cup (A_{i}^{r-1}+2^{r-1})$ and $A_{i+2^{r}}^{r}=A_{i}^{r-1}\cup \left(([2^{r-1}]\setminus A_{i}^{r-1})+2^{r-1} \right)$. Finally, define $\mathcal A^{r} =\{A_1^{r},\ldots, A_{2^{r+1}}^{r}\}$. One can prove by induction on $r$ that this family satisfies property~(\ref{eq:property}).
Now, suppose $\ell = 2^{a}$. Property (\ref{eq:property}) implies that $\mathcal A^{k + a}$ is a strong $k$-wise $\ell$-eventown in $[2^{k+a}] = [2^{k}\ell]$ of size $2^{k+a+1} = 2^{k+1}\ell$. For $j \in [\lfloor n/(2^{k}\ell)\rfloor]$ let $\mathcal B_j = \{A + (j-1)2^{k}\ell : A\in \mathcal A^{k+a}\}$ and define
\[\mathcal B = \left\{\bigcup_{j\in [\lfloor n/(2^{k}\ell)\rfloor]} B_j : B_j \in \mathcal B_j \text{ for } j\in [\lfloor n/(2^{k}\ell)\rfloor]\right\}\]
A moment's thought shows that $\mathcal B$ is a strong $k$-wise $\ell$-eventown in $[n]$ of size $|\mathcal B| = (2^{k+1}\ell)^{\lfloor n/(2^{k}\ell)\rfloor}$.
\medskip
Frankl and Odlyzko conjectured in \cite{FO83} that for any $\ell\in \NN$ there exists $k(\ell)\in \NN$ such that if $k\ge k(\ell)$ then any $k$-wise $\ell$-eventown has size at most $2^{(1+o(1))n/\ell}$ as $n\rightarrow \infty$ (which would be asymptotically tight by Construction~\ref{cons:strongkwiseleventown}). Lemma~\ref{lemma:beatingblockfamilies} implies that if such $k(\ell)$ exists then $k(\ell) \ge (1-o(1))(\log_2\log_2\ell)$, at least when $\ell$ is a power of $2$.
Note that Lemma~\ref{lemma:beatingblockfamilies} shows that at least when $\ell$ is a power of $2$ we have strong $k$-wise $\ell$-eventowns in $[n]$ of size roughly $2^{C(k)\left(\log \ell/\ell\right) n}$, where $C(k)\sim k2^{-k}$. Moreover, if there were an analogue of Lemma~\ref{lemma:stepup} for any $k$ (not just $k$ odd) then for any $\ell \in \NN$ one could start from Frankl and Odlyzko's construction and iterate such lemma $k-2$ times in order to obtain a strong $k$-wise $\ell$-eventown in $[n]$ of size roughly $2^{C(k)\left(\log \ell/\ell\right) n}$ where $C(k) \sim 2^{-k}$. We find it plausible that such families exist for any $k,\ell\in\NN$, provided $n$ is sufficiently large (depending on $k$ and $\ell$).
In Theorem~\ref{thm:ddefectloddtown} we showed that for any $d\in \NN$ and $\ell$ a prime number, any $d$-defect $\ell$-oddtown in the universe $[n]$, for $n$ large, has size at most $(d+1)(n-2(\lceil\log_{2}(d+2)\rceil-1))$, improving Vu's upper bound of $(d+1)n$ described at the beginning of Section~\ref{section:ddefectloddtowns}. Vu~\cite{V99} also showed that there exist $d$-defect $\ell$-oddtowns in $[n]$ of size $(d+1)(n-\ell\lceil \log_{2}(d+1)\rceil)$. These families come from the following construction:
\begin{cons}
\label{cons:ddefectloddtown}
Let $t = \lceil\log_{2}(d+1)\rceil$, $s=\ell t$ and $\mathcal S$ be a collection of $d+1$ subsets of $[t]$. Moreover, let $B_1,\ldots, B_{t}$ be $t$ disjoint subsets of $[s]$, each of size $\ell$. For each $S\in \mathcal S$ let $B_{S}=\cup_{i\in S} B_i$ and define $\mathcal B =\{B_S:S\in\mathcal S\}$. Then, the family $\mathcal A$ defined by
\[\mathcal A=\{B\cup \{i\}:B\in\mathcal B, i\in [n]\setminus [s]\}\]
is a $d$-defect $\ell$-oddtown of size $ |\mathcal A|=(d+1)(n-s)$. Indeed, for $B,B'\in\mathcal B$ and $i,i'\in [n]\setminus [s]$ we have
\[|(B\cup \{i\})\cap (B'\cup \{i'\})|=|B\cap B'|+|\{i\}\cap \{i'\}|\equiv |\{i\}\cap \{i'\}| \pmod \ell\]
and the latter is non-zero modulo $\ell$ if and only if $i= i'$.
\end{cons}
This construction can be improved for some values of $\ell$ and $d$. Notice, that the only relevant property of family $\mathcal B$ in Construction~\ref{cons:ddefectloddtown} is that it is an $\ell$-eventown on the universe $[s]$ of size at least $d+1$. Thus, if there exists an $\ell$-eventown of size $d+1$ in a universe of size smaller than $\ell\lceil \log_{2}(d+1)\rceil$ then we can improve Vu's lower bound on the maximum size of a $d$-defect $\ell$-oddtown. Frankl and Odlyzko's construction mentioned earlier shows that an $\ell$-eventown in the universe $[s]$ of size at least $2^{c\left(\log\ell/\ell\right)s}$ exists for some constant $c>0$ as $s\rightarrow \infty$. Since $2^{c\left(\log \ell /\ell\right)s}\ge d+1$ if $s\ge c^{-1}\left(\ell/\log \ell\right)\log_{2}(d+1)$, this implies that there are $d$-defect $\ell$-oddtowns of size $(d+1)(n-C\left(\ell/\log \ell\right)\log_{2}(d+1))$ for some constant $C>0$ as $n\rightarrow \infty$, provided $d$ is big enough as a function of $\ell$. It is unclear to us whether the maximum size of a $d$-defect $\ell$-oddtown should depend on $\ell$. We remark that for $d=1$, as Theorem~\ref{thm:1defectloddtown} shows, this is not the case.
In \cite{SV05} Szab\'o and Vu considered the related problem of maximizing the size of a $k$-wise oddtown, i.e., a family of odd-sized sets such that the intersection of any $k$ has even size. They showed that if $k-1$ is a power of $2$ then for large $n$ the answer is $(k-1)(n-2\log_{2}(k-1))$. An example of a $k$-wise oddtown of this size is the one in Construction~\ref{cons:ddefectloddtown} with $d = k-2$ and $\ell = 2$. For the natural generalization of this problem modulo $\ell>2$, Szab\'o and Vu believed that Construction~\ref{cons:ddefectloddtown} with $d= k-2$ provided a $k$-wise $\ell$-oddtown in $[n]$ of maximum possible size, namely, $(k-1)(n-\ell\lceil \log_{2}(k-1)\rceil)$. This turns out not to be the case. Indeed, as described in the previous paragraph, by making a more appropriate choice of $\mathcal B$ in Construction~\ref{cons:ddefectloddtown} one can obtain for suitable values of $k$ and $\ell$ a $k$-wise $\ell$-oddtown of size $(k-1)(n-C\left(\ell/\log \ell\right)\log_{2}(k-1))$ for some constant $C>0$ and $n$ sufficiently large.
\medskip
\noindent
\textbf{Acknowledgements.} We would like to thank Shagnik Das for helpful discussions and comments. | 120,219 |
A proposed development on the southwest corner of U.S. 31 and State Road 32 in Westfield could include a four-story hotel and several other commercial buildings.
Westfield Community Investors LLC, led by Randy Zents and Birch Dalton, has big plans for the 7.1-acre site. The property is being sold by Waitt Elevator Co. Inc., which has moved its headquarters to Sheridan.
The land is included in Westfield's planned "Grand Junction" area, which is considered the gateway into Westfield. A master plan for Grand Junction calls for hotels, conference centers, apartments and office buildings, with the rooftops being taller than the elevated U.S. 31.
The developer hopes the project, dubbed “Gateway Southwest,” will attract a medical office, a free-standing restaurant and two retail buildings in addition to the hotel.
The most recent assessment values the property at $276,000, according to Hamilton County records.
Initially, the developer's proposal called for a three-story hotel, but city officials are pushing for a taller building, preferably five stories. The developer cited concerns about being unable to secure an operator for a hotel that size, but agreed to a minimum height of four stories along the highway.
The other buildings could be as small as one-story, 7,000-square-foot structures. The hotel will be closest to U.S. 31, with the four other buildings surrounding it. Outdoor space such as plazas and courtyards, with public art, are also encouraged by the master plan.
Nurseries, mortuaries, religious institutions, garden and lawn centers, outdoor theaters, civic clubs, private clubs and fast-food restaurants (unless part of a mixed-use building) are prohibited from the development. Fast-casual restaurants, which are described as more upscale than fast-food restaurants, without drive-through service, would be allowed.
The Westfield Plan Commission gave the project a favorable recommendation earlier this month, and the Westfield City Council could take a final vote Monday night.
Dalton told the Plan Commission he expects the hotel project to be the first part of the development to move forward. Restaurants should follow.
“I think it’s a pretty good product,” Dalton said at the July 6 Plan Commission meeting.
Dalton and Zentz are also leading the $40 million mixed-use development called "The Junction" on the northeast corner of U.S. 31 and S.R. 32. The project could include a medical office building, family entertainment center and a hotel, in addition to retail . The land previously housed Westfield High School's football and junior varsity baseball fields. | 86,333 |
North Dakota Code > Chapter 1-01 – General Principles and Definitions
Current as of: 2021 | Check for updates | Other versions
Terms Used In North Dakota Code > Chapter 1-01 - General Principles and Definitions
-.
- children: includes children by birth and by adoption. See North Dakota Code 1-01-18
- Common law: The legal system that originated in England and is now in use in the United States. It is based on judicial decisions rather than legislative.
- Devise: To gift property by will.
-: includes administrator and "administrator" includes executor. See North Dakota Code 1-01-49
- Executor: A male person named in a will to carry out the decedent
- means the next preceding or next following chapter or other part. See North Dakota Code 1-01-49
-.
- Individual: means a human being. See North Dakota Code 1-01-49
-.
- Legatee: A beneficiary of a decedent
-.
- new wealth: means revenues generated by a business in this state through the sale of products or services to:
a. See North Dakota Code 1-01-49
- Oath: includes "affirmation". See North Dakota Code 1-01-49
- Oath: A promise to tell the truth.
- Obligation: An order placed, contract awarded, service received, or similar transaction during a given period that will require payments during the same or a future period.
- Organization: includes a foreign or domestic association, business trust, corporation, enterprise, estate, joint venture, limited liability company, limited liability partnership, limited partnership, partnership, trust, or any legal or commercial entity. See North Dakota Code 1-01-49
- paper: means any flexible material upon which it is usual to write. See North Dakota Code 1-01-27
- Partnership: includes a limited liability partnership registered under chapter 45-22. See North Dakota Code 1-01-49
- Partnership: A voluntary contract between two or more persons to pool some or all of their assets into a business, with the agreement that there will be a proportional sharing of profits and losses.
- Person: means an individual, organization, government, political subdivision, or government agency or instrumentality. See North Dakota Code 1-01-49
- Personal property: includes money, goods, chattels, things in action, and evidences of debt. See North Dakota Code 1-01-49
- Personal property: All property that is not real property.
- Plaintiff: The person who files the complaint in a civil lawsuit.
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-
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-.
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\section{Associativity and $\gloo$-modules}\label{s:lemmas}
For the diagram~$T_3$ in Figure~\ref{f:threetetra}
we will see in Section~\ref{s:tetra2} that the contribution
of the associator~$\Phi$ in the computation of~$\Ws(\Zb(T_3))$ is non-trivial.
We will use
Lemma~\ref{l:assobas} and Corollary~\ref{c:assobas}
below for a similar
purpose in this computation as we used Lemma~\ref{l:assotriv} to
deduce equation~(\ref{e:WZL}).
\begin{lemma}\label{l:assobas}
For $t_i=(\lambda_i, \mu_i, \sigma_i)$ as in equations~(\ref{e:lambdai}), (\ref{e:ti}) and
$\kappa=\frac{\lambda_2(\lambda_1+\lambda_2+\lambda_3)}
{(\lambda_1+\lambda_2)(\lambda_2+\lambda_3)}$ we
have two bases
of~$\Hom_{\gloo}(V_{t_8},\Vo\otimes\Vp\otimes\Vpp)$
that are related by
\begin{equation}\label{e:assobas} \left(\begin{array}{l} (\I_{t_1}\otimes \Ydot_{t_2,
t_3})\circ\Ystd_{t_1,t_9}\\ (\I_{t_1}\otimes \Ystd_{t_2,
t_3})\circ\Ydot_{t_1,t_6}\end{array}\right)=
\left(\begin{array}{cc}
\frac{(-1)^{\sigma_2}\lambda_3}{-\lambda_2-\lambda_3} &
\kappa\\
1 & \frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
\end{array}\right)
\left(\begin{array}{l} (\Ydot_{t_1, t_2}\otimes
\I_{t_3})\circ\Ystd_{t_7,t_3}\\ (\Ystd_{t_1,t_2}\otimes \I_{t_3}
)\circ\Ydot_{t_4, t_3}\end{array}\right). \end{equation}
\end{lemma}
\begin{proof}
We compute
\begin{eqnarray*}
& & (\I_{t_1}\otimes\Ydot_{t_2, t_3})(\Ystd_{t_1, t_9}(w_{t_8})) =
(\I_{t_1}\otimes\Ydot_{t_2,t_3})(F\cdot(v_{t_1}\otimes v_{t_9}))\\
& & = w_{t_1}\otimes\Ydot_{t_2, t_3}(v_{t_9})+(-1)^{\sigma_1}
v_{t_1}\otimes\Ydot_{t_2, t_3}(w_{t_9})\\ & &
=\frac{1}{\lambda_2+\lambda_3} w_{t_1}\otimes E\cdot(
w_{t_2}\otimes w_{t_3})+(-1)^{\sigma_1} v_{t_1}\otimes
w_{t_2}\otimes w_{t_3}\\ & &
=\frac{\lambda_2}{\lambda_2+\lambda_3} w_{t_1}\otimes
v_{t_2}\otimes
w_{t_3}-\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}
w_{t_1}\otimes w_{t_2}\otimes v_{t_3}+(-1)^{\sigma_1}
v_{t_1}\otimes w_{t_2}\otimes w_{t_3}
\end{eqnarray*}
and similarly (or simpler)
\begin{eqnarray*}
(\I_{t_1}\otimes\Ystd_{t_2, t_3})(\Ydot_{t_1, t_6}(w_{t_8})) & = &
(-1)^{\sigma_2}w_{t_1}\otimes v_{t_2}\otimes
w_{t_3}+w_{t_1}\otimes w_{t_2}\otimes v_{t_3},\\
(\Ydot_{t_1,t_2}\otimes \I_{t_3})(\Ystd_{t_7, t_3}(w_{t_8})) & = &
\frac{(-1)^{\sigma_2}\lambda_2}{\lambda_1+\lambda_2}
w_{t_1}\otimes v_{t_2}\otimes w_{t_3}+w_{t_1}\otimes
w_{t_2}\otimes v_{t_3}\\\nopagebreak{} & &
-\frac{(-1)^{\sigma_1+\sigma_2}\lambda_1}{\lambda_1+\lambda_2}v_{t_1}\otimes
w_{t_2}\otimes w_{t_3},\\ (\Ystd_{t_1,t_2}\otimes
\I_{t_3})(\Ydot_{t_4, t_3}(w_{t_8})) & = & w_{t_1}\otimes
v_{t_2}\otimes w_{t_3}+(-1)^{\sigma_1} v_{t_1}\otimes
w_{t_2}\otimes w_{t_3}.
\end{eqnarray*}
We see that the vectors
$(\I_{t_1}\otimes\Ydot_{t_2,t_3})(\Ystd_{t_1,t_9}(w_{t_8}))$ and
$(\I_{t_1}\otimes\Ystd_{t_2,t_3})(\Ydot_{t_1, t_6}(w_{t_8}))$ as well
as~$(\Ydot_{t_1, t_2}\otimes
\I_{t_3})(\Ystd_{t_7,t_3}(w_{t_8}))$ and~$(\Ystd_{t_1, t_2}\otimes
\I_{t_3})(\Ydot_{t_4,t_3}(w_{t_8}))$ are linearly independent.
Equation~(\ref{e:threedecomp}) implies
that~$\Hom_{\gloo}(V_{t_8},\Vo\otimes\Vp\otimes\Vpp)$ is
two-dimensional, so we have
found two bases of that space. Verify that equation~(\ref{e:assobas})
holds when we evaluate the morphisms in this equation on~$w_{t_8}$. This
implies the lemma because~$w_{t_8}$ generates $V_{t_8}$.
\end{proof}
Dually to Lemma~\ref{l:assobas} we have the following corollary.
\begin{coro}\label{c:assobas}
With $t_i$ and~$\kappa$ as in Lemma~\ref{l:assobas} two bases
of
$$\Hom_{\gloo}(\Vo\otimes\Vp\otimes\Vpp,
V_{t_8})$$
are related by
$$ \left(\begin{array}{l} \Astd_{t_1, t_9}\circ(\I_{t_1}\otimes
\Adot_{t_2,t_3})\\ \Adot_{t_1, t_6}\circ(\I_{t_1}\otimes
\Astd_{t_2,t_3})\end{array}\right)=\left(\begin{array}{cc}
\frac{(-1)^{\sigma_2}\lambda_1}{-\lambda_1-\lambda_2} & 1\\
\kappa
& \frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}
\end{array}\right)\left(\begin{array}{l} \Astd_{t_7,t_3}\circ(\Adot_{t_1,t_2}\otimes \I_{t_3})\\
\Adot_{t_4,t_3}\circ(\Astd_{t_1,t_2}\otimes
\I_{t_3})\end{array}\right). $$
\end{coro}
\begin{proof}
By Lemma~\ref{l:assobas} linear maps~$f_i$, $g_i$
are related by
$(f_1,
f_2)^T=A(g_1, g_2)^T$ for a certain matrix $A$.
Therefore, maps~$f_i'$, $g_i'$ satisfying
$f_i'\circ f_j=\delta_{ij} \I_{t_8}$
and $g_i'\circ g_j=\delta_{ij} \I_{t_8}$ are related by
$(f_1', f_2')^T=({A^T})^{-1}(g_1', g_2')^T$. This implies the
corollary.
\end{proof}
In the rest of this section
we prepare the analogue of equation~(\ref{e:Phitrivb})
for the computation in Section~\ref{s:tetra2}.
By the definition of the~$\gloo$-module structure
of~$V_{t_1}\otimes V_{t_2}$ we have
\begin{eqnarray}
F\cdot(v_{t_1}\otimes v_{t_2}) & = & w_{t_1}\otimes
v_{t_2}+(-1)^{\sigma_1} v_{t_1}\otimes
w_{t_2}\quad\mbox{and}\label{e:bc1}\\ E\cdot(w_{t_1}\otimes
w_{t_2}) & = & -(-1)^{\sigma_1}\lambda_2 w_{t_1}\otimes
v_{t_2}+\lambda_1 v_{t_1}\otimes w_{t_2}.\label{e:bc2}
\end{eqnarray}
For~$\lambda_1\not=-\lambda_2$
equations~(\ref{e:bc1}) and~(\ref{e:bc2}) are formulas for a
change of bases in the two dimensional eigenspace of~$H$
on~$\Vo\otimes \Vp$ and imply
\begin{eqnarray}
(\lambda_1+\lambda_2)w_{t_1}\otimes v_{t_2} & = & \lambda_1
F\cdot(v_{t_1}\otimes v_{t_2})-(-1)^{\sigma_1}
E\cdot(w_{t_1}\otimes w_{t_2}),\label{e:wvFE}\\
(\lambda_1+\lambda_2)v_{t_1}\otimes w_{t_2} & = &
(-1)^{\sigma_1}\lambda_2 F\cdot(v_{t_1}\otimes v_{t_2})+
E\cdot(w_{t_1}\otimes w_{t_2}).
\end{eqnarray}
\begin{lemma}\label{l:HI}
With $t_i$ and~$\kappa$ as in Lemma~\ref{l:assobas} the following
formulas hold~:
\begin{eqnarray*}
(\Astd_{t_1,t_2}\otimes \I_{t_3})\circ(\I_{t_1}\otimes
\Ystd_{t_2,t_3}) & = &
\Ystd_{t_4,t_3}\circ\Astd_{t_1,t_6}+\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
\Ydot_{t_4,t_3}\circ\Adot_{t_1,t_6},\\ (\Adot_{t_1,t_2}\otimes
\I_{t_3})\circ(\I_{t_1}\otimes \Ydot_{t_2, t_3}) & = &
\Ydot_{t_7,t_3}\circ\Adot_{t_1,t_9}-\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}
\Ystd_{t_7,t_3}\circ\Astd_{t_1,t_9},\\ (\Adot_{t_1,t_2}\otimes
\I_{t_3} )\circ(\I_{t_1}\otimes\Ystd_{t_2, t_3}) & = &
\Ystd_{t_7,t_3}\circ\Adot_{t_1,t_6},
\\
(\Astd_{t_1,t_2}\otimes \I_{t_3})\circ(\I_{t_1}\otimes\Ydot_{t_2,
t_3}) & = & \kappa\Ydot_{t_4,t_3}\circ\Astd_{t_1,t_9}.
\end{eqnarray*}
\end{lemma}
\begin{proof}
We have
\begin{eqnarray*} & & (\Astd_{t_1,t_2}\otimes \I_{t_3})\circ(\I_{t_1}\otimes
\Ystd_{t_2,t_3})(v_{t_1}\otimes v_{t_6})\\ & &
=(\Astd_{t_1,t_2}\otimes \I_{t_3})(v_{t_1}\otimes v_{t_2}\otimes
v_{t_3})=v_{t_4}\otimes v_{t_3}\\ & & =
\left(\Ystd_{t_4,t_3}\circ\Astd_{t_1,t_6}+\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
\Ydot_{t_4,t_3}\circ\Adot_{t_1,t_6}\right)(v_{t_1}\otimes
v_{t_6}).
\end{eqnarray*}
Using equation~(\ref{e:wvFE}) we see that
\begin{eqnarray*} & & (\Astd_{t_1,t_2}\otimes \I_{t_3})\circ(\I_{t_1}\otimes
\Ystd_{t_2,t_3})(w_{t_1}\otimes w_{t_6})\\ & &
=(\Astd_{t_1,t_2}\otimes \I_{t_3})(w_{t_1}\otimes
F\cdot(v_{t_2}\otimes v_{t_3}))\\ & & =(\Astd_{t_1,t_2}\otimes
\I_{t_3})(w_{t_1}\otimes w_{t_2}\otimes
v_{t_3}+(-1)^{\sigma_2}w_{t_1}\otimes v_{t_2}\otimes w_{t_3})\\
& & =(\Astd_{t_1,t_2}\otimes
\I_{t_3})\left(\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
F\!\cdot\!(v_{t_1}\otimes v_{t_2})\otimes w_{t_3}
-\frac{(-1)^{\sigma_1+\sigma_2}}{\lambda_1+\lambda_2}
E\!\cdot\!(w_{t_1}\otimes w_{t_2})\otimes w_{t_3}\right)\\ & &
=\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
w_{t_4}\otimes w_{t_3}\\ & & =
\left(\Ystd_{t_4,t_3}\circ\Astd_{t_1,t_6}+\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
\Ydot_{t_4,t_3}\circ\Adot_{t_1,t_6}\right)(w_{t_1}\otimes
w_{t_6}).
\end{eqnarray*}
Since an element of~$\Hom_\gloo(V_{t_1}\otimes V_{t_6},
V_{t_4}\otimes V_{t_3})$ is determined by the images of
$v_{t_1}\otimes v_{t_6}$ and $w_{t_1}\otimes w_{t_6}$ this implies
the first equation of the lemma. The remaining three equations are
proved similarly.
\end{proof}
The following corollary holds for reasons of symmetry.
\begin{coro}\label{c:HI}
With $t_i$ and~$\kappa$ as in Lemma~\ref{l:assobas} the following
formulas hold~:
\begin{eqnarray*}
(\I_{t_1}\otimes \Astd_{t_2,t_3})\circ(\Ystd_{t_1,t_2}\otimes
\I_{t_3}) & = &
\Ystd_{t_1,t_6}\circ\Astd_{t_4,t_3}+\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}
\Ydot_{t_1,t_6}\circ\Adot_{t_4,t_3},\\ (\I_{t_1}\otimes
\Adot_{t_2,t_3})\circ(\Ydot_{t_1,t_2}\otimes \I_{t_3}) & = &
\Ydot_{t_1,t_9}\circ\Adot_{t_7,t_3}-\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
\Ystd_{t_1,t_9}\circ\Astd_{t_7,t_3},\\
(\I_{t_1}\otimes\Adot_{t_2,t_3})\circ(\Ystd_{t_1,t_2}\otimes
\I_{t_3}) & = & \Ystd_{t_1,t_9}\circ\Adot_{t_4,t_3},\\
(\I_{t_1}\otimes\Astd_{t_2,t_3})\circ(\Ydot_{t_1,t_2}\otimes
\I_{t_3}) & = & \kappa\Ydot_{t_1,t_6}\circ\Astd_{t_7,t_3}.
\end{eqnarray*}
\end{coro}
\begin{proof}
Let~$\tau_{V,W}\in\Hom(V\otimes W, W\otimes V)$ be the linear map induced by
the permutation of tensor factors. When we interchange the labels~$t_{3n-2}$ and~$t_{3n}$
($n=1,2,3$) in the
equations of Lemma~\ref{l:HI},
replace the equations~$X=Y\in\Hom(V\otimes
W,V'\otimes W')$ by~$\tau_{V',W'} X\tau_{V,W}=\tau_{V',W'}
Y\tau_{V,W}$, and apply the properties preceding equation~(\ref{e:symantisym})
and similar equations for~$\Astd$ and~$\Adot$, then
we obtain the equations of the corollary.
\end{proof}
By equation~(\ref{e:threedecomp}) commutators of elements
of~$\End_{\gl(1\vert 1)}(V_{t_1}\otimes V_{t_2}\otimes V_{t_3})$
lie in the subspace~$\End_{\gl(1\vert 1)}(V_{t_8})$
of~$\End_{\gl(1\vert 1)}(V_{t_1}\otimes V_{t_2}\otimes V_{t_3})$.
We make some explicit computations.
\begin{lemma}\label{l:brIII}
With $t_i$ and~$\kappa$ as in Lemma~\ref{l:assobas}
we have
\begin{eqnarray*}
& & [(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2})\otimes \I_{t_3},
\I_{t_1}\otimes (\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3})]\\ & = &
[(\Ydot_{t_1,t_2}\circ\Adot_{t_1,t_2})\otimes \I_{t_3},
\I_{t_1}\otimes (\Ydot_{t_2,t_3}\circ\Adot_{t_2,t_3})]\\ & = &
[\I_{t_1}\otimes(\Ydot_{t_2,t_3}\circ\Adot_{t_2,t_3}),
(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2}\otimes \I_{t_3})]\\ & = &
[\I_{t_1}\otimes (\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3}),
(\Ydot_{t_1,t_2}\circ\Adot_{t_1,t_2})\otimes \I_{t_3}]\\ & = &
\frac{(-1)^{\sigma_2}\lambda_1\kappa}
{\lambda_1+\lambda_2}(\Ystd_{t_1,t_2}\otimes
\I_{t_3})\circ \Ydot_{t_4, t_3}\circ\Astd_{t_7,
t_3}\circ(\Adot_{t_1,t_2}\otimes \I_{t_3})\\ & &
-\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}(\Ydot_{t_1,t_2}\otimes
\I_{t_3})\circ \Ystd_{t_7, t_3}\circ\Adot_{t_4,
t_3}\circ(\Astd_{t_1,t_2}\otimes \I_{t_3}).
\end{eqnarray*}
\end{lemma}
\begin{proof} For the first commutator we compute
\begin{eqnarray*}
& &
[(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2})\otimes \I_{t_3},
\I_{t_1}\otimes (\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3})]
\\ & = & (\Ystd_{t_1,t_2}\otimes \I_{t_3}) \circ\Ystd_{t_4,t_3}\circ \Astd_{t_1,t_6}\circ
(\I_{t_1}\otimes \Astd_{t_2,t_3})\\ & &
+\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
(\Ystd_{t_1,t_2}\otimes \I_{t_3}) \circ\Ydot_{t_4,t_3}\circ \Adot_{t_1,t_6}\circ
(\I_{t_1}\otimes \Astd_{t_2,t_3})\\
& & -(\I_{t_1}\otimes \Ystd_{t_2, t_3}) \circ\Ystd_{t_1,t_6}\circ \Astd_{t_4,t_3}\circ
(\Astd_{t_1,t_2}\otimes \I_{t_3})\\ & &
-\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}(\I_{t_1}\otimes \Ystd_{t_2,t_3})
\circ\Ydot_{t_1,t_6}\circ\Adot_{t_4,t_3}\circ(\Astd_{t_1,t_2}\otimes \I_{t_3})
\\ & = &
\frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}
(\Ystd_{t_1,t_2}\otimes \I_{t_3}) \circ\Ydot_{t_4,t_3}\circ \Adot_{t_1,t_6}\circ
(\I_{t_1}\otimes \Astd_{t_2,t_3})\\
& & -\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}(\I_{t_1}\otimes \Ystd_{t_2,t_3})
\circ\Ydot_{t_1,t_6}\circ\Adot_{t_4,t_3}\circ(\Astd_{t_1,t_2}\otimes \I_{t_3})
\\
& = & \frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}\left(\kappa
(\Ystd_{t_1, t_2}\otimes \I_{t_3})\circ \Ydot_{t_4,t_3}\circ\Astd_{t_7,t_3}\circ
(\Adot_{t_1,t_2}\otimes \I_{t_3})\right.\\
& & +\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}\left.
(\Ystd_{t_1, t_2}\otimes \I_{t_3})\circ \Ydot_{t_4,t_3}\circ\Adot_{t_4,t_3}\circ
(\Astd_{t_1,t_2}\otimes \I_{t_3})\right)\\
& & -\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}\left(
(\Ydot_{t_1, t_2}\otimes \I_{t_3})\circ \Ystd_{t_7,t_3}\circ\Adot_{t_4,t_3}\circ
(\Astd_{t_1,t_2}\otimes \I_{t_3})\right.\\
& & + \frac{(-1)^{\sigma_2}\lambda_1}{\lambda_1+\lambda_2}\left.
(\Ystd_{t_1, t_2}\otimes \I_{t_3})\circ \Ydot_{t_4,t_3}\circ\Adot_{t_4,t_3}\circ
(\Astd_{t_1,t_2}\otimes \I_{t_3})\right)\\
& = & \frac{(-1)^{\sigma_2}\lambda_1\kappa}
{\lambda_1+\lambda_2}(\Ystd_{t_1,t_2}\otimes
\I_{t_3})\circ \Ydot_{t_4, t_3}\circ\Astd_{t_7,
t_3}\circ(\Adot_{t_1,t_2}\otimes \I_{t_3})\\ & &
-\frac{(-1)^{\sigma_2}\lambda_3}{\lambda_2+\lambda_3}(\Ydot_{t_1,t_2}\otimes
\I_{t_3})\circ \Ystd_{t_7, t_3}\circ\Adot_{t_4,
t_3}\circ(\Astd_{t_1,t_2}\otimes \I_{t_3}),
\end{eqnarray*}
where the first equality follows from Lemma~\ref{l:HI} and Corollary~\ref{c:HI}, the second equality
is a consequence of Lemma~\ref{l:assotriv}, and the third equality is implied by
Corollary~\ref{c:assobas}
and Lemma~\ref{l:assobas}.
The remaining equations can be proven similarly.
\end{proof}
The following lemma will be used to express the action of~$\Phi$
on~$(V_{t_1}\otimes V_{t_2}
\otimes V_{t_3})[[h]]$ in terms of the commutators of our basis elements from
Lemma~\ref{l:brIII}.
\begin{lemma}\label{l:WY}
With $t_i=(\lambda_i,\mu_i,\sigma_i)$ as in equations~(\ref{e:lambdai}), (\ref{e:ti}), and
$$
Y\ = \ (-1)^{\sigma_1+\sigma_2+\sigma_3}\vcentre{comm}\in\End_{\Anab}((b_1,b_2,b_3))
\quad \mbox{($b_i=((-1)^{\sigma_i}\lambda_i,(-1)^{\sigma_i}\mu_i,\sigma_i))$}
$$
we have
$$W(Y)=(\lambda_1+\lambda_2)(\lambda_2+\lambda_3)
\left[(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2})\otimes \I_{t_3},
\I_{t_1}\otimes (\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3})\right].$$
\end{lemma}
\begin{proof}
With $\omega=\sum_\nu a_\nu\otimes b_\nu$ as in equation~(\ref{e:defomega}) we compute
\begin{eqnarray*}
2\,\sum_\nu (-1)^{\deg(v_{t_1})\deg(b_\nu)}
a_\nu\cdot v_{t_1}\otimes b_\nu\cdot v_{t_2} & = &
(\lambda_1(\mu_2+1)+\lambda_2(\mu_1+1))
v_{t_1}\otimes v_{t_2},\\
2\, \sum_\nu (-1)^{\deg(w_{t_1})\deg(b_\nu)}
a_\nu\cdot w_{t_1}\otimes b_\nu\cdot w_{t_2}
& = & (\lambda_1(\mu_2-1)+\lambda_2(\mu_1-1))
w_{t_1}\otimes w_{t_2}.
\end{eqnarray*}
Equation~(\ref{e:defWomega}) implies
\begin{eqnarray*}
2\,W\!\!\left((-1)^{\sigma_1+\sigma_2}\vcentre{t12}\right) & \!\!=\!\! &
(a+b)(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2})\otimes\I_{t_3}+
(a-b)(\Ydot_{t_1,t_2}\circ\Adot_{t_1,t_2})\otimes \I_{t_3},\\
2\,W\!\!\left((-1)^{\sigma_2+\sigma_3}\vcentre{t23}\right) & \!\!=\!\! &
(c+d)\I_{t_1}\otimes(\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3})+
(c-d)\I_{t_1}\otimes(\Ydot_{t_1,t_2}\circ\Adot_{t_1,t_2}),
\end{eqnarray*}
where $a=\lambda_1\mu_2+\lambda_2\mu_1$, $b=\lambda_1+\lambda_2$,
$c=\lambda_2\mu_3+\lambda_3\mu_2$, and $d=\lambda_2+\lambda_3$.
By Lemma~\ref{l:brIII} we have
\begin{eqnarray*}
W(Y) & = & \left[W\!\!\left((-1)^{\sigma_1+\sigma_2}\vcentre{t12}\right),
W\!\!\left((-1)^{\sigma_2+\sigma_3}\vcentre{t23}\right)
\right]\\
& = & (1/4)\left((a+b)(c+d)+(a-b)(c-d)
-(a+b)(c-d)\right.\\
& & \left.-(a-b)(c+d)\right) \left[(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2})\otimes \I_{t_3},
\I_{t_1}\otimes (\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3})\right]\\
& = & bd
[(\Ystd_{t_1,t_2}\circ\Astd_{t_1,t_2})\otimes \I_{t_3},
\I_{t_1}\otimes (\Ystd_{t_2,t_3}\circ\Astd_{t_2,t_3})].
\end{eqnarray*}
This completes the proof.
\end{proof} | 92,787 |
Episode 132 – Sally Thompson / Carl Jung
Episode 132 – Sally Thompson / Carl Jung – Sally Thompson reflects on the cultural differences in thinking between Indigenous and European cultures in the United States that she observed as an archeologist. She pairs her reflection with a passage by the founder of psychoanalysis, Carl Jung.
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MotoLearnshare
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Recently, I have started to play some games for my English learning. In a game, I will briefly explain how to play the game and its story in Japanese. I also try to clear some stages. After that, I edit video, put “hiragana” subtitle for Japanese language learners and try to make English subtitle for my English learning. I will leave the Japanese and English subtitles in my blog and also make green highlighted word list that are linked to online dictionary. If you want to use subtitles for your learning language by game, check my blog. I will announce the latest update for video and blog via Twitter.
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☆Script and Word List in Blog☆ | 52,946 |
TITLE: Kinematics: relative velocity
QUESTION [0 upvotes]: The question goes like this:
An aircraft, with velocity 580 km/h, is supposed to follow a straight path in the direction 38,0° in the northeast (measured from the east). The aircraft experiences a northern wind of 72 km/h. In what direction should the aircraft fly? (the answer is an angle).
I've started with naming the angle $\theta$, then the aircraft has a velocity of $580*\cos{\theta}$ in the x-direction and $580*\sin{\theta} - 72$ in the y-direction. Then we can make the equation $\frac{580*\sin\theta-72}{580*\cos\theta} = \frac{580*sin{38°}}{580*\cos{38°}} = \tan{38°}$. But now I'm struggling on finding the solution to the equation ($\theta=\ldots$). I know the answer is 43.6°, and filling this into the equation gives a valid answer.
REPLY [0 votes]: This type of vector problem, where the unknowns are the size of one vector and the direction of another, can be greatly simplified by a rotation of axes. This rotation should be applied to bring the vector whose size is unknown to lie on one of the axes.
In this case, assuming a wind from the North, and angles measured counter-clockwise from the positive x-axis, we can achieve the above rotation by simply subtracting 38 degrees from each vector's direction. The result is the table below:
The first two columns (V and New Angle) simply give the three vectors (Plane, Air, and Resultant velocity in the new rotated coordinates.
The next two columns give the x- and y- components of each vector. Note that the x and y axes are the rotated ones. The effect of the rotation is to remove the unknown $R$ from one of the columns.
Adding the Plane and Air y-components and setting the sum equal to the y-component of the desired velocity over the ground, we get an equation with one unknown:
$$580 \sin (A) +72 \sin (232)=R \sin(0)=0$$that solves to give $A = 5.61$
Adding 38 to unrotate the axes, we get the heading of the plane is 43.61 degrees. | 184,915 |
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\begin{document}
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\newtheorem{corollary}[theorem]{Corollary}
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\begin{abstract}
The purpose of this paper is to consider flocking formations of a second order dynamic system on a sphere with a Cucker-Smale type flocking operator and cooperative control. The flocking operator consists of a weighted control parameter and a natural relative velocity. The cooperative control law is given by a combination of attractive and repulsive forces. We prove that the solution to this system converges to an asymptotic spherical configuration depending on the control parameters of the cooperative control law. All possible asymptotic configurations for this system are classified and sufficient conditions for rendezvous, deployment, and local deployment, respectively, are presented. In addition, we provide several numerical simulations to confirm our analytic results. We numerically verify that the solution to this system converges to a formation flight with a nonzero constant speed if we add a boost term. The flocking operator acts as a stabilizer on the spherical cooperative control laws.
\end{abstract}
\maketitle
\section{Introduction}\label{sec1}
\setcounter{equation}{0}
Simultaneous control of multiple agents attracts much attention of researchers \cite{G-C-F,K-R,M}. With the recent development of hardware, it is necessary to consider a multi-agent control system on the entire Earth's surface \cite{B-M, L-S,M-G,S-S-Z-T}, not just a local range. The spherical surface has a nonzero curvature rather than a flat space, which yields mathematical challenges. The control system using cooperative manipulation on various manifolds is considered in many studies, such as on a sphere \cite{L2, P-D,T-M-B-G}, Stiefel manifold \cite{M-T-G}, and directed topology \cite{S-M-Z-H-H,Z-X-H-M-T}. In particular, the controllers or algorithms for the uniform deployment of homoclinic agents are investigated using single-integrator model \cite{La-S,P-P} or double-integrator model \cite{L-C}. In \cite{La-S}, the exponential asymptotic stability is shown using invariant manifold techniques for the single-integrator model. After the pioneering work in \cite{O}, the double-integrator multi-agent system of control has been extensively studied. However, as the authors in \cite{S} mentioned, the complete characterization of global convergence of the double-integrator system is unsolved and there are still many open problems. The global convergence property over undirected graphs on the $n$-sphere is studied in \cite{M-G,M-T-G2}. The convergence rate problem over topology networks is investigated in \cite{L-C} with the convergence condition for consensus.
The collective motion of the multi-agent system, including flocking, is a phenomenon frequently observed in nature \cite{Alt, F-E,H-C, S-H}. Recently, many researchers have studied this phenomenon because of its wide applicability. Related topics are also being used in various fields \cite{C-F-R-T, D-M1,Kuromoto,T-T,T-B,V-C-B-C-S,Wi}. In particular, the Cucker-Smale model is often studied due to its high utilization, simple structure, and various mathematical phenomena. The basic Cucker-Smale model is the second-order system of ordinary differential equations (ODEs) describing the ensemble of multiple agents in $\bbr^d$, and the control law is composed of the sum of the following weighted internal relaxation forces \cite{C-S2}:
\begin{align*}
\begin{aligned}
\frac{dx_i}{dt} &= v_i, \\
\frac{dv_i}{dt} &= \sum_{j=1}^{N} \frac{\psi_{ij}}{N}(v_j - v_i),
\end{aligned}
\end{align*}
where $x_i$ and $v_i$ are the position and velocity in $\bbr^d$ of the $i$th agent, respectively, and $\psi_{ij}$ is the communication rate between the $i$th and $j$th agents.
Recently, not only collective motion in the flat space, but also this phenomenon in various manifolds has been attracting attention from researchers. In particular, the dynamics on a spherical surface are closely related to our reality. For example, if we consider the movement of migratory birds, the movement of each agent is close to the movement on a spherical surface rather than a flat space of $\bbr^n$. From an engineering point of view, in recent years, battery performance has rapidly improved, and with the development of an aerial vehicle using sunlight, the movement of unmanned aerial vehicles is approaching a global scale \cite{R, S-W-X}. With the development of hardware technologies, the need to consider the entire Earth as a spatial background has increased.
In this paper, we deal with the second-order system of ODEs on $\mathbb{S}^2$ to achieve a formation flight that maintains the desired pattern while multiple agents fly at a constant speed on a spherical surface.
\begin{align}
\begin{aligned}\label{main}
\dot{x}_i&=v_i,\quad i =1, \ldots, N,\\
\dot{v}_i&=-\frac{\|v_i\|^2}{\|x_i\|^2}x_i+\sum_{j=1}^N\frac{\psi_{ij}}{N}\big(R_{x_j \shortrightarrow x_i}(v_j)-v_i\big)+\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i),
\end{aligned}
\end{align}
where $x_i$ and $v_i$ are the position and velocity of the $i$th agent located on $\mathbb{S}^2$ and the flocking operator on $\mathbb{S}^2$ is expressed by the relative velocity $R_{x_j \shortrightarrow x_i}(v_j)-v_i$. The rotation operator $R_{x_1\shortrightarrow x_2}(y)$, introduced in \cite{C-K-S}, is given by
\begin{align*}
R_{x_1\shortrightarrow x_2}(y)=R(x_1, x_2)\cdot y,
\end{align*}
where $y$ is a column vector. For column vectors $x_1,x_2\in \mathbb{S}^2$ with $x_1\ne -x_2$, $R(x_1,x_2)$ is the following $3\times 3$ matrix.
\begin{align}\nonumber R(x_1,x_2) :=\left\{ \begin{aligned}
&\langle x_1 , x_2\rangle I - x_1 x_2^T + x_2 x_1^T + (1- \langle x_1, x_2\rangle) \Big( \frac{x_1 \times x_2}{|x_1 \times x_2|} \Big) \Big( \frac{x_1 \times x_2}{|x_1 \times x_2|} \Big)^T,\quad&\mbox{if}\quad x_1\ne x_2,\\
&I,\quad&\mbox{if}\quad x_1=x_2,
\end{aligned}
\right.\end{align}
where $I$ is the identity matrix in $\mathbb{R}^3$ and $M^T$ is the transpose of a matrix $M$.
The last term in the right-hand side of \eqref{main} is the cooperative control law on $\mathbb{S}^2$ that is expressed as an appropriate combination of well-known attractive and repulsive forces between $i$th and $j$th nodes based on the position of the agents \cite{L-S,PKH10} and $\sigma_{ij}$ is the inter-particle force parameter between $i$th and $j$th agents given by
\begin{align}\label{sigma}
\sigma_{ij}=\sigma(\|x_i-x_j\|^2)
\end{align}
and
\begin{align}\label{sigma_def}
\sigma(x)=\sigma_a-\frac{\sigma_r}{x}, \quad\mbox{for nonnegative real numbers}~ \sigma_a,\sigma_r\geq 0.
\end{align}
Unlike \cite{L-S}, our model includes the flocking operator instead of the damping term. Therefore, we can add a boost term to our model for the formation flight as in \eqref{eqb}. See Section \ref{sec4}. We further assume that $\psi_{ij}$ is the communication rate between $i$th and $j$th nodes and
\begin{align}
\label{eqn:psi_ij}
\psi_{ij}=\psi(\|x_i - x_j\|),\end{align}
and it satisfies
\begin{align}
\label{eqn:psi}
\mbox{$\psi$ is a nonnegative decreasing $C^1$ function on $[0,2]$ with $\psi(2) = 0$ and $\psi'(2)<0$.}
\end{align}
Throughout this paper we suppose the following admissible initial data conditions: for all $i\in \{1,\ldots,N\}$,
\begin{align}
\label{eqn:ini}\langle x_i(0),x_i(0)\rangle=1 \quad \mbox{and}\quad \langle x_i(0),v_i(0)\rangle=0. \end{align}
From the flocking operator on our system, we expect that the solution to \eqref{main} has a velocity alignment property. See \cite{C-K-S,C-K-S2} for the motivation and the detailed properties of the above operators.
Our system consists of the cooperative control law and Cucker-Smale type flocking operator. The cooperative control term is given by an appropriate linear combination of attractive and repulsive forces between the agents in the ensemble. It is known by \cite{L-S} that the cooperative control law combining the damping term can realize steady-state formations on a spherical surface. Due to the damping term, the ensemble of all agents is stabilized, and one can obtain a stationary formation on a sphere. In our case, to implement the formation flight with a nonzero constant speed, one may replace the damping term with the boost term. However, the control design without the damping term does not produce a robust pattern and creates a chaotic ensemble. We observe that the movement of the ensemble becomes very unstable. See Figures \ref{fig0} and \ref{fig11} in Section \ref{sec4} for more details and numerical simulations.
Using flocking and boost terms, we design a control law to obtain the desired stable formation flight on a spherical surface with a nonzero constant speed. Here, the flocking operator acts as a stabilizer for this boost term so that agents in the ensemble stably maintain their formation and fly at a predetermined constant speed. By changing the cooperative control term like \cite{L-S}, various formations can be formed. In what follows, we completely classify these types of limit cycles in the maximal invariant set of this system and provide a rigorous mathematical proof for the corresponding convergence result. Due to the geometric constraint of the spherical surface and the special properties of the flocking operator, the interaction between the cooperative control and the flocking operator allows the agents to converge to various pattern formations even without the damping term.
\begin{definition}
\cite{C-K-S}
If the solution $\{(x_i(t),v_i(t))\}_{i=1}^N $ to \eqref{main} subject to the admissible initial condition in \eqref{eqn:ini} satisfies the following condition:\begin{align}\nonumber
\max_{1\leq i,j\leq N} \|x_i(t) + x_j(t)\| \| R _{x_j(t)\shortrightarrow x_i(t)}(v_j(t)) - v_i(t) \| = 0,
\end{align}
then we say that the system has velocity alignment at $t\geq 0$.
Similarly, we define asymptotic velocity alignment at $t=\infty$, if
\begin{align}\nonumber
\lim_{t\to \infty}\max_{1\leq i,j\leq N} \|x_i(t) + x_j(t)\| \| R _{x_j(t)\shortrightarrow x_i(t)}(v_j(t)) - v_i(t) \| = 0.
\end{align}
\end{definition}
Additionally, by the geometrically constrained control law in \eqref{main}, we consider the following three types of asymptotic behaviors.
\begin{definition}Let $\{(x_i(t),v_i(t))\}_{i=1}^N $ be the solution to \eqref{main}.
\begin{enumerate}
\item
The ensemble $\{(x_i(t),v_i(t))\}_{i=1}^N $ has an asymptotic rendezvous, if
\[\lim_{t\to\infty }\max_{1\leq i,j\leq N}\|x_i(t)-x_j(t)\|=0.\]
\item
The ensemble $\{(x_i(t),v_i(t))\}_{i=1}^N $ has an asymptotic formation configuration, if
\[\lim_{t\to \infty}\rho(t)=r_0>0.\]
\item
The ensemble $\{(x_i(t),v_i(t))\}_{i=1}^N $ has an asymptotic uniform deployment, if
\[\lim_{t\to\infty }\bar{x}(t)=0,\]
where $\bar{x}$ is the centroid of the agents and $\rho$ is the configuration measurement given by
\[\bar{x}(t)=\frac{1}{N}\sum_{i=1}^N x_i(t),\]
and\[\rho(t)=\sum_{i=1}^N\left\|x_i(t)-\frac{\langle \bar{x},x_i \rangle}{\|\bar{x}\|^2}\bar{x}(t)\right\|^2,\quad \mbox{ if
}~\|\bar{x}(t)\|\ne 0.\]
\end{enumerate}
\end{definition}
\begin{remark}
\begin{enumerate}
\item If we consider an axis parallel to $\bar{x}$ passing through to the origin, then $\rho$ is the total sum of squared distances between $x_i$ and the axis.
\item The structure of $\sigma(\cdot)$ can be used to obtain a pattern scale measurement $\rho(\cdot)$ as in \cite{L-S}.
\end{enumerate}
\end{remark}
\begin{definition}For a given phase state vectors $x_i\in\bbs^2$ and $v_i\in T_{x_i}\bbs^2\subset \bbr^3$ with $\langle x_i,v_i \rangle=0$, we let
\[X=\{x_i\}_{i=1}^N, \quad V=\{v_i\}_{i=1}^N,\quad \mbox{and}\quad Z=\{(x_i,v_i)\}_{i=1}^N.\]
We define an energy functional $\E(Z)$ by
\begin{align*}
\E(Z)&= \E_K(V) + \E_C(X)-\E_C^{min}, \end{align*}
where
\begin{align*}
\E_K(V)&:= \frac{1}{2N}\sum_{i=1}^N\|v_i\|^2,
\\
\E_C(X)& := \frac{\sigma_a}{4N^2 } \sum_{i,j=1}^N \|x_i-x_j\|^2-\frac{\sigma_r}{4N^2 } \log\prod_{i,j=1, i\ne j}^N\|x_i-x_j\|^2,
\end{align*}
and
\[\E_C^{min}= \min_{ X\in (\bbs^2)^N}\E_C(X).\]
\end{definition}
We next present our main analytic results in this paper.
\begin{maintheorem}\label{thm:1}
Assume that $\psi_{ij}$ satisfies \eqref{eqn:psi_ij}-\eqref{eqn:psi} and the initial data $\{(x_i(0),v_i(0))\}_{i=1}^N$ satisfy the admissible initial data conditions in \eqref{eqn:ini}. If $\sigma_r= 0$, then there exists a unique global solution $\{(x_i(t),v_i(t))\}_{i=1}^N $ to \eqref{main}.
If $\sigma_r > 0$ and the initial data satisfy \eqref{eqn:ini} and $x_i(0)\ne x_j(0)$ for any $1\leq i\ne j\leq N$, then there exists a unique global solution $\{(x_i(t),v_i(t))\}_{i=1}^N $ to \eqref{main} and for any $t>0$ and $1\leq i\ne j\leq N$,
\[x_i(t)\ne x_j(t).\]
Moreover, for any initial data, the ensemble has velocity alignment at $t=\infty$.
\end{maintheorem}
\begin{maintheorem}\label{thm:2} Let $N\geq 3$ and $Z(t)=\{(x_i(t),v_i(t))\}_{i=1}^N $ be the solution to \eqref{main} with \eqref{eqn:ini}. Additionally, we suppose that $x_i(0)\ne x_j(0)$ for any $1\leq i\ne j\leq N$ when the repulsive force parameter is nonzero: $\sigma_r > 0$.
Then the solution $Z(t)=\{(x_i(t),v_i(t))\}_{i=1}^N $ satisfies the following asymptotic behaviors.
\begin{enumerate}
\item[(i)] If $\sigma_a>0$, $\sigma_r =0$ and $\displaystyle \E(Z(0))<\E_C^0=\frac{\sigma_a(N-1)}{N^2 }$, then the solution has an asymptotic rendezvous.
\item[(ii)]
If $\displaystyle \frac{2N}{N-1}\sigma_a>\sigma_r>0$ and $\E(Z(0))<\E_{C}^1$, then the ensemble has an asymptotic formation configuration, where $\E_{C}^1$ is the minimum energy of stationary solutions given by
\[\qquad \qquad\E_{C}^1:=\inf_{\|x_i\|=1, \bar{x}=0}\bigg\{\E_C(X)-\E_C^{min} : \sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i)=0,\quad 1\leq i\leq N \bigg\} .\]
\item[(iii)] If $\displaystyle\sigma_r \geq \frac{2N}{N-1}\sigma_a> 0$ or $\displaystyle\sigma_r>0=\sigma_a$, then $\{(x_i(t),v_i(t))\}_{i=1}^N$ has an asymptotic uniform deployment.
\end{enumerate}
\end{maintheorem}
\begin{remark}\label{rmk 1.2}
\begin{enumerate}
\item The conditions for the above theorem are almost optimal. Without the above sufficient conditions, the result in Theorem \ref{thm:2} cannot be established, since there are unstable steady-states that do not satisfy the initial constraints on energy. Further, in the case of $N=2$, we cannot expect that the above results hold due to the result in Corollary \ref{propN2}.
\item The $N$-dependence of $\E_C^0$ in Theorem \ref{thm:2} $(i)$ can be removed by adding a restriction of the initial configuration size; see \cite{C-K-S2}.
\item The solution to our model converges to a stable-state, but the speed of agents may not converge to $0$. See Corollary \ref{propN2} and Proposition \ref{prop 1.6}.
\end{enumerate}
\end{remark}
The remaining subject is the formation flight of a multi-agent ensemble on a sphere. As mentioned before, by adding a boost term to \eqref{main}, we will implement the formation flight of multiple agents on a sphere with a nonzero constant speed through numerical simulations. We emphasize that cooperative control law alone cannot implement this nonzero speed formation pattern. In the formation flight with a nonzero constant speed, the formation of agents is very similar to the formation without the boost term in Theorem \ref{thm:2}. In addition, the mathematical results in Theorem \ref{thm:2} are confirmed through numerical simulation, and various formation patterns are implemented through more general control parameters. We will describe detailed numerical simulation results in Section \ref{sec4}.
The rest of this paper is organized as follows. In Section \ref{sec2}, we present elementary properties of the rotation operator and the global well-posedness of \eqref{main}. In Section \ref{sec3}, we prove the convergence results in our main theorems. In Section \ref{sec4}, we provide several numerical simulations that verify our analytical results and we present the robust formation flight by adding a boost term. Finally, Section \ref{sec5} is devoted to the summary of our main results. \newline
{\bf Notation:}
Throughout this paper, we use the following notation.
\begin{itemize}
\item The domain is a unit sphere defined by
\begin{align*}
\mathbb{S}^2 :=\{(a,b,c)^T\in \bbr^3: a^2+b^2+c^2=1 \}.
\end{align*}
\item We set $X=\{x_i\}_{i=1}^N$ is a sequence of column vectors in $\bbs^2$, $V=\{v_i\}_{i=1}^N$ is a sequence of the corresponding velocity vectors with $\langle x_i,v_i \rangle=0$, and $Z=\{(x_i,v_i)\}_{i=1}^N$ be a sequence of phase vectors.
\item We denote the tangent bundle of $\bbs^2$ by $T\bbs^2$.
\item For a given $z_1,z_2 \in \bbr^3$, we use $\langle z_1,z_2\rangle$ to denote the standard inner product in $\bbr^3$ and the standard symbol
\[\|z_1\| = \|z_1\|_2=\sqrt{\langle z_1,z_1\rangle}.\]
\end{itemize}
\section{Elementary properties of the Newtonian dynamics with the flocking operator and its well-posedness}\label{sec2}
\setcounter{equation}{0}
In this section, we briefly review the elementary properties of \eqref{main} and prove the existence and uniqueness of the global-in-time solution to \eqref{main}. The rotation operator $R_{x_i\shortrightarrow x_j}$ has several remarkable properties such as preserving the modulus of the vector and always having the inverse, but it has singularity when $x_i$ and $x_j$ are antipodal points. Thus, to prove the existence of a global-in-time solution, we must deal with this singularity. For this, we assumed that $\psi(\cdot)$ satisfies the admissible conditions in \eqref{eqn:psi}. Using this condition, in \cite{C-K-S}, we proved that the rotation operator $\psi R$ is locally Lipschitz. From this result, we can obtain the global existence of a solution to the model. In addition, we define an energy functional $\E$ and prove that $\E$ is a decreasing function for time $t\geq 0$. From this property and the definition of $\E$, we can obtain various boundedness of the ensemble which play an important role in later proving the asymptotic behaviors of the solution.
\begin{lemma}\label{lemma 2.3} (Lemma 2.4 in \cite{C-K-S})
Let $x_1, x_2 \in \mathbb{S}^2$ be not antipodal points $(x_1 \ne -x_2)$. Then the rotation operator $R_{\cdot \shortrightarrow \cdot}(\cdot)$ satisfies
\begin{align}
\nonumber
R_{x_1\shortrightarrow x_2}(x_1) = x_2, \quad R_{x_1\shortrightarrow x_2}(x_2) = 2\langle x_1, x_2\rangle x_2 - x_1~ \hbox{ and } ~R_{x_1\shortrightarrow x_2} (x_1 \times x_2) = x_1 \times x_2.
\end{align}
Furthermore, we have
\begin{align}
\nonumber
R_{x_1\shortrightarrow x_2}^{T} = R_{x_2\shortrightarrow x_1},\quad
R_{x_1\shortrightarrow x_2}^T \circ R_{x_1\shortrightarrow x_2} = I_{\mathbb{S}^2}.
\end{align}
\end{lemma}
\begin{lemma}\label{lemma 1.1}
We assume that $\psi_{ij}$ are nonnegative bounded functions for all $i,j\in \{1,\ldots,N\}$.
Let $\{(x_i(t),v_i(t))\}_{i=1}^N$ be a solution to \eqref{main} and the initial data satisfy the admissible conditions in \eqref{eqn:ini}. Then for all $i\in \{1,\ldots,N\}$ and $t>0$,
\begin{align*}\langle v_i(t),x_i(t)\rangle=0 \hbox{ and } \quad \|x_i(t)\|= 1.\end{align*}
\end{lemma}
\begin{proof}
We follow the same argument in \cite{C-K-S}. We take the inner product between the second equation in \eqref{main} and $x_i$ to obtain
\begin{align}\begin{aligned}\label{eq 2.0}
\langle \dot{v}_i, x_i\rangle &=-\|v_i\|^2+\sum_{j=1}^N\frac{\psi_{ij}}{N}\big(\langle R_{x_j\shortrightarrow x_i}(v_j), x_i\rangle-\langle v_i, x_i\rangle\big)
\\&\qquad\qquad\qquad+\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}\big(\|x_i\|^2\langle x_j, x_i\rangle - \langle x_i,x_j \rangle \langle x_i,x_i\rangle\big)
\\&=-\|v_i\|^2+\sum_{j=1}^N\frac{\psi_{ij}}{N}\big(\langle R_{x_j\shortrightarrow x_i}(v_j), x_i\rangle-\langle v_i, x_i\rangle\big)
.
\end{aligned}
\end{align}
By Lemma \ref{lemma 2.3}, \eqref{eq 2.0} implies
\begin{align}\begin{aligned}
\label{eq 2.7}
\langle \dot{v}_i, x_i\rangle
&=-\|v_i\|^2+\sum_{j=1}^N\frac{\psi_{ij}}{N}\big(\langle v_j, x_j\rangle-\langle v_i, x_i\rangle\big).
\end{aligned}
\end{align}
The equalities in \eqref{eq 2.7} yield that
\begin{align*}
\frac{d}{dt}\sum_{i=1}^N|\langle v_i,x_i\rangle|^2&=2\sum_{i=1}^N \big( \langle \dot{v}_i,x_i\rangle+\langle v_i,\dot{x}_i\rangle\big)\langle v_i,x_i\rangle \\
&= 2\sum_{i=1}^N\big(\langle \dot{v}_i, x_i\rangle +\|v_i\|^2\big)~\langle v_i,x_i\rangle
\\
&= 2\sum_{i=1}^N \sum_{j=1}^N\frac{\psi_{ij}}{N}\big(\langle v_j, x_j\rangle-\langle v_i, x_i\rangle\big)~\langle v_i,x_i\rangle
\\
&\leq 2\sum_{i=1}^N \sum_{j=1}^N\frac{\psi_{ij}}{N}\langle v_j, x_j\rangle~\langle v_i,x_i\rangle.
\end{align*}
Then we have
\begin{align*}
\frac{d}{dt}\sum_{i=1}^N |\langle v_i(t),x_i(t)\rangle|^2&\leq2\max_{1\leq j,k\leq N}\psi_{jk} \sum_{i=1}^N \big|\langle v_i(t), x_i(t)\rangle \big|^2.
\end{align*}
From Gronwall's inequality and the admissible conditions on the initial data, it follows that
\begin{align*}\sum_{i=1}^N |\langle v_i(t),x_i(t)\rangle|^2\equiv0,\quad \mbox{for}~t>0.\end{align*}
Again by the inner product between $\dot{x}_i$ and $x_i$ and the first equation of \eqref{main}, we have
\begin{align*}
\frac{d}{dt}\| x_i\|^2=2\langle \dot{x}_i,x_i\rangle =2\langle v_i,x_i\rangle.
\end{align*}
Therefore, we have $\|x_i(t)\|\equiv 1$, for $t>0$, $i\in\{1,\ldots N\}$.
\end{proof}
\begin{proposition}\label{prop 2.4}
We assume that $\psi_{ij}$ satisfies \eqref{eqn:psi_ij}-\eqref{eqn:psi}
and $\sigma_{ij}$ is defined by \eqref{sigma}-\eqref{sigma_def}. For a solution $Z(t)=\{(x_i(t),v_i(t))\}_{i=1}^N $ to \eqref{main} satisfying the admissible conditions in \eqref{eqn:ini}, the following energy identity holds.
\begin{align}\label{EDi}
\frac{d\E(Z(t))}{dt}=-\sum_{i,j=1}^N\frac{\psi(\|x_i(t)-x_j(t)\|)}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j(t))-v_i(t)\|^2.\end{align}
\end{proposition}
\begin{proof}
By Lemma \ref{lemma 2.3},
\begin{align*}
\sum_{i,j=1}^N\frac{\psi_{ij}}{N}\big\langle R_{x_j \shortrightarrow x_i}(v_j)-v_i, v_i\big \rangle
&=\sum_{i,j=1}^N\frac{\psi_{ij}}{N}\big\langle v_j-R_{x_i \shortrightarrow x_j}(v_i), R_{x_i \shortrightarrow x_j}(v_i)\big \rangle\\
&=-\sum_{i,j=1}^N\frac{\psi_{ij}}{N}\big\langle R_{x_i \shortrightarrow x_j}(v_i)-v_j, R_{x_i \shortrightarrow x_j}(v_i)\big \rangle.
\end{align*}
If we change the role of the indices $i$ and $j$ in the above, then we obtain that
\begin{align*}
\sum_{i,j=1}^N\frac{\psi_{ij}}{N}\big\langle R_{x_j \shortrightarrow x_i}(v_j)-v_i, v_i\big \rangle
&=-\sum_{i,j=1}^N\frac{\psi_{ji}}{N}\big\langle R_{x_j \shortrightarrow x_i}(v_j)-v_i, R_{x_j \shortrightarrow x_i}(v_j)\big \rangle.
\end{align*}
Since we assume that $\psi_{ij}=\psi_{ji}$ for all $i,j\in \{1,\ldots,N\}$,
\begin{align}\label{lemma 1.2}
\sum_{i,j=1}^N\frac{\psi_{ij}}{N}\big\langle R_{x_j \shortrightarrow x_i}(v_j)-v_i, v_i\big \rangle= - \sum_{i,j=1}^N\frac{\psi_{ij}}{2N}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2.
\end{align}
We multiply the second equation in \eqref{main} by $v_i$ to obtain
\begin{align*}
\frac{1}{2}\frac{d}{dt} \|v_i\|^2 =\langle \dot v_i,v_i\rangle=&-\frac{\|v_i\|^2}{\|x_i\|^2}\langle x_i,v_i\rangle +\sum_{j=1}^N\frac{\psi_{ij}}{N}\big\langle R_{x_j \shortrightarrow x_i}(v_j)-v_i, v_i\big \rangle\\
&\qquad+\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}\Big(\|x_i\|^2\langle x_j, v_i\rangle - \langle x_i,x_j \rangle \langle x_i,v_i \rangle \Big).
\end{align*}
By the properties in Lemma \ref{lemma 1.1},
\begin{align}\label{eq 2.25}
\frac{1}{2}\frac{d}{dt} \sum_{i=1}^N\|v_i\|^2 = \sum_{i,j=1}^N\frac{\psi_{ij}}{N}\big\langle R_{x_j \shortrightarrow x_i}(v_j)-v_i, v_i\big \rangle+\sum_{i,j=1,j\ne i}^N \frac{\sigma_{ij} }{N}\langle x_j, v_i\rangle.
\end{align}
By \eqref{lemma 1.2} and \eqref{eq 2.25}, the following holds.
\begin{align}\begin{aligned}\label{eq 2.26}
\frac{d\E_K}{dt}&=\frac{1}{2N}\frac{d}{dt} \sum_{i=1}^N\|v_i\|^2\\& = -\sum_{i,j=1}^N\frac{\psi_{ij}}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2+\sum_{i,j=1,j\ne i}^N \frac{\sigma_{ij} }{N^2}\langle x_j, v_i\rangle.
\end{aligned}\end{align}
Since $\sigma_{ij}=\sigma_{ji}$ for all indices $1\leq i\leq N$, we have
\begin{align}\begin{aligned}\label{eq 2.27}
\sum_{i,j=1,j\ne i}^N \frac{\sigma_{ij} }{N^2}\langle x_j, v_i\rangle&= \sum_{i,j=1,j\ne i}^N \frac{\sigma_{ij} }{2N^2}(\langle x_j, v_i\rangle+\langle x_i, v_j\rangle)\\
&= -\sum_{i,j=1,j\ne i}^N \frac{\sigma_{ij} }{2N^2}\langle x_i-x_j, v_i-v_j\rangle
\\& = -\frac{d\E_C}{dt}
.
\end{aligned}\end{align}
From \eqref{eq 2.26}-\eqref{eq 2.27}, we can obtain the energy dissipation identity in \eqref{EDi}.
\end{proof}
\begin{lemma}
\label{lem:con}(Lemma 3.5 in \cite{C-K-S})
Let
\[Q:= (\R^3 \setminus \{0\}) \times (\R^3 \setminus \{0\}) \times \R^3\]
and $T(\cdot,\cdot,\cdot) : Q \rightarrow \R^3$ be a function defined by
\begin{align}\label{eqn:1con}
T(x_1,x_2,v) =\left \{
\begin{aligned}
&\frac{\|x_2\|}{\|x_1\|}\psi \Big( \Big\| \frac{x_1}{\|x_1\|} - \frac{x_2}{\|x_2\|}\Big\| \Big) R_{\small \frac{x_2}{\|x_2\|}\shortrightarrow \frac{x_1}{\|x_1\|}}(v),\quad &\hbox{ if } \frac{x_1}{\|x_1\|} + \frac{x_2}{\|x_2\|} \neq 0,\\
&0,\quad &\hbox{ if } \frac{x_1}{\|x_1\|} + \frac{x_2}{\|x_2\|} = 0.
\end{aligned}\right.
\end{align}
If $\psi$ satisfies assumptions in Theorem \ref{thm:1}, then $T(\cdot,\cdot,\cdot)$ is locally Lipschitz continuous in $Q$.
\end{lemma}
\vspace{1em}
\begin{proof}[\bf Proof of Theorem \ref{thm:1}:] The case of $\sigma_r=0$ is similar to the case of $\sigma_r>0$. We only prove this theorem for the case of $\sigma_r>0$. We follow \cite{C-K-S} for the local existence.
We consider the following system of ODEs subject to the initial data $\{(x_i(0),v_i(0))\}_{i=1}^N$ satisfying the admissible conditions in \eqref{eqn:ini} and $x_i(0)\ne x_j(0)$ for any $i,j\in \{1,\ldots,N\}$:
\begin{align}
\begin{aligned}\label{eqn:wel11}
\dot{x}_i&=v_i,\\
\dot{v}_i&=-\frac{\|v_i\|^2}{\|x_i\|^2}x_i+\sum_{j=1}^N\frac{1}{N}\big(T(x_i, x_j,v_j)-\overline{\psi}_{ij}v_i\big)+\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}\big(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i\big),
\end{aligned}
\end{align}
where $T(\cdot, \cdot,\cdot)$ is given in \eqref{eqn:1con} and
\[\overline{\psi}_{ij}= \psi \left( \Big\| \frac{x_i}{\|x_i\|} - \frac{x_j}{\|x_j\|} \Big\|\right).\]
By Lemma~\ref{lem:con}, the right-hand side of \eqref{eqn:wel11} is Lipschitz continuous with respect to $\{(x_i,v_i)\}_{i=1}^N$ in a small neighborhood of $\{(x_i(0),v_i(0))\}_{i=1}^N$ in $\R^{6N}$. By the standard theory of ODEs, a local-in-time solution $Z(t)=\{(x_i(t),v_i(t))\}_{i=1}^N$ of \eqref{eqn:wel11} exists.
By the same argument in the proof of Lemma \ref{lemma 1.1}, we have
\begin{align*}\begin{aligned}
\langle \dot{v}_i, x_i\rangle &=-\|v_i\|^2+\sum_{j=1}^N\frac{\overline{\psi}_{ij}}{N}\left(\frac{\|x_j\|}{\|x_i\|}\Big\langle R_{\frac{x_j}{\|x_j\|}\shortrightarrow \frac{x_i}{\|x_i\|}}(v_j), x_i\Big\rangle-\langle v_i, x_i\rangle\right)
\\
&= -\|v_i\|^2+\sum_{j=1}^N\frac{\overline{\psi}_{ij}}{N}\big(\left\langle v_j, x_j\right\rangle-\langle v_i, x_i\rangle\big)
.
\end{aligned}
\end{align*}
Therefore,
\begin{align*}
\frac{d}{dt}\sum_{i=1}^N|\langle v_i,x_i\rangle|^2
&= 2\sum_{i=1}^N\big(\langle \dot{v}_i, x_i\rangle +\|v_i\|^2\big)~\langle v_i,x_i\rangle
\\
&= 2\sum_{i=1}^N \sum_{j=1}^N\frac{\overline\psi_{ij}}{N}\big(\langle v_j, x_j\rangle-\langle v_i, x_i\rangle\big)~\langle v_i,x_i\rangle
\\
&\leq 2\sum_{i=1}^N \sum_{j=1}^N\frac{\overline\psi_{ij}}{N}\langle v_j, x_j\rangle~\langle v_i,x_i\rangle.
\end{align*}
Similar to the proof of Lemma \ref{lemma 1.1}, we obtain
\begin{align*}\sum_{i=1}^N |\langle v_i(t),x_i(t)\rangle|^2\equiv0,~\mbox{for}~ t>0.\end{align*}
Thus, the admissible initial conditions imply that
\begin{align*}\langle v_i(t),x_i(t)\rangle=0 \hbox{ and } \quad \|x_i(t)\|= 1,\end{align*}
and $Z(t)=\{(x_i(t),v_i(t))\}_{i=1}^N$ is also a solution of \eqref{main}.
To prove the existence of the global-in-time solution to \eqref{main} and its uniqueness, we consider the maximal interval $I_M=[0,t_{M})$ of existence of a solution to \eqref{main}. From the energy inequality in Proposition~\ref{prop 2.4},
\[\E_K(V(t))\leq \E(Z(t))\leq \E(Z(0)).\]
Thus,
\[\|v_i(t)\|^2\leq 2N\E(Z(0)),\quad \mbox{ for all }\quad i\in\{1,\ldots,N\}.\] Similarly, we have
\[\E_C(X(t))\leq \E(Z(t))\leq \E(Z(0))\] and this implies that there is $l>0$ such that
$\{(x_i,v_i)\}_{i=1}^N$ are uniformly bounded and
\[\|x_i(t)-x_j(t)\|>l.\]
This yields that the local solution on $I_M=[0,t_{M})$ can extend more and we conclude that
\[t_{M} = \infty.\]
\end{proof}
\section{Invariance principle and asymptotic behavior of the solution}
\label{sec3}
\setcounter{equation}{0}
In this section, we consider the asymptotic behavior of the solution $Z(t)=\{(x_i(t),v_i(t))\}_{i=1}^N$ to \eqref{main}. The main idea for the proof of Theorem \ref{thm:2} in Section \ref{sec1} is to use LaSalle's invariance principle and classification of the limit cycles in the maximal invariance set. For this, we use the energy functional $\E$ defined in the previous section. To use LaSalle's invariance principle, we need a functional with a negative semi-definite orbital derivative and a corresponding positively compact invariant set $\Omega_M$ consisting phase points $\{(x_i,v_i)\}_{i=1}^N$, where the orbital derivative of the functional on $\Omega_M$ is zero. The largest invariant set contains the $\omega$-limit set of the ensemble. We can prove that it is a velocity-aligned set. This can be obtained by combining the geometric properties of the spherical surface and the rotation operator $R$. The concept of the largest invariant set and LaSalle's invariance principle are described as follows.
\begin{definition}\cite{K-G}
For a given domain $\mathcal{D}$, let $Z(t)\in \mathcal{D}$ be the solution to an autonomous system
\begin{align}\label{system}
\dot Z=f(Z).
\end{align}
For a given functional $\mathcal{V}(\cdot)$, the derivative of $\mathcal{V}$ along the trajectory of \eqref{system} is defined by
\[\dot{\mathcal{V}}(Z)= \langle \nabla \mathcal{V}, f(Z)\rangle. \]
The set $\Omega\subset \mathcal{D}$ is said to be a positively invariant set if $Z_0\in \Omega$ and $t_0\in\bbr$, the following holds.
\[Z(t)\in \Omega, \quad t\geq t_0, \]
where $Z(t)$ is the solution to $\dot{Z}=f(Z)$ subject to $Z(t_0)=Z_0$.
\end{definition}
\begin{theorem}\label{LITHM}[LaSalle's invariance principle] Let $Z(t)$ be the solution to \eqref{system}
with a locally Lipschitz function $f$ and let $\Omega\subset \mathcal{D}$ be a positively invariant compact set with respect to \eqref{system}. Assume that there is a continuously differentiable functional $\mathcal{V}(\cdot)$ defined in the domain $\mathcal{D}$ and it satisfies
\[\dot{\mathcal{V}}(Z)\leq 0,\quad Z\in \Omega.\]
Let $E$ be the set of all points satisfying $\dot{\mathcal{V}}(Z)=0$ and $M$ be the largest invariant set in $E$.
Then $Z(t)$ starting in $\Omega$ tends to the largest positively invariant set $M$ as $t$ goes to $\infty$.
\end{theorem}
\begin{proof}
For the proof of this theorem, see \cite{K-G}.
\end{proof}
We next define the corresponding positively invariant compact set of \eqref{main}:
\begin{definition}
For a given $\E_0>0$, we define a set
\[\Omega_{\E_0}=\Big\{Z=\{(x_i,v_i)\}_{i=1}^N: \E(Z)\leq \E_0 \Big\}\subset ( T\bbs^{2})^N.\]
\end{definition}
By the definition of $\E$ and Proposition \ref{prop 2.4}, $\Omega_{\E_0}$ is a positively invariant compact set with respect to \eqref{main}. Moreover, Proposition \ref{prop 2.4} implies that
\[\dot{\E}=-\sum_{i,j=1}^N\frac{\psi(\|x_i-x_j\|)}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2\leq 0.\]
Therefore, we directly apply LaSalle's invariance principle to obtain the following proposition.
\begin{proposition}\label{prop 3.2}Let $\{(x_i(t),v_i(t))\}_{i=1}^N $ be a solution to \eqref{main} with the admissible conditions in \eqref{eqn:ini}. We assume that $\psi_{ij}$ satisfies \eqref{eqn:psi_ij}-\eqref{eqn:psi}
and $\sigma_{ij}$ is defined by \eqref{sigma}-\eqref{sigma_def}.
If the initial energy is $\E_0$, the all $\omega$-limit set of the solution $\{(x_i(t),v_i(t))\}_{i=1}^N$ are contained in the largest positively invariant set of $\Omega_{\E_0}$ satisfying
\[\dot\E=-\sum_{i,j=1}^N\frac{\psi(\|x_i-x_j\|)}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2=0.\]
\end{proposition}
We next present characteristics of the largest positively invariant set of $\Omega_{\E_0}$. For this, the geometric properties of the unit sphere and the rotation operator are crucially used. We start with a specially designed two-agents system.
\begin{proposition}\label{prop 3.3}We assume that $\psi(\cdot)$ is a nonnegative bounded function satisfying \eqref{eqn:psi} and $\sigma(\cdot)$ is defined by \eqref{sigma_def}.
Let $\{(x_1(t),v_1(t)),(x_2(t),v_2(t))\}$ be the solution to
\begin{align}
\begin{aligned}\label{two_eq}
\dot{x}_1&=v_1,\\
\dot{v}_1&=-\frac{\|v_1\|^2}{\|x_1\|^2}x_1+ \frac{n\sigma(\|x_1-x_2\|^2) }{N}(\|x_1\|^2x_2 - \langle x_1,x_2 \rangle x_1),\\
\dot{x}_2&=v_2,\\
\dot{v}_2&=-\frac{\|v_2\|^2}{\|x_2\|^2}x_2+ \frac{(N-n)\sigma(\|x_1-x_2\|^2) }{N}(\|x_2\|^2x_1 - \langle x_1,x_2 \rangle x_2),
\end{aligned}
\end{align}
where $N$ and $n$ are natural numbers with $n\leq N$.
If $x_1(t)\ne x_2(t)$, $x_1(t)\ne -x_2(t)$ and
\begin{align}\label{eq vel_align}
0=\psi(\|x_1(t)-x_2(t)\|)\| R_{x_2(t) \shortrightarrow x_1(t)}(v_2(t))-v_1(t)\|^2
\end{align} for all $t\geq 0$, then the solution satisfies one of the followings:
\begin{itemize}
\item[(1)]the two nodes move around the same great circle at the same constant speed. Their relative position is antipodal or $\sigma(\|x_1-x_2\|^2)=0$ holds,
\item[(2)]The trajectories of the two nodes form two parallel circles that have the same radius with distance $\displaystyle 2\sin\frac{\theta}{2}$, where $\theta$ is a constant between $0$ and $\pi$. Moreover, their speed is
\[\sqrt{\frac{-\sigma(\|x_1-x_2\|^2)}{2}(1+\cos \theta)}.\]
\end{itemize}
\end{proposition}
\begin{proof}
We assume that $x_1(t)$ and $x_2(t)$ are not antipodal for any $t\geq 0$. Using a coordinate change, it suffices to consider
\begin{align}\label{eq 3.2}
x_1(0)=(1,0,0)^T,\quad v_1(0)=(0,a,b)^T
\end{align}
and
\begin{align}\label{eq 3.3}
x_2(0)=(\cos \theta,\sin \theta,0)^T,\quad v_2(0)=(-a \sin\theta,a\cos\theta,b)^T,
\end{align}
where $\theta$ is a constant satisfying $ 0<\theta<\pi$.
Note that $\psi(\|x_1(t)-x_2(t)\|)\ne 0$ for any $t\geq $, since $x_1(t)$ and $x_2(t)$ are not antipodal for any $t\geq 0$. By the assumption \eqref{eq vel_align},
\begin{align}\label{eq 3.1}
R_{x_2(t) \shortrightarrow x_1(t)}(v_2(t))=v_1(t),\quad R_{x_1(t) \shortrightarrow x_2(t)}(v_1(t))=v_2(t).
\end{align}
Since $(x_1(t),v_1(t))$ and $(x_2(t),v_2(t))$ satisfy \eqref{eq 3.1} for all $t\geq 0$, we have
\[\frac{d}{dt}[R_{x_2(t) \shortrightarrow x_1(t)}(v_2(t))-v_1(t)]=0.\]
Therefore, we have
\begin{align}\label{eq 3.5}\frac{d R_{x_2(t) \shortrightarrow x_1(t)}}{dt}(v_2(t))+R_{x_2(t) \shortrightarrow x_1(t)}(\dot v_2(t))-\dot v_1(t)=0.\end{align}
We use \eqref{two_eq} and \eqref{eq 3.2}-\eqref{eq 3.1} to obtain that
\begin{align}\label{eq dv1}
\dot v_1(0)=-(a^2+b^2)(1,0,0)^T+ \frac{n}{N}\sigma(\|x_1(0)-x_2(0)\|^2)(0,\sin\theta,0)^T
\end{align}
and
\begin{align}\label{eq dv2}
\dot v_2(0)=-(a^2+b^2)(\cos\theta,\sin\theta,0)^T+ \frac{N-n}{N}\sigma(\|x_1(0)-x_2(0)\|^2)(1-\cos^2\theta,-\sin\theta\cos\theta,0)^T.
\end{align}
By \eqref{eq dv1}-\eqref{eq dv2} and the definition of $R(\cdot,\cdot)$,
\begin{align}\label{eq 3.4}
R_{x_2(0) \shortrightarrow x_1(0)}(\dot v_2(0))-\dot v_1(0)=\left(0,~-\sigma(\|x_1(0)-x_2(0)\|^2)\sin\theta,~0\right)^T.\end{align}
Then, by elementary calculation and the definition of the rotation operator $R$, we have
\begin{align}\begin{aligned}\label{eq 3.6}
\frac{d}{dt} R(x_2(t),x_1(t))
&=\langle v_2(t) , x_1(t)\rangle I+ \langle x_2(t) , v_1(t)\rangle I
- v_2(t) x_1(t)^T
\\
&\qquad- x_2(t) v_1(t)^T
+ v_1(t) x_2(t)^T
+ x_1(t) v_2(t)^T
\\&\qquad
+ \frac{d}{dt}\left(\frac{1- \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\right) ( x_2(t) \times x_1(t) ) ( x_2(t) \times x_1(t))^T
\\&\qquad
+\left(\frac{1- \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\right) ( v_2(t) \times x_1(t) ) ( x_2(t) \times x_1(t))^T\\&
\qquad
+\left(\frac{1- \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\right) ( x_2(t) \times v_1(t) ) ( x_2(t) \times x_1(t))^T
\\&\qquad
+\left(\frac{1- \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\right) ( x_2(t) \times x_1(t) ) ( v_2(t) \times x_1(t))^T\\&
\qquad
+\left(\frac{1- \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\right) ( x_2(t) \times x_1(t) ) ( x_2(t) \times v_1(t))^T.\end{aligned}
\end{align}
Clearly, we have
\begin{align}\begin{aligned}\label{eq 3.7}
\frac{d}{dt}\left(\frac{1- \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\right)&=\frac{- \langle v_2(t), x_1(t)\rangle- \langle x_2(t), v_1(t)\rangle}{|x_2(t) \times x_1(t)|^2}\\
&\quad-\frac{2- 2 \langle x_2(t), x_1(t)\rangle}{|x_2(t) \times x_1(t)|^4}\langle v_2(t) \times x_1(t)+x_2(t) \times v_1(t),x_2(t) \times x_1(t)\rangle .
\end{aligned}\end{align}
If we use \eqref{eq 3.2}-\eqref{eq 3.3} and \eqref{eq 3.6}-\eqref{eq 3.7}, then we obtain
\phantom{\eqref{eq 3.6}\eqref{eq 3.4}}
\begin{align}\label{eq 3.10}
\left[\frac{d}{dt} R(x_2(t),x_1(t))\right]\cdot v_2(t)\bigg|_{t=0}=\left(0,~-2b^2\tan\frac{\theta}{2}, ~2 ab\tan\frac{\theta}{2}\right)^T.
\end{align}
From \eqref{eq 3.5},\eqref{eq 3.4} and \eqref{eq 3.10}, it follows that
\[(0,0,0)=\left(0,~-2\Big(b^2+\frac{\sigma_{12}}{2}+\frac{\sigma_{12}}{2}\cos \theta\Big)\tan\frac{\theta}{2},~ 2ab \tan\frac{\theta}{2} \right),\]
where $\sigma_{12}=\sigma(\|x_1(0)-x_2(0)\|^2)$. Therefore, only two cases are possible:
\[b=0,~ \sigma_{12}=0\]
or
\[a=0, ~b=\pm\sqrt{-\frac{\sigma_{12}}{2}(1+\cos \theta)}.\]
From this result, we complete the proof of this proposition.
\end{proof}
The following corollary of Proposition \ref{prop 3.3} gives a special property for the case of $N=2$. From this, it follows that the result in Theorem \ref{thm:2} does not hold for $N=2$. See Remark \ref{rmk 1.2}. This proposition also will be used in the proof of the main theorem.
\begin{corollary}\label{propN2}We assume that $\psi(\cdot)$ is a nonnegative bounded function satisfying \eqref{eqn:psi} and $\sigma(\cdot)$ is defined by \eqref{sigma_def}. If $N=2$ and $\sigma_r>0$, then the largest positively invariant set $\Omega_M$ consists of the points of the following trajectories.
\begin{itemize}
\item[(1)]Their relative position is antipodal.
\item[(2)]The two nodes move around the same great circle at the same constant speed with $\sigma(\|x_1-x_2\|^2)=0$.
\item[(3)]Each node rotates at the speed
\[\sqrt{\frac{-\sigma(\|x_1-x_2\|^2)}{2}(1+\cos \theta)}\]
on one of two parallel circles that have the same radius with distance $\displaystyle 2\sin\frac{\theta}{2}$,
where $\theta$ is a constant between $0$ and $\pi$.
\end{itemize}
\end{corollary}
The third case in Corollary \ref{propN2} is unstable. We next focus on the case of $N\geq 3$. The following simple lemma plays a crucial role in the proof of the main theorem. The geometric properties of $\mathbb{S}^2$ and the rotation operator give the following rigid motions.
\begin{lemma}\label{lemma 1.5}
Assume that $\{(x_1,v_1),(x_2,v_2),(x_3,v_3)\}\subset T\bbs^2$ are velocity aligned and their positions are not antipodal to each other,
i.e,
\[R_{x_j \shortrightarrow x_i}(v_j)-v_i=0,\quad \mbox{for any ~$i,j\in \{1,2,3\}$.}\]
Then $v_1=v_2=v_3=0$ or $\{x_1,x_2,x_3\}$ are located on a great circle.
\end{lemma}
\begin{proof}We assume that $\{x_1,x_2,x_3\}$ are not located on a great circle. Then, we can consider the triangle on $\bbs^2$ with three points $\{x_1,x_2,x_3\}$ as vertices. Their line segment is a part of a great circle connecting two points of $\{x_1,x_2,x_3\}$. Then the sum of the interior angles of any spherical triangle is strictly greater than $\pi$. We assume that $v_1\ne 0$ and
\[R_{x_1 \shortrightarrow x_2}(v_1)=v_2,\quad R_{x_2 \shortrightarrow x_3}(v_2)=v_3,\quad R_{x_3 \shortrightarrow x_1}(v_3)=v_1.\]
That means
\[R_{x_3 \shortrightarrow x_1}\circ R_{x_2 \shortrightarrow x_3}\circ R_{x_1 \shortrightarrow x_2}(v_1)=v_1.\]
However, the above holds only for $v_1=0$ or the sum of the exterior angles of the spherical triangle is $2\pi$. Since the sum of the interior angles of any spherical triangle is strictly greater than $\pi$, the sum of the exterior angles of the spherical triangle is not $2\pi$. Therefore, we obtained the desired result.
\end{proof}
\begin{lemma}\label{lemma 3.5}
We assume that $\psi_{ij}$ satisfies \eqref{eqn:psi_ij}-\eqref{eqn:psi}
and $\sigma_{ij}$ is defined by \eqref{sigma}-\eqref{sigma_def}. Let $M_{\E_0}\subset \Omega_{\E_0}$ be the largest positively invariant set of \eqref{main} satisfying
\begin{align}\label{eq 3.11}
\dot\E=-\sum_{i,j=1}^N\frac{\psi(\|x_i-x_j\|)}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2=0.
\end{align}
If $N\geq 3$ and $\sigma_r>0$, then for any particle trajectory $\{(x_i(t),v_i(t))\}_{i=1}^N$ of \eqref{main} in $M_{\E_0}$, there is $t_0\geq 0$ such that one of the followings holds.
\begin{itemize}
\item[(a)] $v_i(t_0)=0$ for all $i\in \{1,\ldots,N\}$ and $\{x_i(t_0)\}_{i=1}^N$ are not deployed on one great circle,
\item[(b)] $\{x_i(t_0)\}_{i=1}^N$ are deployed on a great circle and their velocities are aligned along the great circle.
\end{itemize}
\end{lemma}
\begin{proof}Let
\[\psi_{ij}(t)=\psi(\|x_i(t)-x_j(t)\|).\]
We divide it into the following three cases:
\[N\geq 5, \quad N=4, \quad N=3.\]
For $N\geq 5$, we take any trajectory $\{(x_i(t),v_i(t))\}_{i=1}^N$ of \eqref{main} in $M_{\E_0}$. We assume that there is $t_0\in \bbr$ such that $(x_i(t_0),v_i(t_0))_{i=1}^N$ are not deployed on a great circle. Since $N\geq 5$, there are three points $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ such that any two points of them are not antipodal. This implies that
\begin{align}\label{eq 3.12}
\psi_{ij}(t_0),\psi_{jk}(t_0), \psi_{ki}(t_0)\ne 0.
\end{align}
By \eqref{eq 3.11}-\eqref{eq 3.12} and Lemma \ref{lemma 1.5},
\begin{align}\label{eq 3.135}
v_i(t_0)=v_j(t_0)=v_k(t_0)=0
\end{align}
or $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ are deployed on a great circle.
If we assume that \eqref{eq 3.135} does not hold, then $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ are deployed on a great circle. For this case, we claim that $\{v_i(t_0), v_j(t_0), v_k(t_0)\}$ are aligned along their great circle.
\begin{proof}[The proof of the claim:]
If one of the velocities is a zero vector, then all velocities are zero since they are not antipodal to each other and velocities are aligned. Then we are done.
Next, we consider the case that all velocities are nonzero. We assume that the velocities are not aligned along the great circle. Then for sufficiently small $\epsilon>0$, $\{x_i(t_0+\epsilon),x_j(t_0+\epsilon),x_k(t_0+\epsilon)\}$ does not located on the great circle but $v_i(t_0+\epsilon),v_j(t_0+\epsilon),v_k(t_0+\epsilon)$ are not zero vector by the continuity of the solution. Therefore, any two vectors of $\{x_i(t_0+\epsilon),x_j(t_0+\epsilon),x_k(t_0+\epsilon)\}$ are not antipodal. Therefore, it does not satisfy \[v_i(t_0+\epsilon)=v_j(t_0+\epsilon)=v_k(t_0+\epsilon)=0\]
nor $\{x_i(t_0+\epsilon),x_j(t_0+\epsilon),x_k(t_0+\epsilon)\}$ are deployed on a great circle. Thus, it is not invariant by Lemma \ref{lemma 1.5}. Therefore, if three points are deployed on a great circle, then their velocities are aligned along the great circle.
\end{proof}
Thus, we have proved that one of the following statements holds.
\begin{itemize}
\item[(a')] $v_i(t_0)=v_j(t_0)=v_k(t_0)=0$ and $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ are not deployed on a great circle,
\item[(b')]$\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ are deployed on a great circle and their velocities are aligned along the great circle.
\end{itemize}
Next, we consider the rest of the agents $\{(x_l,v_l)\}_{1\leq l\leq N, l\ne i, j,k}$ for both cases $(a')$ and $(b')$ in the above. For (a'), we fixed the $l$th agent with position $x_l(t_0)$, $1\leq l\leq N$, $l\ne i, j,k$. Then, at least one of $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ is not antipodal with $x_l(t_0)$. Thus, by \eqref{eq 3.11}, $v_l(t_0)=0$ for all $l\in \{1,\ldots, N\}$. Therefore, in this case, all velocities are the zero vector. For (b'), we also fix the $l$th particle with $(x_l(t_0),v_l(t_0))$. If $x_l(t_0)$ is located on the great circle containing $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$, then $v_l(t_0)$ is aligned along the great circle by the same argument in the above.
If there is an index $l\in \{1,\ldots,N\}-\{i,j,k\}$ such that $x_l(t_0)$ is not located on the great circle containing $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$, then we can find three points in $\{x_i(t_0),x_j(t_0),x_k(t_0), x_l(t_0)\}$ and the three points are not antipodal to each other and they are not located on any great circle. Thus, by Lemma \ref{lemma 1.5}, their velocities are all the zero vectors. If there is $t_0\in \bbr$ such that $\{(x_i(t_0),v_i(t_0))\}_{i=1}^N$ are not deployed on a great circle, then we obtain the result for $N\geq 5$. Otherwise, for all $t\geq 0$, $\{(x_i(t),v_i(t))\}_{i=1}^N\in M_{\E_0}$ are located on the great circle. Similar to the previous case, we can verify that their velocities are aligned along the great circle.
For $N=4$, there is $t_0\in \bbr$ such that $\{(x_i(t_0),v_i(t_0))\}_{i=1}^N$ are not deployed on a great circle or not. We first assume that there is $t_0\in \bbr$ such that $\{(x_i(t_0),v_i(t_0))\}_{i=1}^N$ are not deployed on a great circle. Then we can find three position in $\{(x_i(t_0),v_i(t_0))\}_{i=1}^N$, say
\[S=\{(x_i(t_0),v_i(t_0)),(x_j(t_0),v_j(t_0)),(x_k(t_0),v_k(t_0))\},\]
such that $S$ are not deployed on a great circle and any two agents in $S$ are not antipodal. Then by the same argument in the case of $N\geq 5$, we obtain the result.
The remaining case is that four points $\{(x_i(t_0),v_i(t_0))\}_{i=1}^N$ are located on a great circle for all $t\geq 0$. Then we can divide it into two subcases: they form two pairs of antipodal points or not. If they do not form two pairs of antipodal points, then there are three points $\{x_i(t_0),x_j(t_0),x_k(t_0)\}$ such that any two points of them are not antipodal. Thus, by the previous argument, this lemma holds. Next, we consider two pairs of antipodal points. Take $(x_1(t_0),v_1(t_0))$ in $\{(x_i(t_0),v_i(t_0))\}_{i=1}^4$. Without loss of generality, we let $x_3(t_0)$ be the antipodal points of $x_1(t_0)$. Thus, $x_2(t_0)$ and $x_4(t_0)$ are antipodal each other. Then we have
\begin{align}\label{eq 3.13}
R_{x_1 \shortrightarrow x_2}(v_1)=v_2,\quad R_{x_2 \shortrightarrow x_3}(v_2)=v_3,\quad R_{x_3 \shortrightarrow x_4}(v_3)=v_4,\quad \quad R_{x_4 \shortrightarrow x_1}(v_4)=v_1.
\end{align}
Similar to the case of $N\geq 5$, we assume that the velocities are not aligned along the great circle. Then there is a sufficiently small $\epsilon>0$ such that $\{x_1(t_0+\epsilon),x_2(t_0+\epsilon),x_3(t_0+\epsilon), x_4(t_0+\epsilon)\}$ does not located on a great circle and $\{v_1(t_0+\epsilon),v_2(t_0+\epsilon),v_3(t_0+\epsilon),v_4(t_0+\epsilon)\}$ are not the zero vector by the continuity of the solution. Moreover, by \eqref{eq 3.13}, the four points are located on a hemisphere. Hence, the four points are not located on the same great circle and any two vectors of $\{x_1(t_0+\epsilon), x_2(t_0+\epsilon), x_3(t_0+\epsilon), x_4(t_0+\epsilon)\}$ are not antipodal. Since their velocities are nonzero, it is not invariant. Using the same method in the case of $N\geq 5$, we can prove that the velocities are aligned along the great circle or all velocities are zero vectors.
Finally, we consider $N=3$. If they are not on a great circle, then there is no pair of antipodal points. Therefore, all velocities are zero by Lemma \ref{lemma 1.5}. Thus, it suffices to consider that they are located on a great circle. Then by a similar argument in the case of $N=4$, we can also prove that the velocities are aligned along the great circle or all velocities are zero vectors.
\end{proof}
\begin{proposition}\label{prop 1.6}We assume that $\psi_{ij}$ satisfies \eqref{eqn:psi_ij}-\eqref{eqn:psi}
and $\sigma_{ij}$ is defined by \eqref{sigma}-\eqref{sigma_def}. Let $M_{\E_0}\subset \Omega_{\E_0}$ be the largest positively invariant set of \eqref{main} satisfying
\begin{align*}
\sum_{i,j=1}^N\frac{\psi(\|x_i-x_j\|)}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2=0.
\end{align*}
If $N\geq 3$ and $\sigma_r>0$, then $M_{\E_0}$ consists of the points of the following trajectories:
\begin{itemize}
\item[(1)] a stationary state, i.e., $v_i(t)=0$ for all $i\in \{1,\ldots,N\}$,
\item[(2)] the all nodes moving around the same great circle at the same constant speed.
\end{itemize}
Moreover, for both cases of (1) and (2), the following holds for $i\in \{1,\ldots,N\}$.
\begin{align}\label{eq 3.15}
\sum_{j=1,j\ne i}^N \frac{\sigma(\|x_i-x_j\|^2) }{N}(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i)=0.
\end{align}
\end{proposition}
\begin{proof}
By Lemma \ref{lemma 3.5}, if $N\geq 3$, then for any solution $\{(x_i(t),v_i(t))\}_{i=1}^N \in M_{\E_0}$, there is $t_0\geq 0$ such that
\begin{itemize}
\item $v_i(t_0)=0$ for all $i\in \{1,\ldots,N\}$ and $\{x_i(t_0)\}_{i=1}^N$ are not deployed on a great circle~~~~~
\[\mbox{or}\]
\item $\{x_i(t_0)\}_{i=1}^N$ are deployed on a great circle and their velocities are aligned along the circle.
\end{itemize}
\noindent$\circ$ Case 1: all agents $\{x_i(t_0)\}_{i=1}^N$ are not deployed on one great circle.\\
By the above, $v_i(t_0)=0$ for all $i\in \{1,\ldots,N\}$.
In this case, we claim that for all $t\geq 0$ and $i\in \{1,\ldots,N\}$, $v_i(t)=0$.
Assume not, i.e., there is $t_1\geq 0$ and an index $i_1\in \{1,\ldots,N\}$ such that
\[v_{i_1}(t_1)\ne 0.\]
Without loss of generality, we may assume that $t_1>t_0$. Then there is the maximum interval $[t_0,T]$ such that for all $1\leq i\leq N$,
\[v_i(t)=0, \quad t\in [t_0,T]\]
and there is $i_1'\in\{1,\ldots, N\}$ such that
\[v_{i_1'}(t)\neq 0, \quad t\in (T,T'),\]
for some $T'>0$. Therefore, at $t=T$, $\{x_i(t)\}_{i=1}^N$ are not deployed on a great circle.
By the continuity of the solution, we can choose small $\epsilon>0$ such that
\begin{align}\label{eq 3.14}
v_{i_1'}(T+\epsilon)\neq 0
\end{align}
and $\{x_i(T+\epsilon)\}_{i=1}^N$ are not deployed on an one great circle. Thus, there are three points in $\{x_i(T+\epsilon)\}_{i=1}^N$ such that any two points in the three points are not antipodal. Since $\{(x_i(t),v_i(t))\}_{i=1}^N \in M_{\E_0}$, we have
\[\sum_{i,j=1}^N\frac{\psi(\|x_i(t)-x_j(t)\|)}{2N^2}\| R_{x_j(t) \shortrightarrow x_i(t)}(v_j(t))-v_i(t)\|^2=0\quad \mbox{ at $t=T+\epsilon$.}\]
This implies that $\| R_{x_j(T+\epsilon) \shortrightarrow x_i(T+\epsilon) }(v_j(T+\epsilon) )-v_i(T+\epsilon) \|^2=0$ for these three agents. Thus, by Lemma \ref{lemma 1.5}, their velocities at $t=T+\epsilon$ are all zero vectors. By similar argument in Lemma \ref{lemma 3.5}, other agents also have zero velocity. This is a contradiction to \eqref{eq 3.14}. Since $\{(x_i(t),v_i(t))\}_{i=1}^N \in M_{\E_0}$ is the solution to \eqref{main}, \eqref{eq 3.15} holds for all $t\geq 0$ and $i\in \{1,\ldots,N\}$.\\
\noindent $\circ$ Case 2: all $\{x_i(t_0)\}_{i=1}^N$ are deployed on a great circle and their velocities are aligned along the circle.\\
Their trajectories also keep the same great circle. Therefore, after changing the coordinates, we can represent the solution by
\[x_i=(\cos\theta_i,\sin \theta_i,0),\quad i=1,\ldots,N.\]
Therefore, we have
\[v_i=\dot\theta_i(-\sin\theta_i,\cos \theta_i,0),\quad i=1,\ldots,N\]
and
\[\dot v_i=\ddot\theta_i(-\sin\theta_i,\cos \theta_i,0)-(\dot\theta_i)^2(\cos\theta_i,\sin \theta_i,0),\quad i=1,\ldots,N.\]
Moreover, by the velocity alignment, for any $i,j\in\{1,\ldots,N\}$,
\begin{align}\label{thetadot}
\dot \theta_i=\dot \theta_j.
\end{align}
Since $\psi_{ij}\big(R_{x_j \shortrightarrow x_i}(v_j)-v_i\big)=0$ for any $i,j\in\{1,\ldots,N\}$, the second equation in the main system gives the following equality.
\[\ddot\theta_i(-\sin\theta_i,\cos \theta_i)=\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}\left[(\cos\theta_j,\sin \theta_j) - \cos (\theta_i-\theta_j)(\cos\theta_i,\sin \theta_i)\right].\]
This yields that
\begin{align}\label{eq 3.16}\ddot\theta_i=\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}\sin (\theta_j-\theta_i).\end{align}
The right hand side of the above is a constant for $t>0$ by \eqref{thetadot}. Therefore, if $\ddot\theta_i \ne 0$, then $\dot\theta_i$ goes to $\infty$ or $-\infty$. However, it is impossible since $\E$ is bounded. Thus, we have $\ddot\theta_i = 0$ and it is a constant rigid motion. Finally, \eqref{eq 3.16} and $\ddot\theta_i=0$ implies \eqref{eq 3.15}.
\end{proof}
\begin{lemma}\cite{L-S}\label{lemma 3.6} Let $\{(x_i(t),v_i(t))\}_{i=1}^N $ be a solution to \eqref{main} and $x_i(t)\in \bbs^2$ for all $i\in \{1,\ldots,N\}$.
We assume that $\sigma_{ij}$ is defined by \eqref{sigma} and the following holds.
\begin{align}\label{eq 3.165}
\sum_{j=1,j\ne i}^N \frac{\sigma(\|x_i(t)-x_j(t)\|^2) }{N}(\|x_i(t)\|^2x_j(t) - \langle x_i(t),x_j(t) \rangle x_i(t))=0,\quad \mbox{for all $i\in \{1,\ldots,N\}$}.
\end{align}
Then
\begin{align*}
\Big(N\sigma_a-\frac{(N-1)\sigma_r}{2}\Big)\|\bar{x}(t)\|^2=\sigma_a\sum_{i=1}^N \langle x_i(t),\bar{x}(t) \rangle^2,
\end{align*}
where $\bar{x}$ is the centroid of the agents.
\end{lemma}
\begin{proof}
By \eqref{eq 3.165}, $x_i\in \bbs^2$ and $\sigma_{ij}=\sigma_{ji}$, we have
\begin{align*}
0=\sum_{i,j=1,j\ne i}^N \sigma_{ij} (x_i - \langle x_i,x_j \rangle x_i)=\sum_{i,j=1,j\ne i}^N \left(\sigma_a-\frac{\sigma_r}{\|x_i-x_j\|^2}\right)(x_i - \langle x_i,x_j \rangle x_i).
\end{align*}
Since $\|x_i-x_j\|^2=2-2\langle x_i,x_j\rangle$,
\begin{align*}
0=\frac{1}{2}\sum_{i,j=1,j\ne i}^N \left(\sigma_a\|x_i-x_j\|^2-\sigma_r\right)x_i
=\sum_{i=1}^N \left(N\sigma_a-N\sigma_a\langle x_i,\bar{x} \rangle-\frac{(N-1)\sigma_r}{2}\right)x_i.
\end{align*}
Therefore, we have
\begin{align*}
0=\sum_{i=1}^N \left(\sigma_a-\sigma_a\langle x_i,\bar{x} \rangle-\frac{N-1}{N}\frac{\sigma_r}{2}\right)\langle x_i,\bar{x}\rangle=
\Big(N\sigma_a-\frac{(N-1)\sigma_r}{2}\Big)\|\bar{x}\|^2-\sigma_a\sum_{i=1}^N \langle x_i,\bar{x} \rangle^2.
\end{align*}
\end{proof}
\begin{proof}[\bf Proof of Theorem \ref{thm:2}] We will use Proposition \ref{prop 3.2} and the classification of the largest positively invariant set in Proposition \ref{prop 1.6}.
Let $M_{\E_0}$ be the largest positively invariant set of \eqref{main} satisfying
\begin{align*}
\sum_{i,j=1}^N\frac{\psi(\|x_i-x_j\|)}{2N^2}\| R_{x_j \shortrightarrow x_i}(v_j)-v_i\|^2=0\quad \mbox{and}\quad \E(Z) \leq \E_0\quad \mbox{for}\quad Z\in M_{\E_0}.
\end{align*}
\newline
For $(i)$, we assume that $\sigma_a>0$, $\sigma_r =0$ and $\E_0=\E(Z(0))<\E_C^0$. We claim that for any $\{(x_i,v_i)\}_{i=1}^N\in M_{\E_0}$,
\[x_i=x_j,\quad \mbox{for any}~i,j\in \{1,\ldots,N\}.\] We take any trajectory $\{(x_i(t),v_i(t))\}_{i=1}^N\in M_{\E_0}$ and fix $t\geq 0$. Then the following three cases are possible.
\begin{enumerate}
\item[$\circ$] all vectors in $\{x_1(t),\ldots,x_N(t)\}$ are the same,
\item[$\circ$] there are only two different vectors in $\{x_1(t),\ldots,x_N(t)\}$,
\item[$\circ$] at least, there are three different vectors in $\{x_1(t),\ldots,x_N(t)\}$.
\end{enumerate}
We assume that there are only two different vectors in $\{x_1(t),\ldots,x_N(t)\}$. For simplicity, we may assume that $x_1(t)\ne x_2(t)$. Then we can easily check that $\{(x_1(t),v_1(t)), (x_2(t),v_2(t))\}$ is the solution to \eqref{two_eq}. Thus, we can apply Proposition \ref{prop 3.3} to obtain that $x_1(t)$ and $x_2(t)$ are antipodal since the case (2) in Proposition \ref{prop 3.3} does not occurs by
\[\sigma(\|x_i-x_j\|^2)=\sigma_a> 0.\]
This implies that any two vectors in $\{x_1(t),\ldots,x_N(t)\}$ are the same or antipodal. However, all configurations in this case violate the assumption: $\E(Z) \leq \E_0$.
Assume that at least, there are three different vectors in $\{x_1,\ldots,x_N\}$. By Proposition \ref{prop 1.6},
they are a stationary state or constant speed motions on a great circle and satisfying
\[0=\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i).\]
By Lemma \ref{lemma 3.6},
\begin{align*}
\|\bar{x}\|^2=\frac{1}{N}\sum_{i=1}^N \langle x_i,\bar{x} \rangle^2.
\end{align*}
This means that $x_i=x_j$ for all indices $i,j\in \{1,\ldots,N\}$ and this is a contradiction.
Therefore, all positions in $M_{\E_0}$ satisfy that $x_i=x_j$ for any $i,j\in\{1,\ldots,N\}$ and this implies that the solution has the asymptotic rendezvous property by Theorem \ref{LITHM}.\\
Next we consider $(ii)$. Assume that $\displaystyle\frac{2N}{N-1}\sigma_a>\sigma_r>0$ and $\E(Z(0))<\E_{C}^1$. We take any $\{(x_i(t),v_i(t))\}_{i=1}^N\in M_{\E_0}$ and fix time $t\geq 0$. By Proposition \ref{prop 1.6}, $\{(x_i(t),v_i(t))\}_{i=1}^N\in M_{\E_0}$ is a stationary state or constant speed motions on a great circle and satisfies that for any $i=1,\ldots,N$,
\[0=\sum_{j=1,j\ne i}^N \frac{\sigma_{ij} }{N}(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i).\]
Since $\E(Z(0))<\E_{C}^1$ and $\displaystyle\frac{d\E(Z(t))}{dt} \leq 0$, we have $\bar{x}\neq 0$. By Lemma \ref{lemma 3.6},
\[0=\Big(N\sigma_a-\frac{(N-1)\sigma_r}{2}\Big)\|\bar{x}\|^2-\sigma_a\sum_{i=1}^N \langle x_i,\bar{x} \rangle^2.\]
This implies that
\begin{align*}
(1-\frac{N-1}{N}\frac{\sigma_r}{2\sigma_a})\|\bar{x}\|^2=\frac{1}{N}\sum_{i=1}^N \langle x_i,\bar{x} \rangle^2.
\end{align*}
Note that
\[\rho(t)+\frac{1}{\|\bar{x}\|^2}\sum_{i=1}^N \langle x_i,\bar{x} \rangle^2=N.\]
Therefore, $\displaystyle\rho(t)=\frac{(N-1)\sigma_r}{2\sigma_a}$ for all $t\geq 0$ and we conclude that the ensemble has an asymptotic formation configuration by Theorem \ref{LITHM}.\\
Finally, we deal with $(iii)$. Let $\E_0$ be the initial energy of a fixed solution $\{(x_i,v_i)\}_{i=1}^N$. We assume that $\displaystyle\sigma_r \geq \frac{2N}{N-1}\sigma_a> 0$ or $\displaystyle\sigma_r>0=\sigma_a$. Similar to the previous cases, if $N\geq 3$, then Proposition \ref{prop 1.6} and Lemma \ref{lemma 3.6} imply that
\[\bar{x}=0,\quad \mbox{for any}~\{(x_i,v_i)\}_{i=1}^N\in M_{\E_0}.\]
Thus, the ensemble has an asymptotic uniform deployment by Theorem \ref{LITHM}.
\end{proof}
\begin{remark}
By a symmetric formation, for $N=3$, we can construct an example such that $\rho(t)$ converges to $N$ when $\displaystyle \sigma_r < \frac{2N}{N-1}\sigma_a$. Therefore, the above result is almost optimal.
\end{remark}
\section{Simulation results}\label{sec4}\setcounter{equation}{0}
In this section, by adding a boost term to \eqref{main}, we numerically implement the nonzero-speed formation flight of the solution to the main system. We verify the results in Theorem \ref{thm:2} through numerical simulations. In \cite{L-S}, cooperative control laws were able to achieve various steady-state patterns by interacting with a damping term. However, it is necessary to remove the damping term and add a boost term to keep the formation and a non-zero speed of agents simultaneously. In this case, the desired formation was not constructed by only cooperative control (Figure \ref{fig0}). On the other hand, \eqref{main} derives robust non-stationary formations even when the boost term is added due to interaction with the cooperative control law and the flocking term for the spherical surface. We note that each agent maintains a nonzero constant speed (Figure \ref{fig9}). In particular, it can be seen that the patterns of the flight formation are very similar to the stationary patterns without the boost term. See Figure \ref{fig6} for the comparison between maximal diameters of the ensemble in \eqref{main} and the boosted ensemble in \eqref{eqb}.
\begin{figure}[!ht]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig1a.eps}\\
(a) $t=0$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig1b.eps}\\
(b) $t=1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig1d.eps}\\
(c) $t=10$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig1f.eps}\\
(d) $t=50$
\end{minipage}
\caption{Time evolution when $\psi(x)=1$ and $\sigma_{ij}=0$}
\label{fig1}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig2a.eps}\\
(a) $t=1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig2c.eps}\\
(b) $t=10$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig2e.eps}\\
(c) $t=50$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig2f.eps}\\
(d) $t=100$
\end{minipage}
\caption{Time evolution when $\psi(x)=2(\exp(2-x)-1)$, $\sigma_a=1$ and $\sigma_r=0.5$}
\label{fig2}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3g.eps}\\
(a) $\sigma_r=0$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3d.eps}\\
(b) $\sigma_r=0.01$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3e.eps}\\
(c) $\sigma_r=0.04$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3f.eps}\\
(d) $\sigma_r=0.08$
\end{minipage}
\vfill
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3a.eps}\\
(e) $\sigma_r=0.1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3b.eps}\\
(f) $\sigma_r=0.2$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3c.eps}\\
(g) $\sigma_r=0.3$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig3h.eps}\\
(h) $\sigma_r=0.5$
\end{minipage}
\caption{Patterns of formation when $\sigma_a=1$ and $T=400$}
\label{fig3}
\end{figure}
For the numeric simulations, we consider the flocking formation on $\mathbb{S}^2$ with $N=6$ agents.
The initial position and velocity are randomly chosen in $(x,v)\in T\mathbb{S}^2\cap\left([-1,1]^3\times [-3,3]^3\right)$ as follows:
\begin{align*}
&x_1(0)= (-0.1192,\phantom{-}0.5108 ,-0.8514), && x_2(0)=( \phantom{-}0.8547 , -0.3671, \phantom{-}0.3671),\\
& x_3(0)=(\phantom{-}0.7076 , \phantom{-}0.2235, \phantom{-}0.6704 ),&&
x_4(0)=(\phantom{-} 0.3600,\phantom{-} 0.7364, \phantom{-} 0.5728 ),\\
& x_5(0)=( \phantom{-} 0.8977, -0.4406, \phantom{-}0.0000 ),& &x_6(0)=( \phantom{-} 0.8754, -0.2398, -0.4197),
\end{align*}
and
\begin{align*}
&v_1(0)=( -1.1540, -1.7264, -0.8743), &&v_2(0)=( -1.3068, \phantom{-}0.0568, \phantom{-}3.0000 ),\\ &v_3(0)=( \phantom{-}0.9331, \phantom{-} 1.5789, -1.5113),
&&v_4(0)=( \phantom{-}2.7989, \phantom{-} 0.7976, -2.7848 ), \\&v_5(0)=( -1.0254, -2.0892, \phantom{-} 2.5773 ),&& v_6(0)=( \phantom{-}0.4591, \phantom{-} 0.8375, \phantom{-}0.4789 ).
\end{align*}
Here, all initial data satisfy the admissible conditions:
\[\|x_i(0)\|=1\quad \mbox{ and }\quad\langle x_i(0),v_i(0)\rangle=0.\]
The energy can be totally dissipated when $\sigma=0$ and hence the agents are no longer moving after some time period. See Figure \ref{fig1} (c). To visually express such a phenomenon, in this section, we use a red point for the position $x_i(t)$ of $i$th agent at $t=t_0$ and the blue line in all figures that represents the trajectory of agents for the time interval $[t_0-3, t_0]$. Thus, a red point without a blue line in some figure, for example, Figure \ref{fig1}(d), means the corresponding agent does not move in the time interval $[t-3, t]$.
For the cooperative control law, we first consider the following case.
\[\sigma(x)=\sigma_a-\frac{\sigma_r}{x}.\]
In this case, as seen in Section \ref{sec3}, if the attractive force parameter $\sigma_a$ is less than the repulsive force parameter $\sigma_r>0$, then their configuration is evenly distributed on the entire spherical surface. Therefore, we assume $\psi(x)=2(\exp(2-x)-1)$ and take $\sigma_r<\sigma_a$.
With this decreasing function $\psi$ and $\sigma_a=1$, the pattern formations of \eqref{main} are given in Figure \ref{fig3} for several values of $\sigma_r$.
In Figure \ref{fig3}, if $\sigma_r$ is sufficiently small, that is, the effect of repulsion is small, then agents are clustered together.
As $\sigma_r$ increases, the force to repel each other becomes stronger.
Therefore, the diameter is getting larger. Also, $\rho(t)$ converges $(N-1)\sigma_r/2\sigma_a$ as we proved in Theorem \ref{thm:2}.
See Figure \ref{fig4} and Figure \ref{fig5} for numerical simulations. We note that for all cases except for that their configuration is on a great circle, it keeps the energy of the system in \eqref{main} being dissipated. This leads the agents to no longer move after a period of time as we can see in Figure \ref{fig2}.
\begin{figure}[!ht]
\centering
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig4a.eps}\\
(a) Norm of position $\|x_1\|$
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig4b.eps}\\
(b) Norm of velocity $\|v_1\|$
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig4c.eps}\\
(c) $\rho(t)$
\end{minipage}
\caption{The norm of position, velocity of the first agent and $\rho(t)$ when $\sigma_r=0.01$ and $\sigma_a=1$ }
\label{fig4}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig5a.eps}\\
(a) Norm of position $\|x_1\|$
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig5b.eps}\\
(b) Norm of velocity $\|v_1\|$
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig5c.eps}\\
(c) $\rho(t)$
\end{minipage}
\caption{The norm of position, velocity of the first agent and $\rho(t)$ when $\sigma_r=0.3$ and $\sigma_a=1$ }
\label{fig5}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{fig6.eps}\\
(a) Maximum diameter for \eqref{main}
\vspace{1em}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{fig6b.eps}\\
(b) Maximum diameter for \eqref{eqb} with $b=0.2$
\end{minipage}
\caption{Maximum diameter of agents when $\beta\in [0,4]$ and $T=50$}
\label{fig6}
\end{figure}
Next, we consider more general parameters:
\[\sigma(x)=\sigma_a-\frac{\sigma_r}{x^\beta}.\]
Then the exponent $\beta$ represents the effect of repulsive force.
If $\beta>1$, then the closer the agents are to each other, the stronger the force they repel against each other.
The maximum diameter of agents when $\sigma_a=5$ and $\sigma_r=0.5$ is given in Figure \ref{fig6} and hence we can check that the role of $\beta$.
In addition, in Figure \ref{fig7}, the patterns for the specific values $\beta=0, 0.5, 1, 1.25, 1.26, 1.5, 2, 5$ are displayed. For these cases, we can find that each agent is arranged at the same distance as the adjacent agents. It can be seen that their configuration pattern changes for $\beta$ between $1.25$ and $1.26$. See Figure \ref{fig7}(d) and (e).
\begin{figure}[!ht]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig7a.eps}\\
(a) $\beta=0$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig7b.eps}\\
(b) $\beta=0.5$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig7c.eps}\\
(c) $\beta=1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig8a.eps}\\
(d) $\beta=1.25$
\end{minipage}
\vfill
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig8b.eps}\\
(e) $\beta=1.26$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig7d.eps}\\
(f) $\beta=1.5$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig7e.eps}\\
(g) $\beta=2$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{fig7f.eps}\\
(h) $\beta=5$
\end{minipage}
\caption{Patterns when $\sigma_a=5$ and $\sigma_r=0.5$}
\label{fig7}
\end{figure}
As we can see in the previous examples, the agents could stop depending on their energy. This happens in almost all cases except for a few limited cases. To avoid this phenomenon, we add a boost term in the second equation of \eqref{main}. The flocking model on a sphere with boost term is given by
\begin{align}
\begin{aligned}\label{eqb}
\dot{x}_i&=v_i,\\
\dot{v}_i&=-\frac{\|v_i\|^2}{\|x_i\|^2}x_i+f_i^bv_i+\sum_{j=1}^N\frac{\psi_{ij}}{N}\big(R_{x_j \shortrightarrow x_i}(v_j)-v_i\big)+\sum_{j=1}^N \frac{\sigma_{ij} }{N}(\|x_i\|^2x_j - \langle x_i,x_j \rangle x_i),
\end{aligned}
\end{align}
where $f_i^b$ is a boost parameter.
One example of the boost parameter $f_i^b$ is as follows:
\begin{equation}\label{booster}
f_i^b=\begin{cases}
\displaystyle
-\frac{2}{b}\|v_i\|+2, & \text{if $\|v_i(t)\|\leq b$,}
\\
0, & \text{otherwise,}
\end{cases}
\end{equation}
where $b$ is a positive constant. We note that the boost term increases velocity when the speed falls below a value.
When we consider the boost term such as \eqref{booster}, the agents escape from a sphere after a certain large time because of the accumulation of computation error.
If we want to observe the dynamics of agents for a long time, an additional correction term should be added as in \cite{L-S}.
However, for this boosted case, we observe that the dynamics of the ensemble remains on the sphere. With the same parameter of Figure \ref{fig2}, the time evolution of agents when $b=0.2$ is given in Figure \ref{fig9}.
Comparing to Figure \ref{fig2}, we can check that the agents are moving at the nonzero constant speed without stopping.
See Figure \ref{fig10}.
The velocity is not zero and $\rho(t)$ is perturbed near \[\frac{(N-1)\sigma_r}{2\sigma_a}=1.25\]
because the boost term pushes the agents steadily.
\begin{figure}[!ht]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig9a.eps}\\
(a) $t=1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig9c.eps}\\
(b) $t=30$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig9e.eps}\\
(c) $t=60$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig9f.eps}\\
(d) $t=90$
\end{minipage}
\caption{Time evolution of \eqref{eqb} when $\sigma_a=1$ and $\sigma_r=0.5$}
\label{fig9}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig10a.eps}\\
(a) Norm of position $\|x_1\|$
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig10b.eps}\\
(b) Norm of velocity $\|v_1\|$
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig10c.eps}\\
(c) $\rho(t)$
\end{minipage}
\caption{The norm of position, velocity of the first agent and $\rho(t)$ for \eqref{eqb} with $b=0.2$ }
\label{fig10}
\end{figure}
In \cite{L-S}, the authors consider the following second-order system of ODEs with the cooperative control law on the sphere: for $1\le i\le N$,
\begin{align}
\begin{aligned}\label{L-S}
\frac{dx_i}{dt} &= v_i, \\
\frac{dv_i}{dt} &= -\|v_i\|^2x_i-k_vv_i+u_i+f_i^0,
\end{aligned}
\end{align}
where
\[u_i=\sum_{j\in \mathcal{N}_i}\omega_{ij}\left(k_a- \frac{k_r}{\|x_i-x_j\|^2}\right)(x_j - \langle x_i,x_j\rangle x_i)\quad \mbox{and} \quad f_i^0=-k_0\left(x_i-\frac{x_i}{\|x_i\|}\right).\]
\begin{figure}[ht!]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0a.eps}\\
(a) $t=0$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0b.eps}\\
(b) $t=1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0c.eps}\\
(c) $t=2$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0d.eps}\\
(d) $t=3$
\end{minipage}
\vfill
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0f.eps}\\
(e) $t=5$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0g.eps}\\
(f) $t=8$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0h.eps}\\
(g) $t=10$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0i.eps}\\
(h) $t=200$
\end{minipage}
\caption{Time evolution of \eqref{L-S}}
\label{fig0}
\end{figure}
\begin{figure}[ht!]
\centering
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig0a.eps}\\
(a) $t=0$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11a.eps}\\
(b) $t=1$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11b.eps}\\
(c) $t=2$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11c.eps}\\
(d) $t=3$
\end{minipage}
\vfill
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11d.eps}\\
(e) $t=5$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11e.eps}\\
(f) $t=8$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11f.eps}\\
(g) $t=10$
\end{minipage}
\begin{minipage}{0.243\textwidth}
\centering
\includegraphics[width=\textwidth]{fig11g.eps}\\
(h) $t=200$
\end{minipage}
\caption{Time evolution of \eqref{L-S}when $k_v=0$ and $f_i^b$ is equipped}
\label{fig11}
\end{figure}
Here, agents are controlled by the control law $u_i$ and stayed on the sphere by feedback term $f_i^0$.
The parameters $k_v$, $k_a$, and $k_r$ are a damping, attraction, and repulsion constant, respectively. Due to the damping term $k_vv_i$, the ensemble of all agents is stabilized and one can obtain the stationary formation on the sphere.
Therefore, this model is appropriate for the case when the target area is fixed.
See Figure \ref{fig0}.
This figure is the time evolution of \eqref{L-S} with parameters $(N,k_a,k_r,k_v,k_0)=(6,1,0.1,7,10^4)$ and initial configuration is randomly chosen satisfying $\|x_i(0)\|=1$ and $\langle x_i(0),v_i(0)\rangle=0$ for all index $i\in \{1,\ldots,N\}$.
Note that if there is no damping term, that is, $k_v=0$, then the system \eqref{L-S} yields chaotically moving agents.
If the boost term $f_i^b$ is added in the absence of damping, agents move more chaotically.
See Figure \ref{fig11}. However, for the case of the system of \eqref{main}, even if there is no damping term or there is a boost term, we can obtain a moving formation due to the flocking operator in the model \eqref{main}. Therefore, it is suitable to apply this model for detection or surveillance problems where the target area is not decided or the target area is relatively large.
\section{Conclusion}\label{sec5}\setcounter{equation}{0}
In this paper, we studied a spherical flocking model with attractive and repulsive forces. For any admissible initial conditions, we demonstrate the velocity alignment as well as the global well-posedness of the model in Theorem \ref{thm:1}. In Theorem \ref{thm:2}, we classify all possible asymptotic configurations for this model: rendezvous, formation configuration, and uniform deployment. In particular, our model maintains the desired pattern of a formation flight with nonzero constant speed. We observe that this nonzero speed formation cannot be achieved without the flocking operator, which acts as a stabilizer. Our analytic results were supported by numerical simulations.
\bibliographystyle{amsplain} | 152,564 |
\begin{document}
\title[Asymptotics of $\Ptw$]{Global Asymptotics of the Second Painlev\'e equation in Okamoto's space}
\author{P. Howes}\thanks{This research was supported by an Australian Postgraduate Award. }
\address{School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia\\ Fax: +61 2 9351 4534}
\email{[email protected]}
\author{N. Joshi}\thanks{This research was supported by the Australian Research Council grant \# DP110102001. }
\address{School of Mathematics and Statistics F07, The University of Sydney, NSW 2006, Australia\\ Tel: +61 2 9351 2172\\ Fax: +61 2 9351 4534}
\email{[email protected]}
\begin{abstract}
We study the solutions of the second Painlev\'e equation ($\Ptw$) in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, $x$, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlev\'e equation ($\Pth$). By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repellor for the dynamics in Okamoto's space. Moreover, we show that the limit set of the solutions exists and is compact and connected.
\end{abstract}
\keywords{The second Painlev\'e equation, thirty-fourth Painlev\'e equation, asymptotic behaviour, resolution of singularities, rational surface}
\subjclass[2000]{34M55; 34E05,34M55, 34M30,14E15}
\date{}
\maketitle
\section{Introduction}
The second Painlev\'e equation
\begin{align}\label{painleve2}
\frac{d^2y}{dx^2}&=2y^3+xy+\alpha,
\end{align}
is well known from its application in random matrix theory \cite{tw:94,m:04} amongst many others. For integer values of $\alpha$, it is known that there exist rational solutions, which are expressed as the logarithmic derivative of ratios of successive Yablonskii-Vorob'ev polynomials. In this paper, we study the asymptotic behaviour of all solutions in complex projective space of dimension two for $x\in{\mathbb C}$, with $|x|\to\infty$. Throughout the paper we assume that $\alpha$ is fixed and bounded.
The asymptotic study of $\Ptw$ was initated by Boutroux \cite{bou:14} and has more recently been studied in \cite{joshi:88,joshi:92,kit:94,dei:95,fok:06}. Boutroux provided a change of variables in which the asymptotic behaviours become explicit and studied the equation directly, whilst more recent approaches have centred around the Riemann-Hilbert method. This paper uses the explicit construction of the blown up space of initial conditions in order to give a description of the solution space as the modulus of the independent variable approaches infinity.
It is known through Okamoto's \cite{oka:79} compactification of the space of initial conditions that the solution space is connected. However, until the work of Duistermaat and Joshi \cite{djo:10} in the case of ${\hbox{P}_{\scriptsize\hbox{I}}}$, no completeness and connectedness study of the asymptotic behaviours had been carried out for any of the Painlev\'e equations to our knowledge. In this paper we fill this gap for the second and thirty-fourth Painlev\'e equations.
The main result of this paper is to show compactness and connectedness of the limit sets of solutions to the second and thirty-fourth \cite{inc:56} Painlev\'e equations as the independent variable approaches infinity. A more precise statement is made in Section \ref{statements}. The proof of this statement relies on the construction of a function $d$, which is used to measure the distance of a solution of (\ref{painleve2}) to a set of exceptional lines created in the process of blowing up (\ref{painleve2}) in $\mathbb{P}^2(\mathbb{C})$. As a corollary, we find that all solutions to $\Ptw$ which are not uniformly asymptotically zero must necessarily have infinitely many poles.
The paper is organised as follows: In Section \ref{setup}, we provide a change of variables to make more explicit the asymptotic behaviour of (\ref{painleve2}), as well as provide some definitions and explain the terminology to be used in subsequent sections.
In Section \ref{statements}, we introduce the main result, as well as provide some corollaries to this result.
In Section \ref{specsols}, we show how the special solutions to (\ref{painleve2}) arise naturally by considering the degenerate values of the autonomous energy function described in Section \ref{setup}. Furthermore, we will make some remarks about the resolution of singularities for nonlinear systems, and their dependence on choices of coordinates and asymptotic limits. The blow up (resolution of singularities) for the system (\ref{boutrouxsys})-(\ref{boutrouxsys2}) is carried out in Appendix \ref{app1}, where are the details are given explicitly as the proofs of the statements in Section \ref{statements} require precise asymptotic estimates. In Section \ref{conc} we make some concluding remarks.
\section{Hamiltonian System and Definitions}\label{setup}
The second Painlev\'e equation (\ref{painleve2}) has a Lagrangian $L$ given by
\begin{equation}
\label{p2}
L(Q,\dot{Q},x)=\dfrac{\dot{Q}^2}{2}+\dfrac{Q^4}{2}+x\dfrac{Q^2}{2}+\alpha Q,
\end{equation}
which with the Euler-Lagrange equations implies (\ref{painleve2}) for $Q$. Performing a standard Legendre transformation leads to an associated Hamiltonian
\begin{equation}
\label{preham}
H(Q,P,x)=\dfrac{P^2}{2}-\dfrac{Q^4}{2}-x\dfrac{Q^2}{2}-\alpha Q.
\end{equation}
Whilst this Hamiltonian system implies (\ref{painleve2}) for Q, we make the canonical change of variables with type 3 generating function, $G_3=-pQ+Q^3/3+xQ/2$:
\begin{subequations}
\begin{align}
\label{half}
P &= p-(Q^2+x/2), \\
q &= Q.
\end{align}
\end{subequations}
Under this change the Hamiltonian becomes
\begin{equation}\label{hamiltonian1}
H(q,p;x)=\dfrac{p^2}{2}-(q^2+x/2)p-(\alpha+1/2)q,
\end{equation}
which with Hamilton's equations implies the second Painlev\'e equation (\ref{painleve2}) for $q$ \textit{and} the thirty-fourth Painlev\'e equation for $p$:
\begin{equation}
\label{p34}
\dfrac{d^2p}{dx^2}=\dfrac{1}{2p}\left(\dfrac{dp}{dx}\right)^2+2p^2-xp-\dfrac{\beta}{2p},
\end{equation}
where $\beta=(\alpha-1/2)^2$.
Thus from this point forward, relevant statements made in the analysis of $\Ptw$ hold analogously for $\Pth$. Further benefits of choosing this coordinatisation will be discussed in Subsection \ref{coords}.
\subsection{Boutroux Scaling}
We are interested in studying the limit $x\rightarrow \infty$, $x\in \mathbb{C}$ and so we perform another change of variables (\`a la Boutroux \cite{bou:14}) to make the leading order asymptotic behaviour explicit. The Hamiltonian (\ref{hamiltonian1}) is almost weighted homogeneous, which inspires the following change of variables:
\begin{eqnarray*}
q & = & \lambda u, \\
p & = & \lambda^2 v,\\
x &=& \lambda^2 \zeta ,
\end{eqnarray*}
and then
\begin{equation}
H=\lambda^4\left(\dfrac{v^2}{2}-(u^2v+\dfrac{\zeta v}{2}-\dfrac{(\alpha+1/2)u}{\lambda^3})\right).
\end{equation}
$\zeta=1$ if and only if $\lambda^2=x$. Then if $x=x(z)$ and $\dot{}$ is differentiation with respect to $z$, then
\begin{displaymath}
\dot{u}=\dot{x}\left((v-u^2-1/2)x^{1/2}-\dfrac{u}{2x}\right),
\end{displaymath}
and so if we choose $\dot{x}x^{1/2}=1$, then we have $z=2/3x^{3/2}$. We then have the system:
\begin{eqnarray}\label{boutrouxsys}
\dot{u} & = & v-u^2-1/2-\dfrac{u}{3z}, \\ \label{boutrouxsys2}
\dot{v} & = & 2uv+\dfrac{2\alpha+1}{3z}-\dfrac{2v}{3z} .
\end{eqnarray}
This pair of equations implies the Boutroux forms of $\Ptw$ and $\Pth$ for $u$ and $v$ respectively:
\begin{subequations}
\begin{align}
\label{bp2}
\ddot{u}&=2u^3+u+\dfrac{2\alpha}{3z}-\dot{u}\dfrac{1}{z}+\dfrac{u}{9z^2} \\
\ddot{v}&=\dfrac{\dot{v}^2}{2v}+2v^2-v-\dfrac{\dot{v}}{z}-\dfrac{(2\alpha-1)^2}{18vz^2}+\dfrac{1}{4z^2}. \label{bp32}
\end{align}
\end{subequations}
The Boutroux system (\ref{boutrouxsys})-(\ref{boutrouxsys2}) is an order $z^{-1}$ perturbation of the autonomous system with time independent Hamiltonian given by
\begin{equation}\label{energyfct}
E:=v^2/2-u^2v-v/2,
\end{equation}
where
\begin{equation}
\dot{E}:=\dfrac{-4E}{3z}+\dfrac{4\alpha v-(2\alpha+1)(2u^2+1)}{6z}\,\,.
\end{equation}
\begin{remark}
The second Painlev\'e equation has the B\"acklund transformations
\begin{equation}
\label{bt1}
\mathcal{T}^{{\pm}}:\enskip y(x;\alpha\pm 1)=-y-\dfrac{2\alpha\pm 1}{2y^{2}\pm 2y^{{\prime}}+x}.
\end{equation}
Under the Boutroux change of variables, the B\"acklund transformation becomes
\begin{equation}
\label{bt2}
\mathcal{T}^{{\pm}}:\enskip u(z;\alpha\pm 1)=-u-\dfrac{2\alpha\pm 1}{3zu^{2}\pm u\pm 3z\dot{u}+3z/2}.
\end{equation}
Upon elimination of the derivative term in (\ref{bt1}), one finds the equation referred to as $alt-dP_I$. In the Boutroux setting, the equation becomes a Boutroux form of $alt-dP_I$:
\begin{equation}
\label{baltdp1}
\dfrac{2\alpha+1}{\bar{u}+u}+\dfrac{2\alpha-1}{\underbar{u}+u}+6zu^2+3z=0,
\end{equation}
where $u=u(z,\alpha)$ and $\bar{u}=u(z,\alpha+1)$, $\underbar{u}=u(z,\alpha-1)$. This is an appropriate scaling for the large $z$ limit for $alt-dP_I$.
\end{remark}
\subsection{Notation and Definitions}
The natural setting to study the Painlev\'e equations is in complex projective space, where the poles become zeroes in the coordinate charts near infinity in the affine plane. Following the pioneering work of Okamoto \cite{oka:79}, we study the Painlev\'e system in $\mathbb{P}^2(\mathbb{C})$.
We embed the Boutroux-Painlev\'e system in $\mathbb{P}^2(\mathbb{C})$ and identify the affine coordinates with homogeneous coordinates as
\begin{eqnarray*}
[1:u:v] &= &[u^{-1}:1:vu^{-1}]=[u_{01}:1:v_{01}]\\
&=&[v^{-1}:uv^{-1}:1]=[v_{02}:u_{02}:1]\,.
\end{eqnarray*}
It is known from \cite{oka:79} that every continuous Painlev\'e equation can be regularised (made free from the indeterminacy of the flow through base points) by blowing up $\mathbb{P}^2(\mathbb{C})$ at 9 points. Indeed it is known from Sakai's classification \cite{sak:01} that all (continuous and discrete) Painlev\'e equations can be regularised by a 9-point blow up of $\mathbb{P}^2(\mathbb{C})$.
We denote the line at infinity by $L_0$. Note that it is given by $u_{01}=0$ or $v_{02}=0$. For $0\leq i\leq 8$ corresponding to the $i$-th stage of the blowing up process, we denote the $i$-th base point $b_i$ and the exceptional line attached to the base point by $L_{i+1}$ and the coordinates of the two charts coordinatising $L_{i+1}$ by $(u_{ij},v_{ij})$, $j=1,2$.
Moreover, we denote the proper transform of $L_i$ after the ninth blow up as $L_i^{(9-i)}$ and the $z$-dependent set of lines where the vector field becomes infinite by $I(z):=\bigcup_{i=0, i\neq 6}^{8} L_{i}^{(9-i)}(z)$. This set will be referred to as \textit{the infinity set}. The final space obtained by regularising the Boutroux-Painlev\'e system is denoted $S_9(z)$, but we will often drop the explicit $z$ dependence for ease of notation.
In the proofs of the results, we regularly make use of the Jacobian of the coordinate transformation from $(u,v)$ to the new coordinate system, given by
\begin{displaymath}
w_{ij} = \dfrac{\partial u_{ij}}{\partial u}\dfrac{\partial v_{ij}}{\partial v}-\dfrac{\partial u_{ij}}{\partial v}\dfrac{\partial v_{ij}}{\partial u} .
\end{displaymath}
\section{Statement of results} \label{statements}
In this section we will state the main result: that the limit set of the solutions of the second and thirty-fourth Painlev\'e equations forms a non-empty, compact and connected subset of $S_9(\infty) \backslash I(\infty)$, which is the space $S_9(z) \backslash I(z)$ obtained by replacing all $1/z$ terms by $0$. The result is a consequence of Theorem \ref{theorepel}, which will be proved in Section \ref{proof}. In this theorem, $d(z)$ denotes a distance measure to the infinity set $I(z)$, that is, $d(z)=0$ if and only if the solution to the Painlev\'e system crosses $I(z)$. It is shown in Lemma \ref{lemma2} that a continuous such distance measure exists. The corollaries which follow from Theorem \ref{theorepel} are proved in this section.
\begin{thm}\label{theorepel}
Let $\epsilon_1,\epsilon_2,\epsilon_3$ be given such that $\epsilon_1>0$, $0<\epsilon_2<4/3$, $0<\epsilon_3<1$. Then there exists $\delta\in\mathbb{R}_{>0}$ such that if $|z_0|>\epsilon_1$, and $|d(z_0)|<\delta$, then
\begin{displaymath}
\rho = \sup \left\{r>|z_0|\,\, \mbox{such that}\,\, |d(z)|<\delta\,\, \mbox{whenever}\, |z_0|\leq |z|\leq r\right\}
\end{displaymath} satisfies
\begin{description}
\item[\textrm{(i)}]
\begin{displaymath}
\delta \geq |d(z_0)| \left(\frac{\rho}{z_0}\right)^{4/3-\epsilon_2}(1-\epsilon_3)\,.
\end{displaymath}
\item[\textrm{(ii)}] If $|z_0|\leq |z| \leq \rho$, then
\begin{displaymath}
d(z)=d(z_0)\left(\frac{z}{z_0}\right)^{4/3+\epsilon_2(z)}(1+\epsilon_3(z))\, .
\end{displaymath}
\item[\textrm{(iii)}] If $|z|\geq \rho$ then
\begin{displaymath}
|d(z)|\geq \delta (1-\epsilon_3) \, .
\end{displaymath}
\end{description}
\end{thm}
\begin{remark} Note that in case (iii) of Theorem \ref{theorepel}, we have $d(z)>0$ for all complex times $z$, $|z|\geq|z_0|$ and so the solution never reaches the infinity set.
\end{remark}
The following is a definition of a limit set for complex dynamical systems.
\begin{definition}
For every solution $\mathbb{C} \backslash \{0\} \ni z \mapsto U(z) \in S_9(z) \backslash I(z)$, let $\Omega_U$ denote the set of all $s \in S_9(\infty) \backslash I(\infty)$ such that there exists a sequence $z_j \in \mathbb{C}$ with the property that $z_j \rightarrow \infty$ and $U(z_j) \rightarrow s$ as $j \rightarrow \infty$. The subset $\Omega_U$ of $S_9(\infty) \backslash I(\infty)$ is called the limit set of the solution $U$.
\end{definition}
\begin{thm}\label{corlimitset}
There exists a compact subset $K$ of $S_9(\infty) \backslash I(\infty)$ such that for every solution $U$ the limit set $\Omega_U$ is contained in K. The limit set $\Omega_U$ is a non-empty, compact and connected subset of K, invariant under the flow of the autonomous Hamiltonian system on $S_9(\infty) \backslash I(\infty)$. For every neighbourhood A of $\Omega_U$ in $S_9$ there exists an $r>0$ such that $U(z)\in A$ for every $z\in\mathbb{C}$ such that $|z|>r$. If $z_j$ is any sequence in $\mathbb{C}\backslash \{0\}$ such that $z_j \rightarrow \infty$ as $j\rightarrow \infty$, then there is a subsequence $j=j(k)\rightarrow\infty$ as $k\rightarrow \infty$ and an $s\in \Omega_U$ such that $U(z_{j(k)})\rightarrow s$ as $k\rightarrow \infty$.
\begin{proof}
See Corollary 4.6 in \cite{djo:10}, whose method of proof also applies here.
\end{proof}
\end{thm}
\begin{lemma}\label{invariance}
$\Omega_U$ is invariant under the transformation $(E,u,v)\mapsto (E,-u,v)$.
\begin{proof}
It is known that the solutions $y(x)$ to the second Painlev\'e equation are single valued in the complex plane. The transformation to Boutroux coordinates $y(x)=x^{1/2}u(z)$, $z=2x^{3/2}/3$ is singular at $z=0$ and correspondingly $x=0$. They introduce multivaluedness of the solutions $u(z)$ when $z$ runs around the origin. The single valuedness of $y(x)$ and the relation $u(z)=(3z/2)^{-1/3} y((3z/2)^{2/3})$ implies that the analytic continuation of $u(z)$ along the path $z e^{i\theta}$ becomes $-u(z)$ when $\theta$ runs from 0 to $3\pi$. That is
\begin{align}
&u(ze^{3\pi i})=-u(z),
\end{align}
and from \eqref{boutrouxsys} we have
\begin{align}
&v(ze^{3\pi i}) =v(z).
\end{align}
Recall also from \eqref{energyfct} that $E=v^2/2-u^2v-v/2$. The lemma follows from these results.
\end{proof}
\end{lemma}
\begin{corollary}\label{infpoles}
Every solution $u(z)$ of the second Painlev\'e equation whose limit set is not $\{0\}$ has infinitely many poles.
\begin{proof}
Let $u(z)$ be a solution of the Boutroux-Painlev\'e equation with only finitely many poles. Let $U(z)$ be the corresponding solution of the system in $S_9 \backslash S_{9}(\infty)$, and $\Omega_U$ the limit set of $U$.
According to Theorem \ref{corlimitset}, $\Omega_U$ is a compact subset of $ S_{9}(\infty)\backslash I(\infty)$. If $\Omega_U$ intersects one of the two pole lines $L_6^{(3)}(\infty)$ or $L_9(\infty)$ at a point $p$, then there exists $z$ with $|z|$ arbitrarily large such that $U(z)$ is arbitrarily close to $p$, when the transversality of the vector field to the pole lines implies that $U(\zeta) \in (L_6^{(3)}\cup L_9)$ for a unique $\zeta$ near $z$, which means that $u(z)$ has a pole at $z = \zeta$.
As this would imply that $u(z)$ has infinitely many poles, it follows that $\Omega_U$ is a compact subset of $S_9 (\infty) \backslash (I(\infty)\cup L_6^{(3)}(\infty) \cup L_9(\infty))$. However, $L_6^{(3)}(\infty) \cup L_9(\infty)$ is equal to the set of all points in $S_9(\infty)\backslash I(\infty)$ which project to the line $L_0(\infty)$ in the complex projective plane, and therefore $S_9 (\infty)\backslash (I(\infty)\cup L_6^{(3)}(\infty) \cup L_9(\infty))$ is the affine $(u, v)$ coordinate chart, of which $\Omega_U$ is a compact subset, which implies that $u(z)$ and $v(z)$ remain bounded for large $|z|$.
It follows from the boundedness of $u$ and $v$ that $u(z)$ and $v(z)$ are equal to holomorphic functions of $1/z$ in a neighbourhood of $z=\infty$, which in turn implies that there are complex numbers $u(\infty)$, $v(\infty)$ which are the limit points of $u(z)$ and $v(z)$ as $|z| \rightarrow \infty$. In other words, $\Omega_U = \{(u(\infty), v(\infty))\}$, a one point set. Because the limit set $\Omega_U$ is invariant under the autonomous Hamiltonian system and contains only one point, this point is an equilibrium point of the autonomous Hamiltonian system given by:
\begin{subequations}
\begin{align}
\dot{u}=& v- u^2-1/2 \\
\dot{v}=& 2uv\, .
\end{align}
\end{subequations}
That is, $(u(\infty),v(\infty))\in \{(0,1/2),(i/\sqrt{2},0),(-i/\sqrt{2},0)\}$. By assumption in the corollary, we assume $(u(\infty),v(\infty)) \neq (0,1/2)$, and so $u(\infty)$ must equal one of $\pm i/\sqrt{2}$. But according to lemma \ref{invariance}, $(a,b)\in \Omega_U \iff (-a,b)\in\Omega_U$. This contradicts the deduction that $\Omega_U$ is a one point set.
\end{proof}
\end{corollary}
\section{Special Solutions to $\Ptw$ in the Okamoto-Boutroux space}\label{specsols}
For generic values of $E$, Equation (\ref{energyfct}) defines an elliptic curve with four distinct branch points. A natural question to ask is what happens at degenerate values of the autonomous Hamiltonian energy function $E$. Define
\begin{equation}
\label{degH}
h(u,v,E):=v\left(\dfrac{v}{2}-\dfrac{1}{2}-u^2\right)-E
\end{equation}
For generic $E$ the roots of $h$ are distinct. However, the roots of $h$ are no longer distinct when its gradient vanishes. These \lq\lq singular\rq\rq\ points are given by
\begin{equation}
\label{dis}
\nabla h=(h_u, h_v) = \left( 2v(v^2-v-2E), u^4+u^2+\dfrac{1}{4}+2E\right)=0
\end{equation}
Simultaneous with $h=0$, these equations imply that $E=0$ or $E=-1/8$. In the case of $E=0$, Equation (\ref{energyfct}) gives
\begin{subequations}
\begin{align}
(u,v) &=(u,0) \label{ecase1}\\
\noalign{or}
(u,v) &=(u,2u^2+1 )\label{ecase2}
\end{align}
\end{subequations}
In the case (\ref{ecase1}), then
\begin{align*}
\dot{E}&=\dfrac{-4E}{3z}+\dfrac{4\alpha v-(2\alpha+1)(2u^2+1)}{6z} \\
&=0+\dfrac{(2\alpha+1)(2u^2+1)}{6z}
\end{align*}
and so if $\alpha=-1/2$ then $\dot{E}=0$. An inductive argument on the $n$-th derivative of $E$ gives the following result:
\begin{proposition}
Under the conditions $v=0$ and $\alpha=-1/2$, $E^{(n)}=0$.
\end{proposition}
These conditions are exactly those needed to specify the seed solution from which the family of Airy type solutions are generated. The Airy family of solutions for (\ref{painleve2}) is given as the logarithmic derivative of the general solution to Airy's differential equation in the form
\begin{equation}
\label{airyeqn}
\dfrac{d^2y}{dx^2}+\dfrac{x}{2}y=0.
\end{equation}
Under Boutroux's change of variables, the governing Airy equation becomes
\begin{equation}
\label{airyeqnboutroux}
\dfrac{d^2y}{dz^2}+\dfrac{dy}{dz}\dfrac{1}{3z}+\dfrac{y}{2}=0,
\end{equation}
where $u(z)$, the solution to the Boutroux form of $\Ptw$ is related to $y(z)$ by $u(z)=\dot{y}/y$. The formal solution to this equation near a regular point $z_0$ is given by
\begin{align*}
y(z) &=\sum_{k=0}^{\infty} a_k (z-z_0)^k.
\end{align*}
The first two coefficients $a_0$ and $a_1$ are free, whilst the remaining $a_k$ satisfy
\begin{align}
\nonumber &a_2 =\dfrac{a_0}{4}+\dfrac{a_1}{6z_0},\\
\label{rec} &3(k+2)(k+3)a_{k+3}+(k+2)(3k+4)a_{k+2}+\dfrac{3z_0}{2}a_{k+1}+\dfrac{3}{2}a_k=0, \,\, k\ge0 \, .
\end{align}
A zero of the Airy function at $z=z_0$ corresponds to a pole of $u(z)$, with Laurent expansion given by
\begin{align}
\nonumber
u(z) = &\frac{1}{z-z_0}+\frac{1}{6 z_0}+\frac{\left(-18 z_0^2-19\right) (z-z_0)}{108
z_0^2}+\frac{\left(9 z_0^2+37\right) (z-z_0)^2}{216 z_0^3}\\
\label{lau_airy} &\qquad+\frac{\left(-324
z_0^4-2520 z_0^2-9565\right) (z-z_0)^3}{58320 z_0^4}+{\mathcal O}\left((z-z_0)^4\right).
\end{align}
In this case we have the solution crossing the pole line defined by $u_{92}=0$. In fact, there is a one-to-one correspondence between the Airy type solutions and the solutions which have a pole at $(u_{92},v_{92})=(0,0)$, when $\alpha=-1/2$. That is to say the Airy solutions are the unique solutions which cross the pole line $u_{92}=0$ at $v_{92}=0$ for $\alpha=-1/2$.
The second zero (\ref{ecase2}) of the Hamiltonian energy is also interesting. In this case we find the condition $v=2u^2+1$ is consistent with the equations (\ref{boutrouxsys})-(\ref{boutrouxsys2}) if and only if $\alpha=1/2$. By considering the $z$-derivative of $E$, we find
\begin{align}
\dot{E}&=\dfrac{(2\alpha -1)(2u^2+1)}{6z}.
\end{align}
We have here two new cases for a zero derivative. For $\alpha=1/2$, we have Airy solutions as before, related by the B\"acklund transformation (\ref{bt2}) which shifts $\alpha$ to $\alpha+1$. However, in the case of $u(z)\sim\pm i/\sqrt{2}+O(z^{-1})$, we have a different type of solution. This is the leading order behaviour of the tronqu\'ee type solutions to $\Ptw$.
The other singular value of the autonomous energy $E$ which is of interest is when $E=-1/8$. In this case two of the four distinct branch points of the elliptic curve coalesce and the curve is degenerate. The double point occurs at $u=0$, $v=1/2$. This is the leading order behaviour of the rational solutions to the system (\ref{boutrouxsys})-(\ref{boutrouxsys2}). As seen in Corollary \ref{infpoles}, besides this degeneracy, all other solutions have infinitely many poles. We see that the special solutions of the nonautonomous equation correspond to degenerations in the autonomous flow of the system near infinity.
\subsection{Blowing-up and coordinate systems}\label{coords}
In Section \ref{setup}, a coordinate change was performed to bring the Hamiltonian system for $\Ptw$ to the form (\ref{hamiltonian1}). We could have continued with the system of equations given by Hamilton's equations of motion for the system (\ref{preham}). However, there are several reasons we preferred the system (\ref{hamiltonian1}) over the former. Firstly, it allows us to study both $\Ptw$ and $\Pth$ in tandem. Secondly, it allows the reader to tie in the results with that of Okamoto \cite{oka:79}, who also used this coordinate choice. A third consideration is one which seems to be overlooked in the literature, which is that the number of blowups and the location of the base-points depends on the choice of coordinate system. If one were to study the system defined by (\ref{preham}), we would find the following sequence of base points:
\begin{align}
\nonumber (u_{02},v_{02})&= (u/v,1/v)\xleftarrow{(0,0)} (u_{12},v_{12})\xleftarrow{(0,0)} (u_{21},v_{21}) \\
\label{seq}
&\xleftarrow{(1,0)} (u_{31},v_{31}) \xleftarrow{(0,0)} (u_{41},v_{41}) \xleftarrow{(-1/2,0)} (u_{51},v_{51}) \xleftarrow{(\frac{1-2\alpha}{3z},0)} (u_{61},v_{61}),\\
\nonumber &(u_{21},v_{21})
\xleftarrow{(-1,0)} (u_{71},v_{71}) \xleftarrow{(0,0)} (u_{81},v_{81}) \xleftarrow{(1/2,0)} (u_{91},v_{91}) \xleftarrow{(\frac{1+2\alpha}{3z},0)} (u_{101},v_{101}).
\end{align}
where the label on each arrow represents the centre of the blow up in the preceding coordinate chart. From this one would be naively led to believe that 10 blow ups is the number required to resolve the singularities of this system. We can however bring this number back down to 9 if we consider a blow down of a line as a 'negative' blow-up. In this case, we can blow down the proper transform of the line $u_{01}=0$, a curve which can be considered the blow up of a regular point (having been a curve with self-intersection $-1$). Hence we see that the minimal blow up required is indeed canonical (in this case requiring 9 blow ups). Thus we observe that a 'good' coordinate choice in the beginning makes the calculation simpler and clearer.
\begin{remark}
The process of resolution presented explicitly in the appendix shows that the structure of the blow up (the number and location of the base points and the coordinate charts) of the Boutroux form of the Painlev\'e system is asymptotically close to that of the autonomous limit system, i.e., the system whereby the powers of $1/z$ in (\ref{boutrouxsys})-(\ref{boutrouxsys2}) are replaced with zero, which is solved by elliptic functions. Indeed, the difference between the two systems is only visible in the last two branches of the blow up, at the pole lines.
This is noteworthy because the operations of blowing up and taking limits do not commute in general.
\end{remark}
\section{Proof of Theorem \ref{theorepel}}\label{proof}
In this section we show that the infinity set $I$ is a repeller. This implies that every solution which starts in Okamoto's space of initial conditions remains there for all complex times $z$. The proof of this result requires the construction of a distance measure $d$, used to measure the distance to the infinity set.
Let $\mathcal{S}$ denote the fiber bundle of the surfaces $S_9 = S_9(z)$, $z \in\mathbb{C}\backslash 0$. If $\mathcal{I}$ denotes the union in $\mathcal{S}$ of all $I(z)$, $z \in\mathbb{C} \backslash 0$, then $\mathcal{S}$ $\backslash$ $\mathcal{I}$ is Okamoto's Òspace of initial conditionsÓ, fibered by the surfaces $S_9(z) \backslash I(z)$. We will analyse the asymptotic behaviour for $|z| \rightarrow \infty$ of the solutions of the Painlev\'e system (\ref{boutrouxsys})-(\ref{boutrouxsys2}), by studying the $z$-dependent vector field in the coordinate systems introduced in Section \ref{setup}.
Near the part of the infinity set given by $L_0^{(9)} \cup L_1^{(8)}\cup L_2^{(7)} \cup L_3^{(6)}\cup L_4^{(5)} \cup L_7^{(2)}$ we use the function $1/E$ to measure the distance to the infinity set. Because the lines $L_5^{(4)}$ and $L_8^{(1)}$ contain the lift of base points of $E$, the function $1/E$ is no longer a good measure of distance to the infinity set. These require an alternative measure of distance. Near $L_5^{(4)}$ we use $w_{62}$ and near $L_8^{(1)}$ we use $w_{91}$, where $w_{ij}$ is the Jacobian of the coordinate change from $(u,v)$ to $(u_{ij},v_{ij})$.
\begin{lemma}
The reciprocal of the autonomous Hamiltonian energy function E, and the Jacobians $w_{62}$ and $w_{91}$ are zero on the infinity set.
\end{lemma}
\begin{proof}
The calculations of $1/E$, $w_{62}$ and $w_{91}$ in each coordinate chart of Okamoto's space are provided in Appendix \ref{app1}. For each $0\le i \le 8$, the lines $L_i^{(9-i)}$ in the infinity set are determined by the relation $v_{i1}=0$, with $u_{i1}\in \mathbb{C}$. These show that in the limit as we approach an exceptional line $L_i^{(9-i)}$, we have
\begin{equation}
\label{inf}
\left.
\begin{array}{ll}
1/E &={\mathcal O}\bigl( v_{i1}^{\lambda_i}\bigr)\\
w_{62} &={\mathcal O}\bigl(v_{i1}^{\mu_i}\bigr) \\
w_{91} &={\mathcal O}\bigl( v_{i1}^{\nu_i}\bigr)
\end{array}
\right\} \quad {\rm as}\, v_{i1} \rightarrow 0 \,
\end{equation}
for some some positive integers $\lambda_i, \mu_i, \nu_i$. It follows that all three functions vanish on each $L_i^{(9-i)}\in I$.
\end{proof}
\begin{lemma}\label{lemma1}
Let $I_3:= \cup_{i=0}^3 I_i^{(9-i)}$. For every $\epsilon > 0$, there exists a neighbourhood $U$ of $I^3$ in S such that $|\dot{E}/(E)+4/(3z)|<\epsilon$ in $U$ and for all $z\in\mathbb{C}$\textbackslash $\{0\}$. For every compact subset K of $L_4^{(5)}\backslash L_5^{(4)} \cup L_7^{(2)}\backslash L_8^{(1)}$ there exists a neighbourhood V of K in $S_9$ and a constant $C>0$ such that $|3z\dot{E}/(4E)|$ $\leq C$ in V and for all $z\in \mathbb{C}\backslash \{ 0\}$.
\end{lemma}
\begin{proof}
Because $I_3$ is compact, it suffices to show that every point of $I_3$ has a neighbourhood in which the estimate holds. On $I_3$, the function $r:=\dot{E}/(E)+4/(3z)$ is equal to the following in each coordinate chart:
\begin{align}
\label{firstr}
r_{01} = & \dfrac{u_{01} \left(2 \alpha u_{01}^{2}-4 \alpha u_{01} v_{01}+4 \alpha+u_{01}^{2}+2\right)
}{3 v_{01} z \left(u_{01}^{2}-u_{01} v_{01}+2\right)} \\
r_{02} = & \dfrac{v_{02} \left(4 \alpha u_{02}^{2}+2 \alpha v_{02}^{2}-4 \alpha v_{02}+2 u_{02}^{2}+
v_{02}^{2}\right)}
{3 z \left(2 u_{02}^{2}+v_{02}^{2}-v_{02}\right) } \\
r_{11} =& \dfrac{v_{11} \left(4 \alpha u_{11}^{2} v_{11}+2 \alpha v_{11}-4 \alpha+2 u_{11}^{2} v_{11}+
v_{11}\right)}
{3 z \left(2 u_{11}^{2} v_{11}+v_{11}-1\right)} \\
r_{12} =&\dfrac{u_{12} v_{12} \left(2 \alpha u_{12} v_{12}^{2}+4 \alpha u_{12}-4 \alpha v_{12}+u_{12} v_{12}
^{2}+2 u_{12}\right)}{3 z \left(u_{12} v_{12}^{2}+2 u_{12}-v_{12}\right)} \end{align}
\begin{align}
r_{21}=&\dfrac{u_{21} v_{21}^{2} \left(2 \alpha u_{21} v_{21}^{2}+4 \alpha u_{21}-4 \alpha+u_{21} v_{21}
^{2}+2 u_{21}\right)}{3 z \left(u_{21} v_{21}^{2}+2 u_{21}-1\right)
}\,\\
r_{22}=&\dfrac{u_{22}^{2} v_{22} \left(2 \alpha u_{22}^{2} v_{22}^{2}-4 \alpha v_{22}+4 \alpha+u_{22}
^{2} v_{22}^{2}+2\right)}{3 z \left(u_{22}^{2} v_{22}^{2}-v_{22}+2
\right)}\,\\
r_{31}=&\left({6 z \left(2 u_{31} v_{31}^{2}+4 u_{31}+v_{31}\right)}\right)^{-1} \times \\ \nonumber
& v_{31} \big(8 \alpha u_{31}^{2} v_{31}^{4}+16 \alpha u_{31}^{2} v_{31}^{2}+8 \alpha
u_{31} v_{31}^{3} +2 \alpha v_{31}^{2}-4 \alpha+4 u_{31}^{2} v_{31}^{4}\\ \nonumber
&\hspace{1in}+8 u_{31}^{2} v_{31}^{2}+4 u_{31} v_{31}^{3}+8 u_{31} v_{31}+v_{31}^{2}+2\big) \\
r_{32}=&u_{32} v_{32}^{2} \left({6 z \left(2 u_{32}^{2} v_{32}^{2}+u_{32} v_{32}^{2}+4\right)}\right)^{-1} \times \\ \nonumber
& \big(8 \alpha u_{32}^{4} v_{32}^{2}+8 \alpha u_{32}^{3} v_{32}^{2}+
2 \alpha u_{32}^{2} v_{32}^{2}+16 \alpha u_{32}^{2}-4 \alpha+4 u_{32}^{4} v_{32}^{2}+4 u_{32}
^{3} v_{32}^{2}\\ \nonumber
&\hspace{1in}+u_{32}^{2} v_{32}^{2}+8 u_{32}^{2}+8 u_{32}+2\big)
\\
r_{42}=& v_{42}\left(6 z \left(2 u_{42}^{2} v_{42}^{2}+v_{42}+4\right)\right)^{-1} \times \\ \nonumber
&\big(8 \alpha u_{42}^{6} v_{42}^{4}+8 \alpha u_{42}^{4} v_{42}^{3}+16 \alpha
u_{42}^{4} v_{42}^{2}+2 \alpha u_{42}^{2} v_{42}^{2}-4 \alpha+4 u_{42}^{6} v_{42}^{4}+4 u_{42}
^{4} v_{42}^{3} \\ \nonumber
& \hspace{1in}+8 u_{42}^{4} v_{42}^{2}+u_{42}^{2} v_{42}^{2}+8 u_{42}^{2} v_{42}+2\big)
\\
r_{71}=&\dfrac{u_{71} \left(2 \alpha u_{71}^{2} v_{71}^{2}-4 \alpha u_{71} v_{71}^{2}+4 \alpha+u_{71}
^{2} v_{71}^{2}+2\right)}{3 z \left(u_{71}^{2} v_{71}^{2}-u_{71} v_{71}^{2}
+2\right)}\, .\label{lastr}
\end{align}
The part $L_0^{(9)}\backslash L_1^{(9)}$ is equal to the line $v_{02}=0$, on which $r_{02}=0$. The part $L_1^{(8)}\backslash L_2^{(7)}$ is equal to the line $u_{12}=0$, on which $r_{22}=0$. The part $L_3^{(6)}\backslash L_4^{(5)}$ is equal to the line $u_{32}=0$, on which $r_{32}=0$. The part $L_3^{(6)}\backslash L_2^{(7)}$ is equal to the line $v_{42}=0$ on which $r_{42}=0$. This covers the entire $I_3$ and so the proof for the first part of the lemma is complete.
For the second statement we have the line $L_4^{(5)}\backslash (L_3^{(6)}\cup L_5^{(4)})$ is equal to the line $v_{41}=0$, $u_{41}\neq 1/4$ on which $r_{41}=(1-2\alpha)(3z(4u_{41}+1))$. Therefore for every compact subset $K_1$ of $L_4^{(5)}\backslash L_5^{(4)}$ there exists a neighbourhood $V_1$ such that $r_{41}$ is bounded. Similarly we have the line $L_7^{(2)}\backslash (L_0^{(9)}\cup L_8^{(1)})$ is equal to the line $u_{72}=0$, $v_{72}\neq 0$, on which $r_{72}=(1+2\alpha)/(3v_{72}z)$. Therefore on every compact subset $K_2$ of $L_7^{(2)}\backslash L_8^{(1)}$ there exists a neighbourhood $V_2$ such that $r_{41}$ is bounded. We take $K=K_1 \cup K_2$ and $V=V_1\cup V_2$, which completes the proof of the second statement of the lemma.
\end{proof}
In the following two lemmas we require asymptotic information about the dynamics near the infinity set. From the appendix, we have that the line $L_{5}^{(4)}\backslash L_{4}^{(5)}$ is determined by the relations $v_{62}=0$, $u_{62} \in \mathbb{C}$, while the line $L_{8}^{(1)}\backslash L_{7}^{(2)}$ is determined by the relations $u_{91}=0$, $v_{91} \in \mathbb{C}$. Similarly the lines $L_{4}^{(5)}\backslash L_{3}^{(6)}$ and $L_{7}^{(2)}\backslash L_{0}^{(9)}$ are determined by $v_{52}=0$, $u_{52} \in \mathbb{C}$ and $u_{81}=0$, $v_{81} \in \mathbb{C}$ respectively. Table \ref{asymtable} shows the leading order behaviour of the solutions and the functions used as a measure of distance to the infinity set near these lines.
\begin{center}
\begin{table}
\begin{tabular}{|l|l|}
\hline
&\\
Near $L_5^{(4)}$ as $v_{62}\rightarrow 0\,$: & Near $L_8^{(1)}$ as $u_{91}\rightarrow 0\,$: \\
$\dot{u}_{62} \sim -v_{62}^{-1} $ & $\dot{v}_{91} \sim u_{91}^{-1} $ \\
$w_{62} \sim -v_{62}/4 $ & $w_{91} \sim -u_{91} $ \\
$\dfrac{\dot{w}_{62}}{w_{62}}\sim \dfrac{4}{3z}+O(w_{62}) $ & $\dfrac{\dot{w}_{91}}{w_{91}}\sim \dfrac{4}{3z}+O(w_{91}) $ \\
$Ew_{62} \sim 1 - \dfrac{2\alpha-1}{12u_{62}z}\, .$ & $Ew_{91}\sim 1 - \dfrac{2\alpha+1}{3v_{91}z}\, .$\\
&\\
\hline
&\\
Near $L_4^{(5)}$ as $v_{52}\rightarrow 0\,$: & Near $L_7^{(2)}$: as $u_{81}\rightarrow 0\,$:\\
$\dot{u}_{52} \sim -2v_{52}^{-1} $ & $\dot{v}_{81} \sim 2u_{81}^{-1} $ \\
$\dot{v}_{52} \sim u_{52}^{-1} $ & $\dot{u}_{81} \sim -u_{v_{1}}^{-1} $ \\
$w_{62} \sim -u_{52}v_{52}^2/4 $ & $w_{91} \sim -v_{81}u_{81}^2 $ \\
$Ew_{62} \sim 1 $ & $Ew_{91}\sim 1 $\\
$\dfrac{\dot{E}}{E}\sim -\dfrac{4}{3z}+\dfrac{1-2\alpha}{12u_{52}z}\sim -\dfrac{4}{3z}+\dfrac{1-2\alpha}{12z}\dot{v}_{52}\, .$ & $\dfrac{\dot{E}}{E}\sim -\dfrac{4}{3z}+\dfrac{1+2\alpha}{3v_{81}z}\sim -\dfrac{4}{3z}+\dfrac{1+2\alpha}{12z}\dot{u}_{81}\, .$ \\
&\\
\hline
\end{tabular}
\caption{Comparison of asymptotic behaviour near the infinity set}
\label{asymtable}
\end{table}
\end{center}
\begin{remark}
Note that the behaviour of the Boutroux-Painlev\'e system near the pairs $L_5^{(4)}$ and $L_8^{(1)}$, $L_4^{(5)}$ and $L_7^{(2)}$ are equivalent to leading order, up to multiplicative constants. Due to this observation, the proofs of the following lemmas will only consider one of the two in each pair, while the proof near the other can be inferred from the similar behaviour of its pair.
\end{remark}
\begin{remark}
The solution cannot be simultaneously close to both $L_6^{(3)}$ and $L_9$. This can be seen by the coordinate relation
\begin{displaymath}
v_{91} \propto v_{61}^{-3}\,\, \mbox{as $v_{61} \rightarrow 0$.}
\end{displaymath}
That is, as a solution moves towards one of the pole lines (given by $v_{61}=0$ or $v_{91}=0$), the solution becomes unboundedly distant from the other pole line.
\end{remark}
In the following lemma we show that the three functions $1/E$, $w_{62}$ and $w_{91}$ can be stitched together to form a continuous distance function, which will be later used to show that the infinity set is repelling.
\begin{lemma} \label{lemma2}
Suppose $z$ is bounded away from zero. There exists a continuous complex valued function $d$ on a neighbourhood of $I$ in $S_9$ such that $d=1/E$ in a neighbourhood of $I \backslash (L_{5}^{(4)} \bigcup L_8^{(1)})$ in $S_9$, $d=w_{62}$ in a neighbourhood of $L_{5}^{(4)}\backslash L_{4}^{(5)}$ in $S_9$, and $d=w_{91}$ in a neighbourhood of $L_{8}^{(1)}\backslash L_{7}^{(2)}$ in $S_9$. Moreover, $E\,d\rightarrow 1$, $d/w_{52}\rightarrow 1$ when approaching $L_{5}^{(4)}\backslash L_{4}^{(5)}$, and $E\,d\rightarrow 1$, $d/w_{91}\rightarrow 1$ when approaching $L_{8}^{(1)}\backslash L_{7}^{(2)}$.
If the solution at the complex time $z$ is sufficiently close to a point of $L_{5}^{(5)}\backslash L_{4}^{(6)}$ (resp. $L_{8}^{(1)}\backslash L_{7}^{(2)}$), then there exists a unique $\zeta\in \mathbb{C}$, such that $|z-\zeta|={\mathcal O}(|d(z)||u_{62}(z)|)$ (resp. ${\mathcal O}(|d(z)||v_{91}(z)|))$, where d(z) is small, $u_{62}(z)$ (resp. $v_{91}(z)$) is bounded and $u_{62}(\zeta)=0$ (resp. $v_{91}(\zeta)=0$). That is, the point $z=\zeta$ is a pole of the Boutroux-Painlev\'e system. For large finite $R_5$ $\in \mathbb{R}_{>0}$, the connected component of $\zeta$ in $\mathbb{C}$ of the set of all $z\in \mathbb{C}$ such that $|u_{62}(z)|\leq R_5$ is an approximate disc $D_5$ with centre at $\zeta$ and radius $\sim |\delta | R_5$, and $z\mapsto u_{62}(z)$ is a complex analytic diffeomorphism from $D_5$ onto $\{u_{62}\in \mathbb{C} \bigm | |u_{62}|\leq R_5\}$.
Depending on which pole line the solution is close to, we write $\delta:=d(\zeta)=w_{62}(\zeta)$ or $\delta:=d(\zeta)=w_{91}(\zeta)$. Then we have $d(z)/\delta \sim 1$ as $\delta \rightarrow 0$.
\end{lemma}
\begin{proof}
Without loss of generality, we assume the solution is near the part of the infinity set with which the poles of negative residue are associated. From the explicit details presented in the appendix it follows that $L_{5}^{(5)}\backslash L_{4}^{(6)}$ is determined by the equations $v_{62}=0$, $u_{62} \in \mathbb{C}$. Asymptotically for $v_{62}\rightarrow 0$, and bounded $u_{62}$ and $z^{-1}$, we have
\begin{subequations}
\begin{eqnarray}
\dot{u}_{62} & \sim & -v_{62}^{-1} \\
w_{62} & \sim & -v_{62}/4 \\
\dot{w}_{62}/w_{62} &=&\frac{4}{3z} + {\mathcal O}(v_{62})=\frac{4}{3z}+{\mathcal O}(w_{62}) \label{w62eqn} \\
Ew_{62} &\sim & 1+ (1-2\alpha)/(12u_{62}z)\label{Ew62eqn}
\end{eqnarray}
\end{subequations}
Provided the solution to the Boutroux-Painlev\'e system is close to $L_{5}^{(5)}\backslash L_{4}^{(6)}$ then \eqref{w62eqn} gives
\begin{displaymath}
w_{62}=\left(\frac{z}{\zeta}\right)^{4/3}w_{62}(\zeta)(1+{o}(1))\,.
\end{displaymath}
If $|z-\zeta|\ll |\zeta|$ (iff $z/\zeta\sim 1$) then we have $w_{62}(z)\sim w_{62}(\zeta)$ and so $v_{62}$ is approximately constant ($v_{62}\sim -4w_{62}(\zeta)$) and so
\begin{displaymath}
u_{62}(z)\sim u_{62}(\zeta)-v_{62}^{-1}(\zeta)(z-\zeta).
\end{displaymath}
So $u_{62}$ fills an approximate disc, centred at $u_{62}(\zeta)$ with radius $\sim R$ if $z$ runs over an approximate disc of radius $|v_{62}(\zeta)| R$. If $|v_{62}(\zeta)| \ll 1/|\zeta|$, then the solution at $z$ in an approximate disc $D$, centred at $\zeta$ with radius $\sim |v_{62}|R$ has the properties that $v_{62}(z)/v_{62}(\zeta) \sim 1$ and $z\rightarrow u_{62}(z)$ is a complex analytic diffeomorphism from $D$ onto an approximate disc centred at $u_{62}(\zeta)$ with radius $\sim R$. If $R$ is chosen to be sufficiently large, we have $0\in u_{62}(D)$, that is, the solution to the Boutroux-Painlev\'e system has a pole at a unique point in $D$ (as $u_{62}=0$ corresponds to a pole with residue $-1$). We may shift the centre of the approximate disc so that $u_{62}(\zeta)=0$, that is, shift the disc to be centred at the pole point.
Provided $|z-\zeta |\ll |\zeta|$, we have $d(z)/d(\zeta)=w_{62}(z)/w_{62}(\zeta) \sim 1$, that is, $w_{62}(z)/\delta \sim1$ and so $u_{62}(z) \sim 2^{-2} \delta^{-1}(z-\zeta)$. Then for given $R_5\in\mathbb{R}_{>0}$, the equation $|u_{62}(z)|=R_5$ corresponds to $|z-\zeta|\sim 2^2|\delta |R_5$, which is still small compared to $|\zeta |$ if $|\delta |$ is sufficiently small. It follows the connected component of $D_5$ of the set of all $z\in\mathbb{C}$ such that $|u_{62}(z)|\leq R_5$ is an approximate disc with centre at $\zeta$ and radius $|\delta |R_5$, more precisely, $z\mapsto u_{62}(z)$ is a complex analytic diffeomorphism from $D_5$ onto $\{ u_{62}\in\mathbb{C} \big\| |u_{62}|\leq R_5 \}$, and that $d(z)/\delta\sim 1$ for all $z\in D_5$.
$E$ has a pole at $z=\zeta$, but it follows from the relation \eqref{Ew62eqn} that $Ew_{62}\sim 1$ when $1\gg |z^{-1}u_{62}^{-1}|\sim |\zeta^{-1}\delta(z-\zeta)^{-1}|$, that is, when $|z-\zeta|\gg |\delta|/|\zeta|$. As the approximate radius of $D_5$ is $|\delta|R_5\gg|\delta|/|\zeta|$ as $R_5\gg1/|\zeta|$, we have $Ew_{62}\sim 1$ for $z\in D_5\backslash D_6$, where $D_6$ is a disc centred at $\zeta$ with small radius compared to the radius of $D_5$.
The set $L_{4}^{(6)}\backslash L_5^{(5)}$ is visible in the coordinate system $(u_{52},v_{52})$, where it corresponds to the equation $v_{52}=0$ and is parametrised by $u_{52}\in\mathbb{C}$. The set $L_5^{(5)}$ minus one point corresponds to $u_{52}=0$ and is parametrised by $v_{52}\in\mathbb{C}$. The equations that express $(u_{51},v_{51})$ and $(u_{52},v_{52})$ in terms of $(u_{41},v_{41})$ show
\begin{subequations}
\begin{eqnarray}
u_{52} & = & u_{51}v_{51} \\
v_{52} & = & u_{51}^{-1} \label{coord52} \\
u_{51}&=&u_{62}+(1-2\alpha)/(12z),
\end{eqnarray}
\end{subequations}
which implies that $u_{62}\rightarrow \infty$ if and only if $v_{52}\rightarrow 0$. That is, when the point near $L_5^{(5)}$ approaches the intersection point with $L_4^{(6)}$, then $Ew_{62}\rightarrow 1$.
As remarked earlier, analogous arguments to the above case can be made where the solution of the Boutroux-Painlev\'e system is close to $L_{8}^{(1)}\backslash L_{7}^{(2)}$. This completes the proof of the lemma.
\end{proof}
\begin{lemma}\label{lemma3}
For large finite $R_{4} \in \mathbb{R}_{>0}$, the connected component of $\zeta\in\mathbb{C}$ of the set of all $z\in\mathbb{C}$ such that the solution at the complex time z is close to $L_4^{(5)}\backslash L_{3}^{(6)}$, with $|u_{52}(z)|\leq R_4$, but not close to $L_{5}^{(4)}$, is the complement of $D_{5}$ in an approximate disk $D_4$ with centre at $\zeta$ and radius $\sim |\delta R_4|^{1/2}$. For all $z\in D_4$, the largest approximate disc, we have $|z-\zeta | \ll |\zeta|$ and $d(z)/\delta\sim 1$
The analogous statement holds true in the 7-th and 8-th coordinate charts.
\end{lemma}
\begin{proof}
Asymptotically as $v_{52}\rightarrow 0$, and for bounded $u_{52}$ and $z^{-1}$, we have
\begin{subequations}
\begin{eqnarray}
\dot{u}_{52} & \sim & -2v_{52}^{-1} \label{v52eqns1}\\
\dot{v}_{52} & \sim & u_{52}^{-1} \\
w_{52}&\sim& -u_{52}v_{52}^2 \\
Ew_{52}&\sim& 1 \\
\dot{E}/E &\sim& -4/3z +(1-2\alpha)/(3u_{52}z) \sim -4/3z +(1-2\alpha)/(3z)\dot{v}_{52}\, \label{v52eqns2}.
\end{eqnarray}
\end{subequations}
So
\begin{eqnarray}
\nonumber &\log(E(z_1)/E(z_0))\sim \log(z_1/z_0)^{-4/3}+(1-2\alpha)/3(v_{52}(z_1)/z_1-v_{52}(z_0)/z_0)\\
&\quad +\int_{z_0}^{z_1} z^{-2}v_{52}(z) dz ,
\end{eqnarray}
and hence
\begin{displaymath}
E(z_1)/E(z_0)\sim (z_1/z_0)^{-4/3}(1+o(1))
\end{displaymath}
It follows that $E(z_1)/E(z_0)\sim 1$ if for all $z$ on the segment from $z_0$ to $z_1$ we have $|z/z_0|\sim 1$, that is, $|z-z_0|\ll |z_0|$ and $|v_{52}|\ll |z_0|$.
We chose $z_0$ on the boundary of $D_5$, when $\delta/d(z_0)\sim E(z_0)\delta\sim E(z_0)w_{62}(z_0)\sim 1$, and
\begin{eqnarray}
|u_{62}(z_0)|=R_5& \implies & |\frac{2\alpha-1}{3z}+\frac{1}{v_{52}}|=R_5 \\
& \implies & |v_{52}(z_0)| \sim R_5^{-1} \ll 1\,.
\end{eqnarray}
Furthermore
\begin{displaymath}
|u_{52}(z_0)|\sim |w_{52}v_{52}^{-2}|\sim |\delta| R_5^{-2}
\end{displaymath}
which is small when $\delta$ is sufficiently small. If we have $z=\zeta+r(z_0-\zeta)$, $r\geq1$ we have $|u_{62}(z)|\geq R_5 \gg 1$, and hence $|v_{52}|\ll1$ from \eqref{coord52}. We also have $|z-z_0|/|z_0|=(r-1)|1-\zeta/z_0|\ll1$ if $r-1$ is small.
Now (\ref{v52eqns1})-(\ref{v52eqns2}) and $E\sim \delta^{-1}$ give $v_{52}^{-1}\sim (-u_{52}\delta^{-1})^{1/2}$, and so
\begin{subequations}
\begin{eqnarray}
\frac{d(u_{52}^{1/2})}{dz} & \sim & -i\delta^{-1/2} \\
\implies u_{52}^{1/2}(z) & \sim & u_{52}^{1/2}(z_0) -i\delta^{-1/2}(z-z_0) \\
u_{52}(z)&\sim&-\delta^{-1}(z-z_0)^2
\end{eqnarray}
\end{subequations}
if $|\delta^{-1/2}||z-z_0|\gg u_{52}(z_0)^{1/2}$.
For large $R_4\in\mathbb{R}_{>0}$, the equation $|u_{52}(z)|=R_4$ corresponds to $|z-z_0|\sim |\delta R_4 |^{1/2}$, which is small compared to $|z_0|\sim |\zeta|$, and so
\begin{displaymath}
|z-\zeta|\leq|z-z_0|+|z_0-\zeta|\ll |\zeta|\, .
\end{displaymath}
\end{proof}
We are now in a position to prove Theorem \ref{theorepel}.
\begin{proof}[Proof of Theorem \ref{theorepel}]
Suppose a solution of the Boutroux-Painlev\'e system is near the set $I_3$ at times $z_0$ and $z_1$. It follows from Lemma \ref{lemma3} that we have for every solution close to $I$, the set of complex times $z$ such that the solution is not close to $I_3$ is the union of approximate discs of radius $\sim |d|^{1/2}$ contained within approximate discs of radius $\sim |d|^{1/3}$. It also follows from Lemma \ref{lemma3} that the annular region between these discs is at least of order $|d|^{1/3} - |d|^{1/2} \sim |d|^{1/3}$, as $|d|^{1/3} \gg |d|^{1/2}$.
Hence there if the solution is near $I$ for all complex times $z$ such that $|z_0|\leq |z|\leq |z_1|$, then there exists a path $\gamma$ from $z_0$ to $z_1$ such that the solution is close to $I_3$ \emph{for all} $z\in \gamma$, and $\gamma$ is $\mathbf{C}^1$ close to the path $\left[0,1\right]\ni t \mapsto z_1^t/z_0^{1-t}$.
Then Lemma \ref{lemma1} implies that
\begin{align}
\nonumber {\rm log}(E(z)/E(z_0))&=-\dfrac{4}{3}{\rm log}(z/z_0)\int_0^1 dt +o(1) \\
\Rightarrow\ E(z)&=E(z_0)(z/z_0)^{-4/3(1+o(1))}(1+o(1))
\end{align}
and so we have
\begin{align}\label{dzcor1}
d(z)&=d(z_0)(z/z_0)^{4/3(1+o(1))}(1+o(1))\,.
\end{align}
Then Lemma \ref{lemma2} implies that, as long as we are close to $I$, as the solution moves into the region where $d$ is given by one of the two Jacobians $w_{62}$ or $w_{92}$, the ratio of $d$ remains close to 1.
For the first statement of the theorem, we have
\begin{align*}
\delta \,> & \,d(z)\, \geq d(z_0)\left(\frac{z}{z_0}\right)^{4/3-\epsilon_2}(1-\epsilon_3)
\end{align*}
and so
\begin{align*}
\delta \, \geq & \sup_{z|\,|d(z)|<\delta} d(z_0)\left(\frac{z}{z_0}\right)^{4/3-\epsilon_2}(1-\epsilon_3) \,.
\end{align*}
The second statement follows directly from \eqref{dzcor1}, while the third follows by the assumption on $z$.
\end{proof}
\section{Conclusion}\label{conc}
In this paper we have constructed Okamoto's space of initial conditions for the Boutroux-Painlev\'e system describing the behaviour of the second and thirty-fourth Painlev\'e equations in the asymptotic limit where the independent variable goes to infinity. Not treated here, but of interest, is the case where the parameter $\alpha$ of $\Ptw$ goes to infinity, cf. \cite{joshi:99,kaw:05}. From our explicit construction, we are able to conclude that each respective limit set of solutions to these equations forms a compact, connected and non-empty set of the space $S_9(\infty) \backslash I(\infty)$, which is the elliptic surface corresponding to the space of initial conditions for the autonomous limit system. We also showed that all solutions except those which vanish uniformly at infinity necessarily have infinitely many poles. Moreover, the special solutions to $\Ptw$ were found at singular values of the energy function $E$, where the branch points of the elliptic curve coalesce.
\appendix
\section{Resolution of singularities for (\ref{boutrouxsys})-(\ref{boutrouxsys2})} \label{app1}
In this appendix, we construct Okamoto's space of initial conditions for the Boutroux-Painlev\'e system explicitly. Recall the notation from Section \ref{setup}, where the coordinates $(u_{ij},v_{ij})$ refer to the two coordinates in the $j$-th patch of the $i$-th blow up, $j=1,2$, $i=0,1,...,9$.
The resolution of the Boutroux-Painlev\'e system can be seen in Figure \ref{blowupfig}, and can be summarised by the following diagram, where we omit the coordinate charts which are free from base points:
\begin{align*}
(u_{02},v_{02})&= (u/v,1/v)\xleftarrow{(0,0)} (u_{12},v_{12})\xleftarrow{(0,0)} (u_{21},v_{21}) \\
&\xleftarrow{(1/2,0)} (u_{31},v_{31}) \xleftarrow{(0,0)} (u_{41},v_{41}) \xleftarrow{(-1/4,0)} (u_{51},v_{51}) \xleftarrow{(\frac{1-2\alpha}{12z},0)} (u_{61},v_{61}),\\
(u_{01},v_{01}) &= (1/u,v/u)
\xleftarrow{(0,0)} (u_{72},v_{72}) \xleftarrow{(0,0)} (u_{82},v_{82}) \xleftarrow{(0,\frac{1+2\alpha}{3z})} (u_{91},v_{91}).\\
\end{align*}
Here the label above each arrow represents the base point that is blown up in the preceding coordinate chart.
\begin{center}
\begin{figure}[h!]
\scalebox{.9}
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\usefont{T1}{ptm}{m}{n}
\rput(0.2324707,-0.705){$u$}
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\rput(0.7124707,-1.145){$v$}
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\rput(4.392471,-0.745){$1/u$}
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\rput(2.5324707,1.935){$1/v$}
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\rput(1.8724707,1.955){$u/v$}
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\rput(3.8324707,-1.165){$v/u$}
\usefont{T1}{ptm}{m}{n}
\rput(12.122471,0.775){$L_0^{(9)}$}
\usefont{T1}{ptm}{m}{n}
\rput(8.082471,2.935){$L_1^{(8)}$}
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\rput(10.202471,2.675){$L_2^{(7)}$}
\usefont{T1}{ptm}{m}{n}
\rput(11.22247,3.535){$L_3^{(6)}$}
\usefont{T1}{ptm}{m}{n}
\rput(12.862471,3.475){$L_4^{(5)}$}
\usefont{T1}{ptm}{m}{n}
\rput(13.942471,2.895){$L_5^{(4)}$}
\usefont{T1}{ptm}{m}{n}
\rput(15.082471,3.215){$L_6^{(3)}$}
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\rput(10.882471,-0.945){$L_7^{(2)}$}
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\rput(9.422471,-1.825){$L_8^{(1)}$}
\usefont{T1}{ptm}{m}{n}
\rput(10.952471,-2.845){$L_9$}
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\rput(6.7224708,0.535){$L:u=0$}
\usefont{T1}{ptm}{m}{n}
\rput(7.2824707,-2.445){$L:v=0$}
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\rput(4.771875,1.155){Blow up}
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}
\caption{The 9 point blow up of $\mathbb{P}^2(\mathbb{C})$ showing the configuration of the exceptional curves. The numbers represent the self intersection of the lines they are adjacent to. The configuration of the irreducible divisors (the infinity set) is that of the root lattice $E_7^{(1)}$ (see Figure \ref{E71}). The dashed lines indicating $L_6^{(3)}$ and $L_9$ are the poles lines, where the vector field is transversal to the line and a crossing indicates a pole of residue $\pm 1$ for $u$.}
\label{blowupfig}
\end{figure}
\end{center}
\begin{figure}
\begin{center}
\includegraphics{e71}
\end{center}
\caption{\label{E71}
The Dynkin diagram for $E_{7}^{(1)}$, the numbers $i$ indicate the line $L_i$ which gives rise to the node. The nodes $j$ and $k$ are connected when $L_j^{(9-j)}$ intersects $L_k^{(9-k)}$.}
\end{figure}
\begin{remark}
The following blow up calculations are provided in explicit detail for completeness. The essential information for proofs in the body of the paper can be found in equations (\ref{firstr})-(\ref{lastr}) and Table \ref{asymtable}.
\end{remark}
\subsection{Embedding in $\mathbb{P}^2$}
In the second affine chart:
\begin{eqnarray*}
u_{01} & = & u^{-1} \\
v_{01} & = & vu^{-1}\\
\dot{u}_{01} &=&\frac{3 (u_{01}^{2}+2) z-2 (3 v_{01} z-1) u_{01}}
{6 z}\\
\dot{v}_{01} &=& \frac{2 (2 \alpha u_{01}^{2}+9 v_{01} z-(3 v_{01} z+1) u_{01} v_{01}
)+(3 v_{01} z+2) u_{01}^{2}}{6 u_{01} z}\\
E_{01}&=&\frac{-((u_{01}-v_{01}) u_{01}+2) v_{01}}{2 u_{01}^{3
}}\\
w_{01}&=&-u_{01}^{3}
\end{eqnarray*}
The line at infinity $L_0$ is given by $u_{01}=0$. There is a base point in this chart $b_7$ given by $u_{01}=0$, $v_{01}=0$ which is a base point of both the Painlev\'e vector field and of the autonomous energy function $E$.
In the third affine chart:
\begin{align*}
u_{02} & = uv^{-1} \\
v_{02} & = v^{-1}\\
\dot{u}_{02} &=\frac{-(3 ((v_{02}-2) v_{02}+6 u_{02}^{2}) z+4 \alpha
u_{02} v_{02}^{2}+2 (v_{02}-1) u_{02} v_{02})}{6 v_{02} z
}\\
\dot{v}_{02} &=\frac{-((v_{02}-2) v_{02}+6 u_{02} z+2 \alpha v_{02}^{2})
}{3 z}\\
E_{02}&=\frac{-((v_{02}-1) v_{02}+2 u_{02}^{2})}{2 v_{02}
^{3}}\\
w_{02}&=-v_{02}^{3}
\end{align*}
The line at infinity $L_0$ is given by $v_{02}=0$. The vector field has a base point $b_0$ given by $u_{02}=0$, $v_{02}=0$, which is a base point of both the Painlev\'e vector field and of $E$.
\subsection{Resolution of $b_0$}
We blow up $\mathbb{P}^2$ at $b_0$. In the first chart:
\begin{eqnarray*}
u_{02} & = & u_{11} v_{11} \\
v_{02} & = & v_{11}\\
u&=&u_{11}\\
v&=&v_{11}^{-1}\\
\dot{u}_{11}&=&\frac{-(2 (3 u_{11} z+1) u_{11} v_{11}+3 (v_{11}-2) z
)}{6 v_{11} z}\\
\dot{v}_{11}&=&\frac{-(v_{11}-2+6 u_{11} z+2 \alpha v_{11}) v_{11}}{3 z} \\
w_{11}&=&-v_{11}^2\\
E_{11}&=&\frac{-(v_{11}-1+2 u_{11}^{2} v_{11})}{2 v_{11}^{2}}
\end{eqnarray*}
$v_{11}=0$ defines $L_1$ and the line $L_0^{(1)}$ is not visible in this chart. There are no base points in this chart.
In the second chart:
\begin{eqnarray*}
u_{02} & = & u_{12} \\
v_{02} & = & u_{12}v_{12} \\
u&=& v_{12}^{-1}\\
v&=& u_{12}^{-1} v_{12}^{-1}\\
\dot{u}_{12}&=&\frac{-(2 (u_{12}^{2} v_{12}-3 z+2 \alpha u_{12}^{2} v_{12}) v_{12}+
(3 v_{12}^{2} z-2 v_{12}+18 z) u_{12})}{6 v_{12} z}\\
\dot{v}_{12}&=&\frac{(3 v_{12}^{2} z+2 v_{12}+6 z) u_{12}-6 v_{12} z}{6 u_{12}
z}\\
w_{12}&=&-u_{12}^{2} v_{12}^{3}\\
E_{12}&=&\frac{-u_{12} v_{12}^{2}-2 u_{12}+v_{12}}{2 u_{12}^{2} v_{12}^{3}}
\end{eqnarray*}
$u_{12}=0$ defines $L_1$ in this chart, and $v_{12}=0$ defines the lift of $L_0$=$L_0^{(1)}$. There is a base point $b_1$ of the vector field and of $E$ at $u_{12}=0$, $v_{12}=0$.
\subsection{Resolution of $b_1$}
In the first chart:
\begin{eqnarray*}
u_{12} & = & u_{21}v_{21} \\
v_{12} & = & v_{21} \\
u&=&v_{21}^{-1}\\
v&=&u_{21}^{-1} v_{21}^{-2}\\
\dot{u}_{21}&=&\frac{-(u_{21}^{2} v_{21}^{3}-6 z+2 \alpha u_{21}^{2} v_{21}^{3}+3 (v_{21}^{2}
+4) u_{21} z)}{3 v_{21} z}\\
\dot{v}_{21}&=&\frac{(3 v_{21}^{2} z+2 v_{21}+6 z) u_{21}-6 z}{6 u_{21} z}\\
w_{21}&=&-u_{21}^{2} v_{21}^{4}\\
E_{21}&=&\frac{-(v_{21}^{2}+2) u_{21}+1}{2 u_{21}^{2} v_{21}^{4}}
\end{eqnarray*}
Here, $v_{21}=0$ defines $L_2$ and $u_{21}=0$ defines the proper transform $L_1^{(1)}$ of $L_1$. The proper transform $L_0^{(2)}$ of $L_0^{(1)}$ is not visible in this chart. There is a base point $b_2$ of both the vector field and the function $E$ given by $u_{21}=1/2$, $v_{21}=0$.
\begin{eqnarray*}
u_{12} & = & u_{22} \\
v_{12} & = & u_{22}v_{22} \\
u&=&u_{22}^{-1} v_{22}^{-1} \\
v&=&u_{22}^{-2} v_{22}^{-1}\\
\dot{u}_{22}&=&\frac{-(2 u_{22}^{2} v_{22}+3 u_{22} v_{22} z-2) u_{22} v_{22}+6 (
v_{22}-3) z-4 \alpha u_{22}^{3} v_{22}^{2})}{6 v_{22} z}\\
\dot{v}_{22}&=&\frac{(u_{22}+3 z) u_{22}^{2} v_{22}^{2}-6 (v_{22}-2) z+2
\alpha u_{22}^{3} v_{22}^{2}}{3 u_{22} z}\\
w_{22}&=&-u_{22}^{4} v_{22}^{3}\\
E_{22}&=&\frac{-(u_{22}^{2} v_{22}-1) v_{22}-2}{2 u_{22}^{4} v_{22}^{3}}
\end{eqnarray*}
Here, $u_{22}=0$ defines $L_2$ and $v_{22}=0$ defines the proper transform $L_0^{(2)}$ of $L_0^{(1)}$. The proper transform $L_1^{(1)}$ of $L_1$ is not visible in this chart. There is a base point $b_2$ of both the vector field and the function $E$ given by $u_{22}=0$, $v_{22}=2$.
\subsection{Resolution of $b_2$}
In the first chart we have
\begin{align*}
u_{21}=& u_{31} v_{31}+1/2 \\
v_{21}=&v_{31}\\
u=&v_{31}^{-1}\\
v=&2/(2 u_{31} v_{31}^{3}+v_{31}^{2})\\
\dot{u}_{31}=&-(12 (2 u_{31} v_{31}+1) v_{31} z)^{-1}\\
\times &((8 u_{31}^{3} v_{31}^{4}+12 u_{31}^{2} v_{31}^{3}+36 u_{31}^{2} v_{31}^{
2} z+8 u_{31}^{2} v_{31}+120 u_{31}^{2} z+v_{31}+6 z) v_{31}\\
&+2 (3 v_{31}^{3}+15 v_{31}^{2} z+2 v_{31}+18 z) u_{31}
+2 (2 u_{31} v_{31}+1)^{3} \alpha
v_{31}^{2})\\
\dot{v}_{31}=&(6z (2 u_{31} v_{31}+1) )^{-1}((6 v_{31}^{2} z+4 v_{31}+12 z) u_{31} v_{31}+3 v_{31}^{2} z+2 v_{31}-6 z)\\
E=&(2u31 v31^{2}+4 u31+v31)^{-1}(v31^{3} (-4 u31^{2} v31^{2}-4 u31 v31-1))\\
w_{31}=&-2^{-2}(2 u_{31} v_{31}+1)^{2} v_{31}^{3}
\end{align*}
$v_{31}=0$ defines $L_3$, and $u_{31}v_{31}+1/2=0$ defines $L_1^{(2)}$, the proper transform of $L_1^{(1)}$. There is a base point $b_3$ in this chart given by $u_{31}=0$, $v_{31}=0$.
In the second chart:
\begin{align*}
u_{21}=&u_{32}+1/2\\
v_{21}=&u_{32} v_{32}\\
u=&u_{32}^{-1} v_{32}^{-1} \\
v=&2/(2 u_{32}^{3} v_{32}^{2}+u_{32}^{2} v_{32}^{2}) \\
\dot{u}_{32}=& -(12 v_{32} z)^{-1}\\
&\times(2 (2 u_{32}^{4} v_{32}^{3}+2 u_{32}^{3} v_{32}^{3}+3 u_{32} v_{32}^{2} z
+24 z)+(v_{32}+12 z) u_{32}^{2} v_{32}^{2}+2 (2 u_{32}+1
)^{2} \alpha u_{32}^{2} v_{32}^{3}) \\
\dot{v}_{32}=&(12 (2 u_{32}+1) u_{32} z)^{-1}\\
&\times (2 (2 (2 u_{32}^{5} v_{32}^{3}+3 u_{32}^{4} v_{32}^{3}+9 z)+3
(v_{32}+6 z) u_{32}^{3} v_{32}^{2})+(v_{32}^{2}+30 v_{32} z+8
) u_{32}^{2} v_{32}\\
&+2 (3 v_{32}^{2} z+2 v_{32}+60 z) u_{32}
+2 (2 u_{32}+1)^{3} \alpha u_{32}^{2} v_{32}^{3}) \\
E=& -((2 u_{32}+1)^{2} u_{32}^{3} v_{32}^{4})^{-1}(2 u_{32}^{2} v_{32}^{2}+u_{32} v_{32}^{2}+4)) \\
w_{32}=& -2^{-2}(2 u_{32}+1)^{2} u_{32}^{3} v_{32}^{4}
\end{align*}
$u_{32}=0$, $u_{32}+1/2=0$ and $v_{32}=0$ define $L_3$, $L_1^{(2)}$ and $L_2^{(1)}$ respectively. There are no base points in this chart.
\subsection{Resolution of $b_3$}
In the first chart we have
\begin{align*}
u_{31}=&u_{41} v_{41}\\
v_{31}=&v_{41}\\
u=&v_{41}^{-1}\\
v=& 2/(2 u_{41} v_{41}^{4}+v_{41}^{2})\\
\dot{u}_{41}=&-(12 (2 u_{41} v_{41}^{2}+1) v_{41} z)^{-1}\\
&\times (v_{41}+6 z+8 u_{41}^{3} v_{41}^{7}+4 (3 v_{41}^{3}+12 v_{41}^{2} z+4
v_{41}+36 z) u_{41}^{2} v_{41}^{2}\\
&+2 (3 v_{41}^{3}+18 v_{41}^{2} z+4 v_{41}+12
z) u_{41}+2 (2 u_{41} v_{41}^{2}+1)^{3} \alpha v_{41}) \\
\dot{v}_{41}=&(6z (2 u_{41} v_{41}^{2}+1))^{-1}(2 (3 v_{41}^{2} z+2 v_{41}+6 z) u_{41} v_{41}^{2}+3 v_{41}^{2} z+2 v_{41}
-6 z) \\
E=&-((2 u_{41} v_{41}^{2}+1)^{2} v_{41}^{2})^{-1}(2 u_{41} v_{41}^{2}+4 u_{41}+1) \\
w_{41}=&-2^{-2}(2 u_{41} v_{41}^{2}+1)^{2} v_{41}^{2}
\end{align*}
$v_{41}=0$ defines $L_4$, and $u_{41}v_{41}^2+1/2=0$ defines $L_1^{(3)}$. There is a base point $b_4$ in this chart given by $u_{41}=-1/4$, $v_{41}=0$
In the second chart we have
\begin{align*}
u_{31}=&u_{42} \\
v_{31}=&u_{42} v_{42}\\
u=&u_{42}^{-1} v_{42}^{-1} \\
v=&2/(2 u_{42}^{4} v_{42}^{3}+u_{42}^{2} v_{42}^{2})\\
\dot{u}_{42}=&-(12 (2 u_{42}^{2} v_{42}+1) v_{42} z)^{-1}\\
&\times (2 (2 (2 u_{42}^{3} v_{42}^{2}+3 u_{42} v_{42}+9 z) u_{42}
^{4} v_{42}^{3}+3 (v_{42}+6) z)+(30 u_{42} z+1)
(v_{42}+4) u_{42} v_{42}\\
&+2 (3 v_{42}+4) u_{42}^{3} v_{42}^{2}
+2
(2 u_{42}^{2} v_{42}+1)^{3} \alpha u_{42} v_{42}^{2}) \\
\dot{v}_{42}=&(12z (2 u_{42}^{2} v_{42}+1) u_{42} )^{-1}\\
&\times (2 (2 (2 u_{42}^{3} v_{42}^{2}+3 u_{42} v_{42}+12 z) u_{42}^{4}
v_{42}^{3}+3 (v_{42}+4) z+18 (v_{42}+4) u_{42}^{2} v_{42} z
)\\
&+(v_{42}+8) u_{42} v_{42}+2 (3 v_{42}+8) u_{42}^{3} v_{42}^{2}
+2 (2 u_{42}^{2} v_{42}+1)^{3} \alpha u_{42} v_{42}^{2}) \\
E=&-((2 u_{42}^{2}
v_{42}+1)^{2} u_{42}^{2} v_{42}^{3})^{-1}(2 u_{42}^{2} v_{42}^{2}+v_{42}+4)) \\
w_{42}=&-2^{-2}(2 u_{42}^{2} v_{42}+1)^{2} u_{42}^{2} v_{42}^{3}
\end{align*}
Here, $u_{42}=0$, $u_{42}^2v_{42}+1/2=0$ and $v_{42}=0$ define $L_4$, $L_1^{(3)}$ and $L_3^{(1)}$ respectively. There are no base points in this chart. There is a base point $b_4'$ in this chart given by $u_{42}=0$, $v_{42}=-4$, which is the manifestation of $b_4$ in this chart.
\subsection{Resolution of $b_4$}
In the first chart we have
\begin{align*}
u_{41}=& u_{51} v_{51}-1/4 \\
v_{41}=&v_{51}\\
u=&v_{51}^{-1}\\
v=&4/(4u_{51}v_{51}^5-v_{51}^4+2v_{51}^2) \\
\dot{u}_{51}=&-(48z (v_{51}^{2}-2-4 u_{51} v_{51}^{3}) v_{51} )^{-1}\\
&\times (v_{51}^{6}-6 v_{51}^{4}-24 v_{51}^{3} z+4 v_{51}^{2}+8-64 u_{51}^{3} v_{51}^{9}\\
&+48 (v_{51}^{5}-2 v_{51}^{3}-10 v_{51}^{2} z-4 v_{51}-28 z) u_{51}^{2} v_{51}
^{3}-2 (4 u_{51} v_{51}^{3}-v_{51}^{2}+2)^{3} \alpha\\
&-4 (3 v_{51}^{7}-12 v_{51}^{5}-54 v_{51}^{4} z-8 v_{51}^{3}-72 v_{51}^{2} z+24
v_{51}+24 z) u_{51}) \\
\dot{v}_{51}=&(6z (v_{51}^{2}-2-4 u_{51} v_{51}^{3}))^{-1}(3 v_{51}^{4} z+2 v_{51}^{3}-4 v_{51}+12 z-4 (3 v_{51}^{2} z+2 v_{51}+6 z
) u_{51} v_{51}^{3}) \\
E=&-2 ((v_{51}^{2}-2-4 u_{51} v_{51}^{3})^{2} v_{51})^{-1} (4 u_{51} v_{51}^{2}+8 u_{51}-v_{51}) \\
w_{51}=&-2^{-4}(4 u_{51} v_{51}^{3}-v_{51}^{2}+2)^{2} v_{51}
\end{align*}
$v_{51}=0$ defines $L_5$, and $u_{51}v_{51}^3-v_{51}^2/4+1/2=0$ defines $L_1^{(4)}$. There is a base point $b_5$ in this chart given by $u_{51}=(1-2\alpha)/(12z)$, $v_{51}=0$
In the second chart we have
\begin{align*}
u_{41}=&u_{52}-1/4\\
v_{41}=&u_{52} v_{52}\\
u=&u_{52}^{-1} v_{52}^{-1} \\
v=&4/(4 u_{52}^{5} v_{52}^{4}-u_{52}^{4} v_{52}^{4}+2
u_{52}^{2} v_{52}^{2})\\
\dot{u}_{52}=&-(48 (4 u_{52}^{3} v_{52}^{2}-u_{52}^{2} v_{52}^{2}+2) v_{52} z)^{-1}\\
&\times (4 ((16 u_{52}^{8} v_{52}^{6}-12 u_{52}^{7} v_{52}^{6}+3 u_{52}^{
6} v_{52}^{6}+16) u_{52} v_{52}-2 (v_{52}-24 z)-12 (v_{52}-8 z
) u_{52}^{5} v_{52}^{4}\\
&-(v_{52}+72 z) u_{52}^{2} v_{52}^{2})
-(v_{52}^{2}-96) u_{52}^{6} v_{52}^{5}+2 (3 v_{52}^{2}-96 v_{52} z+64
) u_{52}^{4} v_{52}^{3}\\
&+8 (3 v_{52}^{2} z-2 v_{52}+144 z) u_{52}^{3}
v_{52}^{2}+2 (4 u_{52}^{3} v_{52}^{2}-u_{52}^{2} v_{52}^{2}+2)^{3} \alpha v_{52}
) \\
\dot{v}_{52}=&(48 (4 u_{52}^{3} v_{52}^{2}-u_{52}^{2} v_{52}^{2}+2) u_{52} z)^{-1}\\
&\times (4 ((16 u_{52}^{8} v_{52}^{6}-12 u_{52}^{7} v_{52}^{6}+3 u_{52}^{6} v_{52}
^{6}+24) u_{52} v_{52}-2 (v_{52}-12 z)-12 (v_{52}-10 z)
u_{52}^{5} v_{52}^{4}\\
&-(v_{52}+72 z) u_{52}^{2} v_{52}^{2})-(v_{52}^{
2}-96) u_{52}^{6} v_{52}^{5}+6 (v_{52}^{2}-36 v_{52} z+32) u_{52}^{4}
v_{52}^{3}\\
&+8 (3 v_{52}^{2} z-4 v_{52}+168 z) u_{52}^{3} v_{52}^{2}+2 (4
u_{52}^{3} v_{52}^{2}-u_{52}^{2} v_{52}^{2}+2)^{3} \alpha v_{52}) \\
E=&-2((4 u_{52}^{3} v_{52}^{2}-u_{52}^{2} v_{52}^{2}+2)^{2} u_{52} v_{52}^{2})^{-1} (4 u_{52}^{2} v_{52}^{2}-u_{52} v_{52}^{2}+8) \\
w_{52}=&-2^{-4}(4 u_{52}^{3} v_{52}^{2}-u_{52}^{2} v_{52}^{2}+2)^{2} u_{52} v_{52}^{2}
\end{align*}
Here, $u_{52}=0$, $u_{52}^3v_{52}^2-u_{52}^2v_{52}^2/4+1/2=0$ and $v_{52}=0$ define $L_5$, $L_1^{(4)}$ and $L_4^{(1)}$ respectively. There are no base points in this chart. There is a base point $b_5'$ in this chart given by $u_{52}=0$, $v_{52}=(12z)/(1-2\alpha)$, which is the manifestation of $b_5$ in this chart.
\subsection{Resolution of $b_5$}
In the first chart we have
\begin{align*}
u_{51}=& u_{61} v_{61} +(1-2 \alpha)/(12 z)\\
v_{51}=&v_{61}\\
u=&v_{61}^{-1}\\
v=&12 z/((v_{61}^{3}-3 v_{61}^{2} z+6 z+12 u_{61}
v_{61}^{4} z) v_{61}^{2}-2 \alpha v_{61}^{5})\\
\dot{u}_{61}=&-(432 (v_{61}^{3}-3 v_{61}^{2} z
+6 z+12 u_{61} v_{61}^{4} z-2 \alpha v_{61}^{3}) z^{3})^{-1}\\
&\times (v_{61}^{7}-9 v_{61}^{6} z+27 v_{61}^{5} z^{2}-18 v_{61}^{3} z^{2}-324
v_{61}^{2} z^{3}-756 z^{3}+1728 u_{61}^{3} v_{61}^{10} z^{3}-16 \alpha^{4} v_{61}^{7}
\\
&-9 (3 z^{2}-2) v_{61}^{4} z +8 (2 v_{61}^{3}-9 v_{61}^{2} z+18 z+36
u_{61} v_{61}^{4} z) \alpha^{3} v_{61}^{4}\\
&+72 (9 z^{2}+4) v_{61} z^{2}+
432 (v_{61}^{6}-3 v_{61}^{5} z+6 v_{61}^{3} z+36 v_{61}^{2} z^{2}+16 v_{61} z+96 z
^{2}) u_{61}^{2} v_{61}^{3} z^{2}\\
&+36 (v_{61}^{8}-6 v_{61}^{7} z+9
v_{61}^{6} z^{2}+12 v_{61}^{5} z+30 v_{61}^{4} z^{2}+144 v_{61}^{2} z^{2}-216 v_{61} z
^{3}+96 z^{2}\\
&-4 (45 z^{2}-4) v_{61}^{3} z) u_{61} z\\
&+36 (
v_{61}^{5}-3 v_{61}^{4} z-2 v_{61}^{3}+22 v_{61}^{2} z+16 z-48 u_{61}^{2} v_{61}^{8} z\\
&-4
(v_{61}^{3}-6 v_{61}^{2} z+12 z) u_{61} v_{61}^{4}) \alpha^{2} v_{61} z\\
&-2 (2 v_{61}^{7}-9 v_{61}^{6} z+180 v_{61}^{3} z^{2}-648 v_{61}^{2} z^{3}+432
v_{61} z^{2}-324 z^{3}-1728 u_{61}^{3} v_{61}^{10} z^{3}\\
&+9 (3 z^{2}+2)v_{61}^{4} z +1296 (v_{61}^{2}-2) u_{61}^{2} v_{61}^{6} z^{3}\\
&+36 (v_{61}^{6}-9 v_{61}^{4} z^{2}+102 v_{61}^{2} z^{2}+16 v_{61} z+144 z^{2}) u_{61} v_{61}^{2} z) \alpha) \\
\dot{v}_{61}=&(6 (v_{61}^{3}-3 v_{61}^{2} z+6 z+12 u_{61} v_{61}^{4} z-2
\alpha v_{61}^{3}) z) ^{-1}\\
&\times(-2 v_{61}^{3} (\alpha-6 u_{61} v_{61} z) (3 v_{61}^{2} z+2 v_{61}+
6 z)+3 (v_{61}^{5}+4 v_{61}-12 z) z-(9 z^{2}-2)
v_{61}^{4}) \\
E=&(v_{61} (v_{61}^{3}-3 v_{61}^{2} z+6 z+
12 u_{61} v_{61}^{4} z-2 \alpha v_{61}^{3})^{2})^{-1}\\
&\times(6 (2 \alpha v_{61}^{2}+4 \alpha-12 u_{61} v_{61}^{3} z-24 u_{61} v_{61} z-v_{61}
^{2}+3 v_{61} z-2) z) \\
w_{61}=&-2^{-4}3^{-2} z^{-2}(2 \alpha v_{61}^{3}-12 u_{61} v_{61}^{4} z-v_{61}^{3}+3 v_{61}^{2} z-6 z
)^{2}
\end{align*}
$v_{61}=0$ defines $L_6$, and $u_{61}v_{61}^4-v_{61}^2/4+v_{61}(1-2\alpha)/(12z)+1/2=0$ defines $L_1^{(5)}$. There are no base points in this chart. The vector field is non-zero and transversal to the line $L_6$.
In the second chart
\begin{align*}
u_{51}=& u_{62} +(1-2 \alpha)/(12 z)\\
v_{51}=&u_{62} v_{62}\\
u=&u_{62}^{-1} v_{62}^{-1}\\
v=&(12 z)/((12 u_{62}^{4}
v_{62}^{3} z+u_{62}^{3} v_{62}^{3}-3 u_{62}^{2} v_{62}^{2} z+6 z) u_{62}^{2} v_{62}^{2}
-2 \alpha u_{62}^{5} v_{62}^{5})\\
\dot{u}_{62}=&-(432 (12 u_{62}^{4} v_{62}^{
3} z+u_{62}^{3} v_{62}^{3}-3 u_{62}^{2} v_{62}^{2} z+6 z-2 \alpha u_{62}^{3} v_{62}^{3}
) v_{62} z^{3})^{-1}\\
&\times(4 (4 (27 (4 u_{62}^{11} v_{62}^{9} z+u_{62}^{10} v_{62}
^{9}+6 z^{2}) z^{2}-\alpha^{4} u_{62}^{8} v_{62}^{9})-81 (v_{62}-
16) u_{62}^{3} v_{62}^{3} z^{3})\\
&+(v_{62}-216 z^{2}) u_{62}^{8
} v_{62}^{8}
+36 (v_{62}-36 z^{2}) u_{62}^{9} v_{62}^{8} z-108 (7 v_{62}
-24) u_{62} v_{62} z^{3}\\
&+27 (v_{62}^{2}+16 v_{62}+480 z^{2}) u_{62}^{6
} v_{62}^{5} z^{2}
-9 (v_{62}^{2}-36 v_{62} z^{2}-288 z^{2}) u_{62}^{7}
v_{62}^{6} z\\
&-18 (v_{62}^{2}+324 v_{62} z^{2}-24 v_{62}-2016 z^{2}) u_{62}^{4
} v_{62}^{3} z^{2}+9 (96 (v_{62}
+6) z^{2}-(3 z^{2}-2) v_{62}^{2}) u_{62}^{5} v_{62}^{4} z\\
&+8 (36 u_{62}^{4} v_{62}^{3} z+2 u_{62}^{3} v_{62}^{3}-9 u_{62}^{2}
v_{62}^{2} z+18 z) \alpha^{3} u_{62}^{5} v_{62}^{6}+72 ((9 z^{2}+
4) v_{62}-108 z^{2}) u_{62}^{2} v_{62}^{2} z^{2}\\
&-36 (48 u_{62}^{8} v_{62}^{6} z+4 u_{62}^{7} v_{62}^{6}-24 u_{62}^{6} v_{62}^{5} z-u_{62}
^{5} v_{62}^{5}+2 u_{62}^{3} v_{62}^{3}-22 u_{62}^{2} v_{62}^{2} z-16 z\\
&+3 (v_{62}+16
) u_{62}^{4} v_{62}^{3} z) \alpha^{2} u_{62}^{2} v_{62}^{3} z+2 (2
(864 u_{62}^{10} v_{62}^{7} z^{3}-u_{62}^{7} v_{62}^{7}-216 u_{62} v_{62} z^{2}+162 z
^{3}\\
&-18 (v_{62}+36 z^{2}) u_{62}^{8} v_{62}^{6} z+108 (3 v_{62}-22
) u_{62}^{2} v_{62} z^{3}-18 (5 v_{62}+12) u_{62}^{3} v_{62}^{2} z^{2}
)\\
&+9 (v_{62}^{2}+36 v_{62} z^{2}+288 z^{2}) u_{62}^{6} v_{62}^{4} z-
9 ((3 z^{2}+2) v_{62}+384 z^{2}) u_{62}^{4} v_{62}^{3} z
) \alpha u_{62} v_{62}^{2}) \\
\dot{v}_{62}=&(432 (12 u_{62}^{4} v_{62}^{3} z+u_{62}^{3} v_{62}^{3}-3 u_{62}^{2
} v_{62}^{2} z+6 z-2 \alpha u_{62}^{3} v_{62}^{3}) z^{3})^{-1}\\
&\times ((4 (27 (4 (4 u_{62} z+1) u_{62}^{9} v_{62}^{8}-
(7 v_{62}-32) z) z^{2}-4 \alpha^{4} u_{62}^{7} v_{62}^{8}-81
(v_{62}-16) u_{62}^{2} v_{62}^{2} z^{3})\\
&+(v_{62}-216 z^{2}
) u_{62}^{7} v_{62}^{7}+36 (v_{62}-36 z^{2}) u_{62}^{8} v_{62}^{7} z+
27 (v_{62}^{2}+16 v_{62}+576 z^{2}) u_{62}^{5} v_{62}^{4} z^{2}\\
&-9 (
v_{62}^{2}-36 v_{62} z^{2}-288 z^{2}) u_{62}^{6} v_{62}^{5} z-18 (v_{62}^{2}+360 v_{62} z^{2}-32 v_{62}-2304 z^{2}) u_{62}^{3} v_{62}^{2} z^{2}\\
&+8 (36
u_{62}^{4} v_{62}^{3} z+2 u_{62}^{3} v_{62}^{3}-9 u_{62}^{2} v_{62}^{2} z+18 z)
\alpha^{3} u_{62}^{4} v_{62}^{5}\\
&+72 ((9 z^{2}+4) v_{62}-108 z^{2}
) u_{62} v_{62} z^{2}+9 (24 (5 v_{62}+32) z^{2}-(3 z^{
2}-2) v_{62}^{2}) u_{62}^{4} v_{62}^{3} z\\
&-36 (48 u_{62}^{8} v_{62}^{6}
z+4 u_{62}^{7} v_{62}^{6}-24 u_{62}^{6} v_{62}^{5} z-u_{62}^{5} v_{62}^{5}+2 u_{62}^{3} v_{62}
^{3}-22 u_{62}^{2} v_{62}^{2} z-16 z\\
&+3 (v_{62}+16) u_{62}^{4} v_{62}^{3} z
) \alpha^{2} u_{62} v_{62}^{2} z+2 (2 (864 u_{62}^{10} v_{62}^{7} z^{
3}-u_{62}^{7} v_{62}^{7}-216 u_{62} v_{62} z^{2}+162 z^{3}\\
&+324 (v_{62}-8)
u_{62}^{2} v_{62} z^{3}-18 (v_{62}+36 z^{2}) u_{62}^{8} v_{62}^{6} z\\
&-18 (5 v_{62}+16) u_{62}^{3} v_{62}^{2} z^{2})+9 (v_{62}^{2}+36
v_{62} z^{2}+288 z^{2}) u_{62}^{6} v_{62}^{4} z\\
&-9 ((3 z^{2}+2
) v_{62}+408 z^{2}) u_{62}^{4} v_{62}^{3} z) \alpha v_{62})
v_{62}) \\
E=&((12 u_{62}^{4} v_{62}^{3} z+u_{62}^{3} v_{62}^{3}-3 u_{62}^{2} v_{62}^{2} z+6 z-2 \alpha u_{62}^{3} v_{62}^{3})^{2} u_{62} v_{62})^{-1} \\
&\times (6z (2 \alpha u_{62}^{2} v_{62}^{2}+4 \alpha-12 u_{62}^{3} v_{62}^{2} z-u_{62}^{
2} v_{62}^{2}+3 u_{62} v_{62} z-24 u_{62} z-2) ) \\
w_{62}=& -2^{-4}3^{-2} z^{-2}(2 \alpha u_{62}^{3} v_{62}^{3}-12 u_{62}^{4} v_{62}^{3} z-u_{62}^{3} v_{62}^{3}+3 u_{62}^{2} v_{62}^{2} z-6 z)^{2} v_{62}
\end{align*}
Here, $u_{62}=0$, $u_{62}^4v_{62}^3-u_{62}^2v_{62}^2/4+u_{62}^3v_{62}^3(1-2\alpha)/(12z)+1/2=0$ and $v_{62}=0$ define $L_6$, $L_1^{(5)}$ and $L_5^{(1)}$ respectively. There are no base points in this chart.
We have thus resolved all the singularities of the system which originated from the base point $b_0$. We return to the 01 coordinate chart to resolve the other base point.
\subsection{Resolution of $b_6$}
The base point $b_6$ is from the 01-chart.
In the first chart:
\begin{align*}
u_{01}=&u_{71} v_{71}\\
v_{01}=&v_{71}\\
u=&1/(u_{71} v_{71})\\
v=&1/u_{71}\\
\dot{u}_{71}=&-(3 v_{71} z)^{-1}(u_{71}^{2} v_{71}-2 u_{71} v_{71}+6 z+2 \alpha u_{71}^{2} v_{71}) \\
\dot{v}_{71}=&(6 u_{71} z)^{-1}(2 (2 \alpha u_{71}^{2} v_{71}+9 z-(3 v_{71} z+1) u_{71} v_{71})+(3 v_{71} z+2) u_{71}^{2} v_{71}) \\
E=&-(2 u_{71}^{3} v_{71}^{2})^{-1}((u_{71}-1) u_{71} v_{71}^{2}+2) \\
w_{71}=&-u_{71}^{3} v_{71}^{2}
\end{align*}
Here $v_{71}=0$ and $u_{71}=0$ define the lines $L_7$ and $L_0^{(7)}$ respectively. There are no base points in this chart.
\begin{align*}
u_{01}=&u_{72} \\
v_{01}=&u_{72} v_{72}\\
u=&1/u_{72} \\
v=&v_{72}\\
\dot{u}_{72}=&(6 z)^{-1}(2 (u_{72}+3 z)-3 (2 v_{72}-1) u_{72}^{2} z) \\
\dot{v}_{72}=&(3 u_{72} z)^{-1}(2 (\alpha u_{72}+3 v_{72} z)-(2 v_{72}-1) u_{72})
\\
E=&(2 u_{72}^{2})^{-1}(((v_{72}-1) u_{72}^{2}-2) v_{72}) \\
w_{72}=& -(3 v_{72})^{-1}u_{72}(2 (\alpha u_{72}+3 v_{72}^{2})-(2 v_{72}-1) u_{72}
)
\end{align*}
Here $u_{72}=0$ defines the line $L_7$, the line $L_0^{(7)}$ is not visible in this chart. There is a base point $b_7$ of both the vector field and the function $E$ given by $u_{72}=0$, $v_{72}=0$.
\subsection{Resolution of $b_7$}
\begin{align*}
u_{72}=&u_{81} v_{81} \\
v_{72}=&v_{81}\\
u=&1/(u_{81} v_{81}) \\
v=&v_{81}\\
\dot{u}_{81}=&-(6 v_{81} z)^{-1}(2 (2 \alpha u_{81}+3 z)+3 (2 v_{81}-1) u_{81}^{
2} v_{81}^{2} z-2 (3 v_{81}-1) u_{81}) \\
\dot{v}_{81}=&(3 u_{81} z)^{-1}(2 (\alpha u_{81}+3 z)-(2 v_{81}-1) u_{81}) \\
E=&(2 u_{81}^{2} v_{81})^{-1}((v_{81}-1) u_{81}^{2} v_{81}^{2}-2) \\
w_{81}=&v_{81}^{-1}(-2 (u_{81} v_{81}^{2}-1))
\end{align*}
Here $v_{81}=0$ and $u_{81}=0$ define the lines $L_8$ and $L_7^{(1)}$ respectively. There is a base point $b_{8}'$ of the vector field given by $u_{81}=-3z(1+2 \alpha)^{-1}$, $v_{82}=0$ which is not a base point of the energy function $E$.
In the second chart:
\begin{align*}
u_{72}=&u_{82} \\
v_{72}=&u_{82} v_{82}\\
u=&1/u_{82} \\
v=&u_{82} v_{82}\\
\dot{u}_{82}=&-(6 z)^{-1}(6 u_{82}^{3} v_{82} z-3 u_{82}^{2} z-2 u_{82}-6 z) \\
\dot{v}_{82}=&(6 u_{82} z)^{-1}(3 (2 u_{82}^{2} v_{82} z-u_{82} z-2) u_{82} v_{82}+2 (3 v_{82} z+1
)+4 \alpha) \\
E=&(2 u_{82})^{-1}(((u_{82} v_{82}-1) u_{82}^{2}-2) v_{82}) \\
w_{82}=&-u_{82}
\end{align*}
Here $u_{82}=0$ defines the line $L_8$, the line $L_7^{(1)}$ is not visible in this chart. There is a base point $b_8$ of the vector field given by $u_{82}=0$, $v_{82}=-(1+2 \alpha)(3 z)^{-1}$. There is a base point $b_{8,\infty}$ of the autonomous energy function $E$ at $u_{82}=0$, $v_{82}=0$.
\subsection{Resolution of $b_8$}
In the first chart:
\begin{align*}
u_{82}=&u_{91} v_{91} \\
v_{82}=& v_{91} -(1+2 \alpha)(3 z)^{-1}\\
u=&1/(u_{91} v_{91}) \\
v=&((3 v_{91} z-1) u_{91} v_{91}-2
\alpha u_{91} v_{91})/(3 z)\\
\dot{u}_{91}=&-(18 z^{2})^{-1} u_{91}((2 (3 v_{91} z-1) u_{91} v_{91}-3 z) (6
v_{91} z-1) u_{91}+8 (\alpha^{2} u_{91}^{2} v_{91}-3 z)\\
&-2 (2
(9 v_{91} z-2) u_{91} v_{91}-3 z) \alpha u_{91})) \\
\dot{v}_{91}=&-(18 u_{91} z^{2})^{-1}\\
& \times(2 (9 (u_{91} v_{91}-z) z-4 \alpha^{2} u_{91}^{3} v_{91}
^{2})-(2 (3 v_{91} z-1) u_{91} v_{91}-3 z) (3
v_{91} z-1) u_{91}^{2} v_{91}\\
&+2 (4 (3 v_{91} z-1) u_{91} v_{91}-3 z
) \alpha u_{91}^{2} v_{91}) \\
E=&-(18 u_{91} v_{91} z^{2})^{-1}((2 \alpha u_{91} v_{91}+3 z-(3 v_{91} z-1) u_{91} v_{91}
) u_{91}^{2} v_{91}^{2}+6 z) (3 v_{91} z-1-2 \alpha) \\
w_{91}=&-u_{91}
\end{align*}
Here $v_{91}=0$ and $u_{91}=0$ define the lines $L_9$ and $L_8^{(1)}$ respectively. There are no base points in this chart.
\begin{align*}
u_{82}=&u_{92} \\
v_{82}=& u_{92} v_{92} -(1+2 \alpha)(3 z)^{-1}\\
u=&1/u_{92}\\
v=&(3z)^{-1}((3 u_{92} v_{92} z-1) u_{92}-2 \alpha u_{92})\\
\dot{u}_{92}=&-(6z)^{-1}(6 u_{92}^{4} v_{92} z-2 u_{92}^{3}-3 u_{92}^{2} z-2 u_{92}-6 z-4 \alpha u_{92}
^{3}) \\
\dot{v}_{92}=&(18 z^{2})^{-1}(3 (6 (2 u_{92} v_{92} z-1) u_{92}^{2} v_{92}-(8 v_{92}-1)) z+8 \alpha^{2} u_{92}-2 (9 v_{92} z^{2}-1) u_{92}\\
&-2
(18 u_{92}^{2} v_{92} z-4 u_{92}-3 z) \alpha) \\
E=&-(18 u_{92} z^{2})^{-1}((2 \alpha u_{92}+3 z-(3 u_{92} v_{92} z-1) u_{92})
u_{92}^{2}+6 z) (3 u_{92} v_{92} z-1-2 \alpha) \\
w_{92}=& -1
\end{align*}
Here $u_{92}=0$ defines the line $L_9$, the line $L_8^{(1)}$ is not visible in this chart. There are no base points in this chart.
The Boutroux-Painlev\'e vector field is regular and non-zero on $L_9$, furthermore the vector field is transversal to $L_9$. | 100,996 |
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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import topology.instances.int
/-!
# Topology on the natural numbers
The structure of a metric space on `ℕ` is introduced in this file, induced from `ℝ`.
-/
noncomputable theory
open metric set filter
namespace nat
noncomputable instance : has_dist ℕ := ⟨λ x y, dist (x : ℝ) y⟩
theorem dist_eq (x y : ℕ) : dist x y = |x - y| := rfl
lemma dist_coe_int (x y : ℕ) : dist (x : ℤ) (y : ℤ) = dist x y := rfl
@[norm_cast, simp] theorem dist_cast_real (x y : ℕ) : dist (x : ℝ) y = dist x y := rfl
lemma pairwise_one_le_dist : pairwise (λ m n : ℕ, 1 ≤ dist m n) :=
begin
intros m n hne,
rw ← dist_coe_int,
apply int.pairwise_one_le_dist,
exact_mod_cast hne
end
lemma uniform_embedding_coe_real : uniform_embedding (coe : ℕ → ℝ) :=
uniform_embedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
lemma closed_embedding_coe_real : closed_embedding (coe : ℕ → ℝ) :=
closed_embedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
instance : metric_space ℕ := nat.uniform_embedding_coe_real.comap_metric_space _
theorem preimage_ball (x : ℕ) (r : ℝ) : coe ⁻¹' (ball (x : ℝ) r) = ball x r := rfl
theorem preimage_closed_ball (x : ℕ) (r : ℝ) :
coe ⁻¹' (closed_ball (x : ℝ) r) = closed_ball x r := rfl
theorem closed_ball_eq_Icc (x : ℕ) (r : ℝ) :
closed_ball x r = Icc ⌈↑x - r⌉₊ ⌊↑x + r⌋₊ :=
begin
rcases le_or_lt 0 r with hr|hr,
{ rw [← preimage_closed_ball, real.closed_ball_eq_Icc, preimage_Icc],
exact add_nonneg (cast_nonneg x) hr },
{ rw closed_ball_eq_empty.2 hr,
apply (Icc_eq_empty _).symm,
rw not_le,
calc ⌊(x : ℝ) + r⌋₊ ≤ ⌊(x : ℝ)⌋₊ : by { apply floor_mono, linarith }
... < ⌈↑x - r⌉₊ : by { rw [floor_coe, nat.lt_ceil], linarith } }
end
instance : proper_space ℕ :=
⟨ begin
intros x r,
rw closed_ball_eq_Icc,
exact (set.finite_Icc _ _).is_compact,
end ⟩
instance : noncompact_space ℕ :=
noncompact_space_of_ne_bot $ by simp [filter.at_top_ne_bot]
end nat
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2014-12-18 03:20:14.164652 | 65,316 |
TITLE: Why do some people say the universe is hyperbolic?
QUESTION [0 upvotes]: I thought that it was generally agreed that the universe seems to be flat, according to the Planck data, but every once in a while I'll hear someone say that the universe is hyperbolic. I know this isn't a very formal question, but I'm wondering where this idea comes from? Is it just a common misinterpretation of something, or are they right?
REPLY [0 votes]: Generally the characterization of the geometry of cosmological space (not space-time) is determined by the values of the four parameters in the Friedmann Equation presented in the section:
https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation . These are the $\Omega$ density parameters which in a cosmological model have values for the density of radiation, matter, curvature, and dark energy (also known as the cosmological constant density). The corresponding subscripts are: R, M, k, and $\Lambda$. The sum of these four parameters equals 1. (I will use here just the subscripts to represent the $\Omega$ variables.) If R+M+$\Lambda$ > 0, then k < 0 corresponding to what is called a 3D hyper-spherical space. R+M+$\Lambda$ < 0, then k > 0 corresponding to what is called a 3D hyperbolic space. If k = 0 then the universe shape is called flat or Euclidean. | 202,951 |
TITLE: Prove for every $\alpha > 0$, the function $f'$ changes its sign in $(-\alpha, \alpha)$.
QUESTION [1 upvotes]: Consider the function $$f(x)=\begin{cases}
x^2\sin(\frac{1}{x})+cx, & \text{if }x \neq 0;\\
0, & \text{if }x = 0;\\
\end{cases}
$$ where $0 < c < 1$.
Prove that for every $\alpha > 0$, the derivative $f'$ changes its sign on $(-\alpha, \alpha)$.
I know $f$ is differentiable in $\mathbb{R}$ and $f'(x) = 2x\sin(\frac{1}{x})-\cos(\frac{1}{x})+c$ for $x \neq 0$ and $f'(0)=0$. But don't know how to proceed after.
REPLY [1 votes]: For any given $\alpha>0$, choose an integer $n$ so large that $$n>\frac{1}{2\pi\alpha}.$$ Define
\begin{align*}
x_1&\equiv\frac{1}{2n\pi},\\x_2&\equiv\frac{1}{(2n+1)\pi}.
\end{align*}
Clearly, $x_1$ and $x_2$ are both in $(0,\alpha)$. Moreover,
\begin{align*}
f'(x_1)&=2x_1\sin\left(\frac{1}{x_1}\right)-\cos\left(\frac{1}{x_1}\right)+c\\
&=\frac{1}{n\pi}\underbrace{\sin(2n\pi)}_{=0}-\underbrace{\cos(2n\pi)}_{=1}+c=c-1<0,\\
f'(x_2)&=2x_2\sin\left(\frac{1}{x_2}\right)-\cos\left(\frac{1}{x_2}\right)+c\\
&=\frac{2}{(2n+1)\pi}\underbrace{\sin((2n+1)\pi)}_{=0}-\underbrace{\cos((2n+1)\pi)}_{=-1}+c=c+1>0.
\end{align*}
In fact, this works as long as $-1<c<1$ (instead of $0<c<1$). | 24,800 |
It's Monday and today is release day from NIH/NCI, I have been here since last Wednesday. I underwent the "PHP" surgery on Thursday morning, then spent roughly 28 hours in ICU before being moved back to a regular room. The PHP surgery is for the phase 3 trial. An experimental chemo drug is put right to the organ. As you can imagine the phase 1 and 2 trials were lots of trial and error to see if the drug worked. The unique thing about this trial is the delivery system for the chemo. It is called the Delcath system. I had 2 catheters placed in my groin and 1 catheter on each side of my neck. Specialized Radiologists map the blood veins going to and from the liver, the insert platinum coils in the ones not needed. The catheters are in place to deliver the drug directly to the liver and then be brought back out of the body and the blood is filtered then placed back through the neck catheter, the blood is filtered for 5-10 minutes depending on clotting ability. As most know chemo drugs destroy cells, and apply pressure for 15 minutes to get blood to clot, ii was doing fine then about 30 minutes later my groin started shooting blood out. A moment of concern for sure but pressure applied again and it held. The next 10-14 days are the most critical as the platelet counts in all previous trial patients have fallen drastically, we will be watching this. I come back in 5 weeks to see if this treatment has done anything good. All the nurses and Doctors at NIH
May Update #2
Posted by Adam "Mac" at 6:20 AM
Wonderful blog. You and your family are in my prayers. | 277,443 |
TITLE: Equivalent conditions for a topological vector space to admit an inner-product
QUESTION [4 upvotes]: Let $V$ be a topological vector space.
Inspired by this list
Every inner-product induces a (unique?) norm
Every norm induces a (unique?) metric
Every metric induces a (unique?) uniform structure
Every uniform structure induces a (unique?) topology,
I'm trying to complete the list
$V$ admits an inner-product iff ???
$V$ admits a norm iff it is Hausdorff and has a convex bounded neighborhood of zero (Kolmogorov's theorem)
$V$ admits a metric iff it is Hausdorff and has a countable base of neighborhoods of zero
$V$ admits a (unique) uniform structure, for free
and the list
$V$ is a Euclidean space iff ???
$V$ is a Hilbert space iff it has a complete inner-product and the scalars are $\Bbb R$ or $\Bbb C$
$V$ is a Banach space iff it has a complete norm and the scalars are $\Bbb R$ or $\Bbb C$
$V$ is a ??? iff it has a complete metric
Namely, what are equivalent conditions for $V$ to admit an inner-product? How do Euclidean spaces $\textbf{R}^n$ fit into this scheme? Does $V$ have a special name when it has a complete metric, or are these spaces not important enough? In the first list, is the induced structure always unique?
REPLY [2 votes]: I'll assume TVS includes that $X$ is $T_1$ (so Tychonoff)
Some random relevant facts:
A normed space $(X,||.||)$ has its norm induced by an inner product iff the paralellogram law holds:
$$\forall x,y \in X: 2||x||^2 + 2||y||^2 = ||x-y||^2 +||x+y||^2$$
e.g. see this question and its answers. The inner product is uniquely determined by the norm.
A TVS is normable iff it is locally convex and it has a base at $0$ of bounded sets (in the sense that $A$ is bounded iff for every neighbourhood $V$ of $0$ there is some scalar $c$ with $A \subseteq cV$). See any good textbook. The norm can always be scaled etc. so it's not quite unique.
$V$ is a Euclidean space iff it has finite dimension (if the scalar field is $\mathbb{R}$ or $\mathbb{C}$, of course).
A TVS that has a complete and compatible metric is called a Fréchet space. They can be topologically characterised by $V$ with a countable local base at $0$ which is also uniformly complete in the induced uniformity. See these notes for more info.
The uniformity on $V$ is never unique (but it exists for all $T_1$ TVS's). Spaces with unique uniformities are "almost-compact" and TVS's never are. | 170,352 |
\begin{document}
\title{Three--dimensional Gaussian fluctuations of non--commutative random surfaces along time--like paths}
\author{Jeffrey Kuan}
\maketitle
\abstract{We construct a continuous--time non--commutative random walk on $U(\mathfrak{gl}_N)$ with dilation maps $U(\mathfrak{gl}_N)\rightarrow L^2(U(N))^{\otimes\infty}$. This is an analog of a continuous--time non--commutative random walk on the group von Neumann algebra $vN(U(N))$ constructed in \cite{K}, and is a variant of discrete--time non--commutative random walks on $U(\mathfrak{gl}_N)$ \cite{B,CD}.
It is also shown that when restricting to the Gelfand--Tsetlin subalgebra of $U(\mathfrak{gl}_N),$ the non--commutative random walk matches a (2+1)--dimensional random surface model introduced in \cite{BF}. As an application, it is then proved that the moments converge to an explicit Gaussian field along time--like paths. Combining with \cite{BF} which showed convergence to the Gaussian free field along space--like paths, this computes the entire three--dimensional Gaussian field. In particular, it matches a Gaussian field from eigenvalues of random matrices \cite{Bo}.
\section{Introduction}
Let us review some results in the mathematical and physics literature in order to motivate the problem.
The \textit{Anisotropic Kardar--Parisi--Zhang} (AKPZ) equation, which was introduced in \cite{W} and is a variant of the KPZ equation first considered in \cite{KPZ}, describes a
universal class of random surface growth models. Letting $h(t)$ denote the height of the surface at time $t,$ the equation in two dimensions is
$$
\partial_t h = \nu_x \partial_x^2 h + \nu_y \partial y^2 h + \frac{1}{2}\lambda_x(\partial_x h)^2 + \frac{1}{2}\lambda_y(\partial_y h)^2 + \eta,
$$
where $\eta$ is space--time white noise and $\lambda_x,\lambda_y$ have different signs. (When $\lambda_x$ and $\lambda_y$ have the same sign, the equation is just the usual KPZ equation in two dimensions). Using non--rigorous methods, it was predicted (e.g. \cite{KS}) that the stationary distribution for the AKPZ dynamics would be the Gaussian free field (see \cite{S} for a mathematical approach to the Gaussian free field). The question about the full three--dimensional process across different time variables remained open.
However, the equation is mathematically ill--defined, due to the non--linear term. One mathematical approach is to consider exactly solvable models in the AKPZ universality class. There have been two models considered, an interacting particle system and the eigenvalue process of a random matrix. Both will be described now.
The interacting particle system, studied in \cite{BF}, lives on the lattice $\Z\times\Z_+$. It was shown that along \textit{space--like paths}, the particle system is a determinantal point process. (See Theorem \ref{SpaceLikePath} for the definition of space--like paths). By computing the correlation kernel and taking asymptotics, it was shown that the fluctuations of the height function of the particle system indeed converge to the Gaussian free field. But due to the space--like path restriction, the problem of computing the limiting three--dimensional field remained unsolved.
The random matrix model looks at the eigenvalues of minors of a large random matrix whose entries are evolving as Ornstein--Uhlenbeck processes. By a combinatorial argument,
\cite{Bo} was able to compute the limiting three--dimensional Gaussian field, which has the Gaussian free field as a stationary distribution. The asymptotics at the edge were also computed in \cite{So}. However, one drawback is that the eigenvalues are not Markovian, as shown in \cite{ANv}.
With these two models in mind, it is natural to want to consider an exactly solvable model that ``combines'' both models, and which is both Markovian and allows for the limiting three--dimensional field to be computed. This paper will construct such a model.
Let us outline the body of the paper. First, the model will be constructed as a continuous--time non--commutative random walk, which is a non--commutative version of the usual random walk in classical probability. The ``state space''
is $U(\mathfrak{gl}_N)$, the universal enveloping algebra of the Lie algebra $\mathfrak{gl}_N$ of $N\times N$ matrices. The dilation maps are algebra homomorphisms
$j_n:U(\mathfrak{gl}_N)\rightarrow (M^{\otimes \infty},\omega)$, where $M$ is a von Neumann sub--algebra of the $U(\mathfrak{gl}_N)$--module $L^2(U(N))$ and $\omega$ is a state on $M^{\otimes\infty}$. These $j_n$ are a non--commutative analog of the usual definition of a stochastic process as a family of maps $X_n$ from a probability space $(\Omega,\mathcal{F},\mathbb{P})$ to a state space $S$. It is proved below (Theorem
\ref{QuantumTheorem}) that there is a semigroup of non--commutative Markov operator $\{P_t\}_{t\geq 0}$ on $U(\mathfrak{gl}_N)$ which is consistent with $j_n$.
This model is analogous to a previously constructed non--commutative random walk on the group von Neumann algebra $vN(U(N))$ with dilation maps $vN(U(N))\rightarrow vN(U(N))^{\otimes\infty}$ \cite{K}. Additionally, it preserves the states from \cite{BB}. All of these construction involve a continuous family of characters of the infinite--dimensional unitary group $U(\infty)$. There have also been previous non--commutative random walks using the basic representation of $U(N)$ as input \cite{B,CD}.
It also turns out that $P_t$ preserves $Z:=Z(U(\mathfrak{gl}_N))$, the centre of $U(\mathfrak{gl}_N)$.
This means that $P_t\Big|_Z$ is a Markov operator in the usual (classical) sense. This Markov operator has a natural
description: By using the Harish--Chandra isomorphism which identifies $Z$ with the ring of shifted
symmetric polynomials in $N$ variables, $P_t$ can be identified with the Markov operator $Q_t$ of an
interacting system of $N$ particles on $\Z$. This is shown in Proposition \ref{OneLevel} below. This
interacting system is known as the \textit{Charlier Process}, see \cite{KOR}. In fact, the projection of
the interacting particle system from \cite{BF} onto $\Z\times\{N\}$ is exactly $Q_t$. When restricting our non--commutative
random walk to the Gelfand--Tsetlin subalgebra, which is the subalgebra of $U(\mathfrak{gl}_N)$ generated by the centres $Z(U(\mathfrak{gl}_k)), 1\leq k\leq N$,
it also matches the two--dimensional particle system along space--like paths; see Theorem \ref{TwoLevels} for the precise statement. It is worth mentioning
that the matching most likely does not hold along time--like paths.
We then take asymptotics of certain elements of the Gelfand--Tsetlin subalgebra and prove convergence to jointly
Gaussian random variables. These elements correspond to moments of the random surface. Here, there
is no requirement that the paths be space--like, allowing for convergence to Gaussians along time--like paths as well. The explicit
covariance formula is given in Theorem \ref{TimeLikePath}.
At first glance, it appears to be slightly different from the covariance formula for eigenvalues of random matrices. However, the process
here corresponds to Brownian motion (see e.g. \cite{Bi2,CD}). Indeed, after applying
the usual rescaling from Brownian motion to Ornstein--Uhlenbeck, the covariance from \cite{Bo} is
recovered.
\textbf{Acknowledgments}. The author would like to thank Alexei Borodin,
Alexey Bufetov, Philippe Biane and Ivan Corwin for enlightening discussions.
\section{Preliminaries}
Let us review some background about representation theory and non--commutative random walks. See \cite{B2} for
an introduction to non--commutative random walks.
\subsection{Representation Theory}
The universal enveloping algebra $\U$ is the unital algebra over $\C$ generated by $\{E_{ij},1\leq i,j\leq N\}$ with relations
$E_{ij}E_{kl}-E_{kl}E_{ij}=\delta_{jk}E_{il}-\delta_{il}E_{kj}$. It carries a natural $*$--operation induced from complex conjugation on $\C$. The coproduct
$\Delta: \U\rightarrow\U\otimes\U$ is the algebra morphism sending $E_{ij}$ to $E_{ij}\otimes 1 + 1\otimes E_{ij}$. There is a natural one--to--one correspondence between finite--dimensional $\U$--modules, finite--dimensional Lie algebra representations of $\mathfrak{gl}_N$, and finite--dimensional representations of the Lie group $G:=U(N)$.
Let $L^2(G)$ be the Hilbert space of square--integrable
complex--valued functions on $G$. Recall that by the
Peter--Weyl theorem, this Hilbert space has an orthogonal basis
given by the matrix coefficients of all irreducible
representations of $G$,
i.e.
$$
\{g \mapsto \eta(\pi_{\la}(g)\xi)\},
$$
where $\pi_{\la}$ runs over all irreducible representations of
$G$, $\{\xi\}$ runs over a basis for $V_{\la}$ and $\{\eta\}$
runs overs a basis for $V_{\la}^*$. Denote this basis by
$\{f_{\xi\eta}\}.$ Then there is a non--degenerate pairing
$\langle \cdot,\cdot\rangle$ between $\U$ and $L^2(G)$ given by
$$
\langle X, f_{\xi\eta} \rangle = \eta(X\xi).
$$
This can be heuristically understood as $\langle X,f\rangle = f(X),$ since $f_{\xi\eta}(g)=\eta(g\xi)$. This pairing defines
an injection $\U\hookrightarrow L^2(G)^*$. Let us review the
algebra structure of $L^2(G)^*$.
There is a co--algebra structure on $L^2(G)$ given by the
co--product $\Delta:\LG\rightarrow \LG\otimes\LG \cong
L^2(G\times G)$ defined by $\Delta(f)(x,y)=f(xy)$. The
multiplication $\mu$ on $\LG^*$ is the composition
$$
\LG^*\otimes \LG^* \stackrel{\rho}{\longrightarrow} (\LG\otimes
\LG)^* \stackrel{\Delta^*}{\longrightarrow} \LG^*,
$$
where $\rho(\phi\otimes\psi)(f\otimes h)=\phi(f)\psi(h)$. Use Sweedler's notation to write
$$\Delta(f) = \sum_{(f)} f_{(1)}\otimes f_{(2)}.$$
Evaluating both sides at $(x,y)\in G\times G$ shows
$$
f(xy)=\sum_{(f)} f_{(1)}(x) f_{(2)}(y) \text{ for all } x,y\in
G.$$
Then
$$
\mu(\phi\otimes\psi)(f)=\Delta^*\rho(\phi\otimes\psi)(f)=\rho(\phi\otimes\psi)(\Delta
f) = \sum_{(f)} \phi(f_{(1)})\psi(f_{(2)}).
$$
In particular, if $\phi_x\in \LG^*$ denotes evaluation at $x$,
i.e. $\phi_x(f)=f(x)$, then
$$
(\phi_x\phi_y)(f) = \sum_{(f)} \phi_x(f_{(1)})\phi_y(f_{(2)}) =
\sum_{(f)} f_{(1)}(x)f_{(2)}(y) = f(xy).
$$
So $\phi_x\phi_y=\phi_{xy}$. We also write $\phi_{X}(\cdot)$
for$\langle X,\cdot\rangle.$
With the pairing between $\U$ and $\LG$ above, define the action $\pi$ of $\U$ on $\LG$ by
$$
\pi(a): f \mapsto \langle \mathrm{id} \otimes a , \Delta f\rangle.
$$
The symbol $\pi$ will sometimes be repressed, in the sense that $af$ means $\pi(a)f$. Observe that $\pi$ preserves each summand in the Peter--Weyl decomposition
$\LG=\bigoplus_{\la} V_{\la}^{(1)}\oplus\cdots\oplus
V_{\la}^{(\dim(\la))}$. To see this, suppose we are given some matrix coefficient in an irreducible representation $V_{\lambda}$, that is, an $f\in \LG$ of the form
$$
f(g) = \langle gv,w\rangle \text{ for fixed } v,w\in V_{\lambda}.
$$
Then by the definition of the co--product
$$
\sum_{(f)} f_{(1)}(g_1)f_{(2)}(g_2) = \langle g_1g_2 v,w\rangle.
$$
Since $\langle X, f_{(2)}\rangle = f_{(2)}(X)$, we see that
\begin{equation}\label{Compare}
(\pi(X)f)(g) = \langle g\cdot Xv, w\rangle.
\end{equation}
Thus, $\pi(X)$ is of the form $\langle gv', w\rangle$ for $v',w\in V_{\lambda}$, so the summand is preserved. Letting be the von Neumann algebra consisting of the
elements of $\Hom_{\C}(\LG,\LG)$ which preserve each summand in
the Peter--Weyl decomposition, we have that $\pi$ sends $\U$ to $M$. From the definition of the co--product in $\U$, the $n$--th tensor power $\pi^{\otimes n}:\U\rightarrow M$
is defined by
$$
\pi^{\otimes n}(X) = \sum_{i=1}^n \Id^{\otimes i-1}\otimes \pi(X) \otimes \Id^{\otimes n-i}.
$$
In general, any Lie group $G$ acts on its Lie algebra $\mathfrak{g}$ via the adjoint action
$$
\mathrm{Ad}(g)x = gxg^{-1}, \quad g\in G,x\in\mathfrak{g}.
$$
This action extends naturally to $\U$. For a subgroup $K$ of $G$, let
$\U^K=\{x\in\U: \mathrm{Ad}(g)x=x \text{ for all } g\in K\}$. In particular,
$\U^G=Z(\U)$, the centre of $\U$.
Recall that the Harish--Chandra isomorphism identifies $Z(\U)$ with shifted symmetric polynomials.
Explicitly, each $X\in Z(\U)$ acts as some constant $p_X(\la)$ on the irreducible representation
$V_{\la}$. It turns out that $p_X$ is symmetric in the shifted variables $\la_i-i$.
\subsection{Non--commutative probability}
A non--commutative probability space $(\mathcal{A},\phi)$ is a unital $^*$--algebra
$\mathcal{A}$ with identity $1$ and a state $\phi:\mathcal{A}\rightarrow\C$, that is, a linear map such that $\phi(a^*a)\geq 0$ and $\phi(1)=1$. Elements of $\mathcal{A}$ are
called \textit{non--commutative random variables}.
This generalises a classical probability space, by considering $\mathcal{A}=L^{\infty}(\Omega,\mathcal{F},\mathbb{P})$ with $\phi(X)=\E_{\mathbb{P}}X$. We also need a notion of convergence. For a
large parameter $L$ and $a_1,\ldots,a_r\in\mathcal{A},\phi$ which depend on $L$, as well as a limiting space $(\mathbb{A},\Phi)$, we say that $(a_1,\ldots,a_r)$ converges to $\mathbf{(a_1,\ldots,a_r)}$ with respect to the state $\phi$ if
$$
\phi(a_{i_1}^{\epsilon_1}\cdots a_{i_k}^{\epsilon_k})\rightarrow \Phi(\mathbf{a_{i_1}^{\epsilon_1}\cdots a_{i_k}^{\epsilon_k}})
$$
for any $i_1,\ldots,i_k\in\{1,\ldots,r\},\epsilon_j\in\{1,*\}$ and $k\geq 1$.
There is also a non--commutative version of a Markov chain. If $X_n:(\Omega,\mathcal{F},\mathbb{P})\rightarrow E$ denotes the Markov process with transition operator $Q:L^{\infty}(E)\rightarrow L^{\infty}(E)$, then the Markov property is
$$
\E[Yf(X_{n+1})]=\E[Y Qf(X_n)]
$$
for $f\in L^{\infty}(E)$ and $Y$ a $\sigma(X_1,\ldots,X_n)$--measurable random variable. Letting $j_n:L^{\infty}(E)\rightarrow L^{\infty}(\Omega,\mathcal{F},\mathbb{P})$ be defined by $j_n(f)=f(X_n)$, we can write the Markov property as
$$
\E[j_{n+1}(f)Y]=\E[j_n(Qf)Y]
$$
for all $f\in L^{\infty}(E)$ and $Y$ in the subalgebra of $L^{\infty}(\Omega,\mathcal{F},\mathbb{P})$ generated by the images of $j_0,\dots,j_n$.
Translating into the non--commutative setting, we define a \textit{non--commutative Markov operator} to be a semigroup of
completely positive unital linear maps $\{P_t:t\in T\}$ from a $^*$--algebra $U$ to itself (not necessarily an algebra morphism). The set $T$ indexing time can be either
$\N$ or $\R_{\geq 0}$, that is to say, the Markov process can be either discrete or continuous time.
If for any times $t_0<t_1<\ldots\in T$ there exists algebra morphisms $j_{t_n}$ from $U$ to a non--commutative probability space $(W,\omega)$ such that
$$
\omega(j_{t_n}(f)w)=\omega(j_{t_{n-1}}(P_{t_n-t_{n-1}}f)w)
$$
for all $f\in U$ and $w$ in the subalgebra of $W$ generated by the images of $\{j_t:t\leq t_{n-1}\}$, then $j_t$ is called a \textit{dilation} of $P_t$.
\section{Non--commutative random walk on $\U$}
The first thing that needs to be done is to define states on $\U$. Note that is already has a natural $^*$--algebra structure,
Given any positive type, normalized (sending the identity to
$1$), class function $\kappa\in \LG$, we have the decomposition
$$\kappa = \sum_{\la\in \widehat{G}}\widehat{\kappa}(\la)\frac{\chi_{\la}}{\dim\la},$$ where
$\widehat{G}$ denotes the set of equivalence classes of irreducible representations of $G$,
and $\chi_{\la}$ are the characters corresponding to $\la$. By the orthogonality relations,
$\widehat{\chi_{\la}}(\la)=1$. This
defines a state $\kappa$ on $M$ by
$$
\kappa(X) = \sum_{\la} \widehat{\kappa}(\la)
\sum_{i=1}^{\dim\la} \Tr(X\vert_{V_{\la}^{(i)}}), \quad X\in M.
$$
This naturally pulls back via $\pi:\U\rightarrow M$ to a state $\kappa(\cdot)$ on $\U$.
Recall an equivalent definition of these states from \cite{BB}.
There is a canonical isomorphism
$D:U(\mathfrak{gl}(N))\rightarrow \mathcal{D}(N)$ where
$\mathcal{D}(N)$ is the algebra of left--invariant differential
operators on $U(N)$ with complex coefficients. Then the
state $\langle \cdot \rangle_{\kappa}$ on $U(\mathfrak{gl}(N))$
is defined by $$
\langle X\rangle_{\kappa} = D(X)\kappa(U)\vert_{U=I}.
$$
The state can be computed using the formula (see e.g. page 101
of \cite{kn:V}) for $X=E_{i_1j_1}\cdots E_{i_kj_k}$:
\begin{equation}\label{UsingTheFormula}
D(X)\kappa(U) =
\partial_{t_1}\cdots\partial_{t_k}\kappa\left(U e^{t_1E_{i_1j_1}}\cdots
e^{t_kE_{i_kj_k}}\right)\Big|_{t_1=\cdots=t_k=0}.
\end{equation}
Comparing \eqref{UsingTheFormula} and \eqref{Compare} shows that $D=\pi$.
Here, $e^{tE_{ij}}$ is just the usual exponential of matrices,
which has the simple expression
\begin{equation}\label{SimpleExpression}
e^{tE_{ij}} =
\begin{cases}
Id + tE_{ij}, \ \ i\neq j\\
Id + (e^t -1)E_{ii} \ \ i=j
\end{cases}
\end{equation}
Note that since \eqref{UsingTheFormula} only involves linear
terms in the $t_j$, one can replace $e^{tE_{ii}}$ with $Id +
tE_{ii}$ without changing the value of the right hand side of
\eqref{UsingTheFormula}.
This is a slightly different approach from \cite{BB}, which used the (equivalent) formula
$$
E_{ij} \mapsto \sum_k x_{ik}\partial_{jk}.
$$
It is not hard to see that the two definitions of $\state{\cdot}_{\kappa}$ are equivalent. For each $\la\in \widehat{G}$ and $X=\Eij$, and
letting $v_1,\ldots,v_d$ be a basis of $V_{\la}$,
\begin{eqnarray*}
\state{X}_{\chi_{\la}} &=& \partial_{t_1}\cdots\partial_{t_k}\chi_{\la}\left(e^{t_1E_{i_1j_1}}\cdots
e^{t_kE_{i_kj_k}}\right)\Big|_{t_1=\cdots=t_k=0}.\\
&=& \partial_{t_1}\cdots\partial_{t_k}\sum_{r=1}^d \Big\langle e^{t_1E_{i_1j_1}}\cdots
e^{t_kE_{i_kj_k}}v_r,v_r \Big\rangle\Big|_{t_1=\cdots=t_k=0}\\
&=& \sum_{r=1}^d \Big\langle \Eij v_r,v_r \Big\rangle\\
&=& \Tr\left(X\big|_{V_{\la}}\right)
\end{eqnarray*}
By linearity, this holds for all $\kappa$ and all $X$.
Now that the states have been defined, we define the non--commutative Markov process.
In order to define a continuous--time non--commutative Markov process, there needs to be
a semigroup $\{\kappa_t:t\geq 0\}$ in $\LG$. Indeed, such a semigroup exists: for any $t\geq 0$, let
$$
\kappa_t(U)=e^{t\Tr(U-\Id)}.
$$
Now fix times $t_1<t_2<\ldots$. Let $\mW$ be the infinite tensor product of von Neumann algebras $M^{\otimes\infty}$
with respect to the state $\omega=\kappa_{t_1}\otimes\kappa_{t_2-t_1}\otimes\kappa_{t_3-t_2}\otimes\ldots$.
For $n\geq 1$
define the morphism $j_{t_n}:\U\rightarrow\mW$ to be the map
$j_{t_n}(X)=\pi^{\otimes n}(X)\otimes \mathrm{Id}^{\otimes\infty},$
and let $\mW_n$ be the subalgebra generated by the images of
$j_{t_1},\ldots,j_{t_n}$. Define $P_t:\U\rightarrow\U$ by
$(\mathrm{id}\otimes \kappa_t)\circ\Delta$. Note that $P$ is
linear as a map of complex vector spaces, but is not an algebra
morphism, since the trace is not preserved under multiplication
of matrices.
To simplify notation, write $\state{\cdot}_t$ for $\state{\cdot}_{\kappa_t}$
and $j_n$ for $j_{t_n}$.
\begin{theorem}\label{QuantumTheorem}
(1) The maps $(j_n)$ are a dilation of the
non--commutative Markov operator $P_t$. In other words,
$$
\omega(j_n(X)w)=\omega(j_{n-1}(P_{t_n-t_{n-1}}X)w), \quad X\in
U(\mathfrak{gl}(N)), \quad w\in \mW_{n-1}.
$$
(2) The pullback of $\omega$ under $j_n$ is the state $\langle
\cdot \rangle$ on $U(\mathfrak{gl}(N))$, i.e. $\langle
X\rangle_{t_n} = \omega(j_n(X))$.
(3) For $n\leq m,$ we have
$$
\omega(j_n(X)j_m(Y))=\langle X\cdot P_{t_m-t_n}Y\rangle_{t_n}.
$$
(4) The non--commutative markov operators $P_t$ satisfy the
semi--group property $P_{t+s}=P_{t}\circ
P_{s}$.
(5) For any subgroup $K\subset U(N)$, the restriction of $P_t$ to $\U^K$ is still a
non--commutative transition kernel. In other words, $P_t \U^K\subset \U^K$.
In particular, $P_t Z(\U)\subset Z(\U)$.
\end{theorem}
\begin{proof}
(1) This is essentially identical to Proposition 3.1 from \cite{CD}. It is reproduced here
for completeness.
The left--hand--side is
\begin{multline*}
\omega((\pi^{\otimes n-1}\otimes\pi)\Delta X w) = \sum_{(X)} \omega(\pi^{\otimes n-1}(X_{(1)})\otimes \pi(X_{(2)}) w) \\
= \sum_{(X)} \omega(\pi^{\otimes n-1}(X_{(1)})w) \state{X_{(2)}}_{t_n-t_{n-1}}.
\end{multline*}
The right--hand--side is
$$
\sum_{(X)} \omega(j_{n-1}(\state{X_{(2)}}_{t_n-t_{n-1}}X_{(1)}w) = \sum_{(X)} \omega(j_{n-1}(X_{(1)})w)\state{X_{(2)}}_{t_n-t_{n-1}}.
$$
(2) Let $m_n$ denote the $n$--fold multiplication $\LG^{\otimes
n}\rightarrow\LG$ that sends $f_1\otimes\cdots \otimes f_n$ to
$f_1\cdots f_n$. We will show that
\begin{equation}\label{Eqn1}
m_n(\pi^{\otimes n}(X)(f_1\otimes\cdots\otimes
f_n))=\pi(X)(f_1\cdots f_n).
\end{equation}
The case of general $n$ follows inductively from $n=2$. The left hand side is
$$
\sum_{(X)} (\pi(X^{(1)})f_1) \cdot (\pi(X^{(1)})f_2) = \sum_{(X,f_1,f_2)} \langle X^{(1)}, f_1^{(2)} \rangle f_1^{(1)} \cdot \langle X^{(2)}, f_2^{(2)} \rangle f_2^{(1)}.
$$
The right hand side is
$$
\langle \mathrm{id} \otimes X, \Delta(f_1\cdot f_2) \rangle = \sum_{(f_1,f_2)} \langle X, f_1^{(2)}f_2^{(2)} \rangle f_1^{(1)} \cdot f_2^{(1)}.
$$
So it suffices to show that
$$
\sum_{(X)} \langle X^{(1)}, f_1^{(2)} \rangle \langle X^{(2)}, f_2^{(2)} \rangle = \langle X, f_1^{(2)}f_2^{(2)} \rangle.
$$
But this is just the definition of multiplication in a dual Hopf algebra. So \eqref{Eqn1} is true.
Now recall that if $A$ is a Hopf algebra with co--unit
$\epsilon:A\rightarrow\C$, then the $n$--fold tensor power
$A^{\otimes n}$ is also a Hopf algebra, with co--unit
$\epsilon^{(n)}:A^{\otimes n}\rightarrow \C$ defined by the
composition
$$
A^{\otimes n}\xrightarrow{m_n}A
\stackrel{\epsilon}{\longrightarrow}\C.
$$
In other words,
$$
\epsilon^{(n)}(a_1\otimes \cdots \otimes a_n)=\epsilon(a_1
\cdots a_n) = \epsilon(a_1)\cdots \ep(a_n).
$$
The second equality holds because $\ep$ is a morphism of
$\C$--algebras.
When $A=\LG,$ then $\ep:\LG\rightarrow\C$ is defined by
$\ep(f)=f(Id_G)$. I now claim that for $X\in \U$
\begin{equation}\label{Eqn2}
\omega(\pi^{\otimes n}(X)) =
\ep^{(n)}X(\kappa_{t_n}).
\end{equation}
For $n=1$, this follows immediately from the definitions. For
$n\geq 2$, write as usual
$$
\pi^{\otimes n}(X)=\sum_{(X)} X_{(1)}\otimes\cdots\otimes
X_{(n)} \in M^{\otimes n}.
$$
Then
\begin{eqnarray*}
\omega(\pi^{\otimes n}(X)) &=& \sum_{(X)}
\kappa_{t_1}(X_{(1)})\cdots\kappa_{t_n-t_{n-1}}(X_{(n)}) \\
&=& \sum_{(X)} \ep
X_{(1)}(\kappa_{t_1}) \cdots \ep X_{(n)}(\kappa_{t_n-t_{n-1}}).
\end{eqnarray*}
At the same time,
\begin{eqnarray*}
\ep^{(n)}X(\kappa_{t_n}) &=& \ep^{(n)}X(\kappa_{t_1}\cdots \kappa_{t_n-t_{n-1}}) \\
&=& \ep^{(n)}\sum_{(X)} X_{(1)}(\kappa_{t_1}) \otimes \cdots \otimes
X_{(n)}(\kappa_{t_n-t_{n-1}})\\
&=& \sum_{(X)} \ep X_{(1)}(\kappa_{t_1}) \cdots \ep
X_{(n)}(\kappa_{t_n-t_{n-1}}).
\end{eqnarray*}
So \eqref{Eqn2} is true.
Finally, we can combine the results to obtain
\begin{align*}
\omega(j_n(X))=\omega(\pi^{\otimes n}(X)) &=
\ep^{(n)}\pi^{\otimes n}X(\kappa_{t_1}\otimes\cdots\otimes \kappa_{t_n-t_{n-1}}) \\
&= \ep m_n(\pi^{\otimes
n}X(\kappa_{t_1}\otimes\cdots\otimes \kappa_{t_n-t_{n-1}}))\\
&=\ep \pi(X)(\kappa_{t_1}\otimes\cdots\otimes \kappa_{t_n-t_{n-1}}) =\langle X\rangle_{\kappa_{t_n}}.
\end{align*}
This is exactly what (2) stated.
(3) By repeated applications of (1),
$$\omega(j_n(X)j_m(Y))=\omega(j_n(X)j_n(P_{t_m-t_n}Y)).$$ Since $j_n$
is a morphism of algebras, this equals $\omega(j_n(X\cdot
P_{t_m-t_n}Y))$, which by (2) equals the right--hand--side of (3).
(4) By linearity, it suffices to prove this for monomials of the form
$E=\Eij$. Introduce some notation: let $K=\{1,\cdots,k\}$ and
for any subset $S\subseteq K$, define
$
E_S = \prod_{s\in S} E_{i_sj_s},
$
where the product is taken in increasing order. The term
$E_{\emptyset}$ is understood to be $1$. So, for example, if
$E=E_{13}E_{42}E_{55}E_{12}$ and $S=\{1,2,4\}$ then
$E_S=E_{13}E_{42}E_{12}$. With this notation,
$$
\Delta E = \sum_{S\subseteq K} E_S\otimes E_{K\backslash S}.
$$
And therefore
\begin{align*}
P_{t_2-t_1}E&=\sum_{S\subseteq K} \langle E_{K\backslash
S}\rangle_{t_2-t_1} E_S,\\
\quad P_{t_1}P_{t_2-t_1}E &=
\sum_{\substack{ S\subseteq K \\ R\subseteq S}}
\state{E_{S\backslash R}}_{t_1} \state{E_{K\backslash
S}}_{t_2-t_1} E_R.
\end{align*}
Since
$$
P_{t_2}E = \sum_{R\subseteq K}
\state{E_{K\backslash R}}_{t_2}E_R,
$$
it suffices to show
$$
\state{E_{K\backslash R}}_{t_2} = \sum_{R\subseteq
S\subseteq K} \state{E_{K\backslash S}}_{t_2-t_1}
\state{E_{S\backslash R}}_{t_1} \quad \text{for all }
R\subseteq K,
$$
or equivalently
$$
\state{E_{K}}_{t_2} = \sum_{S\subseteq K}
\state{E_{K\backslash S}}_{t_2-t_1} \state{E_{S}}_{t_1}.
$$
This follows from \eqref{UsingTheFormula} and the general
Leibniz rule applied to the derivatives of the product
$\kappa_{t_1}\cdot\kappa_{t_2-t_1}$.
(5) This is Proposition 4.3 from \cite{CD}. Here is also a bare--bones proof when $K=G$
Let $X\in Z(\U).$ The goal is to show that $P(X)Y=YP(X)$
for all $Y\in \U$. It suffices to show this when
$Y\in\mathfrak{g}$. In this case,
\begin{align*}
&XY=YX \Longrightarrow \Delta(XY)=\Delta(YX) \Longrightarrow
\Delta(X)\Delta(Y)=\Delta(Y)\Delta(X)\\
&\Longrightarrow \sum_{(X)} X_{(1)}Y\otimes X_{(2)} +
X_{(1)}\otimes X_{(2)}Y = \sum_{(X)} YX_{(1)}\otimes X_{(2)} +
X_{(1)}\otimes YX_{(2)}.
\end{align*}
Now apply the linear map $id \otimes \state{\cdot}$ to both
sides to get
$$
\sum_{(X)} \state{X_{(2)}}X_{(1)}Y + \state{X_{(2)}Y}X_{(1)} =
\sum_{(X)} \state{X_{(2)}}YX_{(1)} + \state{YX_{(2)}}X_{(1)}.
$$
Since the state $\state{\cdot}$ is tracial, the second summand
on both sides are equal. The first summand on the
left--hand--side is $P(X)Y$ while the first summand on the
right--hand--side is $YP(X)$, so $P(X)\in Z(\U)$ as needed.
\end{proof}
\section{Connections to classical probability}
In this section, we will show that restricting to the centres $Z(U(\mathfrak{gl}_1),$ $\ldots,Z(U(\mathfrak{gl}_N))$ reduces the non--commutative random walk to a (2+1)--dimensional random surface growth model. First, here is a description of the model, which was introduced in \cite{BF}.
\subsection{Random surface growth}
Consider the two--dimensional lattice $\Z\times\Z_+$. On each horizontal level $\Z\times\{n\}$ there are
exactly $n$ particles, with at most one particle at each lattice site. Let $X^{(n)}_1>\ldots>X^{(n)}_n$
denote the $x$--coordinates of the locations of the $n$ particles. Additionally, the particles need to
satisfy the \textit{interlacting property} $X^{(n+1)}_{i+1} < X^{(n)}_i \leq X^{(n+1)}_i.$
The particles can be viewed as a random stepped surface, see Figure \ref{Figure}. This can be made rigorous by defining the height function at $(x,n)$ to be the number of particles to the right of $(x,n)$.
\begin{figure}[H]
\captionsetup{width=0.8\textwidth}
\centering
\caption{ The particles as a stepped surface. The lattice is shifted to make the visualization easier.}
\includegraphics[height=5cm]{Surface.png}
\label{Figure}
\end{figure}
The dynamics on the particles are as follows. The initial condition is the \textit{densely packed} initial
condition, $\Lambda_i^{(n)}=-i+1,1\leq i\leq n$. Each particle has a clock with exponential waiting time of
rate $1$, with all clocks independent of each other. When the clock rings, the particle attempts to jump
one step to the right. However, it must maintain the interlacing property. This is done by having
particles push particles above it, and jumps are blocked by particles below it. One can think of lower
particles as being more massive. See Figure \ref{Jumps} for an example.
The projection to $\Z\times\{n\}$ is still Markovian, and is known as the \textit{Charlier process}
\cite{KOR}. It can be described by as a continuous--time Markov chain on $\Z^n$ with independent increments
$e_i/n,1\leq i\leq n$, (where $\{e_i\}$ is the canonical basis for $\Z^n$) conditioned to stay in the Weyl chamber $(x_1>x_2>\ldots>x_n)$. Equivalently, the
conditioned Markov chain is the Doob $h$--transform for some harmonic function $h$. There is a nice
description of $h$ in terms of representation theory, namely, $h(x_1,\ldots,x_n)$ is the dimension of the
irreducible representation of $\mathfrak{gl}_n$ with highest weight $(x_1,x_2+1,\ldots,x_n+n-1)$. Explicitly, $$
\dim\la = \prod_{i<j} \frac{\lambda_i-i-(\lambda_j-j)}{j-i}.
$$
Below, let $Q_t^{(N)}$ denote the Markov operator of this Markov chain.
\begin{figure}
\captionsetup{width=0.8\textwidth}
\caption{The red particle makes a jump. If any of the black particles attempt to jump, their jump is
blocked by the particle below and to the right, and nothing happens. White particles are not blocked.}
\centering
\includegraphics[height=6cm]{Jumps.png}
\label{Jumps}
\end{figure}
The construction of the full particle system is based on a general multi--variate construction from
\cite{BF}, which is based on \cite{DF}. Suppose there are two Markov chains with state spaces
$\mathcal{S}, \mathcal{S}^*$ and transition probabilities $P,P^*$. Also assume there is a Markov operator
$\Lambda:\mathcal{S}^*\rightarrow\mathcal{S}$ which intertwines with $P,P^*$ in the sense that $\Lambda P^* = P\Lambda$.
In other words, there is a commutative diagram
\begin{equation}\label{CD}
\begin{CD}
\mathcal{S}^* & @>P^*>> & \mathcal{S}^*\\
@VV\Lambda V & & @VV\Lambda V\\
\mathcal{S} & @>P>> & \mathcal{S}
\end{CD}
\end{equation}
\vspace*{1\baselineskip}
Then the state space is $\{(x^*,x)\in \mathcal{S}^*\times\mathcal{S}: \Lambda(x^*,x)\neq 0\}$ with transition probabilities
$$
\mathrm{Prob}((x^*,x)\rightarrow (y^*,y)) =
\begin{cases}
\frac{P(x,y)P^*(x^*,y^*)\Lambda(y^*,y)}{\Delta(x^*,y)},& \quad \Delta(x^*,y)\neq 0\\
0,& \quad \Delta(x^*,y)=0
\end{cases}
$$
Additionally, if the intial condition is a \textit{Gibbs measure}, that is, a probability distribution of the form $\mathbb{P}(x^*)\Lambda(x^*,x)$, then the dynamics preserves Gibbs measures. All constructions and definitions extend naturally to any finite number of Markov chains.
Here, $Q_t^{(N)}$ and $Q_t^{(N-1)}$ will play the roles of $P^*,P$, and the projection $\Lambda$ is
$$
\Lambda(x_1>\ldots>x_N,y_1>\ldots>y_{N-1}) = \frac{h(y)}{h(x)}.
$$
The construction implies that
\begin{equation}\label{Gibbs}
\mathbb{P}(X^{(N)}(t) = x^{(N)} \vert X^{(M)}(t) = x^{(M)}) = \frac{h(x^{(N)})}{h(x^{(M)})} , \forall N\leq M, t\geq 0
\end{equation}
\begin{multline}\label{Push}
\mathbb{P}(X^{(N)}(t) = x^{(N)} \vert X^{(M)}(s) = y^{(M)},X^{(N)}(s) = y^{(N)}) \\
= \mathbb{P}(X^{(N)}(t) = x^{(N)} \vert X^{(N)}(s) = y^{(N)}), \forall N\leq M,s\leq t
\end{multline}
The intuition behind \eqref{Push} is that since particles on lower levels push and block the particles on higher levels, the evolution of the $N$--th level is independent of the evolution $M$--th level. Equation \eqref{Gibbs} is a mathematical formulation of the statement that the dynamics preserves Gibbs measures.
\subsection{Restriction to centre}
Before continuing, we need to compute the states of certain
observables.
\begin{proposition}\label{Formula}
Let $\Pi$ denote the set of partitions of the set $\{1,\ldots,m\}$, let $\left| \pi\right|$
denote the number of blocks of the partition $\pi\in \Pi$ and let $B\in\pi$ mean that $B$ is
a block in $\pi$. Then
$$
\langle E_{i_1j_1}\cdots E_{i_mj_m}\rangle_t = \sum_{\pi \in\Pi}
t^{\left| \pi\right|}\prod_{\substack{ B\in\pi \\ B=\{b_1\ldots,b_k \}}}
1_{j_{b_1}=i_{b_2},j_{b_2}=i_{b_3},\ldots,j_{b_k}=i_{b_1}}
$$
\end{proposition}
\begin{example}
$$
\state{E_{21}E_{12}E_{21}E_{12}}_t = 2t^2+t
$$
with two contributing partitions having two blocks: $\{1,2\}\cup\{3,4\},\{1,4\}\cup\{2,3\}$,
and one contributing partition having one block $\{1,2,3,4\}$.
\end{example}
\begin{example}
$$
\state{E_{11}^3E_{22}}_t=t^4+3t^3+t^2
$$
with one contributing partition having four blocks:$\{1\}\cup\{2\}\cup\{3\}\cup\{4\},$
three contributing partitions having three blocks: $\{1,2\}\cup\{3\}\cup\{4\},\{1,3\}\cup\{2\}\cup\{4\},\{2,3\}\cup\{1\}\cup\{4\}, $
and one contributing partition having two blocks: $\{1,2,3\}\cup\{4\}.$
\end{example}
\begin{example} For any $m$,
$$
\state{E_{jj}^m}_t = B_m(t)
$$
where $B_m(t)$ is the $m$--th Bell polynomial. These are also the moments of a Poisson
random variable with mean $t$, so under the state $\state{\cdot}_t$, each $E_{jj}$ can be
heuristically understood to be distributed as Poisson(t).
\end{example}
\begin{example} $$
\state{E_{11}E_{12}}_t=0
$$
with no contributing partitions.
\end{example}
\begin{proof}
Recall the defintion of $\state{\cdot}_t$ in \eqref{UsingTheFormula}. By Faa di Bruno formula,
$$
\langle E_{i_1j_1}\cdots E_{i_mj_m}\rangle_t = \sum_{\pi\in \Pi} f^{(|\pi|)}(y)\prod_{B\in\pi}\frac{\partial^{|B|}y}{\prod_{b\in B}\partial x_b}\Bigg|_{x_1=\cdots=x_m=0}
$$
where
$$
f(y)=e^{ty}, \quad y = Tr(e^{x_1E_{i_1j_1}}\cdots e^{x_mE_{i_mj_m}}-Id).
$$
Note that
$$
f^{(|\pi|)}(y)\Bigg|_{x_1=\cdots=x_m=0} = t^{|\pi|}f(y)\Bigg|_{y=0}=t^{|\pi|}.
$$
Since we are taking the derivative with respect to $x_b$ and setting equal to $0$,
we only need the linear terms in $x_b$, so it is equivalent to replace $y$ with
$$
y = Tr((Id+x_1E_{i_1j_1})\cdots (Id+ x_mE_{i_mj_m})-Id).
$$
Here, $E_{ij}$ are the usual $N\times N$ matrices acting on $\C^N$, not the generators of $\U$.
Expanding the parantheses, all terms other than $Tr\left(\prod_{b\in B} x_bE_{i_bj_b}\right)$ do not contribute,
since these do not survive differentiation with respect to $x_b,b\in B$. Finally, since
$$
Tr\left(\prod_{\substack{ b\in B \\ B=\{b_1\ldots,b_k \}}} E_{i_bj_b}\right)=
1_{j_{b_1}=i_{b_2},j_{b_2}=i_{b_3},\ldots,j_{b_k}=i_{b_1}},
$$
the proof is finished.
\end{proof}
In section 7 of \cite{GKLLRT}, explicit generators of the centre $Z(\U)$ were found. See also chapter 7 of \cite{kn:M} for an exposition.
Let $\mathcal{G}_m$ denote the directed graph with vertices and edges
$$
\{1,\ldots,m\} \quad \{(i,j):1\leq i,j\leq m\}.
$$
Let $\Pi_k^{(m)}$ denote the set of all paths in $\mathcal{G}_m$ of length $k$ which start and
end at the vertex $m$. For $\pi\in \Pi_k^{(m)}$ let $r(\pi)$ denote the length of the first
return to $m$. Let $E(\pi)\in U(\mathfrak{gl}_m)$ denote the element with coefficient $r(\pi)$ obtained by taking
the product when labeling
the edge $(i,j)$ with $E_{ij}$ when $i\neq j$, and the edge $(i,i)$ with $E_{ii}-m+1$.
For example, the path
$$
\pi=\{5\rightarrow 3\rightarrow 3\rightarrow 1\rightarrow 5\rightarrow 5\rightarrow 2\rightarrow 5\}
$$
is in $\Pi^{(5)}_7$ with $r(\pi)=4$ and
$$
E(\pi)=4E_{53}(E_{33}-4)E_{31}E_{15}(E_{55}-4)E_{52}E_{25}.
$$
Define the elements
$$
\Psi_k := \sum_{m=1}^N \sum_{\pi\in \Pi_k^{(m)}} E(\pi) \in U(\mathfrak{gl}_N).
$$
For example,
$$
\Psi_1 = \sum_{m=1}^N (E_{mm}-m+1), \quad \Psi_2 = \sum_{m=1}^N (E_{mm}-m+1)^2 + 2\sum_{1\leq l<m\leq N} E_{ml}E_{lm}.
$$
When we wish to emphasize that $\Psi_k\in U(\mathfrak{gl}_N)$, the notation $\Psi_{k}^{(N)}$ will be used.
\footnote{Caution: This notation is consistent with notation from integrable probability but different from
notation in representation theory.}
\begin{theorem}\label{Gelfand} \cite{GKLLRT}
The centre $Z(\U)$ is generated by the elements $1,\{\Psi_k\}_{k\geq 1}$. Furthermore, the Harish--Chandra isomorphism maps $\Psi_k$
to the shifted symmetric polynomial $\sum_{m=1}^N (\la_m-m+1)^k$.
\end{theorem}
\begin{remark} Writing $\mathfrak{gl}_N = \mathfrak{n}_- \oplus \mathfrak{h} \oplus \mathfrak{n}_+$
where $\mathfrak{n}_+,\mathfrak{n}_-$ are the upper and lower nilpotent subalgebras and
$\mathfrak{h}$ is the diagonal subalgebra, the Harish--Chandra homomorphism is the projection
$$
\U = (\mathfrak{n}_-\U + \U\mathfrak{n}_+)\oplus U(\mathfrak{h})\rightarrow U(\mathfrak{h})=S(\mathfrak{h})=\C[\la_1,\ldots,\la_N].
$$
This sends
$$
\Psi_k = \sum_{m=1}^N (E_{mm}-m+1)^k + (\text{other terms}) \mapsto \sum_{m=1}^N (E_{mm}-m+1)^k = \sum_{m=1}^N (\la_m-m+1)^k.
$$
Of course, $\sum_{m=1}^N (E_{mm}-m+1)^k$ is in general not central.
\end{remark}
Now it is time to explicitly state the relationship between the non--commutative random walk and the growing stepped surface. One may be tempted to think that
$$
\begin{CD}
\U & @>P_t>> & \U\\
@AAA & & @AAA\\
U(\mathfrak{gl}_{N-1}) & @>P_t>> & U(\mathfrak{gl}_{N-1})
\end{CD}
$$
\vspace*{1\baselineskip}
\noindent is a non--commutative version of \eqref{CD}. However, care needs to be taken because the inclusion map
does not send $Z(U(\mathfrak{gl}_{N-1}))$ to $Z(\U)$.
A slight change of variables will make statements cleaner. If $p(\lambda)$ is a shifted symmetric
polynomial, then by definition it is symmetric in the variables $x_i=\lambda_i-i+1$, and
let $\bar{p}(x)$ denote the corresponding symmetric polynomial.
\begin{proposition} If $Y\in Z(\U)$ is sent to the symmetric polynomial $p_Y(x)$ by the Harish--Chandra
isomorphism, then
$$
\state{Y}_t = \E \left[\bar{p}_Y(X^{(N)}_1(t),\ldots,X^{(N)}_N(t))\right].
$$
\end{proposition}
\begin{proof}
This is not new, see \cite{BB}, but the proof is similar to
Theorem \ref{TwoLevels} below, so will be repeated for clarity. By a result from \cite{BK},
$$
e^{t\Tr (U-\Id)} = \sum_{\lambda} \mathrm{Prob}(X_i^{(N)}(t) = \lambda_i-i+1,1\leq i\leq N)\frac{\chi_{\la}(U)}{\dim\la}
$$
where $\chi_{\la}$ and $\dim\la$ are the character and dimension of the highest weight representation $\la$.
Thus, by linearity,
\begin{align*}
\state{Y}_t &= \sum_{\la} \mathrm{Prob}(X_i^{(N)}(t) = \lambda_i-i+1,1\leq i\leq N) \frac{\state{Y}_{\chi_{\la}}}{\dim\la}\\
&= \sum_{\la} \mathrm{Prob}(X_i^{(N)}(t) = \lambda_i-i+1,1\leq i\leq N) p_Y(\la_1,\ldots,\la_N)\\
&= \sum_x \mathrm{Prob}(X_i^{(N)}(t) = x_i,1\leq i\leq N)\bar{p}_Y(x_1,\ldots,x_N)
\end{align*}
The last line is simply the right--hand--side of the proposition.
\end{proof}
\begin{proposition}\label{OneLevel} Suppose that $P_t$ and $Q_t$ are two semigroups which preserve $Z(\U)$ and satisfy Theorem \ref{QuantumTheorem}(1),
then $P_tX=Q_tX$ for all $X\in Z(\U)$. In particular, $P_t$ is the Markov operator of the process
$(X_1^{(N)}(t)>\ldots>X_N^{(N)}(t))$.
\end{proposition}
\begin{proof}
Theorem \ref{QuantumTheorem}(1) and (2) imply that $\state{P_tX}_s = \state{Q_tX}_s = \state{X}_{t+s}$
for all $s,t\geq 0$. In order to show $P_tX=Q_tX$, it suffices to show that if $Y\in Z$
satisfies $\state{Y}_t=0$ for all $t\geq 0$, then $Y=0$. Suppose this is not true, and let
$Y$ be a counterexample of minimal degree. But then $\state{P_tY}_s=\state{Y}_s=0$ and
Theorem \ref{Gelfand} implies that $ P_tY = Y + t(\text{lower degree terms})$. By assumption,
$P_tY-Y\in Z$ also satisfies $\state{P_tY-Y}_s$ for all $s\geq 0$. Thus, since
$Y$ is of minimal degree, $P_tY-Y=0$.
If $Y$ has degree $d>1$, then by Theorem \ref{Gelfand} the term $E_{11}^d$ appears in $Y$. Thus
$P_tY$ has a $tdE_{11}^{d-1}$ term which cannot cancel with any term in $Y$. If $Y$ has degree
$1$ then $Y=a_1\Psi_1+a_0$ and $P_tY-Y = a_1tN$. Thus, there is a contradiction,
so no such $Y$ can exist.
The second art of the proposition follows if we show that $Q_t$ preserves shifted symmetric polynomials.
But this follows because the process is the Doob $h$--transform of a random walk which is invariant under
permuting the co--ordinates, and $h$ is anti--symmetric.
\end{proof}
\begin{example}
For $N=2$, one can explicitly compute (after a long calculation)
\begin{multline*}
P_t\Psi_4=\Psi_4 + 4t\Psi_3 + (6t^2+8t)\Psi_2 + 2t\Psi_1^2 \\
+ (4t^3+24t^2+10t)\Psi_1 + (2t^4+24t^3+38t^2+6t).
\end{multline*}
For instance, the only appearance of the monomial $E_{11}E_{22}$ in the right hand side is in $\Psi_1^2$. The only monomial in $\Psi_4$ that can lead to $E_{11}E_{22}$ is
$4E_{22}E_{21}E_{11}E_{12}$. The co--product $\Delta(E_{ij})=1\otimes E_{ij} + E_{ij}\otimes 1$ sends $E_{ij}$ either to the left tensor factor or the right tensor factor. In order to get $E_{11}E_{22}$, we must send $E_{22}E_{11}$ to the left and $E_{21}E_{12}$ to the right. Since $\state{E_{21}E_{12}}_t=t,$ the coefficient of $E_{22}E_{11}$ in $P_t\Psi_4$ must be $4t$. Since the coefficient of $E_{22}E_{11}$ in $\Psi_1^2$ is $2$, this implies that the coefficient of $\Psi_1^2$ in $\Psi_4$ is $2t$. Similar considerations can be applied to produce the other terms.
Now, evaluating this at $(4,2)$ with $t=3$ would predict that $$P_t\Psi_4(4,2)=257+12*65+78*17+6*5^2+354*5 + 1170=5453.$$
And indeed, the explicit determinantal formula from \cite{BF} for $N=2$ yields
$$
\frac{\sum_{b=x}^{\infty}\sum_{a=y}^b (b^k+(a-1)^k)(b-a+1)t^{b+a}\det\left[\begin{array}{cc}
(b-x)!^{-1} & (b-(y-1))!^{-1} \\
(a-1-x)!^{-1} & (a-y)^{-1} \end{array}\right]}{\sum_{b=x}^{\infty}\sum_{a=y}^b (b-a+1)t^{b+a}\det\left[\begin{array}{cc}
(b-x)!^{-1} & (b-(y-1))!^{-1} \\
(a-1-x)!^{-1} & (a-y)^{-1} \end{array}\right]}.
$$
A numerical computation at $(x,y)=(4,2),t=3,k=4$ with the sum from $b=x$ to $b=50\approx\infty$ yields
5452.999999999999999999999999999999418... Note that it is not obvious from the summation that the answer would even be an integer.
\end{example}
\begin{example}For general $N$,
\begin{multline*}
P_t\Psi_1=\Psi_1+tN, \quad P_t\Psi_2=\Psi_2+2t\Psi_1+(t^2+Nt)N, \\
P_t\Psi_3=\Psi_3+3t\Psi_2+3(t^2+Nt)\Psi_1+N(t^3+3t^2N+\frac{1}{2}t(N^2+1)).
\end{multline*}
One can check explicitly that the semigroup property holds.
\end{example}
\begin{example}\label{P4} We wish to take asymptotics $N\approx \eta L$ and $t\approx \tau L$. We would get
\begin{align*}
P_t\Psi_{1,N} &= \Psi_1+\const, \\
P_t\Psi_{2,N} &=\Psi_2+2\tau L\Psi_1+\const, \\
P_t\Psi_{3,N} &= \Psi_3+3\tau L\Psi_2+3(\tau^2+\eta\tau)L^2\Psi_1+\const\\
P_t\Psi_{4,N} &= \Psi_4 + 4\tau L\Psi_3 + (6\tau^2+4\tau\eta)L^2\Psi_2 + (4\tau^3 + 12\tau^2\eta+2\tau\eta^2)L^3\Psi_1 \\
& \quad + 2\tau L \Psi_1^2 + \const\\
P_t\Psi_{1,N}^2 &= \Psi_{1,N}^2 + 2\eta\tau L^2\Psi_{1,N} + \const
\end{align*}
Again, one can check that the semigroup property holds.
\end{example}
\begin{theorem}\label{TwoLevels}
Suppose $Y_1\in Z(U(\mathfrak{gl}_{N_1})),\ldots, Y_r\in Z(U(\mathfrak{gl}_{N_r}))$ are mapped to the
symmetric polynomials $\bar{p}_{Y_1},\ldots\bar{p}_{Y_r}$ under the Harish--Chandra isomorphism. Assume
that $N_1\geq\ldots\geq N_r$ and $t_1\leq \ldots \leq t_r$. Then
$$
\state{Y_1P_{t_2-t_1}Y_2\cdots P_{t_r-t_1}Y_r}_{t_1}=\mathbb{E}\left[\bar{p}_{Y_1}(X^{(N_1)}(t_1))\cdots\bar{p}_{Y_r}(X^{(N_r)}(t_r))\right].
$$
\end{theorem}
\begin{proof}
In order to simplify notation and elucidate the idea of the proof, assume $r=2$. The more general case follows from exactly the same argument.
First prove it for $t_1=t_2$. Assume $N_1=N\geq M=N_2$. Let $m(\la,\mu)$ denote the multiplicity of $\mu$ in the restricted representation $V_{\la}\Big|_{U(M)}$. Use $\bar{m}(\cdot,\cdot)$ to denote the same quantity in the shifted co--ordinates $x_i=\la_i-i+1$. Then by the Gibbs property, that is \eqref{Gibbs},
\begin{align*}
\mathrm{RHS}
&= \sum_{x^{(N)},x^{(M)}} \mathrm{Prob}(X^{(N)}(t)=x^{(N)}_i,X^{(M)}=x^{(M)}_j) \bar{p}_{Y_1}(x^{(N)})\bar{p}_{Y_2}(x^{(M)})\\
&= \sum_{x^{(N)},x^{(M)}} \mathrm{Prob}(X^{(N)}(t)=x^{(N)}_i)\frac{\bar{m}(x^{(N)},x^{(M)})h(x^{(M)})}{h(x^{(N)})} \bar{p}_{Y_1}(x^{(N)})\bar{p}_{Y_2}(x^{(M)})\\
\end{align*}
At the same time,
$$
\state{Y_1Y_2}_t = \sum_{\la^{(N)}} \mathrm{Prob}(X^{(N)}(t)=\la^{(N)}_i-i+1) \frac{1}{\dim \la^{(N)}} \mathrm{Tr}(Y_1Y_2\Big|_{V_{\la^{(N)}}}).
$$
Since $Y_1$ is central, it acts as $p_{Y_1}(\la^{(N)})\Id$ on $V_{\la^{(N)}}$, so this equals
$$
\sum_{\la^{(N)}} \mathrm{Prob}(X^{(N)}(t)=\la^{(N)}_i-i+1) \frac{p_{Y_1}(\la^{(N)})}{\dim \la^{(N)}} \mathrm{Tr}(Y_2\Big|_{V_{\la^{(N)}}}).
$$
By restricting $V_{\la^{(N)}}$ to $U(M)$ and using that $Y_2$ acts as $p_{Y_2}(\la^{(M)})\Id$ on $V_{\la^{(M)}}$, we get
$$
\sum_{\la^{(N)},\la^{(M)}} \mathrm{Prob}(X^{(N)}(t)=\la^{(N)}_i-i+1) \frac{m(\la^{(N)},\la^{(M)})\dim\la^{(M)}}{\dim \la^{(N)}} p_{Y_1}(\la^{(N)}) p_{Y_2}(\la^{(M)}).
$$
This is equal to the right--hand--side from above.
Now consider when $t=t_1\leq t_2=s$. Write $P_{s-t}Y_2$ as a sum over basis elements, that is $P_{s-t}Y_2=\sum_{\rho} c_{\rho}Y_{\rho}$. Then
\begin{align*}
\state{Y_1P_{s-t}Y_2}_t &= \sum_{\rho} c_{\rho}\state{Y_1Y_{\rho}}_t \\
&= \sum_{\rho}c_{\rho}\mathbb{E}\left[\bar{p}_{Y_1}(X^{(N)}(t))\bar{p}_{Y_{\rho}}(X^{(M)}(t))\right]\\
&= \E\left[\bar{p}_{Y_1}(X^{(N)}(t))(P_{s-t}\bar{p}_{Y_2})(X^{(M)}(t)) \right]
\end{align*}
Thus, it suffices to prove that
$$
\E\left[\bar{p}_{Y_1}(X^{(N)}(t))\bar{p}_{Y_2}(X^{(M)}(s)) \right]=\E\left[\bar{p}_{Y_1}(X^{(N)}(t))(P_{s-t}\bar{p}_{Y_2})(X^{(M)}(t)) \right]
$$
We have
\begin{align*}
&\E\left[\bar{p}_{Y_1}(X^{(N)}(t))\bar{p}_{Y_2}(X^{(M)}(s)) \right]\\
&=\sum_{y^{(N)},x^{(M)}} \bar{p}_{Y_1}(y^{(N)})\bar{p}_{Y_2}(x^{(M)}) \mathbb{P}(X^{(M)}(s) = x^{(M)},X^{(N)}(t) = y^{(N)})\\
&=\sum_{y^{(N)},y^{(M)},x^{(M)}} \bar{p}_{Y_1}(y^{(N)})\bar{p}_{Y_2}(x^{(M)}) \mathbb{P}(X^{(M)}(s) = x^{(M)},X^{(N)}(t) = y^{(N)},X^{(M)}(t) = y^{(M)})\\
&=\sum_{y^{(N)},y^{(M)},x^{(M)}} \bar{p}_{Y_1}(y^{(N)})\bar{p}_{Y_2}(x^{(M)}) \mathbb{P}(X^{(M)}(s) = x^{(M)}\vert X^{(N)}(t) = y^{(N)},X^{(M)}(t) = y^{(M)})\\
& \quad \quad \quad \quad \quad \times\mathbb{P}( X^{(N)}(t) = y^{(N)},X^{(M)}(t) = y^{(M)})
\end{align*}
By \eqref{Push} and the fact that $P_t=Q_t$, this then equals
\begin{align*}
&=\sum_{y^{(N)},y^{(M)},x^{(M)}} \bar{p}_{Y_1}(y^{(N)})\bar{p}_{Y_2}(x^{(M)}) \mathbb{P}(X^{(M)}(s) = x^{(M)}\vert X^{(M)}(t) = y^{(M)})\\
&\quad \quad \quad \quad \quad \times \mathbb{P}( X^{(N)}(t) = y^{(N)},X^{(M)}(t) = y^{(M)}))\\
&=\sum_{y^{(N)},y^{(M)}} \bar{p}_{Y_1}(y^{(N)})(P_{s-t}\bar{p}_{Y_2})(y^{(M)}) \mathbb{P}( X^{(N)}(t) = y^{(N)},X^{(M)}(t) = y^{(M)}))\\
&=\E\left[\bar{p}_{Y_1}(X^{(N)}(t))(P_{s-t}\bar{p}_{Y_2})(X^{(M)}(t)) \right]
\end{align*}
\end{proof}
We wrap up this section by giving an example showing that although $P_t=Q_t$ on $Z(\U)$, they are not equal on subalgebras
generated by different $Z(\U)$. The determinantal formula from \cite{BF} yields
$$
Q_1(\Psi_1^{(2)}\Psi_1^{(1)})(\lambda^{(2)},\lambda^{(1)})\approx 2.37\ldots, \text{ when } \lambda^{(2)}=(1,0),\lambda^{(1)}=(0).
$$
However, $$P_{t}(\Psi_1^{(2)}\Psi_1^{(1)})=\Psi_1^{(2)}\Psi_1^{(1)}+2t\Psi_1^{(1)}+t\Psi_1^{(2)}+2t^2+t,$$
and when evaluated at $\lambda^{(2)}=(1,0),\lambda^{(1)}=(0),t=1$ yields $3$.
\section{Covariance Structure}
In this section, it will be shown that the central elements are asymptotically Gaussian with an explicit
covariance that generalizes the Gaussian free field. Let us review some previously known results.
\begin{theorem}\label{SpaceLikePath}
\cite{BB,BF} Suppose $N_j=\lfloor \eta_j L\rfloor, t_j=\tau_j L$ for $1\leq j\leq r$. Assume they lie on a \textit{space--like path}, that is $N_1\geq\ldots\geq N_r$ and $t_1\leq\ldots \leq t_r$. Then as $L\rightarrow\infty$,
$$
\left(\frac{\Psi_{k_1}^{(N_1)}-\state{\Psi_{k_1}^{(N_1)}}_{t_1}}{L^{k_1}},\ldots,\frac{P_{t_r-t_1}\Psi_{k_r}^{(N_r)}-\state{P_{t_r-t_1}\Psi_{k_r}^{(N_r)}}_{t_1}}{L^{k_r}}\right) \rightarrow (\xi_1,\ldots,\xi_r),
$$
where the convergence is with respect to the state $\state{\cdot}_{t_1}$, and $(\xi_1,\ldots,\xi_r)$ is a Gaussian vector with covariance
$$
\displaystyle\E[\xi_i\xi_j]= \left(\frac{1}{2\pi i}\right)^2\iint_{\vert z\vert>\vert w\vert}\limits (\eta_i z^{-1} + \tau_i + \tau_i z)^{k_i} (\eta_j w^{-1} + \tau_j + \tau_j w)^{k_j} (z-w)^{-2}dzdw.
$$
\end{theorem}
The proof uses that the particle system is a determinantal point process along space--like paths. This
condition is necessary due to the construction using \eqref{CD}. In particular, there are no maps going up
from $S$ to $S^*$. A natural question is to ask what happens along time--like paths, that is, $N_1\leq N_2,t_1\leq t_2$. The main theorem
is
\begin{theorem}\label{TimeLikePath}
Suppose $N_j=\lfloor \eta_j L\rfloor, t_j=\tau_j L$ for $1\leq j\leq r$. Assume $\min(\ga_1,\ldots,\ga_r)=\ga_1$. Then as $L\rightarrow\infty$
$$
\left(\frac{\Psi_{k_1}^{(N_1)}-\state{\Psi_{k_1}^{(N_1)}}_{t_1}}{L^{k_1}},\ldots,\frac{P_{t_r-t_1}\Psi_{k_r}^{(N_r)}-\state{P_{t_r-t_1}\Psi_{k_1}^{(N_1)}}_{t_1}}{L^{k_r}}\right) \rightarrow (\xi_1,\ldots,\xi_r),
$$
where the convergence is with respect to the state $\state{\cdot}_{t_1}$, and $(\xi_1,\ldots,\xi_r)$ is a Gaussian vector with covariance
\[
\displaystyle\E[\xi_i\xi_j]=
\begin{cases}
\displaystyle\left(\frac{1}{2\pi i}\right)^2\iint_{\vert z\vert>\vert w\vert}\limits (\eta_i z^{-1} + \tau_i + \tau_i z)^{k_i} (\eta_j w^{-1} + \tau_j + \tau_j w)^{k_j} (z-w)^{-2}dzdw,&\\
\hspace{3in} \eta_i\geq\eta_j,\ga_i\leq\ga_j&\\
\displaystyle\left(\frac{1}{2\pi i}\right)^2\iint_{\vert z\vert>\vert w\vert}\limits (\eta_j\frac{\tau_j}{\tau_i} z^{-1} + \tau_j + \tau_i z)^{k_j} (\eta_i w^{-1} + \tau_i + \tau_i w)^{k_i} (z-w)^{-2}dzdw,&\\
\hspace{3in} \eta_i<\eta_j,\ga_i\leq\ga_j&
\end{cases}
\]
\end{theorem}
\begin{example}\label{Ex}
The double integral can be computed using resides and the Taylor series
$$(z-w)^{-2}=z^{-2}\left(1 + 2\frac{w}{z} + 3\frac{w^2}{z^2} + \ldots\right).$$
So for instance,
\begin{align*}
&\state{ \left(\frac{\Psi_{1}^{(\eta_1 L)}-\state{\Psi_{1}^{(\eta_1 L)}}_{\ga_1 L}}{L^{1}}\cdot\frac{P_{(\ga_2-\ga_1)L}\Psi_{1}^{(\eta_2 L)}-\state{P_{(\ga_2-\ga_1)L}\Psi_{1}^{(\eta_2 L)}}_{\ga_1 L}}{L^{1}}\right) }_{\ga_1 L}\\
&\rightarrow \ga_1\min(\eta_1,\eta_2).
\end{align*}
This can be checked using Proposition \ref{Formula}. Assume without loss of generality that $\eta:=\eta_1\leq \eta_2$ and set $\ga=\ga_1$. Since $\state{E_{ii}E_{jj}}= \state{E_{ii}}\state{E_{jj}}$ for $i\neq j$, then
\begin{align*}
& \quad \lim_{L\rightarrow\infty} L^{-2}\state{\left(\sum_{i=1}^{\lfloor \eta_1 L \rfloor } (E_{ii}-\ga_1 L)\right)\left(\sum_{j=1}^{\lfloor \eta_2 L\rfloor} (E_{jj} - \ga_1 L)\right)}_{\ga_1 L}\\
& =\lim_{L\rightarrow\infty} L^{-2}\state{\left(\sum_{i=1}^{\lfloor \eta_1 L \rfloor } E_{ii}-\eta_1 \ga_1 L^2\right)\left(\sum_{j=1}^{\lfloor \eta_1 L\rfloor} E_{jj} - \eta_1 \ga_1 L^2\right)}_{\ga_1 L}\\
&= \lim_{L\rightarrow\infty} \frac{\left( \eta L(\tau^2L^2 + \tau L) +\eta L(\eta L - 1)(\tau L)^2 -2\eta\tau\cdot\eta\tau L^4 + \eta^2\tau^2 L^4 \right)}{L^2}\\
&= \eta\tau
\end{align*}
\end{example}
The remainder of this section will prove Theorem \ref{TimeLikePath}. By Theorem 5.1 of \cite{BB} and the
fact that $P_t$ preserves the centre, it is immediate that convergence to a Gaussian vector holds. It only
remains to compute the covariance.
From the presence of $\Psi_1^2$ in $P_t\Psi_4$, it is necessary to understand products of $\Psi_k$. Heuristically,
if $\Psi_1 \approx cL^2 + \xi L$, where $\xi$ is a Gaussian random variable, then
$\Psi_1^2 \approx c^2L^4+2c\xi L^3$. Here are two examples which demonstrate this:
\begin{example}
Since
$$
\lim_{L\rightarrow\infty}L^{-2}\state{\Psi_1^{(\eta L)}}_{\tau L} = (\tau \eta -\frac{1}{2} \eta^2),
$$
the heuristics would predict that
\begin{align*}
& \quad \quad \lim_{L\rightarrow\infty} \state{\frac{\left(\Psi_1^{(\eta L)}-\state{\Psi_1^{(\eta L)}}_{\tau L}\right)\left(\left[\Psi_1^{(\eta L)}\right]^2-\state{\left[\Psi_1^{(\eta L)}\right]^2}_{\tau L}\right)}{L^4}}_{\tau L} \\
&=\left(2\tau \eta - \eta^2\right)\lim_{L\rightarrow\infty} \state{\frac{\left(\Psi_1^{(\eta L)}-\state{\Psi_1^{(\eta L)}}_{\tau L}\right)\left(\Psi_1^{(\eta L)}-\state{\Psi_1^{(\eta L)}}_{\tau L}\right)}{L^2}}_{\tau L}\\
&= \left(2\tau \eta - \eta^2\right)\eta \tau
\end{align*}
And indeed, an explicit calcuation yields
\begin{align*}
&\quad \lim_{L\rightarrow\infty} L^{-4}\state{\left(\sum_{i=1}^{\eta L} E_{ii}-\eta\tau L^2\right)\sum_{j,k=1}^{\eta L} E_{jj}E_{kk} + \left(\sum_{i=1}^{\eta L} E_{ii} - \eta\tau L^2 \right)(-\eta^2 L^2)\sum_{j=1}^{\eta L} E_{jj}}_{\tau L}\\
&=\lim_{L\rightarrow\infty} L^{-4}\Big((\eta \tau L^2 + 3L^4\eta^2\tau^2 + \eta^3\tau^3 L^6)-\eta\tau L^2(\eta^2\tau^2L^4 + \eta \tau L^2)\\
& \quad \quad - (\eta^2L^2)(\eta^2\tau^2 L^4 + \eta \tau L^2 - \eta\tau L^2\cdot\eta\tau L^2\Big)\\
&= (2\tau \eta-\eta^2)\eta\tau.
\end{align*}
\end{example}
\begin{example} Consider Theorem \ref{SpaceLikePath} with $r=2,k_1=3,k_2=4$. Using the formula for $P_t \Psi_4$ from Example \ref{P4} and
replacing $\Psi_1^2$ with $(2\tau_1 \eta_2-\eta_2^2)\Psi_1$ yields
\begin{align*}
&\ \quad 12\eta_2\tau_1^2\tau_2(\eta_2^2\tau_1 + \tau_1(3\eta_2\tau_2 + 2\eta_2^2) + \tau_2(\tau_2+3\eta_2)(\tau_1+\eta_1))\\
&=12\eta_2\tau_1^3(\eta_2^2\tau_1 + \tau_1(3\eta_2\tau_1 + 2\eta_2^2) + \tau_1(\tau_1+3\eta_2)(\tau_1+\eta_1))\\
&+ 4(\tau_2-\tau_1)\cdot 3\eta_2\tau_1^2(\eta_2^2\tau_1+6\eta_2\tau_1^2+3\tau_1(\eta_2+\tau_1)(\eta_1+\tau_1))\\
&+(6(\tau_2-\tau_1)^2+4(\tau_2-\tau_1)\eta_2)\cdot 6\eta_2\tau_1^2(\tau_1(\tau_1+\eta_1)+\eta_2\tau_1)\\
&+(4(\tau_2-\tau_1)^3+12(\tau_2-\tau_1)^2\eta_2+2(\tau_2-\tau_1)\eta_2^2)\cdot 3\eta_2\tau_1^2(\tau_1+\eta_1)\\
&+2(\tau_2-\tau_1)\cdot (2\tau_1 \eta_2-\eta_2^2)\cdot 3\eta_2\tau_1^2(\tau_1+\eta_1),
\end{align*}
which can be checked computationally.
\end{example}
Given a partition $\rho=(\rho_1,\ldots,\rho_l)$, let its \textit{weight} $\wt(\rho)$ denote
$\left| \rho \right| + l(\rho)=\rho_1+\ldots+\rho_l+l$, and let
$\Psi_{\rho}=\prod_{i=1}^l \Psi_{\rho_i}$. In the asymptotic limit, we should be able
to replace $\Psi_{\rho}$ with a linear combination of $\Psi_{\rho_i}$. In the examples above,
$\Psi_1^2$ was replaced with $(2\tau \eta - \eta^2)\Psi_1$.
\begin{proposition} Let $\eta,\tau>0$ be fixed.
(1) Set $N=\lfloor \eta L\rfloor $ and $t=\tau L$. Then $\state{\Psi_{\rho,N}}_{t} =\Theta( L^{\wt(\rho)})$.
(2) There exist constants $c'_{k,\rho}(\tau,\eta)$ such that
$$
P_{\tau L}\Psi_{k,N} = \sum_{\rho} (c'_{k,\rho}(\tau,\eta)+o(1))L^{k+1-\wt(\rho)}\Psi_{\rho}.
$$
where the sum is over $\rho$ with weight $\wt(\rho)\leq k+1$.
(3) For any $\tau_1>\tau_0,$ there exist constants $c_{kj}(\tau_1,\tau_0,\eta)$ such that
\begin{multline*}
\lim_{L\rightarrow\infty} \state{\frac{\Psi_m - \state{\Psi_m}_{\tau_0 L}}{L^m} \cdot \frac{P_{(\tau_1-\tau_0) L}\Psi_k
-\state{P_{(\tau_1-\tau_0) L}\Psi_k}_{(\tau_1-\tau_0) L}}{L^k}}_{\tau_0 L} \\
= \lim_{L\rightarrow\infty} \sum_{j=1}^k c_{kj}(\tau_1,\tau_0,\eta)\state{\frac{\Psi_m - \state{\Psi_m}_{\tau_0 L}}{L^m} \cdot \frac{\Psi_j - \state{\Psi_j}_{\tau_0 L}}{L^j}}_{\tau_0 L}
\end{multline*}
\end{proposition}
\begin{proof}
(1) This can be proved from \cite{BB}, but this will be an alternative proof.
By definition,
$$
\Psi_{\rho} = \sum_{m_1=1}^{\rho_1}\cdots \sum_{m_l=1}^{\rho_l} \sum_{\pi_1 \in \Pi_{\rho_1}^{(m_1)}}\cdots \sum_{\pi_l \in \Pi_{\rho_l}^{(m_l)}} E(\pi_1)\cdots E(\pi_l).
$$
Consider the sum over $l$--tuples $(\pi_1,\ldots,\pi_l)$ such that the paths $\pi_1,\ldots,\pi_l$
cross over a total of exactly $\nu$ distinct vertices.
There are $\binom{N}{\nu}=\Theta(L^{\nu})$ such $l$--tuples, so it
remains to estimate $\state{E(\pi_1)\cdots E(\pi_l)}_{\tau L}$. Let $\pi$ be
the union of the paths $\pi_1,\ldots,\pi_l$. Decompose $\pi$ into the union of
$s$ simple cycles. By Proposition \ref{Formula}, $\state{E(\pi_1)\cdots E(\pi_l)}_{\tau L}=\bigO{L^s}$.
Decomposing $\pi_j$ into $s_j$ simple cycles, it is clear that
$s=s_1+\ldots+s_l$. If $\pi_j$ covers exactly $\nu_j$ vertices, then
elementary graph theory gives $s_j=\rho_j-\nu_j+1$. Since
$\nu_1+\ldots+\nu_l\geq \nu$, thus
$$
\state{\Psi_{\rho,N}}_{t} = \bigO{L^{\nu}L^{s_1+\ldots+s_l}}=\bigO{L^{\rho_1+\ldots+\rho_l+l}}.
$$
To get a lower bound, just
observe that the constant term in $\Psi_{\rho,N}$ is $\Theta(L^{\wt(\rho)})$.
(2) By Theorem \ref{QuantumTheorem}(5), $P_{\tau L}\Psi_{k}$ can expressed as a linear combination of $\Psi_{\rho}$. Taking
$\state{\cdot}_{L}$ and using that $\state{P_{\tau L}X}_L = \state{X}_{(1+\tau)L}$, it follows from (1) that only $\wt(\rho)\leq k+1$ terms
have nonzero coefficieints.
(3) First apply (2) to the left--hand--side. Then, by (1),
\begin{multline*}
\Psi_{\rho_1}\ldots \Psi_{\rho_l} - \state{\Psi_{\rho_1}\ldots \Psi_{\rho_l}}_{\tau_0 L} \\
= \sum_{j=1}^l \state{\Psi_{\rho_1}}_{\tau_0 L} \ldots \widehat{\state{\Psi_{\rho_j}}_{\tau_0 L}} \ldots \state{\Psi_{\rho_l}}_{\tau_0 L}\left(\Psi_{\rho_j}-\state{\Psi_{\rho_j}}_{\tau_0 L}\right)
+ \text{smaller order terms}
\end{multline*}
\end{proof}
Given a Laurent polynomial $p(w)$, let $p(w)[w^r]$ denote the cofficient of $w^r$ in $p(w)$. Using the expansion $(z-w)^{-2}=z^{-2}(1+2(w/z)+3(w/z)^2+\ldots)$ in Theorem \ref{SpaceLikePath} and taking residues, one obtains
$$
\sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_2)(\eta_2 w^{-1} + \tau_1 + \tau_1 w)^{l}[w^r] = (\eta_2 w^{-1} + \tau_2 + \tau_2 w)^k[w^r], r\leq -1.
$$
For example, for $k=3$ and $r=-1$, and using the expansion of $P_t\Psi_3$, this says
\begin{equation}\label{SpaceLikeExample}
1\cdot (3\eta_2^2\tau_1 + 3\eta_2\tau_1^2) + 3(\tau_2-\tau_1)\cdot 2\eta_2\tau_1 + 3((\tau_2-\tau_1)^2 + (\tau_2-\tau_1)\eta_2)\cdot\eta_2 = 3\eta_2^2\tau_2 + 3\eta_2\tau_2^2.
\end{equation}
We need a formula for $r\geq 1$. Theorem \ref{TimeLikePath} follows from the proposition below.
\begin{proposition}
For $r\geq 1,$
$$
\sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_1)(\eta_1 z^{-1} + \tau_1 + \tau_1 z)^l[z^r] = (\eta_1\frac{\tau_2}{\tau_1}z^{-1} + \tau_2 + \tau_1 z)^k [z^r].
$$
\end{proposition}
\begin{proof}
We start with an illustrative example. For $k=3$ and $r=1$ we would want to show
\begin{equation}\label{AlreadyKnow}
1\cdot (3\eta_1\tau_1^2 + 3\tau_1^3) + 3(\tau_2-\tau_1)\cdot 2\tau_1^2 + 3((\tau_2-\tau_1)^2 + (\tau_2-\tau_1)\eta_1)\cdot\tau_1 = 3\eta_1\tau_1\tau_2 + 3\tau_1\tau_2^2.
\end{equation}
This can be checked directly, but in general the coefficients $c_{kl}$ are difficult to work with.
Instead, we would like to show that it follows directly from the covariance formula along space--like paths.
Indeed, this can be done just by multiplying \eqref{SpaceLikeExample} by $(\tau_1/\eta_1)^r$. (And recall that
$\eta_2<\eta_1$ in \eqref{SpaceLikeExample} while $\eta_1<\eta_2$ in \eqref{AlreadyKnow}).
Let $S_l^{(r)} = \{ (\epsilon_1,\ldots,\epsilon_l)\in \{-1,0,+1\}^l: \epsilon_1 + \ldots + \epsilon_l=r\}$ and define
$$
\chi(j) =
\begin{cases}
\eta_1, j=-1\\
\tau_1, j=0\\
\tau_1, j=1
\end{cases}
\quad
\chi'_{\text{ti}}(j) =
\begin{cases}
\eta_1\frac{\tau_2}{\tau_1}, j=-1\\
\tau_2, j=0\\
\tau_1, j=1
\end{cases}
\quad
\chi'_{\text{sp}}(j)=
\begin{cases}
\eta_1, j=-1\\
\tau_2, j=0\\
\tau_2, j=1
\end{cases}
$$
With this notation, what we want to show is that
\begin{equation}\label{WTS}
\sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_1)\sum_{\vec{\epsilon}\in S_l^{(r)}} \prod_{j=1}^l \chi(\epsilon_j)
= \sum_{\vec{\epsilon'} \in S_k^{(r)}} \prod_{j=1}^k \chi'_{\text{ti}}(\epsilon_j'), r\geq 1.
\end{equation}
From \eqref{AlreadyKnow},
$$
\sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_1)\sum_{\vec{\epsilon}\in S_l^{(-r)}} \prod_{j=1}^l \chi(\epsilon_j)
= \sum_{\vec{\epsilon'} \in S_k^{(-r)}} \prod_{j=1}^k \chi'_{\text{sp}}(\epsilon_j'), r\geq 1.
$$
By sending $\epsilon_j\mapsto -\epsilon_j$, this is equivalent to
$$
\sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_1)\sum_{\vec{\epsilon}\in S_l^{(r)}} \prod_{j=1}^l \chi(-\epsilon_j)
= \sum_{\vec{\epsilon'} \in S_k^{(r)}} \prod_{j=1}^k \chi'_{\text{sp}}(-\epsilon_j'), r\geq 1.
$$
And since for all $r$,
$$
\sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_1)\sum_{\vec{\epsilon}\in S_l^{(r)}} \prod_{j=1}^l \chi(-\epsilon_j)=
\left(\frac{\eta_1}{\tau_1} \right)^r \sum_{l=1}^k c_{kl}(\tau_2,\tau_1,\eta_1) \sum_{\vec{\epsilon}\in S_l^{(r)}} \prod_{j=1}^l \chi(\epsilon_j)
$$
it thus follows that the left--hand--side of \eqref{WTS} equals
$$
\left(\frac{\tau_1}{\eta_1} \right)^r\sum_{\vec{\epsilon'} \in S_k^{(r)}} \prod_{j=1}^k \chi'_{\text{sp}}(-\epsilon_j'), r\geq 1.
$$
So it suffices to show that
$$
\left(\frac{\tau_1}{\eta_1}\right)^r \prod_{j=1}^k \chi'_{\text{sp}}(-\epsilon_j') = \prod_{j=1}^k \chi_{\text{ti}}'(\epsilon_j'), \quad \text{ for all } \vec{\epsilon}'\in S_k^{(r)}, r\geq 1.
$$
Since $r=\left| \{\epsilon_j=1\}\right| - \left| \{\epsilon_j=-1\} \right|,$ it follows that the left--hand--side is
$$
\left(\frac{\tau_1}{\eta_1}\right)^r \cdot \eta_1^{\left| \epsilon_j=1\right|} \tau_2^{\left| \epsilon_j=0\right|} \tau_2^{\left| \epsilon_j=-1\right|} = \tau_1^r \eta_1^{\left| \epsilon_j=-1\right|} \tau_2^{\left| \epsilon_j=0\right|} \tau_2^{\left| \epsilon_j=-1\right|}.
$$
And similarly, the right--hand--side is
$$
\tau_1^{\left| \epsilon_j=1\right|} \tau_2^{\left| \epsilon_j=0\right|} \left(\eta_1\frac{\tau_2}{\tau_1}\right)^{\left| \epsilon_j=-1\right|}
=\tau_1^r \eta_1^{\left| \epsilon_j=-1\right|} \tau_2^{\left| \epsilon_j=0\right|} \tau_2^{\left| \epsilon_j=-1\right|}.
$$
\end{proof}
The formula in Theorem \ref{TimeLikePath} appears to be different from the formula in \cite{Bo}. In particular, the covariance along space--like paths is different from the covariance along time--like paths.
However, after rescaling from Brownian Motion to Ornstein--Uhlenbeck, i.e. replacing $\ga_i,\ga_j$ with $e^{2\ga_i},e^{2\ga_j}$ and multiplying by $e^{-\ga_j k_j}e^{-\ga_i k_i}$, the formula becomes
\[
\displaystyle\E[\xi_i\xi_j]=
\begin{cases}
\displaystyle-\frac{1}{\pi}\frac{e^{\ga_j}}{e^{\ga_i}}\iint_{\vert z\vert>\vert w\vert}\limits (\eta_i z^{-1} + e^{\tau_i} + z)^{k_i} (\eta_j w^{-1} + e^{\tau_j} + w)^{k_j} (\frac{e^{\ga_j}}{e^{\ga_i}}z-w)^{-2}dzdw,&\\
\hspace{3in} \eta_i\geq\eta_j,\ga_i\leq\ga_j&\\
\displaystyle-\frac{1}{\pi}\frac{e^{\ga_j}}{e^{\ga_i}}\iint_{\vert z\vert>\vert w\vert}\limits (\eta_jz^{-1} + e^{\tau_j} + z)^{k_j} (\eta_i w^{-1} + e^{\tau_i} + w)^{k_i} (\frac{e^{\ga_j}}{e^{\ga_i}}z-w)^{-2}dzdw,&\\
\hspace{3in} \eta_i<\eta_j,\ga_i\leq\ga_j&
\end{cases}
\]
In both expressions, the $z$--contour is larger and corresponds to the higher level ($\eta_i$ in the first
case and $\eta_j$ in the second). Hence, by switching the subscripts $i$ and $j$ in $\eta$, the formula is the same in
both cases. It also matches the formula in \cite{Bo} with the expression $e^{\ga_j-\ga_i}$ playing the
role of $c(t_p,t_q)$.
\bibliographystyle{alpha} | 210,707 |
You can write "wow!" on the iPad all you want. That's no guarantee of "pow!" in iPad sales.Apple
Psst! Did you hear that Apple released a new iPad?
You'd be forgiven if you didn't. After all, Apple didn't bother to hold an event to introduce its newest 9.7-inch tablet, which drops the Air name and gets a $70 price cut along with a faster processor and brighter display. Instead, Apple made the announcement in a press release issued with little fanfare Tuesday morning.
"New customers and anyone looking to upgrade will love this new iPad for use at home, in school, and for work, with its gorgeous Retina display, our powerful A9 chip, and access to the more than 1.3 million apps designed specifically for it," Phil Schiller, Apple's senior vice president of worldwide marketing, said in a release.
We'll see if there are that many customers looking for a new iPad.
This marks the first time the company has unveiled an iPad in such a low-key manner. It underscores just how far tablets have fallen: Global shipments dropped for a ninth consecutive quarter in the last three months of 2016, according to product tracker IDC. Apple has fared even worse -- iPad sales have dropped for 12 straight quarters.
"The market essentially peaked in 2014," said IDC analyst Ryan Reith.
Just look how far the iPad has fallen in the eyes of Apple. An iPad has played a part in at least one of Apple's "special events" each year.
The last time an iPad felt truly special -- where it warranted an event for itself -- was back in 2011 when Steve Jobs came back from medical leave to unveil the iPad 2. But the tablet market was still fresh, and even the minor upgrades (rear- and front-facing cameras, a faster processor) warranted excitement. Remember, this was before competitors flooded the market with cheaper me-too tablets.
Since then, the iPad has shared the spotlight with other Apple products, although notable versions like the iPad Air and the iPad Mini served as the headliner. That wasn't the case in 2015 with the original, supersized iPad Pro, which served as the opening act to the iPhone 6S, and the smaller 9.7-inch iPad Pro, which debuted with the budget-friendlier iPhone SE a year ago.
So now we come to the new iPad, which is simply called iPad. It's just one of several announcements Apple made via press release today, including a special-edition red iPhone 7 and iPhone 7 Plus, an iPhone SE with double the storage and a new video editing and sharing app called Clips.
It appears as if the iPad has been downgraded to the status of new Mac updates, which often get announced via press release, too. Still, Apple CEO Tim Cook said in January that he remains "very bullish" on iPads.
An Apple spokesman declined to comment Tuesday.
Resurgence of tablets? Uh, no.
After a long stretch of silence on the tablet front, consumer tech's two largest names -- Apple and Samsung -- have both introduced new products within two months of each other.
In late February, Samsung unveiled the Galaxy Tab S3, a 9.7-inch tablet device running Google's Android software, and two larger variants of the Galaxy Book, a Surface Pro-like tablet running on Microsoft's Windows 10 software.
Samsung boasted of taking some of the elements of its popular phones -- the bright screen, fast-charging technology and the S-Pen -- and incorporating them into the new tablets.
"We're bringing the best of Galaxy to this area," said Hassan Anjum, senior manager of product marketing for Samsung's tablets.
The Galaxy Tab S3 is one of two tablets Samsung unveiled ahead of the Mobile World Congress trade show last month.Samsung
Does the flurry of new tablets mean something significant for the tablet business? Don't bet on it.
In Samsung's case, the company was looking to keep a little buzz alive with new products, but wasn't quite ready to go with its headliner act, the Galaxy S8 phone. It's telling that the buzz over Samsung's tablets, which were introduced ahead of the Mobile World Congress trade show a few weeks back, disappeared almost immediately thanks to excitement over the nostalgia-heavy Nokia 3310.
Likewise, it might not be a coincidence that Apple is letting its announcements trickle out a week before Samsung's big Galaxy S8 event.
Because while the iPad may have lost much of its steam since the heyday of tablets, a new Apple product still has the ability to turn some heads.
CNET's Shara Tibken contributed to this story.
Does the Mac still matter? Apple execs tell why the MacBook Pro was over four years in the making, and why we should care.
Batteries Not Included: The CNET team shares experiences that remind us why tech stuff is cool.
Apple
- reading•Apple's quiet iPad launch proves no one cares about tablets
-. | 332,768 |
Sneak Peek Sunday
it’s back!
Curious to know where Cassie and her mom are heading? Here’s the next sneak peek into my short prequel, THE OTHER SIDE OF THE MIRROR. This is the tale before the fantasy romance, BROKEN MIRROR. Want to read the first few sneak peaks? Click HERE for week one, and HERE for week two, and HERE for week three. And if you want to read the whole book, check below for links to purchase. I’ll also be serializing it on Wattpad for FREE, but that doesn’t happen until May. Until then, enjoy!
The Other Side of the Mirror
A Fantasy Tale of the Black Court
sneak peak
Her mother’s brave facade crumpled. “She didn’t make it through.”
Images of her baby sister flashed in her memory. Bree sitting in a colorful pen of balls at a fast-food restaurant, her baby face alight with glee. Bree and Cassie hidden under the covers with a flashlight, while Cassie sang protective nursery rhymes to her sister to drive away the nightmares. Bree holding out her hand to Trina when Trina’s parents had been murdered by the Queen and lisping, “You can sleep in my room, I don’t mind.”
A lump the size of the world settled in Cassie’s throat.
“Oh Mama…” There was nothing else to say. It was just the two of them now. And if they stayed here with no water or food, soon there would be no MacElvys left alive.
#
Stepping out from the shelter of the fairy circle was one of the hardest things Cassie had ever done. She looked out at the dark forbidding trees leaning in their threatening circle, swallowed hard, and raised her foot.
“Wait.” Theresa gripped Cassie’s sleeve and tugged her back before she could take that first step out of the guardian pansies. “What if we leave here and Bryanna comes through?”
≈≈≈
Want to read all of chapter one for free?
I’m publishing THE OTHER SIDE OF THE MIRROR in May, starting with chapter one, on Wattpad. Wattpad is a place where you serialize books, or write them chapter by chapter, and members can comment on what they read. And it’s all free! Click HERE to check out Wattpad. Sign up and follow me, Jessica Aspen, and every week, starting in May, you’ll get a chapter of THE OTHER SIDE OF THE MIRROR to read. For free!
But, if you can’t wait, and you want the whole book, you can purchase it in it’s entirety at these fine distributors
Kobo & All Romance Ebooks (pending)
Want to know where it all started? Want to be ready when book three, Broken Mirror comes out? Read The Dark Huntsman,
book one in the Tales of the Black Court.
Dare to enter Jessica Aspen’s world of steamy, fantasy romance in her new twisted fairy tale trilogy: Tales of the Black Court…Start with book one, The Dark Huntsman, and discover your imagination
Buy on All Romance Ebooks
Buy on Barnes and Noble. | 250,699 |
TITLE: Parametric equation of clock hands
QUESTION [3 upvotes]: I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to $12$ etc.) Suppose I know the center (starting point) and the radii of both the hour and the minute hands. In order to locate the endpoints (tip) of the clock hands, what parametric equation I should use? The complication seems to be that the hour hands start from $12$ (or we could assume it's "$0$") at the top of the clock. So, the traditional parametric equation for finding a point around a circle's circumference (see below) needs to be modified a bit here. If I get the hour hand equation down, I think I can adapt it for minute hand. My trig knowledge isn't that good, so if someone could provide me a solution (or even just a clue), that'd be very helpful. Thank you!
The conventional parametric equations of a circle are:
\begin{align}
x &= r \cos(t)\\
y &= r \sin(t)
\end{align}
where $r$ is the radius and $t$ is in radian for above equations.
I found a document that derives how to graph the hour and minute hands, but the equations there (pasted below) doesn't seem to draw at least the hour hand right. For example, from $12$ to $3$ hour, it draws points in the fourth quadrant of a circle...
\begin{align}
x &= r \cos(\frac\pi{360}t)\\
y &= r \sin(\frac\pi{360}t)
\end{align}
where $t$ is time in minutes.
REPLY [1 votes]: $ \theta $ is in radians ;
$$ x_H= H \sin \theta ;\quad y_H=H \cos \theta ; $$
$$ x_M= M \sin \theta ;\quad y_M = M\cos \theta ; $$
$ \theta_{minute\, hand} = \Omega t ; \quad\theta_{hour\, hand} = \omega t ; $ where $t$ is in minutes.
$ \omega =\frac12 ^0 $ per minute, $ \Omega = 6 ^0$per minute | 91,998 |
\section{Introduction}
\subsection{Motivation: Community Structure in Networks}
In the last decade
stochastic social networks have been analyzed mathematically from various
points of view. Understanding such networks sheds light on many questions
arising in biology, epidemology, sociology and large computer networks.
Researchers have concentrated particularly on a few properties
that seem to be common to many networks: the small-world property,
power-law degree distributions,
and network transitivity,
For a broad view on the structure and dynamics of networks,
see \cite{bk:NewmanBarabasiWatts06}.
M. Girvan and M.E.J. Newman, \cite{ar:GirvanNewman02},
highlight another property that is found in many networks,
the property of {\em community structure},
in which network nodes are joined together in tightly knit groups,
between which there are only looser connections.
Motivated by
\cite{ar:Newman04a}, and
the first author's involvement in a project studying social networks,
we were led to study the graph parameter
$q_{ij}\left( G\right) $, the number of vertex subsets $X\subseteq V$ with
$i$ vertices such that $G\left[ X\right] $ has exactly $j$ components.
$q_{ij}\left( G\right) $, counts the number of degenerated communities which consist of $i$
members, and which split into $j$ isolated subcommunities.
The ordinary bivariate generating function associated with
$q_{ij}\left( G\right) $
is the two-variable
graph polynomial
\[
Q\left( G;x,y\right) =\sum_{i=0}^{n}\sum_{j=0}^{n}q_{ij}\left( G\right)
x^{i}y^{j}.
\]
We call $Q\left( G;x,y\right) $ the
\emph{subgraph component polynomial} of $G$.
The coefficient of $y^{k}$ in $Q\left( G;x,y\right) $ is the ordinary
generating function for the number of vertex sets that induce a subgraph of
$G$ with exactly $k$ components.
\subsection{$Q(G;x,y)$ as a Graph Polynomial}
There is an abundance of graph polynomials studied in the literature,
and slowly there is a framework emerging,
\cite{pr:Makowsky06,ar:MakowskyZoo,pr:GodlinKatzMakowsky07},
which allows to compare graph polynomials with respect to their
ability to distinguish graphs, to encode other graph polynomials or
numeric graph invariants, and their computational complexity.
In this paper we study
the {\emph subgraph component polynomial}
$Q\left( G;x,y\right)$ as a graph polynomial in its own right
and explore its properties within this emerging framework.
Like the bivariate Tutte polynomial,
see \cite[Chapter 10]{bk:Bollobas99},
the polynomial
$Q\left( G;x,y\right)$ has several remarkable properties.
However, its distinguishing power is quite different from the
Tutte polynomial and other well studied polynomials.
Our main findings are:
\begin{itemize}
\item
$Q\left( G;x,y\right)$ distinguishes graphs which cannot be distinguished
by the matching polynomial, the Tutte polynomial, the characteristic
polynomial, or the bivariate chromatic polynomial introduced in
\cite{ar:DPT03} (Section \ref{se:distinct}).
\item
Nevertheless, we construct an infinite family of pairs of graphs
which cannot be pairwise distinguished by $Q\left( G;x,y\right)$
(Proposition \ref{prop:inftrees}).
\item
The Tutte polynomial,
satisfies a linear recurrence relation
with respect to edge deletion and edge contraction, and is universal
in this respect.
$Q\left( G;x,y\right)$
also
satisfies a linear recurrence relation, but with respect to three
kinds of vertex elimination, and is universal in this respect.
(Theorems \ref{theo_decom} and \ref{th:universal}).
\item
A graph polynomial in three indeterminates,
$\xi(G;x,y,z)$,
which satisfies a linear recurrence relation
with respect to three kinds of edge elimation, and which is universal
in this respect, was introduced in
\cite{pr:AverbouchGodlinMakowsky07,ar:AverbouchGodlinMakowsky08}.
It subsumes both the Tutte polynomial and the matching polynomial.
For a line graph $L(G)$ of a graph $G$, we have
$Q\left( L(G);x,y\right)$ is a substitution instance of
$\xi(G;x,y,z)$
(Theorem \ref{thm:q_xi}).
\item
For fixed positive integer $n$ the univariate polynomial
$Q(G;x,n)$ can be interpreted as counting weighted homomorphisms,
\cite{ar:DyerGreenhill2000},
and is related to the Widom-Rowlinson model for $n$ particles
(Theorem \ref{thm:hom}).
\item
$Q(G;x,y)$ is reconstructible from its vertex deletion deck
in the sense of \cite{ar:BondyHemminger77,ar:Bondy91}
(Theorem \ref{th:reconstruct}).
\item
$Q(G;x,y)$ can be used (Section \ref{se:random}), to compute the
probability $P_{k}\left( G\right) $ that a vertex induced subgraph of $G$
has exactly $k$ components from the subgraph polynomial.
For $k=1$ this is known as the
{\em residual connectedness reliability}
(Section \ref{se:random}).
\item
Also like for the Tutte polynomial, cf. \cite{ar:JaegerVertiganWelsh90},
$Q(G;x_0,y_0)$ has the {\em Difficult Point Property}, i.e. it
is $\sharp \mathbf{P}$-hard to compute for
all fixed values of $(x_0,y_0) \in \mathbb{R}^2 -E$
where $E$ is a semi-algebraic set of lower dimension
(Theorem \ref{th:dpp}).
In \cite{ar:MakowskyZoo} it is conjectured that
the Difficult Point Property holds for a wide class of graph polynomials,
the graph polynomials definable in Monadic Second Order Logic.
The conjecture has been verified only for special cases,
\cite{ar:BlaeserDell07,ar:BlaeserDellMakowsky08,ar:BlaeserHoffmann08}.
\item
$Q(G;x_0,y_0)$ is fixed parameter tractable in the sense of
\cite{bk:DowneyFellows99} when restricted to
graphs classes of bounded tree-width (Proposition \ref{prop:tw})
or even to classes of bounded clique-width (Proposition \ref{prop:cw}).
For the Tutte polynomial, this is known only for graph classes
of bounded tree-width, \cite{ar:Noble98,ar:Andrzejak97,pr:MRAG06}.
\end{itemize}
\subsection*{Outline of the paper}
The paper is organized as follows:
In Section
\ref{se:Q} we introduce the polynomial $Q(G;x,y)$ and its
univariate versions.
In Seection \ref{se:distinct} we discuss the distinguishing power of
$Q(G;x,y)$ and compare this to other graph polynomials.
In Section \ref{se:combint} we show how certain graph parameters
are definable using $Q(G;x,y)$ and relate it to partition functions
and counting weighted homomorphisms.
In Section \ref{se:decomp} we give a recursive defintion of
$Q(G;x,y)$ using deletion, contraction and extraction of vertices
and show that
$Q(G;x,y)$ is universal. We also compare it to
the universal edge elimination polynomial $\xi(G;x,y,z)$ defined
in \cite{pr:AverbouchGodlinMakowsky07,ar:AverbouchGodlinMakowsky08}
and give a subset expansion formula for
$Q(G;x,y)$.
In Section \ref{se:cliquesep} we prove decomposition formulas
for $Q(G;x,y)$ for clique separators.
In Section \ref{se:reconstruct} we show the reconstructibility of
$Q(G;x,y)$.
In Section \ref{se:random} we discuss its use to compute
the residual connectedness reliability.
In Section \ref{se:complexity} we discuss the complexity of computing
$Q(G;x,y)$.
In Section \ref{se:conclu} we draw conclusions and state open problems.
\section{The Subgraph Component Polynomial $Q(G;x,y)$}
\label{se:Q}
\subsection{The Bivariate Polynomial}
Let $G=\left( V,E\right) $ be a finite undirected graph with $n$ vertices
and let $k\leq n$ be a positive integer. Assume the vertices of $G$ fail
stochastic independently with a given probability $q=1-p$. What is the
probability that a subgraph of $G$ with exactly $k$ components survives? The
solution of this problem leads to the enumeration of vertex induced
subgraphs of $G$ with $k$ components. For a given vertex subset $X\subseteq
V $, let $G\left[ X\right] $ be the \emph{vertex induced subgraph} of $G$
with vertex set $X$ and all edges of $G$ that have both end vertices in $X$.
We denote by $k\left( G\right) $ the number of components of $G$. Let
$q_{ij}\left( G\right) $ be the number of vertex subsets $X\subseteq V$ with
$i$ vertices such that $G\left[ X\right] $ has exactly $j$ components:
\[
q_{ij}\left( G\right) =\left\vert \left\{ X\subseteq V:\left\vert
X\right\vert =i\wedge k\left( G\left[ X\right] \right) =j\right\}
\right\vert
\]
The ordinary generating function for these numbers is the two-variable
polynomial
\[
Q\left( G;x,y\right) =\sum_{i=0}^{n}\sum_{j=0}^{n}q_{ij}\left( G\right)
x^{i}y^{j}.
\]
We call $Q\left( G;x,y\right) $ the
\emph{subgraph component polynomial} of $G$.
Since loops or parallel edges do not contribute to connectedness properties of a
graph, we assume in this paper that all graphs are simple.
\begin{figure}[ht]
\begin{center}
\epsfig{file={k1_3.eps},width=0.5in}
\end{center}
\caption{The star $Star_3=K_{1,3}$} \label{k1_3}
\end{figure}
The star $K_{1,3}$, presented in Figure \ref{k1_3}, has the subgraph
polynomial
\[
Q\left( K_{1,3};x,y\right)
=1+4xy+3x^{2}y+3x^{3}y+x^{4}y+3x^{2}y^{2}+x^{3}y^{3}.
\]
The term $3x^{2}y^{2}$ tell us that there are 3 possibilities to select two
vertices of $G$ that are non-adjacent.
The empty set induces the null graph $N=\left( \emptyset ,\emptyset \right) $
that we consider as being connected, which gives $q_{00}\left( G\right)
=Q\left( G;0,0\right) =1$ for any graph. Substitution of $1$ for $y$ results
in an univariate polynomial that is the ordinary generating function for all
subsets of $V$, i.e. $Q(G;x,1)=\left( 1+x\right) ^{n}$. | 208,451 |
TITLE: Finding the volume of two intersecting cylinders at arbitrary angles
QUESTION [5 upvotes]: Suppose we know the length, radius, position and orientation of two cylinders. Is there a general formula to calculate the volume of space shared by the intersection of the cylinders?
REPLY [3 votes]: This answer is about the particular case in which the axes of the two cylinders meet at a point, both have the same radius, and both cylinders extend to infinity (this last assumption is to avoid complexities regarding truncated cylinder intersections). We can take the radius as $1$ and then re-scale to radius $a$ by inserting a factor of $a^3$.
Let the $x$ axis be the axis of cylinder $A$, and for some $m>0$ let the line $y=mx$ be the axis of cylinder $B$. The (signed) distance between a point $(x,y)$ and the line $y=mx$ is $\frac{y-mx}{r}$ where $r=\sqrt{1+m^2}$ Then the restriction that $(x, y)$ lies under cylinder $B$ is that $-1 \le \frac{y-mx}{r} \le 1,$ while $(x, y)$ lies under cylinder $A$ provided $-1 \le y \le 1$.
Note we're considering the part of the intersection lying above the $xy$ plane; then the height over cylinder $A$ is $\sqrt{1-y^2}$ and the height over cylinder $B$ is $\sqrt{1-\left[\frac{y-mx}{r}\right]^2}.$ These heights are the same over a point $(x, y)$ (i.e. the two cylinders intersect over that point) iff $y^2-\left[\frac{y-mx}{r}\right]^2=0$, which factors into two lines $y=-\frac{mx}{r-1}$ and $y=\frac{mx}{r+1}$. The product of these slopes is $-1$ so the lines are orthogonal.
If $L_B$ is the axis $y=mx$ of cylinder $B$, and for convenience $L_A$ is the $x$-axis, the axis of cylinder $A$, we'll use the notations $L_B^+$ and $L_B^-$ for the lines parallel to $L_B$ and one unit away from it, above and below respectively. Similar notation for $L_A^+$ and $L_A^-.$ The first two of these are the upper and lower boundaries of the projection of cylinder $B$ onto the $xy$ plane, and similarly the second two are the upper and lower boundaries of the projection of cylinder $A.$ Equations for these are
$$L_B^+ : y=mx+r \\ L_B^- : y=mx-r \\ L_A+ : y=1 \\ L_A^- : y=-1$$
The intersections of these lines consist of four points $P, Q, R, S$ which form an equilateral parallelogram. We have
$$P=\left(\frac{1-r}{m}, 1\right), \ Q=\left(\frac{1+r}{m}, 1\right), \ R=\left(\frac{-1+r}{m}, -1\right), \ S=\left(\frac{-1-r}{m}, -1\right)$$
Now if we denote the origin $(0, 0)$ by $O$, we can see there are four triangular regions each having $O$ as one vertex, and two adjacent vertices of the parallelogram as the other two vertices. Above each of these triangles we know which of the two cylinders lies below the other.
It turns out that the contributions to the volume from each of these triangles (i.e. the volume of the intersection of the cylinders lying above that triangle) is the same.
This can be seen by symmetry, since e.g. the part lying above $\triangle OQR$ could be set up with the line $y=mx$ playing the role of the $x$ axis, and taking an orthogonal axis pointing downwards from $O$, so that the "old" $x$ axis is now at the same slope $m$ in this rotated and flipped coordinate system.
It remains now to find the volume over $\triangle OPQ$, which may be described as $0 \le y \le 1$ and $\frac{y(1-r)}{m} \le x \le \frac{y(1+r)}{m}$. Now over this triangle it is cylinder $A$ whose height is lower. This means we're integrating $\sqrt{1-y^2}$ over $\triangle OPQ$. At any specific $y$ the height being constant, we only need multiply by the length of the horizontal section of $\triangle OPQ$ at distance $y$ from the $x$ axis, and integrate. Thus we get
$$ \int_0^1 \frac{(2r)}{m y \sqrt{1-y^2}} dy=(2r/m)\cdot(1/3).$$
Recalling that we have found only the volume above the $xy$ plane, and that there are four triangles each giving the above contribution to the part above the $xy$ plane, we arrive at
$$\frac{16}{3}\cdot \frac{\sqrt{1+m^2}}{m}$$
for the volume of the intersection of the cylinders.
Note as $m \to \infty$ (so that the angle of intersection goes to 90 degrees) this formula approaches the correct $16/3$ (the usual formula $(16/3)a^3$ when $a=1.$) And also as expected it becomes infinite as $m \to 0^+.$
Added note: If $\theta$ is the angle between the two axes, then $m=\tan \theta.$ I used the letter $r$ during the derivation to mean $\sqrt{1+m^2}$ only for convenience. If we reset $r$ to now mean the common radius of the cylinders, and apply a trig identity, we can reexpress the volume in the form
$$V=\frac{16}{3 \sin \theta} \cdot r^3,$$
which is only off by the $sin \theta$ divisor from the result $(16/3)r^3$ when the cylinders are orthogonal. | 113,940 |
TITLE: Understanding information retrieval in distinguishing coin stacks
QUESTION [2 upvotes]: Consider the following problem. There are $10$ stacks of $10$ coins, each visually identical. In $9$ of the stacks, the coins weigh $10$ grams each, while in the other stack the coins weigh $11$ grams each. You have a digital scale but can only use it once to identify the heavier stack. How do you do it?
One answer to this is to to take $k$ coins from the $k$-th stack, for each $1\leq k\leq 10$, and weigh them all at once. The resulting weight then contains enough "information" to tell which stack the heavier coins came from.
Now if we modified the situation so that each stack only has $9$ coins, this method wouldn't work. In particular, there's a sense in which the strategy is able to retrieve less information in this set-up compared to the set-up with 10 coins per stack.
Question: I would like to understand (perhaps using language from information theory) the precise sense in which this strategy yields less information.
Apologies if the question is a bit vague, but I hope it is understandable to someone who knows about information theory. I have no background in information theory, but I am trying to make a conceptual connection to things like entropy.
REPLY [1 votes]: Not sure if this is a useful way to thing about the problem, but maybe it is at least an interesting exercise.
To talk about entropy, we need probability, so suppose each of the 10 stacks is equally likely to be the outlier. Then, the amount of entropy in this information is just $\log(10)$ (for whatever choice of $\log$ base).
Consider a given weighing strategy (for stacks of 8 coins). The result will be 10 grams times the total number of coins weighed plus 1 gram times the number of coins weighed from the outlier stack. No matter what our strategy is, there are at most 9 possible outcomes (from 0 to 8) for the number of coins we weighed from the outlier stack. The entropy of any measurement with 9 (or fewer) outcomes is bounded by the entropy of 9 equally likely outcomes. The entropy of 9 equally likely outcomes is $\log(9)$. This is less than $\log(10)$, so it is impossible to have recovered all the original information.
Note, first, that this is obvious. If there are 9 outcomes for 10 inputs, it must be the case that (at least) two inputs map to the same output. (This is the pigeonhole principle.) This is exactly what makes things impossible — we cannot distinguish between two inputs that have the same output. Probably this corresponds to your ordinary understanding of the problem. (But I guess that doesn't mean that entropy will always be fruitless.)
Also note that there is no weighing strategy that will actually have entropy $\log(9)$. Different strategies will have different entropies based on the distribution of results (which is in turn induced by the distribution we initially chose for the unknown information). For example, weighing nothing (or otherwise weighing the same number of coins from each stack) will always give the same result; this is zero entropy. The highest entropy will be when we take different numbers of coins from every stack except for a single pair of stacks from which we take the same number of coins. We can calculate that this has entropy $-\frac{2}{10}\log(\frac{2}{10})-\frac{8}{10}\log(\frac{1}{10}) = \log(5 \cdot 2^{4/5})$. | 65,563 |
TITLE: Solutions to $a^2+b^2+(ab)^2=c^2$
QUESTION [2 upvotes]: In a comment to the question $n^2+(n+1)^2+n^2\cdot(n+1)^2$ is a perfect square it is proved that:
$(1)\quad b=a+1$ give integer solutions to
$a^2+b^2+(ab)^2=c^2$ for all $a\in\mathbb N$.
In the answers to the question $\forall m\in\mathbb N\exists n>m+1\exists N\in\mathbb N:m^2+n^2+(mn)^2=N^2$ it is proved that:
$(2)\quad b=2a^2$ give integer soulutions for all $a\in\mathbb N$.
Conjecture:
For each $a\in\mathbb N$ there are integer solutions to
$a^2+b^2+(ab)^2=c^2$ that are neither of the types $(1)$ or $(2)$
above.
Prove the conjecture or give a counter-example. It is tested for all $a<1000$.
REPLY [2 votes]: The Pell type equation $$ x^2 - (1 + A^2) y^2 = A^2 $$ has an infinite set of solutions $(x,y).$ The first three predictable types are
$$
\left(
\begin{array}{c}
A \\
0
\end{array}
\right),
$$
$$
\left(
\begin{array}{c}
A^2 - A + 1 \\
A - 1
\end{array}
\right),
$$
$$
\left(
\begin{array}{c}
A^2 + A + 1 \\
A + 1
\end{array}
\right).
$$
After that, an infinite sequence of $(x,y)$ as column vectors may be found by multiplying by the generator of the (oriented) automorphism group of the binary quadratic form $ x^2 - (1 + A^2) y^2. $ That matrix is
$$ M =
\left(
\begin{array}{cc}
2A^2 + 1 & 2 A^3 + 2 A \\
2A & 2 A^2 + 1
\end{array}
\right).
$$
The answer you write as $b = 2 a^2$ comes up as
$$ M =
\left(
\begin{array}{cc}
2A^2 + 1 & 2 A^3 + 2 A \\
2A & 2 A^2 + 1
\end{array}
\right)
\left(
\begin{array}{c}
A \\
0
\end{array}
\right) =
\left(
\begin{array}{c}
2A^3 + A \\
2 A^2
\end{array}
\right).
$$
The bad news is that, for a fixed $A,$ there may be others. These others show up as later entries in the Pell sequences for smaller values of $A.$ I do not have a genuinely sensible two-dimensional description for all answers. | 104,899 |
TITLE: Existence of a fixed point to a vector-valued mapping
QUESTION [3 upvotes]: I have the following system:
$$\begin{cases}
F_1(x_1,x_2) \colon= (c_1 - a_{11}x_1 - a_{21}x_2)^{a_{11}}(c_2 - a_{12}x_1 - a_{22}x_2)^{a_{12}} = x_1 \\
F_2(x_1,x_2) \colon= (c_1 - a_{11}x_1 - a_{21}x_2)^{a_{21}}(c_2 - a_{12}x_1 - a_{22}x_2)^{a_{22}} = x_2
\end{cases}$$
where $c_i, a_{ij}\geq 0$ and $c_1 + c_2 = a_{11}+a_{12} = a_{21} + a_{22} = 1.$ I strongly suspect the above system has a solution $(x_1,x_2)$ with $x_i>0,$ but not quite sure how to prove it.
The Jacobian after taking $\ln$ from both sides is positive definite which can be seen easily. But I am not sure how that would help if we wanted to use Implicit Function Theorem, for instance. I thought about using Banach's fixed point theorem but the map $F = (F_1, F_2)$ does not seem to be a contraction since the derivative will be unbounded due to the exponents $a_{ij}\leq 1.$
I think a vector version of the Intermediate Value Theorem is probably most promising but all I can find is Poincare-Miranda Theorem, which I am not sure is immediately applicable. For what its worth we can assume the domain I am interested is:
$$D = \{(x_1,x_2)\vert\, a_{1i}x_1 + a_{2i}x_2 < c_i \}\subseteq\mathbb{R}^2_{>0}$$
Finally, its one-dimensional version is an immediate consequence of the Intermediate Value Theorem
REPLY [1 votes]: Yes, the function $F$ is not a contraction, so we can't use Banach fixed point theorem. There are of course many generalisations of it but I would rather go for topological fixed point theorems.
First observe that the domain of $F$ is equal to
$$D_F = \{(x_1,x_2)\in \Bbb R^2:c_1 - a_{11}x_1 - a_{21}x_2\geq 0,\ c_2 - a_{12}x_1 - a_{22}x_2\geq 0\}.$$
Let $$M:=\{(x_1,x_2)\in\Bbb R^2:x_1,x_2\geq 0\}\text{ and }K:=M\cap D_F.$$ These are closed convex sets and $K$ is bounded (hence compact). Indeed, if $(x_1,x_2)\in K$ then
$$
(a_{11}+a_{12})x_1 + (a_{21}+a_{22})x_2\leq c_1+c_2,\text{ so }
x_1+x_2\leq 1.
$$
This together with $x_1,x_2\geq 0$ gives boundedness of $K$.
Case $c_1\cdot c_2=0$.
Assume that $c_1=0$ and that there exists a positive fixed point $(x_1,x_2)\in D_F$ of $F$, i.e. $x_1,x_2>0$ and $F(x_1,x_2)=(x_1,x_2)$.
If $a_{11}>0$ or $a_{21}>0$ then
$$0\leq c_1 - a_{11}x_1 - a_{21}x_2< 0,$$ a contradiction. Therefore $a_{11}=0$ and $a_{21}=0$. Therefore $x_1=F_1(x_1,x_2)=0$, a contradiction. This shows that in this case there are no positive fixed points (the point $(0,0)$ is a nonnegative fixed point, though). Similarly we can treat the case $c_2=0$.
Case $c_1,c_2>0$.
Now assume $c_1,c_2>0$.
The idea of what will be done
Having this we can consider many fixed point theorems of functions $F\colon K\to \Bbb R^2$. Unfortunately $K$ isn't invariant, so we can't use a Brouwer fixed point theorem. However there are many theorems that deal with such cases but some additional assumptions are needed (which is obvious, since for example $f\colon [0,1]\to \Bbb R$, $f(x)=x+2$ doesn't possess fixed points). These additional assumptions are mainly about the behaviour of $F$ on the boundary of $K$, for example:
compression / expansion
tangency.
The idea is that we can consider the retraction $r\colon \Bbb R^2\to K$ (there are such retractions for convex closed sets, for example a metric projection) and the function $r\circ F\colon K\to K$, being $K$ invariant, has a fixed point (from Brouwer f.p.t.). Therefore $r(F(x_1,x_2))=(x_1,x_2)$ for some $(x_1,x_2)$. This additional assumption can give us that this can't happen on $\partial K$ and therefore $r(F(x_1,x_2))=F(x_1,x_2)$.
In our case we have a situation of tangency, that is $F(x,y)$ is tangent to $K$, that is $F(x,y)$ is always in the so called Bouligand/Clarke tangent cone of $K$. I'm going to avoid these notions and to solve this problem directly, without using the more sophisticated theorems than Brouwer's.
Back to proof
Of course
$F(M)\subset M$.
$F_1(x,y)=0\iff F_2(x,y)=0$
Let $r\colon M\to K$ be a metric projection, that is to any $(x,y)\in M$ it assigns the closest point in $K$. Therefore $r|_K = \mathrm{id}|_K$ and if $r(p)\neq p\iff p\notin K$ then $r(p)\in\partial_M K$ (boundary with respect to $M$).
Then $r\circ F\colon K\to K$ has a fixed point $(x_1,x_2)$, that is $$r(F(x_1,x_2))=(x_1,x_2).$$
If $r(F(x_1,x_2))=F(x_1,x_2)$ then we get a fixed point of $F$ on $K$.
If $r(F(x_1,x_2))\neq F(x_1,x_2)$ then $(x_1,x_2)=r(F(x_1,x_2))\in \partial_M K$. Therefore $c_1 - a_{11}x_1 - a_{21}x_2=0$ or $c_2 - a_{12}x_1 - a_{22}x_2=0$ and then $F(x_1,x_2)=(0,0)$. This is however impossible, since then $(x_1,x_2)=r(F(x_1,x_2))=r(0,0)=(0,0)$ and then $F(x_1,x_2)\neq (0,0)$.
Now we know that $F$ has a fixed point $(x_1,x_2)$ in $K$. It sufficies to show that $x_1,x_2>0$. If for example $x_1=0$ then $F_1(x_1,x_2)=0$ then $F_2(x_1,x_2)=0$, then $x_2=0$ then $F_1(x_1,x_2)=F_1(0,0)\neq 0$, a contradiction.
Final remarks
Here is the picture of the situation for $a_{11}=0.07$, $a_{22}=0.31$, $c_1=0.37$.
The green polgon is the set $K$. The curved shape is the set $F(K)$. We see that $F(K)\not\subset K$. The cross (red/black) is the value $F(p)$ of the dot $p$ (red/black).
More in Desmos. | 172,339 |
\begin{document}
\begin{center}
{\bf SHARP MOMENT ESTIMATES FOR }\\
\vspace{3mm}
{\bf POLYNOMIAL MARTINGALES }\\
\vspace{4mm}
{\bf E. Ostrovsky, L.Sirota.}\\
\vspace{5mm}
Department of Mathematics, Bar-Ilan University, Ramat-Gan, 59200,Israel.\\
e-mail: \ [email protected] \\
\vspace{3mm}
Department of Mathematics, Bar-Ilan University, Ramat-Gan, 59200,Israel.\\
e-mail: \ [email protected] \\
\vspace{4mm}
{\bf Abstract.} \par
\vspace{3mm}
\end{center}
In this paper non-asymptotic moment
estimates are derived for tail of distribution for discrete time polynomial martingale
by means of martingale differences as a rule in the terms of {\it unconditional} and
{\it unconditional relative} moments and tails of distributions of summands. \par
We show also the exactness of obtained estimations. \par
\vspace{4mm}
{\it Key words:} Random variables and vectors, Iensen, Osekowski, Rosenthal and triangle inequalities, recursion,
martingales, martingale differences, regular varying function, Lebesgue-Riesz and Grand Lebesgue norm and spaces,
lower and upper estimates, moments and relative moments, quadratic characteristic of martingale, filtration, examples,
natural norming, tails of distribution, conditional expectation. \par
\vspace{3mm}
{\it Mathematics Subject Classification (2002):} primary 60G17; \ secondary
60E07; 60G70.\\
\vspace{4mm}
\section{ Introduction. Notations. Statement of problem. Announce.} \par
\vspace{3mm}
Let $ (\Omega,F,{\bf P} ) $ be a probabilistic space, which will be presumed sufficiently
rich when we construct examples (counterexamples),
$ \xi(i,1), \xi(i,2), \ldots,\xi(i,n), \ n \le \infty $ being a {\it family} of a centered
$ ({\bf E} \xi(i,m) = 0, i=1,2,\ldots,n) $ martingale-differences on the basis of the fixed
{\it flow }of $ \sigma - $ fields (filtration) $ F(i): F(0) =
\{\emptyset, \Omega \}, \ F(i) \subset F(i+1) \subset F, \
\xi(i,0) := 0; \ {\bf E} |\xi(i,m)| < \infty, $ and for every $ i \ge 0,
\ \forall k = 0,1,\ldots,i -1 \ \Rightarrow $
$$
{\bf E} \xi(i,m)/F(k) = 0; \ {\bf E}\xi(i,m)/F(i) = \xi(i,m) \ (\mod \ {\bf P}). \eqno(1.0)
$$
\vspace{3mm}
Let also $ I = I(n) = I(d; n) = \{i_1; i_2; \ldots; i_d \} $ be the set of indices of the
form $ I(n) = I(d; n) = \{ \vec{i} \} = \{i \} = \{i_1, i_2, \ldots, i_d \} $ such that $ 1 \le i_1 < i_2 <
i_3 < i_{d-1} < i_d \le n; \ J = J(n) = J(d; n) $ be the set of indices of
the form (subset of $ I(d; n)) \ J(d; n) = J(n) = \{ \vec{ j} \} = \{ j \} = \{j_1; j_2; \ldots; j_{d-1} \} $
such that $ 1 \le j_1 < j_2 \ldots < j_{d-1} \le n- 1; \ b(i) = \{ b(i_1; i_2; \ldots i_d) \} $ be a
$ d $ dimensional numerical non-random sequence symmetrical relative to all argument permutations,
$$
\vec{i} \in I \ \Rightarrow \xi(\vec{i}) \stackrel{def}{=} \prod_{s=1}^d \xi(i_s, s);\eqno(1.1a)
$$
$$
\vec{j} \in J \ \Rightarrow \xi(\vec{j}) \stackrel{def}{=} \prod_{s=1}^{d-1} \xi(i_s, s); \eqno(1.1b)
$$
$$
\sigma^2(i_k,s) := \Var(\xi_{i_k,s}),
$$
$$
\sigma^2(\vec{i}) := \prod_{s=1}^d \sigma^2 ({i_s,s}), \ \vec{i} = \{i_1, i_2, \ldots, i_d \} \in I;
$$
$$
\sigma^2(n,\vec{j}):= \prod_{s=1}^{d-1} \sigma^2 ({j_s,s}), \ \vec{j} = \{j_1, j_2, \ldots, j_{d-1} \} \in J;
$$
$$
Q_d = Q(d,n, \{ \xi(\cdot) \} ) = Q(d,n) = Q(d,n, \vec{b}) = \sum_{\vec{i} \in I(d,n) } b(\vec{i}) \xi(\vec{i}) \eqno(1.2)
$$
being a homogeneous polynomial (random polynomial) of power $ d $ on the
random variables $ \xi(\cdot, \cdot) $ "without diagonal members", (on the other
words, multiply stochastic integral over discrete stochastic martingale
measure), $ n $ be an integer number: $ n = 1, 2, \ldots, $ in the case $ n = \infty $
we will understood $ Q(d, \infty) $ as a limit $ Q(d;\infty) = \lim_{n \to \infty} Q(d; n), $ if
there exists with probability one.\par
Here $ b = \vec{b} = b(i), \ i \in I(d,n) $ be arbitrary non-random numerical sequence, and we denote
$$
||b||^2 = \sum_{i \in I(d,n)} b^2(i); \ b \in B \hspace{6mm} \Leftrightarrow ||b|| = 1. \eqno(1.2b)
$$
We will denote also in the simple case when $ \vec{b} = \vec{b}_d = \vec{1} = \{ 1,1,\ldots,1 \} \ d \ - $ times
$$
R(d) = Q(d,n, \vec{1}) = \sum_{\vec{i} \in I(d,n) } \xi(\vec{i}). \eqno(1.3)
$$
It is obvious that the sequence $ (Q(d; n); F(n)); n = 1, 2, 3, \ldots, $ is a
martingale ("polynomial martingale"). \par
The case of non-homogeneous polynomial is considered analogously. \par
We denote as usually the $ L(p) $ norm of the r.v. $ \eta $ as follows:
$$
|\eta|_p = \left[ {\bf E} |\eta|^p \right]^{1/p}, \ p > 2;
$$
the case $ p = 2 $ is trivial for us.\par
\vspace{4mm}
{\bf We will derive the moment estimations of a Khintchine form }
$$
U(p;d,n) = U(p) \stackrel{def}{=}
\sup_{b \in B} | Q(d,n, b, \{ \xi(\cdot) \}) |_p \le \overline{Q}(p;d,n) = \overline{Q}(p)
$$
{\bf for martingale (and following in the independent case) in the terms of unconditional moments, more
exactly, in the $ L(p) $ norms of summands:}
$$
\mu_m(p) \stackrel{def}{=} \sup_i |\xi(i,m)|_p. \eqno(1.4)
$$
Denote also in the martingale case
$$
V(p) = V_d(p) \stackrel{def}{=} \prod_{m=1}^d \mu_m(d \cdot p). \eqno(1.5a)
$$
and for the independent variables $ \{ \xi(i,m) \} $
$$
W(p) = W_d(p) \stackrel{def}{=} \prod_{m=1}^d \mu_m(p). \eqno(1.5b)
$$
Note that if all the functions $ p \to \mu_m(p), \ m =1,2,\ldots,d $ are regular varying:
$$
\sup_{p \ge 2} [ \mu_m(d \cdot p) /\mu_m(p)] < \infty,
$$
then $ V(p) \asymp W(p). $ \par
\vspace{3mm}
As we knew, the previous result in this direction is obtained in the article \cite{Ostrovsky4}:
$$
U(p;d,n) \le C_1(d) \cdot p^d \ V(p),
$$
and in the independent case, i.e. when all the (two-dimensional indexed) centered r.v. $ \xi(i,m) $
are common independent,
$$
U(p;d,n) \le C_2(d) \cdot (p^d/\ln p) \cdot W(p), \ p \ge 2.
$$
{\it We intend to improve both these estimates to the following non-improvable as } $ p \to \infty: $
$$
U(p;d,n) \le \gamma(d) \cdot \frac{p^d}{(\ln p)^d } \cdot V_d(p), \eqno(1.6a)
$$
in general (martingale) case and
$$
U(p;d,n) \le \kappa(d) \cdot \frac{p^d}{(\ln p)^d } \cdot W_d(p), \eqno(1.6b)
$$
for the independent variables; it will be presumed obviously the finiteness of $ V(p) $ and $ W(p). $ \par
\vspace{3mm}
There are many works about this problem; the next list is far from being complete:
\cite{Burkholder1}, \cite{Burkholder2},
\cite{Fan1}, \cite{Grama1}, \cite{Grama2}, \cite{Hitczenko1}, \cite{Lesign1}, \cite{Liu1}, \cite{Osekowski1}, \cite{Ostrovsky0},
\cite{Ostrovsky7}, \cite{Ostrovsky8}, \cite{Peshkir1}, \cite{Ratchkauskas1}, \cite{Volny1} etc. \par
See also the reference therein.\par
Notice that in the articles \cite{Lesign1}, \cite{Liu1}, \cite{Volny1} and in many others are described some new
applications of these estimates: in the theory of dynamical system, in the theory of polymers etc.\par
\vspace{4mm}
\section{ Main result: moment estimation for polynomial martingales. }
\vspace{4mm}
We must describe some new notations. The following function was introduced by A.Osekowski (up to factor 2)
in the article \cite{Osekowski1}:
$$
Os(p) \stackrel{def}{=} 4 \ \sqrt{2} \cdot \left( \frac{p}{4} + 1 \right)^{1/p} \cdot \left( 1 + \frac{p}{\ln (p/2)} \right). \eqno(2.1)
$$
Note that
$$
K = K_{Os} \stackrel{def}{=} \sup_{p \ge 4} \left[\frac{Os(p)}{p/\ln p} \right] \approx 15.7858, \eqno(2.2)
$$
the so-called Osekowski's constant. \par
Let us define the following numerical sequence $ \gamma(d), \ d = 1,2,\ldots: \ \gamma(1) := K_{Os} = K, $ (initial condition) and
by the following recursion
$$
\gamma(d+1) = \gamma(d) \cdot K_{Os} \cdot \left( 1 + \frac{1}{d} \right)^d. \eqno(2.3)
$$
Since
$$
\left( 1 + \frac{1}{d} \right)^d \le e,
$$
we conclude
$$
\gamma(d) \le K_{Os}^d \cdot e^{d-1}, \ d = 1,2,\ldots. \eqno(2.4)
$$
\vspace{3mm}
{\bf Theorem 2.1.} {\it Let the sequence } $ \gamma(d) $ {\it be defined in (2.3). Then }
$$
U(p;d,n) \le \gamma(d) \cdot \frac{p^d}{(\ln p)^d } \cdot V_d(p) =
\gamma(d) \cdot \frac{p^d}{(\ln p)^d } \cdot \prod_{m=1}^d \mu_m(d \ p). \eqno(2.5)
$$
\vspace{3mm}
{\bf Proof.} \\
\vspace{3mm}
{\bf 0.} We will use the induction method over the "dimension" $ d, $ as in the article of authors \cite{Ostrovsky4},
starting from the value $ d=1. $ \\
{\bf 1.} One dimensional case $ d=1.$ We apply the celebrate result belonging to A.Osekowski \cite{Osekowski1}:
$$
\left|\sum_{k=1}^n \xi_k \right|_p \le C_{Os}(p) \cdot
\left\{ \left| \left( \sum_{k=1}^n {\bf E} \xi_k^2/F(k-1) \right)^{1/2} \right|_p +
\left| \left( \sum_{k=1}^n |\xi_k|^p \right)^{1/p} \right|_p \right\} \stackrel{def}{=} \eqno(2.6)
$$
$$
C_{Os}(p) \left\{ S_1(p) + S_2(p) \right\}, \eqno(2.6a)
$$
in our notations; $ \xi_k = \xi(1,k), \ F(0) = \{ \emptyset, \Omega \}. $ The variable
$$
\theta(n) := \left( \sum_{k=1}^n {\bf E} \xi_k^2/F(k-1) \right)^{1/2}, \eqno(2.7)
$$
so that $ S_1(p) = |\theta(n)|_p, $ is named in \cite{Osekowski1} by "conditional square function"
of our martingale and the variable $ \theta^2(n) \ - $ by
"quadratic (predictable) characteristic" in the review \cite{Peshkir1}. \par
We deduce using Iensen and triangle inequalities taking into account the restriction $ p \ge 4: $
$$
\theta^2(n) = \sum_{k=1}^n {\bf E} \xi_k^2 /F(k-1),
$$
$$
|\theta^2(n)|_{p/2} \le \sum_{k=1}^n | \ {\bf E} \xi_k^2 /F(k-1) \ |_{p/2} \le \sum_k |\xi_k|_p^2 = \sum_{k=1}^n \mu_k^2(p). \eqno(2.8)
$$
Since
$$
|\theta^2(n)|_{p/2} = |\theta(n)|_p^2,
$$
we ascertain
$$
S_1(p) = |\theta(n)|_p \le \sqrt{ \sum_{k=1}^n \mu_k^2(p) }. \eqno(2.9)
$$
\vspace{3mm}
Let us estimate now the value $ S_2(p). $ This evaluate is simple:
$$
S_2^p(p) = {\bf E} \left( \sum_k |\xi_k|^p \right) = \sum_k |\xi_k|_p^p = \sum_{k=1}^n \mu_k^p(p),\eqno(2.10)
$$
$$
S_2(p) \le \left( \sum_{k=1}^n \mu_k^p(p) \right)^{1/p} \le \sqrt{ \sum_{k=1}^n \mu_k^2(p) }. \eqno(2.11)
$$
Thus,
$$
\left| \ \sum_{k=1}^n \xi_k \ \right|_p \le K_{Os} \cdot \frac{p}{\ln p} \cdot \sqrt{ \sum_{k=1}^n \mu_k^2(p) }. \eqno(2.12)
$$
\vspace{3mm}
{\bf 2.} Since the sequence $ \{ b(i) \} $ is non-random, the random sequence $ \{\xi^{(b)}(k) \} := \{ b(k) \cdot \xi(k) \} $ is also
a sequence of martingale differences relative at the same filtration. We apply the last inequality (2.11) for the martingale differences
$ \{\xi^{(b)}(k) \}: $
$$
\left| \ \sum_{k=1}^n b(k) \xi_k \ \right|_p \le K_{Os} \cdot \frac{p}{\ln p} \cdot \sqrt{ \sum_{k=1}^n b^2(k) \mu_k^2(p) }, \eqno(2.13)
$$
and we obtain after taking supremum over $ \vec{b} \in B: $
$$
U(p;1,n) = \sup_{b \in B} | \ Q(1,n, b, \{ \xi(\cdot) \}) \ |_p \le K_{Os} \cdot \frac{p}{\ln p} \cdot \sup_k \mu_k(p),
$$
or equally
$$
U(p;1,n) \le \gamma(1) \cdot \frac{p}{\ln p } \cdot V_1(p). \eqno(2.14)
$$
\vspace{3mm}
{\bf 3. Remark 2.1.} We deduce as a particular case choosing in (2.13) $ b(k) = 1/\sqrt{n}: $
$$
n^{-1/2} \left| \ \sum_{k=1}^n \xi_k \ \right|_p \le K_{Os} \cdot \frac{p}{\ln p} \cdot \sqrt{ n^{-1} \sum_{k=1}^n \mu_k^2(p) }, \eqno(2.15)
$$
which is some generalization of the classical Rosenthal's inequality on the martingale case and in turn is a slight
simplification of the A.Osekovski result. \par
In turn,
$$
\sup_n \left[ n^{-1/2} \left| \ \sum_{k=1}^n \xi_k \ \right|_p \right] \le K_{Os} \cdot \frac{p}{\ln p} \cdot \sup_{k} \mu_k(p). \eqno(2.15b)
$$
\vspace{3mm}
{\bf 4. Induction step } $ d \to d+1 $ is completely analogous to one in the article \cite{Ostrovsky7}, section 3, and may be omitted.\par
\vspace{3mm}
{\bf 5. Remark 2.2; an example.} Suppose $ d = \const \ge 2, \ n \ge d + 1. $
We deduce as a particular case choosing
$$
b(\vec{i}) = 1/\sqrt{n(n-1) \ldots (n-d+1)} \sim n^{-d/2}: \hspace{6mm} n^{-d/2} |R(d)|_p =
$$
$$
n^{-d/2}| Q(d,n, \vec{1})|_p = n^{-d/2} \left| \ \sum_{\vec{i} \in I(d,n) } \xi(\vec{i}) \ \right|_p \le
C(d) \cdot \frac{p^d}{\ln^d p} \cdot \left[ \sup_{i,m} \mu_{i,m}(p) \right]^d. \eqno(2.16)
$$
\vspace{4mm}
\section{Independent case. }
\vspace{4mm}
The reasoning is basically at the same as in the last section. We will use the famous Rosenthal's inequality
(more exactly, a consequence of this inequality) \cite{Rosenthal1} instead the Osekowski's estimate:
$$
n^{-1/2} \left|\sum_{k=1}^n \xi_k \right|_p \le K_{R} \cdot \frac{p}{\ln p} \cdot \sqrt{ n^{-1} \sum_{k=1}^n \mu_k^2(p) }, \eqno(3.1)
$$
where now $ \{\xi_k \} $ is the sequence of the centered independent random variables with finite $ p^{th} $ moment,
$ K_R $ is the Rosenthal's constant. This estimate is non-improvable. \par
The exact value of this constant is obtained in \cite{Ostrovsky9}:
$$
K_R \approx 1.77638/e \approx 0.6535.
$$
\vspace{3mm}
Define the announced sequence $ \kappa = \kappa(d), \ d = 1,2,\ldots $ as follows: $ \kappa(1) := K_R $ and by the following recursion
$$
\kappa(d+1) = \kappa(d) \cdot K_{Os} \cdot \left( 1 + \frac{1}{d} \right)^d. \eqno(3.2)
$$
Since
$$
\left( 1 + \frac{1}{d} \right)^d \le e,
$$
we conclude
$$
\kappa(d) \le K_R \cdot (K_{Os} \cdot e)^{d-1}, \ d = 1,2,\ldots. \eqno(3.3)
$$
\vspace{3mm}
{\bf Theorem 3.1.} {\it Let the sequence } $ \kappa(d) $ {\it be defined in (3.3). Then in the considered here independent case }
$$
U(p;d,n) \le \kappa(d) \cdot \frac{p^d}{(\ln p)^d } \cdot W_d(p) =
\kappa(d) \cdot \frac{p^d}{(\ln p)^d } \cdot \prod_{m=1}^d \mu_m(p). \eqno(3.4)
$$
\vspace{3mm}
{\bf Remark 3.1.} Let us emphasise the difference between martingale and independent cases. This difference is except the coefficient
but in the factors $ V_d(p) = \prod_{m=1}^d \mu_m(d \cdot p) $ and $ W_d(p) = \prod_{m=1}^d \mu_m(p). $ \par
It is clear that there are many examples when $ W_d(p) < \infty $ but $ V_d(p) = \infty. $ \par
\vspace{4mm}
\section{Exponential bounds for tails of polynomial martingales. }
\vspace{4mm}
We intend in in this section to obtain the {\it exponential } bounds for tails of distribution for the r.v. $ Q(d,n) $
through its (obtained) moments estimates. We can consider only the martingale case (section 2). \par
\vspace{3mm}
{\bf Theorem 4.1.} Suppose that the described below sequence of the mean zero martingale differences $ \{ \xi(i,m) \} $ satisfies
the restriction
$$
\sup_{i,m} \max( {\bf P}( \xi(i,m) \ge x ), {\bf P}( \xi(i,m) \le - x ) ) \le \exp \left( - C_1 x^{q} \ (\ln x)^{- q \ r} \right),\eqno(4.1)
$$
$$
x > e, \ C_1 = \const > 0, \ q = \const > 0, \ r = \const. \eqno(4.1a)
$$
Then
$$
\sup_{b \in B} \max( {\bf P}( Q(d,n,b) \ge x ), {\bf P}( Q(d,n,b) \le - x)) \le
$$
$$
\exp \left\{ - C_2 \ x^{q/(dq + 1) } \ (\ln x)^{- q(r-d)/( dq+ 1 ) } \right\}, \ x > e. \eqno(4.2)
$$
\vspace{3mm}
{\bf Proof.} It follows from the theory of the so-called Grand Lebesgue spaces \cite{Kozatchenko1},
\cite{Ostrovsky3}, chapter 1, section 1.8, \cite{Ostrovsky7} that the inequality (4.1) is equivalent to the
finiteness of the following norm
$$
\sup_{i,m} \sup_{p \ge 4} \left[ |\xi(i,m)|_p \cdot p^{-1/q} \cdot \log^{-r} p \right] = C_3 < \infty, \eqno(4.3)
$$
or equally
$$
\sup_{i,m} |\xi(i,m)|_p \le C_3 \ p^{1/q} \ \log^r p. \eqno(4.3a)
$$
We apply the theorem 2.1:
$$
\sup_{b \in B}| Q(d,n,b)|_p \le C_4 \ p^{d + 1/q} \ [\log p]^{r - d},
$$
which is in turn equivalent to the proposition (4.2). \par
Note that other exponential bounds for tail of distribution for the r.v. $ Q(d,n,b) $ under some additional conditions
is obtained in \cite{Ostrovsky4}. \par
\vspace{4mm}
\section{Concluding remarks. }
\vspace{4mm}
{\bf A. Examples of lower estimates.}\\
\vspace{3mm}
Denote in the independent case
$$
K_I(p; d) = \sup_n \sup_{b \in B} \sup_{{\xi(i,m): |\xi(m,i)|_p < \infty}} \left[ \frac{Q(d,p)}{\prod_{m=1}^d \mu_m(p)} \right].\eqno(5.1)
$$
where the last $ "\sup" $ is calculated over all the sequences of the centered {\it independent} variables $ \{ \xi(i,m) \} $
satisfying the condition $ |\xi(i,m)|_p < \infty. $ We obtained
$$
K_I(p; d) \le \frac{C_0(d) \ p^d}{ \ln^d p}, \ C_0(d) = \const > 0.
$$
Our new statement:
$$
K_I(p; d) \ge \frac{C(d) \ p^d}{ \ln^d p }, \ C(d) = \const > 0. \eqno(5.2)
$$
Proof is very simple. The moment estimations are derived in \cite{Kallenberg1} for
the symmetrical polynomials on mean zero independent identical symmetrically
distributed variables, i.e. particular case for us, for which it is proved that
$$
\frac{|Q(d,n)|_p}{ \sqrt{\Var Q(d,n)} \ \mu^d(p)} \ge \frac{C_1(d) \ p^d}{ \ln^d p}, \ C_1(d) = \const > 0. \eqno(5.3)
$$
\vspace{4mm}
Another a more simple example. Let $ n = 1 $ and a r.v. $ \eta $ has a Poisson distribution with unit parameter:
$$
{\bf P} (\eta = k) = e^{-1}/k!,
$$
and define $ \xi = \eta - 1, $ then the r.v. $ \xi $ is centered and
$$
p \to \infty \Rightarrow |\xi|_p \sim p/( e \cdot \ln p).
$$
Let also $ \xi_j, \ j = 1,2,\ldots $ be independent copies of $ \xi. $ Then
$$
\left|\prod_{j=1}^d \xi_j \right|_p \sim e^{-d} \frac{p^d}{(\ln p)^d}, \ p \to \infty. \eqno(5.4)
$$
\vspace{3mm}
{\bf B. Estimations for normed variables.} \\
\vspace{3mm}
Denote
$$
\Psi(b) = \Psi(d,n,b) = \Var( Q(d,n,b)), \ b \in B, \eqno(5.5)
$$
and impose the following condition on the martingale distribution
$$
0 < C_1(d) \le \sup_n \sup_{b \ne 0} \left[ \frac{\Psi(d,n,b)}{||b||^2} \right] \le C_2(d) < \infty. \eqno(5.6)
$$
This condition was introduced and investigated in \cite{Ostrovsky4}. \par
We define also a so - called {\it relative moments} for the r.v. $ \{ \xi(i,m) \} $ under the {\it natural norming:}
$$
\tilde{\mu}_m(p) \stackrel{def}{=} \sup_i \left| \ \xi(i,m)/\sqrt{\Var(\xi(i,m))} \ \right|_p, \eqno(5.7)
$$
Denote also in the martingale case
$$
\tilde{V}(p) = \tilde{V}_d(p) \stackrel{def}{=} \prod_{m=1}^d \tilde{\mu}_m(d \cdot p). \eqno(5.8a)
$$
and for the independent variables $ \{ \xi(i,m) \} $
$$
\tilde{W}(p) = \tilde{W}_d(p) \stackrel{def}{=} \prod_{m=1}^d \tilde{\mu}_m(p). \eqno(5.8b)
$$
{\bf We will derive as before the moment estimations of a form }
$$
\tilde{U}(p;d,n) = \tilde{U}(p) \stackrel{def}{=}
$$
$$
\sup_{b \in B} \left| \ Q(d,n, b, \{ \xi(\cdot) \}) /\sqrt{ \Var ( Q(d,n, b, \{ \xi(\cdot) \}) ) } \ \right|_p \le
$$
$$
\tilde{Q}(p;d,n) = \tilde{Q}(p).\eqno(5.9)
$$
Namely,
$$
\tilde{U}(p;d,n) \le \gamma(d) \cdot \frac{p^d}{(\ln p)^d } \cdot \tilde{V}_d(p), \eqno(5.10a)
$$
in general (martingale) case and
$$
\tilde{U}(p;d,n) \le \kappa(d) \cdot \frac{p^d}{(\ln p)^d } \cdot \tilde{W}_d(p), \eqno(5.10b)
$$
for the independent variables; it will be presumed of course the finiteness of the variables $ \tilde{V}(p) $ and $ \tilde{W}(p). $ \par
\vspace{4mm}
Authors hope that the last two estimates are more convenient for the practical using. \par
\newpage
{\bf C. Possible generalizations.} \\
\vspace{3mm}
It is interest by our opinion to generalize our estimates on the {\it predictable} sequence $ b(\vec{i}). $
A preliminary (one-dimensional) result in this direction see in the article \cite{Ostrovsky0};
see also \cite{Choi1}.\par
\vspace{4mm} | 144,077 |
As FOSCL wraps up another Beach & Bay Cottage Tour, not a step is being missed in gearing up for the group’s Summer Book Sale, Aug. 14-16.
A summer staple of Bethany Beach documented as the “Annual Book Sale” in the 1998 FOSCL newsletter, the event continues to be one that many visitors consider when scheduling their vacation, noted Theo Loppatto, FOSCL Book Sale Committee and FOSCL publicity chair. All proceeds of the event go to the library, to enhance and improve its services and programs to the community.
The book sale kicks off on Thursday, Aug. 14, at 10 a.m. at the South Coastal Library. The sale takes place Thursday, Aug. 14, from 10 a.m. to 8 p.m.; Friday, Aug. 15, from 1 to 5 p.m.; and Saturday, Aug. 16, from 9 a.m. to 3 p.m.
Unique to this fundraiser is its green character, Loppatto said. The materials on sale are donations from the community, and the sale itself is produced and directed by volunteer members of the community the library serves, both full-time residents and summer-only residents and visitors.
“Furthermore, it is not uncommon to see a particular book or other material donated, purchased, re-donated and repurchased; and the green cycle continues,” she said. “The dedicated cadre of volunteers look forward to their stint at the book sale as an opportunity to visit with friends and see neighbors as they move through the sale,” she added.
“Many of the book sale volunteers have been working the sale for years and have their own area of expertise. For example, among our set-up crew, there are volunteers who for years have set up a particular area of the sale room, be it bestsellers, non-fiction, children’s materials, DVDs and CDs, or paperbacks,” said Loppatto. “They know the materials and the most effective way to display them.”
She described the production for the sale itself as being like a finely tuned ballet.
“The Friends of South Coastal Book Sale Committee sets up tables and places signage throughout the room early the day prior to sale,” Loppatto said. “At 10 that same day, a dedicated group of volunteers arrives with trucks and hand trolleys to physically move the merchandise from its storage facility into the library. After they finish, another group of volunteers comes to unpack and arrange the materials in the sale room.
“This process reverses itself at the end of the sale,” she noted. “During the three days of the August sale, another wonderful group of volunteers serve as cashiers and customer service assistants to ensure customer satisfaction.
“This year’s sale promises to be full of wonderful materials judging by donations that continue to come to the library and are accepted up to and throughout the sale,” Loppatto said. “There may even be a few surprise sale events!” | 200,138 |
Otherwise, it will create a negative perception of judiciary's independence, says CJ
KUALA LUMPUR: The Attorney-General must stop leading the Judicial and Legal Service Commission to ensure that the judiciary can be seen as a truly independent body, says the Chief Justice.
The head of the judiciary, Tun Arifin Zakaria, said that it would be a conflict of interest for the A-G to lead both services as he was a member of the Executive, when judicial officers comes under the judiciary.
In a democracy, the three branches of Government – the Legislative, Executive and Judiciary – must remain independent of each other.
“If the A-G continues to lead both services, I worry it would create a negative perception of the judiciary’s independence, an opinion many parties share,” said Arifin in a speech at the Judicial Officers Conference here yesterday.
The Chief Justice’s call is in line with the universal concept of judicial independence, whereby the courts should not be subject to undue influence from other branches of the Government or persons with partisan interests.
In an immediate reaction, Attorney-General Tan Sri Mohd Apandi Ali confirmed that the A-G’s Chambers (AGC) had received the proposal and was still studying it from the point of view of the Constitution and from a historic perspective.
“We will come up with the AGC’s views and discuss it at our next Legal and Judicial Service Commission’s meeting before the end of the year,” he told The Star.
Currently, the Judicial and Legal Service Commission managed the careers – from appointing, promoting, transferring and disciplining – of its members, which includes judicial officers like Sessions Court judges and magistrates, and legal officers like deputy public prosecutors and senior federal counsels.
Later, during a press conference, Arifin said people who disagreed with a judgment might say the magistrates were toeing the line with the A-G’s Chambers as they were effectively the same body.
“Imagine if a senior officer from the AGC or even the A-G himself was prosecuting. Lagi menggeletar (they’ll be even more nervous) to handle the case,” he said.
Arifin said Public Service Circular 6/2010 which made the A-G the chief of the judicial service was a contradiction to an existing decision by the Federal Court and no longer relevant
He pointed out that when the Commission was formed, the two groups were placed together as there were only a few hundred staff members. However, there were now 636 employees in the legal service and 4,787 serving in the judicial service as of April this year.
“The time has come for the judicial service to be lead by an someone from within its ranks,” he said, adding that such a candidate would be better equipped to run the service.
Arifin suggested that the Chief Court Registrar lead the judicial service while the Attorney-General lead the legal service.
The separation would also stop judicial officers and legal officers from being transferred between departments, unless the move is approved in writing by their chiefs.
However, Arifin said transfers should still be allowed, with due process, to ensure staff get experience as both judges and prosecutors.
Chief Registrar Datin Latifah Mohd Tahar, who also attended the conference, told reporters the paper on the proposal had been submitted to the Commission and the matter could be decided on within the year.
In 2006, the then Chief Justice Tun Ahmad Fairuz Sheikh Abdul Halim said the Judiciary intended to propose to the Government to abolish the Judicial and Legal Service Commission.
He added that magistrates and Sessions Court judges should be absorbed into the judiciary, fearing that there would be interference by “unseen hands” if they remain as civil servants.
by Chelsea L.Y. Ng and Qishin Tariq The Star/Asia News Network
It’s about time, says thelegal fraternity of proposal
PETALING JAYA: The legal fraternity applauded the Chief Justice’s proposal for greater separation between judicial and legal services, calling it long overdue.
Former Court of Appeals Justice Mah Weng Kwai (pic) said the proposal finally presented a clear demarcation between the judicial and legal services.
“It has been a combined service for the longest time, since before I joined the service in 1973,” said Mah, who started his career as a magistrate before becoming a deputy public prosecutor and then senior federal counsel.
Responding to the Chief Justice’s suggestion that officers would still be allowed to be transferred between the services, Mah said it should be taken one step further with both services completely independent and non-transferable.
Former magistrate Akbardin Abdul Kader said, if implemented, the move would ensure former DPPs were not biased when they were elevated to the bench.
“Hence, they will remain as DPPs until they retire and so the same for judicial officers,” he said.
Malaysian Bar president Steven Thiru said the Chief Justice’s concerns were valid and deserved due consideration.
He said the fact the Attorney-General was a member of the commission could open the judiciary to questions in any decision in favour of the prosecution.
He noted that the proposal would appear to require a constitutional amendment that would place Sessions Court judges and magistrates under the sole jurisdiction of the judiciary, and no longer under the Commission.
“This strengthens the concept of separation of powers that vests judicial power in the judiciary and requires the exercise of those powers without any influence by the other arms of Government,” he said, adding that the removal of any conflict of interest would inspire more confidence in the decisions of Sessions Court judges and magistrates.
Former Malaysian Bar president Yeo Yang Poh said the Bar had called for the change for decades, adding that from time to time, a Chief Justice of the day would “warm up” to the idea.
In 2006, when Yeo was serving as president, Chief Justice Tun Ahmad Fairuz Sheikh Abdul Halim made a similar call for a separation of the judiciary from the commission.
Yeo added that it was the first time he had heard of a proposal being handed to the commission by the Chief Registrar.
He said having the judicial and legal services combined was not desirable for two reasons: in practical terms, not every one could be fearless; while in theory, even if all legal officers could overcome the pressure, there would still be the perception of impartiality.
“You can’t blame an observer that perceives something is not quite right. A judge could say they would remain impartial even if judging their father; but does it look right?” he asked.
A former officer from the Judicial And Legal Services, who declined to be named, said the risk of transfers were a common reality.
“We used to threaten judges up to the Sessions Court (level), if they misbehave, we will get them transferred as DPPs. A few of them were actually transferred,” he said.
He said though the “threats” were in jest, it shocked him that they were sometimes really carried out, adding that not all moves were sinister, as it was occasionally meant as a lesson for subordinate courts which had made errant judgments.
Jan 22, 2014 ... Stupid fellow ! Dr Ling, former Malaysian Transport Minister slams Attorney- General. UTAR Council Chairman Tun Dr Ling Liong Sik speaking ...
Jul 7, 2015 ... Separate the Attorney-General's powers to correct a flaw in Malaysian legal system ! ... There is a flaw in our system, inherited since before Independence, that may ... Najib has denied any wrongdoing and he is said to be mulling legal .... Dr Ling, former Malaysian Transport Minister slams Attorney-General.
Apr 8, 2016 ... “1MDB relied on debt (bank loans, bonds and sukuk) to form its capital, a chunk of which had been sanctioned or supported by the Government ...
Jun 4, 2015 ... Malaysia's 1MDB's questionable accounts. Summary raises questions over spending. It shows where money went but fails to debunk critics. | 110,512 |
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Project/Program Manager
Mountain Limited
PROFESSIONAL / BUSINESS / MANAGERIAL. Manage/Coordinate/Prioritize a long list of projects & Annual Enrollment.
4 days ago Event Planning Intern Description
United Way of Dane County
Work closely with sponsoring companies in the business community. For more information, or to apply, please email a resume and a cover letter to [email protected]..
18 hours Financial Analyst, Finance Department
Gurtin Fixed Income Management
Gurtin Fixed Income Management, a boutique asset management firm with over $10 billion in assets under management, seeks a financial analyst able to assist with..
7 days ago | 99,840 |
Senior Clinical Data Manager (440768)
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POST DATE 9/13/2016
END DATE 10/12/2016
Advanced Clinical
Richmond, VA
JOB DESCRIPTIONJob Description:
OVERVIEW:Acts as a Lead CDM across programs/studies overseeing all Data Management (DM) activities performed internally or by CROsDM representative in Core Study Team Meetings for assigned Program(s) Primary point of contact for DM at the Global Program Team.Ensures consistency on the design of the trials within program(s) (standard CDASH CRFs, edit checks, clinical data review guidelines, CRF completion guidelines)Coordinate all DM activities performed by CRO/Vendor(S) and internal cross functional team Study Team members. Oversees the study start-up, conduct & finalization for Data Management activities within Program(s)Ensures study conventions, processes, knowledge sharing and best practice exist for program(s)/studies assignedLeads the review and implementation of Clinical Data Review Guidelines Defines requirements for data validation based on metrics generated and lessons learned from cleaning activitiesPlans appropriately to ensure adherence to timelines, including assignment of workloads and identifying resource and timeline issues within program(s)Mentors and trains new DM personnel and/or other team members as assignedSupports & assists junior staff for assigned trials Provides effective input into DM initiatives and innovations for quality, efficiency and continuous improvement High level activities to be performed:Reviews and contributes to preparation of protocols, specifically related to Schedule of Events (SOE) and Data Management sectionParticipates in Investigator/KOM meetings Provides expertise to the team for DM processes and deliverables Performs and obtains input from study team members for DM study start-up activities including preparing or reviewing the eCRFs, eCCGs, DMP, and performing User Acceptance Testing (UAT), database go-live Responsible for ongoing review of data, database maintenance and creation / maintenance of study documentationAccountable and responsible for ensuring data quality and timeliness of study-related DM deliverables internally or from CROsCollaborates with CROs and study team members to identify and resolve issues that may impact deliverables Identifies and communicates data trends to study teamCollaborates with internal team and with CROs to pro-actively manage data availability and integrationCoordinates all DM Data Review activities and any other Clinical Study Team Data ReviewEnsures completion of all activities leading to the Study Database Lock (including medical coding and SAE reconciliation)Liaises with Statistical Programming and Biostatistics for data deliverables for IDMC and SDTM datasets Collaborates in the creation and maintenance of DM SOP/WI(s), Guidelines, and Forms as required Interacts with the study team members to define the necessary listings and reports needed to support study data validation, review and cleaning process. QUALIFICATIONS/EXPERIENCE:Exhibits strong leadership, communication (written and oral), interpersonal communication skills, logical thinking, attention to detail and problem solving abilities. Excellent project management, organizational, prioritization, and multitasking skills. Demonstrated ability to work independently.Experience in effectively collaborating with CRO s and external vendors. Experience with clinical data management systems, proficiency in EDC (preferably RAVE)Experience in basic SAS / SQL programming.Extensive experience of CDISC/CDASH and other industry standards.Experience with reporting tools like J-review is a plus.Full knowledge of the drug development, study conduct processes, regulations related to Data Management and clinical research such as CFR Title 21, ICH Guidelines, EU Directives, GCP and relevant regulatory compliance.Extensive CRO/Vendor management experience including day to day operations.Full knowledge of clinical data flow in clinical trial start-up, conduct, closing and submission.Works well in a cross functional team environment with and ability to work in a team to meet fast paced timelines and achieve deadlines. Education:Bachelor s degree and/or at least 10-12 yrs. of pharmaceutical experience, including data management and clinical trials
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Let Advanced Clinical help you find a career in the clinical research industry! We have built strong relationships with client companies and can match your talents with the right clinical research jobs. | 39,575 |
TITLE: Sum of positive divisors if and only if perfect square
QUESTION [1 upvotes]: let n be a positive odd integer, prove that the sum of the positive divisors of n is odd if and only if n is a perfect square.
I know that based on the prime factorization theory that every integer n can be written as the product of primes, if their sum is odd that means that there are equal pairs of even and odd divisors. Is this enough to conclude that n must be a perfect square?
REPLY [0 votes]: "" if their sum is odd that means that there are equal pairs of even and odd divisors" "
That doesn't actually make sense.
What I think you want to say is that if the sum is odd there must be an odd number of odd terms.
(even + even = even, odd + odd = even, even + odd = odd. By induction $a_1 + a_b + ...... + a_n$ is even if and only if there is an even number of odd terms.)
Since $n$ is odd there are no even terms so all factors are odd. And...
The sum of the factors will be odd if and only if there are an odd number of factors.
So we need to prove:
Prop: If $n$ is odd, then $n$ will have an odd number of factors if and only if $n$ is a perfect square.
And, actually, that will be true whether $n$ is even or odd.
The basic idea is that for any factor $k$ then $\frac nk$ is also a factor and factors com in pairs. If any $n$ is a perfect square then $\sqrt n = \frac n{\sqrt{n}}$ is the only factor that pairs to itself.
A more thorough Proof: Let's imagine listing out all the $m$ factors of $n$ in order $k_1=1 < k_2 < ..... < k_m = n$.
Notice that if $k_i$ is a factor then $k_i' = \frac n{k_i}$ is also a factor and must also be in the list. But where is it on the list.
Well, $k_{i- 1} < k_i < k_{i+1}$ so $k_{i+1}' = \frac n{k_{i+1}} < k_{i}' = \frac n{k_{i}} < k_{i-1}' = \frac n{k_{i-1}}$
The list $k_1 < ..... < k_m$ can be rewritten as $k_m' < ..... < k_1'$ and we know $k_i' = k_{m+1-i}'$ (and with the knowledge that $k_i*k_i' = n$).
If $m$ is odd then the middle term is $k_{\frac {m+1}2} = k_{m+1 - \frac {m+1}2}' = k_{\frac {m+1}2}'$ so $k_{\frac {m+1}2}*k_{\frac {m+1}2}' = k_{\frac {m+1}2}*k_{\frac {m+1}2} = n$ and $n$ is a perfect square.
If $m$ is not odd then there is no middle term and there is no $k_i = k_i'$ , so there is no factor so that $k_i * k_i = n$. (Remember, those lists were all the factors.)
So perfect squares are the only integers with an odd number of factors and all perfect squares have an odd number of factors.. | 185,171 |
India pulses to stay dear, export ban seen toothless
By Rajendra Jadhav
MUMBAI (Reuters) - India's decision to extend an export ban on pulses to ease domestic prices that have soared is largely seen as a toothless move, industry officials said.
India on Monday extended a ban on pulses exports for another 12 months, the second since it was first clamped in June 2006.
"We had been exporting only around three percent of total production. So, an extention of the ban will not have any effect on domestic prices," said Gopal Kogta, president of Pulses Manufactures and Exporters Association of India.
"The market was expecting this decision. Sentiments may change in short term, but prices will remain firm due to the lower production," said Chowda Reddy, an analyst at Karvy Comtrade.
Prices of pulses, a staple food for most Indians, have risen significantly in the spot market in the country that is the biggest producer and consumer of the nutritious commodity.
In the Delhi market, the spot price of chana, a major pulse, has risen 21 percent to 2,625 rupees per 100 kg in last three months. In Mumbai, yellow peas rose 32.3 percent to 2,211 rupees per 100 kg in the last one year.
In a year of surging inflation -- it hit a 14-month high in mid-March -- the winter crop output too is likely to fall by 8.8 percent in 2007/08, against 9.4 million tonnes last year.
The lower ground water level and scarcity of rains would lead to lower yield of winter pulses, said Hari Kishan Periwal, a trader based in Bikaner, Rajasthan. Continued... | 253,478 |
TITLE: $\operatorname{deg}(f+g) \leq \max (\operatorname{deg} f, \operatorname{deg} g)$.
QUESTION [0 upvotes]: Show that $\operatorname{deg}(f \cdot g)=\operatorname{deg} f+\operatorname{deg} g$ and $\operatorname{deg}(f+g) \leq \max (\operatorname{deg} f, \operatorname{deg} g)$.
Shouldn't $\operatorname{deg}(f+g)$ be equal to $\max (\operatorname{deg} f, \operatorname{deg} g)$, as we are multiplying both the polynomials? Please help as I can't figure out ehy it should be true.
REPLY [3 votes]: If $\deg f \neq \deg g$ then $\deg(f + g) = \max\{\deg f, \deg g\}$ but if $\deg f = \deg g$ then $\deg(f + g)$ can be any possible degree $\le \max\{\deg f, \deg g\}$.
This is because if $\deg f = \deg g = n$ and $f(x) = x^n + f_1(x)$ and $g(x) = -x^n + g_1(x)$ then
$$f(x) + g(x) = (x^n + f_1(x)) + (-x^n + g_1(x)) = f_1(x) + g_1(x).$$
And here the degree of $f_1 + g_1$ can be anything.
REPLY [3 votes]: Assuming that you meant to write "... as we are adding both polynomials", the inequality is there because the leading terms (and possibly other terms too) might be opposites of one another.
A quick example: $f(x) = 3x^2 + 5x - 2$ and $g(x) = -3x^2 - 2x + 1$. Then,
$$
(f+g)(x) = f(x) + g(x) = 3x - 1.
$$
Clearly,
$$
\deg(f+g) = 1 < 2 = \max(2, 2) = \max(\deg f, \deg g).
$$
By the way, a similar issue arises if you are considering polynomials with coefficients from a ring with zero divisors, such as $\mathbb{Z}/6\mathbb{Z}$, in which $2 \cdot 3 = 0$. The more general property that still holds is
$$
\deg(f \cdot g) \leq \deg f + \deg g
$$
Assuming that your coefficients are in an integral domain (such as $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{Z}/p \mathbb{Z}$ for $p$ a prime), then the degree function behaves just like the logarithm:
$$
\deg(f \cdot g) = \deg f + \deg g
$$ | 42,014 |
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But like.. where are you suppose to build? There's uneven surfaces everywhere...
I just realised. I've talked to you before. Like, a year ago. Hello ^~^
I am also good. I just got given a lot of clothes and shoes for free and they are nice
I am happy.
And he was glorious in Star Trek hnng.
This is the other. | 361,356 |
A front door is an entryway to greet visitors at. It is the main path through which guests can come inside and also figure out the rest of the property. When it comes to replacing the front door, homeowners should not take it lightly. Even if the entire home is decorated, an uninviting door can create a lousy buzz among your guests. The basic properties of a good front door must be weather resistance, stylish and innovative.
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Wooden doors come in a variety of options to pick from. The type and the rarity of the wood used stands responsible for the manufacturing costs. The most interesting thing is that they are durable but require very less maintenance. The maintenance requirement depends on the quality of the wood and how it has been treated while manufacturing. Wood naturally regulates humidity. So, it can absorb moisture from the air during a wet season as well as release moisture when the weather is dry.
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In a case of a wooden frame, joint problem can easily be solved using wooden beads.
It is important to note that when wooden doors are compared with any other type of door, it can be fathomed that wooden doors require very less maintenance, but deliver uniqueness, longevity and beauty. | 75,644 |
\begin{document}
\title{The genus and the category of configuration spaces}
\author{R.N.~Karasev}
\email{r\_n\[email protected]}
\address{
Roman Karasev, Dept. of Mathematics, Moscow Institute of Physics
and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700}
\thanks{This research was supported by the President of the Russian Federation grant No. MK-1005.2008.1, and partially supported by the Dynasty Foundation.}
\keywords{configuration space, equivariant topology, Lyusternik-Schnirelmann category}
\subjclass[2000]{55R80,57S17,14N20}
\begin{abstract}
In this paper configuration spaces of smooth manifolds are considered. The accent is made on actions of certain groups (mostly $p$-tori) on this spaces by permuting their points. For such spaces the cohomological index, the genus in the sense of Krasnosel'skii-Schwarz, and the equivariant Lyusternik-Schnirelmann category are estimated from below, and some corollaries for functions on configuration spaces are deduced.
\end{abstract}
\maketitle
\section{Introduction}
The main subject of this paper is configuration spaces of smooth manifolds. Let $M$ be a smooth manifold. A particular case of configuration space is the space of all sequences $(x_1, \ldots, x_n)\in M^n$ of pairwise distinct points. This classical configuration space is of particular interest as the space of configurations of $n$ points in classical mechanics. Some topological properties of this space are studied in the book~\cite{fh2001}.
In this paper we are interested in the topology of the obvious action of the symmetric group $\mathfrak S_n$ on the space of $n$-configurations; we also consider the restriction of this action to some subgroups $G\subseteq \mathfrak S_n$. The space of $n$-configurations in $\mathbb R^k$ was studied in \cite{fuks1970,vass1988,rot2008}. The main object of these studies was some measure of complexity of the configuration space, such as the homological index, or the genus in the sense of Krasnosel'skii-Schwarz~\cite{kr1952,schw1957,schw1966}, see Section~\ref{genus-and-cat}. The estimates on the genus of the configuration space of $n$ pairwise distinct points in $\mathbb R^2$ were used to study the ``topological complexity'' of finding complex roots of a polynomial in~\cite{vass1988}, see also~\cite{sma1987}.
Along with the configuration space of pairwise distinct points, we consider more general configuration spaces. By a configuration space for $M$ we mean some subspace of its Cartesian power $M^n$, defined by removing configurations with some multiple point coincidences. For example, some pairs in the configuration may be required to be distinct points, or every $k$ of points in the configuration may be required not to be the same point. One example of such a configuration space was used in the author's paper~\cite{kar2008bil} and the previous studies~\cite{fartab99,far00} of billiards in smooth convex bodies in $\mathbb R^d$. In the cited papers, some estimates on the homological index (see Sections~\ref{eq-cohomology} and \ref{eq-cohomology-p-tori} for definitions) of the configuration space, combined with the Lyusternik-Schnirelmann theory, gave some lower bounds on the number of distinct closed billiard trajectories. Another application of configuration spaces with general coincidence constraints arises in the study of coincidences of maps, see some examples in~\cite{vol1992,vol2005}.
As it was already mentioned, in addition to considering the action of $\mathfrak S_n$ on configuration spaces we also consider the actions of $G\subset \mathfrak S_n$. Let the group $G$ act on the points freely and transitively. In this case the configuration space may be considered as the space of all maps $G\to M$, denote it $\Map(G, M)$. The group $G$ acts on itself by left multiplications, so $G$ acts on $\Map(G, M)$ by right multiplications. We denote this action by
$$
(g, \phi)\in G\times \Map(G, M)\mapsto \phi^g,
$$
by definition put $\phi^g(x) = \phi(gx)$ for any $g,x\in G$.
The paper is organized in the following way. The main tool is considering the equivariant cohomology of configuration spaces. In Section~\ref{eq-cohomology} we collect the general notions on the equivariant cohomology, define a measure of homological complexity (index) of $G$-spaces. In Section~\ref{eq-cohomology-p-tori} we focus on the case of $G$ being a $p$-torus, introduce another index and prove a new result: Theorem~\ref{indind} on estimating the index from below.
In Section~\ref{index-conf} some lower bounds of topological complexity (indexes) of particular configuration spaces with respect to the group action are given.
In Sections~\ref{symm-config} and \ref{genus-and-cat} the main results on configuration spaces are formulated and proved. The results of Section~\ref{symm-config} show the existence of a configuration with certain symmetrical equalities, the author is particularly interested in metric symmetries, but the results are stated for general functions instead of a metric. Results of Section~\ref{genus-and-cat} estimate the genus in the sense of Krasnosel'skii-Schwarz of configuration spaces and may be used to find lower bounds on the number of critical points of a smooth symmetric function using the Lyusternik-Schnirelmann theory.
Note that in this paper Theorems~\ref{constr-conf-gen} and~\ref{pwconstr-gen} and their corresponding Lemmas~\ref{constr-conf-ind} and \ref{pwconstr-ind} (see below) are actually proved for $\mathbb R^d$. It seems plausible that for a closed manifold $M$ the estimate on the index may be larger, as it was for the ``billiard'' configuration space of the sphere in~\cite{kar2008bil}, where the bound was larger by $1$ compared to Theorem~\ref{constr-conf-gen}.
The author thanks A.Yu.~Volovikov for useful discussions and remarks.
\section{Equivariant cohomology of $G$-spaces}
\label{eq-cohomology}
In this section we state some facts on the equivariant cohomology and define a homological measure of complexity of a $G$-space.
We consider topological spaces with continuous action of a finite group $G$ and continuous maps between such spaces that commute with the action of $G$. We call them $G$-spaces and $G$-maps.
The facts in this section are quite well known, see books~\cite{hsiang1975,bart1993}.
We consider the equivariant cohomology (in the sense of Borel) of $G$-spaces, defined as
$$
H_G^*(X, M) = H^*((X\times EG)/G, M),
$$
where some $\mathbb Z[G]$-module $M$ (acted on by the fundamental group of $(X\times EG)/G$) gives the coefficients for the cohomology.
Consider the $G$-equivariant cohomology of the point $H_G^*(M) = H_G^*(\pt, M) = H^*(BG, M)$. For any $G$-space $X$ the natural map $X\to\pt$ induces the natural map of cohomology $\pi_X^* : H_G^*(M)\to H_G^*(X, M)$.
\begin{defn}
The \emph{upper cohomological index} of a $G$-space $X$ with coefficients in $M$ is the maximal $n$ such that the natural map
$$
H_G^n(M)\to H_G^n(X, M)
$$
is nontrivial. Denote the upper index $\uhind_M X = n$. Denote the supremum over all $\mathbb Z[G]$-modules
$$
\uhind_G X = \sup_M \uhind_M X.
$$
\end{defn}
If a $G$-space $X$ has fixed points, its cohomology $H^*_G(X, M)$ contains $H^*_G(M)$ as a summand, thus its upper index is obviously $+\infty$.
The following property is obvious by definition.
\begin{lem}[Monotonicity of index]
If there exists a $G$-map $f:X\to Y$ then $\uhind_M X\le \uhind_M Y$ for any coefficients $M$.
\end{lem}
The $G$-equivariant cohomology is often calculated from the spectral sequence of the fibration $X_G = (X\times EG)/G\to BG$ with fiber $X$. Here we state the lemma from~\cite[Section~11.4]{mcc2001}.
\begin{lem}
\label{specseqeq}
Let $R$ be a ring with trivial $G$-action. There exists a spectral sequence with $E_2$-term
$$
E_2^{x, y} = H^x(BG, \mathcal H^y(X, R)),
$$
that converges to the graded module, associated with the filtration of $H_G^*(X, R)$.
The system of coefficients $\mathcal H^y(X, R)$ is obtained from the cohomology $H^y(X, R)$ by the action of $G = \pi_1(BG)$. The differentials of this spectral sequence are homomorphisms of $H^*(BG, R)$-modules.
\end{lem}
In this lemma we denote the grading in the spectral sequence by $(x, y)$ (not usual), because the letter $p$ is reserved for the prime number throughout this paper.
\section{Equivariant cohomology of $G$-spaces for $G=(Z_p)^k$}
\label{eq-cohomology-p-tori}
In this section we study in greater detail the equivariant cohomology of $G$-spaces in the case $G=(Z_p)^k$.
In this section the cohomology is taken with coefficients $Z_p$, in notations we omit the coefficients. For a group $G=(Z_p)^k$ the algebra $A_G=H_G^*(Z_p)$ has the following structure (see~\cite{hsiang1975}). In the case $p>2$ it has $2k$ multiplicative generators $v_i,u_i$ with dimensions $\dim v_i = 1$ and $\dim u_i = 2$ and relations
$$
v_i^2 = 0,\quad\beta{v_i} = u_i.
$$
We denote $\beta(x)$ the Bockstein homomorphism.
In the case $p=2$ the algebra $A_G$ is the algebra of polynomials of $k$ one-dimensional generators $v_i$.
Consider again the spectral sequence from Lemma~\ref{specseqeq}. For every term $E_r(X)$ of this spectral sequence there is a natural map $\pi^*_r : A_G\to E_r(X)$ (it maps $A_G$ to the bottom row of $E_r(X)$).
\begin{defn}
Denote the kernel of the map $\pi^*_r$ by $\Ind^r_G X$.
\end{defn}
Let us list the properties of $\Ind^r_G X$, that are obvious by the definition. We omit the subscript $G$ when it is clear what group is meant.
\begin{itemize}
\item
(Monotonicity) If there is a $G$-map $f:X\to Y$, then $\Ind^r X\supseteq \Ind^r Y$.
\item
$\Ind^{r+1} X$ may differ from $\Ind^r X$ only in dimensions $\ge r$.
\item
$\bigcup_r \Ind^r X = \Ind X = \ker \pi_X^* : A_G\to H_G^*(X)$.
\end{itemize}
The first property in this list is very useful to prove nonexistence of $G$-maps. Following~\cite{vol2000,vol2005} we define a numeric invariant of this system $\Ind^r X$, that is enough for us.
\begin{defn}
Put
$$
i_G(X) = \max \{r : \Ind_G^r X = 0\}.
$$
\end{defn}
It is easy to see that $i(X)\ge 1$ for any $G$-space $X$, $i(X)\ge 2$ for a connected $G$-space $X$, and $i(X)$ may be equal to $+\infty$. The following properties are quite clear from the definition.
\begin{itemize}
\item
(Monotonicity) If there is a $G$-map $f:X\to Y$, then $i_G(X) \le i_G(Y)$.
\item
If $H^m(X) = 0$ for $m > n$, then either $i_G(X)=+\infty$ or $i_G(X)\le n+1$.
\item
If $\tilde H^m(X) = 0$ for $m <n$, then $i_G(X)\ge n + 1$.
\end{itemize}
The following lemma from~\cite{vol2005} (Lemma~2.1) tells more about the monotonicity.
\begin{lem}
\label{same-i-map}
Let $X, Y$ be connected paracompact $G$-spaces and let $f:X\to Y$ be a $G$-map. Suppose that $i_G(X)=i_G(Y)=n+1$. Then the map $f^*: H^n(Y)\to H^n(X)$ is nontrivial.
\end{lem}
We need the following result to estimate $i_G(X)$ from below. Some special case of it was used in the proof of Theorem~4 in the paper~\cite{kar2008bil}.
\begin{thm}
\label{indind}
Let $G=(Z_p)^k$, let $G$-space $X$ be connected. Suppose the groups $H^m(X, Z_p)$ for $m < n$ are composed of finite-dimensional $Z_p[G]$-modules, induced from proper subgroups $H\subset G$. Then
$$
i_G(X)\ge n + 1.
$$
\end{thm}
A $Z_p[G]$-module $M$ is induced from $Z_p[H]$-module $N$ iff $M=Z_p[G]\otimes_{Z_p[H]} N$. A finite-dimensional module is induced iff it is coinduced: $M=\Hom_{Z_p[H]}(Z_p[G], N)$ (see~\cite{bro1982}, Ch.~III, Proposition~5.9).
\begin{proof}
Put $S_G=A_G$ for $p=2$, and $S_G=Z_p[u_1,\ldots,u_k]$ for $p>2$. This is a subalgebra of polynomials in $A_G$.
If a $G$-module $M$ is coinduced from an $H$-module $N$ then $H_G^*(M) = H_H^*(N)$ (see~\cite{bro1982}, Ch.~III, Proposition~6.2). In this case the Hilbert polynomial of $H_G^*(M)$ has degree no more than $k-1$. If a $G$-module is composed of induced modules, then it follows from the cohomology exact sequence that the Hilbert polynomial of its cohomology has degree $\le k-1$, or otherwise the cohomology exact sequence could not be exact for large cohomology dimensions.
Therefore, the rows $1,\ldots, n-1$ in $E_2$-term of the spectral sequence of Lemma~\ref{specseqeq} have Hilbert polynomials of degree $\le k-1$. In all next terms $E_r$ ($r\ge 2$) this fact remains true, since the dimensions $\dim E_r^{x, y}$ decrease, and the Hilbert polynomial of a row cannot get a larger degree.
Now let us show that the image of the differentials $d_2,\ldots,d_n$ in the bottom row of the spectral sequence is zero (that is equivalent to $i(X)\ge n+1$). Indeed, if an $A_G$-module $L$ has Hilbert polynomial of degree $\le k-1$ then
$$
\Hom_{A_G}(L, A_G) = 0,
$$
since every $l\in L$ is annihilated by some nontrivial element of $S_G$ (or the Hilbert polynomial would have degree $\ge k$), and none of the nonzero elements of $A_G$ is annihilated by nontrivial elements of $S_G$.
\end{proof}
\section{Definitions of configuration spaces}
In this section we give the definitions of different configuration spaces and introduce some notation.
\begin{defn}
Let $\Delta: M\to \Map(G, M)$ be the diagonal map, i.e. $\Delta(x)$ is the constant map of $G$ to $x\in M$.
Denote $\Map_\Delta(G, M) = \Map(G, M)\setminus\Delta(M)$. This is the space of nonconstant maps $G\to M$.
\end{defn}
\begin{defn}
Denote $[n]=\{1,2,\ldots, n\}$, and denote the configuration space
$$
V(n, M) = \Map([n], M).
$$
\end{defn}
\begin{defn}
Let $\mathcal S$ be a nonempty family of subsets in $[n]$. Denote $V\left(n, M, \mathcal S\right)$ the set of maps $f:[n]\to M$ such that every $f|_S$ for $S\in\mathcal S$ is nonconstant. We call the family $\mathcal S$ a \emph{constraint system}. Denote
$$
w(\mathcal S) = \min_{S\in\mathcal S} |S|.
$$
We consider only the nontrivial case $w(\mathcal S)\ge 2$.
\end{defn}
If the configurations are considered as maps $G\to M$, the following definition is needed.
\begin{defn}
Let $\mathcal S$ be a nonempty family of subsets in $G$, and let $\mathcal S$ be invariant with respect to $G$-action on itself. Denote $V\left(G, M, \mathcal S\right)$ the set of maps $f:G\to M$ such that every $f|_S$ for $S\in\mathcal S$ is nonconstant. Again, we call the family $\mathcal S$ a \emph{constraint system}. The number $w(\mathcal S)$ is defined as in the previous definition and is considered to be at least $2$.
\end{defn}
The constraint system $\mathcal S=\{G\}$ gives the space $\Map_\Delta(G, M)$, defined above.
For a finite set $X$ denote $\binom{X}{w}$ the family of all $w$-element subsets of $X$. The classical configuration space, the set of $n$-tuples of pairwise distinct points in $M^n$, is denoted $V\left(n, M, \binom{[n]}{2}\right)$ in our notation. The spaces $V\left(n, M, \binom{[n]}{w}\right)$ are called configuration-like spaces in~\cite{colusk1976,vol2007}, here we call them simply configuration spaces.
\section{The index of configuration spaces}
\label{index-conf}
Now we are ready to state and prove the core results of this paper, the lower bounds on indexes of configuration spaces.
\begin{lem}
\label{conf-space-ind} Let $G=(Z_p)^k$, let $M$ be a smooth oriented (if $p\not=2$) closed manifold of dimension $d$. Then
$$
i_G\left(\Map_\Delta(G, M)\right) \ge d(p^k-1) + 1.
$$
\end{lem}
Note that in~\cite{vol1992,vol2005} the index of $\Map_\Delta(G, M)$ was estimated from above. It was needed to show that (under some additional assumptions) for any continuous map $f:X\to M$ of a $G$-space $X$ to $M$ some $G$-orbit in $X$ is mapped to a point.
\begin{proof}
Denote $q=p^k$. Obviously, $M$ contains some copy of $\mathbb R^d$ and $\Map_\Delta(G, M)$ contains the respective copy of $S=\Map_\Delta(G, \mathbb R^d)$, the latter space is homotopy equivalent to a $d(q-1)-1$-dimensional sphere. For a sphere, it is obvious to see that $i(S) = d(q-1)$, thus $i(\Map_\Delta(G, M)) \ge d(q-1)$.
Denote for brevity $X = \Map(G, M)$ and $X_\Delta = \Map_\Delta(G, M)$.
Consider the contrary: $i(X_\Delta) = d(q-1)$. In this case by Lemma~\ref{same-i-map} the inclusion map $j : S\to X_\Delta$ of the sphere induces a nontrivial map $j^*: H^{d(q-1)-1}(X_\Delta)\to H^{d(q-1)-1}(S)$. Take the fundamental class $y\in H^{d(q-1)-1}(S)$ and consider $x\in H^{d(q-1)-1}(X_\Delta)$ such that $j^*(x) = y$. From the following diagram
\begin{equation}
\begin{CD}
\label{leseq}
H^{d(q-1)}(X) @<{\pi^*}<< H^{d(q-1)}(X, X_\Delta) @<{\delta}<< H^{d(q-1)-1}(X_\Delta) @<{\iota^*}<< H^{d(q-1)-1} (X)\\
@V{j^*}VV @V{j^*}VV @V{j^*}VV @V{j^*}VV\\
H^{d(q-1)}(\mathbb R^{dq}) @<{\pi^*}<< H^{d(q-1)}(\mathbb R^{dq}, S) @<{\delta}<< H^{d(q-1)-1}(S) @<{\iota^*}<< H^{d(q-1)-1} (\mathbb R^{dq})
\end{CD}
\end{equation}
we see that $j^*(\delta(x)) = \delta(y)\not=0$.
Consider a tubular neighborhood $N(\Delta)$ of the diagonal $\Delta(M)$ in $X$. The pair $(X, X_\Delta)$ has the same cohomology as $(B(V), S(V))$, where $V$ is the normal vector bundle of the diagonal, $B(V)$ and $S(V)$ being its unit ball and unit sphere spaces. Since the tangent bundle to $\Delta(M)$ is the same as the tangent bundle of $M$, then $V$ fits to the following exact sequence
$$
0\to T(M) \to \oplus_{g\in G} T(M) \to V\to 0
$$
over $\Delta(M)$, since the restriction of the tangent bundle of $\Map(G, M)$ to $\Delta(M)$ equals $\oplus_{g\in G} TM$. Hence, the group $G$ acts naturally on $\oplus_{g\in G} TM$, and on $V$.
The manifold $M$ is $Z_p$-oriented and the bundle $V$ is $Z_p$-oriented. Then by Thom's isomorphism $H^*(X, X_\Delta)=u H^*(M)$, where $u$ is the $d(q-1)$-dimensional fundamental class of $V$. From $\delta(x)\not=0$ it follows that $\delta(x) = au$ for some $a\in Z_p^*$. The horizontal exactness of diagram~\ref{leseq} shows that $\pi^*(au) = 0$ and $\pi^*(u)=0$.
Note that the map $\pi^*: H^{d(q-1)} (X, X_\Delta) \to H^{d(q-1)} (X)$ has to be injective, this is a consequence of the Poincar\'e duality and injectivity of the natural map $H_d(N(\Delta)) = H_d(M) \to H_d(X)$. Thus $\pi^*(u)\not=0$, that is a contradiction.
\end{proof}
\begin{lem}
\label{constr-conf-ind}
For any constraint system $\mathcal S$ on $G=(Z_p)^k$ and $d$-dimensional smooth manifold $M$
$$
i_G\left(V\left(G, M, \mathcal S\right)\right) \ge (d-1)(p-1)p^{k-1} + w(\mathcal S) - 1.
$$
\end{lem}
Note that in~\cite{colusk1976,vol2005,vol2007} the case $M=\mathbb R^m$ and $\mathcal S = \binom{G}{w}$ was considered and some upper bounds on the index were found. The result of paper~\cite{vol2007} states that
$$
i_G\left(V\left(G, \mathbb R^m, \binom{G}{w}\right)\right) \le (d-1)(p^k-1) + w - 1,
$$
so there is some gap between lower and upper bounds for $k>1$.
\begin{proof}
The index is monotonic, so it suffices to consider the case $M=\mathbb R^d$, which is assumed in this proof.
We are going to apply Theorem~\ref{indind}, so we have to know the cohomology $H^*\left(V(G, M, \mathcal S), Z_p\right)$ first. The coefficients $Z_p$ of the cohomology are omitted in the notation in this proof. The space $V(G, M, \mathcal S)$ is a complement to the set of linear (as subspaces of $\mathbb R^{dp^k}$) subspaces $L_S\subset \Map(G, M)$, here we denote for any $S\in\mathcal S$
$$
L_S = \{f:G\to M\ \text{such that}\ f|_S\ \text{is constant}\}.
$$
For nonempty $\mathcal T\subseteq\mathcal S$ put $L_{\mathcal T} = \bigcap_{S\in\mathcal T} L_S$.
By the result from book~\cite{gormac1988}, cited here by the review~\cite[Corollary~2]{vass2001}, the reduced cohomology of $V(G, M, \mathcal S)$ can be represented as follows
\begin{equation}
\label{gor-mac}
\tilde H^i\left(V(G, M, \mathcal S)\right) = \bigoplus_{\mathcal T\subseteq \mathcal S} H_{qd-i-\dim L_{\mathcal T} - 1} (\Delta(\mathcal T), \partial\Delta(\mathcal T)),
\end{equation}
the sum being taken over distinct subspaces $L_{\mathcal T}$. In~\cite{vass2001} this formula is proved so that the isomorphism may be not natural, so we have to be careful to introduce $G$-action on this formula. Actually this formula becomes natural, if we note that this formula is a particular case of the Leray spectral sequence for the direct image of a sheaf under the inclusion $V(G, M, \mathcal S)\to \Map(G, M)$. Hence, the reduced cohomology $\tilde H^*\left(V(G, M, \mathcal S)\right)$ should be replaced by its associated graded module, obtained from some filtering of the cohomology.
Thus the action of $G$ on the right side of Equation~\ref{gor-mac} describes the action of $G$ on the associated graded module of the cohomology. Let us study this action.
The cohomology to the right in Equation~\ref{gor-mac} is the cohomology of the order complex for the poset of spaces $L_{\mathcal U}\supseteq L_{\mathcal T}$, relative to its subcomplex, spanned by proper inclusions $L_{\mathcal U}\supset L_{\mathcal T}$. The \emph{order complex} of a poset $P$ is a simplicial complex, that has $P$ as the vertex set, and the set of chains in $P$ as the set of simplices.
In this proof, it suffices to note that the dimension of the order complexes in question is no more than $q - \dim L_{\mathcal T}/d - w(\mathcal S) + 1$.
Now consider the $G$-action on the right part of Equation~\ref{gor-mac}. If the space $L_{\mathcal T}$ is not fixed under $G$-action, then the summand, that corresponds to $L_{\mathcal T}$, has $G$-action, induced from the stabilizer of $L_{\mathcal T}$. Theorem~\ref{indind} allows us to ignore such summands. The subspaces $L_{\mathcal T}$ that are fixed under $G$-action have dimension no more than $dp^{k-1}$. Thus, they contribute to they cohomology $\tilde H^i\left(V(G, M, \mathcal S)\right)$ with the following inequality on the dimension
$$
qd - i - \dim L_{\mathcal T} - 1 \le q - \dim L_{\mathcal T}/d - w(\mathcal S) + 1,
$$
then by simple transformations
\begin{multline*}
i\ge q(d-1) - (d-1)\dim L_{\mathcal T}/d + w(\mathcal S) - 2 \ge\\
\ge (d-1)(q-p^{k-1}) + w(\mathcal S) - 2 = (d-1)(p-1)p^{k-1} + w(\mathcal S) - 2.
\end{multline*}
We have proved that in dimensions $i\le (d-1)(p-1)p^{k-1}+w(\mathcal S) - 1$, the right part of Equation~\ref{gor-mac} is composed of induced $Z_p[G]$-modules, but this right part is itself a decomposition of the cohomology $\tilde H^i\left(V(G, M, \mathcal S)\right)$. Thus Theorem~\ref{indind} can be applied to complete the proof.
\end{proof}
Now we are going to study the classical configuration space. Denote by $\pm Z_p$ the $\mathbb Z[\mathfrak S_n]$-module, which is $Z_p$ with the action of $\mathfrak S_n$ by the sign of permutation. In the case $p=2$, put $\pm Z_2=Z_2$.
\begin{lem}
\label{pwconstr-ind}
Let $n=p^k$ be a prime power, let $M$ be a smooth manifold of dimension $d\ge 2$, let the constraint system $\mathcal S\subset 2^n$ consist of all pairs $\mathcal S = \binom{[n]}{2}$. Then under the natural action of $\mathfrak S_n$ on the configuration space we have
$$
\uhind_N V(n, M, \mathcal S) = (d-1)(n-1),
$$
where $N=\pm Z_p$ for even $d$, and $N=Z_p$ for odd $d$.
\end{lem}
\begin{proof} Actually the proof for $d=2$ is given in~\cite{vass1988} and works for arbitrary $d\ge 2$ in the same way. Below we give a short sketch.
We go to the case $M=\mathbb R^d$, consider the configuration space $X=V(n, M, \mathcal S)$, and its one-point compactification $X'=X\cup\pt$. The Poincar\'e-Lefschetz duality tells that there is a natural isomorphism (since $N=(\pm Z_p)^{\otimes d-1}$ and $(\pm Z_p)^{\otimes d}$ gives the orientation of $X$)
$$
H_{dn-k}(X'/\mathfrak S_n, \pt, \pm Z_p) = H^k_{\mathfrak S_n} (X, N).
$$
In~\cite{fuks1970} a certain relative cellular decomposition of the pair $(X', \pt)$ that respects $\mathfrak S_n$-action was constructed. It can be described as follows: consider the graded trees, such that the grades of the vertices are $0,1,\ldots, d$, every vertex of grade $<d$ has children, all the leafs (vertices with no children) have grade $d$, and the number of leafs is $n$. We consider such a tree $T$ along with the following data: for every vertex $v\in T$ its children are ordered, and the leafs have labels $1,2,\ldots, n$ in some order.
Say that the number $i\in[n]$ \emph{belongs to vertex} $v$, if the leaf with mark $i$ is a descendant of $v$. We say that indexes $i$ and $j$ \emph{split on level $k$}, if they belong to the same vertex $v$ of grade $k-1$, but belong to different vertices $v_i$ and $v_j$ of grade $k$. If the vertices $v_i$ and $v_j$ are ordered as $v_i<v_j$ as children of $v$, we write $i<_k j$ to show that $i$ and $j$ split on level $k$ in this given order.
Now define the open cell $C_T$ in $X$, corresponding to $T$, by the following rule:
$$
C_T = \{(x_1, \ldots, x_n)\in M^n : \text{if}\ i<_kj,\ \text{then}\ x_{ik} < x_{jk},\ \text{and}\ \forall l < k\ x_{il}=x_{jl} \},
$$
where $x_{ik}$ is the $k$-th coordinate of $i$-th point.
The cellular decomposition of $X$ can be described informally as follows: every two points $x_i$ and $x_j$ in a configuration $(x_1,\ldots, x_n)$ must be distinct, consider the minimal $k$ such that their coordinates $x_{ik}$ and $x_{jk}$ are different, and if $x_{ik} < x_{jk}$, say that $i<_k j$. The pattern of such relations $i<_k j$ is exactly a tree with $d+1$ levels and $n$ marked leaves at the bottom level.
The cellular decomposition of $(X', \pt)$ is obtained by taking the closures of $C_T$. It is clear from the definition, that the dimension of a cell $C_T$ is equal to the number of vertices in $T$ minus one. It is also clear that $\mathfrak S_n$ permutes the cells, so a cellular decomposition of $(X'/\mathfrak S_n, \pt)$ is induced. On the level of trees it corresponds to forgetting the labels on the leafs.
This cellular decomposition of $(X'/\mathfrak S_n, \pt)$ has only one cell $\sigma$ of minimal dimension $n+d-1$, that corresponds to the tree with $1$ vertex on each of levels $0,\ldots, d-1$, and $n$ vertices on the bottom level. The cells of dimension $n+d$ correspond to the trees with $1$ vertex on levels $0,\ldots, d-2$, $2$ vertices on level $d-1$, and $n$ vertices on level $d$. The coefficients of the boundary operator between $n+d$-dimensional and $n+d-1$-dimensional cells has the form $\pm\binom{n}{k}$ ($k=1,\ldots,n-1$), see~\cite[Theorem~2.5.1]{vass1988}; this is true for coefficients $Z_p$, or $\pm Z_p$, as we need.
If $n$ is a prime power $n=p^k$, then all the coefficients $\pm\binom{n}{k}$ are zero modulo $p$. Hence, the minimal cell gives a nontrivial element of homology $\sigma\in H_{n+d-1} (X'/{\mathfrak S_n}, \pt, \pm Z_p)$ (in fact it is also true for both coefficients $\pm Z_p$ and $Z_p$) and its Poincar\'e-Lefschetz dual $\xi$ is a nontrivial element of $H_{\mathfrak S_n}^{(d-1)(n-1)}(X, N)$.
In fact $\xi$ is an image of the element of $H^*(B\mathfrak S_n, N)$ that is the $d-1$-th power of the Euler class of the standard $n-1$-dimensional irreducible representation $W$ of $\mathfrak S_n$. This can be shown by considering the map $\pi : X\to \mathbb R^{(d-1)n}$, that forgets the last coordinate of every point $x_i$, this map is $\mathfrak S_n$-equivariant, its target space is $W^{d-1}$ as $\mathfrak S_n$-representation, and its zero set $\sigma$ (note that all zeros are nondegenerate) must be dual to the Euler class $e(W)^{d-1}$.
Thus by definition $\uhind_N X = (d-1)(n-1)$.
\end{proof}
\section{Existence of symmetric configurations}
\label{symm-config}
In this section we consider certain equations for a system of functions on some configuration space, for such equations we prove existence of their solutions.
Let us describe the idea more precisely. The results of this section are mostly inspired by the theorems of inscribing regular figures. One example of such theorems is the famous theorem of Schnirelmann~\cite{schn1934} that every simple smooth closed curve in $\mathbb R^2$ has an inscribed square. Some more results on inscribing $Z_p$-symmetric configurations are found in the paper~\cite{mak1989} and other papers of V.V.~Makeev.
Unlike the original problems on inscribing a congruent (or similar) copy of a given configuration, we consider here configurations of points on a manifold and try to find configurations with some metric equalities, which are not required to determine the configuration rigidly. The following particular result has explicit geometric meaning.
\begin{thm}
\label{metric-eq}
Let $p>2$ be a prime, let $M$ be an oriented closed smooth manifold of dimension $d$. Consider a continuous function $\rho:M\times M\to \mathbb R$.
Suppose that we have $d$ elements $g_1,\ldots, g_d$ of the group $G=Z_p$. Then there exists a nonconstant map $\phi: G\to M$ such that (group operation in $G$ is denoted $+$)
$$
\forall g\in G,\ \forall i=1,\ldots, d\quad \rho(\phi(g), \phi(g + g_i)) = \rho(\phi(e), \phi(g_i)).
$$
\end{thm}
In the case when $\rho$ is some continuous metric, we obtain more equalities because $\rho$ is symmetric ($\rho(x, y) = \rho(y, x)$). Moreover, in this case the numbers $\rho(\phi(g), \phi(g + g_i))$ have to be positive.
More specially, in the case ($d=2$, $G=Z_5$) Theorem~\ref{metric-eq} gives the following statement: for any continuous metric $\rho$ on a two-dimensional oriented manifold there are five distinct points $p_1,p_2,\ldots,p_5$ such that
$$
\rho(p_1p_2)=\rho(p_2p_3)=\rho(p_3p_4)=\rho(p_4p_5)=\rho(p_5p_1)
$$
and
$$
\rho(p_1p_3)=\rho(p_3p_5)=\rho(p_5p_2)=\rho(p_2p_4)=\rho(p_4p_1).
$$
In fact, Theorem~\ref{func-eq} (see below) tells, that in the two above equalities we may take two distinct metrics $\rho_1$ and $\rho_2$.
Let us state the results more generally. Theorem~\ref{metric-eq} has the following general form, where the functions $\rho$ are replaced by arbitrary continuous functions on the configuration space.
\begin{thm}
\label{func-eq}
Let $p>2$ be a prime, let $M$ be an oriented closed smooth manifold of dimension $d$, let $G=Z_p$, let $\alpha_i : \Map(G, M)\to \mathbb R$ ($i=1,\ldots,d$) be some continuous functions on the configuration space.
Then there exists a configuration $\phi\in \Map_\Delta(G, M)$ such that for any $i=1,\ldots, d$ the number $\alpha_i(\phi^g)$ does not depend on $g\in G$.
\end{thm}
Now we replace the special constraint system in Theorem~\ref{func-eq} by some arbitrary constraint system and formulate the following result.
\begin{thm}
\label{constr-func-eq}
Let $p>2$ be a prime, let $M$ be a smooth manifold of dimension $d$, let $G=(Z_p)^k$, and let $\mathcal S$ be some constraint system. Put
$$
m=\left\lfloor\frac{(d-1)(p^k-p^{k-1})+ w(\mathcal S) - 2}{p^k-1}\right\rfloor.
$$
Consider some $m$ continuous functions $\alpha_i : V(G, M, \mathcal S)\to \mathbb R$.
Then there exists a configuration $\phi\in V(G, M, \mathcal S)$ such that for every $i=1,\ldots, m$ the number $\alpha_i(\phi^g)$ does not depend on $g\in G$.
\end{thm}
In this theorem the number of functions decreased compared to Theorem~\ref{func-eq}, but the constraint system can be arbitrary and the manifold does not have to be closed or oriented.
\begin{proof}[Proof of Theorem~\ref{func-eq}]
For every $\alpha_i$ in Theorem~\ref{func-eq} consider the $G$-map of the configuration space $f_i: \Map_\Delta(G, M)\to \mathbb R^q$ by the formula
$$
f_i(\phi) = \oplus_{g\in G} \alpha_i(\phi^g).
$$
Consider the diagonal $\Delta(\mathbb R)\in\mathbb R^q$ and the space $\mathbb R^q/\mathbb R = W_i$, $f_i$ induces the map $h_i : \Map_\Delta(G, M) \to W_i$.
We have to prove that the total map $h = h_1\oplus\dots\oplus h_d$ maps some point $\Map_\Delta(G, M)$ to zero in $W = W_1\oplus\dots\oplus W_d$. The latter space has $G$-action, so $h$ can be considered as $G$-equivariant section of a $G$-bundle over $\Map_\Delta(G, M)$.
The considered $G$-bundle over $\Map_\Delta(G, M)$ is a pullback of the $G$-bundle $W$ over $\pt$. The latter bundle (representation) has nonzero Euler class in $H_G^{d(p-1)}(\pt)$ (see~\cite{hsiang1975,vol1992}). Since $i(\Map_\Delta(G, M)) \ge d(p-1) + 1$, the image of this Euler class in $H_G^{d(p-1)} (\Map_\Delta(G, M))$ is nonzero too. This guarantees the existence of a zero and Theorem~\ref{func-eq} is proved.
\end{proof}
Theorem~\ref{constr-func-eq} is deduced from Lemma~\ref{constr-conf-ind} in the similar way.
\section{The genus and the category}
\label{genus-and-cat}
The estimates in the cohomological index of the configuration spaces in Section~\ref{index-conf} give estimates for the genus in the sense of Krasnosel'skii-Schwarz and the equivariant Lyusternik-Schnirelmann category of those spaces.
Let us start from the Lyusternik-Schnirelmann category. We formulate some special cases of definitions from the book~\cite{bart1993}.
\begin{defn}
Let $X$ be a $G$-space, \emph{$G$-category} of $X$ is the minimal size of $G$-invariant open cover (i.e. cover by $G$-invariant open subsets) $\{X_1,\ldots,X_n\}$ of $X$ such that every inclusion map $X_i\to X$ is $G$-homotopic to inclusion of some orbit $G/H\to X$. Denote $G$-category of $X$ by $\Gcat X$.
\end{defn}
This category gives a lower bound on the number of $G$-orbits of critical points for some $G$-invariant $C^2$-smooth function $f$ on $X$, if $f$ is a proper function bounded either from below or from above. One of the main ways to find lower bounds for $G$-category is to use $G$-genus, introduced in~\cite{kr1952,schw1957} for free $G$-actions, in~\cite{schw1966} for fiber bundles, and in~\cite{clp1986,clp1991} for arbitrary $G$-action. In fact there are different types of genus (see also the book~\cite{bart1993} for a detailed discussion), here we use one certain type.
\begin{defn}
Let $\mathcal A$ be some family of $G$-spaces. Let $X$ be a $G$-space, \emph{$\mathcal A$-genus} of $X$ is the minimal size of $G$-invariant open cover (i.e. cover by $G$-invariant open subsets) $\{X_1,\ldots,X_n\}$ of $X$ such that every $X_i$ can be $G$-mapped to some $D\in\mathcal A$. Denote $\mathcal A$-genus of $X$ by $\Agen X$.
\end{defn}
Equivalently (see~\cite{bart1993}), for paracompact spaces and finite groups $G$ the genus can be defined in the following way. Note that in this paper we consider paracompact spaces and finite groups only.
\begin{defn}
Let $X$ be a $G$-space, \emph{$\mathcal A$-genus} of $X$ is the minimal $n$ such that $X$ can be $G$-mapped to a join $D_1*D_2*\dots*D_n$, where $D_i\in\mathcal A$.
\end{defn}
The following theorem estimates the equivariant category by the genus, it follows directly from the definitions.
\begin{thm}
\label{cat-by-genus}
For any $G$-space $X$ denote $\mathcal O_G(X)$ the set of distinct types of orbits $G/H\subseteq X$. Then
$$
\Gcat X\ge \OGXgen X.
$$
\end{thm}
The genus $\OGXgen X$ is usually estimated by the following type of genus.
\begin{defn}
If the family $\mathcal A$ contains only one $G$-space, which is the disjoint union of all nontrivial orbit types
$$
D_G = \bigsqcup_{H\subset G} G/H,
$$
then we denote $\Agen X = \Ggen X$.
\end{defn}
In the paper~\cite{vol2007} this genus is denoted $g_G(X)$. In the sequel we shall mainly use this genus by the following reason. If a $G$-space has $G$-fixed points, then $\OGXgen X=1$. If a $G$-space has no fixed points, then $\OGXgen X\ge \Ggen X$. Still, for free $G$-spaces we use the $\OGXgen X$ itself.
Now let us state some estimates on the genus of certain configuration spaces.
\begin{thm}
\label{nonconst-conf-gen}
Suppose that $G=(Z_p)^k$, $M$ is a smooth oriented (if $p>2$) closed manifold of dimension $d$. Then
$$
\Ggen \Map_\Delta(G, M) \ge d(p^k-1) + 1.
$$
\end{thm}
\begin{thm}
\label{constr-conf-gen}
Suppose that $G=(Z_p)^k$, $\mathcal S$ is a constraint system on $G$. Then for any $d$-dimensional smooth manifold $M$
$$
\Ggen V(G, M, \mathcal S) \ge (d-1)(p-1)p^{k-1} + w(\mathcal S) - 1.
$$
\end{thm}
In order to prove the estimates on the genus, we need some well-known lemmas. In the case of a free $G$-action of arbitrary finite group $G$ the genus can be estimated in the following way (see~\cite{schw1957}).
\begin{lem}
\label{gen-by-index}
If $G$ acts freely on $X$ then
$$
\OGXgen X\ge \uhind_G X + 1.
$$
\end{lem}
It should be mentioned that the above lemma is a simple consequence of the second definition of genus and the fact that $H^*_G(X, M) = H^*(X/G, M)$ for a free $G$-space $X$. The following lemma is clear from the definition of genus and existence of an $H$-map $G\to H$, where $H$ acts on $G$ by left multiplications.
\begin{lem}
\label{gen-group-change}
If $G$ acts freely on $X$ and $H\subset G$ is a subgroup, then
$$
\OGXgen X\ge \OHXgen X.
$$
\end{lem}
There is a good estimation of the genus by $i_G(X)$, we restate Proposition~4.7 from the paper~\cite{vol2005}, noting Remark~4.5 from the same paper.
\begin{lem}
\label{gen-by-i}
Suppose $G=(Z_p)^k$ acts on $X$ without fixed points. Then
$$
\Ggen X\ge i_G(X).
$$
\end{lem}
Now Theorem~\ref{nonconst-conf-gen} is deduced from Lemmas~\ref{gen-by-i} and \ref{conf-space-ind}, and Theorem~\ref{constr-conf-gen} is deduced from Lemmas~\ref{gen-by-i} and \ref{constr-conf-ind}.
Let us state some corollary of Theorem~\ref{constr-conf-gen}.
\begin{cor}
\label{pwconstr-gen0}
Let $G=(Z_p)^k$, let $M$ be a smooth manifold of dimension $d$, let the constraint system $\mathcal S$ consist of all pairs $\mathcal S = \binom{G}{2}$. Then
$$
\Ggen V(G, M, \mathcal S) \ge (d-1)(p-1)p^{k-1}+1.
$$
\end{cor}
\begin{defn}
Denote the group of permutations of $[n]$ by $\mathfrak S_n$.
\end{defn}
In the papers~\cite{fuks1970,vass1988,rot2008} the classical configuration space $V\left(n, M, \binom{[n]}{2}\right)$ was considered, and the genus of $V\left(n, M, \binom{[n]}{2}\right)$ with respect to $\mathfrak S_n$-action was estimated from below. The estimates for $n$ being a prime power in the case $d=2$ and for $n=2^k$ in the case of arbitrary $d$ were better than in Corollary~\ref{pwconstr-gen0}. Here we prove the appropriate result for $\mathfrak S_n$-genus.
\begin{defn}
For a positive integer $n$ and a prime $p$ denote $D_p(n)$ the sum of digits in the $p$-ary representation of $n$.
\end{defn}
\begin{thm}
\label{pwconstr-gen}
Let $M$ be a smooth manifold of dimension $d\ge 2$, let the constraint system $\mathcal S\subset 2^n$ consist of all pairs $\mathcal S = \binom{[n]}{2}$. Denote $X=V(n, M, \mathcal S)$. Then for every prime $p$ we have
$$
\OSnXgen X \ge (d-1)(n-D_p(n)) + 1.
$$
\end{thm}
Here we formulate the result for arbitrary manifold, but actually we give prove for the case $M=\mathbb R^d$, as in the previous works. The author does not know whether the genus
$$
\OSnXgen V(n, M, \mathcal S)
$$
can be replaced by
$$
\Zpkgen V(n, M, \mathcal S)
$$
in this theorem for the case when the number of points is a prime power $n=p^k$, as it is done in the statement of Corollary~\ref{pwconstr-gen0}.
\begin{proof}[Proof of Theorem~\ref{pwconstr-gen}]
The theorem for prime powers is deduced from Lemmas~\ref{gen-by-index} and \ref{pwconstr-ind} directly.
Consider the general case: take some $n$ and represent it as a sum of $D_p(n)$ powers of $p$:
$$
n = \sum_{i=1}^{D_p(n)} p^{k_i}.
$$
As in~\cite{vass1988}, we can select $D_p(n)$ disjoint open subsets $M_1,\ldots, M_{D_p(n)}$ of $M$, each being homeomorphic to $\mathbb R^d$. Consider the subspace of $V(n, M, \mathcal S)$ formed by the following configurations: the first $p^{k_1}$ points lie in $M_1$, the other $p^{k_2}$ points lie in $M_2$ and so on. So we can state that
$$
V(n, M, \mathcal S) \supset \prod_{i=1}^{D_p(n)} V(p^{k_i}, M_i, \mathcal S).
$$
The spaces $X_i=V(p^{k_i}, M_i, \mathcal S)$ have actions of their respective groups $G_i=\mathfrak S_{p^{k_i}}$, and have by Lemma~\ref{pwconstr-ind} some nontrivial cohomology classes in $H_{G_i}^{(d-1)(p^{k_i}-1)}(X_i, N)$ as natural images of some classes in $H^*(BG_i, N)$, $N$ is the same as in Lemma~\ref{pwconstr-ind}. By the K\"unneth formula the product of these classes is a nontrivial natural image of some class from $H^{(d-1)(n-D_p(n))}(BG_1\times\dots\times BG_{D_p(n)}, N)$. Denote $G=G_1\times\dots\times G_{D_p(n)}$, so we see that for the upper $G$-index
$$
\uhind_G V(n, M, \mathcal S)\ge\uhind_G \prod_i X_i\ge (d-1)(n-D_p(n)).
$$
From Lemma~\ref{gen-by-index} it follows that if $X=V(n,M,S)$, then $\OGXgen X\ge (d-1)(n-D_p(n))+1$. Then by Lemma~\ref{gen-group-change} $\OSnXgen X\ge \OGXgen X\ge (d-1)(n-D_p(n))+1$.
\end{proof} | 33,932 |
How to make a Frog Prince costume?
The Frog Prince is not holiday specific. In other words the costume doesn’t have a particular holiday that you have to wear it. Valentine’s Day makes a great holiday for the Frog Prince, though. The Frog Prince story has a frog who can talk, and when kissed will become a human prince to marry his true love. So, if you are looking for a little love on Valentine’s you might want to find your Frog Prince costume!
##Materials Needed:##
* Green fabric
* Pattern
* Sewing machine
* Thread
* Pink fabric
* Light green fabric
##Instructions:##
Before you can begin to follow our steps you will need to do some shopping. The frog prince will require a pattern from either an online store or one in your local area. You can also shop at Wal-Mart stores for costume patterns and fabric.
Step 1: You will need to follow the instructions on the frog pattern to make the main costume. The dark green or regular green color is the largest section of material needed for the body, legs, and arms. The light green fabric is for the stomach patch. The pink fabric can be used for the frog mouth depending on the pattern you find. If it is more like Kermit the Frog you will definitely need some pink.
Step 2: The only other thing you need to complete the costume after you get it sewn, is the crown. You need to look like a prince to secure that kiss! | 95,116 |
Daniel Hayes
Dorothy Foehr Huck and J. Lloyd Huck Chair in Nanotherapeutics and Regenerative Medicine, Associate Professor of Biomedical Engineering
- N242 Millennium Science Complex
University Park, PA
- [email protected]
- 814-865-0780
Research Summary.
Huck Affiliations
Links
Publication TagsStem Cells Nanoparticles Bone Silver Tissue Engineering Scaffolds Tissue Defects Oligonucleotides Scaffolds (Biology) Micrornas Nanocomposites Repair Polycaprolactone Genes Collagen Proteins Gene Expression Molecular Weight Cellulose Osteogenesis Glass Ceramics Orthopedics Hydrogel Hydrogels
Most Recent Papers
Transcriptomic Profiling of Adipose Derived Stem Cells Undergoing Osteogenesis by RNA-Seq
Shahensha Shaik, Elizabeth C. Martin, Daniel J. Hayes, Jeffrey M. Gimble, Ram V. Devireddy, 2019, Scientific reports
Collagen-infilled 3D printed scaffolds loaded with miR-148b-transfected bone marrow stem cells improve calvarial bone regeneration in rats
Kazim K. Moncal, R. Seda Tigli Aydin, Mohammad Abu-Laban, Dong N. Heo, Elias Rizk, Scott M. Tucker, Gregory Lewis, Daniel J. Hayes, Ibrahim Tarik Ozbolat, 2019, Materials Science and Engineering C
Combinatorial Delivery of miRNA-Nanoparticle Conjugates in Human Adipose Stem Cells for Amplified Osteogenesis
Mohammad Abu-Laban, Prakash Hamal, Julien H. Arrizabalaga, Anoosha Forghani, Asela S. Dikkumbura, Raju R. Kumal, Louis H. Haber, Daniel J. Hayes, 2019, Small
Decellularized Adipose Tissue Hydrogel Promotes Bone Regeneration in Critical-Sized Mouse Femoral Defect Model
Omair A. Mohiuddin, Brett Campbell, J. Nick Poche, Michelle Ma, Emma Rogers, Dina Gaupp, Mark A.A. Harrison, Bruce A. Bunnell, Daniel J. Hayes, Jeffrey M. Gimble, 2019, Frontiers in Bioengineering and Biotechnology
Facile synthesis and structural insight of nanostructure akermanite powder
Fariborz Tavangarian, Caleb A. Zolko, Abbas Fahami, Anoosha Forghani, Daniel J. Hayes, 2019, Ceramics International on p. 7871-7877
Thermal performance and surface analysis of steel-supported platinum nanoparticles designed for bio-oil catalytic upconversion during radio frequency-based inductive heating
Jacob Bursavich, Mohammad Abu-Laban, Pranjali D. Muley, Dorin Boldor, Daniel J. Hayes, 2019, Energy Conversion and Management on p. 689-697
Differentiation of Adipose Tissue–Derived CD34+/CD31− Cells into Endothelial Cells In Vitro
Anoosha Forghani, Srinivas Koduru, Cong Chen, Ashley N. Leberfinger, Dino Ravnic, Daniel J. Hayes, 2019, Regenerative Engineering and Translational Medicine
Fabrication and characterization of thiol-triacrylate polymer via Michael addition reaction for biomedical applications
Anoosha Forghani, Leah Garber, Cong Chen, Fariborz Tavangarian, Timothy B. Tighe, Ram Devireddy, John A. Pojman, Daniel Hayes, 2019, Biomedical Materials (Bristol)
Comparison of thermally actuated retro-diels-alder release groups for nanoparticle based nucleic acid delivery
Mohammad Abu-Laban, Raju R. Kumal, Jonathan Casey, Jeff Becca, Daniel LaMaster, Carlos N. Pacheco, Dan G. Sykes, Lasse Jensen, Louis H. Haber, Daniel J. Hayes, 2018, Journal of Colloid And Interface Science on p. 312-321
Comparative proteomic analyses of human adipose extracellular matrices decellularized using alternative procedures
Caasy Thomas-Porch, Jie Li, Fabiana Zanata, Elizabeth C. Martin, Nicholas Pashos, Kaylynn Genemaras, J. Nicholas Poche, Nicholas P. Totaro, Melyssa R. Bratton, Dina Gaupp, Trivia Frazier, Xiying Wu, Lydia Masako Ferreira, Weidong Tian, Guangdi Wang, Bruce A. Bunnell, Lauren Flynn, Daniel J. Hayes, Jeffrey M. Gimble, 2018, Journal of Biomedical Materials Research - Part A on p. 2481-2493
Most-Cited Papers
Electrospun bio-nanocomposite scaffolds for bone tissue engineering by cellulose nanocrystals reinforcing maleic anhydride grafted PLA
Chengjun Zhou, Qingfeng Shi, Weihong Guo, Lekeith Terrell, Ammar T. Qureshi, Daniel J. Hayes, Qinglin Wu, 2013, ACS Applied Materials and Interfaces on p. 3847-3854
Silver nanoscale antisense drug delivery system for photoactivated gene silencing
Paige K. Brown, Ammar T. Qureshi, Alyson N. Moll, Daniel J. Hayes, W. Todd Monroe, 2013, ACS nano on p. 2948-2959
MiR-148b-Nanoparticle conjugates for light mediated osteogenesis ofhuman adipose stromal/stem cells
Ammar T. Qureshi, William T. Monroe, Vinod Dasa, Jeffrey M. Gimble, Daniel J. Hayes, 2013, Biomaterials on p. 7799-7810
Design, synthesis, and biological evaluation of β-lactam antibiotic-based imidazolium- and pyridinium-type ionic liquids
Marsha R. Cole, Min Li, Bilal El-Zahab, Marlene E. Janes, Daniel J. Hayes, Isiah M. Warner, 2011, Chemical Biology and Drug Design on p. 33-41
Photoactivated miR-148b-nanoparticle conjugates improve closure of critical size mouse calvarial defects
Ammar T. Qureshi, Andrew Doyle, Cong Chen, Diana Coulon, Vinod Dasa, Fabio DelPiero, Benjamin Levi, W. Todd Monroe, Jeffrey M. Gimble, Daniel J. Hayes, 2015, Acta Biomaterialia on p. 166-173
Thermoreversible and Injectable ABC Polypeptoid Hydrogels
Sunting Xuan, Chang Uk Lee, Cong Chen, Andrew B. Doyle, Yueheng Zhang, Li Guo, Vijay T. John, Daniel J. Hayes, Donghui Zhang, 2016, Chemistry of Materials on p. 727-737
A novel composite film for detection and molecular weight determination of organic vapors
Bishnu P. Regmi, Joshua Monk, Bilal El-Zahab, Susmita Das, Francisco R. Hung, Daniel J. Hayes, Isiah M. Warner, 2012, Journal of Materials Chemistry on p. 13732-13741
Antimicrobial biocompatible bioscaffolds for orthopaedic implants
Ammar T. Qureshi, Lekeith Terrell, W. Todd Monroe, Vinod Dasa, Marlene E. Janes, Jeffrey M. Gimble, Daniel J. Hayes, 2014, Journal of Tissue Engineering and Regenerative Medicine on p. 386-395
Characterization of novel akermanite
A. S. Zanetti, G. T. Mccandless, J. Y. Chan, J. M. Gimble, Daniel J. Hayes, 2015, Journal of Tissue Engineering and Regenerative Medicine on p. 389-404
Thiol-acrylate nanocomposite foams for critical size bone defect repair
Leah Garber, Cong Chen, Kameron V. Kilchrist, Christopher Bounds, John A. Pojman, Daniel Hayes, 2013, Journal of Biomedical Materials Research - Part A on p. 3531-3541
News Articles Featuring Daniel Hayes
Feb 07, 2020
$2.8M grant to fund bioprinting for reconstruction of face, mouth, skull tissues
Seamlessly correcting defects in the face, mouth and skull is highly challenging because it requires precise stacking of a variety of tissues including bone, muscle, fat and skin. Now, Penn State researchers are investigating methods to 3D bioprint and grow the appropriate tissues for craniomaxillofacial reconstruction.
Full Article | 349,900 |
Sounds like they are going to try to play this like a dumber version of “Dynasty” with Palin and Bachmann trying to be dime store versions of Crystal and Alexis….
Linda Evans and Joan Collins, they ain’t…
And smart they ain’t either….
Still, “Dynasty’s” been gone for many years. I guess it’s time for a pale imitation to emerge as the TV Remake: “Dynasty 2012”.
Everything else is starting to look like the 1980’s (See Wall Street -the Movie), so why not redo this too?
Isn’t that how the entertainment world works? Wait a few years then remake good, original ideas into a second rate remake? Or is that GOP Politics I’m thinking of?
Since Reagan, it’s so hard to separate the two worlds. But, god knows these two women are entertainers and not real politicians….
From Salon.com:
Poor Michele Bachmann! She’d been working the divisive culture warrior routine for years, to great success, when suddenly John McCain transforms a low-profile Alaska governor into the Queen of the Tea Parties. It’s enough to make anyone resentful. So it’s not too surprising that the Bachmann presidential campaign has begun its war with the Sarah Palin pseudo-campaign.
We live in a society that expects and sometimes demands that ambitious women battle each other for the limited number of slots available to them at the top of power structures, so the story of Bachmann “taking on” Palin would’ve been inevitable even if Bachmann strategist Ed Rollins hadn’t begun directly drawing contrasts between the two. (Yes, we also live in a society where Ed Rollins is working to get Michele Bachmann elected president and we just have to deal with that.)
The problem is, as Steve Kornacki explained, Bachmann’s campaign began gearing up when it looked like there was no chance of Palin entering the race. But Palin’s riding a bus across the country now for some reason, so Bachmann’s people need to make sure she remains a pseudo-candidate.
First, Rollins accused Palin of being unserious on talk radio yesterday, and he pointed out that Palin quit her job, while Bachmann is still an elected official. In an interview with Politico’s Ben Smith, Rollins laid out the Bachmann campaign’s official anti-Palin message: Bachmann is smarter than Palin but just as adorable:)
via Oh no, Bachmann and Palin are fighting – 2012 Elections – Salon.com.
One response to “Dynasty 2012: The Michelle Bachmann and Sarah Palin Cat Fight”
Since Salon is a wholly owned subsidiary of the Democrat Party, Salon’s “political analysis” of 2 conservative women amounts to no more than wishful thinking. With our economy in shambles due to Obama’s policies of “spreading the wealth around” to his political cronies, short circuiting normal market signals in a capitalist economy, Obama’s re-election prospects decline with the increasingly dismal economic numbers coming out. | 61,226 |
Our guest today is David Mcraney, an internationally bestselling author, journalist, and lecturer who created the You Are Not So Smart blog, books, and podcast. David, who lives in Mississippi, cut his teeth covering Hurricane Katrina on the Gulf Coast and across the Deep South. Since then, he has been a beat reporter, editor, photographer, voiceover artist and television host. Before that, he had a varied working life, waiting tables, working construction, selling leather coats, building and installing electrical control panels, and owning pet stores. He’s here to talk to us today about his latest book ‘How to Beat your Brain’, an attempt to help us overcome our quirks and make decisions more effectively.
MWS Podcast 76: David Mcraney as audio only:
Download audio: MWS_Podcast_76_David_Mcraney
If you’d like to listen to the full unedited version of the talk in which we go into the Middle Way in a bit more depth and talk a bit more about David’s hopes for the book, you can do so here:
MWS Podcast 76: David Mcraney full version as audio only:
Download audio: MWS_Podcast_76_David_Mcraney_full_version
Here’s also the link to David’s blog post on Brand Loyalty that we talked about
4 thoughts on “The MWS Podcast 76: David Mcraney on how to beat your brain”
A great podcast with lots of good material. I’ve also just been looking at two of David’s books, and can recommend them to anyone who wants a readable introduction to cognitive biases and the like.
I obviously do have some differences of emphasis from David. One is that he discusses cognitive biases and heuristics as phenomena that obviously have both positive and negative features, whereas I am more interested in drawing on this work to more clearly and broadly differentiate the positive from the negative features. I tend to do this by distinguishing between the aspects of cognitive biases that are unavoidable aspects of our embodied situation, and those for which we can take some responsibility because we are capable of changing them. Absolutisation seems to be the distinguishing feature of those aspects of cognitive biases we can do something about.
I also think it’s necessary to grasp the nettle where absolutist or metaphysical beliefs are concerned. There really is no clear difference between absolutisation as it is found in cognitive biases and fallacies and absolutisation as it is found in philosophical and religious traditions – and the practical effects are indistinguishable – so I can see no justification for treating them differently. But that does not mean an anti-religious stance, given that negative metaphysical beliefs are just as absolute as positive ones (so atheism is just as dogmatic as theism) and many other aspects (for example, archetypes) of religion can be identified and worked with aside from absolute beliefs.
I’ve just listened to this podcast, and I’m also halfway through reading David’s book (the one called ‘You Can Beat Your Brain’ – something awful about that UK title). I thought the main thing was ‘should you do anything about these biases?’ with the answer yes, because it can have such a negative impact on ourselves and others, sometimes many others. In thinking about ‘can you do anything about these biases?’ I’ve realised, in my own experience, that one of the reasons we have a formal risk assessment procedure (let’s say, for planning school trips, where the children’s wellbeing is in my hands) is to protect the children from my (mistaken) belief that I will always act in a rational fashion. Perhaps we should be explicitly giving this as a reason why we have to do formal risk assessment procedure (as people who have to do this sort of thing love to complain about ‘health and safety gone mad’ etc.). The ‘sunk cost’ fallacy seems particularly dangerous for things like school trips, especially those that contain ‘hazardous pursuits’ – the trip leader can feel that they’ve spent so much time and effort planning a particular activity that they’ll carry on with it even though circumstances suggest that its not a good idea (pressing on up the mountain in horrendous weather).
Hi Jim,
I do agree that forward planning in general can be one way of working round likely biases. However, bureaucratised forward planning doesn’t strike me as a particularly effective way to do it, because bureaucracy tries to standardise everything, undermines trust and may also undermine our sense of responsibility. I have worked in a college in the past where I had to do risk assessments for trips, and I usually did think they were ridiculous. Perhaps there are some trips where there were appreciable or unusual risks for which prior planning would be useful and, as you say, it would help us avoid the assumption that we’ll always respond adequately. However, I can’t think of any actual trip (given the pretty safe trips I did, like taking students to a Buddhist monastery) where I actually did anything different at all because of the risk assessment. It’s the one-size-fits-all and lack of discretion and lack of trust that seemed to me so undermining about the risk assessment process.
You’ve hit the nail on the head there, about the bureaucracy issues. If the burden of formalising what you would have done anyway is too great then trips don’t even get started, to the detriment of the children who would have gone on them. | 372,679 |
Nokia today announced one of the most significant milestones in its history with the sale of its 1.5 billionth Series 40 mobile phone.
The 1.5 billionth phone was a Nokia Asha 303, which was sold to a female consumer in the Brazilian city of São Paulo and represents a historic moment as Nokia continues its strategy of connecting the next billion.
Nokia Asha 303, A stunning touch screen handset with QWERTY keypad designed for web browsing, social networking and gaming, that features pre-installed entertainment and applications such as Angry Birds Lite.
Company cliams that new Nokia Asha range.
15 Comments on "Nokia Hits 1.5 Billion Sales Of Series 40 Mobile Phones"
@asha
This all things won’t be happen until Microsoft will release a next major version of Windows Phone. May be you have to wait till the May. I have used browsing on Lumia 800 and didn’t find any lag or issue with browsing. Yeah but dual core will probably do all stuffs at more speed.
@Vikas Patidar
i am already using htc titan which has 1.5 ghz processor,but compared to android and ios web experience is very slow.though ui of windows is truely amazing,what lacks is dual core technology and applications.
hope windows will cope with that soon.cause this year is of quad core phones which even makes dual core technology microsopic.
@Tremerin D’souza Hi, you just wants to see screenshots of the apps or running it on phones. If you need screenshots then you can find all those on windows phone marketplace site but if you need running it on phones then I’ll post it later. | 60,478 |
Discussion in 'Contests' started by Void, Jun 27, 2018.
Hmm, kinda long period of time, but still thanks for replying.
Be patient on the medals! The staff have a lot to do. [/mandatory72hrpost]
Scrap.tf has a sort of 'policy' of making the price of a key $1.96 due to a fee or something of some sort. Anyway, that would be enough. 3 keys all being $1.96 would be $5.88. And $5 was the required amount.
Never got a medal in my life before, how do you know when you get it? Does it alert you automatically or do you have to check your profile to see if you got a medal?
There most likely will be some post about medals being handed out on Discord or forums.Also, there will be a normal "new item" alert in-game once the medals are given out.
you also dont need to even start the game. steam will also give you an aleart that you have 1 new item in your inventory
Every time I get a notification on steam, I get excited and think it’s the medal, only to find out it’s not
Actually, I do believe you need to log into the game to get the item since it's not from a trade. From my past experiences, steam never notified me about donator/participation medals until I went into TF2 and clicked the new item notify.
1 and a half week passed. ||| ||| ||| |.
Hey. Be patient.
I finished sorting through charity donors last night. Over 500 individual donors had to be retrieved, checking that their Steam info was correctly input and such. Jam entries are also currently being sorted through and are almost done. You can rest easy knowing that you'll have medals soon enough.
Again
(If anyone remembers the Scorcher trailers in Tropic Thunder
oh man,good movie (at least in my opinion)
pathetic, lmao
well, that's over and done with.
Was pretty fun TBH, even though it was my first event.
just jokes brah no need to be rude
Forgot I submitted art for this, hopefully I get my badge. Best contest I entered so far
The jam countdown clock now shows a white block and says "Unable to connect" fwiw.
Press F to pay respects.
F
Separate names with a comma. | 89,497 |
Have been looking at a lot on the net and woundered what you might think wouls be a good 1st machine.
The 35p Ohler is said to be good and then there is the F-1 "Basic" and the "Master" units that have a 16 foot cord to run back to the shooting table, so you can make adjustments while the range is "Hot".
Any ideas? | 395,896 |
Does anyone do stuff really out of the ordinary for Thanksgiving dinner? I know a handful of families that cook something than Turkey as their main course, but I’m mostly interested in what sorts of side dishes your family/friends prepare that are out of the ordinary. Let’s share unique recipes and salivate over the catatonic gluttony of tomorrow.
Some interesting turkey and thanksgiving factoids after the break…
- Did you know that the president officially pardoned the turkey in 2001?
- In the US, we’ve doubled our per capita consumption of turkey since 1970 to about 17.5 pounds per person.
- 271 million turkeys will be raised (and slaughtered) in 2008!
- Virginia is the fourth largest produces of turkeys. We produce over 21 million turkeys per year.
Thanksgiving Dinner More Expensive Due to Turkey and Cranberry Prices
[stats] [pic from xybermatthew]Tagged as: factoid, Holidays, thanksgiving, turkey
Did you know the President has officially pardoned a turkey every year since 1989.
@1, no, because i have the brain the size of a pea.
So does GWB… make your point Thor.
If he pardoned THE turkey as opposed to A turkey, then we’re all in trouble. Since 2001.
I like green bean casserole, stuffing, biscuits, oysters on the half-shell, shrimp cocktail, mashed potatoes, ham biscuits, gravy, Pinot Noir and taking off my trousers after it’s all done.
I read the entire label on the deep-fry-your-turkey kit at Kroger. The kit consists of one 3 gallon jug of cottonseed oil. Luckily, the oil can be re-used after Thanksgiving (optional filter not included), because “cottonseed oil does not absorb the flavor of foods cooked in it.”
So, people, more proof that cottonseed oil is NOT FOOD. It is clothing. Do not eat your shorts.
Deep fried turkey is great. Deep fried snickers are even better!
Deep fried
snickersknickers are even better!
FTFY
Don’t support the turkicide. Eat a Turk’y instead.
@9: So you’re saying we shouldn’t eat the turkey we kill? Well, there’s a pile of bird carcasses going to waste. Can we still give the gibblets to the hounds?
shen, Sarah Palin would not approve. Without turkey killing she wouldn’t have a backdrop to talk about the economy.
Sarah Palin can go drink a Crystal Pepsi and STFU.
It’s at about 4:00
but it has all the banality of evil, not really worth watching.
colfer, I got to 2:32 and paused the video. Since I will be eating turkey tomorrow, would it be wise for me to go passed this point?
Don’t be a pussy, echo. If you’re gonna eat it, you should see how it gets to your table.
I watched it. It’s not nearly as bad as they made it sound.
That’s cause you can’t hear the turkeys screaming.
Shen, the worst part of Thanksgiving is the turkey… it’s all the other quality food items that makes me happy!
The YouTube I linked to fuzzes out the worst parts, and that place is not even a factory farm.
The last vegetarian Tgiving I went to was great, but we had to peel tiny legumes for about four hours beforehand. Chai was actually introduced to a big city via our home kitchen. My roommate distributed it in liquid form back then. Tranks for the meowmeries.
To each their own, but that’s only because you all have never raised your own meat before…
/Waiting for it…waiting for it… Don’t disappoint me, ya’ll. I know it’s killing you to not say something.
The president is on track to pardon many turkeys before his term expires.
My mother has enslaved me in the kitchen this afternoon to prepare our Thanksgiving feast: honey-brined turkey, stuffing (which yes, we bake in the turkey), mashed potatoes, roasted butternut squash, green bean casserole (the one with French’s onions!), Waldorf salad, and cranberry sauce. Oh, and a pumpkin pie, an apple pie and a serious load of pumpkin cookies. Let it never be said that Italians eat wop food on the holidays.
omfg, i want pumpkin cookies!!
omfg, i want pumpkin cookies!!
(see, i want pumpkin cookies really badly.)
Salted the turkey today, will roast it tomorrow. We also will have sweet potatoes baked with bourbon, brown sugar and butter. Mashed potatoes. Brussel sprouts sauteed with turkey bacon. Yeast rolls. Stuffing both with and without oysters. Pecan and pumpkin pies. Cranberry sauce from the can and fresh cranberry relish. Corn pudding with roasted chantrelle mushrooms. Sage gravy.
Mostly, I’ll have a little turkey and a lot of oyster stuffing with lots of gravy a roll or two and a fair bit of both potato dishes. Then a big slice of pumpkin pie later in the day. My favorite part is the hot turkey sandwiches on Friday which I always have on toasted sourdough bread because when I was a kid my mom’s parents lived in San Francisco and would bring fresh sourdough with them when they came to visit for Thanksgiving every year. My mom always made hot turkey sandwiches on the sourdough the next day and it stuck.
ok, only because all the “old” ladies in my family died, I have “inherited” all the holiday dinners from my mom’s side of the family. Therefore, cooking for 20, we are having roasted turkey with dressing (INSIDE the friggin’ bird), mashed potatoes, sauteed fresh green beans, harvard beets, squash casserole, yeast rolls, jellied, whole and fresh with orange cranberry sauces, corn pudding, pumpkin pie, chocolate pie, carrot cake and some sort of extremely exotic apps brought by my foodie cousins, usually involving some sort of seafood like smoked salmon or crab. I plan to be comotose tomorrow night, but only because of the combination of rich foods with too much alcohol consumed because of relative stress… as opposed to too much alcohol consumed because of usual stress…
All I care about today is a green bean casserole getting IN MY BELLY! Along with pie. Must. MUST have pie.
This might be the first food thread that does not mention that rancid kitty cat semen called Aioli. Looks like jerked off dolphin.
Thurston….are you the guy that got arrested at Seaworld a few years ago?
My mom has banned the green bean casserole (I’ll make one for myself when I get home) but we use 2T browned butter, 2T soy sauce and 2T balsamic vinegar for sauce on green beans and it is way yummy. My family is uber-traditional for this holiday. Have a good one y’all!
@34: I am sad to hear that green bean casserole has been banned by your mother, wish we could share ours.
Luckily, since I’ve been cooking, I’ve been force-fed screwdrivers and champagne, so I am really excited about pie! Also, I named our turkey Sviatoslav, and everyone thinks I am out of my mind. Apparently telling them that I am a grad student is not a sufficient response.
@23 HAHA. I just got that.
The movie shown above says that Thanksgiving dinner for 10 costs $45? My turkey alone cost that.
Everything’s traditional up here at chez-of-the-parents-in-law. I did make a gluten-free/soy-free coconut cream pie, bittersweet chocolate tart, and wild mushroom/gf sourdough dressing. And in a few, I’ll go down and make myself some gravy.
The turkey is brined and smoked. Buttered corn, garlicky green beans, roasted yams and apples, mashed potatoes with extra garlic and herbs . . . typical fare. The rest of the group will have the traditional sausage and italian bread dressing, along with a very rich turkey/bacon gravy.
In the past, I’d provide vegetarian entrees, but I eat the turkey these days.
Soon, very soon, the drinking will start. Wooohah!
i had aquila d’oro chianti and a pint of chocolate ice cream for thanksgiving dinner.
i have too much to be genuinely concerned about
(not me) for being so full.
@37 - we started with bloody marys at 11:30 am. Why wait?
The best part about thanksgiving is the football. Oh and the sausage dip (Hot sausage, Rotel and cream chesse with melted cheddar on top) to go with it. | 415,252 |
TITLE: How are overtones produced by plucking a string?
QUESTION [20 upvotes]: I read the following from wikipedia:
When a string is plucked normally, the ear tends to hear the
fundamental frequency most prominently, but the overall sound is also
colored by the presence of various overtones (frequencies greater than
the fundamental frequency).
When I pluck a string, I just notice a node at each end and an antinode at the middle. How can we have overtones in addition to the fundamental frequency? It seems counterintuitive for me.
REPLY [5 votes]: You start with a triangular form, which has its fourier series. Let's say the initial shape is $f(x)$:
$$f(x)=\sum_n a_n \sin \frac{\pi n x}{L}$$
where $n$ counts the modes ($1$ is fundamental). So initial shape determines the harmonic content $a_n$ of certain modes. If you pluck in the middle, you will put more of the fundamental into the initial spectrum, and in every odd mode, but no even modes at all. If you pluck near the end of the string (like on a guitar), you get all modes, with higher modes still less prominent.
This shape then evolves in time. Every mode has a frequency, related to the wavelength:
$$f(x,t)=\sum_n a_n \sin \frac{\pi n x}{L}\cos \omega_n t$$
$\omega_n$ is proportional to $n$ (determined by $\omega=kc$, where $k=2\pi/\lambda$ the wavevector of the mode and $c$ is the speed of propagation of waves on the string).
This would make motion completely periodic and the shape would return to initial form after one cycle of the fundamental frequency. However, vibration is always damped: some small amount by radiating sound into the air, and mostly, by conversion into heat (the string not perfectly elastic). Higher frequency modes are always much more damped: if you demand very fine-detailed curly vibrations, this will get damped a lot because curvature of the string is high. Usually, you get something like that:
$$f(x,t)=\sum_n a_n e^{-t/\tau_n} \sin \frac{\pi n x}{L}\cos \omega_n t$$
where $\tau_n$ is the characteristic damping time of $n$th mode. You could say that the intensity of spectral components falls off faster for higher overtones and the tone is getting more sinusoidal. Very high harmonics associated by the sharp triangular kink at the plucking position just produce a "plunk" sound and disappear almost instantly.
Damping higher frequencies always has a smoothing effect on the shape: with time, the shape gets rounded, without sharp corners. | 30,403 |
Cabaret at Le Monde
Cabaret and Variety (comedy, variety)
- Venue 47Le Monde - Venue @ Le Monde
- 21:00
- Aug 25
- 1 hour
- Country: United Kingdom - Scotland
- Group: Blond Ambition Productions
- Warnings and additional info: Strong Language/Swearing
- Babes in arms policy: Babies are not allowed in the venue
- Policy applies to: Children under 2 years
DescriptionThe fabulous Bruce Devlin. 'Hugely enjoyable hour!' **** (FreshAir.org.
Edinburgh Spotlight 50. | 282,700 |
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Amplify is a New York based education company that creates K-12 curriculum, assessment and other programs for 21st century students. In 2017 Dominick was asked to illustrate a serious of openings for their digital science chapters on Microbiome and Metabolism. The illustrations had to function in a variety of formats across multiple digital devices. | 44,568 |
Belinda - welcoming first-time participants to Write! Canada 2008
Dear Friends,
She postively glowed as she took her place at the microphone. Her passion for writing, for encouraging others in their calling, and for ministering "belonging" to people, began to flow.
"I want to welcome you!" she said. There was no doubt as to her sincerity.
A little later in her talk...
"I am an encourager. That's what I love to do. So if you need any encouragement this weekend, come and see me. I'll give you some! It's what I do best..." Her joyful enthusiasm erupted into laughter and rippled across the room as other joined her. Any ice in the room, at that point, was instantly melted.
My mind is so full of all that God is saying and doing! So much so that I can't sit and force myself to concentrate on the very thing I came here to focus on this weekend - writing. But I did want to share this snapshot of the conference with you. I'm SURE Belinda will have more to share in the coming days. If not directly about the conference, you're sure to see the fruit of it as she moves forward in this aspect of God's calling on her life.
Special blessing are sent to those who would have loved to be here but couldn't make it. We're bringing home C.D.'s!
Blessings all! Susan.
3 comments:
Susan omitted to mention that on the evaluation forms, one person said that the part they enjoyed most was her testimony, where she shared how coming to this conference, Write! Canada, 8 years ago for the first time, has impacted her life. She added pizzaz and joy to the welcome to other writers.
Susan,
Thanks for the tribute to Belinda!
Interesting she speaks of herself as an encourager! She certainly is. She would be a great one to welcome first-time participants and to minister belonging. That's what is so dear about o¨r Belinda.
Way to go Susan and way to go Belinda. We look forward to hearing all the little gold nuggets you bring back. I know you'll learn tons!
Love to you both in Guelph!
Love right back to you Joyful, and Ang, and all our friends at home! What a wonderful time we are having; Bonnie too.
I have ordered the CD's of the sessions I was at, and will share! | 126,728 |
Watch Cave In perform with Kyle Scofield and Nate Newton on bass
Posted by Zenae Zukowski on June 14, 2018
Last night (13th), Cave in performed at The Royale in Boston, MA in memory of their former bandmate Caleb Scofield, who passed away this past March in a horrific car accident. During the show, Caleb’s brother Kyle joined the stage to perform six songs on bass while Converge etc. bassist, Nate Newton played the remaining […] | 309,301 |
If you're looking for a unique crochet piece to make, one that can be dressed up or down, and can even go to the office, look no farther than the Bette Bolero with Scarf, made in Heartland® yarn.
Check out the Bette Bolero with Scarf in action below. It's a stylish piece that's still classic, so it will be a useful part of your wardrobe for years to come.
Trouble viewing video? Click here. | 258,872 |
loans are smart ways to borrow money for a limited time. We offer the best services for all kinds of pay loan needs, with cash offerings between $100 and $1000. We are ready offer pay day loans for all reasons and needs. […] – This help video will assist Cape Coral Florida residents find a payday loan in their area. Some of the names mentioned include Advance America, Cash Plus, Check N Go and the Check Cashing Store. When looking for online payday loans remember that there are many different website available. Before taking out any type of bad credit payday loan remember to check the APR at this website: cfsaa.com The current maximum amount charged in the state of Florida is $15 for every $100 borrowed. This is the equivalent to an APR of 391.07%. Check out installment loans, unsecured personal loans or high interest credit cards as alternatives. […]
CLICK FOR CASH 818quote.com MORE INFO http Payday loans no credit check is the best option today! With our helpful and secure web at yo can get some extra money for all your need’s, such as; bills, car repair, rent, school money or anything alse. This is 100% your money so you can spend it your way. To get your no credit check payday loans you should be 18 years old and older, a valid bank account for money deposit and proof of income minimum $900. If you met your criteria go to payday loans no credit check no credit check payday loans. […] … […]
Apply at for Payday Loans Vancouver! Some Useful Facts about Payday Loans in Vancouver There are normally no credit checks because the only repayment is pre-scheduled and secured through electronic debit from a specified account that is verified in the beginning. Payday… […] […] Customer testimonial video where the customer describes how easy it was for him to apply for his second loan as a returning customer. PayDayOne.com is a state licensed payday loan lender which specializes in online no fax payday loans. http […] | 32,311 |
Sanders Delivers the Block 3 EW Software for USAF's F-22 Raptor
NASHUA, NH, 11-AUG-00 -- Sanders, a Lockheed Martin Company, has delivered a critical component for the U.S. Air Force's F-22 Raptor air dominance fighter. The company's F-22 Electronic Warfare (EW) team today delivered the Block 3 EW Operational Flight Program software to the Avionics Integration Lab (AIL) in Seattle, Wash., ahead of its Aug. 15 due date. The team also provided upgraded hardware to the AIL and the Boeing Flying Test Bed in preparation for the aircraft's flight certification testing. Don Donovan, Sanders' Vice President and General Manager for F-22 and JSF Programs, said the delivery supports the F-22 Team testing required to obtain the Low Rate Initial Production (LRIP) decision expected by the Pentagon later this year. The LRIP decision is crucial to the entire F-22 program and we are very pleased to have met this important milestone in support of the Lockheed Martin/Boeing Team. We set very high objectives for our Block 3 hardware and software systems in order to ensure we meet the LRIP decision criteria. We have met or exceeded all of those objectives with the content and quality of this product. I couldn't be more proud of our team, he said.
The Block 3 software, which completed extensive static and dynamic lab testing at Sanders' facilities in Nashua, N.H., is a significant portion of the overall Integrated Avionics software, scheduled to fly onboard the F-22 in December. The Sanders delivery represents approximately 15 percent of the Operational Flight Program software scheduled to fly on the F-22 fighter.
Bob Rearden, Lockheed Martin's Vice President for F-22 Programs, said, The Sanders team has provided a comprehensive EW suite that is a critical element of the integrated avionics capability of the F-22. This integrated avionics capability will be a main focus of our flight testing which begins later this year with the flight of Block 3.0 software in the F-22. The EW system has been performing extremely well to date at the AIL and on the Flying Test Bed, and Sanders' Block 3 software will provide a significant increase in capabilities available for testing.
The Block 3 delivery also included countermeasures and missile launch detection hardware and software for the F-22. Sanders' EW suite is an integral part of the aircraft's advanced sensor suite which will enable Raptor pilots to achieve air dominance over any potential adversary well into the 21st century. In addition to its EW suite, Sanders has previously provided the fighter's airborne videotape recorders, Communication, Navigation and Identification (CNI) antennae, graphic processor video interface, operational debrief system, stores management and modules, and mission support software. The value of these Sanders products for the Raptor during the production and support programs could exceed $2 billion through the year 2013. | 146,932 |
\begin{document}
\begin{frontmatter}
\title{A Nonlinear Perron-Frobenius Approach \\ for Stability and Consensus \\ of Discrete-Time Multi-Agent Systems\thanksref{footnoteinfo}}
\thanks[footnoteinfo]{This work was supported in part by the Italian Ministry of Research and Education (MIUR) with the grant ``CoNetDomeSys", code RBSI14OF6H, under call SIR 2014 and by Region Sardinia (RAS) with project MOSIMA, RASSR05871, FSC 2014-2020, Annualita' 2017, Area Tematica 3, Linea d'Azione 3.1.}
\author[Cagliari]{Diego Deplano}\ead{[email protected]},
\author[Cagliari]{Mauro Franceschelli}\ead{[email protected]},
\author[Cagliari]{Alessandro Giua}\ead{[email protected]}
\address[Cagliari]{Department of Electrical and Electronic Engineering, University of Cagliari, Italy}
\begin{keyword}
Multi-agent systems;
Consensus;
Nonlinear Perron-Frobenius theory;
Order-preserving maps;
Stability analysis.
\end{keyword}
\begin{abstract}
In this paper we propose a novel method to establish stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) evolving according to nonlinear heterogeneous local interaction rules which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. This paper directly generalizes results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on the structure of local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis.
\end{abstract}
\end{frontmatter}
\pagebreak
\section{Introduction}
The study of complex systems where local interactions between individuals give rise to a global collective behavior has aroused much interest in the control community. Such complex systems are often called Multi-Agent Systems (MAS), consisting of multiple interacting agents with mutual interactions among them.
A topic that captured the attention of many researchers is the consensus problem \cite{Olfati2007}, where the objective is to design local interaction rules among agents such that their state variables converge to the same value, the so called agreement or consensus state.
A MAS can be modeled as a dynamical system. In the discrete time linear case, classical Perron-Frobenius Theory is crucial in the convergence analysis. Indeed, in one of the most popular works in this topic (\cite{jadbabaie2003coordination}), the authors established criteria for convergence to a consensus state for MAS whose global dynamics can be represented by linear time-varying systems with non-negative row-stochastic state transition matrices, which are object of study of the classical Perron-Frobenius Theory.
The notable aspect of this work was to exploit such theory and graph theory instead of Lyapunov theory, allowing to study systems for which finding a common Lyapunov function to establish convergence is difficult or even impossible. Particularly, as it later became clear by the work in \cite{Olshevsky2008}, it allowed studying switched linear systems for which there does not exist a common quadratic Lyapunov function.
Along this line of thought, in this paper we aim to exploit nonlinear Perron-Frobenius theory \cite{LemmensNussbaum2012}, a generalization of non-negative matrix theory, to address nonlinear interactions in MASs without Lyapunov based arguments. It follows that a MAS modeled by a non-negative row-stochastic matrix is a particular case of the proposed general theory. The literature on nonlinear consensus problems is vast. It is mostly composed by particular nonlinear consensus protocols which offer advantages such as finite-time convergence \cite{Fran2015,Fran2017}, resilience to non-uniform time-delays \cite{Sun2009} and many more. These protocols are usually proved to converge to the consensus state via ad hoc Lyapunov functions. Most of approaches which aim to establish convergence to consensus for some class of nonlinear MAS falls in the general convexity theory of \cite{Moreau05}, i.e., each agent's next state is strictly inside the convex hull spanned by the state value of its neighbors. We mention the work in \cite{ZhiyunLin2007}, which is the continuous-time counterpart to the result of Moreau in \cite{Moreau05}, where the authors identify a class of non-linear interactions denoted as \emph{sub-tangent} and establish necessary and sufficient conditions on the network topology for convergence to consensus.
Our approach sharply differs from the previous literature. We identify a class of functions (which comprises also non-negative row-stochastic matrices) which we prove to have a special convergence properties in the positive orthant $\mathbb{R}^n_{\geq 0}$. In particular, we take inspiration from nonlinear Perron-Frobenius theory and considered order-preserving and sub-homogeneous nonlinear maps. Furthermore, the approach presented in this paper differs significantly from the preliminary results presented in \cite{Deplano18} in both the statement of the theorems, lemmas, and their proof.
The \textbf{main contribution} of this paper is threefold. First, we provide sufficient conditions for stability of a class of nonlinear discrete-time systems represented by positive, sub-homogeneous, type-K order preserving maps. Second, we propose a sufficient condition on the structure of heterogeneous local interaction rules among agents which guarantees that the global model of the MAS falls into the considered class of nonlinear discrete-time systems. Third, we propose a sufficient condition which links the topology of the network and the structure of the local interaction rules to guarantee the achievement of a consensus state, i.e., the network state in which all state variables have the same value. Our results are a generalization to nonlinear discrete-time system of non-negative matrix theory applied to multi-agent systems, in so doing our results do not exploit Lyapunov function arguments.
This paper is organized as follows. In Section \ref{sec:background} we present our notation and background material on multi-agent systems, order-preserving and sub-homogenoeus maps and recall the concept of periodic fixed-points. In Section \ref{sec:mainresults} we state our main results which consists in the statement of three main theorems. In Section \ref{shlsop} we discuss the proof of our results, we first discuss and list the required technical lemmas and then present the proof of each theorem in separate subsections. In Section \ref{sec:examples} we present examples of application of our theoretical results. Finally, in Section \ref{conclusion} we give our concluding remarks.
\section{Background} \label{sec:background}
In this work we propose novel tools to perform stability analysis (consensus as a special case) of MASs whose state update is represented by \emph{positive}, \emph{order-preserving} and \emph{sub-homogeneous} maps. In this section we define a model of autonomous nonlinear MASs in discrete-time, its associated graph and present the above mentioned properties which define the class of MAS under study.
\subsection{Multi-agent systems}
We consider a MAS composed by a set of agents $V=\left\{1,\ldots,n\right\}$, which are modeled as autonomous discrete-time dynamical systems with scalar state in $\mathbb{R}_{\geq 0}=\{x\in\mathbb{R}:x\geq 0\}$. Agents are interconnected and update their state as follows
\begin{equation}\label{eq:locdiscretesystem}
\begin{array}{c}
x_{1}(k+1)=f_1\left(x_1(k),\ldots,x_n(k)\right)\\
\vdots\\
x_{n}(k+1)=f_n\left(x_1(k),\ldots,x_n(k)\right)
\end{array}\quad,
\end{equation} where $k\in\{0,1,2,\ldots\}$ is a discrete-time index. Introducing the
aggregate state $x =\left[x_{1},\ldots,x_{n}\right]^T\in \mathbb{R}^n$, system \eqref{eq:locdiscretesystem} can be written as
\begin{equation}\label{eq:globdiscretesystem}
x(k+1) = f(x(k))
\end{equation} where $f:\mathbb{R}_{\geq 0}^n\rightarrow \mathbb{R}_{\geq 0}^n$ is differentiable. Hence, in this work we consider \emph{positive} systems \cite{Valcher18}. Positivity is a term with different meanings in different contexts, here by positive system we denote a system (and the associated map) with state that evolves in $\mathbb{R}_{\geq 0}^n$.
\begin{defn}[\textbf{Positive systems and maps}]\label{def:positivesystem}
~
System \eqref{eq:globdiscretesystem} is called positive if $f$ maps non-negative vectors into non-negative vectors, i.e., $f : \mathbb{R}_{\geq 0}^n \rightarrow \mathbb{R}_{\geq 0}^n$. Correspondingly, map $f$ is said to be positive. \hfill$\blacksquare$
\end{defn}
We now associate to the map $f$ a graph $\mathcal{G}(f)$ which captures the pattern of interactions among agents and denote it as \emph{inference graph} \cite{Liu13}.
Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a graph where $\mathcal{V}=\left\{1,\ldots, n\right\}$ is the set of nodes representing the agents and $\mathcal{E}\subseteq \mathcal{V} \times \mathcal{V}$ is a set of directed edges. A directed edge $(i,j)\in \mathcal{E}$ exists if node $i$ sends information to node $j$. To each agent $i$ is associated a set of nodes called neighbors of agent $i$ defined as $\mathcal{N}_i=\left\{j\in \mathcal{V}: (j,i)\in \mathcal{E}\right\}$. A \emph{directed path} between two nodes $p$ and $q$ in a graph is a finite sequence of $m$ edges $e_k=(i_k,j_k)\in E$ that joins node $p$ to node $q$, i.e., $i_1=p$, $j_m=q$ and $j_k=i_{k+1}$ for $k=1,\ldots,m-1$. A node $j$ is said to be \emph{reachable} from node $i$ if there exists a directed path from node $i$ to node $j$. A node is said to be \emph{globally reachable} if it is reachable from all nodes $i\in \mathcal{V}$.
\begin{defn}[\textbf{Inference graph}] \label{def:inferencegraph}
~
Given a map $f$ its inference graph $\mathcal{G}(f) = (\mathcal{V}, \mathcal{E})$ is defined by a set of nodes $\mathcal{V}$ and a set of directed edges $\mathcal{E}\subseteq \left\{\mathcal{V}\times \mathcal{V}\right\}$. An edge $(i,j)\in \mathcal{E}$ from node $i$ to node $j$ exists if
\begin{equation*}
\frac{\partial f_i(x)}{\partial x_j}\neq 0\quad x \in \mathbb{R}_{\geq 0}^n \setminus S,
\end{equation*}
where $S$ is a set of measure zero in $\mathbb{R}^n$. \hfill $\blacksquare$
\end{defn}
\subsection{Order-preserving maps}\label{Intro:nonlinearPerron}
The set of $\mathbb{R}^n_{\geq 0}$ is a partially ordered set with respect to the natural order relation $\leq$. For $u,v\in \mathbb{R}^n_{\geq 0}$, we can write the partial ordering relations as follows
\begin{align*}
u\leq v &\Leftrightarrow u_i\leq v_i \quad \forall i\in \mathcal{V}, \\
u\lneq v &\Leftrightarrow u\leq v \text{ and } u\neq v,\\
u< v &\Leftrightarrow u_i< v_i \quad \forall i \in \mathcal{V}.
\end{align*}
The partial ordering $\leq$ yields an equivalence relation $\sim$ on $\mathbb{R}^n_{\geq 0}$, i.e., $x$ is equivalent to $y$ ($x \sim y$) if there exist $\alpha,\beta \geq 0$ such that $x\leq \alpha y$ and $y\leq \beta x$. The equivalence classes are called \emph{parts} of the cone of non-negative real vectors and the set of all parts is denoted by $\mathcal{P}$. It can be shown (see \cite{GaubertAkian2006}) that the cone $\mathbb{R}_{\geq 0}^n$ has exactly $2^n$ parts, which are given by $$P_I=\{x\in\mathbb{R}_{\geq 0}^n\:|x_i>0,\:\forall i\in I \text{ and } x_i=0 \text{ otherwise} \}\:,$$ with $I\subseteq \{1,\ldots,n\}$. We define a partial ordering on the set of parts $\mathcal{P}$ given by $P_{I_1} \preceq P_{I_2}$ if $I_1\subseteq I_2$.
Maps which preserve such a vector order are said to be \emph{order-preserving}. Next, we provide a formal definition of three kinds of order-preserving maps present in the current literature.
\begin{defn}[\textbf{Order-preservation}]\label{def:ssop}
~
A positive map $f$ is said to be
\begin{itemize}
\item Order-preserving, if $\forall x,y \in \mathbb{R}^n_{\geq 0}$ it holds $$x\leq y \Leftrightarrow f(x)\leq f(y).$$
\item Strictly order-preserving, if if $\forall x,y \in \mathbb{R}^n_{\geq 0}$ it holds $$x\lneq y \Leftrightarrow f(x)\lneq f(y).$$
\item Strongly order-preserving, if $\forall x,y \in \mathbb{R}^n_{\geq 0}$ it holds
\begin{equation*}
x\lneq y \Leftrightarrow f(x)< f(y).\tag*{$\blacksquare$}
\end{equation*}
\end{itemize}
\end{defn}
The next remark is in order to clarify the context of the contribution of this paper.
\begin{rem}
~
For linear maps, order-preservation and positivity are equivalent properties and correspond to non-negative matrices. Since this is not the case for general nonlinear maps, in this work we consider positive nonlinear maps which are also order-preserving.\hfill $\blacksquare$
\end{rem}
Now, we are ready to introduce the definition of \emph{type-K order-preserving} maps, shown next, which plays a pivotal role in the characterization of the class of nonlinear systems in which we are interested and which will be discussed at length in the proofs of our results.
\begin{defn}[\textbf{Type-K Order-preservation}]\label{def:lsop}
~
A positive map $f$ is said to be type-K order-preserving if for any $x,y\in \mathbb{R}_{\geq 0}^n$ and $x\lneq y$ it holds
\begin{enumerate}[label=$(\roman*)$]
\item $x_i = y_i \Rightarrow f_i(x)\leq f_i(y)$ ,
\item $x_i< y_i \Rightarrow f_i(x) < f_i(y)$ ,
\end{enumerate}
for all $i=1,\ldots,n$, where $f_i$ is the $i$-th component of $f$.\hfill$\blacksquare$
\end{defn}
As it will be shown later, such a property is sufficient but not necessary for classical order-preservation. However, since it is easily identifiable from the sign structure of the Jacobian matrix, it allows to easily establish order-preservation of a given function. Furthermore, it constraints the behavior of the system, preventing the system from evolving with periodic trajectories and thus helping in proving convergence to a steady state.
\subsection{Sub-homogeneous maps}
Order-preserving dynamical systems and nonlinear Perron-Frobenius theory are closely related.
In the theory of order-preserving dynamical systems, the emphasis is placed on strong order-preservation. For discrete-time strongly order-preserving dynamical systems one has generic convergence to periodic trajectories under appropriate conditions \cite{Polacik1992}. An extensive overview of these results was given by Hirsch and Smith \cite{Hirsch2006}.
On the other hand, in nonlinear Perron-Frobenius theory one usually considers discrete-time dynamical systems that need not be strongly order-preserving, but satisfy an additional concave assumption and obtain similar results regarding periodic trajectories \cite{Lemmens2006nlp}. The concave assumption of interest in this paper is sub-homogeneity.
\begin{defn}[\textbf{Sub-homogeneity}]\label{def:sub-homogeneity}
~
A positive map is said to be sub-homogeneous if $$\alpha f(x) \leq f(\alpha x)$$ for all $x\in \mathbb{R}_{\geq 0}^n$ and $\alpha \in [0,1]$.\hfill $\blacksquare$
\end{defn}
Order-preserving and sub-homogeneous maps arise in a variety of applications, including optimal control and game theory \cite{GaubertAkian2003}, mathematical biology \cite{Rosenberg2001}, analysis of discrete event systems \cite{Gunawardena2003} and so on.
\subsection{Periodic points}
Concluding this section, we recall some basic concepts on periodic points which are instrumental to state our main results.
Consider the state trajectory of the system in eq. \eqref{eq:globdiscretesystem}. A point $x\in \mathbb{R}^n$ is called a \emph{periodic point} of map $f$ if there exist an integer $p\geq 1$ such that $f^p(x)=x$. The minimal such $p\geq 1$ is called the \emph{period} of $x$ under $f$.
If $f(x)=x$, we call $x$ a fixed point of $f$. A \emph{fixed point} is a periodic point with period $p=1$. Fixed points of a map are equilibrium points for a dynamical system. We denote $F_f=\{x\in X:f(x)=x\}$ the set of all fixed points of map $f$.
The \emph{trajectory} of the system in eq. \eqref{eq:globdiscretesystem} with initial state $x$ is given by $\mathcal{T}(x,f)=\{f^k(x):k\in \mathbb{Z}\}$.
If $f$ is clear from the context, we simply write $\mathcal{T}(x)$ to denote its trajectory, where $x$ is the initial state. If $x$ is a periodic point, we say that $\mathcal{T}(x)$ is a periodic trajectory.
We denote the limit set of a point $x$ of map $f$ as $\omega(x,f)$ (or simply $\omega(x)$ if $f$ is clear from the context), which is defined as $$\omega(x)=\bigcap_{k\geq 0} cl\left(\{f^m(x):m\geq k\}\right),$$ with $cl(\cdot)$ denoting the closure of a set, i.e., the set together with all of its limit points. If $x$ is a fixed point it follows that the set $\omega(x)$ is a singleton, i.e., a set containing a single point.
\section{Main results} \label{sec:mainresults}
In this section we state and clarify the main results of this paper, while the following sections are dedicated to their proof.
For positive maps which are also order-preserving and sub-homogeneous, existing results (see next section for insights) do not provide any condition to ensure convergence to a fixed point, but only to periodic points \cite{Lemmens2006nlp} when the initial state is strictly positive, i.e., $x\in\mathbb{R}^n_+$. Furthermore, to the best of our knowledge, no result provides any information about trajectories whose initial state lies in the boundary of $\mathbb{R}_{\geq 0}^n$.
Our aim is thus to fill this void considering the previously defined class of order-preserving maps, called \emph{type-K order-preserving}, for which we prove convergence to a fixed point for any initial state $x\in\mathbb{R}_{\geq 0}^n$ (and not only for $x\in\mathbb{R}^n_+$). This result is given in next theorem.
\begin{thm}[\textbf{Convergence}]\label{th:supnormembed}
~
Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If $f$ has at least one positive fixed point in $\mathbb{R}^n_+$ then all periodic points are fixed points, i.e., the set $\omega(x)$ is a singleton and $\forall x\in \mathbb{R}_{\geq 0}^n: \lim_{k \rightarrow \infty} f^k(x)=\bar{x}$ where $\bar{x}$ is a fixed point of $f$.\hfill$\blacksquare$
\end{thm}
Using this technical result, another of our contributions is a sufficient condition on the heterogeneous local interaction rules under which a MAS is stable, i.e., its state converges to a fixed point. Fixed points are synonymous for equilibrium points, while in the literature the term fixed points is widely used in the context of iterated maps, the term equilibrium point is usually preferred in the context of discrete-time dynamical systems. This result is given in Theorem \ref{th:nonlinearconvergence}, whose statement is shown next.
\begin{thm}[\textbf{Stability}]\label{th:nonlinearconvergence}
~
Consider a MAS as in \eqref{eq:globdiscretesystem} with at least one positive equilibrium point. If the set of differentiable local interaction rules $f_i$, with $i= 1,\ldots,n$, satisfies the next conditions:
\begin{enumerate}[label=$(\roman*)$]
\item $f_i(x) \in \mathbb{R}_{\geq 0}$ for all $x\in \mathbb{R}_{\geq 0}^n$;
\item $\partial f_i/\partial x_i> 0$ and $\partial f_i/\partial x_j\geq 0$ for $i\neq j$;
\item $\alpha f_i(x) \leq f_i(\alpha x)$ for all $\alpha\in[0,1]$ and $x\in K$;
\end{enumerate}
then the MAS converges to one of its equilibrium points for any positive initial state $x(0)\in\mathbb{R}_{\geq 0}^n$.
\hfill$\blacksquare$
\end{thm}
As a special case, we also study the consensus problem for the considered class of MAS. We propose a sufficient condition based on the result in Theorem \ref{th:nonlinearconvergence} so that, for any initial state in $\mathbb{R}^n_+$, the MAS asymptotically reaches the consensus state, i.e., all state variable converge to same value. The proposed sufficient condition is graph theoretical and based on the inference graph $\mathcal{G}(f)$. The condition is satisfied if there exists a globally reachable node in graph $\mathcal{G}(f)$ and the consensus state is a fixed point for the considered MAS. This result is given in the next theorem.
\begin{thm}[\textbf{Consensus}]\label{th:nonlinearconsensus}
~
Consider a MAS as in \eqref{eq:globdiscretesystem}. If the set of differentiable local interaction rules $f_i$, with $i= 1,\ldots,n$, satisfies the next conditions:
\begin{enumerate}[label=$(\roman*)$]
\item $f_i(x)\in \mathbb{R}_{\geq 0}$ for all $x\in \mathbb{R}_{\geq 0}^n$;
\item $\partial f_i/\partial x_i> 0$ and $\partial f_i/\partial x_j\geq 0$ for $i\neq j$;
\item $\alpha f_i(x) \leq f_i(\alpha x)$ for all $\alpha\in[0,1]$ and $x\in \mathbb{R}^n_{\geq 0}$;
\item $f_i(x)=x_i$ if $x_i=x_j$ for all $j\in\mathcal{N}_{i}^{in}$;
\item Inference graph $\mathcal{G}(f)$ has a globally reachable node;
\end{enumerate}
then, the MAS converges asymptotically to a consensus state for any initial state $x(0)\in\mathbb{R}_{\geq 0}^n$.\hfill$\blacksquare$
\end{thm}
In the remainder of the paper, we discuss the proof of our main results in Theorem \ref{th:supnormembed}, \ref{th:nonlinearconvergence} and \ref{th:nonlinearconsensus}.
\section{Proof of main results} \label{shlsop}
We begin by clarifying the relationships among the different kinds of order-preservation.
\begin{rem}\label{rem:lorderpreservation}
~
Strong order-preservation $\Rightarrow$ Type-K order-preservation $\Rightarrow$ strict order-preservation $\Rightarrow$ order-preservation.\hfill$\blacksquare$
\end{rem}
Every converse relationship in Remark \ref{rem:lorderpreservation} does not hold. Given $x,y\in\mathbb{R}$, let $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$, we have the following counter-examples:
\begin{itemize}
\item $f(x,y)=[1,1]^T$ is order-preserving but not strictly;
\item $f(x,y)=[y,x]^T$ is strictly order-preserving but not type-K;
\item $f(x,y)=[\sqrt{x}+y,y]^T$ is type-K order-preserving but not strongly.
\end{itemize}
Usually, to verify order-preservation is not an easy task. For differentiable continuous-time systems $\dot{x}=f(x)$ a sufficient condition to ensure order-preservation is given by Kamke \cite{Smith88,Kamke32}. The Kamke condition usually exploited in the analysis of continuous time systems is shown next.
\begin{lem}[\textbf{Kamke Condition}]\label{th:ifflsop}
~\cite{Smith88,Kamke32}~
The map $f$ of a continuous-time system $$\dot{x}=f(x)$$ is order-preserving if its Jacobian matrix is Metzler, i.e.,
\begin{equation*}
\partial f_i/\partial x_j\geq 0\text{ for }i\neq j\:.\tag*{$\blacksquare$}
\end{equation*}
\end{lem}
As a counterpart to Lemma \ref{th:ifflsop}, for discrete-time systems we propose a sufficient condition to ensure type-K order-preservation of a map $f$, instrumental to the analysis of discrete-time systems, which we denote \emph{Kamke-like} condition.
\begin{prop}[\textbf{Kamke-like condition}]\label{th:ifflsoplike}
~
The map $f$ of a discrete-time system $$x(k+1)=f(x(k))$$ is type-K order-preserving if its Jacobian matrix is Metzler with strictly positive diagonal elements, i.e., if
\begin{equation} \label{condition}
\partial f_i/\partial x_i> 0\text{ and } \partial f_i/\partial x_j\geq 0\text{ for }i\neq j\:.
\end{equation}
\end{prop}
\begin{pf*}{\textbf{Proof}.}
Let $x\in \mathbb{R}^n$ and, without lack of generality, $y=x+\varepsilon e_j$ where $\varepsilon >0$ and $e_j$ denotes a canonical vector with all zero values but the $j$-th which is $1$, thus $x\lneq y$. If \eqref{condition} holds, then
\begin{enumerate}
\item If $i\neq j$ then $y_i=x_i+\varepsilon 0=x_i$ and $$\frac{\partial f_i(x)}{\partial x_j}= \lim_{\varepsilon\rightarrow 0} \frac{f_i(x+\varepsilon e_j)-f_i(x)}{\varepsilon}\geq0\:,$$ which implies that $f_i(x)\leq f_i(x+\varepsilon e_j)=f_i(y)$, i.e., condition $(i)$ of Definition \ref{def:lsop}.
\item If $i = j$ then $y_i=x_i+\varepsilon 1>x_i$ and $$\frac{\partial f_i(x)}{\partial x_i}= \lim_{\varepsilon\rightarrow 0} \frac{f_i(x+\varepsilon e_i)-f_i(x)}{\varepsilon}>0\:,$$ which implies that $f_i(x)<f_i(x+\varepsilon e_i)=f_i(y)$, i.e., condition $(ii)$ of Definition \ref{def:lsop}.
\end{enumerate}
Since $1)\Rightarrow \eqref{condition}$ and $2) \Rightarrow \eqref{condition}$, the proof is complete.\hfill $\square$
\end{pf*}
Having clarified how to verify the type-K order-preserving property for a discrete-time system, we move on in the next subsection to discuss a significant property of order-preserving maps which are also sub-homogeneous, i.e., non-expansivness with respect to the so-called Thompson's metric.
\subsection{Non-expansive maps}
Dynamical systems defined by order-preserving and sub-homogeneous maps are non-expansive under the Thompson's metric. Here, we introduce the concepts of \emph{non-expansiveness} and \emph{Thompson's metric} and give a few useful lemmas.
\begin{defn}[\textbf{Thompson's metric} \cite{Thompson64}]\label{def:thompson}
~
For $x,y\in\mathbb{R}^n_{\geq 0}$ define $$M(x/y)=\inf\{\alpha\geq 0: y\leq \alpha x\}\:,$$ with $M(x/y)=\infty$ if the set is empty. By mean of function $M(y/x)$, Thompson's metric $d_T:\mathbb{R}^n\times\mathbb{R}^n\rightarrow[0,\infty]$ is defined for all $(x, y)\in (\mathbb{R}^n\times \mathbb{R}^n)\setminus (0,0)$ as follows $$d_T(x,y)=\log(\max\{M(x/y),M(y/x)\})$$ with $d_T(0, 0) = 0$. \hfill $\blacksquare$
\end{defn}
\begin{defn}[\textbf{Non-expansiveness}]\label{def:non-expansiveness}
~
A positive map $f$ is called non-expansive with respect to a metric $d:\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R}_{\geq 0}$, if $$d(f(x),f(y))\leq d(x,y)$$ for all $x,y\in\mathbb{R}_{\geq 0}^n$. \hfill $\blacksquare$
\end{defn}
The next result taken from \cite{GaubertAkian2006} but stated in reference to positive cones $K$. In this paper we always consider as a particular case the cone of non-negative vectors, i.e., $K=\mathbb{R}^n_{\geq0}$, which is a solid, closed and convex cone.
\begin{lem} \label{lem:non-expansiveness}
~\cite{GaubertAkian2006}~
Let $K$ be a solid closed convex cone\footnote{A set $K\in\mathbb{R}^n$ is called a \emph{convex cone} if $\alpha K \subseteq K$ for all $\alpha \geq 0$ and $K \cap (-K) = \{0\}$. The convex cone $K$ is \emph{closed} if it is a closed set in $\mathbb{R}^n$ and it is solid if it has a non-empty interior.} in $\mathbb{R}^n$. If $f:K\rightarrow K$ is an order-preserving map, then it is sub-homogeneous if and only if it is non-expansive with respect to Thompson's metric $d_T$. \hfill $\blacksquare$
\end{lem}
Such a property allows one to prove detailed results concerning the behavior of dynamical systems, see \cite{GaubertAkian2006,Lemmens2005,Nussbaum1990} and also Chapter~8 in \cite{LemmensNussbaum2012} and reference therein.
Thompson's metric $d_T$ and the sup-norm $\left\lVert{\cdot}\right\rVert_{\infty}$ defined by $$\left\lVert{x}\right\rVert_{\infty}=\max_i \left\|x_i\right\|\:,$$ are closely related thanks to the following lemma.
\begin{lem} \label{lem:isometry}
~\cite{Thompson64}~
The coordinate-wise logarithmic function $L:\mathbb{R}^n_{+}\rightarrow\mathbb{R}^n$ is an isometry from $(\mathbb{R}^n_{+},d_T)$ to $(\mathbb{R}^n,\left\lVert\cdot\right\rVert_{\infty})$, with $\mathbb{R}_+=\{x\in \mathbb{R}:x>0\}$.\hfill$\blacksquare$
\end{lem}
By Lemma \ref{lem:isometry}, if a positive map $f$ can be restricted to $\mathbb{R}_+^n=int(\mathbb{R}_{\geq 0}^n)$ and if it is order preserving and sub-homogenous, then $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$ given by $g = log \circ f \circ exp$, is a sup-norm non-expansive map that has the same dynamical properties as $f$. The dynamics of sup-norm non-expansive maps is widely known. In fact, there exists the following result, which is the simplified version of Theorem 4.2.1 in \cite{LemmensNussbaum2012}.
\begin{lem}\label{lem:boundedness}
~\cite{Thompson64}~
If $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a sup-norm non-expansive then only one of the next cases can occur:
\begin{enumerate}[label=$(\roman*)$]
\item $\forall x\in\mathbb{R}^n$ trajectories $\mathcal{T}(x)$ are unbounded;
\item $\forall x\in\mathbb{R}^n$ trajectories $\mathcal{T}(x)$ are bounded.\hfill $\blacksquare$
\end{enumerate}
\end{lem}
\subsection{Proof of Theorem \ref{th:supnormembed}}
As pointed out in the previous section, for positive maps $f$ which are also order-preserving and sub-homogeneous it is possible to establish the boundedness of any trajectory with initial states $x(0)\in \mathbb{R}_+^n$ and entirely enclosed in $\mathbb{R}_+^n$. On the contrary, there are still no results for trajectories with initial state $x(0)$ in the boundary of $\mathbb{R}_{\geq 0}^n$.
When $f$ satisfies the additional property of type-K order preservation, Theorem \ref{th:supnormembed} gives a sufficient condition for the boundedness of any trajectory with initial state $x(0)\in\mathbb{R}_{\geq 0}^n$ and convergence of such trajectories to a fixed point of map $f$.
The proof of Theorem \ref{th:supnormembed} requires the preliminary discussion of several lemmas which are needed to prove that type-K order-preservation as opposed to simple order-preservation, together with other properties, is sufficient to extend result to trajectories starting at any point in $\mathbb{R}^n_{\geq 0}$ and exclude the existence of periodic trajectories.
\begin{lem}\label{lem:zeropattern}
~
Let a positive map $f$ be type-K order-preserving. For all $x\in\mathbb{R}_{\geq 0}^n$ it holds that $f_i^k(x)>0$ for all $i$ such that $x_i>0$ and $k\geq 1$.
\end{lem}
\begin{pf*}{\textbf{Proof}.}
For any $x \in\mathbb{R}_{\geq 0}^n$ let $I(x)\subset\{1,\ldots, n\}$ be such that $x_i>0$ for $i\in I(x)$ and $x_i=0$ otherwise. Since $\mathbf{0}\leq x$, by type-K order-preservation of $f$ follows $f(\mathbf{0})\leq f(x)$. More precisely it holds $f_i(x)> f_i(\mathbf{0}) \geq 0$ for $i\in I(x)$ and $f_i(x)\geq f_i(\mathbf{0}) \geq 0$ otherwise, implying $I(x)\subseteq I(f(x))$. By induction, $I(x)\subseteq I(f^k(x))$, i.e., $f^k_i(x)>0$ for all $i\in I(x)$, completing the proof. \hfill $\square$
\end{pf*}
\begin{lem}\label{lem:partsordering}
~
Let a positive map $f$ be sub-homogeneous and type-K order-preserving. For all $x\in\mathbb{R}_{\geq 0}^n$ there exists a part $P\in\mathcal{P}(\mathbb{R}_{\geq 0}^n)$ and an integer $k_0\in \mathbb{Z}$ such that $f^{k}(x)\in P$ for all $k\geq k_0$.
\end{lem}
\begin{pf*}{\textbf{Proof}.}
Since by Lemma \ref{lem:non-expansiveness} $f$ is non-expansive under the Thompson's metric $d_T$, then $x\sim y$ implies $f(x)\sim f(y)$. This can be easily proved by noticing that $d_T(f(x),f(y))\leq d_T(x,y)<\infty$ since $x\sim y$.
This means that $f$ maps parts into parts, i.e., for all $x\in\mathbb{R}_{\geq 0}^n$ and $x'\in [x]=P_{I_0}$ it holds $f(x')\in[f(x)] = P_{I_1}$. By Lemma \ref{lem:zeropattern} it follows $P_{I_0}\preceq P_{I_1}$ and therefore $[x]\preceq [f(x)]$.
Generalizing, we say that $f^k(x)\in P_{I_k}$ with $k\in \mathbb{Z}$ and $I_{k}\subseteq I_{k+1}\subseteq\{1,\ldots,n\}$. There exists $k_0\in \mathbb{Z}$ such that $I_k=I_{k_0}$ for all $k> k_0$ and thus $P_k=P_{k_0}$. This completes the proof.\hfill $\square$
\end{pf*}
\begin{lem}\label{lem:boundedTrajectory}
~
Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If $f$ has a positive fixed point $\bar{x}\in\mathbb{R}_+^n$, then for all $x\in\mathbb{R}_{\geq 0}^n$ the trajectory $\mathcal{T}(x)$ is bounded.
\end{lem}
\begin{pf*}{\textbf{Proof}.}
By Lemma \ref{lem:isometry} it follows that $g=\log\circ f \circ \exp$ is a sup-norm non-expansive map that has the same dynamical properties as $f$ for all $x\in \mathbb{R}_+^n$. By Lemma \ref{lem:boundedness} we know that one of the two cases can occur:
\begin{enumerate}[label=$(\roman*)$]
\item all trajectories $\mathcal{T}(\log(x),g)$ are unbounded;
\item all trajectories $\mathcal{T}(\log(x),g)$ are bounded.
\end{enumerate}
Since $f$ has a fixed point $x_f\in \mathbb{R}_+^n$, such that $f(x_f)=x_f$, then $x_g=\log(x_f)$ is a fixed point of $g$, i.e., $g(x_g)=x_g$. The trajectory $\mathcal{T}(\log(x_f),g)$ is obviously bounded and therefore case $(ii)$ holds.
By Lemma \ref{lem:partsordering}, we can partition $\mathbb{R}_{\geq 0}^n$ in two disjoint sets $S_1$, $S_2$ such that if for $x$ there exists $k_0\in\mathbb{Z}$ such that $f^{k_0}(x) \in \mathbb{R}_+^n$, then $x\in S_1$, otherwise $x\in S_2$. We analyze these two cases.
1) For all $x\in S_1$, by Lemma \ref{lem:partsordering}, it holds that $f^{k}(x) \in \mathbb{R}_+^n$ for all $k\geq k_0$. Let $x_0 = f^{k_0}(x)$. Since case $(ii)$ holds $\mathcal{T}(\log(x_0),g)$ is bounded, because of the isometry also $\mathcal{T}(x_0,f)$ is bounded, and therefore also $\mathcal{T}(x,f)$. We conclude that for all $x\in S_1$ trajectories $\mathcal{T}(x,f)$ are bounded.
2) For all $x\in S_2$, by Lemma \ref{lem:partsordering}, there exists $k_0\in\mathbb{Z}$ such that $f^k(x)\in P_I$ with $I(x)\subset N = \{1,\ldots,n\}$ for all $k\geq k_0$.
Without loss of generality, here we assume $I=\{1,\ldots,m\}$, where $m<n$.
Let $x =[z_1^T,z_2^T]^T$ with $z_1\in \mathbb{R}^m_{\geq 0}$ and consider the following $m$-dimensional map $f^*:\mathbb{R}_+^m\rightarrow \mathbb{R}_+^m$ defined by $$f^*_i(z_1)=f_i(z_1,z_2),\quad z_2=\mathbf{0}\:,$$ with $i\in I(x)$.
It is not difficult to check that $f^*$ is still sub-homogeneous and type-K order preserving. Accordingly, $g^*=\log\circ f^* \circ \exp$ is a sup-norm non-expansive map that has the same dynamical properties as $f^*$ for all $x\in \mathbb{R}_+^m$.
The main point now is to prove that if $(ii)$ occurs then all trajectories $\mathcal{T}(\log(z_1),g^*)$ are also bounded. To this aim, we first need to show that for all $i\in I(x)$ it holds
\begin{equation}\label{eq:gis}
g^*_i(z_1)\leq g_i(z_1,z_2).
\end{equation}
Since both the exponential and the logarithmic functions are strictly increasing, \eqref{eq:gis} is equivalent to
\begin{equation}\label{eq:fis}
f^*_i(z_1)\leq f_i(z_1,z_2).
\end{equation}
By definition, \eqref{eq:fis} holds if $z_2 = 0$. If $z_2\neq 0$, for any $x=[z_1^T,z_2^T]^T$ consider $\bar{x}=[z_1^T,\bar{z}_2^T]^T$ such that $\bar{z}_2=\mathbf{0}$. Since $f$ is order-preserving, for all $i\in I$ it holds that $f_i(\bar{x})\leq f_i(x)$, which is equivalent to write $f_i(z_1,\bar{z}_2)\leq f_i(z_1,z_2)$ . By definition, $f_i^*(z_1)=f_i(z_1,\bar{z}_2)$. Therefore, $f_i^*(z_1) \leq f_i(z_1,z_2)$ for all $z_2\neq 0$, i.e., \eqref{eq:fis} and \eqref{eq:gis} hold.
Suppose that $(ii)$ occurs and there exist $\hat{z}_1\in \mathbb{R}_+^m$ such that $\mathcal{T}(\log(\hat{z}_1),g^*)$ is unbounded. By \eqref{eq:gis} it is clear that given $\hat{x}=[\hat{z}_1^T,z_2^T]^T$ the trajectory $\mathcal{T}(\log(\hat{x}),g)$ is also unbounded, contradicting $(ii)$.
Let $x_0 = f^{k_0}(x)$. Since all trajectories $\mathcal{T}(\log(x_0),g)$ are bounded, because of the isometry also $\mathcal{T}(x_0,f)$ is bounded, and therefore also $\mathcal{T}(x,f)$. We conclude that for all $x\in S_2$ trajectories $\mathcal{T}(x,f)$ are bounded.\hfill $\square$
\end{pf*}
\begin{lem}\label{lem:intsingleton}
~\cite{Jiang1996}~
Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If for all $x\in K$ the trajectory $\mathcal{T}(x)$ is bounded, then for all $x\in \mathbb{R}_{\geq 0}^n$, $\omega(x)$ is a singleton.\hfill$\blacksquare$
\end{lem}
Finally, after restating the Theorem \ref{th:supnormembed} for convenience of the reader, we present a compact proof based on the results presented in this section and Lemma 3.1.3 in \cite{LemmensNussbaum2012}.
\textbf{Theorem 7 (Convergence)} $\text{ }$ \emph{Let a positive map $f$ be sub-homogeneous and type-K order-preserving. If $f$ has at least one positive fixed point in $\mathbb{R}^n_+$ then all periodic points are fixed points, i.e., the set $\omega(x)$ is a singleton and $\forall x\in \mathbb{R}_{\geq 0}^n: \lim_{k \rightarrow \infty} f^k(x)=\bar{x}$ where $\bar{x}$ is a fixed point of $f$}.
\begin{pf*}{\textbf{Proof of Theorem \ref{th:supnormembed}}.}
~
By Lemma~\ref{lem:boundedTrajectory} it follows that all trajectories $\mathcal{T}(x)$ are bounded for all $x\in\mathbb{R}_+^n$. By Lemma \ref{lem:intsingleton} it follows that all periodic points are fixed points, i.e., the set $\omega(x)$ is a singleton. By Lemma 3.1.3 in \cite{LemmensNussbaum2012}, since $f$ is continuous all trajectories are bounded and all periodic points are fixed points it holds $\lim_{k \rightarrow \infty} f^k(x)=\bar{x}$ where $\bar{x}$ is a fixed point of $f$.\hfill $\square$
\end{pf*}
\subsection{Proof of Theorem \ref{th:nonlinearconvergence}}
By means of the technical result in Theorem \ref{th:supnormembed} we prove our second main result, a sufficient condition on the structure of the heterogeneous local interaction rules of the MAS under consideration so that the global map (possibly unknown due to an unknown network topology) is positive, type-K order preserving and sub-homogeneous map, thus falling within the class of systems considered in Theorem \ref{th:ifflsoplike}. We also restate the theorem for convenience of the reader.
\textbf{Theorem 8 (Stability)} $\text{ }$ \emph{Consider a MAS as in \eqref{eq:globdiscretesystem} with at least one positive equilibrium point. If the set of differentiable local interaction rules $f_i$, with $i= 1,\ldots,n$, satisfies the next conditions:
\begin{enumerate}[label=$(\roman*)$]
\item $f_i(x) \in \mathbb{R}_{\geq 0}$ for all $x\in \mathbb{R}_{\geq 0}^n$;
\item $\partial f_i/\partial x_i> 0$ and $\partial f_i/\partial x_j\geq 0$ for $i\neq j$;
\item $\alpha f_i(x) \leq f_i(\alpha x)$ for all $\alpha\in[0,1]$ and $x\in K$;
\end{enumerate}
then the MAS converges to one of its equilibrium points for any positive initial state $x(0)\in\mathbb{R}_{\geq 0}^n$.}
\begin{pf*}{Proof of Theorem \ref{th:nonlinearconvergence}.}
We start the proof by establishing equivalence relationships between the properties $(i)-(iii)$ of the local interaction rules of the MAS listed in the statement of Theorem \ref{th:nonlinearconvergence} and properties (a)-(c) shown next:
\begin{enumerate}[label=$(\alph*)$]
\item $f$ is positive;
\item $f$ is type-K order-preserving;
\item $f$ is sub-homogeneous;
\end{enumerate}
We now prove all equivalences one by one.
\begin{itemize}
\item $[(i)\Leftrightarrow (a)]$ Condition $(i)$ implies that $f$ maps a point of $\mathbb{R}_{\geq 0}^n$ into $\mathbb{R}_{\geq 0}^n$ and is, therefore, a positive map (see Definition \ref{def:positivesystem}).
\item $[(ii)\Rightarrow (b)]$ due to Proposition \ref{th:ifflsoplike} (Kamke-like condition).
\item $[(iii)\Leftrightarrow (c)]$ by Definition \ref{def:sub-homogeneity} of a sub-homogeneous map, sub-homogeneity can be verified element-wise for map $f$, thus the equivalence follows.
\end{itemize}
Thus, if conditions $(i)$ to $(iii)$ hold true for all local interaction rules $f_i$ with $i=1,\ldots,n$, since by assumption map $f$ has at least one positive fixed point, we can exploit the result in Theorem \ref{th:supnormembed} to establish that for all positive initial conditions, the state trajectories of the MAS converge one of its positive equilibrium points. \hfill $\square$
\end{pf*}
\subsection{Proof of Theorem \ref{th:nonlinearconsensus}}
To prove our third main result, namely Theorem \ref{th:nonlinearconsensus}, we need to introduce two technical lemmas. The first lemma, shown next, states sufficient conditions under which the elements along the rows of the Jacobian matrix of a map $f$ computed at a consensus point $c \mathbf{1}$ sum to one.
\begin{lem}\label{lem:stochasticJ}
~
Let a map $f$ be positive and differentiable. If the set of fixed points $F_f$ of map $f$ satisfies $F_f\supseteq\{c\mathbf{1},\quad c\in\mathbb{R}_{\geq 0}\}$, i.e., the set of fixed points contains at least all positive consensus states, then $$J_f(c\mathbf{1})\mathbf{1}=\mathbf{1}\quad \forall c \in \mathbb{R}_+\:,$$ where $J_f(c\mathbf{1})$ denotes
the Jacobian of $f$ evaluated in $c\mathbf{1}$.
\end{lem}
\begin{pf*}{\textbf{Proof}.} Since $f$ is differentiable, we can apply directly the definition of directional derivative in a point $x\in \mathbb{R}^n_{\geq 0}$ along a vector $v\in\mathbb{R}^n$ obtaining $$J_f(x)v = \lim_{h\rightarrow 0}\frac{f(x+hv)-f(x)}{h}\:.$$ Now we evaluate this expression in a consensus point $x=c\mathbf{1}\in F_f$, and along the direction $v=\mathbf{1}$ which is an invariant direction of $f$. We obtain
\begin{align}
J_f(c\mathbf{1})\mathbf{1} &= \lim_{h\rightarrow 0}\frac{f(c\mathbf{1}+h\mathbf{1})-f(c\mathbf{1})}{h}\:, \nonumber \\
&= \lim_{h\rightarrow 0}\frac{\cancel{c\mathbf{1}}+h\mathbf{1}-\cancel{c\mathbf{1}}}{h}= \mathbf{1}\:,\nonumber
\end{align}
thus proving the statement. \hfill $\square$
\end{pf*}
Next, we introduce a critical lemma needed to prove our third main result. In particular, we show that if there exists a fixed point of map $f$ different from a consensus point, then there exists a consensus point such that the Jacobian of map $f$ computed at that consensus point has a unitary eigenvalue with multiplicity strictly greater than one.
\begin{lem} \label{lem:invdir}
~Let $f$ be positive, sub-homogeneous, type-K order-preserving and have a set of fixed points $F_f$ such that $$F_f\supseteq\{c\mathbf{1}, \ \ c\in \mathbb{R}_{\geq 0}\}\:.$$
If there exists a fixed point $\bar{x}\in\mathbb{R}^n_{\geq 0}$ such that $$\bar{x}\neq c\mathbf{1}\:,\quad \forall c\in\mathbb{R}_{\geq 0}$$ then there exists $\bar{c}(\bar{x})>0$ such that the Jacobian matrix $J_f(\bar{c}(\bar{x})\mathbf{1})$ of map $f$ computed at $\bar{c}(\bar{x})\mathbf{1}$ has a unitary eigenvalue with multiplicity strictly greater than one.
\end{lem}
\begin{pf*}{\textbf{Proof}.}
Let $\bar{x}=\left[\bar{x}_1,\ldots,\bar{x}_n\right]^T\in\mathbb{R}^n_{\geq 0}$ be a fixed point of map $f$ and let $c_1,c_2\in\mathbb{R}_{\geq 0}$ be such that
\begin{align*}
c_1 & = \min_{i=1,\ldots, n} \bar{x}_i\:,\\
c_2 & = \max_{i=1,\ldots, n} \bar{x}_i\:.
\end{align*}
We define three sets
\begin{align*}
I_{min}(\bar{x}) & =\{i:\bar{x}_i = c_1\},\\
I_{max}(\bar{x}) & =\{i:\bar{x}_i = c_2\},\\
I(\bar{x}) & =\{i:\bar{x}_i\neq c_1,c_2\}.
\end{align*}
Consider a point $y$ such that the $i$-th component is defined by
\begin{equation}\label{defy}
y_i=
\begin{cases}
c_1 & \text{if } i\in I_{min}(\bar{x}) \\
c_3 & \text{otherwise}
\end{cases}
\end{equation}
and such that
\begin{gather}\label{eq:res1}
c_1\mathbf{1}\lneq y \lneq \bar{x}\lneq c_2\mathbf{1}\:.
\end{gather}
By \eqref{defy} and \eqref{eq:res1} it follows that
\begin{equation}\label{eq:res2}
y\leq c_3\mathbf{1}.
\end{equation}
$\bullet$ Now, we prove that
\begin{equation}\label{eq:ordflow}
f(y)\leq y.
\end{equation}
Since map $f$ is type-K order preserving, from \eqref{eq:res1} it follows $c_1\leq f_i(y) \leq \bar{x}_i$ and from \eqref{eq:res2} $f_i(y) \leq c_3$ for $i=1,\ldots,n$. For $i \in I_{min}(\bar{x})$, by definition $\bar{x}_i=c_1$ and thus $f_i(y)=c_1$, otherwise for $i \in I(\bar{x})\cup I_{max}(\bar{x})$ by \eqref{eq:res1} $\bar{x}_i\geq y_i = c_3$ and it follows $c_1\leq f_i(y)\leq c_3$. Thus, \eqref{eq:ordflow} holds.
$\bullet$ Now, we prove that
\begin{equation}\label{eq:constr2}
\frac{c_3}{c_2}\bar{x}_i\leq f_i(y)\leq c_3,\quad i=1,\ldots,n\:.
\end{equation}
Since $f$ is order-preserving and sub-homogeneous, then $f$ is non-expansive under the Thompson's metric (see Definition \ref{def:thompson}) by Lemma \ref{lem:non-expansiveness}. Now, by exploiting the definition of non-expansive map, we compute an upper bound to $d_T(\bar{x},f(y))$. It holds
\begin{align}
d_T(\bar{x},f(y)) &\leq d_T(\bar{x},y) \nonumber\\
&=\log\left(\max \left\{ M(\bar{x}/y), M(y/\bar{x}) \right\}\right) \nonumber
\end{align}
where
\begin{align}
M(\bar{x}/y)= &\inf\{\alpha\geq 0: y\leq \alpha \bar{x}\}= \max_i \frac{y_i}{x_i}=1, \nonumber\\
M(y/\bar{x})= &\inf\{\alpha\geq 0: \bar{x}\leq \alpha y\} = \max_i \frac{x_i}{y_i}\leq \frac{c_2}{c_3}. \nonumber
\end{align}
Since $c_2\geq c_3$, it holds
\begin{equation}\label{upper}
d_T(\bar{x},f(y))\leq \frac{c_2}{c_3}.
\end{equation}
Now, we compute a lower bound to $d_T(\bar{x},f(y))$, where
\begin{align}
d_T(\bar{x},f(y))=\log\left(\max \left\{ M(\bar{x}/f(y)), M(f(y)/\bar{x}) \right\}\right) \nonumber
\end{align}
and
\begin{align}
M(\bar{x}/f(y))= &\inf\{\alpha\geq 0: f(y)\leq \alpha \bar{x}\}= \max_i \frac{f_i(y)}{x_i}=1, \nonumber\\
M(f(y)/\bar{x})= &\inf\{\alpha\geq 0: \bar{x}\leq \alpha f(y)\} = \max_i \frac{x_i}{f_i(y)}\geq \frac{c_2}{c_3}. \nonumber
\end{align}
Thus
$\max \left\{ M(\bar{x}/f(y)), M(f(y)/\bar{x}) \right\} \geq \frac{c_2}{c_3},$
therefore
\begin{equation}\label{lower}
d_T(\bar{x},f(y))\geq \frac{c_2}{c_3}.
\end{equation}
By inequalities \eqref{upper} and \eqref{lower} it follows that $$d_T(\bar{x},f(y))= \max_i \frac{x_i}{f_i(y)}= \frac{c_2}{c_3},$$ thus proving that the inequality in \eqref{eq:constr2} holds true.
$\bullet$ Due to Theorem \ref{th:supnormembed} it holds $\displaystyle \lim_{k\rightarrow\infty} f^k_i(y)=\bar{y}_i$. Now, we prove that
\begin{equation} \label{13}
\bar{y}_i=
\begin{cases}
c_1 & \text{if } i\in I_{min}(\bar{x}),\\
c_3 & \text{if } i\in I_{max}(\bar{x}),\\
c_1 & \text{if } i\in I(\bar{x})\text{ and }\\ &\exists k^*:f^{k^*}_i(y)<f_i^{k^*-1}(y),\\
c_3 & \text{otherwise}.
\end{cases}
\end{equation}
In \eqref{13} three cases may occur:
\begin{enumerate}
\item If $i \in I_{min}(\bar{x})$ then $\bar{x}_i=c_1$ and by \eqref{eq:res1} it follows $f_i(y)=c_1$.
\item If $i \in I_{max}(\bar{x})$ then $\bar{x}_i=c_2$ and by \eqref{eq:constr2} it follows $f_i(y)=c_3$.
\item If $i \in I(\bar{x})$, by \eqref{eq:ordflow} two cases may occur:
\begin{enumerate}
\item There exists $k^*>0$ such that $f^{k^*}_i(y)<y_i$. In this case, by type-K order-preservation it holds that $f^k_i(y)<f_i^{k-1}(y)\quad \forall k\geq k^*+1$ and therefore $$\lim_{k\rightarrow \infty} f_i^k(y)=c_1.$$
\item Otherwise $f^k_i(y)=f^{k-1}(y)\quad \forall k>0$ and therefore $$\lim_{k\rightarrow \infty} f_i^k(y)=y_i=c_3.$$
\end{enumerate}
\end{enumerate}
Thus, by \eqref{13} we proved that for any fixed point $\bar{x}$ different from a consensus point $c\mathbf{1}$ there exists a fixed point $\bar{y}$ with elements corresponding to either $c_1$ or $c_3$ and such that $I(\bar{y})=\emptyset$.
Now, consider a point $z$ such that its $i$-th component is defined as follows
\begin{equation}\label{eq:newpoint}
z_i=
\begin{cases}
c_1 & \text{if } i \in I_{min}(\bar{y}) \\
c_4 & \text{if } i \in I_{max}(\bar{y})
\end{cases}
\end{equation}
with $c_4 \in \left[c_1,c_3\right]$.
By \eqref{13} and \eqref{eq:newpoint}, we can conclude that $z$ is fixed point, i.e., $f(z)=z$, for all values of $c_4$ in the interval $c_4\in \left[c_1, c_3\right]$.
Now, let $v(\bar{x})$ be a vector such that
\begin{equation}\label{defV}
v_i(\bar{x}) =
\begin{cases}
0 & \text{if } i\in I_{min}(\bar{x})\\
1 & \text{if } i\in I_{max}(\bar{x})\\
0 \; or \; 1 & \text{if } i\in I(\bar{x})
\end{cases}\:,
\end{equation}
Thus, by \eqref{defV} the point $c_1\mathbf{1}+hv(\bar{x})$ is a fixed point of map $f$ for all $h\in[0, c_3-c_1]$. Thus, it follows that $$f(c_1\mathbf{1}+hv(\bar{x})) = c_1\mathbf{1}+hv\:,\quad h\in[0, c_3-c_1]\:.$$
Since $v(\bar{x}) \neq \mathbf{1}$, it holds (by reasoning along the lines of Lemma \ref{lem:stochasticJ}) that the Jacobian of map $f$ computed at $c_1\mathbf{1}$ has a right eigenvector equal to $v(\bar{x})$, i.e., $J_f(c_1\mathbf{1})v(\bar{x})=v(\bar{x})$. By Lemma \ref{lem:stochasticJ} it holds that the Jacobian of $f$ satisfies $J_f(c\mathbf{1})\mathbf{1}=\mathbf{1}$ for all $c>0$. Thus, if there exists $\bar{x}\neq c \mathbf{1}$ then there exists $\bar{c}(\bar{x})=\displaystyle \min_{i=1,\ldots,n} \bar{x}_i=c_1$ such that matrix $J_f(\bar{c}(\bar{x})\mathbf{1})$ has a untiray eigenvalue with multiplicity strictly greater than one, thus proving the statement of this lemma. \hfill $\square$
\end{pf*}
Finally, we recall for convenience of the reader and detail a proof of our third and last main result, i.e., Theorem \ref{th:nonlinearconsensus}, which provides sufficient conditions for asymptotic convergence to the consensus state.
\textbf{Theorem 9 (Consensus)} $\text{ }$ \emph{Consider a MAS as in \eqref{eq:globdiscretesystem}. If the set of differentiable local interaction rules $f_i$, with $i= 1,\ldots,n$, satisfies the next conditions:
\begin{enumerate}[label=$(\roman*)$]
\item $f_i(x)\in \mathbb{R}_{\geq 0}$ for all $x\in \mathbb{R}_{\geq 0}^n$;
\item $\partial f_i/\partial x_i> 0$ and $\partial f_i/\partial x_j\geq 0$ for $i\neq j$;
\item $\alpha f_i(x) \leq f_i(\alpha x)$ for all $\alpha\in[0,1]$ and $x\in \mathbb{R}^n_{\geq 0}$;
\item $f_i(x)=x_i$ if $x_i=x_j$ for all $j\in\mathcal{N}_{i}^{in}$;
\item Inference graph $\mathcal{G}(f)$ has a globally reachable node;
\end{enumerate}
then, the MAS converges asymptotically to a consensus state for any initial state $x(0)\in\mathbb{R}_{\geq 0}^n$}.
\begin{pf*}{\textbf{Proof of Theorem \ref{th:nonlinearconsensus}}}
We start the proof by establishing the relations between properties $(i)-(v)$ and the following:
\begin{enumerate}[label=$(\alph*)$]
\item $f$ is positive;
\item $f$ is type-K order-preserving;
\item $f$ is sub-homogeneous;
\item $F_f=\{c\mathbf{1}: c\in \mathbb{R}_{\geq 0}\} $.
\end{enumerate}
We go through all equivalences one by one.
\begin{enumerate}
\item $[(i)\Leftrightarrow (a)]$ See Proof of Theorem \ref{th:nonlinearconvergence}.
\item $[(ii)\Leftrightarrow (b)]$ See Proposition \ref{th:ifflsoplike}.
\item $[(iii)\Leftrightarrow (c)]$ See Proof of Theorem \ref{th:nonlinearconvergence}.
\item $[(i-v)\Rightarrow (d)]$ The proof of this implication is given below.
\end{enumerate}
Condition $(iv)$ implies that the consensus space $c \mathbf{1}$ is a subset of the set of fixed points $F_f$ of map $f$, i.e., $$F_f\supseteq\{c\mathbf{1}: c\in \mathbb{R}_{\geq 0}\}.$$
By Theorem \ref{lem:stochasticJ} the Jacobian matrix $J_f(c\mathbf{1})$ evaluated at a consensus point is row-stochastic, i.e., $J_f(c\mathbf{1})\mathbf{1}=\mathbf{1}$.
By the definition of inference graph (see Definition \ref{def:inferencegraph}), it holds that $\mathcal{G}(f)=\mathcal{G}(J_f(c\mathbf{1}))$. Thus $\mathcal{G}(J_f(c\mathbf{1}))$ has a globally reachable node by hypothesis and is aperiodic because condition $(ii)$ ensures a self-loop at each node.
Now, we are ready to prove by contradiction that $[(i-v)\Rightarrow (d)]$. In particular, if there exists a fixed point $\bar{x}\neq c \mathbf{1}$, then by Lemma \ref{lem:invdir} the Jacobian matrix $J_f(c\mathbf{1})$ has a unitray eigenvalue with multiplicity strictly greater than one. On the other hand, by the widely known Theorem 5.1 in \cite{Bullo18}, if $\mathcal{G}(J_f(c\mathbf{1}))$ has a globally reachable node and is aperiodic then $J_f(c\mathbf{1})$ has a simple unitary eigenvalue with corresponding eigenvector equal to $\mathbf{1}$, unique up to a scaling factor $c$.
This is a contradiction, therefore it does not exist a fixed point $\bar{x}$ such that $\bar{x}\neq c \mathbf{1}$ with $c> 0$. Thus, we conclude that the set of fixed points of map $f$ satisfies
$$F_f = \{c\mathbf{1}, \ \ c\in \mathbb{R}_{\geq 0}\}.$$
Finally, if conditions $(a)$ to $(c)$ are satisfied, then Theorem \ref{th:nonlinearconvergence} the MAS converges to its set of fixed points $F_f$. If $(d)$ is satisfied, the $F_f$ contains only consensus points and thus the MAS in \eqref{eq:globdiscretesystem} converges to a consensus state for all $x\in\mathbb{R}^n_{\geq 0}$.
\hfill $\square$
\end{pf*}
\section{Examples} \label{sec:examples}
In this section we provide examples to corroborate our theoretical analysis of the convergences properties of discrete-time, nonlinear, positive, type-K order-preserving and sub-homogeneous MAS.
\begin{exmp} \normalfont
As a first example we consider a susceptible-infected-susceptible (SIS) epidemic model \cite{Allen94} described by the following
\begin{align}
x_i(k+1)&=f_i(x(k))\label{eq:SISmodel} \\
& = \displaystyle x_i + h\left[\delta_i(1-x_i)-x_i\sum_{j\in\mathcal{N}_{i}}\beta_{ij}(1-x_j)\right]\:.\nonumber
\end{align}
Such model was originally derived to describe the propagation of an infectious diseases over a group of individuals. Each group is subdivided according to susceptible and infectious. Individuals can be cured and reinfected many times, there is not an immune group. Given $n$ groups, let $x_i(k)$, $y_i(k)$ be the portion of, respectively, susceptibles and infectious of group $i$ at time $k$, it is clear that $x_i(k),y_i(k)\geq 0$ and $x_i(k)+y_i(k)=1$ for any $k$. Thus, it is sufficient to consider the dynamics of one of them to completely describe the system. In model \eqref{eq:SISmodel} variables have the following meaning:
\begin{itemize}
\item $\beta_{ij}\geq 0$ are the infectious rates;
\item $\delta_i\geq 0$ is the healing rate;
\item $h\geq 0$ is the sampling rate.
\end{itemize}
We now evaluate conditions $(i)-(iv)$ of Theorem \ref{th:nonlinearconvergence} to establish the convergence of the associated MAS to a positive fixed point. Due to space limitations, we omit all steps and give directly conditions under which the theorem holds.
\begin{itemize}
\item First we notice that $x_i(k)$ belongs to $[0,1]$ for all $k$. It is guaranteed that for all $x_i(k)\in[0,1]$ also $x_i(k+1)+\in[0,1]$ if and only if \eqref{eq:pos1} and \eqref{eq:pos2} hold,
\begin{align}
&h\delta_i\leq 1\:,\quad h\sum_{j\neq i}\beta_{ij}\leq 1\:,\label{eq:pos1}\\
&h\beta_{ii}\leq\left[\sqrt{1-h\sum_{j\neq i}\beta_{ij}}+\sqrt{h\delta_i}\right]^2\:.\label{eq:pos2}
\end{align}
We conclude that for any $x\in [0,1]^n\subset \mathbb{R}_{\geq 0}^n$ then $f_i(x)\in[0,1]\subset \mathbb{R}_{\geq 0}$, thus proving that condition $(i)$ holds.
\item Condition $(ii)$ holds if and only if the next inequality holds
\begin{equation}\label{eq:typek}
h\beta_{ii}< 1-h\delta_i-h\sum_{j\neq i}\beta_{ij}.
\end{equation}
\item Condition $(iii)$ holds if and only if the next inequality holds
\begin{equation}\label{eq:subhomo}
\delta_i\geq \sum_{j\neq i}\beta_{ij}.
\end{equation}
\item Condition $(iv)$ is satisfied since $\bar{x}=\mathbf{1}\in \mathbb{R}^n_{\geq 0}$ is a positive fixed point.
\end{itemize}
One can prove that \eqref{eq:pos1}, \eqref{eq:pos2}, \eqref{eq:typek}, \eqref{eq:subhomo} are equivalent to \eqref{eq:total}. Let $\beta = \sum_{j\neq i}\beta_{ij}$,
\begin{align}\label{eq:total}
h\delta_i+h\beta< 1-h\beta_{ii}\:,\quad h\beta\leq 0.5\:.
\end{align}
If \eqref{eq:total} holds, then conditions of Theorem~\ref{th:nonlinearconvergence} are satisfied, and we conclude that the MAS converges to a fixed point for all $x\in[0,1]^n$. For a MAS described by graph $\mathcal{G}_1$ in Figure~\ref{fig:graph_SIS}, a numerical simulation is given in Figure~\ref{fig:ev_SIS}.
\end{exmp}
\begin{figure}[b!]
\begin{center}
\begin{subfigure}{0.24\textwidth}
\begin{center}
\includegraphics[width=0.75\textwidth]{Graph_SIS.pdf}
\caption{Graph $\mathcal{G}_1$.}\label{fig:graph_SIS}
\end{center}
\end{subfigure}
~
\begin{subfigure}{0.24\textwidth}
\begin{center}
\includegraphics[width=0.75\textwidth]{Graph_CON.pdf}
\caption{Graph $\mathcal{G}_2$.}\label{fig:graph_CON}
\end{center}
\end{subfigure}
\caption{Graphs of Examples 1 and 2.}\label{fig:graphs}
\end{center}
\end{figure}
\begin{figure}[b!]
\begin{center}
\begin{subfigure}[t]{0.25\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Plot_SIS.pdf}
\caption{Parameters: $h=0.5$, $\beta_{ij} = 0.5$, $\delta_i=0.5$ for all $i$ and $j\in\mathcal{N}_i$.}\label{fig:ev_SIS}
\end{center}
\end{subfigure}
~
\begin{subfigure}[t]{0.25\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{Plot_CON.pdf}
\caption{Parameters: $\varepsilon=0.1$.}\label{fig:ev_CON}
\end{center}
\end{subfigure}
\caption{Evolutions of Examples 1 and 2.}\label{fig:evs}
\end{center}
\end{figure}
\begin{exmp} \label{ex:arctan} \normalfont Consider a MAS described by graph $\mathcal{G}_2$ in Figure~\ref{fig:graph_CON} and nonlinear local interaction rule
\begin{align}
x_i(k+1)&=f_i(x(k))\label{eq:CONmodel} \\
& = \displaystyle x_i(k)+\varepsilon_i \sum_{j\in\mathcal{N}_i} \text{atan}(x_j(k)-x_i(k))\:.\nonumber
\end{align}
We now evaluate conditions $(i)-(v)$ of Theorem \ref{th:nonlinearconsensus} to establish the convergence of the associated MAS to consensus state.
\begin{itemize}
\item Condition $(i)$ holds $\forall x\in\mathbb{R}^n_{\geq 0}$.
\item Condition $(ii)$ holds if and only if $\varepsilon_i \in \left(0,\left|\mathcal{N}_i\right|^{-1}\right]$ for $i=1,\ldots,n$.
\item Condition $(iii)$ holds $\forall x\in\mathbb{R}^n_{\geq 0}$.
\item Condition $(iv)$ is satisfied since $\bar{x}=c\mathbf{1}$ with $c\in\mathbb{R}_{\geq 0}$ is a solution for $x = f(x)$.
\item Condition $(v)$ is satisfied since graph $\mathcal{G}$ has a globally reachable node.
\end{itemize}
Thus, the conditions of Theorem~\ref{th:nonlinearconsensus} are satisfied, and we conclude that the MAS in \eqref{eq:CONmodel} converges to a consensus state. A numerical simulation is given in Figure~\ref{fig:ev_CON}.
\end{exmp}
\section{Conclusions and future works} \label{conclusion}
In this paper we presented three main results related to a class of nonlinear discrete-time multi-agent systems represented by a state transition map which is positive, sub-homogeneous and type-K order preserving.
The first result establishes that a general discrete-time dynamical system converges to one of its equilibrium points asymptotically if its corresponding state transition map is positive, sub-homogeneous and type-K order preserving.
The second result provides sufficient conditions for a set of nonlinear, discrete-time heterogeneous local interaction rules which define the MAS to establish stability of the MAS, independently of its graph topology (which is considered unknown) by exploiting our first main result.
Finally, the third result provides sufficient conditions for a set of nonlinear, discrete-time heterogeneous local interaction rules which define the MAS to establish asymptotic convergence to a consensus state if the inference graph of the MAS has a globally reachable node.
This paper generalizes results for discrete-time linear MAS whose state transition matrix is stochastic to the nonlinear case thanks to nonlinear Perron-Frobenius theory. Examples are provided to show the effectiveness of the stability analysis of a MAS based on our method.
Future work will consider MAS represented by a time-varying set of heterogeneous local interaction rules.
\bibliographystyle{plain}
\bibliography{autosam}
\appendix
\end{document} | 64,380 |
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\begin{document}
\maketitle
\abstract{We develop the theory of Griffiths period map, which relates the classification of smooth projective varieties to the associated Hodge structures, in the framework of Derived Algebraic Geometry. We complete the description of the local period map as a morphism of derived deformation functors, following the path marked by Fiorenza, Manetti and Martinengo. In the end we show how to lift the local period map to a (non-geometric) morphism of derived stacks, in order to construct a global version of that.}
\tableofcontents{}
\section*{Introduction}
Let\blfootnote{This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/I004130/1].} $X$ be a smooth complex projective variety of dimension $d$ and consider a family of deformations $\mathcal X\rightarrow S$ of $X$ over some contractible base $S$; in 1968 Griffiths observed that any such Kuranishi family induces canonically a variation of Hodge structures on $X$. More formally let
\begin{equation*}
0=F^{k+1}H^k\left(X\right)\hookrightarrow F^kH^k\left(X\right)\hookrightarrow\cdots\hookrightarrow F^1H^k\left(X\right)\hookrightarrow F^0H^k\left(X
\right)=H^k\left(X\right)
\end{equation*}
be the Hodge filtration on cohomology and set $b^{p,k}:=\mathrm{dim}\,F^pH^k\left(X\right)$; define
\begin{equation*}
\mathrm{Grass}\left(H^*\left(X\right)\right):=\prod_k\mathrm{Grass}\left(b^{p,k},H^k\left(X\right)\right)
\end{equation*}
which is a complex projective variety as so are the Grassmannians involved. Griffiths constructed the morphism
\begin{eqnarray} \label{intr G lpm}
\mathcal P^p:&S&\xrightarrow{\hspace*{0.5cm}}\mathrm{Grass}\left(H^*\left(X\right)\right) \nonumber \\
&t&\longmapsto \prod_kF^pH^k\left(X_t\right)
\end{eqnarray}
where $X_t$ is the fibre of the family $\mathcal X\rightarrow S$ over $t\in S$; map \eqref{intr G lpm} is said to be the \emph{$p$\textsuperscript{th} local period map} associated to $\mathcal X\rightarrow S$. In \cite{Grif} Griffiths proved that such a map is well-defined and holomorphic; he also computed its differential and showed that it is the same as the contraction map on the space $H^1\left(X,\mathscr T_X\right)$ of first-order deformations of $X$. Moreover it is possible to use map \eqref{intr G lpm} to derive some constraints on the obstructions of $X$. It is important to notice that, despite connecting two algebraically defined objects, map \eqref{intr G lpm} is not algebraic: as a matter of fact one can show that its image lies in the so-called \emph{period domain} (see \cite{Vo} Section 10.1.3), which is a subspace of the Grassmannian determined in general by transcendental equations. \\
The existence and holomorphicity of the local period map says that for any given Kuranishi family of a projective manifold $X$ there is a canonical way to construct a variation of its Hodge structures; moreover such a correspondence seems to be compatible with the general deformation theory of the variety $X$:
prompted by this observation, in 2006 Fiorenza and Manetti described Griffiths period map in terms of deformation functors. Let
\begin{eqnarray*}
\mathrm{Def}_X:&\mathfrak{Art}_{\mathbb C}&\xrightarrow{\hspace*{2cm}}\mathfrak{Set} \\
&A&\mapsto\frac{\left\{\text{deformations of }X\text{ over }A\right\}}{\text{isomorphism}}
\end{eqnarray*}
be the functor of Artin rings parametrising the deformations of the variety $X$ and recall that such a deformation functor is isomorphic to the deformation functor associated to the Kodaira-Spencer dgla $KS_X:=\mathbb R\Gamma\left(X,\mathscr T_X\right)$; in a similar way for all non-negative $p$ define the functor of Artin rings
\begin{eqnarray*}
\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)}:&\mathfrak{Art}_{\mathbb C}&\xrightarrow{\hspace*{3.5cm}}\mathfrak{Set} \\
&A&\mapsto\frac{\left\{A\text{-deformations of }F^pH^*\left(X\right)\text{ inside } H^*\left(X\right)\right\}}{\text{isomorphism}}
\end{eqnarray*}
which describes the deformations of the complex $F^pH^*\left(X\right)$ as a subcomplex of $H^*\left(X\right)$: this functor precisely encodes variations of Hodge structures on $X$. In \cite{FMan1}, \cite{FMan2} and \cite{FMan3} Fiorenza and Manetti proved the following facts:
\begin{itemize}
\item $\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)}$ is a deformation functor (in the sense of Schlessinger) and
\begin{equation*}
\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)}\simeq\mathrm{Def}_{C_{\chi}}
\end{equation*}
where $C_{\chi}$ is the $L_{\infty}$-algebra defined as the cone of the inclusion of dgla's
\begin{equation*}
\chi:\mathrm{End}^{F^p}\left(H^*\left(X\right)\right)\hookrightarrow\mathrm{End}^*\left(H^*\left(X\right)\right)
\end{equation*}
with
\begin{equation*}
\mathrm{End}^{F^p}\left(H^*\left(X\right)\right):=\left\{f\in\mathrm{End}^*\left(H^*\left(X\right)\right)\big|f\left(F^pH^*\left(X\right)\right)\subseteq F^pH^*\left(X\right)\right\};
\end{equation*}
\item the map
\begin{eqnarray*}
\mathrm{FM}^p:&KS_X&\xrightarrow{\hspace*{0.75cm}}C_{\chi} \\
&\xi&\longmapsto\left(l_{\xi},i_{\xi}\right)
\end{eqnarray*}
where $i$ is the contraction of differential forms with vector fields and $l$ stands for the holomorphic Lie derivative, is a $L_{\infty}$-morphism, thus it induces a morphism of deformation functors
\begin{equation*}
\mathrm{FM}^p:\mathrm{Def}_{KS_X}\longrightarrow\mathrm{Def}_{C_{\chi}};\footnote{Here, by a slight abuse of notation, the symbol $\mathrm{FM}^p$ is denoting both the $L_{\infty}$-map and the induced morphism of deformation functors.}
\end{equation*}
\item the natural transformation
\begin{eqnarray} \label{FM lpm intr}
\mathcal P^p:\mathrm{Def}_X\quad\;&\xrightarrow{\hspace*{0.5cm}}&\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)} \\
\forall A\in\mathfrak{Art}_{\mathbb C}\quad\left(\mathscr O_A\overset{\xi}{\rightarrow}\mathscr O_X\right)&\longmapsto& \quad F^pH^*\left(X,\mathscr O_A\right) \nonumber
\end{eqnarray}
is a morphism of deformation functors extending Griffiths period map\footnote{This is why, by a slight abuse of notation, we are using the same symbol for both.} and the two morphisms $\mathrm{FM}^p$ and $\mathcal P^p$ are canonically isomorphic.
\end{itemize}
Fiorenza and Manetti's work shows that the $p$\textsuperscript{th} local period map is actually a \emph{morphism of deformation theories}, thus it commutes with all deformation-theoretic constructions: in particular all results of Griffiths about the differential of map \eqref{intr G lpm} are easily recovered as purely formal corollaries of the preceding statements. Moreover Fiorenza and Manetti's construction works for any proper smooth scheme of dimension $d$ over a field $k$ of characteristic $0$.\\
As we are able to interpret Griffiths period map as a natural transformation of deformation functors, the next step would be to look at it in the context of Derived Deformation Theory: more formally one could ask whether there exist derived enhancements of the functors $\mathrm{Def}_X$ and $\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)}$ for which it is possible to find some natural derived extension of morphism \eqref{FM lpm intr}. In 2012 Fiorenza and Martinengo approached this problem, tackling it from an entirely algebraic viewpoint. As a matter of fact they observed that the contraction of differential forms with vector fields $i$ (seen in the most general way, i.e. as a morphism of complexes of sheaves over $X$) and the Lie derivative $l$ give rise to a morphism of differential graded Lie algebras
\begin{equation} \label{FMM htpy fibre}
\mathrm{FMM}:\mathbb R\Gamma\left(X,\mathscr T_X\right)\overset{\left(l,e^i\right)}{\xrightarrow{\hspace*{1cm}}}\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)\doublerightarrow{\mathrm{incl.}}{0}\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)\right)
\end{equation}
where $\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)$ is the dgla made of non-negatively graded maps of the complex $\mathbb R\Gamma\left(X,\Omega^*_X\right)$ in itself, which can be interpreted as the dgla of all filtration-preserving maps. Notice also that the codomain of map \eqref{FMM htpy fibre} is nothing but the homotopy fibre over $0$ of the inclusion of $\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)$ into $\mathrm{End}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)$. In \cite{FMar} Fiorenza and Martinengo showed that map \eqref{FMM htpy fibre} induces a morphism of derived deformation functors
\begin{equation} \label{FMM lpm}
\mathrm{FMM}:\mathbb R\mathrm{Def}_{\mathbb R\Gamma\left(X,\mathscr T_X\right)}\overset{\left(l,e^i\right)}{\xrightarrow{\hspace*{1cm}}}\mathbb R\mathrm{Def}_{\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)\doublerightarrow{\mathrm{incl.}}{0}\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)\right)}
\end{equation}
whose $0$-truncation $\mathrm{FM}$ is very close to $\mathrm{FM}^p$ (actually $\mathrm{FM}$ is even more interesting than $\mathrm{FM}^p$ as it does not depend on the degree of the filtration, so it can be interpreted as a universal version of Griffiths period map). \\
The ultimate goal of this paper is to lift Fiorenza, Manetti and Martinengo's work to a global level, i.e. to find a morphism of derived stacks whose restriction to formal neighbourhoods gives back map \eqref{FMM lpm}. A first crucial step in order to do so consists of finding a more geometric description of such a map, thus we will define the morphism of derived deformation functors
\begin{eqnarray} \label{geom FMM}
&\scriptstyle{\mathcal P:\;\mathbb R\mathrm{Def}_X}&\scriptstyle{\longrightarrow\;\,\mathrm{hoFlag}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right),F^{\bullet}\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)} \nonumber \\
\scriptstyle{\forall A\in\mathfrak{dgArt}_k^{\leq0}}&\scriptstyle{\left[\mathscr O_{A,*}\overset{\varphi}{\longrightarrow}\mathscr O_X\right]\approx\left[\vcenter{\xymatrix{\ar@{} |{\Box^h}[dr]X\ar@{^{(}->}[r]\ar[d] & \mathcal X\ar[d] \\
\mathrm{Spec}\left(k\right)\ar[r] & \mathbb R\mathrm{Spec}\left(A\right)}}\right]}&\scriptstyle{\mapsto\qquad\left[\left(\left(\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right),F^{\bullet}\right),\tilde{\varphi}\right)\right]}
\end{eqnarray}
and prove that it is naturally isomorphic to map \eqref{FMM lpm}; in formula \eqref{geom FMM} $\mathbb R\mathrm{Def}_X$ parametrises derived deformations of the scheme $X$ (i.e. homotopy flat families of derived schemes deforming the underived scheme $X$), while $\mathrm{hoFlag}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right),F^{\bullet}\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)$ encodes derived deformations of the filtered complex $\left(\mathbb R\Gamma\left(X,\Omega^*_X\right),F^{\bullet}\right)$. Although it is intuitively quite clear what such functors should be, giving a careful definition of them reveals to be non-trivial at all and has actually lead us to develop the notions of affine differential graded category and affine simplicial category, which are probably interesting objects themselves to study. $\mathbb R\mathrm{Def}_X$ and $\mathrm{hoFlag}\left(\mathbb R\Gamma\left(X,\Omega^*_X\right),F^{\bullet}\mathbb R\Gamma\left(X,\Omega^*_X\right)\right)$ are by construction formal neighbourhoods of interesting derived stacks: the former is the formal neighbourhood at $X$ of the derived (non-geometric) stack $\mathcal{DS}ch_{d/k}$ of derived schemes of dimension $d$, which has been recently studied by Pridham in \cite{Pr3}, while the latter is the formal neighbourhood at $\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right),F^{\bullet}\right)$ of the homotopy flag variety $\mathcal{DF}lag_k\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)$, which is studied in \cite{dN2}. \\
The above tools provide us with a path towards a sensible global version of map \eqref{geom FMM}, i.e. a geometric morphism of derived geometric stacks inducing the latter on formal neighbourhoods. A very partial answer to such a question is discussed in the end of this paper, where we consider the non-geometric morphism of derived stacks
\begin{eqnarray} \label{non-geom derived Griff}
&\scriptstyle{\underline{\mathbb R\mathcal P}:\;\mathcal{DS}ch_{d/k}\times^h_{\mathbb R\mathcal Perf_k}\left\{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right\}}&\scriptstyle{\longrightarrow\;\;\mathcal{DF}lag_k\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)} \nonumber \\
\scriptstyle{\forall A\in\mathfrak{dgAlg}_k^{\leq0}}&\scriptstyle{\left[Y,\theta:\mathbb R\Gamma\left(\pi^0 Y,\Omega^*_{Y/A}\right)\overset{\sim}{\rightarrow}\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A\right]}&\scriptstyle{\longmapsto\;\;\;\,\left(\mathbb R\Gamma\left(\pi^0 Y,\Omega^*_{Y/A}\right),F^{\bullet}\right)}
\end{eqnarray}
and observe that it actually induces map \eqref{geom FMM} on formal neighbourhoods at $\left(X,\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)$ and $\underline{\mathbb R\mathcal P}\left(\left(X,\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\right)=\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right),F^{\bullet}\right)$. However it is important to notice that map \eqref{non-geom derived Griff} cannot be geometric for at least two important reasons:
\begin{enumerate}
\item a global geometric morphism lifting Griffiths period map to the derived world cannot be algebraic as the classical underived map is already non-algebraic in general;
\item if we move the problem to the context of the newly-born Derived Analytic Geometry, we will have to take care of monodromy issues: in fact, given a family $p:\mathcal X\rightarrow\mathcal S$ of derived schemes (or even derived analytic spaces) globally deforming $X$, the presence of a (possibly higher) monodromy group acting non-trivially on the fibres of $\mathbb Rp_{*}\Omega^*_{\mathcal X/\mathcal S}$ may represent a serious obstruction to make map \eqref{non-geom derived Griff} into a derived analytic morphism of derived analytic stacks.
\end{enumerate}
The problem of constructing a derived geometric global period map in the context of Derived Analytic Geometry seems to be very interesting: it is now being jointly studied by Holstein and the author and will appear in \cite{dNH}. \\
$\mathbf{Acknowledgements}$ --- The author does wish to thank his PhD supervisor Jonathan P. Pridham for suggesting the problem and for his constant support and advise along all the preparation of this paper. The author is also deeply indebted to Domenico Fiorenza, Ian Grojnowski, Julian V. S. Holstein, Donatella Iacono, Dominic Joyce, Marco Manetti and Elena Martinengo for several inspiring discussions about the period mapping in Deformation Theory.
\section{The Period Map as a Holomorphic Function}
Let $X$ be a compact connected complex K\" ahler manifold of dimension $d$ and consider a family of deformations $\varphi:\mathcal X\rightarrow S$, i.e. a proper holomorphic submersion of complex manifolds (where the base $S$ is contractible) admitting a distinguished fibre $\varphi^{-1}\left(0\right)=:X_0\simeq X$. Recall that a famous result due to Ehresmann says that any such family is $C^{\infty}$-trivial, i.e. there exists a diffeomorphism
\begin{equation} \label{Ehr}
T:\mathcal X\overset{\sim}{\longrightarrow}X_0\times S\simeq X\times S
\end{equation}
over $S$ (see \cite{Vo} Theorem 9.3). For all $t\in S$ let $X_t:=\varphi^{-1}\left(t\right)$: Ehresmann's trivialisation \eqref{Ehr} clearly induces a diffeomorphism $X_t\simeq X$ for all $t$, thus we can think of the morphism $\varphi$ as a collection of complex structures over the $C^{\infty}$-manifold underlying the complex variety $X$. This situation is the prototypical example of all deformation problems and was originally studied by Kodaira and Spencer. \\
A very natural question to ask is how the standard Hodge structures over $X$ vary with respect to the family $\varphi$; more formally, consider the cohomology algebra of $X$
\begin{equation*}
H^*\left(X,\mathbb C\right):=\bigoplus_{0\leq k\leq d}H^{k}\left(X,\mathbb C\right)
\end{equation*}
and recall that each cohomology group $H^k\left(X,\mathbb C\right)$ is endowed with a Hodge structure of weight $k$ defined by the Hodge decomposition
\begin{equation*}
H^k\left(X,\mathbb C\right)=\bigoplus_{p+q=k}H^{p.q}\left(X\right) \qquad\qquad H^{p,q}\left(X\right)\simeq H^q\left(X,\Omega^p_X\right)
\end{equation*}
or, equivalently, by the Hodge filtration
\begin{equation*}
0=F^{k+1}H^k\left(X,\mathbb C\right)\hookrightarrow F^kH^k\left(X,\mathbb C\right)\hookrightarrow\cdots\hookrightarrow F^1H^k\left(X,\mathbb C\right)\hookrightarrow F^0H^k\left(X,\mathbb C\right)=H^k\left(X,\mathbb C\right)
\end{equation*}
where
\begin{equation*}
F^mH^k\left(X,\mathbb C\right):=\bigoplus_{p\geq m}H^{p,q}\left(X\right).
\end{equation*}
The question we want to address is whether the family $\varphi$ induces any interesting structure on the cohomology of the fibres.
\subsection{Griffiths Period Map}
Observe that Ehresmann's trivialisation \eqref{Ehr} provides us with a diagram of isomorphisms of vector spaces
\begin{equation*}
\xymatrix{H^k\left(\mathcal X,\mathbb C\right)\ar[r]^{\sim}\ar@{=}[d] & H^k\left(X,\mathbb C\right)\ar[d]_{\wr} \\
H^k\left(\mathcal X,\mathbb C\right)\ar[r]^{\sim} & H^k\left(X_t,\mathbb C\right)}
\end{equation*}
which commutes for all $k\geq 0$ and $\forall t\in S$; actually much more is true, as for all $k\geq 0$ the sheaf $\mathbb R^k\varphi_*\underline{\mathbb C}$ -- where $\varphi_*:\mathfrak{Sh}\left(\mathcal X\right)\rightarrow\mathfrak{Sh}\left(S\right)$ is the push-forward functor -- is seen to be a local system over $S$ isomorphic to the constant sheaf $\underline{H^k\left(X,\mathbb C\right)}$ (see \cite{Vo} Section 9.2 for a more detailed explanation), thus the diagram above does not depend on the choice of the trivialisation. Denote
\begin{equation*}
h^k:=\mathrm{dim}\,H^k\left(X,\mathbb C\right)\qquad\qquad h^{p,q}:=\mathrm{dim}\,H^{p,q}\left(X\right)\qquad\qquad b^{p,k}:=\mathrm{dim}\,F^pH^k\left(X,\mathbb C\right).
\end{equation*}
A standard argument based on the $E_1$-degeneration of the Hodge-to-De Rham spectral sequence of $X$ shows that there exists a neighbourhood of $0\in S$ such that
\begin{equation*}
\mathrm{dim}\,H^{p,q}\left(X_t\right)=\mathrm{dim}\,H^{p,q}\left(X\right)=:h^{p,q}
\end{equation*}
thus the Hodge numbers of $X$ are invariant under (infinitesimal) deformation; moreover this immediately implies that the Hodge-to-De Rham spectral sequence of such fibres degenerates at its first page\footnote{Up to shrinking the base $S$, the fibres of $\varphi$ are K\" ahler manifolds themselves (see \cite{Vo} Theorem 9.23).}, as well (see \cite{Vo} Proposition 9.20).
\begin{defn} (Griffiths) In the above notations define the \emph{$(p,k)$\textsuperscript{th} local period map} to be
\begin{eqnarray} \label{G plpm}
\mathcal P^{p,k}:&S&\xrightarrow{\hspace*{0.5cm}}\mathrm{Grass}\left(b^{p,k},H^k\left(X,\mathbb C\right)\right) \nonumber \\
&t&\longmapsto F^pH^k\left(X_t,\mathbb C\right).
\end{eqnarray}
\end{defn}
Since $H^k\left(X_t,\mathbb C\right)$ is canonically isomorphic to $H^k\left(X,\mathbb C\right)$ and the Hodge numbers of $X$ are invariant under deformation, map \eqref{G plpm} is well-defined; the following is a famous result of Griffiths.
\begin{thm} \emph{(Griffiths)}
The $(p,k)$\textsuperscript{th} local period map \eqref{G plpm} is holomorphic $\forall p\leq k$.
\end{thm}
\begin{proof}
See \cite{Vo} Theorem 10.9.
\end{proof}
\subsection{The Differential of the Period Mapping}
Griffiths deeply studied the differential of map \eqref{G plpm}, as well: in order to state his result let us review what the \emph{contraction of differential forms with vector fields} and the \emph{(holomorphic) Lie derivative} are. Recall that the tangent sheaf $\mathscr T_X$ is endowed with a natural structure of sheaf of Lie algebras (which can be considered as dgla's concentrated in degree $0$) induced by the canonical isomorphism $\mathscr T_X\simeq\mathcal Der\left(\mathscr O_X,\mathscr O_X\right)$, while $\mathcal End^*\left(\Omega_X^*\right)$ comes with a structure of sheaf of differential graded Lie algebras through the standard differential on $\mathrm{Hom}$ complexes and the standard Lie bracket. Now the contraction morphism is defined to be the {} ``shifted'' map of sheaves of differential graded Lie algebras
\begin{eqnarray} \label{contr map}
i:&\mathscr T_X&\xrightarrow{\hspace*{0.25cm}}\mathcal End^*\left(\Omega_X^*\right)\left[-1\right] \nonumber \\
&\xi&\longmapsto i_{\xi}\text{ such that }i_{\xi}\left(\omega\right):=\xi\lrcorner\,\omega\qquad\qquad\text{(on local sections)}
\end{eqnarray}
while the differential of map \eqref{contr map} (as an element of the complex $\mathrm{Hom}^*\left(\mathscr T_X,\mathcal End^*\left(\Omega_X^*\right)\left[-1\right]\right)$) is by definition the Lie derivative
\begin{eqnarray} \label{Lie deriv}
l:&\mathscr T_X&\xrightarrow{\hspace*{0.25cm}}\mathcal End^*\left(\Omega_X^*\right) \nonumber \\
&\xi&\longmapsto l_{\xi}\text{ such that }l_{\xi}\left(\omega\right):=d\left(\xi\lrcorner\,\omega\right)+\xi\lrcorner\,\left(d\omega\right)\quad\qquad\text{(on local sections)}
\end{eqnarray}
which is a genuine morphism of sheaves of dgla's\footnote{By a slight abuse of notation we will tend to denote by $i$ and $l$ the morphisms that maps \eqref{contr map} and \eqref{Lie deriv} induce on global sections and derived global sections, as well.}.
\begin{thm} \emph{(Griffiths)}
The differential $d\mathcal P^{p,k}$ of map \eqref{G plpm} factors through the (cohomology) contraction map
\begin{equation} \label{dG plpm}
i:H^1\left(X,\mathscr T_X\right)\longrightarrow\mathrm{Hom}\left(F^pH^k\left(X,\mathbb C\right),\frac{H^k\left(X,\mathbb C\right)}{F^pH^k\left(X,\mathbb C\right)}\right).
\end{equation}
Moreover map \eqref{dG plpm} actually takes values in $\mathrm{Hom}\left(F^pH^k\left(X,\mathbb C\right),\frac{F^{p-1}H^k\left(X,\mathbb C\right)}{F^pH^k\left(X,\mathbb C\right)}\right)$.\footnote{This last property is generally known as \emph{Griffiths transversality}.}
\end{thm}
\begin{proof}
The theorem has been stated in relatively modern terms, but a complete proof of it is given in \cite{Vo} Proposition 10.12, Lemma 10.19 and Theorem 10.21.
\end{proof}
The $(p,k)$\textsuperscript{th} local period map \eqref{G plpm} depends by definition on two parameters, a cohomology one -- that is $k$ -- and a filtration one -- that is $p$; we would like to encode all cohomological information about the variations of Hodge structures induced by the family $\phi$ in a single morphism.
\begin{defn}
In the above notations define the \emph{$p$\textsuperscript{th} local period map} to be
\begin{eqnarray} \label{G lpm}
\mathcal P^p:&S&\xrightarrow{\hspace*{0.5cm}}\mathrm{Grass}\left(H^*\left(X,\mathbb C\right)\right) \nonumber \\
&t&\longmapsto \prod_kF^pH^k\left(X_t,\mathbb C\right).
\end{eqnarray}
\end{defn}
Notice that map \eqref{G lpm} is holomorphic and that its differential $d\mathcal P^p$ still factors through a contraction morphism
\begin{eqnarray} \label{per diff}
i:H^1\left(X,\mathscr T_X\right)\longrightarrow\bigoplus_k\mathrm{Hom}\left(F^pH^k\left(X,\mathbb C\right),\frac{H^k\left(X,\mathbb C\right)}{F^pH^k\left(X,\mathbb C\right)}\right).
\end{eqnarray}
\section{The Period Map as a Morphism of Deformation Functors}
The work of Griffiths which has been described in Section $1$ relates deformations of a complex smooth projective variety (or more generally complex K\" ahler manifold) to variations of its Hodge structures. Unfortunately the local period map \eqref{G lpm} is not really a morphism of deformation theories, as it depends on a given deformation of a complex variety $X$, nonetheless its differential \eqref{per diff} is very {} ``deformation-theoretic'' in nature, as it relates the space $H^1\left(X,\mathscr T_X\right)$, i.e. the tangent space to the deformation functor parametrising all deformations of $X$, to another cohomological invariant which depends only on $X$ rather than the special Kuranishi family over $X$ determining map \eqref{G lpm}. Observations like these led Fiorenza and Manetti to believe that Griffiths period map could be described as a morphism of deformation functors (in the sense of Schlessinger) whose induced tangent mapping coincided with map \eqref{per diff}.
\subsection{Deformations of $k$-Schemes}
Let $k$ be any (non-necessarily algebraically closed) field of characteristic $0$ and consider a smooth proper scheme $X$ of dimension $d$: these assumptions over $X$ just algebraically resemble the analytic framework in which Griffiths studied map \eqref{G lpm}, while the fact that the theory we are about to summarise works for any field of characteristic $0$ is a consequence of Deligne's views on Hodge Theory (for more details see \cite{Del1}, \cite{Del2} and \cite{Del3}). Notice also that by \cite{Del1} Theorem 5.5 the Hodge-to-De Rham spectral sequence of the scheme $X$ degenerates at its first page: such a property will be used several times in this paper. \\
Recall that the \emph{functor of deformations of $X$} is the functor of Artin rings
\begin{eqnarray} \label{Def_X}
\mathrm{Def}_X:&\mathfrak{Art}_k&\xrightarrow{\hspace*{2cm}}\mathfrak{Set} \nonumber \\
&A&\mapsto\frac{\left\{\text{deformations of }X\text{ over }A\right\}}{\text{isomorphism}}
\end{eqnarray}
where a deformation\footnote{From now on by deformation we will always mean infinitesimal deformation, i.e. a deformation over an Artinian base.} of $X$ over $A$ is a Cartesian diagram in $\mathfrak{Sch}_k$
\begin{equation*}
\xymatrix{\ar@{} |{\Box}[dr]X\ar@{^{(}->}[r]^i\ar[d] & \mathcal X\ar[d]^p \\
\mathrm{Spec}\left(k\right)\ar[r] & \mathrm{Spec}\left(A\right)}
\end{equation*}
with $i$ a closed immersion and $p$ flat and proper; equivalently a deformation of $X$ over $A$ can be viewed as a morphism of sheaves of $A$-algebras $\mathscr O_A\rightarrow\mathscr O_X$ such that $\mathscr O_A$ is flat over $A$ and $\mathscr O_A\otimes_Ak\simeq \mathscr O_X$. Of course, two $A$-deformations $\mathcal X_1\rightarrow\mathrm{Spec}\left(A\right)$ and $\mathcal X_2\rightarrow\mathrm{Spec}\left(A\right)$ of $X$ are said to be isomorphic if there is an isomorphism $\mathcal X_1\tilde{\longrightarrow}\mathcal X_2$ of schemes over $A$ inducing the identity on $X$: it is well-known that functor \eqref{Def_X} is a deformation functor in the sense of Schlessinger (see \cite{Man2} or \cite{Schl} for a definition). \\
Now let $\left(\mathfrak l,\langle\cdots\rangle_n\right)_{n>0}$ be a $L_{\infty}$-algebra over $k$ (see \cite{Man2} for a definition) and recall that the \emph{deformation functor associated to $\left(\mathfrak l,\langle\cdots\rangle_n\right)_{n>0}$} is defined to be
\begin{eqnarray*}
\mathrm{Def}_{\mathfrak l}:&\mathfrak{Art}_k&\xrightarrow{\hspace*{1.5cm}}\mathfrak{Set} \\
&A&\mapsto\frac{\mathrm{MC}_{\mathfrak l}\left(A\right)}{\text{homotopy equivalence}}
\end{eqnarray*}
where
\begin{equation*}
\mathrm{MC}_{\mathfrak l}\left(A\right):=\left\{x\in\mathfrak l^0\left[1\right]\otimes\mathfrak m_A\text{ s.t. }\sum_{n\geq 1}\frac{\left\langle x^{\odot n}\right\rangle_n}{n!}=0\right\}
\end{equation*}
is the set of solutions of the \emph{Maurer-Cartan equation} and two elements $x_0,x_1\in\mathrm{MC}_{\mathfrak l}\left(A\right)$ are said to be \emph{homotopy equivalent} if there exists a {} ``path'' $x\left(t,dt\right)\in\mathrm{MC}_{\mathfrak l\left[t,dt\right]}\left(A\right)$ such that $x\left(0\right)=x_0$ and $x\left(1\right)=x_1$; again, it is not hard to verify that $\mathrm{Def}_{\mathfrak l}$ is a deformation functor in the sense of Schlessinger. Notice that, if the higher products $\langle\cdots\rangle_n=0$ for all $n\geq3$, i.e. if the $L_{\infty}$-algebra is actually a differential graded Lie algebra (see \cite{dN1} or \cite{Man2} for a definition), we recover the more classical notion of deformation functor associated to a dgla. \\
A fundamental fact in Deformation Theory -- essentially due to Kodaira, Kuranishi and Spencer and developed in many ways by several other authors -- states that the functor of deformations $\mathrm{Def}_X$ associated to a scheme $X$ which satisfies the above conditions is isomorphic to the deformation functor associated to the \emph{Kodaira-Spencer dgla} of $X$, which is defined to be the differential graded Lie algebra $\left(KS_X,\left[-,-\right],D\right)$ where
\begin{eqnarray} \label{KS Lie struct}
&KS_X:=\mathbb R\Gamma\left(X,\mathscr T_X\right)\simeq\Gamma\left(X,\mathscr A^{0,*}_X\left(\mathscr T_X\right)\right)& \nonumber \\
&\left[fd\bar z_I\frac{\partial}{\partial z_i},gd\bar z_J\frac{\partial}{\partial z_j}\right]:=d\bar z_I\wedge d\bar z_J\left(f\frac{\partial g}{\partial z_i}\frac{\partial}{\partial z_j}-g\frac{\partial f}{\partial z_j}\frac{\partial}{\partial z_i}\right).& \nonumber \\
&D\left(\omega\frac{\partial}{\partial z_i}\right):=-\bar{\partial}\left(\omega\right)\frac{\partial}{\partial z_i}&
\end{eqnarray}
\begin{war} \label{KS resol}
In this paper the algebra $KS_X$ will always correspond to the specific resolution $\Gamma\left(X,\mathscr A^{0,*}_X\left(\mathscr T_X\right)\right)$ computing $\mathbb R\Gamma\left(X,\mathscr T_X\right)$, equipped with the Lie structure \eqref{KS Lie struct}. This becomes very relevant in comparisons with the work of Fiorenza, Manetti and Martinengo.
\end{war}
Now consider the natural transformation
\begin{eqnarray} \label{Def_X=Def_KS}
\mathscr O:\mathrm{Def}_{KS_X}&\xrightarrow{\hspace*{1cm}}&\quad\mathrm{Def}_X \nonumber \\
\forall A\in\mathfrak{Art}_k\quad\mathrm{Def}_{KS_X}\left(A\right)\ni\xi\qquad&\longmapsto&\left(\mathscr O_{\xi}\rightarrow\mathscr O_X\right)\in\mathrm{Def}_X\left(A\right).
\end{eqnarray}
where for all open $U\subseteq X$
\begin{equation*}
\mathscr O_{\xi}\left(U\right):=\left\{f\in \mathscr A^{0,0}_X\left(U\right)\otimes A\text{ s.t. }\bar{\partial}f=\xi\lrcorner\partial f\right\}
\end{equation*}
and the map $\mathscr O_{\xi}\rightarrow\mathscr O_X$ is induced by the projection $\mathscr A^{0,0}_X\otimes A\rightarrow\mathscr A^{0,0}_X$.
\begin{thm} \emph{(Kodaira-Spencer, Kuranishi, [...])} \label{Donatella}
In the above notations, map \eqref{Def_X=Def_KS} is an isomorphism of deformation functors.
\end{thm}
\begin{proof}
There is a variety of different proofs of this result in the literature: we refer to \cite{Iac1} Theorem II.7.3 for a very detailed algebraic one; see also \cite{Iac2} Theorem 3.4.
\end{proof}
\subsection{Mapping Cones and Deformations of Filtered Complexes}
The functor $\mathrm{Def}_X$ is the most natural candidate for the domain of a purely {}``deformation-theoretic'' version of Griffiths period map; now we wish to understand what the codomain of such a morphism should be, i.e. we seek a deformation functor which parametrises variations of Hodge structures over $X$. \\
Let $\left(V,d\right)$ be a differential graded $k$-vector space and $\left(W,d\right)$ a subcomplex of its; for any $A\in\mathfrak{Art}_k$, consider the groups\footnote{In this section, by a slight abuse of notation, the symbol $d$ may indifferently denote the differential of the complex $V$, the differential of the twisted complex $V\otimes A$ and the differential of the endomorphism complex $\mathrm{End}\left(\left(V,d\right)\right)$.}
\begin{eqnarray*}
\mathrm{Aut}^V\left(A\right)&:=&\left\{f\in\mathrm{Hom}^0_A\left(V\otimes A,V\otimes A\right)\text{ s.t. } f\equiv\mathrm{Id}_{\left(V,d\right)}\;\left(\mathrm{mod}\;\mathfrak m_A\right)\right\} \\
\mathrm{Aut}^{\left(V,d\right)}\left(A\right)&:=&\left\{f\in\mathrm{Aut}^V\left(A\right)\text{ s.t. }fd=df\right\} \\
\mathrm{Aut}^{W,V}\left(A\right)&:=&\left\{f\in\mathrm{Aut}^V\left(A\right)\text{ s.t. }f\left(W\otimes A\right)=W\otimes A\right\} \\
\tilde{\mathrm{Aut}}^{\left(V,d\right)}\left(A\right)&:=&\left\{f\in\mathrm{Aut}^{V,d}\left(A\right)\text{ s.t. }H^*\left(f\right)\text{ is the identity on }H^*\left(V\otimes A,d\right)\right\}
\end{eqnarray*}
and define the functor of deformations of $\left(W,d\right)$ inside $\left(V,d\right)$ to be the functor of Artin rings
\begin{eqnarray} \label{Grass func}
\mathrm{Grass}_{W,V}:&\mathfrak{Art}_k&\xrightarrow{\hspace*{3cm}}\mathfrak{Set} \nonumber \\
&A&\mapsto\frac{\left\{f\in\mathrm{Aut}^V\left(A\right)\text{ s.t. }df\left(W\otimes A\right)\subseteq f\left(W\otimes A\right)\right\}}{\tilde{\mathrm{Aut}}^{\left(V,d\right)}\left(A\right)\times\mathrm{Aut}^{W,V}\left(A\right)}.
\end{eqnarray}
\begin{rem}
Formula \eqref{Grass func} is the original definition of the functor of deformations of the subcomplex $\left(W,d\right)$ as we find it in \cite{FMan1}; although it is quite elegant, it may not seem very intuitive, as there is no explicit reference to what a deformation of $\left(W,d\right)$ over a local Artinian $k$-algebra $A$ should be. Anyway a more careful look at it immediately shows that a deformation of $\left(W,d\right)$ over $A$ inside $\left(V,d\right)$ is a complex of free $A$-modules $\left(V\otimes A,d_A\right)$ such that its residue modulo $\mathfrak m_A$ equals $\left(V,d\right)$ and $d_A\left(W\otimes A\right)\subseteq W\otimes A$ (this is exactly what the {} ``numerator'' in formula \eqref{Grass func} parametrises); on the other hand two such deformations $\left(V\otimes A,d_A\right)$ and $\left(V\otimes A,d_A'\right)$ are isomorphic if there exists an isomorphism of cochain complexes $\varphi$ between them such that $\varphi\left(W\otimes A,d_A\right)=\varphi\left(W\otimes A,d_A'\right)$ and $H^i\left(\varphi\right)=\mathrm{Id}_{H^i\left(V\otimes A,d\right)}$ for all $i$ (this is exactly what the {} ``denominator'' in formula \eqref{Grass func} parametrises).
\end{rem}
Now consider the graded vector spaces
\begin{eqnarray*}
\mathrm{End}^*\left(\left(V,d\right)\right)&:=&\mathrm{Hom}^*\left(\left(V,d\right),\left(V,d\right)\right) \\
\mathrm{End}^{W}\left(\left(V,d\right)\right)&:=&\left\{f\in\mathrm{End}^*\left(\left(V,d\right)\right)\text{ s.t. }f\left(W\right)\subseteq W\right\}.
\end{eqnarray*}
They are endowed with natural structures of differential graded Lie algebras and there is an obvious inclusion
\begin{equation*}
\chi_{W,V}:\mathrm{End}^W\left(\left(V,d\right)\right)\hookrightarrow\mathrm{End}^*\left(\left(V,d\right)\right)
\end{equation*}
which is a morphism of dgla's; recall also that the \emph{mapping cone} $\left(C_{\chi_{W,V}},\delta\right)$ of the morphism $\chi_{W,V}$ is defined to be its homotopy cokernel, i.e. the complex
\begin{equation*}
\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^W\left(\left(V,d\right)\right)\doublerightarrow{\chi_{W,V}}{0}\mathrm{End}^*\left(\left(V,d\right)\right)\right).
\end{equation*}
More concretely, the mapping cone is given by the formulae
\begin{eqnarray} \label{cone formula}
C_{\chi_{W,V}}^i&:=&{\mathrm{End}^{W}\left(\left(V,d\right)\right)}^i\oplus{\mathrm{End}^{i-1}\left(\left(V,d\right)\right)} \nonumber\\
\delta\left(\left(f,g\right)\right)&:=&\left(df,\chi\left(f\right)-dg\right).
\end{eqnarray}
\begin{prop} \label{mapping cone} \emph{(Fiorenza-Manetti)}
In the above notations, there is a canonical $L_{\infty}$-structure on the mapping cone $C_{\chi_{W,V}}$.
\end{prop}
\begin{proof}
See \cite{FMan2} Section 4 and Section 5.
\end{proof}
\begin{rem}
Fiorenza and Manetti gave two different proofs of Proposition \ref{mapping cone}: the first one is a very elegant but non-constructive proof based on the \emph{Homotopy Transfer Theorem} (see \cite{KS} and \cite{LV}, while \cite{Va} provides a gentler introduction), while the second proof relies on a careful explicit description of all the higher products defining the $L_{\infty}$-structure of $C_{\chi_{W,V}}$; anyway, we are not reporting such formulae since they are not really needed for the sake of this paper.
\end{rem}
Consider the natural transformation
\begin{eqnarray*} \label{Def_C=Grass}
\Psi_{\chi_{W,V}}:&\mathrm{Def}_{C_{\chi_{W,V}}}&\xrightarrow{\hspace*{1cm}}\mathrm{Grass}_{W,V} \\
\forall A\in\mathfrak{Art}_k\qquad\mathrm{Def}_{C_{\chi_{W,V}}}\left(A\right)\ni&\eta&\longmapsto \left(\eta\left(W\otimes A\right),d\right)\in\mathrm{Grass}_{W,V}\left(A\right).
\end{eqnarray*}
\begin{thm} \label{Grass=Def_C} \emph{(Fiorenza-Manetti)}
In the above notations, map \eqref{Def_C=Grass} is an isomorphism of deformation functors; in particular $\mathrm{Grass}_{W,V}$ is a deformation functor in the sense of Schlessinger.
\end{thm}
\begin{proof}
See \cite{FMan1} Proposition 9.2.
\end{proof}
\subsection{Cartan Homotopies and Period Maps}
The work of Fiorenza and Manetti, especially Theorem \ref{Grass=Def_C}, suggests that a good candidate for the codomain of a purely deformation-theoretic version of Griffiths $p$\textsuperscript{th} local period map should be the functor $\mathrm{Grass}_{F^pH^*\left(X,k\right),H^*\left(X,k\right)}$, where
\begin{equation*}
H^*\left(X,k\right):=\mathbb H^*\left(X,\Omega^*_{X/k}\right)
\end{equation*}
is the algebraic De Rham cohomology of the scheme $X$ and $F^{\bullet}$ is the Hodge filtration over it. Now we are almost ready to describe the actual morphism that Fiorenza and Manetti constructed in order to translate Griffiths period map in terms of deformation functors.
\begin{defn} \label{Cart htpy} (Fiorenza-Manetti)
Let $\left(\mathfrak g,d,[-,-]\right)$ and $\left(\mathfrak l,d,[-,-]\right)$ be two differential graded Lie algebras over $k$; a linear map $i\in\mathrm{Hom}^{-1}\left(\mathfrak g,\mathfrak l\right)$ is said to be a \emph{Cartan homotopy} if
\begin{equation*}
\forall a,b\in\mathfrak g\qquad\qquad i\left(\left[a,b\right]\right)=\left[i\left(a\right),di\left(b\right)\right]\quad\text{and}\quad\left[i\left(a\right),i\left(b\right)\right]=0.\footnote{Again, we are denoting by the same symbol the differential and the bracket of the dgla's $\mathfrak g$, $\mathfrak l$ and $\mathrm{Hom}^*\left(\mathfrak g,\mathfrak l\right)$.}
\end{equation*}
\end{defn}
\begin{rem} \label{rem Cart htpy}
The following facts directly follow from Definition \ref{Cart htpy}
\begin{enumerate}
\item The differential of a Cartan homotopy is a morphism of differential graded Lie algebras (i.e. it preserves grading and differentials);
\item The notion of Cartan homotopy is stable under composition with a dgla map and under tensorisation with a differential graded commutative algebra; \item The notion of Cartan homotopy generalises to maps of sheaves of dgla's;
\item Let $i:\mathfrak g\rightarrow\mathfrak l\left[-1\right]$ be a Cartan homotopy and $l:\mathfrak g\rightarrow\mathfrak l$ its differential: $e^{i}$ is an homotopy between $l$ and the zero dgla morphism $0$.
\end{enumerate}
\end{rem}
\begin{example}
The contraction map associated to the scheme $X$ is a Cartan homotopy of sheaves of dgla's (see Section 1.2 for a definition in the context of complex manifolds), while its derived globalisation provides us with an honest Cartan homotopy of dgla's: the latter will be a key ingredient of this paper (see Section 3.2).
\end{example}
The reason why we are interested in Cartan homotopies is that they behave very well with respect to mapping cones.
\begin{prop} \emph{(Fiorenza-Manetti)}
In the notations of Definition \ref{Cart htpy}, let $l:=di$; then the linear map
\begin{eqnarray*}
\check i:&\mathfrak g&\xrightarrow{\hspace*{0.75cm}}C_{l} \\
&a&\longmapsto\left(a,i\left(a\right)\right)
\end{eqnarray*}
is a $L_{\infty}$-morphism; in particular it induces a morphism between the associated deformation functors.
\end{prop}
\begin{proof}
See \cite{FMan1} Proposition 7.4.
\end{proof}
Now, in the notations of Proposition \ref{mapping cone}, for all $p>0$ set
\begin{equation*}
V:=\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\qquad\qquad W:=F^p\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\qquad\qquad\chi^p:=\chi_{V,W}
\end{equation*}
while denote by $i$ the contraction map associated to $X$ and by $l$ its differential, i.e. the Lie derivative.
\begin{war} \label{RGamma resol}
As we did in Warning \ref{KS resol} in the case of the Kodaira-Spencer dgla, we will always fix a specific choice of functor for $\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)$, i.e. the one given by the Dolbeaut resolution: in other words throughout the paper we will have
\begin{equation} \label{RGamma}
\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\simeq\Gamma\left(X,\mathscr A_X^{*,*}\right).
\end{equation}
\end{war}
\begin{thm} \emph{(Fiorenza-Manetti)}
The linear map
\begin{eqnarray*}
\mathrm{fm}^p:&KS_X&\xrightarrow{\hspace*{0.75cm}}C_{\chi^p} \\
&\xi&\longmapsto\left(l_{\xi},i_{\xi}\right)
\end{eqnarray*}
is a $L_{\infty}$-morphism; in particular it induces a morphism of deformation functors
\begin{equation} \label{fm}
\mathrm{fm}^p:\mathrm{Def}_{KS_X}\xrightarrow{\hspace*{0.75cm}}\mathrm{Def}_{C_{\chi^p}}.\footnote{Here, by a slight abuse of notation, the symbol $\mathrm{fm}^p$ is denoting both the $L_{\infty}$-map and the induced morphism of deformation functors.}
\end{equation}
\end{thm}
\begin{proof}
See \cite{FMan1} Theorem 12.1.
\end{proof}
\begin{rem} \label{formal}
Recall that, as a consequence of the $E_1$-degeneration of the Hodge-to De Rham spectral sequence of $X$, the canonical inclusion of complexes $F^p\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\hookrightarrow\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)$ descends to cohomology, i.e. the induced linear map $H^*\left(F^p\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\rightarrow H^*\left(X,k\right)$ is injective. This is equivalent to say that for all $p$ there is a quasi-isomorphism of complexes between $F^p\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)$ and $F^pH^*\left(X,k\right)$\footnote{Notice that the case $p=0$ is trivial.}, which in turn induces a quasi-isomorphism of dgla's between $\mathrm{End}^{F^p\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)$ and $\mathrm{End}^{F^pH^*\left(X,k\right)}\left(H^*\left(X,k\right)\right)$.
\end{rem}
Now denote $\hat{\chi}^p:=\chi_{H^*\left(X,k\right),F^pH^*\left(X,k\right)}$: Remark \ref{formal} entails in particular the existence of a homotopy equivalence of $L_{\infty}$-algebras
\begin{equation*}
h: C_{\chi^p}\longrightarrow C_{\hat{\chi}^p}
\end{equation*}
which induces, by the \emph{Basic Theorem of Deformation Theory} (see \cite{Man2}), an isomorphism
\begin{equation} \label{h}
h: \mathrm{Def}_{C_{\chi^p}}\longrightarrow \mathrm{Def}_{C_{\hat{\chi}^p}}\footnote{Again, the symbol $h$ is denoting both the $L_{\infty}$-map and the induced morphism of deformation functors.}
\end{equation}
between the corresponding deformation functors. In the same fashion, the natural transformation
\begin{equation} \label{H Grass}
H^*:\mathrm{Grass}_{F^p\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right),\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}\xrightarrow{\hspace*{0.75cm}}\mathrm{Grass}_{F^pH^*\left(X,k\right),H^*\left(X,k\right)}.
\end{equation}
induced by the algebraic De Rham cohomology functor is an isomorphism: for a proof see \cite{FMan1} Theorem 10.6.
\begin{defn} \label{alg FM}
For all $p>0$ define the \emph{algebraic $p$\textsuperscript{th} Fiorenza-Manetti local period map} to be the morphism
\begin{equation*}
\mathrm{FM}^p:\mathrm{Def}_{KS_X}\xrightarrow{\hspace*{0.75cm}}\mathrm{Def}_{C_{\hat{\chi}^p}}
\end{equation*}
given by the composition of maps \eqref{h} and \eqref{fm}.
\end{defn}
\begin{defn} \label{geom FM}
For all $p>0$ define the \emph{geometric $p$\textsuperscript{th} Fiorenza-Manetti local period map} to be the morphism
\begin{eqnarray*} \label{FM lpm}
\mathcal P^p:\mathrm{Def}_X\qquad&\xrightarrow{\hspace*{0.5cm}}&\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)} \\
\forall A\in\mathfrak{Art}_k\qquad\mathrm{Def}_X\left(A\right)\ni\left(\mathscr O_A\overset{\xi}{\rightarrow}\mathscr O_X\right)&\longmapsto& \quad F^pH^*\left(X,\mathscr O_A\right)\in\mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)}\left(A\right).
\end{eqnarray*}
\end{defn}
Now we are finally ready to lift Griffiths period map to a morphism of deformation functors.
\begin{thm} \emph{(Fiorenza-Manetti)} \label{Fiorenza-Manetti}
There is a natural isomorphism between maps $\mathrm{FM^p}$ and $\mathcal P^p$, meaning that the diagram
\begin{equation*}
\xymatrix{\mathrm{Def}_{KS_X}\ar[rr]^{\mathrm{FM}^p}\ar[d]_{\mathscr O}^{\wr} & & \mathrm{Def}_{C_{\hat{\chi}^p}}\ar[d]^{\Psi_{\hat{\chi}^p}}_{\wr} \\
\mathrm{Def}_X\ar[rr]^{\mathcal P^p} & & \mathrm{Grass}_{F^pH^*\left(X\right),H^*\left(X\right)}}
\end{equation*}
commutes. Moreover the tangent morphism to the functor $\mathcal P^p$ is the same as map \eqref{per diff}.
\end{thm}
\begin{proof}
See \cite{FMan1} Theorem 12.3 and Corollary 12.5.
\end{proof}
\subsection{Flag Functors and the Fiorenza-Manetti Period Map}
Both $\mathrm{FM^p}$ and $\mathcal P^p$ depend on a filtration parameter: we would like to get rid of it, in order to define universal versions of the algebraic and geometric Fiorenza-Manetti period map. \\
Observe that the target functor of any universal version of the geometric Fiorenza-Manetti period map should not be simply the product of the deformation functors $\mathrm{Grass}_{F^pH^*\left(X,k\right),H^*\left(X,k\right)}$, because the only deformations of the sequence $\left(F^pH^*\left(X,k\right)\right)_p$ of subcomplexes of $H^*\left(X,k\right)$ which may belong to its image are those preserving the property that $F^{\bullet}$ is a filtration. \\
For this reason, let $\left(V,\mathcal F^{\bullet}\right)$ be a filtered complex and define the \emph{flag functor associated to $\left(V,\mathcal F^{\bullet}\right)$} to be
\begin{eqnarray*}
\mathrm{Flag}_V^{\mathcal F^{\bullet}}:&\mathfrak{Art}_k&\xrightarrow{\hspace*{4.5cm}}\mathfrak{Set} \\
&A&\mapsto\left\{\left(\left(U,\mathcal G^p\right)\right)_p\text{ s.t. }\left( U,\mathcal G^p\right)\in\mathrm{Grass}_{\mathcal F^pV,V}\left(A\right),\mathcal G^p U\hookrightarrow\mathcal G^{p-1} U\right\}.
\end{eqnarray*}
Consider the complex
\begin{equation} \label{filt-pres dgla definition}
\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right):=\underset{p}{\bigcap}\mathrm{End}^{\mathcal F^pV}\left(V\right)
\end{equation}
which may be seen as the subcomplex of $\mathrm{End}\left(V\right)$ made of filtration-preserving endomorphisms.
\begin{example}
Notice that if our base filtered complex $\left(V,\mathcal F^{\bullet}\right)$ is the algebraic De Rham complex (or cohomology) of a scheme $X$ equipped with the Hodge filtration, than the complex of filtration-preserving endomorphisms is nothing but the complex of non-negatively graded endomorphisms. In other words, in the notations of formula \eqref{filt-pres dgla definition} we have that
\begin{equation*}
\mathrm{End}^{F^{\bullet}}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)=\mathrm{End}^{\geq0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right).
\end{equation*}
\end{example}
Again $\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right)$ is endowed with a natural structure of differential graded Lie algebra: this comes with a natural inclusion of dgla's
\begin{equation*}
\chi:\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right)\hookrightarrow\mathrm{End}\left(V\right).
\end{equation*}
and let
\begin{equation*}
C^{\mathcal F^{\bullet}}_V:=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right)\doublerightarrow{\chi}{0}\mathrm{End}^*\left(V\right)\right).
\end{equation*}
be its homotopy cokernel.
\begin{prop} \label{gen und Flag dgla} \emph{(Fiorenza-Martinengo)}
In the above notations there is an isomorphism of functors
\begin{equation*}
\mathrm{Flag}_V^{\mathcal F^{\bullet}}\simeq\mathrm{Def}_{C^{\mathcal F^{\bullet}}_V}
\end{equation*}
In particular $\mathrm{Flag}_V^{\mathcal F^{\bullet}}$ is a deformation functor.
\end{prop}
\begin{proof}
See \cite{FMar} Section 5 and Section 6.
\end{proof}
Now consider the functors $\mathrm{Flag}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}^{F^{\bullet}}$ and $\mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}}$: the same arguments used to deal with map \eqref{H Grass} imply that the morphism
\begin{equation} \label{flag htpy stability}
H^*:\mathrm{Flag}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}^{F^{\bullet}}\xrightarrow{\hspace*{0.75cm}}\mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}}
\end{equation}
is well-defined and an isomorphism. \\
\begin{rem} \label{Martinengo}
In the language of \cite{FMar} a pair of differential graded Lie algebras $\left(\mathfrak g,\mathfrak l\right)$ is said to be a \emph{formal pair} if there is an inclusion of dgla's $\mathfrak g\hookrightarrow\mathfrak l$ inducing an injective morphism $H^*\left(\mathfrak g\right)\hookrightarrow H^*\left(\mathfrak l\right)$ on cohomology: in particular by Remark \ref{formal} we have that for any smooth proper $k$-scheme $X$ and for all $p\geq 0$ the pair $\left(\mathrm{End}^{F^p}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right),\mathrm{End}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\right)$ is formal. Moreover The formality argument of Remark \eqref{formal} is uniform in $p$, therefore there is a filtered quasi-isomorphism between $\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right),F^{\bullet}\right)$ and $\left(H^*\left(X,k\right),F^{\bullet}\right)$, providing in turn a week equivalence between the dgla's $\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)$ and $H^*\left(\mathcal End^{\geq 0}\left(\Omega^*_{X/k}\right)\right)$. It follows that the pair $\left(\mathrm{End}^{\geq0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right),\mathrm{End}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\right)$ is formal as well.
\end{rem}
It is easy to see that if $\left(\mathfrak g,\mathfrak l\right)$ is a formal pair of dgla's then the homotopy fibre $\underset{\longleftarrow}{\mathrm{holim}}\left(\mathfrak g\doublerightarrow{\mathrm{incl.}}{0}\mathfrak l\right)$ is quasi-abelian and in fact a model for it is given by the complex $\nicefrac{\mathfrak l}{\mathfrak g}\left[-1\right]$ endowed with the trivial bracket: see \cite{FMar} Section 5 for a more detailed explanation. In particular, if we apply this to the De Rham-theoretic case we have that
\begin{eqnarray} \label{tgt dgrass}
C_{\mathbb R\Gamma\left(X,F^p\Omega^*_{X/k}\right),\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}&\simeq&\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{F^p}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\doublerightarrow{\quad}{\quad}\mathrm{End}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\right) \nonumber \\
&\simeq&\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{F^p}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right] \nonumber \\
&\simeq&\frac{H^*\left(\mathcal End^*\left(\Omega^*_{X/k}\right)\right)}{H^*\left(\mathcal End^{F^p}\left(\Omega^*_{X/k}\right)\right)}\left[-1\right]
\end{eqnarray}
and
\begin{eqnarray} \label{tgt hoflag}
C^{F^{\bullet}}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}&\simeq&\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{\geq0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\doublerightarrow{\quad}{\quad}\mathrm{End}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\right) \nonumber \\
&\simeq&\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right] \nonumber \\
&\simeq&\frac{H^*\left(\mathcal End^*\left(\Omega^*_{X/k}\right)\right)}{H^*\left(\mathcal End^{\geq0}\left(\Omega^*_{X/k}\right)\right)}\left[-1\right]
\end{eqnarray}
where $\mathcal End^*\left(\Omega^*_{X/k}\right)$, $\mathcal End^{\geq0}\left(\Omega^*_{X/k}\right)$ and $\mathcal End^{F^p}\left(\Omega^*_{X/k}\right)$ denote respectively the endomorphism sheaf of $\Omega^*_{X/k}$, the sheaf of its non-negatively graded endomorphisms and the sheaf of those endomorphisms preserving the $p^{\mathrm{th}}$ piece of the Hodge filtration. \\
Summing up the preceding considerations, we obtain a very explicit description of the flag functor associated to the algebraic De Rham complex of $X$.
\begin{cor} \label{und Flag dgla}
There is a chain of isomorphism of deformation functors
\begin{equation} \label{chain}
\mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}}\simeq\mathrm{Flag}_{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}^{F^{\bullet}}\simeq\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}.
\end{equation}
\end{cor}
\begin{proof}
The first isomorphism in chain \eqref{chain} is map \eqref{flag htpy stability}, the other one follows combining Proposition \ref{gen und Flag dgla} and formula \eqref{tgt hoflag}.
\end{proof}
Now we are ready to define some universal version of the Fiorenza-Manetti morphism.
\begin{defn} \label{univ geom Fio-Man}
Define the \emph{universal geometric Fiorenza-Manetti period map} to be the natural transformation
\begin{eqnarray*}
\mathcal P:&\mathrm{Def}_X&\xrightarrow{\hspace*{0.75cm}}\mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}} \\
\forall A\in\mathfrak{Art}_k\quad\quad&\left(\mathscr O_A\overset{\xi}{\rightarrow}\mathscr O_X\right)&\longmapsto\left(\mathcal P^p\left(\left(\mathscr O_A\overset{\xi}{\rightarrow}\mathscr O_X\right)\right)\right)_p.
\end{eqnarray*}
\end{defn}
Notice that Definition \ref{geom FM} ensures that $\mathcal P$ is a well-defined morphism of functors. \\
Map $\mathcal P$ is a good universal version of the geometric Fiorenza-Manetti period map; we would like to complete the picture with a natural universal version of the algebraic Fiorenza-Manetti map, that is we would like to construct a morphism of differential graded Lie algebras
\begin{equation*}
\mathrm{FM}:KS_X\longrightarrow\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]
\end{equation*}
such that the diagram
\begin{equation*}
\xymatrix{\mathrm{Def}_{KS_X}\ar[rrr]^{\mathrm{FM}}\ar[d]^{\wr} & & & \mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}\ar[d]_{\wr} \\
\mathrm{Def}_X\ar[rrr]^{\mathcal P} & & & \mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}}}
\end{equation*}
commutes: we will construct it in Section 3.2.
\section{The Period Map as a Morphism of $\infty$-Groupoids}
Theorem \ref{Fiorenza-Manetti} attests two very interesting facts: the first one is that Definition \ref{alg FM} and Definition \ref{geom FM} are naturally equivalent (and this enables us to simply talk about the \emph{Fiorenza-Manetti local period map}, dropping any further adjective) and the second one is that map \eqref{FM lpm} really extends the period mapping \eqref{G lpm} to a morphism of deformation theories, as the tangent maps are the same. In this perspective, the period map is seen to play a remarkable unifying role in Deformation Theory and Hodge Theory: as a matter of fact a number of highly non-trivial classical results such as \emph{Kodaira Principle} and \emph{Bogomolov-Tian-Todorov Theorem} are recovered as corollaries of Theorem \ref{Fiorenza-Manetti} (see \cite{FMan1}, \cite{FMan3}, \cite{FMar} and \cite{IacMan} for more details). \\
Anyway the contemporary viewpoint on Deformation Theory claims that Schlessinger's deformation functors are not the most suitable tools in order to study general local moduli problems, as they are often unable to capture most of the hidden geometry of such problems. As a matter of fact Schlessinger's functors do not generally take into account automorphisms and higher autoequivalences of the objects they classify and in most cases they do not give a proper description of obstructions, either. Moreover the correspondence between differential graded Lie algebras and deformation functors in the context of classical Deformation Theory is not fully satisfying\footnote{Notice that an instance of such a drawback has already appeared in Section 2.2, since the mapping cone \eqref{cone formula} is endowed with a non-trivial $L_{\infty}$-structure.}, but the most important drawback of Schlessinger's functors for the sake of this paper is that in general they are not formal neighbourhoods of any global moduli space; this is precisely the case of the functor $\mathrm{Def}_X$ defined in Section 2.1: there does not exist any (classical) moduli space of proper smooth schemes of dimension $d>1$, thus for a general choice of the scheme $X$ the functor $\mathrm{Def}_X$ cannot be describing infinitesimally any algebraic space.
\subsection{Quick Review of Derived Deformation Theory}
The critical aspects we have briefly listed above mark some of the reasons that have been leading to the development of Derived Deformation Theory: the rough idea behind this subject is that Deformation Theory is not really a {} ``categorical'' subject, but rather an {} ``$\left(\infty,1\right)$-categorical'' one, meaning that its constructions and invariants should be homotopical (or derived) in nature. In particular the basic objects of Derived Deformation Theory should be homotopy analogues of Schlessinger's functors -- i.e. functors defined over (some subcategory of) $\mathfrak{dgArt}_k$ rather than $\mathfrak{Art}_k$ -- satisfying homotopical versions of Schlessinger's axioms and preserving the homotopical structure of the category of Artinian dg-algebras. Foundational work on Derived Deformation Theory includes \cite{Getz}, \cite{Hin2}, \cite{Kon}, \cite{Lu1}, \cite{Lu3}, \cite{Man1}, \cite{Pr1} and \cite{TV}, while a gentle introduction to the subject can be found in \cite{dN1}: here we quickly review some of the main concepts just to fix notations. \\
There are several different ways to enhance a classical deformation functor to a derived one, giving rise to various consistent derived deformation theories; in \cite{Pr1} Pridham proved that all these variants are homotopy equivalent\footnote{All approaches to Derived Deformation Theory are described by a well-defined $\left(\infty,1\right)$-category: Pridham proved that all such $\left(\infty,1\right)$-categories are equivalent; for more details see \cite{Pr1}.}, thus in this paper by derived deformation functor we will always mean a \emph{Hinich derived deformation functor}\footnote{In the literature people also refer to such functors as \emph{formal moduli problems} or \emph{formal stacks}.}. The latter is a functor
\begin{equation*}
\mathbf F:\mathfrak{dgArt}_k^{\leq 0}\longrightarrow\mathfrak{sSet}
\end{equation*}
satisfying weaker versions of Schlessiger's axioms for classical deformation problems: for a precise definition see \cite{dN1} or the original paper \cite{Hin2}, but essentially $\mathbf F$ is required to be \emph{homotopic} -- i.e. to map quasi-isomorphisms in $\mathfrak{dgArt}_k^{\leq 0}$ to weak equivalences in $\mathfrak{sSet}$ -- and \emph{homotopy-homogeneous} -- i.e. such that for all surjections $A\twoheadrightarrow B$ and all maps $C\rightarrow B$ in $\mathfrak{dgArt}_k^{\leq 0}$ the natural map
\begin{equation*}
\mathbf F\left(A\times_B C\right)\longrightarrow\mathbf F\left(A\right)\times^{h}_{\mathbf F\left(B\right)}\mathbf F\left(C\right)\footnote{The symbol $-\times_-^h-$ denotes the homotopy fibre product in $\mathfrak{sSet}$.}
\end{equation*}
is a weak equivalence. In case $\mathbf F$ is only \emph{homotopy-surjecting} -- i.e. for all tiny acyclic extension $A\rightarrow B$ in $\mathfrak{dgArt}_k^{\leq 0}$ the induced map $\pi_0\left(\mathbf F\left(A\right)\right)\rightarrow\pi_0\left(\mathbf F\left(B\right)\right)$ is surjective -- we will say that it is a \emph{derived pre-deformation functor}. \\
All the geometry of Hinich functors is captured by certain cohomological invariants which generalise tangent spaces and obstruction theories for classical deformation functors: let us briefly recall how to construct them. Given a derived deformation functor $\mathbf F:\mathfrak{dgArt}_k^{\leq 0}\longrightarrow\mathfrak{sSet}$, consider as in \cite{Pr1} Section 1.6 the functor
\begin{eqnarray*}
\mathrm{tan}\,\mathbf F:&\mathfrak{dgVect}_k^{\leq 0}&\xrightarrow{\hspace*{0.50cm}}\mathfrak{sVect}_k \\
&V&\longmapsto\mathbf F\left(k\oplus V\right)
\end{eqnarray*}
and recall that the \emph{$j$-th generalised tangent space} of $\mathbf F$ is said to be the group
\begin{equation*}
H^j\left(\mathbf F\right):=\pi_i\left(\mathrm{tan}\,\mathbf F\left(k\left[-n\right]\right)\right)\qquad\qquad\text{where }n-i=j
\end{equation*}
and the definition is well-given because of \cite{Pr1} Corollary 1.46. Generalised tangent spaces extend the underived notions of tangent and obstruction spaces in the sense that if $\mathbf F$ is a derived deformation functor, the group $H^{j}\left(\mathbf F\right)$ parametrises infinitesimal $j$-automorphisms associated to it; in particular $H^0\left(\mathbf F\right)$ encodes first-order derived deformations and $H^1\left(\mathbf F\right)$ encodes second-order derived deformations, i.e. all obstructions (see \cite{Pr1} Section 1.6).\\
One of the properties of derived deformation functors which are most interesting to us is that they provide the right notion of formal stack, i.e. that they describe derived geometric stacks infinitesimally. Foundational work on higher stacks and Derived Algebraic Geometry includes \cite{Lu1}, \cite{Lu2}, \cite{Pr2}, \cite{Toe} and \cite{TV}: here we only recall that given a (possibly non-geometric) derived stack over $k$
\begin{equation*}
\mathcal F:\mathfrak{dgAlg}_k^{\leq 0}\rightarrow\mathfrak{sSet}
\end{equation*}
a point $x$ over it, the \emph{formal neighbourhood} of $F$ at $x$ is defined as
\begin{eqnarray*}
\hat{\mathbf F}_x:&\mathfrak{dgArt}^{\leq 0}_k&\xrightarrow{\hspace*{1cm}}\mathfrak{sSet}\\
&A&\longmapsto \mathcal F\left(A\right)\times_{\mathcal F\left(k\right)}^h\left\{x\right\}.
\end{eqnarray*}
A well-known folklore result in Derived Algebraic Geometry is that $\hat{\mathbf F}_x$ is a derived deformation functor: a proof of it is hidden somewhere in \cite{Lu1} and \cite{TV}; see also \cite{Toe} and \cite{Pr1}.\\
Now denote by $\Omega^*_{\mathrm{DR}}\left(\Delta^*\right)$ the simplicial differential graded commutative algebra of \emph{polynomial differential forms}, given in simplicial level $n$ by
\begin{equation*}
\Omega^*_{\mathrm{DR}}\left(\Delta^n\right):=\frac{k\left[x_0,x_1,\ldots,x_n,dx_0,dx_1,\ldots,dx_n\right]}{\sum x_i=1,\sum dx_i=0}
\end{equation*}
where $x_0,x_1,\ldots,x_n$ live in cochain degree $0$ and $dx_0,dx_1,\ldots,dx_n$ in cochain degree 1; more generally, given a simplicial set $S$, the symbol $\Omega^*_{\mathrm{DR}}\left(S\right)$ will stand for the simplicial differential graded commutative algebra of \emph{polynomial differential forms on} $S$, which is defined in dg level $p$ by
\begin{equation*}
\Omega^p_{\mathrm{DR}}\left(S\right):=\mathrm{Hom}_{\mathfrak{sSet}}\left(S,\Omega^p_{\mathrm{DR}}\left(\Delta^*\right)\right).
\end{equation*}
Also recall that the \emph{Hinich nerve} of a dgla $\mathfrak g$ is defined to be the derived deformation functor
\begin{align*}
\mathbb R\mathrm{Def}_{\mathfrak g}:\mathfrak{dg}&\mathfrak{Art}^{\leq 0}_k\xrightarrow{\hspace*{1.25cm}}\mathfrak{sSet} \\
&\phantomarrow{\mathfrak{dgArt}^{\leq 0}_k}{A} \mathbb R\mathrm{MC}_{\mathfrak g\otimes\Omega^*_{\mathrm{DR}}\left(\Delta^*\right)}\left(A\right)
\end{align*}
where $\mathbb R\mathrm{MC}_{\mathfrak g\otimes\Omega^*_{\mathrm{DR}}\left(\Delta^*\right)}\left(A\right)$ is the simplicial set determined in level $n$ by the set
\begin{equation*}
\mathbb R\mathrm{MC}_{\mathfrak g\otimes\Omega^*_{\mathrm{DR}}\left(\Delta^n\right)}\left(A\right):=\left\{x\in \left(\mathfrak g\otimes\Omega^*_{\mathrm{DR}}\left(\Delta^n\right)\otimes\mathfrak m_A\right)^1\text{ s.t. } d\left(x\right)+\frac{1}{2}\left[x,x\right]=0\right\}.
\end{equation*}
\begin{thm} \label{compare def theories} \emph{(Hinich, Lurie, Pridham)}
The functor
\begin{eqnarray*}
\mathbb R\mathrm{Def}:&\mathfrak{dgLie}_k&\xrightarrow{\hspace*{0.5cm}}\mathfrak{Def}^{\mathrm{Hin}}_k \\
&\mathfrak g&\longmapsto\mathbb R\mathrm{Def}_{\mathfrak g}
\end{eqnarray*}
is an equivalence of $\left(\infty,1\right)$-categories, thus it induces an equivalence on the homotopy categories
\begin{equation*}
\mathrm{Ho}\left(\mathfrak{dgLie}_k\right)\simeq\mathrm{Ho}\left(\mathfrak{Def}^{\mathrm{Hin}}_k\right).
\end{equation*}
\end{thm}
\begin{proof}
See \cite{Pr1} Corollary 4.56.
\end{proof}
Despite its great theoretical properties, the Hinich nerve is seldom handy enough to make concrete computations. For this reason, recall that the \emph{(derived) Deligne groupoid} associated to a differential graded Lie algebra $\mathfrak g$ is defined to be the formal groupoid
\begin{eqnarray*}
\mathrm{Del}_{\mathfrak g}:&\mathfrak{dgArt}^{\leq 0}&\xrightarrow{\hspace*{1cm}}\mathfrak{Grpd} \\
&A&\longmapsto\left[\nicefrac{\widetilde{\mathrm{MC}}_{\mathfrak g}\left(A\right)}{\widetilde{\mathrm{Gg}}_{\mathfrak g}\left(A\right)}\right]
\end{eqnarray*}
where
\begin{eqnarray} \label{MC}
\widetilde{\mathrm{MC}}_{\mathfrak g}:&\mathfrak{dgArt}_k^{\leq 0}&\xrightarrow{\hspace*{3cm}}\mathfrak{Set} \nonumber \\
&A&\mapsto\left\{x\in \left(\mathfrak g\otimes\mathfrak m_A\right)^1\quad\text{s.t.}\quad d\left(x\right)+\frac{1}{2}\left[x,x\right]=0\right\}
\end{eqnarray}
\begin{eqnarray} \label{Gg}
\widetilde{\mathrm{Gg}}_{\mathfrak g}:&\mathfrak{dgArt}_k^{\leq 0}&\xrightarrow{\hspace*{1cm}}\mathfrak{Grp} \nonumber \\
&A&\mapsto\mathrm{exp}\left(\left(\mathfrak g\otimes \mathfrak m_A\right)^0\right)
\end{eqnarray}
and let
\begin{eqnarray*}
\mathrm{BDel}_{\mathfrak g}:\mathfrak{dgArt}_k^{\leq 0}\longrightarrow\mathfrak{sSet}
\end{eqnarray*}
denote its nerve.
\begin{rem}
Notice that formula \eqref{MC} and formula \eqref{Gg} are just straightforward generalisations of the notions of Maurer-Cartan and gauge functor in underived Deformation Theory; these objects are used to define \emph{extended deformation functors} in the sense of Manetti (see \cite{Man1} or \cite{Man2}). In \cite{Pr1} Pridham also proved that there is an equivalence of $\left(\infty,1\right)$-categories between $\mathfrak{Def}_k^{\mathrm{Man}}$ and $\mathfrak{Def}_k^{\mathrm{Hin}}$.
\end{rem}
\begin{war}
The nerve of the Deligne groupoid associated to a differential graded Lie algebra is a derived pre-deformation functor but not a derived deformation functor: as a matter of fact it is not homotopic in general. Moreover, although it might be a bit confusing, we will tend to refer to both $\mathrm{Del}_{\mathfrak g}$ and $\mathrm{BDel}_{\mathfrak g}$ as the Deligne groupoid associated to the differential graded Lie algebra $\mathfrak g$.
\end{war}
Fix $\mathfrak g\in\mathfrak{dgLie}_k$: we can define the functor
\begin{eqnarray*}
&\underline{\mathrm{BDel}}_{\mathfrak g}:\mathfrak{dgArt}_k^{\leq 0}\xrightarrow{\hspace*{5cm}}\mathfrak{sSet}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad& \\
&\qquad\qquad\; A\longmapsto\mathrm{diag}\left(\xymatrix{\mathrm{BDel}_{\mathfrak g}\left(A\right)\ar[r] & \mathrm{BDel}_{\mathfrak g\otimes\Omega^*\left(\Delta^1\right)}\left(A\right)\ar@<2.5pt>[l]\ar@<-2.5pt>[l]\ar@<2.5pt>[r]\ar@<-2.5pt>[r] & \mathrm{BDel}_{\mathfrak g\otimes\Omega^*\left(\Delta^2\right)}\left(A\right)\ar[l]\ar@<5pt>[l]\ar@<-5pt>[l]\ar[r]\ar@<5pt>[r]\ar@<-5pt>[r] & \cdots\ar@<-7.5pt>[l]\ar@<-2.5pt>[l]\ar@<2.5pt>[l]\ar@<7.5pt>[l]}\right)&
\end{eqnarray*}
which is sometimes called the \emph{simplicial Deligne groupoid} of $\mathfrak g$.
\begin{thm} \label{Del grpd} \emph{(Pridham)}
Let $\mathfrak g$ be a differential graded Lie algebra concentrated in non-negative degrees; we have that
\begin{itemize}
\item the functor $\underline{\mathrm{BDel}}_{\mathfrak g}$ is a derived deformation functor;
\item the functor $\underline{\mathrm{BDel}}_{\mathfrak g}$ is the universal derived deformation functor under $\mathrm{BDel}_{\mathfrak g}$;
\item the functors $\underline{\mathrm{BDel}}_{\mathfrak g}$ and $\mathbb R\mathrm{Def}_{\mathfrak g}$ are weakly equivalent.
\end{itemize}
\end{thm}
\begin{proof}
See \cite{Pr4} Section 3 for the proof of the first two claims, while the last statement is proved in \cite{Hin4} Section 3.
\end{proof}
As a consequence of Theorem \ref{Del grpd} we have that all geometric and homotopy-theoretic information concerning the Hinich nerve of a differential graded Lie algebra $\mathfrak g$ are completely determined by its associated Deligne groupoid, which is a much more down-to-earth object as it is essentially a formal groupoid. Unfortunately, as $\mathrm{BDel}_{\mathfrak g}$ does not map quasi-isomorphisms to weak equivalences, the description of higher tangent spaces we gave above in this section is no longer valid; nonetheless Pridham found a coherent way to define good cohomology theories for derived pre-deformation functors. As a matter of fact fix a derived pre-deformation functor $\mathbf F:\mathfrak{dgArt}_k^{\leq 0}\rightarrow\mathfrak{sSet}$ and define as in \cite{Pr2bis} Section 3.3 the $j$-th generalised tangent space of $\mathbf F$ to be
\begin{equation*}
H^j\left(\mathbf F\right):=
\begin{cases}
\pi_{-j}\left(\mathbf F\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)\right) & \text{ if }j\leq 0 \\
\nicefrac{\pi_0\left(\mathrm{tan}\left(\mathbf F\left(k\left[i\right]\right)\right)\right)}{\pi_0\left(\mathrm{tan}\left(\mathbf F\left(\mathrm{cone}\left(k\left[i\right]\right)\right)\right)\right)} & \text{ otherwise }
\end{cases}
\end{equation*}
which is seen to be consistent with the definition given above in this section in case $\mathbf F$ is also homotopic (see \cite{Pr2bis} Lemma 3.15). \\ Now fix $\mathfrak g$ to be a differential graded Lie algebra over $k$ concentrated in non-negative degrees and apply the above definitions to its Deligne groupoid. We have that
\begin{eqnarray*}
&H^{-1}\left(\mathrm{BDel}_{\mathfrak g}\right)=\pi_1\left(\mathrm{BDel}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)\right)=\pi_1\left(\left[\nicefrac{\widetilde{\mathrm{MC}}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}{\widetilde{\mathrm{Gg}}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\right]\right)=& \\
&\pi_1\left(\left[\nicefrac{\mathrm{MC}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}{\mathrm{Gg}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\right]\right)\simeq\mathrm{Stab}_{\mathrm{Gg}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\left(0\right)&
\end{eqnarray*}
but
\begin{eqnarray*}
&\mathrm{MC}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)=\left\{x\otimes\varepsilon\in\mathfrak g^1\otimes\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\text{ s.t. }d\left(x\right)=0\right\}=Z^1\left(\mathfrak g\right)\varepsilon& \\
&\mathrm{Gg}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)=\mathrm{exp}\left(\mathfrak g^0\otimes\left(\varepsilon\right)\right)\simeq\mathrm{Id}+\mathfrak g^0\varepsilon&
\end{eqnarray*}
and notice that the gauge action just reduces to
\begin{eqnarray} \label{gauge level 0}
\mathrm{Id}+\mathfrak g^0\varepsilon\times Z^1\left(\mathfrak g\right)\varepsilon&\overset{*}{\xrightarrow{\hspace*{1cm}}}& \quad Z^1\left(\mathfrak g\right)\varepsilon \nonumber \\
\left(\mathrm{Id}+a\varepsilon,x\varepsilon\right)\quad&\longmapsto& \left(x+d\left(a\right)\right)\varepsilon
\end{eqnarray}
therefore we get
\begin{eqnarray*}
&\mathrm{Stab}_{\mathrm{Gg}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\left(0\right)=\left\{\left(\mathrm{Id}+a\varepsilon\right)\in\mathrm{Id}+\mathfrak g^0\varepsilon\,\text{ s.t. }\left(\mathrm{Id}+a\varepsilon\right)*0=0\right\}\simeq& \\
&\left\{a\in\mathfrak g^0\text{ s.t. }d\left(a\right)=0\right\}=Z^0\left(\mathfrak g\right)\simeq H^0\left(\mathfrak g\right)&
\end{eqnarray*}
where the last identification follows from the fact that $\mathfrak g$ lives in non-negative degrees. \\
Similarly we see that
\begin{eqnarray*}
&H^0\left(\mathrm{BDel}_{\mathfrak g}\right)=\pi_0\left(\mathrm{BDel}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)\right)=&\\
&\pi_0\left(\left[\nicefrac{\widetilde{\mathrm{MC}}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}{\widetilde{\mathrm{Gg}}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\right]\right)= \nicefrac{\mathrm{MC}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}{\mathrm{Gg}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\simeq\nicefrac{Z^1\left(\mathfrak g\right)\varepsilon}{\mathrm{Id}+\mathfrak g^0\varepsilon}&
\end{eqnarray*}
thus the quotient of $\mathrm{MC}_{\mathfrak g}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)$ under the gauge action \eqref{gauge level 0} is given by $H^1\left(\mathfrak g\right)$. \\
At last observe that $\mathrm{cone}\left(k\oplus k\left[j-1\right]\right)$ is a path object for $k\oplus k\left[j\right]$, so the same kind of computation gives us that for all $j\geq 0$
\begin{eqnarray*}
&H^j\left(\mathrm{BDel}_{\mathfrak g}\right)=\nicefrac{\pi_0\left(\mathrm{tan}\left(\mathrm{BDel}_{\mathfrak g}\left(k\left[i\right]\right)\right)\right)}{\pi_0\left(\mathrm{tan}\left(\mathrm{BDel}_{\mathfrak g}\left(\mathrm{cone}\left(k\left[i\right]\right)\right)\right)\right)}=&\\
&\nicefrac{Z^0\left(\mathfrak g\otimes\left(k\oplus k\left[j\right]\right)\right)}{Z^0\left(\mathfrak g\otimes\left(k\oplus\mathrm{cone}\left(k\oplus k\left[j-1\right]\right)\right)\right)}=\nicefrac{Z^{j+1}\left(\mathfrak g\right)}{\mathfrak g^j}&
\end{eqnarray*}
and again $\mathfrak g^j$ acts on $Z^{j+1}\left(\mathfrak g\right)$ by differentials, so the quotient is $H^{j+1}\left(\mathfrak g\right)$.
\begin{rem} \label{Del grpd coho}
Let $\mathfrak g$ be any differential graded Lie algebra; by combining Theorem \ref{Del grpd} and the above observations we have that
\begin{eqnarray*}
&H^i\left(\mathbb R\mathrm{Def}_{\mathfrak g}\right)\simeq H^i\left(\mathrm{BDel}_{\mathfrak g}\right)=\overset{\overset{\quad d}{\quad\curvearrowleft}}{\nicefrac{Z^{i+1}}{g^i}}\simeq H^{i+1}\left(\mathfrak g\right)&\qquad\forall i\geq 0. \\
&H^{-1}\left(\mathbb R\mathrm{Def}_{\mathfrak g}\right)\simeq H^{-1}\left(\mathrm{BDel}_{\mathfrak g}\right)\simeq\mathrm{Stab}_{\mathrm{Gg}_{\mathfrak g^0}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\left(0\right)\simeq H^0\left(\mathfrak g\right)&
\end{eqnarray*}
\end{rem}
\subsection{The Algebraic Fiorenza-Manetti-Martinengo Period Map}
The above considerations give many motivations to try to lift the period map from a morphism of classical deformation functors to the context of Derived Deformation Theory; Fiorenza and Martinengo started to address such a question, tackling it from an entirely algebraic viewpoint. \\
Let $X$ still be a proper smooth scheme of dimension $d$ over a field $k$ of characteristic $0$ and, again, take the Cartan homotopy defined by the contraction of differential forms with vector fields
\begin{equation*}
i:\mathscr T_X\longrightarrow\mathcal End^*\left(\Omega^*_{X/k}\right)\left[-1\right]
\end{equation*}
and the Lie derivative
\begin{equation*}
l:\mathscr T_X\longrightarrow\mathcal End^*\left(\Omega^*_{X/k}\right)
\end{equation*}
which corresponds to the differential of $i$ in the $\mathrm{Hom}$ complex. Now consider the linear map of dgla's
\begin{equation*}
\tilde i:KS_X\simeq\mathbb R\Gamma\left(X,\mathscr T_X\right)\longrightarrow\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\left[-1\right]
\end{equation*}
defined as the composition of $\mathbb R\Gamma\left(X,i\right)$ with the map
\begin{equation*}
\mathbb R\Gamma\left(X,\mathcal End^*\left(\Omega^*_{X/k}\right)\right)\rightarrow\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)
\end{equation*}
induced by the action of derived global sections of the endomorphism sheaf of $\Omega^*_{X/k}$ on derived global sections of $X$: this is still a Cartan homotopy; denote by $\tilde l$ the associated morphism of dgla's, which is essentially the derived globalisation of the Lie derivative. Recall that $\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X, \Omega^*_{X/k}\right)\right)$ is the differential graded Lie subalgebra of $\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)$ consisting of non-negatively graded endomorphisms of the (derived global sections of the) algebraic De Rham complex and that this is the same as the subalgebra of endomorphisms preserving the Hodge filtration. Notice also that the image of $\tilde l$ is contained in $\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)$, therefore in the end there is a diagram of dgla's
\begin{equation*}
\xymatrix{KS_X\simeq\mathbb R\Gamma\left(X,\mathscr T_X\right)\ar[r]^{\tilde l} & \mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\ar@<0.75ex>[r]^{\text{incl.}}\ar@<-0.75ex>[r]_0 & \mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}
\end{equation*}
where $0$ stands for the zero map. Since $\tilde i$ is a Cartan homotopy, by Remark \ref{rem Cart htpy} $e^{\tilde i}$ gives an homotopy between $\tilde l$ and the zero map, thus there is an induced morphism of dgla's to the homotopy fibre
\begin{equation} \label{FMM dgla}
KS_X\overset{\left(\tilde l,e^{\tilde i}\right)}{\xrightarrow{\hspace*{1cm}}}\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\doublerightarrow{\quad}{\quad}\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\right)
\end{equation}
as observed in \cite{FMar} Section 6; moreover, recall from formula \eqref{tgt hoflag} that a model for the above homotopy fibre is the abelian dgla
\begin{equation*}
\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right].
\end{equation*}
\begin{defn} \label{alg Fio-Man}
Define the \emph{universal algebraic Fiorenza-Manetti local period map} to be the morphism of deformation functors
\begin{equation*}
\mathrm{FM}:\mathrm{Def}_{KS_X}\xrightarrow{\hspace*{1cm}}\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}
\end{equation*}
induced by map \eqref{FMM dgla}.
\end{defn}
\begin{defn} \label{alg Fio-Man-Mar}
Define the \emph{(universal) algebraic Fiorenza-Manetti-Martinengo local period map} to be the morphism of derived deformation functors
\begin{equation*}
\mathrm{FMM}:\mathbb R\mathrm{Def}_{KS_X}\xrightarrow{\hspace*{1cm}}\mathbb R\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}
\end{equation*}
induced by map \eqref{FMM dgla}.
\end{defn}
In Section 2.4 we described a universal version of the geometric period map (see Definition \ref{univ geom Fio-Man}), but we did not construct its Lie-theoretic counterpart: Fiorenza and Martinengo showed that this is precisely given by map \eqref{FMM dgla}.
\begin{thm} \label{Fiorenza-Martinengo} \emph{(Fiorenza-Martinengo)}
The diagram
\begin{equation*}
\xymatrix{\mathbb R\mathrm{Def}_{KS_X}\ar[rrr]^{\mathrm{FMM}}\ar[d]_{\pi^0\pi_{\leq 0}} & & & \mathbb R\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_X/k\right)\right)}\left[-1\right]}\ar[d]^{\pi^0\pi_{\leq 0}} \\
\mathrm{Def}_{KS_X}\ar[rrr]^{\mathrm{FM}}\ar[d]^{\wr} & & & \mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}\ar[d]_{\wr} \\
\mathrm{Def}_X\ar[rrr]^{\mathcal P} & & & \mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}}}
\end{equation*}
is well-defined and commutes.
\end{thm}
\begin{proof}
See \cite{FMar} Section 6.
\end{proof}
\subsection{Affine DG$_{\geq 0}$-Categories and the Dold-Kan Correspondence}
Theorem \ref{Fiorenza-Martinengo} says that morphism $\mathrm{FMM}$ is the correct derived enhancement of the universal Fiorenza-Manetti local period map; however the geometric interpretation of such a result is somehow indirect, thus it would be worth to find an equivalent morphism of derived deformation functors having more evident geometric meaning. Of course the key step in order to do this consists of finding the right domain and codomain for such a morphism, i.e. defining two derived deformation functors $\mathbb R\mathrm{Def}_X$ and $\mathrm{hoFlag}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}^{F^{\bullet}}$ such that
\begin{itemize}
\item $\mathbb R\mathrm{Def}_X$ is weakly equivalent to $\mathbb R\mathrm{Def}_{KS_X}$ and similarly $\mathrm{hoFlag}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}^{F^{\bullet}}$ is weakly equivalent to $\mathbb R\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}$;
\item $\mathbb R\mathrm{Def}_X$ and $\mathrm{hoFlag}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}^{F^{\bullet}}$ are derived enhancements of $\mathrm{Def}_X$ and $\mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}}$, respectively.
\end{itemize}
In order to construct such functors we need some homotopy-theoretic background.
\begin{war}
In this section we will deal with non-negatively graded differential graded chain structures rather than non-positively graded cochain ones, though the pictures they provide are largely equivalent; the reason for this lies in the fact that -- at least in the framework of this paper -- the codomain of a derived deformation functor is the simplicial model category of simplicial sets, which is more directly related to chain structures than cochain ones.
\end{war}
First of all, recall that the \emph{normalisation} of a simplicial $k$-vector space $\left(V_{\bullet},\partial_i,\sigma_j\right)$ is defined to be the non-negatively graded chain complex of $k$-vector spaces $\left(\mathbf NV,\delta\right)$ where
\begin{equation} \label{norm}
\left(\mathbf NV\right)_n:=\bigcap_i\mathrm{ker}\left(\partial_i:V_n\rightarrow V_{n-1}\right)
\end{equation}
and $\delta_n:=\left(-1\right)^n\partial_n$. Moreover, given a map $f:V_{\bullet}\rightarrow W_{\bullet}$ of simplicial $k$-vector spaces, we can define the chain map
\begin{equation*}
\mathbf N\left(f\right):\mathbf NV_{\bullet}\longrightarrow\mathbf NW_{\bullet}
\end{equation*}
identified by the relation $\mathbf N\left(f\right)_n:=f_n|_{\mathbf NV_n}$; notice that this construction gives us a well-defined morphism of chain complexes. In the end, there is a normalisation functor
\begin{equation*}
\mathbf N:\mathfrak{sVect}_k\longrightarrow\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right).
\end{equation*}
at our disposal. \\
On the other hand, let $V$ be a chain complex of $k$-vector spaces and recall that its \emph{denormalisation} is defined to be the simplicial vector space $\left(\left(\mathbf KV\right)_{\bullet},\partial_i,\sigma_j\right)$ given in level $n$ by the vector space
\begin{equation*}
\left(\mathbf KV\right)_n:=\underset{\eta\text{ surjective}}{\underset{\eta\in\mathrm{Hom}_{\Delta}\left(\left[p\right],\left[n\right]\right)}{\prod}}V_p\left[\eta\right]\qquad\qquad\qquad \left(V_p\left[\eta\right]\simeq V_p\right).
\end{equation*}
\begin{rem} \label{denorm}
Notice that
\begin{equation*}
\left(\mathbf KV\right)_n\simeq V_0\oplus V_1^{\oplus n}\oplus V_2^{\oplus\binom{n}{2}}\oplus\cdots\oplus V_k^{\oplus\binom{n}{k}}\oplus\cdots\oplus V_n^{\oplus\binom{n}{n}}.
\end{equation*}
\end{rem}
In order to complete the definition of the denormalisation of $V$ we need to define face and degeneracy maps: we will describe a combinatorial procedure to determine all of them. For all morphisms $\alpha:\left[m\right]\rightarrow\left[n\right]$ in $\Delta$, we want to define a linear map $\mathbf K\left(\alpha\right):\left(\mathbf KV\right)_n\rightarrow\left(\mathbf KV\right)_m$; this will be done by describing all restrictions $\mathbf K\left(\alpha,\eta\right):V_p\left[\eta\right]\rightarrow\left(\mathbf KV\right)_m$, for any surjective non-decreasing map $\eta\in\mathrm{Hom}_{\Delta}\left(\left[p\right],\left[n\right]\right)$. \\
For all such $\eta$, take the composite $\eta\circ\alpha$ and consider its epi-monic factorisation\footnote{The existence of such a decomposition is one of the key properties of the category $\Delta$.} $\epsilon\circ\eta'$, as in the diagram
\begin{equation*}
\xymatrix{\left[m\right]\ar[r]^{\alpha}\ar[d]_{\eta'} & \left[n\right]\ar[d]^{\eta} \\
\left[q\right]\ar[r]^{\epsilon} & \left[p\right].}
\end{equation*}
Now
\begin{itemize}
\item if $p=q$ (in which case $\epsilon$ is just the identity map), then set $\mathbf K\left(\alpha,\eta\right)$ to be the natural identification of $V_p\left[\eta\right]$ with the summand $V_p\left[\eta'\right]$ in $\left(\mathbf KV\right)_m$;
\item if $p=q+1$ and $\epsilon$ is the unique injective non-decreasing map from $\left[p\right]$ to $\left[p+1\right]$ whose image misses $p$, then set $\mathbf K\left(\alpha,\eta\right)$ to be the differential $d_p:V_p\rightarrow V_{p-1}$;
\item in all other cases set $\mathbf K\left(\alpha,\eta\right)$ to be the zero map.
\end{itemize}
The above constructions determine the whole of the simplicial vector space $\left(\left(\mathbf KV\right)_{\bullet},\partial_i,\sigma_j\right)$. As done for normalisation, for any chain map $f:V\rightarrow W$ we can define a morphism of simplicial $k$-vector spaces
\begin{equation*}
\mathbf K\left(f\right):\mathbf KV\longrightarrow\mathbf KW
\end{equation*}
by setting
\begin{eqnarray*}
&V_0\times V_1^{\oplus n}\times V_2^{\oplus\binom{n}{2}}\times\cdots\times V_n^{\oplus\binom{n}{n}}&\overset{\mathbf K\left(f\right)_n}{\xrightarrow{\hspace*{1cm}}}\qquad\;\; W_0\times W_1^{\oplus n}\times W_2^{\oplus\binom{n}{2}}\times\cdots\times W_n^{\oplus\binom{n}{n}} \\
&\left(v_0,\left(v_1^i\right)_i,\left(v_2^j\right)_j,\ldots,v_n\right)&\longmapsto\qquad\left(f_0\left(v_0\right),\left(f_1\left(v_1^i\right)\right)_i,\left(f_2\left(v_2^j\right)\right)_j,\ldots,f_n\left(v_n\right)\right).
\end{eqnarray*}
Again, there is a denormalisation functor
\begin{equation*}
\mathbf K:\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)\longrightarrow\mathfrak{sVect}_k.
\end{equation*}
at our disposal.
\begin{thm} \label{Dold-Kan} \emph{(Dold, Kan)}
The functors $\mathbf N$ and $\mathbf K$ form an equivalence of categories between $\mathfrak{sVect}_k$ and $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)$.
\end{thm}
\begin{proof}
See \cite{GJ} Corollary 2.3 or \cite{We} Theorem 8.4.1.
\end{proof}
The Dold-Kan correspondence described in Theorem \ref{Dold-Kan} is known to induce a number of very interesting $\infty$-equivalences; for instance the Eilenberg-Zilber shuffle product and the Alexander-Whitney map, which we will discuss in more details later in this section, allow us to extend normalisation and denormalisation to a pair of functors
\begin{equation*}
\mathbf N:\mathfrak{sAlg}_k\rightleftarrows\mathfrak{dg}_{\geq 0}\mathfrak{Alg}_k:\mathbf K
\end{equation*}
which is seen to be a Quillen equivalence. Moreover recall that
\begin{itemize}
\item a $\mathrm{dg}_{\geq 0}$-category over $k$ is a category enriched in $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)$;
\item a $k$-simplicial category is a category enriched in $\mathfrak{sVect}_k$;
\item a simplicial category is a category enriched in $\mathfrak{sSet}$;
\item a simplicial groupoid is a simplicial object in $\mathfrak{Grpd}$: equivalently a simplicial groupoid is a simplicial category in which all $1$-morphisms are invertible.
\end{itemize}
All the above structures form well-understood model categories; furthermore it is well-known in the homotopy-theoretic folklore that Theorem \ref{Dold-Kan} induces a Quillen equivalence
\begin{equation} \label{DKcat}
\mathbf N:\mathfrak{sCat}_k\rightleftarrows\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k:\mathbf K.\footnote{There is some abuse of notation in this statement.}
\end{equation}
Tabuada also constructed an explicit Quillen equivalence between $\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k$ and $\mathfrak{sCat}$ (see \cite{Tab}, where an explicit proof of formula \eqref{DKcat} can be found, as well). \\
We will use slightly more general versions of the Dold-Kan correspondence provided by Theorem \ref{Dold-Kan} and its corollaries, so we need to develop a few technical tools. \\
Define $\mathfrak{Aff}_k$ to be the category whose objects are $k$-vector spaces and whose morphisms are affine maps between $k$-vector spaces, i.e.
\begin{equation*}
\mathrm{Hom}_{\mathfrak{Aff}_k}\left(V,W\right):=\left\{v\mapsto f\left(v\right)+b\text{ s.t. }f\text{ linear},b\in W\right\}\simeq\mathrm{Hom}_{\mathfrak{Vect}_k}\left(V,W\right)\times W.
\end{equation*}
$\mathfrak{Aff}_k$ can be thought of as the category of affine spaces over $k$ and affine maps. Given $V,W\in\mathfrak{Aff}_k$, define their tensor product to be
\begin{equation} \label{tens aff}
V\tilde{\otimes}W:=V\oplus W\oplus\left(V\otimes W\right)
\end{equation}
where the tensor product $V\otimes W$ in the right-hand side of formula \eqref{tens aff} is just the tensor product as vector spaces; in a similar way, given two affine maps
\begin{eqnarray*}
&\phi\in\mathrm{Hom}_{\mathfrak{Aff}_k}\left(V,W\right)\quad\text{where}\quad\phi\left(v\right):=f\left(v\right)+b& \\
&\psi\in\mathrm{Hom}_{\mathfrak{Aff}_k}\left(U,Z\right)\quad\text{where}\quad\psi\left(u\right):=g\left(u\right)+d&
\end{eqnarray*}
the tensor product map is given by
\begin{eqnarray} \label{und tens morph aff}
\phi\otimes\psi:&V\oplus W\oplus \left(V\otimes W\right)&\longrightarrow\quad U\oplus Z\oplus \left(U\otimes Z\right)\nonumber \\
&\left(u,v,x\otimes y\right)&\mapsto\left(u+b,v+d,f\left(x\right)\otimes g\left(y\right)\right).
\end{eqnarray}
Formula \eqref{tens aff} and formula \eqref{und tens morph aff} determine a monoidal structure on $\mathfrak{Aff}_k$: we will be more precise about this a little bit further in this section, when dealing with dg$_{\geq 0}$-affine spaces.
\begin{defn} \label{quasi dg vect}
Define a \emph{(chain) differential graded affine space} over $k$ in non-negative degrees (\emph{$dg_{\geq 0}$-affine space} for short) to be a pair $\left(A_0,V\right)$ where
\begin{equation*}
V:\qquad V_0\overset{d}{\longleftarrow} V_1\overset{d}{\longleftarrow} V_2\overset{d}{\longleftarrow}\cdots\overset{d}{\longleftarrow} V_n\overset{d}{\longleftarrow}
\end{equation*}
is a non-negatively graded chain complex of $k$-vector spaces and $A_0$ is an affine space over $k$ whose difference vector space is $V_0$.
\end{defn}
If $\left(A_0,V\right)$ and $\left(B_0,W\right)$ are $\mathrm{dg}_{\geq 0}$-affine spaces over $k$, a morphism $\phi:\left(A_0,V\right)\rightarrow \left(B_0,W\right)$ will be a chain map which is affine in degree $0$ and linear in higher degrees: more formally the set of morphisms between $\left(A_0,V\right)$ and $\left(B_0,W\right)$ is defined to be
\begin{equation*}
\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)}\left(\left(A_0,V\right),\left(B_0,W\right)\right):=\left\{\underline v\mapsto f\left(\underline v\right)+b\text{ s.t. }f\in\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)}\left(V,W\right),b\in W_0\right\}.
\end{equation*}
In the end we have a well-defined category of $\mathrm{dg}_{\geq 0}$-affine spaces over $k$, which we will denote as $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$.
\begin{rem}
We have defined the objects of $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$ as pairs where the first term is an affine space and the second term is a chain complex of vector spaces just to make the affine structure explicit; an equivalent and more compact characterisation of $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$ is
\begin{eqnarray*}
&\mathrm{Ob}\left(\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)\right):=\mathrm{Ob}\left(\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)\right)& \\
&\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)}\left(\left(A_0,V\right),\left(B_0,W\right)\right)\simeq\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)}\left(V,W\right)\times W_0.&
\end{eqnarray*}
In particular $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$ is a $k$-linear category.
\end{rem}
The category $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$ is both complete and cocomplete: limits and colimits are constructed from those in $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)$. For example if $\left(A_0,V\right)$ and $\left(B_0,W\right)$ are $\mathrm{dg}_{\geq 0}$-affine spaces their product will be just $\left(A_0\times B_0,V\times W\right)$, where $V\times W$ is the product of $V$ and $W$ in $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)$ and $A_0\times B_0$ is the affine space over $k$ whose difference vector space is $V_0\times W_0$. \\
We can also put a tensor structure over $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$: given two $\mathrm{dg}_{\geq 0}$-affine spaces $\left(A_0,V\right)$ and $\left(B_0,W\right)$, define their tensor product $\left(A_0,V\right)\otimes\left(B_0,W\right)$ to be the $\mathrm{dg}_{\geq 0}$-affine space determined by the chain complex
\begin{equation} \label{tens struc aff}
V\oplus W\oplus \left(V\otimes W\right).
\end{equation}
Similarly, given
\begin{eqnarray*}
&\phi\in\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)}\left(\left(A_0,V\right),\left(B_0,W\right)\right)\quad\text{where}\quad\phi\left(\underline v\right):=f\left(\underline v\right)+b& \\
&\psi\in\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)}\left(\left(C_0,U\right),\left(D_0,Z\right)\right)\quad\text{where}\quad\psi\left(\underline u\right):=g\left(\underline u\right)+d&
\end{eqnarray*}
the tensor product map is given by
\begin{eqnarray} \label{tens morph aff}
\phi\otimes\psi:&V\oplus W\oplus \left(V\otimes W\right)&\longrightarrow\quad U\oplus Z\oplus \left(U\otimes Z\right)\nonumber \\
&\left(\underline u,\underline v,\underline x\otimes \underline y\right)&\mapsto\left(\underline u+b,\underline v+d,f\left(\underline x\right)\otimes g\left(\underline y\right)\right).
\end{eqnarray}
Formula \eqref{tens struc aff} and formula \eqref{tens morph aff} determine a monoidal structure on $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$: in particular the unit is given by the object $\left(\{*\},0\right)$, the associator is induced by the monoidal structure on $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)$ and the unitors are simply given by
\begin{equation*}
\left(V\oplus 0\oplus\left(V\otimes 0\right)\right)\overset{\sim}{\longrightarrow}V
\end{equation*}
\begin{equation*}
\left(0\oplus V\oplus\left(0\otimes V\right)\right)\overset{\sim}{\longrightarrow}V.
\end{equation*}
The reader can check that the above definitions verify the pentagon and the triangle identities.
\begin{rem}
Let $\left(A_0,V\right)$ and $\left(B_0,W\right)$ be dg$_{\geq 0}$-affine spaces: notice that
\begin{equation} \label{coh}
\left(A_0,V\right)\otimes\left(B_0,W\right)\simeq A_0\tilde{\otimes}B_0
\end{equation}
where formula \eqref{coh} is a canonical identification in $\mathfrak{Aff}_k$; an analogous coherence statement holds for morphisms.
\end{rem}
\begin{defn}
Define a \emph{simplicial affine space} over $k$ to be just a simplicial object in $\mathfrak{Aff}_k$.
\end{defn}
Let $\mathfrak{sAff}_k$ be the category of simplicial affine spaces over $k$, i.e.
\begin{equation*}
\mathfrak{sAff}_k:=\mathfrak{Aff}_k^{\Delta^{\mathrm{op}}}.
\end{equation*}
\begin{rem} \label{sadj}
There is a natural linearisation functor
\begin{equation*}
\mathbf L:\mathfrak{sAff}_k\longrightarrow\mathfrak{sVect}_k
\end{equation*}
which just deletes the non-linear part in the face and degeneracy maps defining a simplicial affine space, as well as the non-linear part of morphisms between simplicial affine spaces; in the same fashion there is a forgetful functor
\begin{equation*}
\mathbf U:\mathfrak{sVect}_k\longrightarrow\mathfrak{sAff}_k
\end{equation*}
which just takes (maps of) simplicial vector spaces and looks at them as (maps of) simplicial affine ones.
\end{rem}
\begin{war}
The pair of functors $(\mathbf U,\mathbf L)$ does not provide an adjunction between $\mathfrak{sAff}_k$ and $\mathfrak{sVect}_k$.
\end{war}
The category $\mathfrak{sAff}_k$ has all small limits and colimits, which are just taken levelwise; moreover define the tensor product in $\mathfrak{sAff}_k$ to be constructed by simply taking the tensor product in $\mathfrak{Aff}_k$ in all levels: it is straightforward to check that this equips such a category with a monoidal structure.\\
Now define the normalisation of a $\mathrm{dg}_{\geq 0}$-affine space over $k$ to be the functor
\begin{eqnarray*}
\breve{\mathbf N}:&\mathfrak{sAff}_k&\xrightarrow{\hspace*{1cm}}\quad\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right) \\
&A_{\bullet}&\longmapsto\qquad\left(A_0,\mathbf N\left(\mathbf L\left(A_{\bullet}\right)\right)\right) \\
&\vcenter{\xymatrix{A_{\bullet}\ar[d]^{\phi} \\
B_{\bullet}}}&\mapsto
\left(\vcenter{\xymatrix{A_0\ar[d]^{\phi_0} \\
B_0}}\;\;,\;\;\vcenter{\xymatrix{\mathbf N\left(\mathbf L\left(A_{\bullet}\right)\right)\ar[d]^{\mathbf N\left(\mathbf L\left(\phi\right)\right)} \\ \mathbf N\left(\mathbf L\left(B_{\bullet}\right)\right)}}\right)
\end{eqnarray*}
and observe that such a definition is well-given as the $0$-th term of the chain complex $\mathbf N\left(\mathbf L\left(A_{\bullet}\right)\right)$ is precisely the difference vector space of $A_0$; in other words, the normalisation of a simplicial vector space does not affect the object in degree $0$, as follows from formula \eqref{norm}. \\
Analogously, define the denormalisation of a simplicial affine space over $k$ to be the functor
\begin{eqnarray*}
\breve{\mathbf K}:&\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)&\xrightarrow{\hspace*{3.5cm}}\mathfrak{sAff}_k \\
&\left(A_0,V\right)&\longmapsto\xymatrix{A_0\ar[r] & A_0\times V_1\ar@<2.5pt>[l]\ar@<-2.5pt>[l]\ar@<2.5pt>[r]\ar@<-2.5pt>[r] & A_0\times V_1^{\oplus 2}\times V_2 \ar[l]\ar@<5pt>[l]\ar@<-5pt>[l]\ar[r]\ar@<5pt>[r]\ar@<-5pt>[r] & \cdots\ar@<-7.5pt>[l]\ar@<-2.5pt>[l]\ar@<2.5pt>[l]\ar@<7.5pt>[l]}
\end{eqnarray*}
where the maps involving $A_0$ and $A_0\times V_1$ are
\begin{eqnarray*}
&A_0\times V_1\ni\left(a,v\right)\mapsto a+d\left(v\right)\in A_0& \\
&A_0\times V_1\ni\left(a,v\right)\mapsto a\in A_0& \\
&A_0\ni a\mapsto\left(a,0\right)\in A_0\times V_1&
\end{eqnarray*}
and all other faces and degeneracies -- which do not involve the affine space $A_0$, but rather only the vector spaces $V_i$ -- are defined as done for classical denormalisation (see Remark \ref{denorm} and subsequent discussion). \\
In a similar way, given
\begin{equation*}
\phi\in\mathrm{Hom}_{\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)}\left(\left(A_0,V\right),\left(B_0,W\right)\right)\quad\text{where}\quad\phi\left(\underline v\right):=f\left(\underline v\right)+b
\end{equation*}
the morphism $\breve{\mathbf K}\left(\phi\right)$ of simplicial affine spaces is defined in level $n$ by the affine map
\begin{eqnarray*}
&A_0\times V_1^{\oplus n}\times V_2^{\oplus\binom{n}{2}}\times\cdots\times V_n^{\oplus\binom{n}{n}}&\longrightarrow \qquad\;\; B_0\times W_1^{\oplus n}\times W_2^{\oplus\binom{n}{2}}\times\cdots\times W_n^{\oplus\binom{n}{n}} \\
&\left(a_0,\left(v_1^i\right)_i,\left(v_2^j\right)_j,\ldots,v_n\right)&\mapsto\left(f_0\left(a_0\right)+b,\left(f_1\left(v_1^i\right)\right)_i,\left(f_2\left(v_2^j\right)\right)_j,\ldots,f_n\left(v_n\right)\right)
\end{eqnarray*}
We are ready to describe the generalisation of Theorem \ref{Dold-Kan} we mentioned before.
\begin{prop} \label{ext Dold-Kan}
The functors $\breve{\mathbf N}$ and $\breve{\mathbf K}$ form an equivalence of categories between $\mathfrak{sAff}_k$ and $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$.
\end{prop}
\begin{proof}
The arguments used in \cite{We} Theorem 8.4.1 to prove the classical Dold-Kan correspondence given by Theorem \ref{Dold-Kan} carry over to this context.
\end{proof}
As follows for instance from the discussion in \cite{SchShi} Section 2.3, the normalisation functor $\breve{\mathbf N}$ can be made into a lax monoidal functor via the Eilenberg-Zilber shuffle map, which is the natural transformation
\begin{equation*}
\mathrm{EZ}:\breve{\mathbf N}\left(-\right)\otimes\breve{\mathbf N}\left(-\right)\longrightarrow\breve{\mathbf N}\left(-\otimes-\right)
\end{equation*}
determined for all $A_{\bullet},B_{\bullet}\in\mathfrak{sAff}_k$ by the morphisms
\begin{eqnarray*}
\mathrm{EZ}^{p,q}_{A_{\bullet},B_{\bullet}}:&\breve{\mathbf N}\left(A_{\bullet}\right)_p\otimes\breve{\mathbf N}\left(B_{\bullet}\right)_q&\xrightarrow{\hspace*{1.5cm}}\breve{\mathbf N}\left(A_{\bullet}\otimes B_{\bullet}\right)_{p+q} \\
&a\otimes b&\longmapsto\underset{\left(\mu,\nu\right)}{\sum}\mathrm{sign}\left(\mu,\nu\right)\,\sigma_{\nu}\left(a\right)\otimes\sigma_{\mu}\left(b\right)
\end{eqnarray*}
where the sum runs over all $(p,q)$-shuffles
\begin{equation*}
\left(\mu,\nu\right):=\left(\mu_1,\ldots,\mu_p,\nu_1,\ldots,\nu_q\right)
\end{equation*}
and the corresponding degeneracy maps are
\begin{equation*}
s_{\mu}:=s_{\mu_p}\circ\ldots s_{\mu_1}\qquad\qquad s_{\nu}:=s_{\nu_q}\circ\ldots s_{\nu_1}.
\end{equation*}
In the same fashion, again from \cite{SchShi} Section 2.3, the denormalisation functor $\tilde{\mathbf K}$ can also be made into a lax monoidal functor by means of the Alexander-Whitney map. The latter is defined to be the natural transformation
\begin{eqnarray*}
\mathrm{AW}:\tilde{\mathbf N}\left(-\otimes -\right)\longrightarrow\tilde{\mathbf N}\left(-\right)\otimes\tilde{\mathbf N}\left(-\right)
\end{eqnarray*}
given for all $A_{\bullet},B_{\bullet}\in\mathfrak{sAff}_k$ by the morphisms
\begin{eqnarray} \label{AW K}
\mathrm{AW}_{A_{\bullet},B_{\bullet}}^{n}:&\tilde{\mathbf N}\left(A_{\bullet}\otimes B_{\bullet}\right)_{n}&\xrightarrow{\hspace*{1.5cm}}\left(\tilde{\mathbf N}\left(A_{\bullet}\right)\otimes\tilde{\mathbf N}\left(B_{\bullet}\right)\right)_n \nonumber \\
&a\otimes b&\longmapsto\underset{p+q=n}{\bigoplus}\left(\tilde d^p\left(a\right)\otimes d_0^q\left(b\right)\right)
\end{eqnarray}
where the {} ``front face'' $\tilde d^p$ and the {} ``back face'' $d_0^q$ are induced respectively by the injective monotone maps $\tilde{\delta^p}:\left[p\right]\rightarrow\left[p+q\right]$ and $\delta_0^p:\left[q\right]\rightarrow\left[p+q\right]$; in particular the Alexander-Whitney map makes the normalisation functor $\tilde{\mathbf N}$ into a comonoidal one (again, see \cite{SchShi} Section 2.3, whose considerations adapt to these context). Notice also that by setting $A':=\tilde{\mathbf N}\left(A\right)$ and $B':=\tilde{\mathbf N}\left(A\right)$ in formula \eqref{AW K} and using the equivalence provided by Proposition \ref{ext Dold-Kan}, we get a version of the Alexander-Whitney transformation
\begin{equation*}
\mathrm{AW}:\tilde{\mathbf K}\left(-\right)\otimes\tilde{\mathbf K}\left(-\right)\longrightarrow\tilde{\mathbf K}\left(-\otimes-\right)\footnote{There is some abuse of notation in this formula}
\end{equation*}
which makes the denormalisation $\tilde{\mathbf K}$ into a lax monoidal functor. Also we have that the composite $\mathrm{AW}\circ\mathrm{EZ}$ is the same as the identity, while the transformation $\mathrm{EZ}\circ\mathrm{AW}$ is chain homotopic to the identity: in particular the Dold-Kan equivalence provided by Proposition \ref{ext Dold-Kan} is lax monoidal. \\
Now we are ready to introduce the notions of affine $\mathrm{dg}_{\geq 0}$-category and affine simplicial category, which will be crucial technical tools to develop a good derived version of the period map.
\begin{defn}
An \emph{affine differential graded category} over $k$ $\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)$ (\emph{affine dg$_{\geq 0}$-category} for short) is a category $\mathfrak C$ enriched over $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$.
\end{defn}
We will denote by $\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k^{\mathfrak{Aff}}$ the $\infty$-category of affine $\mathrm{dg}_{\geq 0}$-categories.\\
Let $\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)$ be an affine $\mathrm{dg}_{\geq0}$-category and denote by $H_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)$ the (honest) category defined by the relations
\begin{eqnarray*}
\mathrm{Ob}\left(H_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)\right)&:=&\mathfrak C \\
\forall X,Y\in\mathfrak C\qquad\mathrm{Hom}_{H_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)}\left(X,Y\right)&:=&H_0\left(\underline{\mathfrak C}_{\bullet}\left(X,Y\right)\right).
\end{eqnarray*}
\begin{defn}
An \emph{affine differential graded groupoid} over $k$ (\emph{affine dg$_{\geq 0}$-groupoid} for short) will be an affine $\mathrm{dg}_{\geq 0}$-category $\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)$ such that the category $H_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)$ is a groupoid.
\end{defn}
We will denote by $\mathfrak{dg}_{\geq 0}\mathfrak{Grpd}_k^{\mathfrak{Aff}}$ the $\infty$-category of affine $\mathrm{dg}_{\geq 0}$-groupoids.
\begin{rem} \label{obsss}
The notion of $\mathrm{dg}_{\geq 0}$-affine space allows us to define a notion of $\infty$-groupoid in the differential graded context: as a matter of fact a more naive notion of $\mathrm{dg}_{\geq 0}$-groupoid -- intended as a $\mathrm{dg}_{\geq 0}$-category where all morphisms in level $0$ are isomorphism -- would not really make sense as every $\mathrm{dg}_{\geq 0}$-category comes with a zero morphism, which is seldom an isomorphism.
\end{rem}
\begin{defn}
An \emph{affine simplicial category} over $k$ $\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)$ is a category $\mathfrak C$ enriched over $\mathfrak{sAff}_k$
\end{defn}
We will denote by $\mathfrak{sCat}_k^{\mathfrak{Aff}}$ the $\infty$-category of affine simplicial categories. \\
Let $\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)$ be an affine simplicial category and denote by $\pi_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)$ the (honest) category defined by the relations
\begin{eqnarray*}
\mathrm{Ob}\left(\pi_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)\right)&:=&\mathfrak C \\
\forall X,Y\in\mathfrak C\qquad\mathrm{Hom}_{\pi_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)}\left(X,Y\right)&:=&\pi_0\left(\underline{\mathfrak C}_{\bullet}\left(X,Y\right)\right).
\end{eqnarray*}
\begin{defn}
An \emph{affine simplicial groupoid} over $k$ will be an affine simplicial category $\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)$ such that the category $\pi_0\left(\left(\mathfrak C,\underline{\mathfrak C}_{\bullet}\right)\right)$ is a groupoid.
\end{defn}
We will denote by $\mathfrak{sGrpd}_k^{\mathfrak{Aff}}$ the $\infty$-category of affine simplicial groupoids.
\begin{rem}
The notion of simplicial affine space allows us to define a notion of $\infty$-groupoid in the $k$-simplicial context, just like $\mathrm{dg}_{\geq 0}$-affine spaces give rise to a good notion of differential graded groupoid, as observed in Remark \ref{obsss}
\end{rem}
Of course any simplicial affine space has an underlying simplicial set, so an affine simplicial category over $k$ is in particular a simplicially enriched category: more formally, there is a natural forgetful functors from $\mathfrak{sCat}_k^{\mathfrak{Aff}}$ to $\mathfrak{sCat}$. \\
The slightly extended version of the Dold-Kan equivalence given by Proposition \ref{ext Dold-Kan} induces a pair of functors
\begin{equation} \label{aff cat DK}
\breve{\mathbf N}:\mathfrak{sCat}_k^{\mathfrak{Aff}}\rightleftarrows\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k^{\mathfrak{Aff}}:\breve{\mathbf K}\footnote{There is some abuse of notation in this formula.}
\end{equation}
where
\begin{eqnarray*}
&\breve{\mathbf N}:\qquad\mathfrak{sCat}_k^{\mathfrak{Aff}}\xrightarrow{\hspace*{3cm}}\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k^{\mathfrak{Aff}}\qquad& \\
&\mathfrak C\xmapsto{\hspace*{4cm}}\mathfrak C& \\
\forall P,Q\in\mathfrak C&\quad\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\xmapsto{\hspace*{2.5cm}}\breve{\mathbf N}\left(\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\right)& \\
\forall P,Q,R\in\mathfrak C&\qquad\left(\vcenter{\xymatrix{\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\otimes\underline{\mathfrak C}_{\bullet}\left(Q,R\right)\ar[d]^{\circ} \\ \underline{\mathfrak C}_{\bullet}\left(P,R\right)}}\right)\mapsto\left(\vcenter{\xymatrix{\breve{\mathbf N}\left(\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\right)\otimes\breve{\mathbf N}\left(\underline{\mathfrak C}_{\bullet}\left(Q,R\right)\right)\ar[d]^{\mathrm{EZ}_{\underline{\mathfrak C}_{\bullet}\left(P,Q\right),\underline{\mathfrak C}_{\bullet}\left(Q,R\right)}} \\ \breve{\mathbf N}\left(\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\otimes\underline{\mathfrak C}_{\bullet}\left(Q,R\right)\right)\ar[d]^{\breve{\mathbf N}\left(\circ\right)} \\ \breve{\mathbf N}\left(\underline{\mathfrak C}_{\bullet}\left(P,R\right)\right)}}\right)&
\end{eqnarray*}
and
\begin{eqnarray*}
&\breve{\mathbf K}:\qquad\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k^{\mathfrak{Aff}}\xrightarrow{\hspace*{3cm}}\mathfrak{sCat}_k^{\mathfrak{Aff}}\qquad& \\
&\mathfrak C\xmapsto{\hspace*{4cm}}\mathfrak C& \\
\forall P,Q\in\mathfrak C&\quad\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\xmapsto{\hspace*{2.5cm}}\breve{\mathbf K}\left(\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\right)& \\
\forall P,Q,R\in\mathfrak C&\qquad\left(\vcenter{\xymatrix{\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\otimes\underline{\mathfrak C}_{\bullet}\left(Q,R\right)\ar[d]^{\circ} \\
\underline{\mathfrak C}_{\bullet}\left(P,R\right)}}\right)\mapsto\left(\vcenter{\xymatrix{\breve{\mathbf K}\left(\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\right)\otimes\breve{\mathbf K}\left(\underline{\mathfrak C}_{\bullet}\left(Q,R\right)\right)\ar[d]^{\mathrm{AW_{\underline{\mathfrak C}_{\bullet}\left(P,Q\right),\underline{\mathfrak C}_{\bullet}\left(Q,R\right)}}} \\
\breve{\mathbf K}\left(\underline{\mathfrak C}_{\bullet}\left(P,Q\right)\otimes\underline{\mathfrak C}_{\bullet}\left(Q,R\right)\right)\ar[d]^{\breve{\mathbf K}\left(\circ\right)} \\
\breve{\mathbf K}\left(\underline{\mathfrak C}_{\bullet}\left(P,R\right)\right)}}\right)&
\end{eqnarray*}
Notice also that the $\infty$-equivalence given by formula \eqref{aff cat DK} restricts to an $\infty$-equivalence
\begin{equation*}
\breve{\mathbf N}:\mathfrak{sGrpd}_k^{\mathfrak{Aff}}\rightleftarrows\mathfrak{dg}_{\geq 0}\mathfrak{Grpd}_k^{\mathfrak{Aff}}:\breve{\mathbf K}.
\end{equation*}
At last, let us recall that there is a natural functor
\begin{equation} \label{bar W}
\bar W:\mathfrak{sCat}\longrightarrow\mathfrak{sSet}
\end{equation}
given by the right adjoint to Illusie's Dec functor; we are not describing it explicitly as its construction is slightly technical and not really needed for the sake of this paper: the definition of $\bar W$ can be found in \cite{GJ} Section V.7 or \cite{Pr3} Section 1. Moreover in \cite{CR} Cegarra and Remedios proved that $\bar W$ is weakly equivalent to the diagonal of the simplicial nerve functor. Functor \eqref{bar W} is also known to induce a right Quillen equivalence
\begin{equation*}
\bar W:\mathfrak{sGrpd}\longrightarrow\mathfrak{sSet}
\end{equation*}
and -- as a corollary of the results in \cite{Pr3} Section 1 -- we also have that functor \eqref{bar W} restricts to an equivalence
\begin{equation*}
\bar W:\mathfrak{sGrpd}_k^{\mathfrak{Aff}}\longrightarrow\mathfrak{sSet}.
\end{equation*}
In Section 3.4 we will apply the functor $\bar W$ to interesting affine simplicial groupoids in order to define rigorously the derived deformation functor $\mathrm{hoFlag}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}^{F^{\bullet}}$, while the functor $\mathbb R\mathrm{Def}_X$ will be constructed in Section 3.5 by using different techniques.
\subsection{Derived Deformations of Filtered Complexes}
In this section we will study in detail how to enhance the formal Grassmannian and the flag functor analysed in Section 2.2 and Section 2.4 to the world of derived deformation functors; in particular we will define the notions of derived formal total Grassmannian and formal homotopy flag variety and make comparisons with the corresponding derived stacks studied in \cite{dN2}. \\
Throughout this section fix $V$ to be a complex of $k$-vector spaces, $S$ a subcomplex of it and $\mathcal F^{\bullet}$ a filtration on $V$. Also recall from \cite{dN2} that for any $R$ in $\mathfrak{Alg}_k$ or even $\mathfrak{dgAlg}^{\leq 0}_k$ there is a model structure on $\mathfrak{FdgMod}_R$ modelled on the projective model structure over $\mathfrak{dgMod}_R$.
\begin{defn} \label{der def V}
A \emph{derived deformation of $V$ over $A\in\mathfrak{dgArt}_k^{\leq 0}$} is the datum of a cofibrant complex of $A$-modules $\mathscr V_A$ and a surjective quasi-isomorphism $\varphi:\mathscr V_A\otimes_Ak\rightarrow V$.
\end{defn}
\begin{defn} \label{der def W}
A \emph{derived deformation of $S$ inside $V$ over $A\in\mathfrak{dgArt}_k^{\leq 0}$} is the datum of a derived deformation $\left(\mathscr V_A,\varphi\right)$ of $V$ and a subcomplex $\mathscr S_A\subseteq\mathscr V_A$ such that
\begin{equation*}
\varphi\big|_{\mathscr S_A}:\mathscr S_A\otimes_Ak\longrightarrow S
\end{equation*}
is still a surjective quasi-isomorphism.
\end{defn}
\begin{defn} \label{der def filt cmplx}
A \emph{derived deformation of $\left(V,\mathcal F^{\bullet}\right)$ over $A\in\mathfrak{dgArt}_k^{\leq 0}$} is a pair $\left(\left(\mathscr V_A, \mathcal F^{\bullet}_A\right),\varphi\right)$, where $\left(\mathscr V_A,\mathcal F^{\bullet}_A\right)$ is a cofibrant filtered complex of $A$-modules and $\varphi:\left(\mathscr V_A, \mathcal F^{\bullet}_A\right)\rightarrow\left(V,\mathcal F^{\bullet}\right)$ is a surjective morphism such that the maps $\mathcal F_A^p\mathscr V_A\otimes_Ak\rightarrow \mathcal F^pV$ induced by $\varphi$ are quasi-isomorphisms for all $p$.
\end{defn}
\begin{rem} \label{fib-cof flag}
Notice that the key technical assumptions in Definition \ref{der def V}, Definition \ref{der def W} and Definition \ref{der def filt cmplx} are
\begin{enumerate}
\item restricting to cofibrant objects while defining derived deformations of (filtered) derived modules;
\item requiring that the structure morphism $\varphi$ is surjective.
\end{enumerate}
Assumption (1) allows us to use tensor products instead of derived tensor products. Assumption (2) makes objects in te slice category $\nicefrac{\mathfrak{dgMod}_A}{V}$ (or $\nicefrac{\mathfrak{FdgMod}_A}{\left(V,\mathcal F^{\bullet}\right)}$) fibrant. Actually in the main geometric example we will consider, that is derived deformations of (filtered) perfect complexes of modules over a scheme, the perfectness condition will imply cofibrancy wherever relevant.
\end{rem}
Derived deformation of complexes, subcomplexes and filtrations are governed by nice derived deformation functors, already hidden in the Derived Deformation Theory folklore: we now propose a rigorous way to define them via the language of affine $\mathrm{dg}_{\geq 0}$-categories. \\
In the above notations, let
\begin{equation*}
\left(hDef_V\left(A\right),\underline{hDef_V}\left(A\right)_{\bullet}\right)
\end{equation*}
be the affine $\mathrm{dg}_{\geq 0}$-category defined by the formulae
\begin{equation*}
hDef_V\left(A\right):=\left\{\text{derived }A\text{-deformations of }V\right\}
\end{equation*}
and for all $\left(\mathscr V_A,\varphi\right),\left(\mathscr W_A,\phi\right)\in hDef_V\left(A\right)$
\begin{eqnarray*}
&\underline{hDef_V}\left(A\right)_0:=\left\{\Psi\in\mathrm{Hom}^0\left(\mathscr V_A,\mathscr W_A\right)\text{ s.t. }\phi\circ\Psi=\varphi\right\}& \\
&\underline{hDef_V}\left(A\right)_1:=\left\{\Psi\in\mathrm{Hom}^{-1}\left(\mathscr V_A,\mathscr W_A\right)\text{ s.t. }\phi\circ\Psi=0\right\}& \\
&\vdots& \\
&\underline{hDef_V}\left(A\right)_n:=\left\{\Psi\in\mathrm{Hom}^{-n}\left(\mathscr V_A,\mathscr W_A\right)\text{ s.t. }\phi\circ\Psi=0\right\}& \\
&\vdots&
\end{eqnarray*}
with the differential induced by the standard differential on $\mathrm{Hom}$ complexes.
Similarly, let
\begin{equation*}
\left(hDef_{S,V}\left(A\right),\underline{hDef_{S,V}}\left(A\right)_{\bullet}\right)
\end{equation*}
be the affine $\mathrm{dg}_{\geq 0}$-category defined by the formulae
\begin{equation*}
hDef_{S,V}\left(A\right):=\left\{\text{(derived) }A\text{-deformations of } S \text{ inside } V\right\}
\end{equation*}
and for all $\left(\left(\mathscr S_A, \mathscr V_A\right),\varphi\right),\left(\left(\mathscr T_A, \mathscr W_A\right),\phi\right)\in hDef_{S,V}\left(A\right)$
\begin{equation*}
\underline{hDef_{S,V}}\left(A\right)_0:=\left\{\Psi\in\mathrm{Hom}^0\left(\left(\mathscr S_A,\mathscr V_A\right),\left(\mathscr T_A,\mathscr W_A\right)\right)\text{ s.t. }\phi\circ\Psi=\varphi\right\}
\end{equation*}
\begin{eqnarray*}
&\underline{hDef_{S,V}}\left(A\right)_1:=\left\{\Psi\in\mathrm{Hom}^{-1}\left(\left(\mathscr S_A,\mathscr V_A\right),\left(\mathscr T_A, \mathscr W_A\right)\right)\text{ s.t. }\phi\circ\Psi=0\right\}& \\
&\vdots& \\
&\underline{hDef_{S,V}}\left(A\right)_n:=\left\{\Psi\in\mathrm{Hom}^{-n}\left(\left(\mathscr S_A,\mathscr V_A\right),\left(\mathscr T_A, \mathscr W_A\right)\right)\text{ s.t. }\phi\circ\Psi=0\right\}& \\
&\vdots&
\end{eqnarray*}
with the differential induced by the standard differential on $\mathrm{Hom}$ complexes. \\
Finally let
\begin{equation*}
\left(hDef_V^{\mathcal F^{\bullet}}\left(A\right),\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_{\bullet}\right)
\end{equation*}
be the affine $\mathrm{dg}_{\geq 0}$-category defined by the formulae
\begin{equation*}
hDef_V^{\mathcal F^{\bullet}}\left(A\right):=\left\{\text{(derived) }A\text{-deformations of }\left(V,\mathcal F^{\bullet}\right)\right\}
\end{equation*}
and for all $\left(\left(\mathscr V_A, \mathcal F^{\bullet}_A\right)\varphi\right),\left(\left(\mathscr W_A,\mathcal G^{\bullet}_A\right),\phi\right)\in hDef_V^{\mathcal F^{\bullet}}\left(A\right)$
\begin{eqnarray*}
&\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_0:=\left\{\Psi\in\mathrm{Hom}^0\left(\left(\mathscr V_A,\mathcal F^{\bullet}_A\right),\left(\mathscr W_A,\mathcal G^{\bullet}_A\right)\right)\text{ s.t. }\phi\circ\Psi=\varphi\right\}& \\
&\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_1:=\left\{\Psi\in\mathrm{Hom}^{-1}\left(\left(\mathscr V_A,\mathcal F^{\bullet}_A\right),\left(\mathscr W_A,\mathcal G^{\bullet}_A\right)\right)\text{ s.t. }\phi\circ\Psi=0\right\}& \\
&\vdots& \\
&\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_n:=\left\{\Psi\in\mathrm{Hom}^{-n}\left(\left(\mathscr V_A,\mathcal F^{\bullet}_A\right),\left(\mathscr W_A,\mathcal G^{\bullet}_A\right)\right)\text{ s.t. }\phi\circ\Psi=0\right\}& \\
&\vdots&
\end{eqnarray*}
with the differential induced by the standard differential on $\mathrm{Hom}$ complexes.
\begin{rem}
The $2$-out-of-$3$ property implies that morphisms in $\underline{hDef_V}\left(A\right)_0$, $\underline{hDef_{S,V}}\left(A\right)_0$ and $\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_0$ are all weak equivalences; in particular
\begin{itemize}
\item $H_0\left(\left(hDef_V\left(A\right),\underline{hDef_V}\left(A\right)_{\bullet}\right)\right)$
\item $H_0\left(\left(hDef_{S,V}\left(A\right),\underline{hDef_{S,V}}\left(A\right)_{\bullet}\right)\right)$
\item $H_0\left(\left(hDef_V^{\mathcal F^{\bullet}}\left(A\right),\underline{hDef_V^{F^{\bullet}}}\left(A\right)_{\bullet}\right)\right)$
\end{itemize}
are groupoids, so the affine $\mathrm{dg}_{\geq 0}$-categories
\begin{itemize}
\item $\left(hDef_V\left(A\right),\underline{hDef_V}\left(A\right)_{\bullet}\right)$
\item $\left(hDef_{S,V}\left(A\right),\underline{hDef_{S,V}}\left(A\right)_{\bullet}\right)$
\item $\left(hDef_V^{\mathcal F^{\bullet}}\left(A\right),\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_{\bullet}\right)$
\end{itemize}
are really affine $\mathrm{dg}_{\geq 0}$-groupoids.
\end{rem}
Now define the derived deformation functors
\begin{eqnarray*}
\mathbb R\mathrm{Def}_V:&\mathfrak{dgArt}_k^{\leq 0}&\xrightarrow{\hspace*{3cm}}\mathfrak{sSet} \\
&A&\longmapsto\bar W\left(\breve{\mathbf K}\left(\left(hDef_V\left(A\right),\underline{hDef_V}\left(A\right)_{\bullet}\right)\right)\right).
\end{eqnarray*}
\begin{eqnarray*}
\mathbb R\mathrm{Def}_{S,V}:&\mathfrak{dgArt}_k^{\leq 0}&\xrightarrow{\hspace*{3cm}}\mathfrak{sSet} \\
&A&\longmapsto\bar W\left(\breve{\mathbf K}\left(\left(hDef_{S,V}\left(A\right),\underline{hDef_{S,V}}\left(A\right)_{\bullet}\right)\right)\right).
\end{eqnarray*}
\begin{eqnarray*}
\mathbb R\mathrm{Def}_V^{\mathcal F^{\bullet}}:&\mathfrak{dgArt}_k^{\leq 0}&\xrightarrow{\hspace*{3cm}}\mathfrak{sSet} \\
&A&\longmapsto\bar W\left(\breve{\mathbf K}\left(\left(hDef_V^{\mathcal F^{\bullet}}\left(A\right),\underline{hDef_V^{\mathcal F^{\bullet}}}\left(A\right)_{\bullet}\right)\right)\right).
\end{eqnarray*}
\begin{lemma} \label{Def_End}
In the above notations we have:
\begin{enumerate}
\item $\mathbb R\mathrm{Def}_V$ is weakly equivalent to the derived deformation functor $\mathbb R\mathrm{Def}_{\mathrm{End}^*\left(V\right)}$ associated to the dgla $\mathrm{End}^*\left(V\right)$;
\item $\mathbb R\mathrm{Def}_{S,V}$ is weakly equivalent to the derived deformation functor $\mathbb R\mathrm{Def}_{\mathrm{End}^S\left(V\right)}$ associated to the dgla $\mathrm{End}^S\left(V\right)$;
\item $\mathbb R\mathrm{Def}_V^{\mathcal F^{\bullet}}$ is weakly equivalent to the derived deformation functor $\mathbb R\mathrm{Def}_{\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right)}$ associated to the dgla $\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right)$;
\end{enumerate}
\end{lemma}
\begin{proof}
This result is well-known in the Derived Deformation Theory folklore: we just recall the morphism
\begin{equation*}
\mathbb R\mathrm{Def}_{\mathrm{End}^*}\left(V\right)\longrightarrow\mathbb R\mathrm{Def}_V
\end{equation*}
giving Claim (1). \\
Note that by Theorem \ref{Del grpd} it suffices to determine such a map on the (derived) Deligne groupoid $\mathrm{BDel}_{\mathrm{End}^*\left(V\right)}$ associated to the dgla $\mathrm{End}^*\left(V\right)$, so define
\begin{eqnarray*} \label{nu}
&\qquad\qquad\qquad\qquad\nu:\mathrm{BDel}_{\mathrm{End}^*\left(V\right)}\xrightarrow{\hspace*{1cm}}\mathbb R\mathrm{Def}_V& \\
&\text{for all }A\in\mathfrak{dgArt}^{\leq 0}_k\quad\widetilde{\mathrm{MC}}_{\mathrm{End}^*\left(V\right)}\left(A\right)\ni \sigma\longmapsto\left[\left(V\otimes A,d+\sigma\right)\right]\quad\qquad& \\
&\qquad\qquad\qquad\widetilde{\mathrm{Gg}}_{\mathrm{End}^*\left(V\right)}\left(A\right)\ni\xi\longmapsto\left[\vcenter{\xymatrix{\left(V\otimes A,d+\sigma_1\right)\ar[d]_{e^{\xi}} \\ \left(V\otimes A,d+\sigma_2\right)}}\right] &
\end{eqnarray*}
Map $\nu$ is known to be a weak equivalence: a very rigorous but quite abstract proof can be found in \cite{Pr0} Section 4.1, while a simpler one can be found in \cite{Man2}; see also \cite{FMar} Section 6. Claim (2) and Claim (3) are proved in an entirely analogous way.
\end{proof}
Now we are ready to define coherent derived version of formal Grassmannians and flag functors.
\begin{defn} \label{holim def cmplx grass}
Define the \emph{derived formal total Grassmannian associated to $S$ inside $V$} to be the functor
\begin{equation*}
\mathrm{hoGrass}_{S,V}=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathbb R\mathrm{Def}_{S,V}\doublerightarrow{\text{forgetful map}}{0}\mathbb R\mathrm{Def}_V\right).
\end{equation*}
In particular $\mathrm{hoGrass}_{S,V}$ is a well-defined derived deformation functor.
\end{defn}
\begin{defn} \label{holim def cmplx}
Define the \emph{formal homotopy flag variety associated to $\left(V,\mathcal F^{\bullet}\right)$} to be the functor
\begin{equation*}
\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathbb R\mathrm{Def}_V^{\mathcal F^{\bullet}}\doublerightarrow{\text{forgetful map}}{0}\mathbb R\mathrm{Def}_V\right).
\end{equation*}
In particular $\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}$ is a well-defined derived deformation functor.
\end{defn}
\begin{prop} \label{hoFlag dgla}
In the above notation we have that:
\begin{enumerate}
\item The functors $\mathrm{hoGrass}_{S,V}$ and $\mathbb R\mathrm{Def}_{C_{S,V}}$ are weakly equivalent.\footnote{Recall from formula \eqref{tgt dgrass} that $$C_{S,V}:=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{S}\left(V\right)\doublerightarrow{\chi}{0}\mathrm{End}^*\left(V\right)\right).
$$}
\item The functors $\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}$ and $\mathbb R\mathrm{Def}_{C_V^{\mathcal F^{\bullet}}}$ are weakly equivalent;\footnote{Recall from formula \eqref{tgt hoflag} that $$C^{\mathcal F^{\bullet}}_V:=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^{\mathcal F^{\bullet}}\left(V\right)\doublerightarrow{\chi}{0}\mathrm{End}^*\left(V\right)\right).
$$}
\end{enumerate}
\end{prop}
\begin{proof}
We only prove Claim (1) as Claim (2) is proved in an entirely similar way.\\
We want to show that the functor $\mathrm{hoGrass}_{S,V}$ is weakly equivalent to the Hinich nerve of
\begin{equation*}
C_{S,V}:=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^S\left(V\right)\doublerightarrow{\mathrm{\text{incl.}}}{0}\mathrm{End}^*\left(V\right)\right).
\end{equation*}
As proved in \cite {Getz} and \cite{Hin1}, the functor $\mathbb R\mathrm{Def}$ (homotopically) commutes with homotopy limits, thus we have that
\begin{equation*}
\mathbb R\mathrm{Def}_{\underset{\longleftarrow}{\mathrm{holim}}\left(\mathrm{End}^S\left(V\right)\doublerightarrow{\quad}{\quad}\mathrm{End}^*\left(V\right)\right)}
\end{equation*}
is weakly equivalent to
\begin{equation*}
\underset{\longleftarrow}{\mathrm{holim}}\left(\mathbb R\mathrm{Def}_{\mathrm{End}^S\left(V\right)}\doublerightarrow{\quad}{\quad}\mathbb R\mathrm{Def}_{\mathrm{End}^*\left(V\right)}\right).
\end{equation*}
By Lemma \ref{Def_End} we have that $\mathbb R\mathrm{Def}_V$ is weakly equivalent to $\mathbb R\mathrm{Def}_{\mathrm{End}^*\left(V\right)}$, while the functor $\mathbb R\mathrm{Def}_{S,V}$ is weakly equivalent to $\mathbb R\mathrm{Def}_{\mathrm{End}^S\left(V\right)}$, thus the statement follows by applying Definition \ref{holim def cmplx grass}.
\end{proof}
As a consequence of Proposition \ref{hoFlag dgla} we have that functors $\mathrm{hoGrass}_{S,V}$ and $\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}$ are really derived versions of $\mathrm{Grass}_{S,V}$ and $\mathrm{Flag}_V^{\mathcal F^{\bullet}}$: this is exactly the content of the next result.
\begin{prop} \label{Flag der enh}
In the above notations we have that:
\begin{enumerate}
\item the derived formal total Grassmannian $\mathrm{hoGrass}_{S,V}$ is a derived enhancement of the functor $\mathrm{Grass}_{S,V}$, i.e.
\begin{equation*}
\pi^0\pi_{\leq 0}\mathrm{hoGrass}_{S,V}\simeq\mathrm{Grass}_{S,V};
\end{equation*}
\item the formal homotopy flag variety $\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}$ is a derived enhancement of the formal flag variety $\mathrm{Flag}_V^{\mathcal F^{\bullet}}$, i.e.
\begin{equation*}
\pi^0\pi_{\leq 0}\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}\simeq\mathrm{Flag}_V^{\mathcal F^{\bullet}}.
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
Again, we only prove Claim (1) as Claim (2) is proved in an entirely similar way.\\
Theorem \ref{Grass=Def_C} and Proposition \ref{hoFlag dgla}.1 state that the (homotopy) dgla $C_{S,V}$ represents both the deformation functor $\mathrm{Grass}_{S,V}$ and the derived deformation functor $\mathrm{hoGrass}_{S,V}$; Theorem \ref{compare def theories} implies that the latter has to be a derived enhancement of the former.
\end{proof}
At last recall from \cite{dN2} that there exist two (locally geometric) derived stacks $\mathcal{DG}rass_k\left(V\right)$ and $\mathcal{DF}lag_k\left(V\right)$ which respectively enhance the total Grassmannian variety $\mathrm{Grass}\left(H^*\left(V\right)\right)$ and the (total) flag variety $\mathrm{Flag}\left(H^*\left(V\right)\right)$ to the derived world (see \cite{dN2} Theorem 2.42 and \cite{dN2} Theorem 2.45). As a conclusion of this section we want to show that functors $\mathrm{hoGrass}_{S,V}$ and $\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}$ are formal neighbourhoods of the above derived stacks.
\begin{prop} \label{comp flag fin}
In the above notations we have that:
\begin{enumerate}
\item the derived deformation functor $\mathrm{hoGrass}_{S,V}$ is the formal neighbourhood of the derived stack $\mathcal{DG}rass_k\left(V\right)$ at $\left[S\hookrightarrow V\right]$;
\item the derived deformation functor $\mathrm{hoFlag}_V^{\mathcal F^{\bullet}}$ is the formal neighbourhood of the derived stack $\mathcal{DF}lag_k\left(V\right)$ at $\left(V,\mathcal F^{\bullet}\right)$.
\end{enumerate}
\end{prop}
\begin{proof}
Again, we only prove Claim (1) as Claim (2) is proved in an entirely similar way.\\
As a first step, recall that -- by \cite{dN2} Proposition 2.41 -- the stack $\mathcal{DG}rass_k\left(V\right)$ is an open derived substack of the \emph{big total derived Grassmannian}
\begin{equation*}
\mathcal{DGRASS}_k\left(V\right):=\underset{\longleftarrow}{\mathrm{holim}}\left(\mathbb R\mathcal Sub_k\doublerightarrow{\left[U\hookrightarrow W\right]\mapsto W}{\mathrm{const}_V}\mathbb R\mathcal Perf_k\right)
\end{equation*}
where
\begin{equation*}
\mathbb R\mathcal Perf_k=\underset{n}{\bigcup}\mathbb R\mathcal Perf_k^n\qquad\qquad\qquad\mathbb R\mathcal Sub_k=\underset{n}{\bigcup}\mathbb R\mathcal Sub_k^n
\end{equation*}
are respectively the locally geometric derived stack of perfect complexes and the locally geometric derived stack of perfect subcomplexes: for more details see \cite{dN2} Section 2.2 and 2.3. \\
As a result, it suffices to show that $\mathrm{hoGrass}_{S,V}$ is the formal neighbourhood of $\mathcal{DGRASS}_k\left(V\right)$ at $\left[S\hookrightarrow V\right]$, which by Definition \ref{holim def cmplx grass} amounts to prove that $\mathbb R\mathrm{Def}_{S,V}$ is the formal neighbourhood of $\mathbb R\mathcal Sub_k$ at $\left[S\hookrightarrow V\right]$ and $\mathbb R\mathrm{Def}_V$ is the formal neighbourhood of $\mathbb R\mathcal Perf_k$ at $V$; we show only the first assertion, as the other one is proved analogously. \\
Observe from \cite{dN2} Remark 2.34, \cite{dN2} Corollary 2.32 and \cite{dN2} Theorem 2.33 that the derived stack $\mathbb R\mathcal Sub^n_k$ is obtained as follows: given for all $A\in\mathfrak{Alg}_k$ the functorial simplicial category
\begin{eqnarray*}
&\mathbf M^n_{\mathrm{sub},k}\left(A\right):=&\text{full simplicial subcategory of $\mathfrak{FdgMod}_A$ made of pairs $(\mathscr E,\mathscr C)$} \\
&&\text{of perfect complexes of length at most $n$ and for which $E\subseteq C$}
\end{eqnarray*}
construct for all $A\in\mathfrak{dg}_b\mathfrak{Nil}^{\leq 0}_k$ the functorial simplicial category
\begin{eqnarray*}
&\tilde{\mathbf M}^n_{\mathrm{sub},k}\left(A\right):=&\text{full simplicial subcategory of $\mathcal W\left(Fd\mathrm{CART}_k\left(A\right)\right)$ \footnotemark} \\ &&\text{made of pairs $(\mathscr E,\mathscr C)$ such that $(\mathscr E\otimes^{\mathbb L}_AH^0\left(A\right),\mathscr C\otimes^{\mathbb L}_AH^0\left(A\right))$} \\
&&\text{is weakly equivalent to an object in $\mathbf M^n_{\mathrm{sub},k}\left(H^0\left(A\right)\right)$ }
\end{eqnarray*}
\footnotetext{For the precise meaning of $Fd\mathrm{CART}_k$ see \cite{dN2} formula (2.29).}and end up with a functor $\bar W\tilde{\mathbf M}^n_{\mathrm{sub},k}$ which turns out to be via Lurie-Pridham Representability (see \cite{Lu1} or \cite{Pr2bis} for more details) the restriction to $\mathfrak{dg}_b\mathfrak{Nil}^{\leq 0}_k$ of the derived geometric stack $\mathbb R\mathcal Sub^n_k$, which is fully determined then.
On the other hand recall from the beginning of this section that the derived deformation functor $\mathbb R\mathrm{Def}_{S,V}$ is obtained from the formal affine $\mathrm{dg}_{\geq 0}$-groupoid
\begin{equation*}
\left(hDef_{S,V}\left(A\right),\underline{hDef_{S,V}}\left(A\right)_{\bullet}\right)\qquad\qquad A\in\mathfrak{dgArt}^{\leq 0}_k
\end{equation*}
by applying functor $\bar W$ and (affine) Dold-Kan denormalisation $\breve{\mathbf K}$. \\
Now fix $A\in\mathfrak{dgArt}^{\leq 0}_k$; clearly we have
\begin{equation} \label{neigh}
\tilde{\mathbf M}^n_{\mathrm{sub},k}\left(A\right)\times^{h}_{\tilde{\mathbf M}^n_{\mathrm{sub},k}\left(k\right)}\left\{\left(S,V\right)\right\}\simeq \breve{\mathbf K}\left(\left(hDef_{S,V}\left(A\right),\underline{hDef_{S,V}}\left(A\right)_{\bullet}\right)\right)\quad\quad A\in\mathfrak{dgArt}^{\leq 0}_k
\end{equation}
thus the equivalence of $\mathbb R\mathrm{Def}_{S,V}$ and the formal neighbourhood of $\mathbb R\mathcal Sub^n_k\subseteq\mathbb R\mathcal Sub_k$ at $\left(S,V\right)$ follows by applying functor $\bar W$ to formula \eqref{neigh}.
\end{proof}
\subsection{Derived Deformations of $k$-Schemes}
Now we want to describe the functor $\mathbb R\mathrm{Def}_X$ which parametrises derived deformations of the scheme $X$: the idea consists of deforming the scheme $X$ through derived schemes instead of ordinary schemes. There are a variety of equivalent definitions of derived scheme (in particular see \cite{Lu1} Definition 4.5.1 and \cite{TV} Chapter 2.2 for the two most standard ways to look at it); the one we are about to recall probably is not the most elegant, but it is definitely the handiest one to make actual computations. As a matter of fact, by \cite{Pr2} Theorem 6.42 a \emph{derived scheme} $S$ over $k$ can be seen as a pair $\left(\pi^0S,\mathscr O_{S,*}\right)$, where $\pi^0S$ is an honest $k$-scheme and $\mathscr O_{S,*}$ is a presheaf of differential graded commutative algebras in non-positive degrees on the site of affine opens of $\pi^0S$ such that:
\begin{itemize}
\item the (cohomology) presheaf $\mathcal H^0\left(\mathscr O_{S,*}\right)\simeq\mathscr O_{\pi^0S}$;
\item the (cohomology) presheaves $\mathcal H^n\left(\mathscr O_{S,*}\right)$ are quasi-coherent $\mathscr O_{\pi^0S}$-modules.
\end{itemize}
Also, recall from \cite{Pr2} that a morphism $f:A\rightarrow B$ in $\mathfrak{dgAlg}_k^{\leq 0}$ is said to be \emph{homotopy flat} if
\begin{equation*}
H^0\left(f\right):H^0\left(A\right)\longrightarrow H^0\left(B\right)
\end{equation*}
is flat and the maps
\begin{equation*}
H^i\left(A\right)\otimes_{H^0\left(A\right)}H^0\left(B\right)\longrightarrow H^i\left(B\right)
\end{equation*}
are isomorphisms for all $i$; moreover a very useful characterisation says that $f$ is homotopy flat if and only if $B\otimes^{\mathbb L}_AH^0\left(A\right)$ is (weakly equivalent to) a discrete flat $H^0\left(A\right)$-algebra: for a proof see \cite{Pr3} Lemma 3.13.\footnote{\cite{Pr3} and \cite{Pr2} actually deal with homotopy flatness in terms of simplicial and $\mathrm{dg}_{\geq 0}$ chain algebras; nevertheless all definitions and arguments readily adapt to cochain algebras in non-positive degrees.}\\
Now define a derived deformation of the scheme $X$ over $A\in\mathfrak{dgArt}^{\leq 0}$ to be a homotopy pull-back diagram of derived schemes
\begin{equation*}
\xymatrix{\ar@{} |{\Box^h}[dr]X\ar@{^{(}->}[r]^i\ar[d] & \mathcal X\ar[d]^p \\
\mathrm{Spec}\left(k\right)\ar[r] & \mathbb R\mathrm{Spec}\left(A\right)}
\end{equation*}
where the map $p$ is homotopy flat; equivalently such a deformation can be seen as a morphism $\mathscr O_{A,*}\rightarrow\mathscr O_X$ of presheaves of differential graded commutative algebras over $A$ such that:
\begin{enumerate}
\item $\mathscr O_{A,*}$ is homotopy flat;
\item the induced $k$-linear morphism $\mathscr O_{A,*}\otimes_A^{\mathbb L} k\rightarrow\mathscr O_X$ is a weak equivalence;
\item the morphism $\mathscr O_{A,*}\rightarrow\mathscr O_X$ is surjective;
\item $\mathscr O_{A,*}$ is cofibrant.
\end{enumerate}
\begin{rem} \label{fib-cof X}
In the above notations, Condition (1) and Condition (2) are proper deformation-theoretic conditions, which resemble the ones characterising underived deformations of schemes (see Section 2.1), while Condition (3) and Condition (4) are fibrancy-cofibrancy conditions, which are needed in order to ensure that certain maps of derived deformation functors which will arise in the rest of the paper are well-defined.
\end{rem}
Now consider the formal groupoid
\begin{eqnarray*}
\mathrm{Del}_X:&\mathfrak{dgArt}_k^{\leq 0}&\xrightarrow{\hspace*{1cm}}\mathfrak{Grpd} \\
&A&\longmapsto\mathrm{Del}_X\left(A\right)
\end{eqnarray*}
defined by the formulae
\begin{equation*}
\mathrm{Del}_X\left(A\right):=\left\{\text{(derived) deformations of }X\text{ over }A\right\}
\end{equation*}
and for all $\left(\mathscr O_{A,*}\overset{\varphi}{\rightarrow}\mathscr O_X\right),\left(\mathscr O'_{A,*}\overset{\phi}{\rightarrow}\mathscr O_X\right)\in\mathrm{Del}_X\left(A\right)$
\begin{equation} \label{Del_X}
\mathrm{Hom}_{\mathrm{Del}_X\left(A\right)}\left(\varphi,\phi\right):=\left\{\Psi\in\mathrm{Hom}_{A}^0\left(\mathscr O_{A,*},\mathscr O'_{A,*}\right)\text{ s.t. }\phi\circ\Psi=\varphi,\Psi\equiv\mathrm{Id}\;\left(\mathrm{mod}\,\mathfrak m_A\right)\right\}
\end{equation}
\begin{rem}
In the notations of formula \eqref{Del_X}, notice that the condition $\Psi\equiv\mathrm{Id}\;\left(\mathrm{mod}\,\mathfrak m_A\right)$ ensures that $\mathrm{Del}_X\left(A\right)$ is a groupoid for all $A\in\mathfrak{dgArt}_k^{\leq 0}$; roughly speaking, the formal groupoid $\mathrm{Del}_X$ can be thought as some sort of (derived) Deligne groupoid associated to the scheme $X$, meaning that its role is intended to formally resemble the one played by the (derived) Deligne groupoid associated do a differential graded Lie algebra, which we described in Section 3.1.
\end{rem}
Now consider the functor
\begin{equation*}
\mathrm{BDel}_X:\mathfrak{dgArt}_k^{\leq 0}\xrightarrow{\hspace*{1cm}}\mathfrak{sSet}
\end{equation*}
given by the nerve of $\mathrm{Del}_X$ and define
\begin{equation*}
\mathbb R\mathrm{Def}_X:\mathfrak{dgArt}_k^{\leq 0}\xrightarrow{\hspace*{1cm}}\mathfrak{sSet}
\end{equation*}
to be the right derived functor of $\mathrm{BDel}_X$.
The definition of $\mathrm{Del}_X$ implies immediately that this is a derived pre-deformation functor, thus -- by \cite{Pr2bis} Theorem 3.16 -- $\mathbb R\mathrm{Def}_X$ turns to be a derived deformation functor. \\
Now let us briefly look at global derived moduli of schemes; consider the assignment
\begin{eqnarray*}
\mathbf{Stack}_{n/k}^0:&\mathfrak{Alg}_k&\xrightarrow{\hspace*{1cm}}\mathfrak{sCat} \\
&A&\longmapsto\mathbf{Stack}_{n/k}^0\left(A\right):=\text{ simplicial category of algebraic $n$-spaces over } A
\end{eqnarray*}
which is the simplicial (underived) moduli functor classifying (underived) $0$-stacks of dimension $n$ over $k$. Pridham has shown that such a functor induces a derived stack $\mathcal D\mathcal Sch_{n/k}$ parametrising derived schemes over $k$ of dimension $n$: see \cite{Pr3} Example 3.36 for a detailed construction. Unfortunately the stack $\mathcal D\mathcal Sch_{n/k}$ is far too large to be geometric, nonetheless Pridham constructed many interesting geometric substacks of its: see \cite{Pr3} Section 3 for more details.
\begin{rem} \label{DSch}
The derived deformation functor $\mathbb R\mathrm{Def}_X$ is the formal neighbourhood of the derived stack $\mathcal D\mathcal Sch_{d/k}$ at $X$; in particular, it follows -- using either \cite{Pr2} Theorem 8.8 or \cite{Pr3} Theorem 10.8 -- that
\begin{equation*}
H^i\left(\mathrm{BDel}_X\right)\simeq H^i\left(\mathbb R\mathrm{Def}_X\right)\simeq\mathrm{Ext}^{i+1}_{\mathscr O_X}\left(\mathbb L_{X/k},\mathscr O_X\right)
\end{equation*}
\end{rem}
\begin{thm} \label{main}
The functors $\mathbb R\mathrm{Def}_X$ and $\mathbb R\mathrm{Def}_{KS_X}$ are weakly equivalent.
\end{thm}
\begin{proof}
We want to construct a natural transformation
\begin{equation*}
\mathbb R\mathrm{Def}_{KS_X}\xrightarrow{\hspace*{1cm}}\mathbb R\mathrm{Def}_X
\end{equation*}
providing a weak equivalence between such derived deformation functors, i.e. an isomorphism on the level of homotopy categories. \\
Again, by Theorem \ref{Del grpd} and the definition of $\mathbb R\mathrm{Def}_X$ it is enough to define such a morphism on $\mathrm{BDel}_{KS_X}$, thus define the map
\begin{eqnarray} \label{mu}
&\qquad\qquad\mu:\mathrm{BDel}_{KS_X}&\xrightarrow{\hspace*{1.5cm}}\mathrm{BDel}_X \nonumber \\
\text{for all }A\in\mathfrak{dgArt}^{\leq 0}_k&\mathrm{MC}_{KS_X}\left(A\right)\ni x&\longmapsto\left[\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\rightarrow\tau^{\leq 0}\mathbb R\mathscr O_k\left(x\right)\simeq\mathscr O_X\right] \nonumber \\
&\mathrm{Gg}_{KS_X}\left(A\right)\ni\xi&\longmapsto\left[\vcenter{\xymatrix{\tau^{\leq 0}\mathbb R\mathscr O_A\left(x_1\right)\ar[dd]_{e^{\xi}}\ar[dr] & \\ & \mathscr O_X \\ \tau^{\leq 0}\mathbb R\mathscr O_A\left(x_2\right)\ar[ur] &} }\right]
\end{eqnarray}
where $\mathbb R\mathscr O_A\left(x\right):=\left(\mathscr A^{0,*}_X\otimes A,\bar{\partial}+l_{x}\right)$, $l$ being the Lie derivative (i.e. the differential of the contraction map) and the map $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\rightarrow\tau^{\leq 0}\mathbb R\mathscr O_k\left(x\right)$ is induced by $A\twoheadrightarrow \nicefrac{A}{\mathfrak m_A}\simeq k$. Notice that the complex $\mathbb R\mathscr O_A\left(x\right)$ is cofibrant, since it is bounded above and the underlying graded module $\mathscr A^{0,*}_X\otimes A$ is projective. Furthermore observe that the surjectivity of the natural map $A\twoheadrightarrow \nicefrac{A}{\mathfrak m_A}$ together with the surjectivity of the canonical morphism $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\twoheadrightarrow H^0\left(\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\right)$ -- due in turn to the fact that the complex $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)$ lives in non-positive degrees -- ensures that the morphism $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\rightarrow\mathscr O_X$ is surjective.\\
In order to show that map \ref{mu} is well-defined, we have to check that
\begin{equation*}
\left[\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\rightarrow\tau^{\leq 0}\mathbb R\mathscr O_k\left(x\right)\simeq\mathscr O_X\right]
\end{equation*}
actually determines a derived deformation of the scheme $X$, i.e. we need to prove that $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)$ is homotopy flat over $A$ and $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\otimes^{\mathbb L}_Ak$ is weakly equivalent to $\mathscr O_X$ as complexes of presheaves of differential graded commutative $k$-algebras: this essentially means to verify that $\mathbb R\mathscr O_A\left(x\right)\otimes^{\mathbb L}_A H^0\left(A\right)$ is flat over $H^0\left(A\right)$. \\
Let us first prove that $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)$ is weakly equivalent to $\mathbb R\mathscr O_A\left(x\right)$. Filter the latter complex by powers of the maximal ideal $\mathfrak m_A$ of $A$, i.e. define the filtered complex $\left(\mathbb R\mathscr O_A\left(x\right),\mathcal F^{\bullet}\right)$ through the relation
\begin{equation*}
\mathcal F^p\mathbb R\mathscr O_A\left(x\right):=\mathfrak m_A^p\mathbb R\mathscr O_A\left(x\right)
\end{equation*}
and take the associated graded object
\begin{equation} \label{assoc grad}
\mathrm{Gr}^p\left(\mathcal F^{\bullet}\right):=\frac{\mathfrak m_A^p\mathbb R\mathscr O_A\left(x\right)}{\mathfrak m_A^{p+1}\mathbb R\mathscr O_A\left(x\right)}.
\end{equation}
Notice that formula \eqref{assoc grad}, so
\begin{equation*}
\mathrm{Gr}^p\left(\mathcal F^{\bullet}\right)\simeq\mathscr A^{0,*}_X\otimes\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}.
\end{equation*}
Now consider the spectral sequence
\begin{equation}\label{spec seq}
H^{p+q}\left(\mathscr A_X^{0,*}\otimes\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}\right)\simeq\bigoplus_{i+j=p+q}\left(H^i\left(\mathscr A^{0,*}_X\right)\otimes H^j\left(\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}\right)\right)\Longrightarrow H^{p+q}\left(\mathbb R\mathscr O_A\left(x\right)\right)
\end{equation}
which converges by the \emph{Classical Convergence Theorem} (see \cite{We} Theorem 5.5.1); note that $H^j\left(\frac{\mathfrak m_A^p}{\mathfrak m_A^ {p+1}}\right)=0$ when $j>0$ and, since the {} ``Dolbeaut'' resolution $\mathscr A^{0,*}_X\hookleftarrow \mathscr O_X$ provides a weak equivalence between $\mathscr A^{0,*}_X$ and $\mathscr O_X$ in the category of $\mathscr O_X$-modules in complexes, also $H^i\left(\mathscr A^{0,*}_X\right)=0$ when $i>0$: this means that at least one of these two terms vanishes whenever $i+j>0$, so the convergence of spectral sequence \eqref{spec seq} implies that
\begin{equation*}
H^n\left(\mathbb R\mathscr O_A\left(x\right)\right)=0\qquad\forall n>0.
\end{equation*}
In particular $\mathbb R\mathscr O_A\left(x\right)$ and $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)$ are weakly equivalent. \\
Now we want to prove that $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\otimes^{\mathbb L}_AH^0\left(A\right)$ is flat over $H^0\left(A\right)$; first notice that
\begin{equation*}
\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\otimes^{\mathbb L}_AH^0\left(A\right)\approx\mathbb R\mathscr O_A\left(x\right)\otimes^{\mathbb L}_AH^0\left(A\right)\approx\mathbb R\mathscr O_A\left(x\right)\otimes_AH^0\left(A\right)\footnote{The symbol $\approx$ stands for {}``weakly equivalent''.}
\end{equation*}
so it is enough to show that $\mathbb R\mathscr O_A\left(x\right)\otimes_AH^0\left(A\right)$ is flat over $H^0\left(A\right)$. In order to prove this let $M$ be any $H^0\left(A\right)$-module, consider the complex
\begin{equation*}
\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AH^0\left(A\right)\right)\otimes_{H^0\left(A\right)}M\simeq\mathbb R\mathscr O_A\left(x\right)\otimes_AM.
\end{equation*}
and filter it by powers of the maximal ideal $\mathfrak m_{H^0\left(A\right)}$ of $H^0\left(A\right)$, i.e. define the filtered complex $\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM,\mathcal F^{\bullet}\right)$ through the relation
\begin{equation*}
\mathcal F^p\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM\right):=\mathfrak m_{H^0\left(A\right)}^p\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM\right).
\end{equation*}
As before, the associated graded object is
\begin{equation*}
\mathrm{Gr}^p\left(\mathcal F\right):=\frac{\mathfrak m_{H^0\left(A\right)}^p\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM\right)}{\mathfrak m_{H^0\left(A\right)}^{p+1}\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM\right)}\simeq\mathscr A^{0,*}_X\otimes\frac{\mathfrak m^p_{H^0\left(A\right)}M}{\mathfrak m^{p+1}_{H^0\left(A\right)}M}
\end{equation*}
and there is a spectral sequence
\begin{equation} \label{spec seq 2}
H^{p+q}\left(\mathscr A^{*,0}_X\otimes\frac{\mathfrak m_{H^0\left(A\right)}^pM}{\mathfrak m_{H^0\left(A\right)}^{p+1}M}\right)\Longrightarrow H^{p+q}\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM\right)
\end{equation}
which still converges because of the Classical Convergence Theorem. Of course
\begin{equation*}
H^{p+q}\left(\mathscr A^{*,0}_X\otimes\frac{\mathfrak m_{H^0\left(A\right)}^pM}{\mathfrak m_{H^0\left(A\right)}^{p+1}M}\right)\simeq\bigoplus_{i+j=p+q}\left(H^i\left(\mathscr A^{0,*}_X\right)\otimes H^j\left(\frac{\mathfrak m_{H^0\left(A\right)}^pM}{\mathfrak m_{H^0\left(A\right)}^{p+1}M}\right)\right)
\end{equation*}
and $H^j\left(\frac{\mathfrak m_{H^0\left(A\right)}^pM}{\mathfrak m_{H^0\left(A\right)}^{p+1}M}\right)=0$ for all $j\neq 0$, thus the convergence of spectral sequence \eqref{spec seq 2} implies
\begin{equation*}
\mathrm{Tor}_n^A\left(\mathbb R\mathscr O_A\left(x\right),M\right)\simeq H^{-n}\left(\mathbb R\mathscr O_A\left(x\right)\otimes_AM\right)=0\qquad\forall n\neq0
\end{equation*}
which gives us the flatness of $\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\otimes^{\mathbb L}_AH^0\left(A\right)$ over $H^0\left(A\right)$. Notice that the same computation also ensures that the map
\begin{equation*}
\left[\tau^{\leq 0}\mathbb R\mathscr O_A\left(x\right)\rightarrow\tau^{\leq 0}\mathbb R\mathscr O_k\left(x\right)\simeq\mathscr O_X\right]
\end{equation*}
is quasi-smooth.\\
Now we want to prove that map \ref{mu} is a weak equivalence of derived deformation functors; by \cite{Pr1} Corollary 1.49 it suffices to check that such a map induces isomorphisms on generalised tangent spaces, so consider the morphisms
\begin{equation*}
H^i\left(\mu\right):H^i\left(\mathrm{BDel}_{KS_X}\right)\longrightarrow H^i\left(\mathrm{BDel}_X\right)\qquad\quad i\geq -1
\end{equation*}
and notice that higher tangent maps in larger negative degrees vanish as $KS_X$ lives only in non-negative degrees. \\
For all $i\geq 0$ we have the chain of canonical identifications
\begin{equation*}
H^i\left(\mathbb R\mathrm{Def}_{KS_X}\right)\simeq H^i\left(\mathrm{BDel}_{KS_X}\right)\simeq H^{i+1}\left(KS_X\right)\simeq H^{i+1}\left(X,\mathscr A_X^{0,*}\left(\mathscr T_X\right)\right)\simeq H^{i+1}\left(X,\mathscr T_X\right)
\end{equation*}
where the first and the second isomorphism come from Remark \ref{Del grpd coho}, the third one is true just by definition and the last one is given by the Dolbeaut Theorem. In the same fashion, there is also a chain of canonical isomorphisms
\begin{eqnarray*}
&H^i\left(\mathrm{BDel}_X\right)\simeq\mathrm{Ext}^{i+1}_{\mathscr O_X}\left(\mathbb L_{X/k},\mathscr O_X\right)\simeq\mathrm{Hom}_{\mathrm D\left(X\right)}\left(\Omega^1_{X/k}\otimes \mathscr O_X,\mathscr O_X\left[-i-1\right]\right)\simeq& \\
&\mathrm{Hom}_{\mathrm D\left(X\right)}\left(\mathscr O_X,\mathcal Hom\left(\Omega^1_{X/k},\mathscr O_X\right)\left[-i-1\right]\right)\simeq\mathrm{Hom}_{\mathrm D\left(X\right)}\left(\mathscr O_X,\mathscr T_X\left[-i-1\right]\right)\simeq\mathrm{Ext}_{\mathscr O_X}^{i+1}\left(\mathscr O_X,\mathscr T_X\right)&
\end{eqnarray*}
where the first isomorphism -- as we discussed before -- comes from the fact that $\mathbb R\mathrm{Def}_X$ is the formal neighbourhood of a derived stack of schemes, the third one is true by adjunction, while all the other ones directly follow from definitions. \\
Finally, for all $i\geq 0$ we see that the map $H^i\left(\mu\right)$ is
\begin{eqnarray} \label{H^i}
H^i\left(\mu\right):&H^{i+1}\left(X,\mathscr T_X\right)&\xrightarrow{\hspace*{0.3cm}}\quad\;\mathrm{Ext}^{i+1}_{\mathscr O_X}\left(\mathscr O_X,\mathscr T_X\right) \nonumber \\
&\xi&\longmapsto\left(\mathscr O_X\overset{\xi}{\longrightarrow}\mathscr T_X\left[-i-1\right]\right)
\end{eqnarray}
where the (cohomology class of the) degree $i$ morphism $\mathscr O_X\overset{\xi}{\longrightarrow}\mathscr T_X\left[-i-1\right]$ is nothing but the map\footnote{Of course, there is some abuse of notation in this sentence.} induced in $\mathrm D\left(X\right)$ by the cocycle $\xi$;
on the other hand -- again by using Remark \ref{Del grpd coho} -- the map $H^{-1}\left(\mu\right)$ turns out to be
\begin{eqnarray} \label{H^-1}
H^{-1}\left(\mu\right):&H^0\left(KS_X\right)\simeq\mathrm{Stab}_{\mathrm{Gg}_{\Gamma\left(X,\mathscr A^{0,0}_X\left(\mathscr T_X\right)\right)}\left(\frac{k\left[\varepsilon\right]}{\left(\varepsilon^2\right)}\right)}\left(0\right)&\xrightarrow{\hspace*{0.3cm}} H^0\left(X,\mathscr T_X\right)\simeq\mathrm{Ext}^1_{\mathscr O_X}\left(\mathscr O_X,\mathscr T_X\right)\nonumber \\
&\qquad\qquad\qquad\quad\mathrm{Id}+\xi&\longmapsto \qquad\;\xi.
\end{eqnarray}
Both map \eqref{H^-1} and map \eqref{H^i} are clearly isomorphisms, so this completes the proof.
\end{proof}
\begin{rem} \label{Def_X der enh}
$\mathbb R\mathrm{Def}_X$ is a derived enhancement of $\mathrm{Def}_X$, i.e.
\begin{equation*}
\pi^0\pi_{\leq 0}\mathbb R\mathrm{Def}_X\simeq\mathrm{Def}_X.
\end{equation*}
\end{rem}
\subsection{The Geometric Fiorenza-Manetti-Martinengo Period Map}
Now we have all the ingredients to give a geometric interpretation of the map $\mathrm{FMM}$ described in Definition \ref{alg Fio-Man-Mar}.
\begin{defn} \label{g FMM map}
Define the \emph{(universal) geometric Fiorenza-Manetti-Martinengo local period map} to be the morphism of derived deformation functors
\begin{equation*}
\mathbb R\mathcal P:\mathbb R\mathrm{Def}_X\xrightarrow{\hspace*{0.75cm}}\mathrm{hoFlag}^{F^{\bullet}}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}
\end{equation*}
identified by the universal morphism of derived deformation functors under the map of derived pre-deformation functors given for all $A\in\mathfrak{dgArt}^{\leq 0}_k$ by
\begin{eqnarray*}
&\mathrm{BDel}_X&\xrightarrow{\hspace*{0.75cm}}\qquad\mathrm{hoFlag}^{F^{\bullet}}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)} \\
&\left[\mathscr O_{A,*}\overset{\varphi}{\longrightarrow}\mathscr O_X\right]\approx\left[\vcenter{\xymatrix{\ar@{} |{\Box^h}[dr]X\ar@{^{(}->}[r]\ar[d] & \mathcal X\ar[d] \\
\mathrm{Spec}\left(k\right)\ar[r] & \mathbb R\mathrm{Spec}\left(A\right)}}\right]&\mapsto \left[\left(\left(\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right),F^{\bullet}\right),\tilde{\varphi}\right)\right] \\
&\left[\vcenter{\xymatrix{\mathscr O_{A,*}'\ar[dd]_{\Psi}\ar[dr]^{\varphi'} & \\ & \mathscr O_X \\ \mathscr O_{A,*}''\ar[ur]_{\varphi''} &} }\right]& \mapsto\left[\vcenter{\xymatrix{\left(\left(\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X'/A}\right),F^{\bullet}\right),\tilde{\varphi'}\right)\ar[d]^{\tilde{\Psi}} \\ \left(\left(\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X''/A}\right),F^{\bullet}\right),\tilde{\varphi''}\right)}}\right]
\end{eqnarray*}
where
\begin{itemize}
\item $\tilde{\varphi}$ is the derived globalisation of the natural $A$-linear map extending $\varphi$ to the algebraic De Rham complex;
\item $\tilde{\Psi}$ is constructed by using the same universal property;
\item the complex
\begin{equation*}
\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right):=\underset{i}{\prod}\left(\bigwedge^i\mathbb L_{\mathcal X/A}\right)
\end{equation*}
is sometimes known as \emph{derived de Rham complex} and the Hodge filtration over it is just
\begin{equation*}
F^p\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right):=\underset{i\geq p}{\prod}\left(\bigwedge^i\mathbb L_{\mathcal X/A}\right).
\end{equation*}
\end{itemize}
\end{defn}
\begin{rem}
The fibrant-cofibrant replacement properties pointed out in Remark \ref{fib-cof flag} and Remark \ref{fib-cof X} ensure that the geometric Fiorenza-Manetti-Martinengo local period map described in Definition \ref{g FMM map} is well-defined.
\end{rem}
In the end all constructions and results we have discussed so far sum up in the following theorem.
\begin{thm} \label{univ lpm ddf}
The diagram of derived deformation functors and (Schlessinger's) deformation functors
\begin{equation*}
\xymatrix{& & \mathbb R\mathrm{Def}_{KS_X}\ar[rrr]^{\mathrm{FMM}\qquad\quad}\ar[dll]^{\sim}\ar[ddd]_{\pi^0\pi_{\leq 0}} & & & \mathbb R\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}\ar[dl]^{\sim}\ar[ddd]^{\pi^0\pi_{\leq 0}} \\
\mathbb R\mathrm{Def}_X\ar[rrrr]^{\mathbb R\mathcal P}\ar[ddd]_{\pi^0\pi_{\leq 0}} & & & & \mathrm{hoFlag}^{F^{\bullet}}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}\ar[ddd]^{\pi^0\pi_{\leq 0}} & \\
& & & & &\\
& &\mathrm{Def}_{KS_X}\ar[rrr]^{\mathrm{FM}\qquad\quad}\ar[dll]^{\sim} & & & \mathrm{Def}_{\frac{\mathrm{End}^*\left(H^*\left(X,k\right)\right)}{\mathrm{End}^{\geq 0}\left(H^*\left(X,k\right)\right)}\left[-1\right]}\ar[dl]^{\sim} \\
\mathrm{Def}_X\ar[rrrr]^{\mathcal P} & & & & \mathrm{Flag}_{H^*\left(X,k\right)}^{F^{\bullet}} &
}
\end{equation*}
commutes up to isomorphism; in particular the morphisms $\mathbb R\mathcal P$ and $\mathrm{FMM}$ are equivalent.
\end{thm}
\begin{proof}
Notice that:
\begin{itemize}
\item the commutativity of the bottom diagram follows from Theorem \ref{Fiorenza-Manetti} and Theorem \ref{Fiorenza-Martinengo};
\item the commutativity (up to isomorphism) of the back diagram corresponds to Theorem \ref{Fiorenza-Martinengo};
\item the commutativity (up to isomorphism) of the front diagram follows immediately from Remark \ref{Def_X der enh}, Proposition \ref{Flag der enh}.2 and the definitions of the maps $\mathcal P$ and $\mathbb R\mathcal P$;
\item the commutativity (up to isomorphism) of the left hand diagram is obtained by combining Theorem \ref{Donatella}, Theorem \ref{main} and Remark \ref{Def_X der enh};
\item the commutativity (up to isomorphism) of the right hand diagram is obtained by combining Corollary \ref{und Flag dgla}, Proposition \ref{hoFlag dgla}.2 and Proposition \ref{Flag der enh}.2.
\end{itemize}
As regards the top diagram, again by Theorem \ref{Del grpd} it suffices to verify its commutativity up to isomorphism on $\mathrm{BDel}_{KS_X}$; moreover Proposition \ref{hoFlag dgla}.2 and Definition \ref{holim def cmplx} say that the derived deformation functors $\mathbb R\mathrm{Def}_{\frac{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\left[-1\right]}$ and $\mathrm{hoFlag}^{F^{\bullet}}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}$ are homotopy fibres, so it is enough to check that the diagrams\footnote{here maps $\mu$ and $\nu$ are the morphisms defined in Theorem \ref{main} and Lemma \ref{Def_End} respectively.}
\begin{equation} \label{diagr1}
\xymatrix{\mathrm{BDel}_{KS_X}\ar[rrrr]^{\mathbb R\mathrm{Def}\left(l\right)\qquad}\ar[d]^{\wr}_{\mu} & & & & \mathrm{BDel}_{\mathrm{End}^*\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\ar[d]_{\wr}^{\nu} \\
\mathbb R\mathrm{Def}_X\ar[rrrr]^{\left(X,\mathscr O_{A,*}\right)=:\mathcal X\mapsto\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right)\qquad\qquad} & & & & \mathbb R\mathrm{Def}_{\mathbb R\Gamma\left(X,\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)} }
\end{equation}
and
\begin{equation} \label{diagr2}
\xymatrix{\mathrm{BDel}_{KS_X}\ar[rrrrr]^{\mathbb R\mathrm{Def}\left(l\right)\qquad}\ar[d]^{\wr}_{\mu} & & & & & \mathrm{BDel}_{\mathrm{End}^{\geq 0}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)}\ar[d]_{\wr}^{\nu} \\
\mathbb R\mathrm{Def}_X\ar[rrrrr]^{\left(X,\mathscr O_{A,*}\right)=:\mathcal X\mapsto\left(\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right),F^{\bullet}\right)\qquad\qquad\qquad\qquad} & & & & & \mathbb R\mathrm{Def}_{F^{\bullet}\mathbb R\Gamma\left(X,\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right),\mathbb R\Gamma\left(X,\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)} } \qquad\quad
\end{equation}
commute up to isomorphism. We are only going to show the commutativity of diagram \eqref{diagr1}, as the commutativity of diagram \eqref{diagr2} is verified by a similar argument.\\
Let us walk along its arrows: for all $A\in\mathfrak{dgArt}_k$ an element $x\in\mathrm{MC}_{KS_X}\left(A\right)$ maps through $\mu$ to $\left[\mathcal X\rightarrow\mathbb R\mathrm{Spec}\left(A\right)\right]$ -- where $\mathcal X=\left(X,\mathbb R\mathscr O_A\left(x\right)\right)$ -- and in turn this is sent to the complex
\begin{equation} \label{cmplx 1}
\mathbb R\Gamma\left(\pi^0\mathcal X,\Omega^*_{\mathcal X/A}\right)\approx\mathbb R\Gamma\left(X,\Omega^*_{\mathbb R\mathscr O_A\left(x\right)/A}\right)
\end{equation}
which is an honest derived deformation over $A$ of the algebraic De Rham complex $\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)$; on the other side, the vector $x$ is sent to the derivation $l_x$ and -- proceeding down along map $\nu$ -- this determines the complex
\begin{equation} \label{cmplx 2}
\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A,d+l_x\right).
\end{equation}
We claim that complexes \eqref{cmplx 1} and \eqref{cmplx 2} are quasi-isomorphic: more precisely, we assert that the natural zig-zag
\begin{equation} \label{resol}
\xymatrix{\mathbb R\Gamma\left(X,\Omega^*_{\mathbb R\mathscr O_A\left(x\right)/A}\right) & \left(\Gamma\left(X,\mathscr A^{*,*}_X\otimes A\right),\partial + \left(\bar{\partial}+l_x\right)\right)\ar[l] & \left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A,d+l_x\right)\ar@{=}[l] }
\end{equation}
is a chain of quasi-isomorphisms. The right-hand morphism in diagram \eqref{resol} is essentially given by the resolution \eqref{RGamma}. As regards the left-hand one, this is constructed in the following way: consider the standard Dolbeaut resolution $\mathscr A^{0,*}_X\hookleftarrow\mathscr O_X$ and twist it through the derivation $l_x$, so get a map $\mathbb R\mathscr O_A\left(x\right)\leftarrow\mathscr O_X\otimes A$ and hence a morphism $\Omega^*_{\mathbb R\mathscr O_A\left(x\right)/A}\leftarrow\Omega^*_X\otimes A$; now just recall that $\mathscr A^{*,*}_X\simeq\Omega^*_X\otimes_{\mathscr O_X}\mathscr A^{0,*}_X$: this provides us with a natural map $\Omega^*_{\mathbb R\mathscr O_A\left(x\right)}\leftarrow\mathscr A^{*,*}_X\otimes A$, whose globalisation finally gives us the left-hand map in diagram \eqref{resol}.\\
Now denote
\begin{equation*}
\mathbb R\mathscr O_A\left(x\right)\left(n\right):=\left(\mathscr A^{n,*}_X\otimes A,\bar{\partial}+l_x\right)
\end{equation*}
and observe that to show that the zig-zag \eqref{resol} is really a chain of quasi-isomorphisms it suffices to prove that the complexes $\mathbb R\mathscr O_A\left(x\right)\left(n\right)$ and $\Omega^n_{\mathbb R\mathscr O_A\left(x\right)}$ are weakly equivalent. As already done in the proof of Theorem \ref{main}, filter them by powers of the maximal ideal $\mathfrak m_A$, i.e. consider the filtrations\footnote{There is some abuse of notation in these formulae.}
\begin{eqnarray*}
&\mathcal F^p\left(\mathbb R\mathscr O_A\left(x\right)\left(n\right)\right):=\mathfrak m_A^p\mathbb R\mathscr O_A\left(x\right)\left(n\right)\quad\Rightarrow\quad\mathrm{Gr}^p\left(\mathcal F\right)\simeq\mathscr A^{n,*}_X\otimes\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}& \\
&\mathcal F^p\left(\Omega^n_{\mathbb R\mathscr O_A\left(x\right)}\right):=\mathfrak m_A^p\Omega^n_{\mathbb R\mathscr O_A\left(x\right)}\quad\Rightarrow\quad\mathrm{Gr}^p\left(\mathcal F\right)\simeq\Omega^n_{\mathscr A^{0,*}_X}\otimes\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}&
\end{eqnarray*}
which kill the twisting $l_x$. Now observe that
\begin{equation*}
\Omega^n_{\mathscr A^{0,*}}\approx\Omega^n_{\mathscr O_X}\approx\mathscr A_X^{n,*}
\end{equation*}
where the first quasi-isomorphism is induced by the Dolbeaut resolution $\mathscr A^{0,*}_X\hookleftarrow\mathscr O_X$ and the second one is true basically by definition of $\Omega^{n}_{\mathscr O_X}$, so in particular $H^m\left(\Omega^n_{\mathscr A^{0,*}_X}\right)=H^m\left(\mathscr A^{n,*}_X\right)$ for all $m$. Finally, look at the induced spectral sequences: we have
\begin{equation*}
H^{p+q}\left(\mathscr A^{n,*}_X\otimes\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}\right)\simeq\bigoplus_{i+j=p+q}\left(H^i\left(\mathscr A^{n,*}_X\right)\otimes H^j\left(\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}\right)\right)\Longrightarrow H^{p+q}\left(\mathbb R\mathscr O_A\left(x\right)\left(n\right)\right)
\end{equation*}
and
\begin{equation*}
H^{p+q}\left(\Omega^n_{\mathscr A^{0,*}_X}\otimes\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}\right)\simeq\bigoplus_{i+j=p+q}\left(H^i\left(\mathscr A^{n,*}_X\right)\otimes H^j\left(\frac{\mathfrak m_A^p}{\mathfrak m_A^{p+1}}\right)\right)\Longrightarrow H^{p+q}\left(\Omega^n_{\mathbb R\mathscr O_A\left(x\right)}\right)
\end{equation*}
so the complexes $\Omega^n_{\mathbb R\mathscr O_A\left(x\right)}$ and $\mathbb R\mathscr O_A\left(x\right)\left(n\right)$ are quasi-isomorphic as their cohomologies are computed by the same spectral sequence. \\
Now let us look at diagram \eqref{diagr1} on the level of morphisms; a gauge element $\xi$ in the Kodaira-Spencer differential graded Lie algebra associated to $X$ maps through $\mathbb R\mathrm{Def}\left(l\right)$ to $l_{\xi}$, which in turn induces by $\nu$ the morphism of complexes
\begin{equation*}
\mathbb R\Gamma\left(X,e^{l_{\xi}}\right):\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A,d+l_{x_1}\right)\longrightarrow\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A,d+l_{x_2}\right).
\end{equation*}
In a similar way, the gauge $\xi$ determines via $\mu$ the morphism of complexes
\begin{equation*}
e^{\xi}:\mathbb R\mathscr O_A\left(x_1\right)\longrightarrow\mathbb R\mathscr O_A\left(x_2\right)
\end{equation*}
which in turn induces through the bottom arrow in diagram \eqref{diagr1} the morphism
\begin{equation*}
\mathbb R\Gamma\left(X,\Omega^*_{e^{\xi}}\right):\mathbb R\Gamma\left(X,\Omega^*_{\mathbb R\mathscr O_A\left(x_1\right)/A}\right)\longrightarrow\mathbb R\Gamma\left(X,\Omega^*_{\mathbb R\mathscr O_A\left(x_2\right)/A}\right)
\end{equation*}
therefore we end up with a diagram
\begin{equation} \label{diagr3}
\xymatrix@C=1em{ \mathbb R\Gamma\left(X,\Omega^*_{\mathbb R\mathscr O_A\left(x_1\right)/A}\right)\ar[d]^{\mathbb R\Gamma\left(X,\Omega^*_{e^{\xi}}\right)} & \left(\Gamma\left(X,\mathscr A_X^{*,*}\otimes A\right),\partial+\left(\bar{\partial}+l_{x_1}\right)\right)\ar[l]\ar[d]^{\Gamma\left(X,e^{l_{\xi}}\right)} & \left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A,d+l_{x_1}\right)\ar[l]\ar[d]^{\mathbb R\Gamma\left(X,e^{l_{\xi}}\right)} \\
\mathbb R\Gamma\left(X,\Omega^*_{\mathbb R\mathscr O_A\left(x_2\right)/A}\right) & \left(\Gamma\left(X,\mathscr A_X^{*,*}\otimes A\right),\partial+\left(\bar{\partial}+l_{x_2}\right)\right)\ar[l] & \left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A,d+l_{x_2}\right).\ar[l] }
\end{equation}
Notice that the right hand square of diagram \eqref{diagr3} commutes because the morphism $\Gamma\left(X,e^{l_{\xi}}\right)$ is induced by $\mathbb R\Gamma\left(X,e^{l_{\xi}}\right)$ via the standard Dolbeaut resolution; as regards the left hand square, consider for all $n$ the unglobalised diagram
\begin{equation} \label{diagr4}
\xymatrix{\Omega^n_{\mathbb R\mathscr O_A\left(x_1\right)}\ar[d]^{\Omega^n_{e^{\xi}}} & \left(\mathscr A_X^{n,*}\otimes A,\bar{\partial}+l_{x_1}\right)\ar[l]\ar[d]^{e^{l_{\xi}}} \\
\Omega^*_{\mathbb R\mathscr O_A\left(x_2\right)} & \left(\mathscr A_X^{n,*}\otimes A,\bar{\partial}+l_{x_2}\right)\ar[l] }
\end{equation}
and again filter all complexes by powers of the maximal ideal $\mathfrak m_A$ in order to kill the derivations $l_{x_1}$, $l_{x_2}$ and hence the gauge $l_{\xi}$: we end up with a sequence of commutative diagrams
\begin{equation*}
\xymatrix{\Omega^n_{\mathscr A^{0,*}_X}\otimes\frac{\mathbf m_A^p}{\mathbf m_A^{p+1}}\ar@{=}[d] & \left(\mathscr A_X^{n,*}\otimes\frac{\mathbf m_A^p}{\mathbf m_A^{p+1}},\bar{\partial}\right)\ar[l]\ar@{=}[d] \\
\Omega^n_{\mathscr A^{0,*}_X}\otimes\frac{\mathbf m_A^p}{\mathbf m_A^{p+1}} & \left(\mathscr A_X^{n,*}\otimes\frac{\mathbf m_A^p}{\mathbf m_A^{p+1}},\bar{\partial}\right)\ar[l] }
\end{equation*}
therefore diagram \eqref{diagr4} has to commute and so does diagram \eqref{diagr3}, as well. This observation completes the proof.
\end{proof}
\section{The Period Map as a Morphism of Derived Stacks}
Theorem \ref{univ lpm ddf} gives the ultimate picture of the local period map as a deformation-theoretic morphism, since it explains how the Fiorenza-Manetti map lifts naturally to the context of Derived Deformation Theory. Anyway, despite being entirely canonical, the Fiorenza-Manetti-Martinengo map\footnote{Again, Theorem \ref{univ lpm ddf} allows us to drop any further adjective.} is still a local morphism: concretely this means that it provides a fully satisfying description of the behaviour of {} ``derived variations of the Hodge structures'' associated to some nice $k$-scheme $X$ with respect to the infinitesimal derived deformations of the scheme itself, but this map is not able to give us any global information, i.e. it does not provide significant relations between the associated global (derived) moduli stacks. In this section we will set a path towards a coherent notion of global derived period map, which will be further studied in \cite{dNH}.
\subsection{A Non-Geometric Global Period Map}
Fix again $X$ to be a smooth proper scheme over $k$ of dimension $d$ and define the (non-geometric) derived stack
\begin{equation}\label{new sch attempt}
\mathcal{DS}ch^X_{d/k}:=\mathcal{DS}ch_{d/k}\times^h_{\mathbb R\mathcal Perf_k}\left\{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right\}
\end{equation}
where the map defining the homotopy fibre product in formula \eqref{new sch attempt} is
\begin{eqnarray*}
&\mathcal D\mathcal Sch_{d/k}&\xrightarrow{\hspace*{1cm}}\mathbb R\mathcal Perf_k \nonumber\\
&Y&\longmapsto \quad\quad\mathbb R\Gamma\left(\pi^0 Y,\Omega^*_{Y/A}\right)
\end{eqnarray*}
for all $A\in\mathfrak{dgAlg}_k^{\leq 0}$.
\begin{defn}
Define the \emph{non-geometric (universal) global period map} to be the morphism of derived stacks
\begin{eqnarray} \label{final mapping}
\underline{\mathbb R\mathcal P}:&\mathcal D\mathcal Sch^X_{d/k}&\xrightarrow{\hspace*{1cm}}\mathcal{DF}lag^0_k\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right) \nonumber\\
&\left[Y,\theta:\mathbb R\Gamma\left(\pi^0 Y,\Omega^*_{Y/A}\right)\overset{\sim}{\rightarrow}\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\otimes A\right]&\longmapsto\quad\quad\left(\mathbb R\Gamma\left(\pi^0 Y,\Omega^*_{Y/A}\right),\mathcal F\right)
\end{eqnarray}
for all $A\in\mathfrak{dgAlg}_k^{\leq 0}$.
\end{defn}
\begin{prop} \label{very final}
Consider the diagram of derived stacks and derived deformation functors
\begin{equation*}
\xymatrix{\mathcal D\mathcal Sch^X_{d/k}\ar[r]^{\underline{\mathbb R\mathcal P}\qquad\qquad} & \mathcal{DF}lag^0_k\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right) \\
\mathbb R\mathrm{Def}_X\ar[u]\ar[r]^{\mathbb R\mathcal P\quad\quad} & \mathrm{hoFlag}^{\mathcal F}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}\ar[u]}
\end{equation*}
where the right-hand-side vertical arrow is the formal neighbourhood inclusion and the left-hand-side one is the composite
\begin{equation*}
\resizebox{.85\hsize}{!}{\xymatrix{ \mathbb R\mathrm{Def}_X\ar[rrrr]^{\left(\mathrm{Id}_{\mathbb R\mathrm{Def}_X},\mathrm{const}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}\right)\qquad\qquad\qquad\quad} & & & & \mathbb R\mathrm{Def}_X\times^h_{\mathrm{RDef}_{\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)}}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/k}\right)\right)\ar[rrrr]^{\;\;\quad\qquad\qquad{\quad\scriptstyle{formal\;neighbourhood\;inclusion}}} & & & & \mathcal D\mathcal Sch^X_{d/k}.} }
\end{equation*}
Then the above square is well-defined and commutes.
\end{prop}
\begin{proof}
The fact that the diagram is well-defined is precisely the content of Remark \ref{DSch} and Theorem \ref{comp flag fin}.2; the commutativity is readily verified just walking along the arrows, as done in the proof of Theorem \ref{univ lpm ddf}.
\end{proof}
\subsection{Towards a Derived Analytic Period Mapping}
Proposition \ref{very final} is certainly a first step towards a globalisation of Griffiths period map, but it is certainly not sufficiently satisfying since map \eqref{final mapping} is highly non-geometric; let us outline here some of the ideas that are to be developed in \cite{dNH} and which should allow us to identify a reasonable global derived period map in the complex-analytic context. \\
Fix $k$ to be the field of complex numbers $\mathbb C$ and the base scheme $X$ to be smooth and projective; consider a family $p:\mathcal X\rightarrow \mathcal S$ of derived schemes globally deforming it, i.e. a homotopy pull-back diagram
\begin{equation*}
\xymatrix{\ar@{} |{\Box^h}[dr]X\ar@{^{(}->}[r]^i\ar[d] & \mathcal X\ar[d]^p \\
\mathrm{Spec}\left(\mathbb C\right)\ar[r] & \mathcal S}
\end{equation*}
where $p$ is flat and $H^0\left(p\right)$ is smooth. For any point $s\in\mathcal S$, we can consider the (homotopy) fibre of $p$ at $s$, which is a derived scheme we will denote as $\mathcal X_s$. Unlike the infinitesimal case, the associated filtered (derived) De Rham complex $\left(\mathbb R\Gamma\left(\pi^0\mathcal X_s,\Omega^*_{\mathcal X_s/\mathbb C}\right),F^{\bullet}\right)$ does not determine a point in the homotopy flag variety associated to $\mathbb R\Gamma\left(X,\Omega^*_{X/\mathbb C}\right)$, because the group $\Omega\left(\pi^0\mathcal S\right)$ -- where $\Omega\left(\pi^0\mathcal S\right)$ stands for the (simplicial) loop group attached to the topological space $\pi^0\mathcal S$ -- acts by (higher) monodromy on the fibres of $\mathbb Rp_{*}\Omega_{\mathcal X/S}$ and thus on $\mathcal{DF}lag_{\mathbb C}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/\mathbb C}\right)\right)$. Therefore we would like to define the global period map associated to the family $p$ as a morphism of derived stacks
\begin{eqnarray} \label{finalissima}
&\mathcal S&\longrightarrow\nicefrac{\mathcal{DF}lag_{\mathbb C}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/\mathbb C}\right)\right)}{\Omega\left(\pi^0S\right)} \nonumber \\
&s&\longmapsto\left[\left(\mathbb R\Gamma\left(\pi^0\mathcal X_s,\Omega^*_{\mathcal X_s/\mathcal S}\right),F^{\bullet}\right)\right].
\end{eqnarray}
However it is not clear that the quotient $\nicefrac{\mathcal{DF}lag_{\mathbb C}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/\mathbb C}\right)\right)}{\Omega\left(\pi^0\mathcal S\right)}$ exists as a derived geometric stack in the algebraic sense -- and in general we do not expect it to do -- but it is very likely to exist in the analytic setting. As a matter of fact in the underived case the monodromy action on the Grassmannian (or flag variety) is properly discontinuous and this ensures that the quotient has a complex structure; it is then reasonable to expect that the action of $\Omega\left(\pi^0\mathcal S\right)$ on the {} ``analytification'' of $\mathcal{DF}lag_{\mathbb C}\left(\mathbb R\Gamma\left(X,\Omega^*_{X/\mathbb C}\right)\right)$ (or a suitable open substack of it recovering the notion of period domain in the derived setting) should be regular enough for the quotient to exist as a derived analytic stack and map \eqref{finalissima} to be (derived) holomorphic. Thus the ultimate study of the global derived period mapping involves massively the newly-born theory of Derived Analytic Geometry, as well as a good notion of analytification functor in the derived context (see \cite{Lu4}, \cite{Porta1}, \cite{Porta2}, \cite{Porta3}, \cite{PortaYue} for foundational work). The goal of \cite{dNH} is to analyse thoroughly the above ideas.
\section*{Notations and conventions}
\begin{itemize}
\item If $i\geq 0$ $\Delta^i$ is the $i$-th standard simplicial simplex
\item $\mathrm{diag}\left(-\right)=$ diagonal of a bisimplicial set
\item $k=$ fixed field of characteristic $0$, unless otherwise stated
\item If $A$ is a (possibly differential graded) local Artin ring, $\mathfrak m_A$ will be its unique maximal (possibly differential graded) ideal
\item $R=$ fixed (possibly differential graded) commutative unital $k$-algebra, unless otherwise stated
\item If $R$ is a commutative unital ring then $\underline R$ is the constant sheaf of stalk $R$
\item If $\left(V^*,d\right)$ is a cochain complex (in some suitable category) then $\left(V\left[n\right]^*,d_{\left[n\right]}\right)$ will be the cochain complex such that $V\left[n\right]^j:=V^{j+n}$ and $d_{\left[n\right]}^j=d^{j+n}$
\item $\mathbb G_m=$ multiplicative group scheme over $k$
\item $X=$ smooth proper scheme over $k$ of finite dimension, unless otherwise stated
\item $\mathscr O_X=$ structure sheaf of $X$
\item $\mathscr T_X=$ tangent sheaf of $X$
\item $\mathscr A^{0,*}_X=$ {}``Dolbeaut'' complex of $X$
\item $\mathscr A^{*,*}_X=$ double complex of {}``$\bar k$''-valued forms on $X$
\item $\Omega^*_{X/k}=$ algebraic De Rham complex of $X$
\item $F^{\bullet}=$ Hodge filtration on $\Omega^*_{X/k}$ or cohomology, unless otherwise stated
\item $\mathbb L_{X/k}=$ (absolute) cotangent complex of $X$ over $k$
\item $\mathfrak{Sh}\left(X\right)=$ category of sheaves of abelian groups over $X$
\item $\mathrm D\left(X\right)=$ derived category of $X$
\item $\Delta=$ category of finite ordinal numbers
\item $\mathfrak{Alg}_k=$ category of commutative associative unital algebras over $k$
\item $\mathfrak{Aff}_k=$ category of (linear) affine spaces over $k$
\item $\mathfrak{Art}_k=$ category of local Artin algebras over $k$
\item $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Vect}_k\right)=$ model category of chain complexes of vector spaces over $k$ in non-negative degrees
\item $\mathfrak{Ch}_{\geq 0}\left(\mathfrak{Aff}_k\right)$ model category of (chain) $\mathrm{dg}_{\geq 0}$-affine spaces over $k$
\item $\mathfrak{Def}^{\mathrm{Hin}}_k=$ $\infty$-category of Hinich derived deformation functors (over $k$)
\item $\mathfrak{Def}^{\mathrm{Man}}_k=$ $\infty$-category of Manetti extended deformation functors (over $k$)
\item $\mathfrak{dg}_{\geq 0}\mathfrak{Alg}_k=$ model category of (chain) differential graded commutative algebras over $k$ in non-negative degrees
\item $\mathfrak{dg}_{\geq 0}\mathfrak{Cat}_k=$ model category of (chain) differential graded categories over $k$
\item $\mathfrak{dg}_{\geq 0}\mathfrak{Cat}^{\mathfrak{Aff}}_k=$ $\infty$-category of affine (chain) differential graded categories over $k$
\item $\mathfrak{dg}_{\geq 0}\mathfrak{Grpd}^{\mathfrak{Aff}}_k=$ $\infty$-category of affine (chain) differential graded groupoids over $k$
\item $\mathfrak{dgAlg}^{\leq 0}_k=$ model category of (cochain) differential graded commutative algebras over $k$ in non-positive degrees
\item $\mathfrak{dgArt}_k=$ model category of (cochain) differential graded local Artin algebras over $k$
\item $\mathfrak{dgArt}^{\leq 0}_k=$ model category of (cochain) differential graded local Artin algebras over $k$ in non-positive degrees
\item $\mathfrak{dgLie}_k=$ model category of (cochain) differential graded Lie algebras over $k$
\item $\mathfrak{dgMod}_R=$ model category of $R$-modules in (cochain) complexes
\item $\mathfrak{dg}_b\mathfrak{Nil}^{\leq 0}_k=$ $\infty$-category of bounded below differential graded commutative $k$-algebras in non-positive degrees such that the canonical map $A\rightarrow H^0\left(A\right)$ is nilpotent
\item $\mathfrak{dgVect}_k^{\leq 0}=$ model category of (cochain) differential graded vector spaces over $k$ in non-positive degrees
\item $\mathfrak{FdgMod}_R=$ model category of filtered $R$-modules in (cochain) complexes
\item $\mathfrak{Grpd}=$ 2-category of groupoids
\item $\mathfrak{Set}=$ category of sets
\item $\mathfrak{Sch}_k=$ category of schemes over $k$
\item $\mathfrak{sAff}_k=$ model category of simplicial affine spaces over $k$
\item $\mathfrak{sAlg}_k=$ model category of simplicial commutative associative unital algebras over $k$
\item $\mathfrak{sCat}=$ model category of simplicial categories
\item $\mathfrak{sCat}_k=$ $\infty$-category of $k$-simplicial categories over $k$
\item $\mathfrak{sCat}_k^{\mathfrak{Aff}}=$ $\infty$-category of affine simplicial categories over $k$
\item $\mathfrak{sGrpd}=$ model category of simplicial groupoids
\item $\mathfrak{sGrpd}_k^{\mathfrak{Aff}}=$ $\infty$-category of affine simplicial groupoids over $k$
\item $\mathfrak{sSet}=$ simplicial model category of simplicial sets
\item $\mathfrak{sVect}_k=$ model category of simplicial vector spaces over $k$
\item $\mathfrak{Vect}_k=$ category of vector spaces over $k$
\end{itemize} | 16,340 |
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Evening in Brazil
The “Evening in Brazil” ensemble debuted in 2009 in a performance at Utah State University’s new state-of-the-art Performance Hall playing Brazilian music from the Bossa Nova movement. The show was very well received and has now become a yearly late winter, sold-out event.
Performers include Mike Christiansen, Professor Emeritus of Music and Former Director of Guitar studies in the Music Department at Utah State University. He averages over 130 performances annually as a soloist, with Lightwood Duo and with the band, Phase II. Mike is the author/co-author of forty-one books (thirty-seven of which have been published by Mel Bay Publications), has recorded thirty-five CDs, and appears on twenty-eight instructional DVDs. He studied in Brazil with Antonio Adolfo and Thiago Trajano.
Saxophonist and clarinetist Eric Nelson performs regularly with the Lightwood Duo and with the rock/jazz band Phase II. He is active as a freelance musician in the Salt Lake City area, where he performs with the Utah Chamber Artists, Ballet West Orchestra, Contemporary Music Consortium, and the Utah Symphony, and in New York with the Riverside Trio and Wagner/Nelson Chamber Jazz. He also teaches music in the public schools.
Christopher Neale, (formerly Professor of Irrigation Engineering at Utah State University) is now Director of Research at the Daugherty Water for Food Institute at the University of Nebraska. He is a native of Brazil where he completed degrees in Civil Engineering and Classical Guitar. He studies contemporary Brazilian music and provides the Brazilian guitar beat to the ensemble’s music.
Linda Linford was born in Boston, and grew up on the East Coast in a bi-lingual household before coming to school at Utah State University. She has been singing and playing piano since she was 6. In high school, she participated in classical vocal competitions and had the opportunity to sing in Carnegie Hall. She came to USU to continue studying vocal performance and pedagogy and graduated in 2013.
The ensemble also includes Lars Yorgason, a highly sought-after freelance bassist in the state of Utah. He has performed with many of the all-time jazz greats in America. Lars has taught jazz at the University of Utah, Weber State University, Utah State University and 15 years at Brigham Young University.
Don Keipp, on percussion, is Professor Emeritus of Music at Weber State University where he directed numerous bands and ensembles. His musical studies have taken him to many countries allowing him to expand his collection of world percussion instruments. As a freelance performer, Keipp has performed with the Salt Lake City Ballet Orchestra, the New American Symphony, the Utah Symphony, the Ogden Concert Band, and the Crestmark Orchestra in Logan. He performs regularly with the Joe McQueen Quartet and the Junction City Big Band.
Travis Taylor is an Ecologist employed by a local environmental consulting company in Cache Valley. He grew up in Ogden, UT where he studied drums with many renowned northern Utah percussion instructors, including EIB’s own Dr. Don Keipp. Travis has occupied the drum throne for a variety of well-known northern Utah area bands, including the regional bluegrass/rock/jam band Tanglewood, which toured throughout the inter-mountain west, west coast and Pacific Northwest. After moving to Logan, UT in 2003 to study Ecology, he was fortunate enough to perform with many Utah State University jazz ensembles and combos including Dr. Larry Smith’s Jazz Kicks Band. He continues to perform regularly with a number of talented musicians and musical educators and is always quick to say yes to the next gig.
The evening’s program will include a number of classic works by Antônio Carlos Jobim (1927-1994) the Grammy Award-winning Brazilian songwriter, composer, arranger, singer, pianist and guitarist. Music from other composers from the Bossa Nova era will also be presented as well as from other contemporary Brazilian composers, exploring different regional rhythms of Brazil.PURCHASE VIP TICKETS
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The Beacon Middle School girls' track team, coming off a second-place finish at the April 20 Blades invitational, rolled west to Bridgeville to take on Phillis Wheatley at the Woodbridge Sports Complex. The Beacon girls won nine events and pulled away late in the meet to win 72-37 while the Phillis Wheatley team won both relays and the long and high jumps.
Sixth-grader Logan Shuttleworth again led the way with a three-win performance of 15 points, taking the 1,600 meters in 5:55, the 800m in 2:51, and the triple jump in 27-feet-2-inches. Sprinter Miquaine Hollins had a big day, winning the 100m in 13.1 and the 200m in 28.5, and came right back in the 4-by-200m relay, running a tough third leg as the relay team set a season best of 1:58.4. Allison Palmer won the 400m in 1:08.6 to remain unbeaten all season, while also finishing second in the long jump in 13-1-1/2 . Annie Judge won the 55m hurdles in 10.1, while Mozella Matthews won both the shot put and the discus events.
The Beacon boys lost to Phillis Wheatley by a close score of 56-53 as they were led by Mike Williams and Dylan Alves, who scored 13 points each. Alves broke the school record in the 110m hurdles, running 15.1; he also won the triple jump in 31-11 and finished second at 200m in 26.3. Williams won the shot put in 37-3, the discus in 99-6, and was second in the 100m in 12.2. Ben Bamforth easily won the 1,600m in 5:33, while Seth DePrince captured the 800 meters in 2:35.
Beacon will host Dover Central on the Cape track Wednesday, May 2, before prepping for the Cape Invitational May 4 at Legends Stadium.
Blades Invitational Highlights - Mariner Middle School athlete Suprenia West-Burton won the triple jump in the 31-foot range and broke the school record in the process. Mike Williams of Beacon finished second in the 100m in a school record of 12.1. Sixth-grader Logan Shuttleworth of Beacon smashed the meet record in the 1,600m by nearly 20 seconds and lowered her own school record with one of the nation's top sixth-grade times of 5:38. Allison Palmer continued her 400m dash win streak as she ran away with the event in a time of 1:06.5. The girls' team from Beacon was second overall with 103 points, while the boys claimed fifth place with 52 points.
Mud Run - The annual Quest Fitness Mud Run, one of the most popular local events in the area with nearly 500 participants expected, will take place Sunday, April 29, at The Farm, just off Route 16 heading west toward Ellendale. Action begins at 9 a.m. There will be male, female and coed team races as well as individual races for males and females, and a youth race to close out the day. Registration is closed and packet pickup is scheduled for 5 to 8 p.m., Saturday, April 28, at Quest Fitness, Villages of Five Points, Lewes. There will be no race-day registration. Finish medals will be presented to all athletes, while awards in 10-year age groups will also be presented to the athletes in individual divisions. Parking is very limited, and runners are encouraged to carpool as much as possible to the event. The gate to The Farm will open for athletes at 7:30 a.m. See you at The Mud!
Upcoming races
8 a.m., Saturday, April 28 -12th Oxford Day 10K Run, Oxford, Md.
9 a.m., Sunday, April 29 - Second Quest Fitness Mud Run, The Farm, Milton.
9 a.m., Saturday, May 5 - Second Roady 5K Run/Walk, Delaware Tech, Georgetown.
7 p.m., Saturday, May 5 - First Shadow Series El Cinco 5-Miler; Cape Henlopen State Park,... | 335,077 |
\begin{document}
\title{\bf Convergence in probability of an ergodic and conformal multi-symplectic numerical scheme for a damped stochastic NLS equation}
\author{
{ Jialin Hong\footnotemark[1], Lihai Ji\footnotemark[2], and Xu Wang\footnotemark[3]}\\
}
\maketitle
\footnotetext{\footnotemark[1]\footnotemark[3]Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R.China. J. Hong and X. Wang are supported by National Natural Science Foundation of China (NO. 91530118, NO. 91130003, NO. 11021101 and NO. 11290142).}
\footnotetext{\footnotemark[2]Institute of Applied Physics and Computational Mathematics, Beijing 100094, P.R.China. L. Ji is supported by National Natural Science Foundation of China (NO. 11471310, NO. 11601032).}
\footnotetext{\footnotemark[3]Corresponding author: [email protected].}
\begin{abstract}
In this paper, we investigate the convergence order in probability of a novel ergodic numerical scheme for damped stochastic nonlinear Schr\"{o}dinger equation with an additive noise. Theoretical analysis shows that our scheme is of order one in probability under appropriate assumptions for the
initial value and noise. Meanwhile, we show that our scheme possesses the unique ergodicity and preserves the discrete conformal multi-symplectic conservation law. Numerical experiments are given to show the longtime behavior of the discrete charge and the time average of the numerical solution, and to test the convergence order, which verify our theoretical results. \\
\textbf{AMS subject classification: }{\rm\small37M25, 60H35, 65C30, 65P10.}\\
\textbf{Key Words: }{\rm\small}Stochastic nonlinear Schr\"{o}dinger equation, fully discrete scheme, ergodicity, conformal multi-symplecticity, charge exponential evolution, convergence order
\end{abstract}
\section{Introduction}
We consider the following weakly damped stochastic nonlinear Schr\"odinger (NLS) equation with an additive noise (see also \cite{CHW16,DO05})
\begin{equation}\label{model}
\left\{
\begin{aligned}
&d\psi-\bi (\Delta\psi+\bi\alpha \psi+\lambda|\psi|^2\psi)dt=\epsilon QdW,\quad t\ge0,\;x\in[0,1]\subset\R,\\
&\psi(t,0)=\psi(t,1)=0,\\
&\psi(0,x)=\psi_0(x)
\end{aligned}
\right.
\end{equation}
with a complex-valued Wiener process $W$ defined on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},\P)$, a linear positive operator $Q$ on $L^2:=L^2(0,1)$ and $\lambda=\pm1$. $\alpha>0$ is the absorption coefficient, $\epsilon\ge0$ describes the size of the noise. In addition, $Q$ is assumed to commute with $\Delta$ and satisfies $Qe_k=\sqrt{\eta_k}e_k$, which $\{e_k\}_{k\ge1}$ is an eigenbasis of $\Delta$ with Dirichlet boundary condition in $L_0^2:=L^2_0(0,1)$. Throughout the paper, the subscript $0$ represents the homogenous boundary condition. Some additional assumptions on $Q$ will be given bellow. The Karhunen--Lo\`eve expansion yields
\begin{align*}
QW(t,x)=\sum_{k=0}^{\infty}\sqrt{\eta_k} e_k(x)\beta_k(t),\quad t\geq0,\quad x\in[0,1],
\end{align*}
where $\beta_k=\beta_k^1+\bi\beta_k^2$ with $\beta_k^1$ and $\beta_k^2$ being its real and imaginary parts, respectively. In addition, we assume that $\{\beta_k^i\}_{k\ge1,i=1,2}$ is a family of $\R$-valued independent identified Brownian motions.
As is well known, for the deterministic cubic NLS equation, the charge of the solution is a constant, i.e., $\|\psi(t)\|_{L^{2}}=\|\psi_{0}\|_{L^{2}}$. However, for \eqref{model} with a damped term and an additive noise in addition, the charge is no longer preserved and satisfies (see Proposition 2.1 in Section 2)
\begin{align}\label{ergodic}
\E\|\psi(t)\|_{L^2}^2=e^{-2\alpha t}\E\|\psi_0\|_{L^2}^2+\frac{\epsilon^{2}\eta}{\alpha}(1-e^{-2\alpha t}),
\end{align}
where $\eta:=\sum_{k=1}^{\infty}\eta_k<\infty$. From this equation, it can be seen that the damped term is necessary to ensure the uniform boundedness of the solution in stochastic case, which also ensures the existence of invariant measures. Indeed, if $\alpha=0$ and $\epsilon\neq0$, the $L^2$ norm grows linearly in time. Moreover, if there exists a unique invariant measure $\mu$ which satisfies
\begin{align}
\lim_{T\to\infty}\frac1T\int_0^T\E f(\psi(s))ds=\int_{H_0^1}fd\mu,\quad\forall~ f\in C_b(H_0^1)
\end{align}
with $H^1_0:=H^1_0(0,1)$, we say that the random process $\psi(t)$ is uniquely ergodic \cite{DO05}. The interested readers are referred to \cite{daprato,DO05} and references therein for the study of ergodicity with respect to the exact solution of stochastic PDEs. We also refer to \cite{abdulle,mattingly,talay,T02} for the study of ergodicity as well as approximate error with respect to numerical solutions of stochastic ODEs and to \cite{brehier,brehier2,brehier3} for those of parabolic stochastic PDEs. For damped stochastic NLS equation \eqref{model}, the authors in \cite{DO05} prove both the existence and the uniqueness of the invariant measure, and \cite{CHW16} presents an ergodic fully discrete scheme which is of order $2$ in space for $\lambda=0$ or $\pm1$ and order $\frac12$ in time for $\lambda=0$ or $-1$ in the weak sense.
The main goal of this work is to construct a fully discrete scheme of \eqref{model} which could inherit both the unique ergodicity and some other internal properties of the original equation, e.g., conformal multi-symplectic property (see \cite{MNS13} for a detailed description in deterministic case), and to give the optimal convergence order of the proposed scheme in probability. To this end, we first apply the central finite difference scheme to \eqref{model} in spatial direction to get a semi-discretized equation, whose solution is shown to be symplectic and uniformly bounded. Then, a splitting technique is used to discretize the semi-discretized equation and obtain an explicit fully discrete scheme. We show that the proposed scheme possesses a conformal multi-symplectic conservation law with its solution uniformly bounded. Thanks to the non-degeneracy of the additive noise, the numerical solution is also shown to be irreducible and strong Feller, which yields the uniqueness of the invariant measure.
Due to the fact that the nonlinear term of \eqref{model} is not global Lipschitz, it is particularly challenging and difficulty to analyze the convergence order of the proposed scheme. Motivated by \cite{BD06,L13}, we construct a truncated equation with a global Lipschitz nonlinear term such that the proposed scheme applied to the truncated equation shows order one in mean-square sense under appropriate hypothesis on initial value and noise. We then construct a submartingale based on which we finally derive convergence order one in probability for the original equation in temporal direction. To the best of our knowledge, there has been no work in the literature which constructs schemes with both ergodicity and conformal multi-symplecticity to \eqref{model}.
The rest of the paper is organized as follows. We show the conformal multi-symplecticity and the charge evolution for \eqref{model} in section 2. In section 3, we construct a fully discrete scheme, which could inherit both the ergodicity and the conformal multi-symplecitcity of the original system. In section 4, we introduce a truncated equation, based on which we derive convergence order one in probability for the proposed scheme. Numerical experiments are carried out in section 5 to verify our theoretical results.
\section{Damped stochastic NLS equation}
This section is devoted to investigate the internal properties of \eqref{model}. We define the space-time white noise $\dot{\chi}=\frac{dW}{dt}$, set $\psi=p+\bi q$, $\dot{\chi}=\dot{\chi}_1+\bi\dot{\chi}_2$ with $p$, $q$, $\dot{\chi}_1=\frac{dW_1}{dt}$ and $\dot{\chi}_2=\frac{dW_2}{dt}$ being real-valued functions, and rewrite \eqref{model} as
\begin{equation}\label{pq}
\left\{
\begin{aligned}
\begin{split}
p_t+q_{xx}+\alpha p+\lambda(p^2+q^2)q&=\epsilon Q\dot{\chi}_1,\\[2mm]
-q_t+p_{xx}-\alpha q+\lambda(p^2+q^2)p&=-\epsilon Q\dot{\chi}_2.
\end{split}
\end{aligned}
\right.
\end{equation}
Denoting $v=p_x$, $w=q_x$, $z=(p,q,v,w)^T$, above equations can be transformed into a compact form
\begin{align}\label{multisym}
Md_tz+K\partial_xzdt=-\alpha Mzdt+\nabla S_0(z)dt+\nabla S_1(z)\circ dW_1+\nabla S_2(z)\circ dW_2,
\end{align}
where
\begin{equation*}M=
\left(
\begin{array}{cccc}
0&-1&0&0\\
1&0&0&0\\
0& 0&0 &0\\
0&0&0&0\\
\end{array}
\right),\quad K=
\left(
\begin{array}{cccc}
0&0&1&0\\
0&0&0&1\\
-1& 0&0 &0\\
0&-1&0&0\\
\end{array}
\right)
\end{equation*}
and
\begin{equation*}
S_0(z)=-\frac{\lambda}4(p^2+q^2)^2-\frac12(v^2+w^2),\quad
S_1(z)=\epsilon Qq,\quad S_2(z)=-\epsilon Qp.
\end{equation*}
In the sequel, we use the notations $L^2:=L^2(0,1)$, $H^p:=H^p(0,1)$ and denote the domain of operators $\Delta^{\frac{p}2}$ with Dirichlet boundary condition by
\begin{align*}
\dot{H}^p:=D(\Delta^{\frac{p}2})=\left\{u\in L^2_0\bigg{|}\|u\|_{\dot{H}^p}:=\|\Delta^{\frac{p}2}u\|_{L^2}=\sum_{k=1}^{\infty}(k\pi)^p|(u,e_k)|^2\le\infty\right\},\quad p\ge1
\end{align*}
with $(u,v):=\int_0^1u(x)\overline{v}(x)dx$ for all $u,v\in L^2$ and $e_k(x)=\sqrt{2}\sin(k\pi x)$. Furthermore, we denote the set of Hilbert-Schimdt operators from $L^2$ to $\dot{H}^p$ by $\mathcal{L}_2^p$ with norm $$\|Q\|_{\mathcal{L}_2^p}:=\sum_{k=1}^{\infty}\|Qe_k\|^2_{\dot{H}^p},\quad p\ge1.$$
Without pointing out, the equations below hold in the sense $\P$-a.s. We now prove that \eqref{model} possesses the stochastic conformal multi-symplectic structure, whose definition is also given in the following theorem.
\begin{tm}
Eq. \eqref{model} is a stochastic conformal multi-symplectic Hamiltionian system, and preserves the stochastic conformal multi-symplectic conservation law
\begin{align*}
d_t\omega(t,x)+\partial_x\kappa(t,x)dt=-\alpha\omega(t,x)dt,
\end{align*}
which means
\begin{equation}\label{2forms}
\begin{split}
\int_{x_0}^{x_1}\omega(t_1,x)dx
-\int_{x_0}^{x_1}\omega(t_0,x)dx
&+\int_{t_0}^{t_1}\kappa(t,x_1)dt
-\int_{t_0}^{t_1}\kappa(t,x_0)dt\\
&=-\int_{x_0}^{x_1}\int_{t_0}^{t_1}\alpha\omega(t,x)dtdx,
\end{split}
\end{equation}
where $\omega=\frac12dz\wedge Mdz$ and $\kappa=\frac12dz\wedge Kdz$ are two differential 2-forms associated with two skew-symmetric matrices $M$ and $K$.
\end{tm}
\begin{proof}
To simplify the proof, we denote $(z_1,z_2,z_3,z_4):=(p,q,v,w)=z^T$ and $(z_l)_t^x:=z_l(t,x)$ for $l=1,2,3,4.$
Noticing that
$\omega=dz_2\wedge dz_1$
and
$\kappa=dz_1\wedge dz_3+dz_2\wedge dz_4$,
thus we have
\begin{equation}\label{H}
\begin{split}
&\int_{x_0}^{x_1}\omega(t_1,x)dx
-\int_{x_0}^{x_1}\omega(t_0,x)dx\\
=&\int_{x_0}^{x_1}\Big{[}d(z_2)_{t_1}^x\wedge d(z_1)_{t_1}^x-d(z_2)_{t_0}^x\wedge d(z_1)_{t_0}^x\Big{]}dx\\
=&\int_{x_0}^{x_1}\Bigg{[}\left(\sum_{l=1}^4\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}d(z_l)_{t_0}^{x_0}\right)\wedge\left(\sum_{i=1}^4\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}d(z_i)_{t_0}^{x_0}\right)\\
&-\left(\sum_{l=1}^4\frac{\partial(z_2)_{t_0}^x}{\partial(z_l)_{t_0}^{x_0}}d(z_l)_{t_0}^{x_0}\right)\wedge\left(\sum_{i=1}^4\frac{\partial(z_1)_{t_0}^x}{\partial(z_i)_{t_0}^{x_0}}d(z_i)_{t_0}^{x_0}\right)\Bigg{]}dx\\
=&\sum_{l=1}^4\sum_{i=1}^4\left[\int_{x_0}^{x_1}\left(\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}
-\frac{\partial(z_2)_{t_0}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_0}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dx\right]
d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}\\
=&:\sum_{l=1}^4\sum_{i=1}^4\mathcal{H}_{l,i}(t_1,x_1)
d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0},
\end{split}
\end{equation}
where $\mathcal{H}_{l,i}(t_1,x_1)=\int_{x_0}^{x_1}\left(\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}
-\frac{\partial(z_2)_{t_0}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t_0}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dx$.
Similarly, we obtain
\begin{equation}\label{M}
\begin{split}
&\int_{t_0}^{t_1}\kappa(t,x_1)dt
-\int_{t_0}^{t_1}\kappa(t,x_0)dt\\
=&\sum_{l=1}^4\sum_{i=1}^4\Bigg{[}\int_{t_0}^{t_1}\Bigg{(}
-\frac{\partial(z_1)_{t}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t}^{x_1}}{\partial(z_l)_{t_0}^{x_0}}
+\frac{\partial(z_1)_{t}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}\\
&-\frac{\partial(z_2)_{t}^{x_1}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}
+\frac{\partial(z_2)_{t}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}\Bigg{)}dt\Bigg{]}
d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}\\
=&:\sum_{l=1}^4\sum_{i=1}^4\mathcal{M}_{l,i}(t_1,x_1)d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}
\end{split}
\end{equation}
and
\begin{align}
\int_{x_0}^{x_1}\int_{t_0}^{t_1}\alpha\omega(t,x)dtdx
=&2\alpha\sum_{l=1}^4\sum_{i=1}^4\left[\int_{x_0}^{x_1}\int_{t_0}^{t_1}\left(\frac{\partial(z_2)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_1)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dtdx\right]
d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}\nonumber\\\label{N}
=&:2\alpha\sum_{l=1}^4\sum_{i=1}^4\mathcal{N}_{l,i}(t_1,x_1)d(z_l)_{t_0}^{x_0}\wedge d(z_i)_{t_0}^{x_0}.
\end{align}
Adding \eqref{H}, \eqref{M} and \eqref{N} together, we can find out that equation \eqref{2forms} holds if
\begin{align}\label{HMN}
\mathcal{H}_{l,i}(t_1,x_1)+\mathcal{M}_{l,i}(t_1,x_1)+2\alpha\mathcal{N}_{l,i}(t_1,x_1)=0
\end{align}
for any $l,i=1,2,3,4$, $t_1\in\R_+$ and $x_1\in\R$.
In fact, rewritting \eqref{pq} as
\begin{equation*}
\left\{
\begin{aligned}
d_{t}z_1=&-\partial_x(z_4)dt-\alpha z_1dt+\frac{\partial S_0(z)}{\partial z_2}dt+\epsilon QdW_1,\\
d_{t}z_2=&\partial_x(z_3)dt-\alpha z_2dt-\frac{\partial S_0(z)}{\partial z_1}dt+\epsilon QdW_2
\end{aligned}
\right.
\end{equation*}
and taking partial derivatives with respect to $(z_i)_{t_0}^{x_0}$ and $(z_l)_{t_0}^{x_0}$ respectively, we have
\begin{equation*}
\left\{
\begin{aligned}
d_{t}\frac{\partial(z_1)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}=&-\frac{\partial}{\partial x}\frac{\partial(z_4)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}dt-\alpha \frac{\partial(z_1)_{t}^x}{\partial(z_i)_{t_0}^{x_0}}dt+\sum_{l=1}^4\frac{\partial S_1(z)}{\partial (z_2)_{t}^x\partial(z_l)_{t}^{x}}\frac{\partial(z_l)_t^x}{\partial(z_i)_{t_0}^{x_0}}dt,\\
d_{t}\frac{\partial(z_2)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}=&\frac{\partial}{\partial x}\frac{\partial(z_3)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}dt-\alpha \frac{\partial(z_2)_{t}^x}{\partial(z_l)_{t_0}^{x_0}}dt-\sum_{i=1}^4\frac{\partial S_1(z)}{\partial (z_1)_t^x\partial(z_i)_{t}^{x}}\frac{\partial(z_i)_{t}^{x}}{\partial(z_l)_{t_0}^{x_0}}dt.
\end{aligned}
\right.
\end{equation*}
Furthermore,
\begin{align*}
d_{t_1}\mathcal{H}_{l,i}(t_1,x_1)=&\int_{x_0}^{x_1}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}d_{t_1}\left(\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\right)+\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}d_{t_1}\left(\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dx\\
=&\int_{x_0}^{x_1}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\left(\frac{\partial}{\partial x}\frac{\partial(z_3)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}-\alpha \frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\right)dt_1dx\\
&-\int_{x_0}^{x_1}\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}\left(\frac{\partial}{\partial x}\frac{\partial(z_4)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}+\alpha \frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\right)dt_1dx\\
=&\frac{\partial(z_1)_{t_1}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t_1}^{x_1}}{\partial(z_l)_{t_0}^{x_0}}
-\frac{\partial(z_1)_{t_1}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_3)_{t_1}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}
-\frac{\partial(z_2)_{t_1}^{x_1}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t_1}^{x_1}}{\partial(z_i)_{t_0}^{x_0}}\\
&+\frac{\partial(z_2)_{t_1}^{x_0}}{\partial(z_l)_{t_0}^{x_0}}\frac{\partial(z_4)_{t_1}^{x_0}}{\partial(z_i)_{t_0}^{x_0}}
-2\alpha\int_{x_0}^{x_1}\frac{\partial(z_1)_{t_1}^x}{\partial(z_i)_{t_0}^{x_0}}\frac{\partial(z_2)_{t_1}^x}{\partial(z_l)_{t_0}^{x_0}}dxdt_1\\
=&-d_{t_1}\mathcal{M}_{l,i}(t_1,x_1)-2\alpha d_{t_1}\mathcal{N}_{l,i}(t_1,x_1),
\end{align*}
which together with the fact that $\mathcal{H}_{l,i}(t_0,x_1)+\mathcal{M}_{l,i}(t_0,x_1)+2\alpha\mathcal{N}_{l,i}(t_0,x_1)=0$
yields \eqref{HMN}.
We hence complete the proof.
\end{proof}
Next, we show that the charge of the solution $\psi(t)$, although it is not conserved anymore, satisfies an exponential type evolution law.
\begin{prop}\label{exactmoment}
Assume that $\E\|\psi_0\|_{L^2}^2<\infty$, then the solution of \eqref{model} is uniformly bounded with
\begin{align}\label{bounded}
\E\|\psi(t)\|_{L^2}^2=e^{-2\alpha t}\E\|\psi_0\|_{L^2}^2+\frac{\epsilon^{2}\eta}{\alpha}(1-e^{-2\alpha t}).
\end{align}
\end{prop}
\begin{proof}
The It\^o's formula applied to $\|\psi(t)\|_{L^2}$ yields
\begin{align*}
d\|\psi(t)\|^2_{L^2}=-2\alpha\|\psi(t)\|^2_{L^2}dt+2\epsilon\Re\left[\int_0^1\overline{\psi}QdxdW\right]+2\epsilon^{2}\eta dt,
\end{align*}
where $\Re[\cdot]$ denotes the real part of a complex value.
Taking expectation on both sides of above equation and solving the ordinary differential equation, we derive
\begin{equation*}
\begin{split}
\E\|\psi(t)\|^2_{L^2}
=&e^{-2\alpha t}\left(\int_0^t2\epsilon^{2}\eta e^{2\alpha s}ds+\E\|\psi_0\|^2_{L^2}\right)\\
&=e^{-2\alpha t}\E\|\psi_0\|_{L^2}^2+\frac{\epsilon^{2}\eta}{\alpha}(1-e^{-2\alpha t}).
\end{split}
\end{equation*}
This concludes the proof.
\end{proof}
We hence get the conclusion that, the damped stochastic NLS equation \eqref{model} possesses the
stochastic conformal multi-symplectic conservation law \eqref{2forms} with its solution being uniformly bounded \eqref{bounded}, as well as the unique ergodicity \cite{DO05}. A natural question is how to construct numerical
schemes which could inherit the properties of \eqref{model} as many as possible, such as, stochastic conformal multi-symplectic conservation law, uniform boundedness of the solution and the unique ergodicity.
\section{Ergodic fully discrete scheme}
We now focus on the construction of schemes which could inherit the stochastic conformal multi-symplecticity and the unique ergodicity.
\subsection{Spatial semi-discretization}
We apply central finite difference scheme to \eqref{model} and obtain
\begin{equation}\label{fenliang}
\left\{
\begin{aligned}
&d\psi_j-\bi \left(\frac{\psi_{j+1}-2\psi_j+\psi_{j-1}}{h^2}+\bi\alpha\psi_j+\lambda|\psi_j|^2\psi_j\right)dt=\epsilon\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)d\beta_k(t),\\
&\psi_0(t)=\psi_{J+1}(t)=0,\\
&\psi_j(0)=\psi_0(x_j),
\end{aligned}
\right.
\end{equation}
where $h$ is the uniform spatial step and $\psi_j:=\psi_j(t)$ is an approximation of $\psi(x_j,t)$ with $x_j=jh$, $j=1,2,\cdots,J$ and $(J+1)h=1$. With notations $\Psi=(\psi_1,\cdots,\psi_J)^T\in\C^J$, $\beta=(\beta_1,\cdots,\beta_P)^T\in\C^P$, $F(\Psi)=
\text{diag}\{|\psi_1|^2,\cdots,|\psi_J|^2\}$, $\Lambda=\text{diag}\{\sqrt{\eta_1^{}},\cdots,\sqrt{\eta_P^{}}\}$,
\begin{equation*}
A=\left(
\begin{array}{cccc}
-2&1 & & \\
1&-2&1 & \\
&\ddots &\ddots &\ddots \\
& &1 &-2\\
\end{array}
\right)\quad\text{and}\quad
\sigma=
\left(
\begin{array}{ccc}
e_1^{}(x_1^{})&\cdots &e_P^{}(x_1^{})\\
\vdots& &\vdots\\
e_1^{}(x_J^{})&\cdots&e_P^{}(x_J^{})
\end{array}
\right),
\end{equation*}
we rewrite \eqref{fenliang} into a finite dimensional stochastic differential equation
\begin{equation}\label{space}
\left\{
\begin{aligned}
&d\Psi-\bi \left(\frac1{h^2}A\Psi+\bi\alpha\Psi+\lambda F(\Psi)\Psi\right)dt=\epsilon\sigma\Lambda d\beta,\\
&\Psi(0)=(\psi_0(x_1^{}),\cdots,\psi_0(x_J^{}))^T.
\end{aligned}
\right.
\end{equation}
In the sequel, we denote the 2-norm for vectors or matrices by $\|\cdot\|$, i.e., $\|v\|=\left(\sum_{j=1}^J|v_j|^2\right)^{1/2}$ for a vector $v=(v_1,\cdots,v_J)^T\in\C^J$ and $\|A\|$=`the square root of the maximum eigenvalues of $A^TA$' for a matrix A. The solution of \eqref{space} is uniformly bounded, which is stated in the following proposition.
\begin{prop}\label{semimoment}
Assume that $\E\|\psi_0\|_{L^2}^2<\infty$, then the solution $\Psi$ of \eqref{space} is uniformly bounded with
\begin{align}
h\E\|\Psi(t)\|^2\le e^{-2\alpha t}h\E\|\Psi(0)\|^2+\frac{2\epsilon^2\eta^{(P)}}{\alpha}(1-e^{-2\alpha t}),
\end{align}
where $\eta^{(P)}:=\sum_{k=1}^P\eta_k^{}$.
\end{prop}
\begin{proof}
Similar to the proof of Proposition \ref{exactmoment}, we apply It\^o's formula to $\|\Psi(t)\|^2$ and obtain
\begin{equation}\label{semiito}
\begin{split}
d\|\Psi(t)\|^2&=2\Re[\overline{\Psi}^Td\Psi]+(\epsilon\sigma\Lambda d\overline{\beta})^T(\epsilon\sigma\Lambda d\beta)\\[1mm]
&=-2\alpha\|\Psi(t)\|^2dt+2\Re[\overline{\Psi}^T\epsilon\sigma\Lambda d\beta]\\
&+\epsilon^2\sum_{j=1}^J\left[\left(\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)d\overline{\beta_k}\right)^T\left(\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)d{\beta}_k\right)\right].
\end{split}
\end{equation}
Taking expectation on both sides of above equation leads to
\begin{align*}
d\E\|\Psi(t)\|^2=&-2\alpha \E\|\Psi(t)\|^2dt+2\epsilon^2\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)dt.
\end{align*}
Thus, multiplying above equation by $he^{2\alpha t}$ and taking integral from $0$ to $t$ leads to
\begin{align*}
\int_0^the^{2\alpha t}d\E\|\Psi(t)\|^2+\int_0^t2\alpha he^{2\alpha t}\E\|\Psi(t)\|^2dt
=2\epsilon^2h\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)\int_0^te^{2\alpha t}dt.
\end{align*}
Based on the fact that $\sum_{j=1}^Je^2_k(x_j)\le2J\le2h^{-1}$, we have
\begin{equation}\label{charge}
\begin{split}
e^{2\alpha t}h\E\|\Psi(t)\|^2-h\E\|\Psi(0)\|^2
=&\frac{\epsilon^2h}{\alpha}(e^{2\alpha t}-1)\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)\\
\le& \frac{2\epsilon^2\eta^{(P)}}{\alpha}(e^{2\alpha t}-1)
\end{split}
\end{equation}
which completes the proof.
In addition, noticing that for $h=1/J$ and $x_j=jh,$ $j=1,\cdots,J$,
\begin{equation*}
\begin{split}
\E\|\psi_0\|_{L^2}^2=&\E\sum_{j=1}^J\int_{x_{j-1}}^{x_j}|\psi_0(x)|^2dx\\
&=\E\sum_{j=1}^J|\psi_0(x_j)|^2h+O(h)=h\E\|\Psi(0)\|^2+O(h),
\end{split}
\end{equation*}
we get the uniform boundedness under the assumption $\E\|\psi_0\|_{L^2}^2<\infty$.
\end{proof}
\begin{rk}\label{symeuler}
Scheme \eqref{fenliang} is equivalent to the symplectic Euler scheme applied to \eqref{multisym}, i.e.,
\begin{equation*}\label{space2}
\left\{
\begin{aligned}
&p_{j+1}-p_j=hv_{j+1},\\[4mm]
&q_{j+1}-q_j=hw_{j+1},\\
&v_{j+1}-v_j=h(q_j)_t+\alpha hq_j-h((p_j)^2+(q_j)^2)p_j-\epsilon\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)\frac{d\beta^2_k(t)}{dt},\\
&w_{j+1}-w_j=-h(p_j)_t-\alpha hp_j-h((p_j)^2+(q_j)^2)q_j+\epsilon\sum_{k=1}^P\sqrt{\eta_k}e_k(x_j)\frac{d\beta^1_k(t)}{dt}.
\end{aligned}
\right.
\end{equation*}
\end{rk}
\subsection{Full discretization}
To construct a fully discrete scheme, which could inherit the properties of \eqref{model}, we are motivated by splitting techniques. We drop the linear terms and stochastic term for the moment and consider the following equation
\begin{align}\label{nonlinear}
d\Psi(t)-\bi \lambda F(\Psi(t))\Psi(t)dt=0
\end{align}
first. Multiplying $\overline{F(\Psi(t))}$ to both sides of \eqref{nonlinear} and taking the imaginary part, we obtain $\|\Psi(t)\|^2=\|\Psi(0)\|^2$, which implies that $F(\Psi(t))=F(\Psi(0))$. Thus, \eqref{nonlinear} is shown to possess a unique solution $\Psi(t)=e^{\bi\lambda F(\Psi(0))t}\Psi(0)$.
For linear equation
\begin{align*}
d\Psi(t)-\bi \left(\frac1{h^2}A\Psi(t)+\bi\alpha\Psi(t)\right)dt=\epsilon\sigma\Lambda d\beta,
\end{align*}
a modified mid-point scheme is applied to obtain its full discretization.
Now we can define the following splitting schemes initialized with $\Psi^0=\Psi(0)$,
\begin{align}\label{full1}
&\Psi^{n+1}=e^{-\frac12\alpha\tau}\tilde{\Psi}^n+\bi\frac{\tau}{h^2}A\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}\tilde{\Psi}^n}2-\frac12\alpha\tau\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}\tilde{\Psi}^n}2+\epsilon\sigma\Lambda\delta_{n+1}\beta,\\\label{full2}
&\tilde{\Psi}^n=e^{\bi\lambda F(\Psi^n)\tau}\Psi^n,
\end{align}
where $\Psi^n=(\psi_1^n,\cdots,\psi_J^n)^T\in\C^{J}$, $\tau$ denotes the uniform time step, $\delta_{n+1}\beta=\beta(t_{n+1})-\beta(t_n)$ and $t_n=n \tau$, $n\in\N$.
Noticing that schemes \eqref{full1}--\eqref{full2} can be rewritten as
\begin{equation}\label{full}
\begin{split}
\Psi^{n+1}-e^{f(\Psi^n)}\Psi^n=&\bi\frac{\tau}{2h^2}A\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)\\
&-\frac14\alpha\tau\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)+\epsilon\sigma\Lambda\delta_{n+1}\beta,
\end{split}
\end{equation}
which can also be expressed in the following explicit form
\begin{equation}\label{explicitscheme}
\begin{split}
\Psi^{n+1}=&\left(I-\frac{\bi\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\left(I+\frac{\bi\tau}{2h^2}A-\frac14\alpha\tau I\right)e^{f(\Psi^n)}\Psi^n\\
&+\left(I-\frac{\bi\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\epsilon\sigma\Lambda\delta_{n+1}\beta
\end{split}
\end{equation}
with $I$ denoting the identity matrix and $f(\Psi^n)=\left(-\frac12\alpha I+\bi\lambda F(\Psi^n)\right)\tau$. Thus, there uniquely exists a family of $\{\mathcal{F}_{t_n}\}_{n\ge1}$ adapted solutions $\{\Psi^n\}_{n\ge1}$ of \eqref{full} for sufficiently small $\tau$.
As for the proposed splitting schemes, \eqref{full2} coincides with the exact solution of the Hamiltonian system $d\Psi(t)-\bi \lambda F(\Psi(t))\Psi(t)dt=0$. It then suffices to show that \eqref{full1} possesses the conformal multi-symplectic conservation law, which is stated in the following theorem.
\begin{tm}
Scheme \eqref{full1} possesses the discrete conformal multi-symplectic conservation law
\begin{align*}
&e^{-\alpha\tau}\frac{dz_j^{n+1}\wedge Mdz_j^{n+1}-dz_j^{n}\wedge Mdz_j^{n}}{\tau}+\frac{dz_j^{n+\frac12}\wedge(K_1dz_{j+1}^{n+\frac12}-K_2dz_{j-1}^{n+\frac12})}{h}\\
=&-\frac12\alpha dz_j^{n+\frac12}\wedge Mdz_j^{n+\frac12}
\end{align*}
with $z_j^n=(p_j^n,q_j^n,v_j^n,w_j^n)^T$, $z_j^{n+\frac12}=\frac12(z_j^{n+1}+e^{-\frac12\alpha\tau}z_j^n)$,
\begin{equation*}K_1=
\left(
\begin{array}{cccc}
0&0&1&0\\
0&0&0&1\\
0& 0&0 &0\\
0&0&0&0\\
\end{array}
\right),\quad K_2=
\left(
\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
-1& 0&0 &0\\
0&-1&0&0\\
\end{array}
\right)
\end{equation*}
and $K_1+K_2=K$.
\end{tm}
\begin{proof}
We denote $\tilde{\Psi}^n$ by $\Psi^n$ for convenience in this proof, and we have
\begin{align*}
\Psi^{n+1}=e^{-\frac12\alpha\tau}{\Psi}^n+\bi\frac{\tau}{h^2}A\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}{\Psi}^n}2-\frac12\alpha\tau\frac{\Psi^{n+1}+e^{-\frac12\alpha\tau}{\Psi}^n}2+\epsilon\sigma\Lambda\delta_{n+1}\beta
\end{align*}
with $\Psi^n=\left(\psi_1^n,\cdots,\psi_J^n\right)^T\in\C^J$.
Denote $\psi_j^n:=p_j^n+\bi q_j^n$ with its real part $p_j^n$ and imaginary part $q_j^n$, $\delta_{n+1}\beta=\delta_{n+1}\beta^1+\bi\delta_{n+1}\beta^2$, $v_{j+1}^{n}:=(p_{j+1}^{n}-p_j^n)h^{-1}$ and $w_{j+1}^{n}:=(q_{j+1}^{n}-q_j^n)h^{-1}$.
Noticing that the $j$-th component of $h^{-2}A\Psi^n$ can be expressed as $h^{-1}(v_{j+1}^n-v_j^n)+\bi h^{-1}(w_{j+1}^n-w_j^n)$, we decompose \eqref{full1} with its real and imaginary parts respectively and derive
\begin{equation*}\label{theoformula}
\left\{
\begin{aligned}
\frac{p_j^{n+1}-e^{-\frac12\alpha\tau}p_j^n}{\tau}+&\frac{w_{j+1}^{n+1}-w_j^{n+1}}{2h}+e^{-\frac12\alpha\tau}\frac{w_{j+1}^{n}-w_j^{n}}{2h}\\
=&-\frac14\alpha(p_j^{n+1}+e^{-\frac12\alpha\tau}p_j^n)+\epsilon\sigma\Lambda\delta_{n+1}\beta^1,\\
\frac{q_j^{n+1}-e^{-\frac12\alpha\tau}q_j^{n}}{\tau}-&\frac{v_{j+1}^{n+1}-v_j^{n+1}}{2h}-e^{-\frac12\alpha\tau}\frac{v_{j+1}^{n}-v_j^{n}}{2h}\\
=&-\frac14\alpha(q_j^{n+1}+e^{-\frac12\alpha\tau}q_j^n)+\epsilon\sigma\Lambda\delta_{n+1}\beta^2.
\end{aligned}
\right.
\end{equation*}
Combining formulae $v_{j+1}^{n}=(p_{j+1}^{n}-p_j^n)h^{-1}$, $w_{j+1}^{n}=(q_{j+1}^{n}-q_j^n)h^{-1}$ with above equations, we get
\begin{align*}
M\frac{z_j^{n+1}-e^{-\frac12\alpha\tau}z_j^n}{\tau}+K_1\frac{z_{j+1}^{n+\frac12}-z_j^{n+\frac12}}{h}
+K_2\frac{z_{j}^{n+\frac12}-z_{j-1}^{n+\frac12}}{h}
=-\frac12\alpha Mz_j^{n+\frac12}+\xi_j^{n+\frac12},
\end{align*}
where $\xi_j^{n+\frac12}:=(-\epsilon\sigma\Lambda\delta_{n+1}\beta^2,\epsilon\sigma\Lambda\delta_{n+1}\beta^1,-v_j^{n+\frac12},-w_j^{n+\frac12})^T$.
Taking differential in phase space on both sides of above equation, and performing wedge product with $dz_j^{n+\frac12}$ respectively, we show the discrete conformal multi-symplectic conservation law based on the symmetry of matrix $-K_1+K_2$ and the fact $dz_j^{n+\frac12}\wedge(-K_1+K_2)dz_j^{n+\frac12}=0$, $dz_j^{n+\frac12}\wedge d\xi_j^{n+\frac12}=0$.
\end{proof}
\begin{rk}
It is also feasible to show that schemes \eqref{full1}--\eqref{full2} are conformal symplectic in temporal direction, which together with Remark \ref{symeuler}, yields the conformal multi-symplecticity of the fully discrete scheme \eqref{full}.
\end{rk}
\begin{prop}\label{fullmoment}
Assume that $\E\|\psi_0\|_{L^2}^2<\infty$, $Q\in\mathcal{HS}(L^2,\dot{H}^2)$ and $P\le C_*(J+1)$ for some constant $C_*\ge1$, then the solution $\{\Psi^n\}_{n\ge1}$ of \eqref{full} is uniformly bounded, i.e.,
\begin{align}
h\E\|\Psi^n\|^2\le e^{-\alpha t_n}h\E\|\Psi^0\|^2+C
\end{align}
with $t_n=n\tau$ and constant $C$ depending on $\alpha,\epsilon,Q$ and $C_*$.
\end{prop}
\begin{proof}
We multiply $\overline{\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)}^T$ to \eqref{full}, take the real part and expectation, and obtain
\begin{equation}\label{moment}
\begin{split}
&\E\|\Psi^{n+1}\|^2-e^{-\alpha\tau}\E\|\Psi^n\|^2\\
=&-\frac14\alpha\tau\E\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2+\E\left[\Re\left[\overline{\left(\Psi^{n+1}-e^{f(\Psi^n)}\Psi^n\right)}^T\epsilon\sigma\Lambda\Delta_{n+1}\beta\right]\right]\\
=&-\frac14\alpha\tau\E\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2
+\E\Bigg{[}\Re\Bigg{[}\bigg{(}-\bi\frac{\tau}{2h^2}A\overline{\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)}\\
&-\frac14\alpha\tau\overline{\left(\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\right)}+\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta}\bigg{)}^T\epsilon\sigma\Lambda\Delta_{n+1}\beta\Bigg{]}\Bigg{]}\\
\le&-\frac14\alpha\tau\E\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2+\frac18\alpha\tau\E\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2\\
&+C\tau\E\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
+\frac18\alpha\tau\E\|\Psi^{n+1}+e^{f(\Psi^n)}\Psi^n\|^2\\
&+C\tau\E\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+\E\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2.
\end{split}
\end{equation}
For the smooth functions $e_k(x)$, we have
\begin{align*}
\left|\Delta e_k(x_j)-\frac{e_k(x_{j+1})-2e_k(x_j)+e_k(x_{j-1})}{h^2}\right|\le Ck^4h^2\le Ck^2,\quad k\ge1
\end{align*}
based on the fact $kh\le P(J+1)^{-1}\le C_*$.
Thus,
\begin{equation}\label{sto1}
\begin{split}
&\E\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
=\epsilon^2\sum_{j=1}^J\E\left|\sum_{k=1}^P\sqrt{\eta_k^{}}\frac{e_k(x_{j+1})-2e_k(x_j)+e_k(x_{j-1})}{h^2}\Delta_{n+1}\beta_k\right|^2\\
&\le2\epsilon^2\sum_{j=1}^J\sum_{k=1}^P\eta_k^{}\left(|\Delta e_k(x_j)|+Ck^2\right)^2\tau
\le CJ\tau\sum_{k=1}^Pk^4\eta_k
\le Ch^{-1}\tau.
\end{split}
\end{equation}
In the last step, we have used the fact $\sum\limits_{k=1}^Pk^4\eta_k\le C\|Q\|_{\mathcal{L}_2^2}\le C$. Similarly,
\begin{align}\label{sto2}
\E\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
=\epsilon^2\sum_{j=1}^J\E\left|\sum_{k=1}^P\sqrt{\eta_k^{}}e_k(x_j)\Delta_{n+1}\beta_k\right|^2
\le CJ\eta\tau
\le Ch^{-1}\tau.
\end{align}
Substituting \eqref{sto1} and \eqref{sto2} into \eqref{moment} and multiplying the result by $h$, we get
\begin{align*}
h\E\|\Psi^{n+1}\|^2\le e^{-\alpha\tau}h\E\|\Psi^n\|^2+C\tau
\le e^{-\alpha t_{n+1}}h\E\|\Psi^0\|^2+C\tau\frac{1-e^{-\alpha t_n}}{1-e^{-\alpha\tau}},
\end{align*}
which, together with the fact $\frac{1-e^{-\alpha t_n}}{1-e^{-\alpha\tau}}\le\frac1{1-(1-\alpha\tau)}=\frac1{\alpha\tau}$,
completes the proof.
\end{proof}
\iffalse
\begin{prop}
Under the assumptions in Proposition \ref{fullmoment} and $\psi_0\in\dot{H}^2$, we have in addition that
\begin{align}\label{2norm}
h\E\left\|\frac1{h^2}A\Psi^{n}\right\|^2\le e^{-\alpha t_n}h\|\psi_0\|^2_{\dot{H}^2}+C\tau^{-1}.
\end{align}
\end{prop}
\begin{proof}
In this proof, we denote $a:=\Psi^{n+1}$ and $b:=e^{f(\Psi^n)}\Psi^n$ for convenience. Multiplying $\frac1{h^2}A\overline{\left(a-b\right)}$ to the transpose of \eqref{full} and taking both the imaginary part and the expectation, we obtain
\begin{align}
&\E\left\|\frac1{h^2}Aa\right\|^2-\E\left\|\frac1{h^2}Ab\right\|^2
=\E\left[\Im\left[ \frac{\alpha}2\left(a+b\right)^T\frac1{h^2}A\left(\overline{a-b}\right)-\frac2{\tau}\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^T\frac1{h^2}A\left(\overline{a-b}\right)\right]\right]\nonumber\\
=&\E\left[\frac{\alpha}2\Im\left[(a+b)^T\frac1{h^2}A\left(-\bi\frac{\tau}{2h^2}A\left(\overline{a+b}\right)-\frac{\alpha\tau}4\left(\overline{a+b}\right)+\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta}\right)\right]\right]\nonumber\\
&-\E\left[\frac{2}{\tau}\Im\left[\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^T\frac1{h^2}A\left(-\bi\frac{\tau}{2h^2}A\left(\overline{a+b}\right)-\frac{\alpha\tau}4\left(\overline{a+b}\right)+\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta}\right)\right]\right]\nonumber\\
=&-\frac{\alpha\tau}4\E\left\|\frac1{h^2}A(a+b)\right\|^2
+\E\left[\frac{\alpha}{2h^2}\Im\left[(a+b)^TA(\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta})\right]\right]\nonumber\\
&+\E\left[\Re\left[\frac1{h^4}\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA^2\left(\overline{a+b}\right)\right]\right]
+\E\left[\frac{\alpha}{2h^2}\Im\left[\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA\left(\overline{a+b}\right)\right]\right]\nonumber\\\label{eq1}
\le&\frac1{\alpha\tau}\E\left\|\frac1{h^2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\right\|^2\le Ch^{-1}.
\end{align}
based on the symmetry of matrix $A$ and \eqref{sto1}, where $\Im[\cdot]$ denotes the imaginary part of a complex number and we have used the fact that
\begin{align*}
\Im\left[(a+b)^TA(\overline{\epsilon\sigma\Lambda\Delta_{n+1}\beta})+\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA\left(\overline{a+b}\right)\right]=0
\end{align*}
and
\begin{align*}
\Re\left[\frac1{h^4}\left(\epsilon\sigma\Lambda\Delta_{n+1}\beta\right)^TA^2\left(\overline{a+b}\right)\right]
\le \frac{\alpha\tau}4\left\|\frac1{h^2}A(a+b)\right\|^2
+\frac1{\alpha\tau}\left\|\frac1{h^2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\right\|^2.
\end{align*}
Noticing that
\begin{align*}
\left\|\frac1{h^2}Ab\right\|^2
=e^{-\alpha\tau}\left\|\frac1{h^2}Ae^{\bi\lambda\tau F(\Psi^n)}\Psi^n\right\|^2
\le e^{-\alpha\tau}\left\|Ae^{\bi\lambda\tau F(\Psi^n)}A^{-1}\right\|^2\left\|\frac1{h^2}A\Psi^n\right\|^2,
\end{align*}
where {\color{red}$\left\|Ae^{\bi\lambda\tau F(\Psi^n)}A^{-1}\right\|^2\le\|A\|^2\left\|e^{\bi\lambda\tau F(\Psi^n)}\right\|_2^2\|A^{-1}\|^2=$}, and that
\begin{align*}
\left\|\frac1{h^2}A\Psi^0\right\|^2,
\end{align*}
so we conclude from \eqref{eq1} that
\begin{align*}
\left\|\frac1{h^2}A\Psi^{n+1}\right\|^2
\le e^{-\alpha\tau}\left\|\frac1{h^2}A\Psi^n\right\|^2+Ch^{-1}
\le e^{-\alpha t_{n+1}}\|\psi_0\|^2_{\dot{H}^2}+Ch^{-1}\tau^{-1},
\end{align*}
which complete the proof.
\end{proof}
\fi
\begin{tm}
Under the assumptions in Proposition \ref{fullmoment} and $\eta_k>0$ for $k=1,\cdots,P$. The solution $\{\Psi^n\}_{n\ge1}$ of \eqref{full} is uniquely ergodic with a unique invariant measure, denoted by $\mu^{\tau}_h$, satisfying
\begin{align}
\lim_{N\to\infty}\frac1N\sum_{n=0}^{N-1}\E f(\Psi^n)=\int_{\C^J}fd\mu_h^{\tau},\quad\forall\;f\in C_b(\C^J).
\end{align}
\end{tm}
\begin{proof}
To show the existence of the invariant measures, we'll use a useful tool called Lyapunov function. For any fixed $h>0,$ we choose $V(\cdot):=h\|\cdot\|^2$ as the Lyapunov function, which satisfies that the level sets $K_c:=\{u\in\C^J:V(u)\le c\}$ are compact for any $c>0$ and $\E[V(\Psi^n)]\le V(\Psi^0)+C$ for any $n\in\N$. Thus, the Markov chain $\{\Psi^n\}_{n\in\N}$ possesses an invariant measure (see Proposition 7.10 in \cite{daprato}).
Now we show that $\{\Psi^n\}_{n\in\N}$ is irreducible and strong Feller (also known as the minorization condition in Assumption 2.1 of \cite{mattingly}), which yields the uniqueness of the invariant measure. In fact, for any $u,v\in\C^J$, we can derive from \eqref{full} that $\delta_1\beta$ can be chosen as
\begin{align*}
\epsilon\sigma\Lambda\delta_{1}\beta=v-e^{f(u)}u-\bi\frac{\tau}{2h^2}A\left(v+e^{f(u)}u\right)+\frac14\alpha\tau\left(v+e^{f(u)}u\right)
\end{align*}
such that $\Psi^0=u,\Psi^1=v$, where we have used the fact that $\sigma$ is full rank and $\Lambda$ is invertible.
Thus, we can conclude based on the homogenous property of the Markov chain $\{\Psi^n\}_{n\in\N}$ that the transition kernel
$P_n(u,A):=\P(\Psi^n\in A|\Psi^0=u)>0$, which implies the irreducibility of the chain.
On the other hand, as $\delta_1\beta$ has $C^{\infty}$ density, it follows from \eqref{explicitscheme} that $\Psi^1$ also has $C^{\infty}$ density for any deterministic initial value $\Psi^0=\Psi(0)$. Then explicit construction shows that $\{\Psi^n\}_{n\in\N}$ possesses a family of $C^{\infty}$ density and is strong Feller.
\end{proof}
The theorems above are evidently consistent with the continuous results \eqref{ergodic}, \eqref{2forms} and \eqref{bounded}, respectively. The next result concerns the error estimation of the proposed scheme, where the truncation technique will be used to deal with the non-global Lipschitz nonlinearity.
\section{Convergence order in probability}
In this section, we focus on the approximate error for the proposed scheme in temporal direction. As the nonlinear term is not global Lipschitz, we consider the following truncated function first
\begin{align}\label{truncate}
d\Psi_R-\bi\left(\frac1{h^2}A\Psi_R+\bi\alpha\Psi_R+\lambda F_R(\Psi_R)\Psi_R\right)dt=\epsilon\sigma\Lambda d\beta,
\end{align}
with $\Psi_R:=\Psi_R(t)=(\psi_{R,1}^{}(t),\cdots,\psi_{R,J}^{}(t))^T$ and initial value $\Psi_R(0)=\Psi(0)$. Here $F_R(v)=\theta\left(\frac{\|v\|}{R}\right)F(v)$ for any vector $v\in\C^J$ and a cut-off function $\theta\in C^{\infty}(\R)$ satisfying $\theta(x)=1$ for $x\in[0,1]$ and $\theta(x)=0$ for $x\ge2$ (see also \cite{BD06,L13}). In addition, we have
\begin{align*}
\|F_R(\Psi_R)\|=\theta\left(\frac{\|\Psi_R\|}{R}\right)\max_{1\le j\le J}|\psi_{R,j}^{}|^2
\le\theta\left(\frac{\|\Psi_R\|}{R}\right)\|\Psi_R\|^2
\le4R^2.
\end{align*}
As a result, the nonlinear term $F_R(\Psi_R)\Psi_R$ is global Lipschitz with respect to the norm $\|\cdot\|$.
The proposed scheme \eqref{explicitscheme} applied to the truncated equation \eqref{truncate} yields the following scheme
\begin{equation}\label{truncatescheme}
\begin{split}
\Psi_R^{n+1}=&\left(I-\frac{\bi\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\left(I+\frac{\bi\tau}{2h^2}A-\frac14\alpha\tau I\right)e^{f_R(\Psi_R^n)}\Psi_R^n\\
&+\left(I-\frac{\bi\tau}{2h^2}A+\frac14\alpha\tau I\right)^{-1}\epsilon\sigma\Lambda\Delta_{n+1}\beta,
\end{split}
\end{equation}
where $f_R(\Psi_R^n)=\left(-\frac12\alpha I+\bi\lambda F_R(\Psi_R^n)\right)\tau$ and $\Psi_R^n=(\psi_{R,1}^n,\cdots,\psi_{R,J}^n)^T$.
\begin{tm}\label{trunerror}
Consider Eq. \eqref{truncate} and the scheme \eqref{truncatescheme}. Assume that $\E\|\psi_0\|_{L^2}^2<\infty$, $Q\in\mathcal{HS}(L^2,\dot{H}^2)$, $\alpha\ge\frac12$ and $\tau=O(h^4)$. For $T=N\tau$, there exists a constant $C_R$ which depends on $\alpha,\epsilon,R,Q,\psi_0$ and is independent of $T$ and $N$ such that
\begin{align*}
h\E\|\Psi_R(T)-\Psi_R^N\|^2\le C_R\tau^2.
\end{align*}
\end{tm}
\begin{proof}
Denote semigroup operator $S(t):=e^{Bt}$ which is generated by the linear operator $B:=\bi\frac1{h^2}A-\frac{\alpha}2I$, then the mild solution of \eqref{space} is
\begin{equation}\label{psit}
\begin{split}
\Psi_R(t_{n+1})=&S(\tau)\Psi(t_n)+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\bi\lambda F_R(\Psi_R(s))\Psi_R(s)ds\\
&-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac{\alpha}2\Psi_R(s)ds+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\epsilon\sigma\Lambda d\beta(s).
\end{split}
\end{equation}
Subtracting \eqref{truncatescheme} from \eqref{psit}, we obtain
\begin{equation*}\label{strongerror}
\begin{split}
&\Psi_R(t_{n+1})-\Psi_R^{n+1}\\
=&S(\tau)\Psi_R(t_n)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)e^{f_R(\Psi_R^n)}\Psi_R^n\\
&+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\bi\lambda F_R(\Psi_R(s))\Psi_R(s)ds-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac{\alpha}2\Psi_R(s)ds\\
&+\int_{t_n}^{t_{n+1}}\left(S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\right)\epsilon\sigma\Lambda d\beta(s)\\
=&S(\tau)\left(\Psi_R(t_n)-\Psi_R^n\right)+\left[S(\tau)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\right]\Psi_R^n\\
&+\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\left(\Psi_R^n-e^{f_R(\Psi_R^n)}\Psi_R^n\right)\\
&+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\bi\lambda F_R(\Psi_R(s))\Psi_R(s)ds-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac12\alpha\Psi_R(s)ds\\
&+\int_{t_n}^{t_{n+1}}\left(S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\right)\epsilon\sigma\Lambda d\beta(s)\\
=&:\uppercase\expandafter{\romannumeral1}+\uppercase\expandafter{\romannumeral2}+\uppercase\expandafter{\romannumeral3}+\uppercase\expandafter{\romannumeral4}+\uppercase\expandafter{\romannumeral5}+\uppercase\expandafter{\romannumeral6}.
\end{split}
\end{equation*}
To show the strong convergence order of \eqref{truncatescheme}, we give the estimates of above terms, respectively.
For terms $\uppercase\expandafter{\romannumeral1}$ and $\uppercase\expandafter{\romannumeral2}$, we have
\begin{align}\label{term1}
\E\|\uppercase\expandafter{\romannumeral1}\|^2
=\E\left\|e^{(\bi\frac1{h^2}A-\frac{\alpha}2I)\tau}(\Psi_R(t_n)-\Psi_R^n)\right\|^2
=e^{-\alpha\tau}\E\|\Psi_R(t_n)-\Psi_R^n\|^2
\end{align}
and
\begin{equation}\label{term2}
\begin{split}
\E\|\uppercase\expandafter{\romannumeral2}\|^2
\le C\E\|(B\tau)^3\Psi_R^n\|^2
&\le C\tau^6\|B^3\|^2\E\|\Psi_R^n\|^2\\[2mm]
&\le Ch^{-13}\tau^6\|A\|^6
\le Ch^{-13}\tau^6
\end{split}
\end{equation}
based on $\left|e^x-(1-\frac{x}2)^{-1}(1+\frac{x}2)\right|=O(x^3)$ as $x\to0$ and Proposition \ref{fullmoment}. In the last step of \eqref{term2}, we also used the fact that $\|A\|$ is uniformly bounded for any dimension $J$, whose proof is not difficult and is given in the Appendix for readers' convenience.
For term $\uppercase\expandafter{\romannumeral6}$,
Taylor expansion yields that
\begin{equation}
\begin{split}
\E\|\uppercase\expandafter{\romannumeral6}\|^2
&\le2\E\left\|\int_{t_n}^{t_{n+1}}\left(S(t_{n+1}-s)-S(\tau)\right)\epsilon\sigma\Lambda d\beta(s)\right\|^2\\
&+2\E\left\|\left(S(\tau)-\left(I-\frac12B\tau\right)^{-1}\right)\epsilon\sigma\Lambda\Delta_{n+1}\beta\right\|^2\\[3mm]
&\le C\tau^2\E\|B\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2
\le Ch^{-1}\tau^3.
\end{split}
\end{equation}
It then remains to estimate terms $\uppercase\expandafter{\romannumeral3}$, $\uppercase\expandafter{\romannumeral4}$ and $\uppercase\expandafter{\romannumeral5}$.
We can obtain the following equation in the same way as that of \eqref{moment}
\begin{align*}
\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\le C\tau\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+C\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2.
\end{align*}
Multiplying above equation by $\|\Psi_R^{n+1}\|^2$, we get
\begin{equation*}
\begin{split}
&\|\Psi_R^{n+1}\|^4+\left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)^2-e^{-2\alpha\tau}\|\Psi_R^n\|^4\\
\le& C\tau\left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+C\tau e^{-\alpha\tau}\|\Psi_R^n\|^2\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2\\
&+C\left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2+C e^{-\alpha\tau}\|\Psi_R^n\|^2\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^2\\
\le& \left(\|\Psi_R^{n+1}\|^2-e^{-\alpha\tau}\|\Psi_R^n\|^2\right)^2
+\tau e^{-2\alpha\tau}\|\Psi_R^n\|^4+C\tau\|h^{-2}A\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^4+\frac{C}{\tau}\|\epsilon\sigma\Lambda\Delta_{n+1}\beta\|^4.
\end{split}
\end{equation*}
Based on \eqref{sto1} and \eqref{sto2}, we take expectation of above equation and derive
\begin{align*}
\E\|\Psi_R^{n+1}\|^4
&\le(1+\tau)e^{-2\alpha\tau}\E\|\Psi_R^n\|^4+Ch^{-2}\tau\\[2mm]
&\le(1+\tau)^{n+1}e^{-2\alpha\tau (n+1)}\E\|\Psi_R^0\|^4+Ch^{-2}
\le Ch^{-2}
\end{align*}
for $\alpha\ge\frac12$, where in the last step we have used the fact that $\E\|\Psi_R^0\|^4\le(\E\|\Psi_R^0\|^2)^2\le Ch^{-2}$ and $(1+\tau)e^{-2\alpha\tau}<1$ for $\alpha\ge\frac12$.
Similarly, we derive
$\E\|\Psi_R^n\|^8\le Ch^{-4},$
which implies that
\begin{align*}
\E\|F_R(\Psi_R^n)\|^4=\E\left(\sum_{j=1}^J\left|\psi^n_{R,j}\right|^4\right)^2\le\E\|\Psi_R^n\|^8\le Ch^{-4},\quad\forall~n\in\N.
\end{align*}
Thus, by Taylor expansion, we have
\begin{equation}\label{345}
\begin{split}
&\uppercase\expandafter{\romannumeral3}+\uppercase\expandafter{\romannumeral4}+\uppercase\expandafter{\romannumeral5}=\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\Big{(}-f_R(\Psi_R^n)+O(f(\Psi_R^n)^2)\Big{)}\Psi_R^n\\
&+\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\bi\lambda F_R(\Psi_R(s))\Psi_R(s)ds
-\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\frac12\alpha\Psi_R(s)ds\\
=&\bi\lambda \int_{t_n}^{t_{n+1}}\left[S(t_{n+1}-s)F_R(\Psi_R(s))\Psi_R(s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)F_R(\Psi_R^n)\Psi_R^n\right]ds\\
&-\frac12\alpha\int_{t_n}^{t_{n+1}}\left[S(t_{n+1}-s)\Psi_R(s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\Psi_R^n\right]ds\\
&+\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)O(f_R(\Psi_R^n)^2)\Psi_R^n\\
:=&\tilde{\uppercase\expandafter{\romannumeral3}}+\tilde{\uppercase\expandafter{\romannumeral4}}+\tilde{\uppercase\expandafter{\romannumeral5}}.
\end{split}
\end{equation}
Now we estimate above terms respectively. For $\tilde{\uppercase\expandafter{\romannumeral3}}$, we have
\begin{align*}
\tilde{\uppercase\expandafter{\romannumeral3}}
=& \bi\lambda\int_{t_n}^{t_{n+1}}\left[S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\right]F_R(\Psi_R^n)\Psi_R^nds\nonumber\\
&+ \bi\lambda\int_{t_n}^{t_{n+1}}S(t_{n+1}-s)\Big{[}F_R(\Psi_R(s))F_R(\Psi_R(s))-F_R(\Psi_R^n)\Psi_R^n\Big{]}ds\nonumber\\
=:&\tilde{\uppercase\expandafter{\romannumeral3}}_1+\tilde{\uppercase\expandafter{\romannumeral3}}_2,
\end{align*}
which satisfies
\begin{align*}
\E\|\tilde{\uppercase\expandafter{\romannumeral3}}_1\|^2
\le& \tau\int_{t_n}^{t_{n+1}}\E\left\|\left[S(t_{n+1}-s)-\left(I-\frac12B\tau\right)^{-1}\left(I+\frac12B\tau\right)\right]F_R(\Psi_R^n)\Psi_R^n\right\|^2ds\\
\le& Ch^{-12}\tau^8\E\|F_R(\Psi_R^n)\Psi_R^n\|^2
\le Ch^{-12}\tau^8\E\left[\sum_{j=1}^J\left|\psi_{R,j}^n\right|^6\right]
\le Ch^{-15}\tau^8
\end{align*}
and
\begin{align*}
\E\|\tilde{\uppercase\expandafter{\romannumeral3}}_2\|^2
\le& \tau\int_{t_n}^{t_{n+1}}\E\Big{\|}S(t_{n+1}-s)\Big{[}F_R(\Psi_R(s))F_R(\Psi_R(s))-F_R(\Psi_R^n)\Psi_R^n\Big{]}\Big{\|}^2ds\\
\le&C\tau\int_{t_n}^{t_{n+1}}\E\|\Psi_R(s)-\Psi_R(t_n)\|^2ds+C\tau^2e^{-\alpha\tau}\E\|\Psi_R(t_n)-\Psi_R^n\|^2.
\end{align*}
Noticing that
\begin{align*}
\E\|\Psi_R(s)-\Psi_R(t_n)\|^2
=&\E\bigg{\|}\int_{t_n}^sS(s-r)\bi\lambda F_R(\Psi_R(r))\Psi_R(r)dr-\int_{t_n}^sS(s-r)\frac{\alpha}2\Psi_R(r)dr\\
&+\int_{t_n}^sS(s-r)\epsilon\sigma\Lambda d\beta(r)\bigg{\|}^2
\le h^{-3}\tau^2,
\end{align*}
thus
\begin{align}
\E\|\tilde{\uppercase\expandafter{\romannumeral3}}\|^2
\le Ch^{-15}\tau^8+Ch^{-3}\tau^4+C\tau^2e^{-\alpha\tau}\E\|\Psi_R(t_n)-\Psi_R^n\|^2.
\end{align}
Term $\tilde{\uppercase\expandafter{\romannumeral4}}$ can be estimated in the same way as the estimation of $\tilde{\uppercase\expandafter{\romannumeral3}}$. Term $\tilde{\uppercase\expandafter{\romannumeral5}}$ turns to be
\begin{equation}\label{term5}
\begin{split}
\E\|\tilde{\uppercase\expandafter{\romannumeral5}}\|^2
\le& C\E\left\|f_R(\Psi_R^n)^2\Psi_R^n\right\|^2
\le C\tau^4\E\left[\sup_{1\le j\le J}\left|-\frac12\alpha+\bi\lambda|\psi_{R,j}^n|^2\right|^4\|\Psi_R^n\|^2\right]\\
\le& C\tau^4\left(\E\left(\sum_{j=1}^J\left|\psi^n_{R,j}\right|^2\right)^8\right)^{\frac12}\left(\E\|\Psi_R^n\|^4\right)^{\frac12}
\le Ch^{-5}\tau^4.
\end{split}
\end{equation}
From \eqref{term1}--\eqref{term5}, we conclude
\begin{align*}
&h\E\|\Psi_R(t_{n+1})-\Psi_R^{n+1}\|^2\\[2mm]
\le& h(1+C\tau^2)e^{-\alpha\tau}\E\|\Psi_R(t_{n})-\Psi_R^{n}\|^2+C\tau^3+Ch^{-4}\tau^4+Ch^{-12}\tau^6+Ch^{-14}\tau^8\\[2mm]
\le& C\tau^{2}+Ch^{-4}\tau^{3}+Ch^{-12}\tau^5+Ch^{-14}\tau^7
\le C\tau^2,
\end{align*}
where in the last two steps we have used the fact that $(1+C\tau^2)e^{-\alpha\tau}<1$ for $\tau$ is sufficiently small and $\tau=O(h^4)$.
\end{proof}
Based on the estimates on truncated equation and its numerical scheme, we are now in the position to give the approximate error between $\Psi(t)$ and $\Psi^n$. The proof of following theorem is motivated by \cite{BD06,L13} and holds for any fixed $T>0$ without other restrictions.
\begin{tm}\label{probability}
Consider Eq. \eqref{space} and scheme \eqref{explicitscheme}. Assume that $\E\|\psi_0\|_{L^2}^2<\infty$, $Q\in\mathcal{HS}(L^2,\dot{H}^2)$, $\alpha\ge\frac12$ and $\tau=O(h^4)$. For any $T>0$, we derive convergence order one in probability, i.e.,
\begin{align}
\lim_{K\to\infty}\P\left(\sup_{1\le n\le[T/\tau]}\sqrt{h}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)=0.
\end{align}
\end{tm}
\begin{proof}
For any $\gamma\in(0,1)$,
we define
$n_{\gamma}:=\inf\{1\le n\le[T/\tau]:\|\Psi(t_n)-\Psi^n\|\ge\gamma\}$ and then deduce that
\begin{align*}
&\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\\
\subset&\Bigg{[}\left(\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|\ge R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\right)\\
&\cup\left(\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\right)\Bigg{]}\\
\subset&\Bigg{[}\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|\ge R-1\right\}\\
&\cup
\left(\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\right)\Bigg{]}.
\end{align*}
If $\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}$ happens, it is easy to show that
$\|\Psi^k\|\le\|\Psi(t_k)-\Psi^k\|+\|\Psi(t_k)\|<R-1+\gamma<R$, $F_R(\Psi_R^k)=F(\Psi_R^k)$, $\Psi_R^k=\Psi^k$ for $k=0,1,\cdots,n_\gamma-1$ and $\Psi_R(t_n)=\Psi(t_n)$ for $0\le n\le n_\gamma$.
Furthermore, comparing scheme \eqref{truncatescheme} with \eqref{explicitscheme} and noticing that
\begin{equation}
\begin{split}
f_R(\Psi_R^{n_\gamma-1})&=\left(-\frac12\alpha I+\bi\lambda F_R(\Psi_R^{n_\gamma-1})\right)\tau\\[2mm]
&=\left(-\frac12\alpha I+\bi\lambda F(\Psi^{n_\gamma-1})\right)\tau=f(\Psi^{n_\gamma-1}),
\end{split}
\end{equation}
we have $\Psi_R^{n_\gamma}=\Psi^{n_\gamma}$,
which implies
$$\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|=\|\Psi(t_{n_\gamma})-\Psi^{n_\gamma}\|\ge\gamma.$$
We conclude that for any $\gamma\in(0,1)$, there exists $n_\gamma\in\N$ such that
\begin{align*}
&\left\{\sup_{0\le n\le n_\gamma}\|\Psi(t_n)\|<R-1\right\}\cap\left\{\sup_{1\le n\le[T/\tau]}\|\Psi(t_{n})-\Psi^{n}\|\ge\gamma\right\}\\
\subset&\{\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge\gamma\}.
\end{align*}
\iffalse
Thus, for some constants $K,K_1>0$, choosing $\gamma=\sqrt{h^{-1}}K\tau$ and $R=\sqrt{(n_\gamma+1)K_1}$, we deduce
\begin{align*}
&\P\left(\sup_{n\ge1}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)\\
\le&\P\left(\sup_{0\le n\le n_\gamma}\sqrt{h}\|\Psi(t_n)\|\ge \sqrt{(n_\gamma+1) K_1}\right)+\P\left(\sqrt{h}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge K\tau\right)\\
\le&\sum_{n=0}^{n_\gamma}\P\left(\sqrt{h}\|\Psi(t_n)\|\ge\sqrt{(n_\gamma+1)K_1}\right)+\P\left(\sqrt{h}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge K\tau\right)\\
\le&(n_\gamma+1)\frac{h\E\|\Psi(t_n)\|^2}{(n_\gamma+1)K_1}+\frac{h\E\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|^2}{K^2\tau^2}
\le\frac{C}{K_1}+\frac{C_R}{K^2},
\end{align*}
where we have used the Chebyshev's inequality, Proposition \ref{semimoment} and Theorem \ref{trunerror}.
As $C_R$ depend on $K_1$
\fi
Thus, for some constants $K,K_1>0$, choosing $\gamma=\sqrt{h^{-1}}K\tau$ and $R=\sqrt{h^{-1}}K_1$, we deduce
\begin{equation}\label{error}
\begin{split}
&\P\left(\sup_{1\le n\le[T/\tau]}\sqrt{h}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)\\
\le&\P\left(\sup_{0\le n\le n_\gamma}\sqrt{h}\|\Psi(t_n)\|\ge K_1\right)+\P\left(\sqrt{h}\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|\ge K\tau\right)\\
\le&\frac{h\E\Big[\sup\limits_{0\le n\le n_{\gamma}}\|\Psi(t_n)\|^2\Big]}{K_1^2}+\frac{h\E\|\Psi_R(t_{n_\gamma})-\Psi_R^{n_\gamma}\|^2}{K^2\tau^2}.
\end{split}
\end{equation}
We claim that $e^{2\alpha t}\|\Psi(t)\|^2$ is a submartingale, which ensures that
\begin{equation*}
\begin{split}
h\E\left[\sup_{0\le n\le n_{\gamma}}\|\Psi(t_n)\|^2\right]
&\le h\E\left[\sup_{0\le n\le n_{\gamma}}e^{2\alpha t_n}\|\Psi(t_n)\|^2\right]\\[2mm]
&\le e^{2\alpha T}h\E\left[\|\Psi(t_{n_\gamma})\|^2\right]
\le Ce^{2\alpha T}
\end{split}
\end{equation*}
based on a martingale inequality and Proposition \ref{semimoment}.
In fact, denoting $C_{J,P}:=\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)$ and
applying It\^o's formula to $e^{2\alpha t}\|\Psi(t)\|^2$ similar to \eqref{semiito},
we derive
\begin{align*}
e^{2\alpha t}\|\Psi(t)\|^2=\|\Psi(0)\|^2+2\int_0^te^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]+\frac{C_{J,P}}{\alpha}\left(e^{2\alpha t}-1\right)
\end{align*}
with $2\int_0^te^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]$ a martingale.
Apparently, we have
\begin{align*}
\E\left[e^{2\alpha t}\|\Psi(t)\|^2|\mathcal{F}_r\right]
&=\|\Psi(0)\|^2+2\int_0^re^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]+\frac{C_{J,P}}{\alpha}\left(e^{2\alpha t}-1\right)\\[2mm]
&\ge\|\Psi(0)\|^2+2\int_0^re^{2\alpha s}\Re\left[\overline{\Psi}(s)\epsilon\sigma\Lambda d\beta(s)\right]+\frac{C_{J,P}}{\alpha}\left(e^{2\alpha r}-1\right)\\[2mm]
&=e^{2\alpha r}\|\Psi(r)\|^2
\end{align*}
for $r\le t$, which completes the claim.
Hence, based on above claim and Theorem \ref{trunerror}, inequality \eqref{error} turns to be
\begin{align*}
\P\left(\sup_{1\le n\le[T/\tau]}\sqrt{h}\|\Psi(t_{n})-\Psi^{n}\|\ge K\tau\right)
\le\frac{Ce^{2\alpha T}}{K_1^2}+\frac{C_R}{K^2},
\end{align*}
which approaches to $0$ as $K_1,K\to+\infty$ properly for any $T>0$.
\end{proof}
\section{Numerical experiments}
In this section, we provide several numerical experiments to illustrate the accuracy and capability of the fully discrete scheme \eqref{full}, which can be calculated explicitly. We investigate the good performance in longtime simulation of the proposed scheme and check the temporal accuracy by
fixing the space step.
In the sequel, we take $\lambda=1,\,\alpha=0.5$, truncate the infinite series of Wiener process till $P=100$ and choose 500 realizations to approximate the expectation.
\begin{figure}[H]
\centering
\subfigure[$\epsilon=0$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{charge_1.pdf}
\end{minipage}
}
\subfigure[$\epsilon=1$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{charge_2.pdf}
\end{minipage}
}
\caption{Evolution of the discrete charge $h\E\|\Psi^n\|^2$ with $t=n\tau$ for (a) $\epsilon=0$ and (b) $\epsilon=1$ ($h=0.1,\, \tau=2^{-6},\, T=35$).}\label{charge00}
\end{figure}
{\it Charge evolution.} For the semidiscretization, the charge of the solution satisfies the evolution formula \eqref{charge}. To investigate the recurrence relation for the discrete charge of the fully discrete scheme, Fig. \ref{charge00} plots the discrete charge for different values of $\epsilon$ with initial value $\psi_0(x)=\sin(\pi x)$, $\eta_k=k^{-6}$, $h=1/J=0.1$, $\tau=2^{-5}$ and $T=35$. We can observe that the discrete charge inherits the charge dissipation law without the noise term, i.e., $\epsilon=0$, and preserves the charge dissipation law approximately with a limit $\frac{\epsilon^2h}{\alpha}\sum_{j=1}^J\sum_{k=1}^P\eta_ke_k^2(x_j)$ calculated through \eqref{charge} for $\epsilon=1$.
\begin{figure}[H]
\centering
\subfigure[$f=\exp(-\|\Psi\|^2)$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{temporal_average_1.pdf}
\end{minipage}
}
\subfigure[$f=\sin(\|\Psi\|^2)$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{temporal_average_2.pdf}
\end{minipage}
}
\caption{The temporal averages $\frac{1}{N}\sum_{n=1}^{N-1}\E[f(\Psi^n)]$ started from different initial values for bounded functions (a) $f=\exp(-\|\Psi\|^2)$ and (b) $f=\sin(\|\Psi\|^2)$ ($h=0.1,\,\epsilon=1,\,\tau=2^{-6},\,T=350$).}\label{temporal_average}
\end{figure}
{\it Ergodicity.}
Based on the definition of ergodicity, if numerical solution $\Psi^n$ is ergodic, its temporal averages $\frac{1}{N}\sum_{n=1}^{N-1}\E[f(\Psi^n)]$ started from different initial values will converge to the spatial average $\int_{\C^J}fd\mu_h^\tau$.
To verify this property, Fig. \ref{temporal_average} shows the
temporal averages of the fully discrete scheme started from five different initial values $initial(1)=(1,~0,~\cdots,~0)^{T}$, $initial(2)=(0.0003 \bi,~0,~\cdots,~0)^{T}$, $initial(3)=(\sin\Big(\frac{1}{101}\pi\Big),~\sin\Big(\frac{2}{101}\pi\Big),~\cdots,~\sin\Big(\frac{100}{101}\pi\Big))^{T}$, $initial(4)=\frac{2+\bi}{20}(1,~2,~\cdots,~100)^{T}$ and $initial(5)=(\exp(-\frac{\bi}{50}),~\exp(-\frac{2\bi}{50}),~\cdots,~\exp(-\frac{100\bi}{50}))^{T}$. From Fig. \ref{temporal_average}, it can be seen that the time averages of \eqref{full} started from different initial values converge to the same value for two continuous and bounded functions $f$, when time $T=250$ is sufficiently large.
\begin{figure}[H]
\centering
\subfigure[$\tau=2^{-8}$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{strong_error_1.pdf}
\end{minipage}
}
\subfigure[$\tau=2^{-10}$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{strong_error_2.pdf}
\end{minipage}
}
\caption{The mean-square convergence error $\Big(h\mathbb{E}\|\Psi(T)-\Psi^N\|^{2}\Big)^{\frac{1}{2}}$ for step sizes (a) $\tau=2^{-8}$ and (b) $\tau=2^{-10}$ ($h=0.25,\,\epsilon=1,T=10^3$). }\label{strong_error}
\end{figure}
{\it Time-independent error.}
As stated in Theorem \ref{trunerror} and \ref{probability}, the mean-square convergence error $\left(h\E\|\Psi_R(T)-\Psi_R^N\|^2\right)^{\frac12}$ with respect to the truncated equation \eqref{truncate} is independent of time $T$, and convergence in probability sense with respect to the original equation is also independent of time $T$. To clarify this property, by defining the mean-square convergence error as
\begin{equation*}
\mathcal{E}_{h,\tau}:=\Big(h\mathbb{E}\|\Psi(T)-\Psi^N\|^{2}\Big)^{\frac{1}{2}},~T=N\tau,
\end{equation*}
Fig. \ref{strong_error} displays the error $\mathcal{E}_{h,\tau}$ over long time $T=10^3$ for different time step sizes: (a) $\tau=2^{-8}$ and (b) $\tau=2^{-10}$ with $h=0.25$, and shows that the mean-square convergence error is independent of time interval, which coincides with our theoretical results.
\begin{figure}[H]
\centering
\subfigure[$\epsilon=0$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{order_0.pdf}
\end{minipage}
}
\subfigure[$\epsilon=1$]{
\begin{minipage}[t]{0.48\linewidth}
\includegraphics[height=6.0cm,width=6.0cm]{order_1.pdf}
\end{minipage}
}
\caption{Rates of convergence of \eqref{full} for (a) $\epsilon=0$ and (b) $\epsilon=1$, respectivly ($h=0.1,\,T=1,\,\tau=2^{-l},\,11\le l\le14$). }\label{order}
\end{figure}
{\it Convergence order.}
We investigate the mean-square convergence order in temporal direction of the proposed method \eqref{full} in this experiment. Let $h=0.1$, $T=1$ and initial value $\psi_0(x)=\sin(\pi x)$. We plot
$\mathcal{E}_{h,\tau}$ against $\tau$ on a log-log scale with various combinations of $(\alpha,\epsilon)$ and take the method \eqref{full} with small time stepsize $\tau=2^{-16}$ as the reference solution. We then compare it to
the method \eqref{full} evaluated with time steps $(2^{2}\tau, 2^{3}\tau, 2^{4}\tau, 2^{5}\tau)$ in order to show the rate of convergence.
Fig. \ref{order} presents the mean-square convergence order for the error $\mathcal{E}_{h,\tau}$ with various sizes of $\epsilon$. Fig. \ref{order} shows that the proposed scheme \eqref{full} is of order 2 for the deterministic case, i.e., $\epsilon=0$, and of order 1 for the stochastic case with $\epsilon=1$, which coincides with the theoretical analysis.
\section{Appendix}
\begin{proof}[Proof of uniform boundedness of $\|A\|$]
Based on the definition of $\|A\|$, we only need to show that
the maximum eigenvalue $\lambda_*:=\max\{\lambda:\det(\lambda I-\hat{A})=0\}$ of positive definite matrix
$\hat{A}:=-A\in\R^{J\times J}$ is uniformly bounded with respect to dimension $J$. Let $x:=\lambda-2>-2$. Then
$\lambda I-\hat{A}$ turns to be
\begin{equation*}
\left(
\begin{array}{cccc}
x&1 & & \\
1&x&1 & \\
&\ddots &\ddots &\ddots \\
& &1 &x\\
\end{array}
\right)\\
\cong\left(
\begin{array}{cccc}
x&1 & & \\
&x-\frac1x&1 & \\
& & x-\frac1{x-\frac1x}& \\
& & &\ddots\\
\end{array}
\right)=:X.
\end{equation*}
We define
$a_1(x):=x$, $a_{n+1}(x):=x-\frac1{a_{n}(x)}$ and $X_n(x)=\prod_{i=1}^na_i(x)$ for $n\ge1$, and deduce that
\begin{align}\label{Xn}
X_{n+2}(x)=xX_{n+1}(x)-X_{n}(x).
\end{align}
Noticing that
$X_2(x)=x^2-1>0$ and $X_2(x)-X_1(x)=x^2-x-1>0$ for any $x\ge2$.
We assume that $X_{j+1}(x)>0$ and $X_{j+1}(x)-X_{j}(x)>0$
for any $x\ge2$ and $1\le j\le n$, which contributes to
\begin{align*}
X_{n+2}(x)-X_{n+1}(x)=(x-2)X_{n+1}(x)+(X_{n+1}(x)-X_{n}(x))>0
\end{align*}
and $X_{n+2}(x)>X_{n+1}(x)>0$
based on \eqref{Xn}. Then the induction yields that $X_n(x)>0$ for any $x\ge2$ and $n\in\N$, which implies that $\lambda_*=\max\{x:X_J(x)=0\}+2\le 4.$
\end{proof}
\nocite{*}
\bibliography{wangxu}
\bibliographystyle{plain}
\end{document} | 166,169 |
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Proszę pooglądać cudowne wideo! Натисніть на назву та подивіться чудове відео про зимовий Краків!!!
неділя, 31 січня 2016 р.
Краків і Різдво / Boże Narodzenie
Краків і Різдво / Boże Narodzenie.
в)
четвер, 14 січня 2016 р.
Happy New 2016 Year!
Happy New 2016ідписатися на: Дописи (Atom) | 113,826 |
Greetings Family and Friends,
I am camping and biking in a location about an hour north of Pittsburgh, PA.
The mile into the campground was on a wavy road that had me and the trailer rocking side to side. Even at 4 mph it felt too fast. And the hills I had to climb! Between Kittanning and where I left in Tionesta there were a lot of long steep hills that had my engine roaring at times.
It feels like I am in a different area of the country. The campground is full of seasonal campers that go home for the work week. It is very very quiet and beautiful.
Getting into my site my van spun its wheels on the wet grass and mud. I sure hope it doesn’t rain between now an Thursday so I can pull out without assistance!
I am testing out using my iPhone to blog because that is all I will have on the Gap and C&O trails. Now I remember that they all’s show up at the top. Maybe the answer is to load the Pictures first.
Also on our first long tour George and I would rate the day from 1=bad to 5=great. I will use that system again too. But maybe I will ad a Zero = I quit!
I am writing this in a coffee shop in East Bank, PA. I am content with food and coffee along my ride on the Armstrong Trail.
More later | 18,689 |
(GREEK NEWS AGENDA)
Greece’s foreign policy is one of self-confidence and open horizons, Foreign Minister Dora Bakoyannis said on Monday (25.5.2009) while addressing the 13th Economist Roundtable with the Greek Government conference in Athens. “We have an opportunity in the framework of re-ordering the international environment to replace outdated policies and views. But in order to exercise such a policy one needs political courage and political will by all governments. We call friends, partners and allies to act in this direction,” she said, during a round-table discussion on “Transatlantic ties, EU, US, Russia and stabilisation in the Balkans.” The minister repeated Greece’s positions regarding the Euro-Atlantic prospects of Balkan countries, saying that Athens was a steadfast supporter of her neighbours’ accession course and backed them with specific steps, “such as the liberalisation of visa requirements for the EU.”
Filed under: Cyprus, Government, Greece, International Relations, Turkey Tagged: | foreign, Government, Greece, Greek, Policy | 223,594 |
TITLE: Definition of a category as a monoid
QUESTION [2 upvotes]: This is my first time writing here. If you want me to write my questions in another way, please let me know because I have no experience in doing that at all.
I am using the book of MacLane introducing the concept of categories.
Let $O$ be a set of objects and $A$ a set of arrows.
A category can be seen as a monoid $(A, \circ)$ by using the following “product over the set O” which is the set of all composable pairs $$A \times_O A = \{\langle f,g \rangle : f,g \in A \text{ with } dom(g) = cod(f) \}.$$
As set we do have the set of arrows. As inner binary operation we do have the composition $\circ$. Associativity is given by the common definition of a category (MacLane).
I would like to show properly the condition of the existence of the neutral element. We know that there exists for each arrow $f: a \to b$ the identity arrows $id_a$ and $id_b$ so that $$f \circ id_a = id_b \circ f = f.$$ But it should be rewritten in a way so that it makes sense with the definition of a neutral element and we can write a real triple $(A,\circ, e)$.
Could you give me some hints? Thank you in advance.
REPLY [1 votes]: In one line, what MacLane is saying is that (small) categories are monads in the bicategory of spans of sets.
More precisely: monoids are a notion that makes sense in any monoidal category. Fix the set $O$ and consider the category $\mathbf G_O$ whose objects are the triples $(A,s,t)$ with $s,t$ maps $A \to O$ and whose mrophisms $(A,s,t) \to (A',s',t')$ are the map $F:A \to A'$ such that $s=s'F$ and $t=t'F$. Then there is a monoidal product on $\mathbf G_O$ defined as:
$$ (A,s,t) \otimes (A',s',t') := (\{(f,f') \in A\times A' : t(f) = s'(f')\}, (f,f')\mapsto s(f), (f,f') \mapsto t'(f')) $$
The unit for this monoidal product is $(O,\mathrm{id}_O,\mathrm{id}_O)$. The claim is that a monoid in this monoidal category $\mathbf G_O$ is precisely a (samll) category with set of objects $O$. In particular, the unit of such a monoid is expressed as a map $e : (O,\mathrm{id}_O,\mathrm{id}_O) \to (A,s,t)$ in $\mathbf G_O$, meaning a map $e:O \to A$ such that $s=se$ and $t=te$. | 70,386 |
Reports
Social Mobility and Stronger Private Sector Role are Keys to Growth in the Arab World
In spite of unprecedented improvements in technological readiness, the Arab World continues to struggle to innovate and create broad-based opportunities for its youth. Government-led investment alone will not suffice to channel the energies of society toward more private sector initiative, better education and ultimately more productive jobs and increased social mobility. The Arab World Competitiveness Report 2018 published by the World Economic Forum and the World Bank Group outlines recommendations for the Arab countries to prepare for a new economic context.
The gap between the competitiveness of the Gulf Cooperation Council (GCC) and of the other economies of the region, especially the ones affected by conflict and violence, has further increased over the last decade. However, similarities exist as the drop in oil prices of the past few years has forced even the most affluent countries in the region to question their existing social and economic models. Across the entire region, education is currently not rewarded with better opportunities to the point where the more educated the Arab youth is, the more likely they are to remain unemployed. Financial resources, while available through banks, are rarely distributed out of a small circle of large and established companies; and a complex legal system limits access to resources locked in place and distorts private initiative.
At the same time, a number of countries in the region are trying out new solutions to previously existing barriers to competitiveness.
- In ten years, Morocco has nearly halved its average import tariff from 18.9 to 10.5 percent, facilitated trade and investment and benefited from sustained growth.
- The United Arab Emirates has increased equity investment in technology firms from 100 million to 1.7 billion USD in just two years.
- Bahrain is piloting a new flexi-permit for foreign workers to go beyond the usual sponsorship system that has segmented and created inefficiencies in the labour market of most GCC countries.
- Saudi Arabia has committed to significant changes to its economy and society as part of its Vision 2030 reform plan, and Algeria has tripled internet access among its population in just five years.
.”
With a few exceptions, such as Jordan, Tunisia and Lebanon, most Arab countries have much less diversified economies than countries in other regions with a similar level of income. For all of them, the way toward less oil-dependent economies is through robust macroeconomic policies that facilitate investment and trade, promotion of exports, improvements in education and initiatives to increase innovation and technological adoption among firms.
Entrepreneurship and broad-based private sector initiative must be a key ingredient to any diversification recipe.
The Arab Competitiveness Report 2018 also features country profiles, available here: Algeria, Bahrain, Egypt, Jordan, Kuwait, Lebanon, Morocco, Oman, Qatar, Saudi Arabia, Tunisia, United Arab Emirates.
> | 213,496 |
\begin{document}
\title{\huge A Note on Curve Counting Scheme in an Algebraic Family and
The Admissible Decomposition Classes}
\author{Ai-Ko Liu\footnote
{email address: [email protected], Current Address:
Mathematics Department of U.C. Berkeley}}
\maketitle
\newtheorem{theo}{Theorem}
\newtheorem{lemm}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{rem}{Remark}
\newtheorem{cor}{Corollary}
\newtheorem{mem}{Examples}
\newtheorem{defin}{Definition}
\newtheorem{axiom}{Axiom}
\newtheorem{conj}{Conjecture}
\newtheorem{exam}{Example}
\newtheorem{assum}{Assumption}
\bigskip
In this paper, we discuss the generalized scheme for curve
counting in the family Seiberg-Witten theory.
Even though the original motivation is to study the
Mcduff's proposal in $b^+_2=1$ category of symplectic four manifolds,
we will formulate our scheme in an algebraic(Kahler) set up. The
material considered in this paper will be relatively
elementary. Nevertheless, the theory discussed here has played an essential
role in the long paper [Liu1].
As a major application of the discussion, one may apply our scheme
in the proof of the G${\ddot o}$ttsche's conjecture about counting of
holomorphic curves[Liu1]. The current
scheme is motivated from the discussion [Mc] of
pseudo-holomorphic curves in symplectic four manifolds. However we
will restrict our discussion to the algebraic varieties here.
One can translate our scheme to pseudo-holomorphic category
by replacing holomorphic curves to pseudo-holomorphic curves.
\medskip
Given an algebraic family of algebraic surfaces ${\cal X}\mapsto B$,
the family Seiberg-Witten invariant ${\cal AFSW}$ (or $FSW$ in the smooth
category) enumerates the algebraic curves (or pseudo-holomorphic curves
in the symplectic category) within the family dual to a given cohomology class.
The basic phenomenon we will study is that not only smooth curves may
appear in the enumeration, curves contain multiple coverings of the so-called
exceptional curves may also occur. The general question we are interested
at is
\noindent{\bf Question}: How to relate the contribution from the smooth
curves to the original family invariant?
\medskip
The general strategy to answer the question is to subtract the
contributions from the various configurations containing multiple coverings of
exceptional curves. The purpose of the
current paper is to provide a skeleton of the curve counting scheme.
\section{\bf The family scheme of Algebraic Surfaces}\label{section; AS}
\bigskip
In this section, consider ${\cal X}\mapsto B$ to be a relatively
smooth algebraic fibration over a smooth base.
For simplicity, the field of definition
can be taken to be ${\bf C}$. The same scheme will work for any
algebraic closed field of characteristic zero as well. Even though
we do not aim at general symplectic four manifolds, we will
recall the phenomena in the symplectic set up from time to time.
The fibers of the fibration are taken to be smooth projective
surfaces. In the following, we denote $dim_{\bf C}B$ to be the complex
dimension of the base even though the scheme work well even when
$B$ does not carry complex structures.
First we introduce certain notations.
Let $C$ denote the cohomology class represented by holomorphic curves.
Let $d_{GT}(C)={C^2-c_1(K_X)\cdot C\over 2}$
denote the complex Gromov Taubes dimension [T3] in Taubes theory.
Then the family Gromov-Taubes dimension
of $C$ is defined axiomatically to be $d_B(C)=d_{GT}(C)+dim_{\bf C} B$.
Given the class $C$ as the sum of $C_1$ and $C_2$, one has the
following equality between their formal family Gromov-Taubes dimensions.
$$d_B(C)=d_B(C_1)+d_B(C_2)-dim_{\bf C}B.$$
This equality reflects that the family moduli space associated to
$C$ is the fiber product of those of $C_1$ and $C_2$.
One can easily generalize the equality to
more than two $C_i$ and the new equality is
$$d_B(\sum_{i\in I} C_i)=\sum_{i\in I} d_B(C_i)-(|I|-1)dim_{\bf C}B,$$
with $|I|$ being the cardinality of the index set $I$.
\medskip
\begin{defin}\label{defin; exception}
A cohomology class $e\in H^2(M, {\bf Z})$ is said to be an exceptional class
if
(i). $e$ is a primitive element in the lattice $H^2(M, {\bf Z})$.
(ii). $e^2=e\cup e[M]<0$.
A pseudo-holomorphic curve in an almost complex four-manifold
$M$ is said to be an exceptional curve if
it is poincare dual to an exceptional class $e$.
\end{defin}
We have the following proposition regarding irreducible exceptional curves,
\begin{prop} \label{prop; unique}
Let $\Sigma_1, \Sigma_2$ be two Riemann surfaces and let $f_1:\Sigma_1\mapsto M$
$f_2:\Sigma_2\mapsto M$ be two (pseudo-)holomorphic maps into the almost complex
four-manifold $M$ with $(f_i)_{\ast}[\Sigma_i]$ dual to an exceptional class
$e\in H^2(M, {\bf Z})$, then $\Sigma_1=\Sigma_2$ and
$f_1$ and $f_2$ coincide.
Namely, $f_i(\Sigma_i), i=1, 2$ coincide and is the only irreducible
(pseudo-)holomorphic curve dual to $e$ in $M$.
\end{prop}
\medskip
\noindent Proof of the proposition: Because $e$ is primitive, the maps
$f_i$ from $\Sigma_i$ to $f_i(\Sigma_i)$ are of degree $1$. From [Mc],
$f_i(\Sigma_i), i=1, 2$ have
at most a finite number of isolated singularities and
$f_i$ are immersions away from the singularities in the images.
Because $\Sigma_i$ are irreducible, $f_i(\Sigma_i)$ are irreducible, too.
We argue that $f_1(\Sigma_1)=f_2(\Sigma_2)$. If not, the
sets $f_1(\Sigma_1)=\cap f_2(\Sigma_2)$ is of a finite cardinality.
However, again by [Mc2] each intersection point contributes positively
to the total intersection number, the total intersection number should
be non-negative. On the other hand, $f_1(\Sigma_1)\cap f_2(\Sigma_2)
=(f_1)_{\ast}[\Sigma_1]\cap (f_2)_{\ast}[\Sigma_2]=e\cup e[M]<0$.
This gives us the necessary contradiction. Thus $f_1(\Sigma_1)=f_2(\Sigma_2)$.
Once we know $f_1(\Sigma_1)=f_2(\Sigma_2)$ and both pseudo-holomorphic
maps are of degree one,
Both $\Sigma_1, \Sigma_2$ can be re-constructed as the normalization of the
complex curve $f_1(\Sigma_1)=f_2(\Sigma_2)$. Therefore $\Sigma_1=\Sigma_2$ and
$f_1=f_2$. $\Box$
\medskip
Even though the ``irreducible'' curve dual to $e$ is always unique, there can be
two or more reducible pseudo-holomorphic curves dual to $e$
with more than one irreducible component. In this case, the conclusion is
weaker and one
can only deduce that two curves share at least one irreducible component
and the fundamental class of this irreducible component has a negative
self-intersection number in the four-manifold $M$.
Recall that Taubes theory [T3] asserts the equivalence of Seiberg-Witten
invariant
and a version of Gromov invariant for symplectic four
manifolds. It indicates that the diffeomorphism invariants $SW$ is
equivalent to the symplectic invariant $Gr$. Despite of the simplicity of
the statement, the actual proof [T1], [T2], [T3]
involves sophisticated analysis and
an amount of new ideas.
It is less well known that the equivalence of $SW$ and $Gr$ fails
for the general symplectic four manifolds with $b^+_2=1$.
Originally Taubes asserted his theorem for $b^+_2>1$ category.
Later he extended his theorem to $b^+_2=1$ case with some additional
assumption. In the mean time, it was discovered experimentally by
the current author in [LL] that the assertion would not be true
without the additional assumption. This motivated Mcduff to
change the original definition of the Gromov invariant in order to match up
with $SW$.
\medskip
\begin{defin}\label{defin; Taubes'}
In defining the Seiberg-Witten invariants of $b^+_2=1$ symplectic
four-manifold $M$, the space of generic Riemannian metrics $g$ and self-dual
two forms $\mu$ on $M$ are divided into chambers. Given a symplectic two form
$\omega$ on $M$ and a Riemannian metric $g$ such that $\omega$ is self-dual
with respect to $g$, the $(g, r\omega=\mu)$, $r\mapsto \infty$ determines
a unique chamber, called the Taubes' chamber in the following discussion.
\end{defin}
\medskip
Let us recall Taubes theorem for $b^+_2=1$ symplectic manifolds.
\begin{theo}(Taubes) \label{theo; =1}
Let $C$ be a cohomology class in $H^2(M, {\bf Z})$, with $M$ being
a symplectic four manifold with $b^+_2=1$. Assume additionally
that $C\cdot S\geq -1$ for all spherical class $S$ with $S^2=-1$.
Then the statement $SW(2C-K_M)=Gr(C)$ holds for $C$, where $SW(2C-K_M)$ is
evaluated in the Taubes' chamber.
\end{theo}
By a spherical class $S$ with $S^2=-1$ one means that the class
is represented by a $S^2$ with self intersection number $-1$.
In the following, we give a simple example that the theorem does not
hold without modification for the classes violating this condition.
\begin{mem}
Consider $M$ to be ${\bf CP}^2\sharp \overline{{\bf CP}^2}$, the symplectic
four manifold constructed by ${\bf CP}^2$ by blowing up one point in
${\bf CP}^2$.
Let $H\in H^2(M, {\bf Z})$
denote the (pull-back of)
the hyperplane class and let $E\in H^2(M, {\bf Z})$ denote the exceptional class.
Take $C=3H+2E$,and a simple calculation shows that $d_{GT}(C)=9-1=8$.
Through the calculation of wall crossing formula it is easy to see
that $SW(9H+3E)=\pm 1$. On the other hand, we argue that
a reasonable definition of
Gromov invariant would be $Gr(3H+2E)=0$. The class
$3H+2E$ can not be represented by irreducible curves. The
representatives are the disjoint union of a cubic curve dual to $3H$
along with the double covering of the exceptional curve $E$.
Even though curves dual to $3H$ gives rise to nonzero invariant, the multiple
covering of $E$ has a negative Gromov dimension $d_{GT}(2E)=-1$. Thus the
total Gromov-Taubes invariant should be $Gr(3H)\times 0=0$.
\end{mem}
\medskip
This simple example illustrates the subtlety to Taubes' theory.
Even though the $SW(9H+3E)\not=0$ in the Taubes' chamber and the analysis in
$SW\mapsto Gr$ [T1] still implies the existence of
pseudo-holomorphic curves in $C$, numerically the curves are counted zero
in the Gromov-Taubes theory. As similar type of phenomena appears for
any non-minimal symplectic four manifolds with $b^+_2=1$, it becomes
the major topological obstruction to identify $SW$ and $Gr$.
The way Taubes dealed with this problem is to rule out the classes $C$
which potentially can be represented as disjoint unions of
pseudo-holomorphic curves and some multiple coverings of exceptional $-1$
curves. This explains the extra condition $C\cdot S\geq -1$ in Taubes'
theorem.
Even though it is not completely obvious. this ill symptom is closely
related to the fact that $b^+_2=1$ symplectic manifolds are not
of simple type in Taubes' chamber. In the fundamental paper of
Taubes, symplectic four manifolds with $b^+_2>1$ were proved to be
of simple type. Thus, the ill symptom does not occur to them.
On the other hand, it is a consequence of the wall crossing formula
[LL] that the non-simple type of $b^+_2=1$ symplectic manifolds is
directly related to the non-vanishing of the wall crossing numbers.
The primitive goal to develop the family Seiberg-Witten theory is to
discuss the family Seiberg-Witten theory in Taubes' chambers (defined by
a large perturbation of fiberwise self-dual symplectic forms).
As a similar application of the family wall crossing formula implies
the non-simple type-ness, one would expect that the similar failure
of $SW=Gr$ would occur. Unlike the $B=pt$ case that $-1$ curves are
the only pseudo-holomorphic curves which persist, the topological
types of the exceptional
curves which persist in the family are less restricted.
Viewed from a different angle, the major distinction between the
standard Gromov-Witten theory and Gromov-Taubes theory lies in the
fact that the latter theory does not restrict the topological types
of the pseudo-holomorphic maps.
The combination of these two issues make the identification between
$FSW$ and $FGr$ extremely difficult.
I.e./ within a given family of symplectic manifolds and a fiberwise
monodromy invariant class $C$,
there can be a whole 'zoo' of exceptional
curves which may appear in some pseudo-holomorphic curve
representations of the class $C$. To make sense of Gromov-Taubes
invariants, one has to deal with these exotic objects in a more
systematical way.
\bigskip
\begin{mem} \label{mem; explain}
For the readers with a background of Gromov-Witten theory, it should be
cautious not to think of Gromov-Taubes invariants as identical to the
standard Gromov-Witten invariants. Besides the question of allowing
dis-connected domain curves, the expected Gromov-Taubes
dimensions of multiple covering
of exceptional curves are different from the dimension formulae of
Gromov-Witten invariants viewed as multiple covering pseudo-holomorphic maps
into the symplectic manifold.
For simplicity, let $e\in H^2(M, {\bf Z})$
be an exceptional class $e^2=-k, c_1(K_M)\cdot e=-2+k$ representing an
exceptional sphere. Let $m\in {\bf N}$
be a positive integer, the expected Gromov-Taubes dimension of the class
$me$ is given by $(-m^2-m)k+2m$.
On the other hand, the expected dimension of the pseudo-holomorphic maps
from $S^2$ to $M$, dual to $me$, (modulo diffeomorphisms on $S^2$) is given by
$$2\{c_1(M)\cdot C-(g-1)\cdot dim_{\bf C}M+3g-3\}=2\{-c_1(K_M)\cdot C+2-3\}
=-2mk+2m-2.$$
The former is quadratic with respect to $m$, while the usual Gromov-Witten
expected dimension is linear in $m$.
Thus, even though we will still call pseudo-holomorphic curves dual to $me$ a
multiple covering of exceptional curves in $e$, we advise the readers not
to confuse them with the multiple covering in the sense of maps. Instead, it
is wiser to think of it as $m$ copies of pseudo-holomorphic curves dual to $e$
sitting on top of each other.
\end{mem}
\medskip
\section{\bf The Pointwise Calculation of Family Dimension}
\label{section; point}
\medskip
Firstly, let us review the original dimension count argument of Taubes.
Let $(M, \omega)$ be a symplectic four-manifold with a compatible almost
complex structure and $C\in H^2(M, {\bf Z})$ be a cohomology class with
a positive energy $C\cdot \omega>0$. In Taubes' theory, he allows the
pseudo-holomorphic curves to have more than one irreducible component.
Suppose that there is a pseudo-holomorphic curve poincare dual to $C$.
Then there is a finite collection of Riemann surfaces $\Sigma_i, 1\leq i\leq k$
and the pseudo-holomorphic maps $f_i:\Sigma_i\mapsto M$ such that
$\sum_{i\leq k} (f_i)_{\ast}[\Sigma_i]\in H_2(M, {\bf Z})$ is poincare dual
to $C$.
In case the map $f_i$ is of degree $m_i$, we may write
$PD((f_i)_{\ast}[\Sigma_i])$ as $m_ie_i$, $m_i\in {\bf N}$. Then we have
the following equality
$$C=\sum_{i\leq k} m_i e_i.$$
Conversely, if $C$ is written as $\sum_{i\leq k} m_i e_i$ and each of
$e_i$ is represented by an irreducible pseudo-holomorphic curve, then one
takes the union of them (counting multiplicity) and represent $C$ as a
pseudo-holomorphic curve in $M$.
Taubes would like to study all the possible decompositions of $C$ into
different $e_i$ which will survive under generic compatible
almost complex structures
perturbation of $M$.
One makes three additional assumptions on $e_i$,
\medskip
(i). $e_i\cdot \omega>{\cal E}(M, \omega)>0$ for some manifold dependent
lower bound of harmonic energy.
\medskip
(ii). The expected Gromov-Taubes dimension
$d_{GT}(e_i)={e_i^2-e_i\cdot c_1({\bf K}_M)\cdot e\over 2}\geq 0$.
\medskip
(iii). $e_i\cdot e_j\geq 0$ for all $i\not= j$.
\medskip
(iv). $e_i^2+c_1({\bf K}_M)\cdot e_i=2g_{arith}(e_i)-2\geq -2$.
\medskip
\begin{defin}\label{defin; decomposition}
Let $e_i, i\leq k$ be a finite number of classes in $H^2(M, {\bf Z})$
which satisfy $(i)., (ii)., (iii).$ and $(iv).$
The expression $C=\sum_{i\leq k} m_i e_i, m_i\in {\bf N}$ is called
a (cohomological) decomposition of $C$.
\end{defin}
\medskip
The reason that one imposes $(i)$ is because a pseudo-holomorphic curve dual to
$e_i$ always has a positive energy.
If an irreducible irreducible
curve dual to $e_i$ has negative Gromov-Taubes dimension
$d_{GT}={e_i^2-c_1({\bf K}_M)\cdot e_i\over 2}<0$, then by Fredholm theory
this type of curves may disappear after a generic perturbation of compatible
almost complex structures on $M$, which does not have chance to
contribute to the Gromov invariant defined by Taubes [T3].
From [Mc2], one knows that two distinct irreducible pseudo-holomorphic curves
in an almost complex four-manifold $M$
intersect positively. Thus $e_i\cdot e_j$, the sum of all the local
intersection contribution, should be non-negative as well.
Because $e_i$ is represented by an irreducible pseudo-holomorphic curve on
an almost complex four-manifold $M$, it satisfies the adjunction formula
with $g_{arith}$ being the arithmetic genus of the curve.
Let $C$ be written as
$\sum_{i\leq k} m_i e_i$ with multiplicity
$m_i\geq 1$ satisfying $(i).$, $(ii).$, $(iii).$ and $(iv)$.
Then $$2d_{GT}(C)={C^2-C\cdot c_1({\bf K}_M)}=
\bigl(\sum_{i\leq k} (e_i^2-e_i\cdot c_1({\bf K}_M))$$
$$+2\sum_{i\not= j} m_i m_j e_i\cdot e_j+
\sum_{i\leq k}((m^2_i-1)e_i^2+(1-m_i)e_i\cdot c_1({\bf K}_M))\bigr)$$
$$=2\sum_{i\leq k}d_{GT}(e_i)+2\sum_{i\not= j} m_i m_j e_i\cdot e_j+
\sum_{i\leq k}\bigl((m^2_i-m_i)e_i^2+
(m_i-1)(e_i^2-e_i\cdot c_1({\bf K}_M))\bigr).$$
By the assumption (iv). $e_i^2+c_1({\bf K}_M)\cdot e_i\geq -2$ and
by $(ii).$ $2d_{GT}(e_i)=e_i^2-c_1({\bf K}_M)\cdot
e_i\geq 0$. Then we know that $e_i^2\geq -1$. Suppose $e_i^2=-1$. From
$(ii).$ again we get $-1=e_i^2\geq c_1({\bf K}_M)\cdot e_i$. Then
$-2\geq e_i^2+c_1({\bf K}_M)\cdot e_i$ and $c_1({\bf K}_M)\cdot e_i=-1$
as well. If it is the case, $g_{arith}(e_i)=0$ and one can argue that
the pseudo-holomorphic curve representing $e_i$ must be a so-called $-1$ curve.
Because Taubes' goal is to develop a version of Gromov invariant
which can be identified with $SW(2C-c_1({\bf K}_M))$ (the $spin^c$ class
$2C-c_1({\bf K}_M)$ is in an additive notation), he is able to
use the Seiberg-Witten simple type-ness condition on $b^+_2>1$ symplectic
four-manifolds and the blowup formula of Seiberg-Witten invariants
to deduce $m_i=1$ for all such $-1$ classes $e_i^2=-1$.
Thus, one find that the last term in the expansion of $2d_{GT}(C)$ is
always non-negative.
In order to count pseudo-holomorphic curves dual to $C$, one imposes
$d_{GT}(C)$ number of generic points and require the pseudo-holomorphic
curves dual to C to pass through these generic points.
From above we find that $$d_{GT}(C)-\sum_{i\leq k}d_{GT}(e_i)\geq 0.$$
If the difference $d_{GT}(C)-\sum_{i\leq k}d_{GT}(e_i)$
is strictly positive, then by dimension reason
there can be no pseudo-holomorphic
curves in $\sum_{i\leq k} e_i$ which pass through all these $d_{GT}(C)$ points,
after one adopts the Fredholm argument to perturb the almost complex structures
of $M$. In order $\sum m_i e_i$ contributes to the Gromov invariant, the
non-negative
sum $2\sum_{i\not= j} m_i m_j e_i\cdot e_j+
\sum_{i\leq k}\bigl((m^2_i-m_i)e_i^2+
(m_i-1)(e_i^2-e_i\cdot c_1({\bf K}_M))\bigr)$ has to vanish term by term.
That is to say,
$(a).$ $m_i=1, \forall i\leq k$.
$(b).$ $e_i\cdot e_j=0, \forall i\not=j$.
$(c).$ Each curve dual to $e_i$ must be smooth.
\medskip
In other words, distinct $e_i$ cannot intersect. Each irreducible
pseudo-holomorphic curve
appears with multiplicity one.
One may develop the following 'philosophical' idea which helps to
explain what happens.
\medskip
(a)'. If $m_i>1$ for some $e_i^2>0$, ideally one may choose two
distinct pseudo-holomorphic curves dual to $e_i$ and they intersect
positively. Then the smoothing of all these
intersection singular points produces an irreducible curve dual to $2e_i$, whose
dimension $d_{GT}(2e_i)>d_{GT}(e_i)+d_{GT}(e_i)$. Continue in this fashion,
Seiberg-Witten theory is expected to count irreducible curves in $m_ie_i$
rather than $m_i$ copies of curves dual to $e_i$, which formally can be viewed as
a degeneration from an irreducible multiplicity one curve dual to $m_ie_i$.
\medskip
(b)'. If $e_i\cdot e_j>0$ for some $i\not j$,
one may think of the smoothing of the $e_i\cdot e_j$
intersection points (counted with multiplicity) and consider (formally) a
curve dual to $e_i+e_j$. Then the union of the curves dual to $e_i$ and $e_j$
can be thought as a degeneration of some irreducible curve dual to $e_i+e_j$
and $d_{GT}(e_i+e_j)>d_{GT}(e_i)+d_{GT}(e_j)$.
\medskip
(c)'. If a curve dual to $e_i$ develops certain singularities, it can
be thought of a degeneration of the smooth curves dual to $e_i$ satisfying
the adjunction equality $e_i^2+c_1({\bf K}_M)\cdot e_i=2g(e_i)-2$. The
curves with singularities are of lower expected dimension than the expected
Gromov-Taubes dimension $d_{GT}(e_i)$.
\medskip
In the $b^+_2=1$ category, $m_i\geq 1$ for the $-1$ classes $e_i^2=-1$.
Then the same argument breaks down.
One way to remedy is to impose extra condition on $C$, as was done by
Taubes (see the statement of theorem. \ref{theo; =1}).
Mcduff introduces a different way to remedy the situation. She (see [Mc])
has shown that
\begin{prop}(Mcduff) \label{prop; mcduff}
Let $M$ be a $b^+_2=1$ symplectic four-manifold and let $C\in H^2(M, {\bf Z})$
be a class satisfying $d_{GT}(C)={C^2-C\cdot c_1({\bf K}_M)\over 2}\geq 0$,
$C\cdot \omega>0$. Then there exists a finite number of $-1$ classes,
$e_i$, $e_i^2=-1$ satisfying the following conditions:
(i). Each $e_i$ is represented by a $-1$ pseudo-holomorphic curve.
\medskip
(ii). $e_i\cdot e_j=0$ for $i\not=j$.
\medskip
(iii). $C\cdot e_i=-n_i<0$.
Then one may re-write $C=(C-\sum n_ie_i)+\sum n_i e_i$ and $C-\sum n_ie_i=C_{red}$
is perpendicular to all these $e_i$, namely $C_{red}\cdot e_i=0$.
\end{prop}
\medskip
Mcduff proposed [Mc] to define $Gr(C)$ using the class $C_{red}$ instead of $C$.
\medskip
\begin{defin}\label{defin; taubes}
Let $C=\sum_i m_i e_i$ be a cohomological decomposition of $C$. The
decomposition is said to be of Taubes' type if
(a). $e_i\cdot e_j=0$ for $i\not j$.
\medskip
(b). $m_i=1$ for all $i$.
\end{defin}
\medskip
In the following, we generalize the Mcduff's proposal to the
family case. It turns out all the different possibilities of
decompositions of curves appearing in the
$B=pt$ case can also appear in the family case. Moreover, in the family
case there are many new possible decompositions of curve classes
which are absent in
the $B=pt$ case due to dimension reason.
Let $\pi:{\cal X}\mapsto B$ be a fiber bundle of symplectic four-manifolds with
the relative symplectic form $\omega_{{\cal X}/B}$.
Similar to the condition $(i). (ii). (iii).$ and $(iv).$ for the $B=pt$ cases,
one imposes the following conditions on the classes $e_i$.
\medskip
(Fi).$e_i\cdot \omega_{{\cal X}/B}>{\cal E}({\cal X}, \omega_{{\cal X}/B})>0$ for
some fiber bundle dependent lower bound of harmonic energy.
\medskip
(Fii).The expected family Gromov-Taubes dimension
$dim_{\bf C}B$+$d_{GT}(e_i)={e_i^2-e_i\cdot c_1({\bf K}_M)\cdot e\over 2}\geq 0$.
\medskip
(Fiii). $e_i\cdot e_j\geq 0$ for all $i\not=j$.
\medskip
(Fiv). $e_i^2+c_1({\bf K}_{{\cal X}/B})\cdot e_i=2g_{arith}(e_i)-2\geq -2$.
\medskip
At this moment we are doing the pointwise analysis for different $b\in B$,
we do not take into account
the monodromy action of $\pi_1(B, b_0)\mapsto H^2(\pi^{-1}(b_0),
{\bf Z})$. At times (e.g. for the universal families $M_{l+1}\mapsto M_l$),
the monodromy representation is completely trivial and we can ignore
it. Otherwise, we have to consider the equivalent classes of $e_i$
or decompositions under the action of
$\pi_1(B, b_0)\mapsto H^2(\pi^{-1}(b_0), {\bf Z})$.
Let us discuss how does the ordinary Taubes' dimension count argument
generalized to the family case. Because we will use the relative
canonical bundle ${\bf K}_{{\cal X}/B}$ throughout the discussion, we
will skip the subscript ${\cal X}/B$ and denote it by ${\bf K}$.
Using the family dimension formula,
$$ dim_{\bf R}B+2d(C)=dim_{\bf R}B+{C^2-C\cdot c_1({\bf K})}
=(\sum_{i\leq k} (dim_{\bf R}B+e_i^2-e_i\cdot c_1({\bf K}))-(k-1)dim_{\bf R}B$$
$$+2 \sum_{i\not= j}
m_i m_j e_i e_j+\sum_i((m^2_i-1)e_i^2+(1-m_i)e_i\cdot K)).$$
\medskip
In a given family, the condition $(Fii).$ is weaker than the condition
$(ii).$ at the $B=pt$ case. Thus, there may be some $e_i$ with $e^2_i<-1$.
Then the last expression $\sum_i((m^2_i-1)e_i^2+(1-m_i)e_i\cdot K))$ may be
negative for $m_i\not=1$. In other words, the formal dimension
expected from family Seiberg-Witten theory (equal to $dim_{\bf R}B+
2d_{GT}(C)$ can be smaller than the actual
formal dimension on the Gromov-Taubes side, $(\sum_{i\leq k}
(dim_{\bf R}B+e_i^2-e_i\cdot c_1({\bf K}))-(k-1)dim_{\bf R}B$.
\bigskip
Motivated from Mcduff's proposal [Mc] and Taubes' theorem \ref{theo; =1},
let us denote $P$ as the index subset $P\subset \{1, 2, \cdots, k\}$
such that $e_i\cdot C<0, i\in
P$, with
$e_i^2<0$. Then we can always regroup the decomposition of $C$ as
$$C=F+E,F=\sum_{i\notin P}m_i e_i; E=\sum_{j\in P}m_j e_j.$$
Namely, one can view $F$ as a whole without going into
the details of the decomposition of the class $F$.
Then the previous expression can be expanded easily into the following
$$dim_{\bf R}B+2d_{GT}(C)=F^2-c_1({\bf K})\cdot F+2F\cdot E+E^2-c_1({\bf K})
\cdot E+dim_{\bf B} B.$$
Then we rewrite the term $F\cdot E$ term into $F \cdot
E=(C-E)\cdot E$, then we have
$$dim_{\bf R}B+2d_{GT}(C)
=F^2-c_1({\bf K})\cdot F+2(C-E)\cdot E+E^2-c_1({\bf K})\cdot E+dim_{\bf R} B.$$
Let us collect $F^2-c_1({\bf K})\cdot F+dim_{\bf R}B+
\sum_{i\in P} (e_i^2-c_1({\bf K})\cdot e_i)$ into
a single term. It is the family dimension of $F$, along with
the all the $e_i, i\in P$.
The sum of the left-over terms, called the family dimension discrepancy,
has the following form
$$\Delta_C(E)=2C\cdot E-E^2-c_1({\bf K})\cdot E-
\sum_{i\in P}(e_i^2-c_1({\bf K})\cdot e_i).$$
To study how do the multiplicities $m_i, i\in P$ in $E=\sum m_i E_i$
affect the family dimension, we have to introduce some combinatorial language.
combinatorial language.
\medskip
\begin{defin} \label{defin; exc}
Let $C$ be a monodromy invariant fiberwise cohomology class of
$\pi: {\cal X}\mapsto B$. Given a point $b\in B$,
Let $e_i, i\in P$ be all the classes represented by irreducible
pseudo-holomorphic curves over $b$ with $C \cdot e_i<0, e_i^2<0$. Then define
the exceptional cone of $C$ over $b$ to be
${\cal EC}_{b}(C)=\sum_i{\bf R}^{\geq 0} e_i\in H^2(\pi^{-1}(b), {\bf R})$.
\end{defin}
\medskip
As $E$ is written as$\sum m_i e_i$ with $m_i\geq 0$. We can view
E as an element in the cone ${\cal EC}_b(C)$
generated by these $e_i$.
Given such a cone ${\cal EC}_{b}(C) \in H^2(\pi^{-1}(b),{\bf R})$
with an indefinite intersection
form, it is possible to define the dual cone ${\cal EC}_b^{*}(C)$ to be
the elements in $H^2(X,{\bf R})$ which have non-negative intersection
pairings with elements
in ${\cal EC}_b(C)$. As ${\cal EC}_b(C)$ is usually not a top dimensional
cone in $H^2$,
the dual cone ${\cal EC}_b^{\ast}(C)$ usually is the direct sum of a vector space
cone with a reduced dual cone in the minimal subspace containing ${\cal EC}_b(C)$.
As a preparation, we want to prove some simple lemma characterizing the cone
${\cal EC}_b(C)$.
Let us review some definitions.
\begin{defin} \label{defin; adm}
Let ${\cal EC}_b(C)$ be an exceptional cone as
described before. Then ${\cal EC}_b(C)$ is said to be admissible over $b$
if $(C-{\cal EC}_b(C))\cap {\cal EC}_b(C)^{\ast}$ contains at least
one lattice point.
\end{defin}
\medskip
The following proposition clarifies the relation between the intersection
form on $H^2(\pi^{-1}(b), {\bf R})$ with the admissible cone ${\cal EC}_b(C)$.
\begin{prop}\label{prop; negative}
Suppose that the exceptional cone over $b$, ${\cal EC}_b(C)$, is
admissible. Then the restriction of the
intersection quadratic form on the cone ${\cal EC}_b(C)$ is negative definite.
\end{prop}
\medskip
This proposition implies that the term $-E^2$ in $\Delta_C(E)$ always
contributes positively when $E\in {\cal EC}_b(C)$.
Notice that we do not claim that the intersection form is negative
definite on the whole minimal vector space in $H^2$
containing ${\cal EC}_b(C)$. If the fibration $\pi:{\cal X}\mapsto B$
is algebraic. Then all the $e_i$ are of type $(1, 1)$.
Recall that Hodge index theorem asserts the intersection form has only one
positive eigenvalue. In this case
this proposition asserts that the exceptional cone
is disjoint to the forward and backward light cones.
\medskip
The proposition is a generalization of Mcduff's proposal.
\medskip
\begin{rem}\label{rem; mc}
When $B=pt$, the only exceptional curves satisfying $(ii).$ are $-1$ curves.
Suppose $e_i, e_j\in {\cal EC}_{pt}(C)$ are two different $-1$ curves.
Then the proposition implies that $e_i+e_j\in {\cal EC}_{pt}(C)$ is
of negative square. In other words, $(e_i+e_j)^2=-2+2e_ie_j<0$. Then $e_i\cdot
e_j$ must be $0$. This recovers Mcduff's proposal that all $-1$ curves
$e_i, e_i\cdot C<0$ must be perpendicular to each other.
\end{rem}
\medskip
\noindent Proof of the Proposition: Set $|P|=n$.
As ${\cal EC}_b(C)$ is admissible, then there must be some tuple
$(m_1,m_2,\cdots, m_n)\in {\bf N}^{n}$
such that $(C-\sum_{1\leq i\leq n}m_i e_i)\cdot e_j \geq 0$ for all
$j\in P$.
For simplicity let us re-scale and use the $\underline{e}_i=
m_i e_i$ as the new generators.
Notice that it is related to the old one by a positive scaling.
Then we have the following inequality
$$(\underline{e}_1+\underline{e}_2+\cdots +\underline{e}_n)
\cdot \underline{e}_i\leq C\cdot \underline{e}_i<0,$$
for all $i\in P$. Now we use that
$\underline{e}_i\cdot
\underline{e}_j\geq 0$
for $i\not= j$.
Then we must have more inequalities of the similar
type. Let $S\subset P$ be any nonempty subset of the index set $P$.
Then $$(\sum_{i\in S} \underline{e}_i)\cdot \underline{e}_j
<-(\sum_{i\in P-S}\underline{e_i}\cdot \underline{e_j})<0, j\in S.$$
\begin{lemm}
Given any two index subsets $A, B\subset P$ such that one includes the
other. Then it follows that $\{\sum_{i\in A}\underline{e}_i\} \cdot
\{\sum_{j\in B}\underline{e}_j\}<0$.
\end{lemm}
\medskip
\noindent Proof of the Lemma: By symmetry we may assume $A\supset B$. Then
we take $A=S$ in the above discussion and we have
$$(\sum_{i\in A} \underline{e}_i)\cdot \underline{e}_j<0, j\in A.$$
We may choose $j\in B\subset A$ and sum up all these inequalities for
$j$ running through $B$, we get
$$(\sum_{i\in A} \underline{e}_i)\cdot (\sum_{j\in B}\underline{e}_j)<0.$$ $\Box$
Now we are ready to prove the statement in the proposition.
Let $x$ be any element in the cone ${\cal EC}_b(C)$. Then it can be written as
$\sum_{i\in P} c_i \underline{e}_i$ with $c_i\in {\bf R}+$.
For convenience let us rearrange the indexes in $P$ in such a way
that the coefficients $c_i$ are monotonically decreasing for increasing $i$.
After such a permutation of indexes,
let us consider elements of the form $f_a=\sum_{a\geq j\geq 1}
\underline{e}_j$.
Then the same element $x$ can be written alternatively as
$$E=\sum_{l\leq n}(c_l-c_{l+1})f_l,$$
where we have set $c_{n+1}=0$. Therefore $E$ is written as an effective
(because $c_l-c_{l+1}\geq 0$ expression over the new generators $f_ls$.
On the other hand, the index sets involve in defining $f_l$ form a
monotonic chain. They are of the form
$\{1\}\subset \{1, 2\}\subset \{1, 2, 3\}\cdots $.
Therefore $f_l\cdot f_{l'}<0$ for all pairs of
$(l,l')$.
Then it is easy to see that
$$E\cdot E=\sum_{l, l'\leq n}(c_l-c_{l+1})(c_{l'}-c_{l'+1})f_l\cdot f_{l'}<0.$$
The equality
can hold only when $c_l-c_{l+1}=0$ for all $l$. As $c_{n+1}$ is defined
to be zero, all the $c_l$ must vanish. In other words, $E$ is the
zero element of the cone ${\cal EC}_b(C)$. $\Box$
\medskip
\begin{prop}\label{prop; sim}
The cone ${\cal EC}_b(C)$ is simplicial. Namely, the generators
$e_i, i\in P$ are all linear independent.
\end{prop}
\medskip
\noindent Proof of prop. \ref{prop; sim}
Suppose that $e_i$ are linearly dependent and there is a linear
equation $\sum a_i e_i=0$. We can move all the terms with negative $a_i$
to the right hand side and rewrite the equation as
$$\sum_{i\in B} b_i e_i=\sum_{j\in B'} b_j e_j, B\cap B'=\emptyset.$$
This tells us that a single element in ${\cal EC}_b(C)$ has more than
one expression in terms of the generators $e_i$.
By prop. \ref{prop; negative}, we can calculate
$$0>(\sum_{i\in B}b_i e_i)\cdot (\sum_{i\in B} b_ie_i)
=(\sum_{j\in B'}b_j e_j)\cdot (\sum_{i\in B} b_ie_i)\geq 0,$$
as $B\cap B'=\emptyset$. Contradiction! $\Box$
The proposition \ref{prop; negative}
implies that ${\cal EC}_b(C)\cap {\cal EC}_b^{\ast}(C)
=\{ 0 \}$. Namely the dual cone is completely disjoint with the
original exceptional cone. On the other hand it is easy to see that
$(C-{\cal EC}_b(C))\cap {\cal EC}_b(C)^{\ast}\not=\emptyset$
if and only if $$({\cal EC}_b(C)-C)\cap (-{\cal EC}_b^{\ast}(C))\not=\emptyset$$
if and only if $${\cal EC}_b(C)\cap( C-{\cal EC}_b^{\ast}(C))\not= \emptyset $$
We will study the subset of
lattice points in ${\cal EC}_b(C)\cap( C-{\cal EC}_b^{\ast}(C))$
closely.
\begin{defin}\label{defin; lambda}
Define the discrete set $\Lambda_b(C)$ to be
the the lattice points in ${\cal EC}_b(C)\cap( C-{\cal EC}_b^{\ast}(C))$.
\end{defin}
The elements in $\Lambda_b(C)$ are the expression $\sum_{i\in P}m_ie_i$,
$m_i\in {\bf Z}^{\geq 0}$,
such that $(C-\sum_{i\in P}m_ie_i)\cdot e_j\geq 0, j\in P$.
Let us list the basic properties of the set
${\cal EC}_b(C)\cap( C-{\cal EC}_b^{\ast}(C))$ in the following simple
proposition,
\begin{prop}
Let $E(C)$ be the intersection ${\cal EC}_b(C)\cap( C-{\cal EC}_b(C)^{\ast})$
of two different translated cones. As a proper
subset of an affine space,
$E(C)$ is convex as well as unbounded. In fact, given any point $z$ in
$E(C)$,
we consider the ray $tz, t\geq 1$, then
this ray is entirely contained in $E(C)$.
\end{prop}
\medskip
\noindent Proof: The statement regarding convexity is trivial.
Suppose $z\in E(C)$. Then $(C-z)\cdot e_i>0$ for all $i\in P$.
Then $z\cdot e_i<C\cdot e_i<0$ and we can rewrite
$(C-tz)=(C-z)-(t-1)z$ and $(C-tz)\cdot e_i=(C-z)\cdot e_i-(t-1)z\cdot e_i>0$
for all $i\in P$. Then $tz\in E(C)$. $\Box$
Because $$2d_{GT}(C)+dim_{\bf R}B=F^2-c_1({\bf K})\cdot F+dim_{\bf R}B+
\sum_{i\in P} (e_i^2-c_1({\bf K})\cdot e_i)+\Delta_C(E),$$
we are interested at knowing when does $\Delta_C(E)$ take non-positive
values.
\medskip
\begin{defin}\label{defin; allow}
An lattice element $E$ in $\Lambda_b(C)\subset {\cal EC}_b(C)$ is
said to be allowable with respect to $C$
if the function value $\Delta_C(E)$ is non-positive.
\end{defin}
The corresponding decomposition $(C-E, E=\sum m_i e_i)$ is said to be an allowable
decomposition.
Let us look at the formula
$$ \Delta_C(E)=2C\cdot E-E^2-c_1({\bf K})\cdot E-\sum_{i\in P}(e_i^2-
c_1({\bf K})\cdot e_i).$$
If we view $E$ as a moving variable in $\Lambda_b(C)$ and take an
element $z$
in $\Lambda_b(C)$, then $nz,n\geq 1$ form a sequence of lattice points in
$\Lambda_b(C)$. Using the fact that the term $E\cdot E$ is negative
definite, we find that the leading quadratic term in $n$, $n^2z\cdot z$ is always
positive. Thus it dominates all the other linear or constant
terms for the large enough
$n$.
Thus, even though the set $\Lambda_b(C)$ is unbounded, the
function $\Delta_C(E)$ is bounded below in the un-compact end. As a result,
the function $\Delta_C(E): \Lambda_b(C)\mapsto {\bf Z}$
must attain its absolute minimum somewhere.
Before discussing the geometric meaning, let us point out
that the locations of the minimums only depend on the numerical
data and is universally independent to the geometric data of
the fiber $\pi^{-1}(b)$ itself.
In principle, the elements $nz$, $n\mapsto \infty$ in the rays will not give
rise to effective decomposition $C=(C-nz)+nz$ (because
$(C-nz)\cdot \omega_{{\cal X}/B}\mapsto -\infty$)
for large enough $n$. It gives an alternative reason to discard the
non-compact end of $\Lambda_b(C)$ as one discusses the function
$\Delta_C(E)$. However, the non-effectiveness (not being representable by
pseudo-holomorphic curves) depends on the
geometric data of the fiber and is not as numerical as the dimension
constraint.
In principle, the location of the actual minimums relies on the
data of the various intersection numbers $C\cdot e_i, e_i^2,
e_i\cdot e_j$.
To characterize these
lattice points geometrically we have to give them a suitable
interpretation.
To study the geometric meaning of the minimums of $\Delta_C(E)$,
let us start with the
$|P|=1$ case first. Let us assume that ${\cal EC}_b(C)$ is a one dimensional cone
generated by a single $e$ with $e^2<0$.
One suppose that $e$ is represented
by an irreducible (pseudo)-holomorphic curve above $b\in B$.
Therefore we write
$$e^2+c_1({\bf K})\cdot e=2g-2,$$
where $g$ is the arithmetic genus of $e$ and is usually bigger than the
geometric genus of $e$ unless the curve dual to $e$ is smooth.
Any lattice point in ${\cal EC}_b(C)$
can be written as $m\cdot e, m\in {\bf N}$. Then the function
$\Delta_C(E):\{{\bf N} e\}\mapsto {\bf Z}$ can be simplified to
$$\Delta_C(m)=2m(e\cdot C)-m^2e^2-m(2g-2-e^2)-(2e^2-2g-2).$$
It is easy to find the value $m_0\in {\bf N}$
for which $\Delta_C(m)$ is minimized. We replace
$m$ by a real variable $x$ and the minimum is
achieved when the derivative is zero.
In other words, one has $$2 e\cdot C-2x\cdot e^2-2g+2+e^2=0.$$
The real number $x$ satisfying this equation is rational. To see the
geometric meaning of the solution, let us consider the expression
$e\cdot C-g$, which is always a negative number.
From simple arithmetic means, it is always possible to re-write
$eC-g$ as $l\cdot e^2+r$ with $l,r$ non-negative, with $r$ being the remainder
,$0\leq r<-e^2$. It is easy to
see that $m=l$ is the closest lattice point to the actual minimum of the
function $\Delta_C(E)$. It is due to the fact that
$f(x)=e\cdot C-xe^2-g+1+{e^2\over 2}$ has the property $f(x+1)=f(x)-e^2$ and
${e^2\over 2}< f(l)=r+1+{e^2\over 2}\leq {-e^2\over 2}$.
Therefore $l\cdot e$ is the unique lattice point
in this range. Moreover it has the crucial property that the function
$d(m)=\Delta_C(me)$ is monotonically decreasing for $m\geq m_{cri}=l$.
In the actual application of the curve counting scheme
in the enumerative application, the $g=0$ case is the most interesting situation.
The significant simple property of the function $\Delta_C(E)$
in the one dimensional
case will be used frequently later. Let us summarize it as a lemma.
\begin{lemm}(Moving lemma)\label{lemm; moving}
Suppose that all the classes $e_i$ satisfy $g(e_i)=0$.
Then the function $\Delta_C(E):\Lambda_b(C)\mapsto {\bf Z}$
is monotonically decreasing if one moves
from $E$ to $E+e_i$, for all $i$.
\end{lemm}
\medskip
\noindent Proof: Suppose that $e_i$ has been fixed in our discussion.
First we notice that $E$ can be rewritten as $E=E_0+m_ie_i$ where
$E_0=\sum_{j\not= i} m_j e_j$.
Then we compare $\Delta_C(E)$ and $\Delta_{C-E_0}(m_ie_i)$ and see
$$\Delta_C(E)-\Delta_{C-E_0}(m_ie_i)=2C\cdot E-E^2-c_1({\bf K})\cdot E-
2(C-E_0)\cdot (m_ie_i)+(m_ie_i)^2+c_1({\bf K})\cdot (m_ie_i)$$
$$=2E_0\cdot (m_ie_i)-(E_0+m_ie_i)^2+(m_ie_i)^2-c_1({\bf K})\cdot E_0=
-E_0^2+c_1({\bf K})\cdot E_0,$$ is a constant independent of
$m_i$. To show that $\Delta_C(E)$ is monotonically decreasing,
it suffices to show that $\Delta_{C-E_0}(m_ie_i)$ is monotonically
decreasing in $m_i$ for $m_ie_i$ in $\Lambda_b(C-E_0)$. This reduces
the problem to the one dimensional case, which has been discussed earlier.
$\Box$
\begin{rem}\label{rem; reallife}
It should be emphasized that in the single $e$ with $g(e)=0$ case, the integer $l$
making $\Delta_C(E)$ minimum is the first positive integer $m$ such that
$(C-me)$ lies in the dual cone ${\cal EC}_b^{\ast}(C)$.
If $C$ is represented by (pseudo) holomorphic curves, then the curve dual to
$C$
can never be irreducible. As $C\cdot e<0$, $C$ must split off as as a
curve dual to $C-me$ and one dual to $me$ with multiplicity $m$.
Symbolically we may write $C=(C-me)+me$ as a decomposition of the
cohomology classes to represent the splitting of curves.
The only general requirement upon $m$ is that
$C-me$ and $e$ exist simultaneously as pseudo-holomorphic curves above
the fiber of $b$. Thus
$(C-me)\cdot e\geq 0$ and it follows that $m\geq {C\cdot
e \over e^2}\geq l$. On the other hand, there is no a priori constraint
about $m$ other than the previous inequality.
On the other hand, for rational $e$, with $e^2+c_1({\bf K})\cdot e=-2$,
the minimal choice
$m=l$ to make $(C-me)\cdot e\geq 0$
has the additional nice property that it makes the dimension discrepancy
function
$\Delta_C(E)$ minimized. In other words, the curve in the class $C$ tends to split
off(bubbling off) a certain curves dual to $m$ multiple of $e$
such that $C-me$ has nonnegative intersection with $e$. The minimum
amount $m=l$ also makes the family moduli space dimension
$dim_{\bf R}B+d_{GT}(e)+d_{GT}(C-me)$ largest.
\end{rem}
\medskip
However this nice topological interpretation does not hold for higher
genera case. In fact, if the arithmetic genus
$g$ is larger than $0$, then the role of $e\cdot C$ is replaced by
$e\cdot C-g$ and it is $e\cdot C-g$ instead of $e\cdot C$ which
is represented as the form $le^2+r$. Thus the minimum of
$\Delta_C(E)$ usually takes place for a larger integer than what
the naive topological
constraint predicts. In other words, the largest expected family dimension
happens for the multiplicity $m_{cri}$ larger than the topological
constraint by the integer $[{g\over -e^2}]$ or $[{g\over -e^2}]+1$. \label{genera}
It is desirable to identify where can the minimum values of
$\Delta_C(E)$ occur in $\Lambda_b(C)$.
Let us consider the translated
cones $C-{\cal EC}_b(C)^{\ast}+e_i$, the translate of
the shifted dual cone $C-{\cal EC}_b^{\ast}(C)$ by the canonical
basis elements $e_i$. Then we have the
translated version of the lattice points $\Lambda_b(C)$ in the
corresponding convex set ${\cal EC}_b(C)\cap C-{\cal EC}_b^{\ast}(C)$, denoted by
$\Lambda_b(C)_i$. Next we consider the set
$$M_b(C)=\Lambda_b(C)-\cup_i(\Lambda_b(C)_i).$$
The following proposition asserts that $M_b(C)$ is a non-empty finite set.
\begin{prop}\label{prop; finite}
Let $M(C)$ be the set as defined above, then the set
$M(C)$ is non-empty and finite.
\end{prop}
\medskip
\noindent Proof:
Suppose that $M_b(C)$ is empty, then it follows that
$\Lambda_b(C)\subset \cup_i
(\Lambda_b(C)_i)$. Let us pick an arbitrary element $\lambda^{(0)}=\lambda$ in the
left hand side. Then it must be in one of the $\Lambda_b(C)_i$.
In other words, $C-(\lambda-e_i)$ also lies in ${\cal C}_E^{\ast}$.
However this implies that $\lambda^{(1)}=\lambda-e_i$ is also in $\Lambda(C)$.
From here we conclude that for any element $\lambda^{(0)}=\lambda$
in $\Lambda(C)$, there must
be some $i$ such that $\lambda^{(2)}=\lambda-e_i$ still lies in the set $\Lambda_b(C)$.
Then by induction, one may use $\lambda-e_i$ instead of $\lambda$
and conclude $\lambda^{(2)}=(\lambda-e_i)-e_j$ is still in $\Lambda_b(C)$.
However it is
impossible as it implies that given an element $\lambda^{(0)}=\lambda$, one can
indefinitely shifts it backward and get another lattice point. Because
there are only a finite number $e_i$,
we must get a lattice point $\lambda^{h}\in \Lambda_b(C)\subset {\cal EC}_b(C)$
, $h\gg 0$
with at least a negative coordinate entry with respect to some $e_i, i\in P$.
Contradiction!
Thus the existence of the lattice point
in $M_b(C)$ has been derived.
To show that $M_b(C)$ is finite, firstly
we notice that $M_b(C)$ is a discrete subset in
$H^2(\pi^{-1}(b), {\bf R})$.
\begin{lemm}\label{lemm; cpt}
\end{lemm}
The set $\overline{{\cal EC}_b(C)\cap (C-{\cal EC}_b^{\ast}(C))\cap
\bigl(\cup_i(C-{\cal EC}_b^{\ast}(C)
+e_i)\bigr)^c}$ is compact in $H^2(\pi^{-1}(b), {\bf R})$.
\medskip
\noindent Proof of the lemma:
Apparently the set is closed, we only need to check it is a bounded
subset of ${\cal EC}_b(C)\subset H^2(\pi^{-1}(b), {\bf R})$.
Suppose that $E\in \overline{{\cal EC}_b(C)\cap (C-{\cal EC}_b^{\ast}(C))\cap
\bigl(\cup_i(C-{\cal EC}_b^{\ast}(C)
+e_i)\bigr)^c}$, then $E\in {\cal EC}_b(C)\cap (C-{\cal EC}_b^{\ast}(C))$
but $E$ is not in the interior of
${\cal EC}_b(C)\cap (C-{\cal EC}_b^{\ast}(C)+e_i)$ for
all $i\in P$.
Thus, $(C-E)\cdot e_i\geq 0, i\in P$ but $(C-E+e_j)\cdot e_i\leq 0, i, j\in P$.
This implies that the values $E\cdot e_i, i\in P$ are bounded by
$$min_{j\in P}(C+e_j)\cdot e_i\leq E\cdot e_i\leq C\cdot e_i, i\in P.$$
This implies that the pairing functionals $e_i\cdot \circ$ take bounded
values on $E$. On the other hand, $e_i, i\in P$ form a basis of the
minimal vector space containing ${\cal EC}_b(C)$, the pairing functional
$e_i\cdot \circ$ is a linear coordinate system. As $E$ has bounded
coordinates, such $E$ forms a bounded subset in ${\cal EC}_b(C)$. $\Box$
From the fact that
$\overline{{\cal EC}_b(C)\cap (C-{\cal EC}_b^{\ast}(C))\cap
(\cup_i(C-{\cal EC}_b^{\ast}(C)
+e_i))^c}$ is compact and $M_b(C)$ is a discrete subset,
it follows that $M_b(C)$ has to be a finite set. $\Box$
Usually it is not clear whether the
lattice points in the finite set $M_b(C)$ are unique or not.
The following proposition clarifies the significance of the lattice points
in $M_b(C)$.
\begin{prop}\label{prop; max}
Let $E=\sum_i m_i e_i, m_i\in {\bf N}$ be in
$\Lambda_b(C)\subset {\cal EC}_b(C)\cap (C-{\cal EC}_b^{\ast}(C))$.
Then there exists at least one element $z$ in $M_b(C)$
and a finite sequence of elements $z_p\in \Lambda_b(C)$, $1\leq p\leq N$
such that
\medskip
(i). $z_1=z$, $z_N=E$.
(ii). If $p\not=1$, then $z_p-z_{p-1}=e_i$ for some $1\leq i\leq |P|$.
\end{prop}
\medskip
\noindent Proof of the proposition: $\Box$
Given an element $E\in \Lambda_b(C)$, if $E$ also lies in $M_b(C)$, we
are done. We set $N=1$ and $z_1=E=z\in M_b(C)$.
Otherwise if $x_0=E\not\in M_b(C)$, then $E\in \Lambda_b(C)_i$
for some $1\leq i\leq |P|$. In other words, there exists an $x_1\in \Lambda_b(C)$
such that $x_0=E=x_1+e_i$ for some $1\leq i\leq |P|$.
Take $x_1\in \Lambda_b(C)$ and repeat the argument, either one gets
$x_2\in \Lambda_b(C)$ such that $x_1=x_2+e_i$ for some $1\leq i\leq |P|$
or $x_2\in M_b(C)$. By induction, one may get a sequence of points
$x_n\in \Lambda_b(C)$. One argues that for a large enough $x_n$,
we must have $x_n\in M_b(C)$, or the induction process never stops and
one gets an infinite sequence $x_n\in \Lambda_b(C), n\in {\bf N}$. Because
there are a finite number of $e_i, 1\leq i\leq |P|$,
such an $x_n=\sum_{1\leq i\leq |P|}\underline{m}_ie_i$ must has a negative
entry $m_j<0$ for some $j\leq |P|$ and
falls out of ${\cal EC}_b(C)$. Contradiction!
Then we take $z_1=z=x_n\in M_b(C)$ and rename the sequence $x_p, 1\leq p\leq n$
by $z_q=x_{n+1-q}, q\leq n+1=N$.
Such a finite sequence of lattice elements satisfies
(i). $z_1=z\in M_b(C)$, $z_N=x_0=E\in \Lambda_b(C)$. $z_q\in \Lambda_b(C)$
(ii). For $p>1$, $z_p-z_{p-1}=e_i$ for some $i\leq |P|$.
\medskip
We are done. $\Box$
\medskip
We assume that $g(e_i)=0$ for all $1\leq i\leq |P|$ in the following
remark.
\medskip
\begin{rem} \label{rem; geometry}
Suppose that $C$ is represented by a pseudo-holomorphic curve over the
point $b\in B$, then one
argues that the curve must contain irreducible curves dual to the various
$e_i, i\leq |P|$. It is because $e_i$ is known to be irreducible
and pseudo-holomorphic above $b\in B$, then $C\cdot e_i\geq 0$ if all the
irreducible components of the curve dual to $C$ are distinct from the one
dual to $e_i$.
Thus, one may write
$C=(C-\sum_i m_i e_i)+(\sum_i m_i e_i)$ with $m_i\in {\bf N}$, where
$(C-\sum_i m_i e_i)$ is dual to a curve disjoint from all the exceptional
curves dual to $e_i$. Then we must have $(C-\sum_i m_i e_i)\cdot e_j\geq 0$
for all $j\leq |P|$. If we take $E=\sum_i m_i E_i\in {\cal EC}_b(C)$,
we must have $(C-E)\in {\cal EC}_b^{\ast}(C)$. In other words,
$E\in \Lambda_b(C)\subset {\cal EC}_b(C)\cap \bigl(C-{\cal EC}_b^{\ast}(C)\bigr)$.
We have use an $E$ to resemble the pattern $C$ splits into different multiples
of $e_i, i\leq |P|$.
If one imagines a pseudo-holomorphic curve dual to $C-E_0$ splits into a curve
dual to $C-E_0-e_i$ and a curve dual to $e_i$ as a degeneration process (known
as the bubbling off phenomenon in symplectic geometry), then
the lattice move $E_0\mapsto E_0+e_i$ in $\Lambda_b(C)$ is equivalent to
``bubbling off'' an unit of $e_i$ from $C$.
The path from $z$ to $E$ by a finite sequence of lattice moves as in the
condition $(ii)$ of prop. \ref{prop; max} indicates that the curve dual to
the sum of $(C-E)$ and $E$ can be viewed formally as degenerated from a curve
dual to $C-z$ and $z$ by a finite number of bubbling offs of $e_i, i\leq |P|$.
\medskip
(a). In other words, if we consider the moduli space (and its bubbling off)
of curves dual to $C$ splitting
into one dual to $C-z$ and a combination of multiple coverings of exceptional
curves dual to $z$, it contains the given curve dual to the sum of $C-E$ and $E$.
\medskip
(b). Given any pseudo-holomorphic curve dual to $C$, one may assign an unique
$E\in \Lambda_b(C)$ to it. By prop. \ref{prop; max}, one may associate
the point $z\in M_b(C)$ to $E$ with the properties,
\medskip
(i). The decomposition $C$ into $(C-z)+z$ has the highest expected
family Gromov-Taubes dimension among all
the decompositions into distinct pseudo-holomorphic curves
related by the elementary moves.
\medskip
(ii). The curve dual to $(C-E)+E$ can be thought as a degenerated version of
curves dual to $(C-z)+z$ by bubbling off a few rational exceptional curves
dual to the $e_i$.
\end{rem}
When one applies the family switching formula of rational exceptional curves,
one is able to rewrite some mixed family invariant of $C-\sum e_i$ over
a locus over which all the exceptional curves in $e_i, i\leq |P|$ co-exist, (
with all the multiplicities $m_i\equiv 1$) in terms of the family invariants
of $C-z$, $z=\sum_{1\leq i\leq |P|} n_i e_i\in M_b(C)$. Combine with remark
\ref{rem; geometry}, this tells us that formally the mixed family
invariant can be interpreted as the counting of curves in the class
$C-\sum_{1\leq i\leq |P|}n_ie_i$.
\medskip
Returning to the pointwise discussion over $b\in B$,
any two different lattice points within $M_b(C)$ cannot be related
by each other by effective translation. i.e.\ shifting from one to another by
$\sum_i c_i e_i,c_i\leq 0$. In reality the geometric meaning of
shifting toward the right means degeneration or bubbling off phenomena(in the
case $g(e_i)=0$). In this way, we may give the lattice points in
$\Lambda_b(C)$ a partial ordering and different elements in $M_b(C)$ are the
maximum elements (not necessarily the greatest element) of the partial ordering.
\begin{defin}
Let $\lambda_1$ and $\lambda_2$ be two different
lattice elements in $\Lambda_b(C)$.
We say that $\lambda_1$ is greater than $\lambda_2$, denoted by $\lambda_1
\sqsupset \lambda_2$
if $lambda_2$ can be gotten from $\lambda_1$ by a finite sequence
of effective lattice translations, i.e.\ there exists
$\lambda_2=z_n, z_{n-1}, z_{n-2}, \cdots, \lambda_1=z_1 \in \Lambda_b(C)$ such
that $z_p-z_{p-1}=n_pe_{i_p}$ for some $n_p\in {\bf N}$, $1\leq i_p\leq |P|$.
It is apparent that
this relation $\sqsupset$ is transitive.
We say that an element $\lambda$ is a maximal
element if there is no other element in $\Lambda_b(C)$ which is greater than
it under the partial ordering $\sqsupset$.
\end{defin}
\medskip
Then the set $M_b(C)$ consists of the maximum elements in
$\Lambda_b(C)$. This justifies the notation $M_b(C)$ and its dependence on
the class $C$.
From the previous lemma \ref{lemm; moving}
one finds that if $\lambda_1\sqsupset \lambda_2$, then
$\Delta_C(\lambda_1)<\Delta_C(\lambda_2)$. Therefore, in the rational case,
$M_b(C)$ also represents the lattices points of $\Lambda_b(C)$
whose $\Delta_C(E)$ values
are smallest. Even though the values of $\Delta_C(E)$ are different for
the different
lattice points, it is no use to compare their value as they are not linked
to each other by effective lattice shifting. On the other hand, our
discussion does not rule out the possibility that by different
effective shiftings one can reach from two different maximum elements $\in
M_b(C)$ to the same lattice point $\lambda\in \Lambda_b(C)$.
As effective shiftings correspond to pseudo-holomorphic curve
degenerations, the single decomposition $(C-\lambda, \lambda)$,
$\lambda\in \Lambda_b(C)$ can possibly be
degenerated from two distinct maximal decompositions
$(C-\lambda_1, \lambda_1), (C-\lambda_2, \lambda_2)$, $\lambda_1, \lambda_2
\in M_b(C)$, $\lambda_1\sqsupset \lambda, \lambda_2\sqsupset \lambda$.
In the latter part of the paper,
we would like to discuss how does the degeneration process affect the
relative
obstruction bundles.
The family switching formula assigns a relative obstruction bundle
to a degeneration of decompositions such that
$\Delta_C(E)-\Delta_C(E+\sum n_ie_i)$ is directly related to the rank of
the bundle.
It turns out that not only on the numerical level do
the partial ordering organizes the lattice points
of $\Lambda_b(C)$ in a nice way, but they also co-relate
the relative obstruction bundles.
This will be the main focus of the next section.
\medskip
\subsection{\bf The Irrational, $g(e_i)>0$, Cases} \label{subsection; irr}
\medskip
In the previous discussion, we have focused upon the $g(e_i)=0$ case.
Let us consider the general situation that
arithmetic genera $g(e_i), i\leq |P|$
may be non-zero. As we discuss earlier (see page \pageref{genera}),
the appearance of the genus term
shifts the minimum value of $\Delta_C(E)$. This happens
even when the exceptional effective cone is of one dimension.
Let us make this simplifying assumption temporally.
Once we have a closer look at the formula
$-x\cdot e^2+e\cdot C-g+1+e^2/2$,
we may collect $e\dot C-g+1$ into a single term with a particular topological
meaning. Imagining that $e$ is represented by a smooth pseudo-holomorphic curve in
$\pi^{-1}(b)$. Suppose $C$ is the first Chern class of a holomorphic line
bundle ${\bf F}$ on the Riemann surface $\Sigma\subset \pi^{-1}(b)$.
Then the expression $e\cdot C-g+1=\int_{\Sigma}c_1({\bf F})-g(\Sigma)+1$
resembles the holomorphic Euler number of the holomorphic line bundle ${\bf F}$.
The appearance of the special number indicates that the tangent-obstruction
complex would contain a term associated to
$H^0(\Sigma, {\bf F})- H^1(\Sigma, {\bf F})$,
$e=PD[\Sigma]$.
In general, consider the intersection matrix $I_{i,j}=e_i\cdot
e_j$, $1\leq i, j\leq |P|$.
Let us take the dual basis $e_i^{\ast}$ with respect to $e_i$ such that
$e_i\cdot e^{\ast}_j=I_{i,j}\delta_{i,j}$,
and the normalized dual basis ${e^{\ast}_j\over
e_j\cdot e_j}=\hat {e}_j$. Then we have $e_i\cdot \hat {e}_j=\delta_{i,j}$ for
$1\leq i, j\leq |P|$.
As the classes $C$ and $c_1({\bf K}_{{\cal X}/B})$
may not lie in the cone ${\cal EC}_b(C)$, we
project them orthogonally into the subspace ${\cal EC}_b(C)
\otimes {\bf R}$.
Denote ${\hat C}:=\sum_{1\leq i\leq |P|}(C\cdot e_i){\hat e_i}$ and
${\hat K}:=\sum_{1\leq i \leq |P|}(c_1({\bf K}_{{\cal X}/B})\cdot e_i)
{\hat e_i}$, then we must have
$$c_1({\bf K}_{{\cal X}/B})\cdot e_j={\hat K}\cdot e_j,
C\cdot e_j={\hat C}\cdot e_j.$$
The ${\hat C}$ and ${\hat K}$ lie in the subspace ${\cal
EC}_b(C)\otimes {\bf R}
\subset H^2(M, {\bf R})$ and are the projection of $C$ and
$c_1({\bf K}_{{\cal X}/B})$ to the subspace. These elements
${\hat C}$ and ${\hat K}$ are rational points of the minimal subspace
${\cal EC}_b(C)\otimes {\bf R}$.
Differentiating $\Delta_C(E)|_{E=\sum m_i e_i}$
with respect to the variable $m_i$, one derives that
$$-2E\cdot e_i+2{\hat C}\cdot e_i-{\hat K}\cdot e_i=0,1\leq i\leq |P|.$$
Then one concludes that the rational point $E=-{\hat C}+{1\over 2}{\hat K}$
is the minimum of the quadratic function $\Delta_C(E)$.
By using the adjunction equality one may rewrite
$${\hat K}=\sum_i(2g(e_i)-2-e_i^2){\hat e_i},$$
$${\hat C}-{1\over 2}{\hat K}=\sum_i(e_i\cdot C-g(e_i)
+1+{e_i^2\over 2}){\hat e_i}.$$
If the arithmetic genera $g(e_i)$
change from zero to positive values, then we find the
minimum of the function $\Delta_C(E)$ are shifted from their original value by
the rational vector
$-g(E):=-\sum_{1\leq i \leq |P|}g(e_i){\hat e_i}$. It is easy to see that
$g(E)\in {\cal EC}_b(C)\otimes {\bf R}$
is a rational element in ${\cal C}_E^{\ast}$;
the dual cone ${\cal EC}_b^{\ast}(C)$. As we have proved (in prop.
\ref{prop; negative}) that
${\cal EC}_b(C)\cap {\cal EC}_b^{\ast}(C)=\{0 \}$,
so the element $g(E)$ never lies in
the original ${\cal EC}_b(C)$.
But it is not clear from the definition whether $-g(E)$ can be in
${\cal EC}_b(C)$ or not.
\medskip
\section{\bf The Global Discussion and The Admissible Decomposition Classes}
\label{section; decomposition}
\bigskip
In the previous section, we have finished the pointwise discussion on the
exceptional cones of $C$ and the family dimensions.
We would like to patch the local discussion over $b\in B$
together and introduce some additional structure on the base space $B$ of
the fibration $\pi:{\cal X}\mapsto B$.
\medskip
Suppose that $e$ is an exceptional class in the sense of definition
\ref{defin; exception}.
Suppose that at a given point $b\in B$ the class $e$ has been
represented by an irreducible holomorphic curve in ${\cal X}_b$, one may
consider the locus $S_e\subset B$ over which the class $e$ is effective.
It is well known that $S_e$ must be a compact subset of the compact set $B$.
By adjunction equality we have
$e^2+c_1({\bf K}_{{\cal X}/B})\cdot e=2g_{arith}(e)-2$. On the other hand, the
Gromov theory predicts that the 'expected' dimension of the set $S_e$ is
$dim_{\bf C}B+{e^2-c_1({\bf K}_{{\cal X}/B})\cdot e\over 2}$.
In order for the curve to be generic, i.e. expected dimension of
$S_e\geq dim_{\bf C}B$, both inequalities
$$e^2-c_1({\bf K}_{{\cal X}/B})\cdot e\geq 0,
e^2+c_1({\bf K}_{{\cal X}/B})\cdot e\geq -2$$
have to be satisfied. In particular, $e^2=-1$ and $e$ is represented by
a genus zero smooth curve. Thus, we may conclude that the only generic
exceptional curve within a family $\pi:{\cal X}\mapsto B$ has to be
a smooth $-1$ curve.
For simplicity, let us assume that the arithmetic genus of the curve dual
to $e$,
$g_{arith}(e)=0$, in the following
discussion. Suppose that $e^2=-n$,
the irreducible rational curve representing $e$ is called a $-n$ rational
(exceptional) curve. According to the dimension formula, the expected
complex dimension of $S_e$ is $dim_{\bf C}B+{d_{GT}(e)\over 2}=dim_{\bf C}B+1-n$.
This indicates when the self-intersection number $e\cdot e$ is more negative,
the expected dimension of $S_e$ would be much lower.
\medskip
In general an irreducible holomorphic curve may degenerate and breaks into
more than one irreducible component. Then there is a subset
$S_e^{sm}\subset S_e$ over which the curve representing $e$ is
smooth and irreducible. The set $S_e-S_e^{sm}$ is the locus over which
the curve representing $e$ breaks into more than one component. In an
ideal situation, $S_e^{sm}$ is dense in $S_e$. Then $S_e-S_e^{sm}$ can be
thought as the ``boundary points'' of $S_e^{sm}$ collecting all the
degenerated configurations of $e$.
\medskip
When more than one $e$, say $e_1, e_2, \cdots, e_k$ are represented by
holomorphic curves at the same $b\in B$, the set of all such $b$ is
nothing but $S_{e_1}\cap S_{e_2}\cdots\cap S_{e_k}=\cap_{i\leq k}S_{e_i}$.
In terms of intersection theory, the expected dimension of $\cap_{i\leq k}S_{e_i}$
is $dim_{\bf C}B+{1\over 2}\sum_{i\leq k}d_{GT}(e_i)$.
\medskip
Let us list all the basic assumptions on a ``perfect''
set of $C$-exceptional classes.
\medskip
\begin{defin}\label{defin; perfect}
Let $C$ be a $(1, 1)$ class on ${\cal X}$ which restricts to non-trivial
class on an algebraic fibration $\pi:{\cal X}\mapsto B$.
A finite set of exceptional classes $Q$ on $\pi:{\cal X}\mapsto B$ is said to
be perfect if they satisfy the following list of basic assumptions.
\begin{assum}\label{assum; perfect}
\noindent (i). $e\cdot C<0$.
\noindent (ii). Either $S_e=\emptyset$ or $S_e$ is a
$dim_{\bf C}B+{1\over 2}d_{GT}(e)$ dimensional closed sub-variety of $B$.
\noindent (iii). The set $S_e^{sm}$ is dense in $S_e$ consisting smooth points
of $S_e$.
\noindent (iv). If ${\bf e}$ is a holomorphic curve in ${\cal X}_b, b\in B$
representing $e$, then all the irreducible components of ${\bf e}$ represent
exceptional classes. That is to say,
none of the irreducible components has a non-negative
self-intersection number. Moreover, if this exceptional class also
satisfies (i). then
it must also be in the original collection $Q$.
\noindent (v). Let ${\bf e}$ be a smooth irreducible holomorphic curve
in ${\cal X}_b$ representing $e$ and $b\in S_e^{sm}$. Then the
composite morphism
$${\bf T}_bB\mapsto H^1({\cal X}_b, \Theta{\cal X}_b)
\mapsto H^1({\cal X}_b, {\cal O}_{{\cal X}_b}({\bf e}))\mapsto H^1({\bf e},
{\cal N}_{\bf e}{\cal X}_b)$$
induces an isomorphism
${\bf N}_{S_e^{sm}}B|_b\mapsto H^1({\bf e}, {\bf N}_{\bf e}{\cal X}_b)$.
\noindent (vi). Suppose $e_1, e_2, \cdots, e_k$, $e_i\cdot e_j\geq 0$, $i\not= j$
are within the
collection $Q$ and are satisfying all (i).-(v)., then the locus of co-existence of
$e_1, e_2, \cdots, e_k$, $\cap_{i\leq k}S_{e_i}$, is either empty or
is a $dim_{\bf C}+{1\over 2}\sum_{i\leq k}d_{GT}(e_i)$ sub-variety in $B$.
\noindent (vii). The set $\cap_{i\leq k}S_{e_i}^{em}$ is dense
in $\cap_{i\leq k}S_{e_i}$ containing the smooth points in it.
\end{assum}
\end{defin}
\medskip
Under the condition (ii)-(iii) in the assumption
\ref{assum; perfect}, each $S_e$ defines an algebraic cycle class $[S_e]\in
{\cal A}_{dim_{\bf C}B+{1\over 2}d_{GT}(e)}(B)$ and can be
used to define the algebraic family Seiberg-Witten invariant of $e$.
The cycle class $[S_e]$ can be viewed as the moduli cycle of the
exceptional class $e$.
Likewise, the locus of co-existence $\cap_{i\leq k}S_{e_i}$ of various
$e_i, 1\leq i\leq k$, also
defines an algebraic cycle class $[\cap_{i\leq k}S_{e_i}]$, which is
identical to the intersection cycle class $\cap_{i\leq k}[S_{e_i}]\in
{\cal A}_{dim_{\bf C}B+{1\over 2}d_{GT}(e_i)}(B)$.
\medskip
\begin{rem}\label{rem; v}
In condition (v), the sheaf cohomology $H^1({\cal X}_b, \Theta{\cal X}_b)$
parametrizes the infinitesimal Kodaira-Spencer
deformations of complex structures on ${\cal X}_b$. Then
${\bf T}_bB\mapsto H^1({\cal X}_b, \Theta{\cal X}_b)$ is the tautological
map induced by the infinitesimal deformation of the family of algebraic surfaces
${\cal X}\mapsto B$.
The morphism $H^1({\cal X}_b, \Theta {\cal X}_b)\mapsto
H^1({\bf e}, {\cal N}_{\bf e}{\cal X}_b)$ is induced by the splitting
$\Theta {\cal X}_b|_{\bf e}={\cal T}_{\bf e}\oplus {\cal N}_{\bf e}{\cal X}_b$
along the holomorphic curve ${\bf e}\subset {\cal X}_b$.
\end{rem}
\medskip
Let $b\in B$ and let ${\cal EC}_b(C)$ be as defined in definition \ref{defin; exc}.
By proposition \ref{prop; sim} the cone ${\cal EC}_b(C)$ is a simplicial cone
generated by a collection of $e_i$ represented by irreducible
exceptional curves in ${\cal X}_b$.
\begin{defin}\label{defin; Qcone}
Define ${\cal EC}_b(C; Q)$ be the simplicial sub-cone of
${\cal EC}_b(C)$ generated by the elements $e_i\in Q$.
\end{defin}
\medskip
As $b$ moves on $B$, the cone ${\cal EC}_b(C; Q)$ often
changes along with $b$.
It makes sense to ask the following question,
\medskip
\noindent {\bf Question:} Describe the pattern of the variations of
${\cal EC}_b(C)$, $b\in B$, in terms of algebraic geometric datum on $B$.
\medskip
Let ${\cal C}$ be a simplicial cone generated by elements in $Q$.
\begin{defin}\label{defin; stratum}
Define $S_{\cal C}\subset B$ to be the set of all $b\in B$ such that
${\cal EC}_b(C; Q)\equiv {\cal C}$.
\end{defin}
For certain ${\cal C}$ there can be no such $b\in B$ and $S_{\cal C}$ is empty.
We are only interested at those ${\cal C}$ with a non-trivial $S_{\cal C}$ and
consider the pair $(S_{\cal C}, {\cal C})$.
\medskip
\begin{prop}\label{prop; const}
Let $Q$ be a perfect finite set of exceptional classes.
Let $(S_{\cal C}, {\cal C})$ be a pair with $S_{\cal C}\not=\emptyset$.
Then $S_{\cal C}$ is a locally closed subset of $B$ and
$S_{\cal C}\subset \overline{S_{\cal C}}$ consists of smooth points of
$\overline{S_{\cal C}}$.
\end{prop}
\noindent Proof of proposition \ref{prop; const}:
Suppose that ${\cal C}$ is generated by $e_i\in Q, 1\leq i\leq k$.
Because $Q$ is perfect, by assumption \ref{assum; perfect} (ii)., (iii).,
(vi)., (vii)., $\cap_{1\leq i\leq k}S_{e_i}$ is a
$dim_{\bf C}B+{1\over 2}\sum_{i\leq k}d_{GT}(e_i)$ dimensional sub-variety in $B$.
By our definition of ${\cal EC}_b(C; Q)$ as a subcone of ${\cal EC}_b(C)$, each
$e_i\in {\cal C}\equiv {\cal EC}_b(C; Q)$ has to be represented by a smooth and
irreducible exceptional curve above $b$. Thus,
$S_{\cal C}\subset \cap_{1\leq i\leq k}S_{e_i}^{sm}$.
The smoothness of $S_{\cal C}$ follows from condition (vii) of
assumption \ref{assum; perfect}.
We plan to argue that $S_{\cal C}$ is open and dense
in the closed set $\cap_{1\leq i\leq k}S_{e_i}$. Then
$\overline{S_{\cal C}}=\cap_{1\leq i\leq k}S_{e_i}$.
Once this is achieved, the local closeness of $S_{\cal C}$ follows.
Firstly, by deformation theory of smooth curves, the set $S_{e_i}^{sm}$ is
open in $S_{e_i}$. Thus $\cap_{1\leq i\leq k}S_{e_i}^{sm}$ is open in
$\cap_{1\leq i\leq k}S_{e_i}$.
Denote the difference $\cap_{1\leq i\leq k}S_{e_i}^{sm}-S_{\cal C}$ as $A$.
We argue that its closure $\overline{A}$ is a higher codimension subset in
$\cap_{1\leq i\leq k}S_{e_i}$.
At all $b\in A$, the class $e_i$, $1\leq i\leq k$ are
represented by irreducible exceptional curves. On the other hand,
$A\cap S_{\cal C}=\emptyset$. Thus, the cones ${\cal EC}_b(C; Q)$, $b\in A$
have to be strictly larger than ${\cal C}$. In other words, for every
$b\in A$, there must
be some additional exceptional classes $\tilde{e}\in Q$, effective and
irreducible over $b$. One may collect all such $\tilde{e}\in Q\in
{\cal EC}_b(C; Q)-{\cal C}$ into a set $E$ when $b$ runs through
all the points in $A$.
It is apparent that $A=\cap_{1\leq i\leq k}S_{e_i}^{sm}-S_{\cal C}\subset
\cup_{\tilde{e}\in E}(\cap_{1\leq i\leq k}S_{e_i}\cap S_{\tilde{e}})$.
By condition (vii) of assumption \ref{assum; perfect},
$(\cap_{1\leq i\leq k}S_{e_i}\cap S_{\tilde{e}})$ is of dimension
$dim_{\bf C}B+{1\over 2}(\sum_{1\leq i\leq k}d_{GT}(e_i)+d_{GT}(\tilde{e}))$ and
is of lower dimension than $\cap_{1\leq i\leq k}S_{e_i}$ because
$d_{GT}(\tilde{e})<0$.
Because the set $E$ is finite,
$\cup_{\tilde{e}\in E}(\cap_{1\leq i\leq k}S_{e_i}\cap S_{\tilde{e}})$,
a finite union of closed subsets of $\cap_{1\leq i\leq k}S_{e_i}$ is
closed.
Thus, one may write $S_{\cal C}$ as
$$\cap_{1\leq i\leq k}S_{e_i}^{sm}-
\cup_{\tilde{e}\in E}(\cap_{1\leq i\leq k}S_{e_i}\cap S_{\tilde{e}})$$
$$=\cap_{1\leq i\leq k}S_{e_i}-(\cap_{1\leq i\leq k}S_{e_i}-
\cap_{1\leq i\leq k}S_{e_i}^{sm})-
\cup_{\tilde{e}\in E}(\cap_{1\leq i\leq k}S_{e_i}\cap S_{\tilde{e}}),$$
and apparently is
open in $\cap_{1\leq i\leq k}S_{e_i}$. $\Box$
\medskip
Because $Q$ is a finite set, there are finitely many possible ${\cal C}$ with
non-empty $S_{\cal C}$. By its definition,
$S_{{\cal C}_1}\cap S_{{\cal C}_2}=\emptyset$ if ${\cal C}_1\not={\cal C}_2$.
\medskip
\begin{prop}\label{prop; closure}
Let $(S_{\cal C}, {\cal C})$ be a pair with $S_{\cal C}\not=\emptyset$.
The boundary set of $S_{\cal C}$, $\overline{S_{\cal C}}-S_{\cal C}$,
is contained inside
a disjoint union of different $S_{{\cal C}'}$ with ${\cal C}\subset {\cal C}'$.
\end{prop}
\medskip
\noindent Proof of proposition \ref{prop; closure}: Take an arbitrary
$b\in \overline{S_{\cal C}}-S_{\cal C}$, there are three different
possibilities.
\medskip
\noindent (A). All the $e_i$, $1\leq i\leq k$ which generates ${\cal C}$
still represent irreducible curves in ${\cal X}_b$. But some other new
exceptional class in $Q$ becomes effective over $b$ and becomes
a generator of ${\cal EC}_b(C; Q)$.
\medskip
\noindent (B). The curves representing
some or all of the $e_i$, $1\leq i\leq k$ break into
more than one irreducible components.
\medskip
\noindent (C). Some new exceptional class $\tilde{e}$, $\tilde{e}\cdot e_i\geq 0$
becomes effective over $b$, while the curves representing some of the $e_i$
break into more than one component. (The mixture of (A). and (B).)
\medskip
Suppose that $\tilde{e}_j\in Q, 1\leq j\leq l$ are the new exceptional class(es)
in case (A)., consider the cone ${\cal C}'$ generated by $e_i, 1\leq i\leq k$ and
$\tilde{e}_j, 1\leq j\leq \tilde{k}$.
Such points $b$ are contained in the set $S_{{\cal C}'}$ and apparently
${\cal C}\subset {\cal C}'$.
In case (B)., suppose that, say $e_1$, has broken into components,
$e_1=\sum_{1\leq r\leq r_1} e_{1;r}$. Then for at least one $e_{1; r}$,
$e_{1; r}$ pairs negatively with $C$, i.e. $e_{1; r}\cdot C<0$. If not,
$$0>e_1\cdot C=(\sum_{1\leq r\leq r_1}e_{1;r})\cdot C=\sum_{1\leq r\leq r_1}
e_{1; r}\cdot C\geq 0,$$
contradicting to the condition (i). of the perfectness assumption.
On the other hand, the condition (iv) of perfectness assumption
\ref{assum; perfect} implies that $e_{1;r}\in Q$ if $e_{1;r}\cdot C<0$.
So ${\cal EC}_b(C; Q)\not=\emptyset$.
Such a $b$ is in $S_{{\cal C}'}$ with ${\cal C}'={\cal EC}_b(C; Q)$.
Because $b$ is in the closure of
$S_{\cal C}$ over which the classes in ${\cal C}$ are
effective, by the degeneration argument
all classes in ${\cal C}$ remain effective over $b$ as well.
This implies that ${\cal C}\subset {\cal EC}_b(C; Q)={\cal C}'$.
The discussion for the case (C). is similar and we leave it to the reader. $\Box$
\medskip
The proposition motivates us to define a partial ordering among
different $(S_{\cal C}, {\cal C})$.
\medskip
\begin{defin}\label{defin; partial}
The pair $(S_{\cal C}, {\cal C})$ is said to be greater than
$(S_{\cal C}', {\cal C}')$ under $\succ$, denoted as
$(S_{\cal C}, {\cal C})\succ (S_{\cal C}', {\cal C}')$,
if ${\cal C}\subset {\cal C}'$.
\end{defin}
Notice that $\succ$ is a necessary condition for $\overline{S_{\cal C}}\cap
S_{{\cal C}'}$ to be non-empty.
\medskip
\subsection{\bf The Admissible Decomposition Classes over $S_{\cal C}$}
\label{subsection; adm}
\bigskip
Having addressed the structure of $S_{\cal C}$, we move ahead to
address the decomposition classes.
Consider the disjoint union $\coprod_{{\cal C}, S_{\cal C}\not=\emptyset}
S_{\cal C}$. Either it is equal to the whole $B$, or it is a closed
subset (by proposition \ref{prop; closure}) of $B$.
In the first case, some $S_{\cal C}$ is of top dimension in $B$ and
${\cal C}$ is generated by a finite number of $-1$ classes in ${\cal C}$.
If it is the case, any effective curve dual to $C$ in the fibers of the
family ${\cal X}\mapsto B$ over $S_{\cal C}$
must break off a certain multiples of $-1$ curves and the family
theory suffers the same symptom as in Mcduff's proposal [Mc] ($B=pt$).
The more interesting situation is when
$\coprod_{{\cal C}, S_{\cal C}\not=\emptyset} S_{\cal C}\subset B$
is a proper closed subset.
In such a situation, any effective curve dual to $C$ over
$b\in \coprod_{{\cal C}, S_{\cal C}\not=\emptyset} S_{\cal C}$ has to
break off certain multiples of exceptional curves in ${\cal EC}_b(C; Q)$.
What types of exceptional curves it has to break off depends on which
$S_{\cal C}$ does $b$ lie in.
\medskip
A sketch of the general enumerative application of our curve counting scheme
to the family algebraic Seiberg-Witten invariants
${\cal AFSW}_{{\cal X}\mapsto B}(1, C)$ can be outlined as the following.
(1). Break the whole base space $B$ into different $S_{\cal C}$ and
$B-\coprod_{{\cal C}; S_{\cal C}\not=\emptyset} S_{\cal C}$.
This set level decomposition gives a stratification of $B$ and
$B-\coprod_{{\cal C}, S_{\cal C}\not=\emptyset} S_{\cal C}$ is the only
top dimensional stratum. The dimension of each $S_{\cal C}$ can
be calculated through the argument of proposition \ref{prop; const}.
\medskip
(2). Attach a local family invariant contribution to each
$\overline{S_{\cal C}}
\not=\emptyset$ depending on the generators $e_i, 1\leq i\leq k$
of ${\cal C}$ and the breaking of $C$ into $C-\sum_{1\leq i\leq k}e_i$
and $\sum_{1\leq i\leq k}e_i$. (see [Liu1] and [Liu3] for
some examples)
The invariant contribution is some mixed family invariant of the form
${\cal AFSW}_{{\cal X}\times_B \overline{S_{\cal C}}\mapsto \overline{S_{\cal C}}}
(\cdot, C-\sum_{1\leq i\leq k}e_i)$.
We expect the local family invariant contribution to be nonzero only if
$dim_{\bf C}B+{1\over 2}
d_{GT}(C-\sum_{1\leq i\leq k}e_i)+{1\over 2}\sum_{1\leq i\leq k}d_{GT}(e_i)\geq
dim_{\bf C}B+d_{GT}(C)$.
\medskip
(3). Apply the family switching formula [Liu3]
to the local family invariant contribution
and identify it with certain mixed family invariant of $C-\sum_{1\leq i\leq k}
n_ie_i$ over $\overline{S_{\cal C}}$. The changing of the
multiplicities of $e_i$, $(1, 1, 1, \cdots, 1)$ to
$(n_1, n_2, n_3, \cdots, n_k)$ is called the switching process which
has been discussed in section \ref{section; point}.
\medskip
(4). As different $\overline{S_{\cal C}}$ may intersect, the naive
subtraction of all the local family invariant contributions from
the original ${\cal AFSW}_{{\cal X}\mapsto B}(1, C)$ leads to over-subtraction.
\medskip
Inductively, one has to define a version of modified family invariant
(see section 5.3. of [Liu1] for some explicit examples in the
differentiable category) for each $S_{\cal C}$.
Schematically it involves subtracting the modified family invariants
(which have already been defined by the induction hypothesis) of
$S_{{\cal C}'}$, $\overline{S_{\cal C}}\supset S_{{\cal C}'}$ to define
the modified family invariant of $S_{\cal C}$.
\medskip
(5). The inductive scheme passes through $B-\coprod_{\cal C}S_{\cal C}$ and
finally
one may define a modified family invariant of $B-\coprod_{\cal C}S_{\cal C}$
by subtracting all the modified family invariants of $S_{\cal C}$ defined.
In the algebraic category, it involves the usage of residual intersection
theory in [F] in arguing that the whole family moduli space of $C$ over $B$,
${\cal M}_C\mapsto B$, can be separated into the subscheme over
$\coprod_{{\cal C}}S_{\cal C}$ and the residual portion (which
definition involves blowing ups inductively). The
modified invariant of $C$ is actually equal to the intersection number
defined by the residual intersection theory. (See [Liu1] for a
discussion in the ${\cal C}^{\infty}$ category.)
\medskip
\begin{defin}\label{defin; admissible}
Let $({\cal C}, S_{\cal C})$ be a pair and let $e_i$ be the generators of
${\cal C}$. Suppose that the decomposition
$(C-\sum_{i\leq k}e_i, \sum_{i\leq k}e_i)$ of $C$ satisfies the
dimension inequality
$$d_{GT}(C-\sum_{i\leq k}e_i)+\sum_{i\leq k}d_{GT}(e_i)\geq d_{GT}(C),$$
then we consider all the $k-$tuples $(n_1, n_2, \cdots, n_k)$, $n_i\in {\bf N}$
such that
$$d_{GT}(C-\sum_{i\leq k}n_ie_i)+\sum_{i\leq k}d_{GT}(e_i)\geq d_{GT}(C).$$
Moreover,
there exists at least one $i_0$ with $1\leq i_0\leq k$, $n_{i_0}\geq 2$ such that
$$d_{GT}(C-\sum_{i\leq k; i\not=i_0}n_ie_i-(n_{i_0}-1)e_{i_0})+
\sum_{i\leq k}d_{GT}(e_i)\geq d_{GT}(C).$$
All such $k-$tuples define decompositions $(C-\sum_{i\leq k}n_ie_i, \sum_{i\leq k}
n_ie_i)$ which can be derived from $(C-\sum_{i\leq k}e_i, \sum_{i\leq k}e_i)$
through elementary effective moves maintaining their family dimensions above
the lower bound ${1\over 2}d_{GT}(C)+p_g+dim_{\bf C}B$.
The set of all such decompositions forms an equivalence class called the
admissible decomposition class associated to $({\cal C}, S_{\cal C})$.
\end{defin}
\medskip
The family switching formula [Liu3]
allows us to relate the family invariants of
different $(C-\sum_{i\leq k}n_ie_i, \sum_{i\leq k}n_ie_i)$ over
$\overline{S_{\cal C}}$. That is why we
view these different decompositions as equivalent.
\medskip
In the published long paper [Liu1], one takes the universal spaces projection
$f_n:M_{n+1}\mapsto
M_n$ of an algebraic surface $M$ as the fiber bundle ${\cal X}\mapsto B$.
Let $C$ be a $(1, 1)$
class on $M$, then the set $Q$ has been implicitly chosen to be the set
of all type $I$ exceptional classes $e$ with $e\cdot (C-{\bf M}(E)E)<0$.
A discussion parallel to the five steps above has been developed
in the ${\cal C}^{\infty}$ category. Please consult [Liu1] for the
notations and the details.
\bigskip
\begin{rem}\label{rem; nonideal}
We have made the simplifying assumptions \ref{assum; perfect} to study the
stratification of the base space $B$. If the conditions (ii)-(v) in
assumption \ref{assum; perfect} are not satisfied, the existence loci
of exceptional curves may not be of the right dimension and one has to
work with the Kuranishi models of the exceptional class to construct
the algebraic cycle class representing the moduli cycles.
Moreover, the violation of the conditions (vi)-(vii) indicates that
the moduli spaces of different exceptional
classes may not intersect transversally in $B$. In the
algebraic category we may use the intersection cycle class
of the moduli cycle classes to represent the co-existence of different
exceptional classes. The theory will be developed else where.
\end{rem}
\bigskip | 168,856 |
\begin{document}
\title{A two-dimensional electrostatic model of interdigitated comb drive in longitudinal mode}
\author{Antonio Gaudiello\footnote{Dipartimento di Matematica e Applicazioni "Renato Caccioppoli",
Universit\`a degli Studi di Napoli Federico II, Via Cintia, Monte S.
Angelo, 80126 Napoli, Italia. e-mail: [email protected]} and
Michel Lenczner\footnote{ UBFC / UTBM / FEMTO-ST, 26, Chemin de
l'Epitaphe, 25030 Besan\c{c}on, France. e-mail:
[email protected] }}\date{ } \maketitle
\begin{abstract}
A periodic homogenization model of the electrostatic equation is constructed
for a comb drive with a large number of fingers
and whose mode of operation is in-plane and longitudinal. The model is
obtained in the case where the distance between the rotor and the stator is
of an order $\varepsilon ^{\alpha }$, $\alpha \geq 2$, where $\varepsilon$ denotes the period of distribution of the fingers. The
model derivation uses the two-scale convergence technique.\ Strong convergences are also established. This allows us to find, after a proper scaling,
the limit of the electrostatic force applied
to the rotor in the longitudinal direction.
\noindent Keywords: {Comb drive, electrostatic forces, MEMS, homogenization }
\medskip
\par
\noindent2010 \AMSname: 35J05, 35B27
\end{abstract}
\section{Introduction}
The technology of Micro-Electro-Mechanical Systems, or MEMS, includes both
mechanical and electronic components on a single chip built with micro
fabrication techniques. The main MEMS parts are sensors, actuators, and
microelectronics. Many types of micro actuation techniques are available,
the most common of which are piezoelectric, magnetic, thermal,
electrochemical, and electrostatic actuation. The latter is clearly the most
widespread because of its compatibility with microfabrication technology,
its ease of integration and its low energy consumption. In particular,
electrostatic comb drives, introduced in \cite{tang1989laterally,
tang1990electrostatic} to enable large travel range at low driving voltage,
are among the most used electrostatically actuated devices in
microelectromechanical systems containing movable mechanical structures.
\begin{figure}[h]
\centering
\includegraphics[scale=0.25]{combDriveScheme1.png}
\caption{The comb drive}
\label{Fig1}
\end{figure}
A comb drive is a deformable capacitor consisting of conductive
stator and rotor, each one composed of parallel fingers, that are
interdigitated, and whose number may exceed one hundred. The stator is
clamped and the rotor is suspended on elastic springs. The elastic
suspension is designed to allow the rotor to move in one of the desired
directions: longitudinal direction, i.e. parallel to the fingers, or in one of the two perpendicular
directions. From the electrical point of view, the stator is grounded and
the rotor is subjected to an electric potential $V$. The difference in
voltage induces an electrostatic force between the stator and the rotor
which causes a displacement of the rotor and therefore restoring forces in
the suspension. The equilibrium state is reached when the mechanical
restoring forces balance the electrostatic force.
The advantages of using electrostatic comb drive actuator approach include
low power dissipation, simple electronic control, and easy capacity-based
sensing mechanism. These devices are intended for applications in mechanical sensors, RF
communication, microbiology, mechanical power transmission,
long-range actuation, microphotonics, and microfluids
\cite{tang1990electrostatic, yeh1999integrated, kim1992silicon,
geiger1998new}.
To achieve considerable electrostatic forces without
reverting to excessively high driving voltages, the freespace gap between the
electrodes must be minimal. With the advances of microfabrication
technology, thinner fingers and smaller gaps can be micromachined. This can
allow for a denser spacing of fingers and thus increase the power density of
comb drive actuators.
Design of complex MEMS involving multiple comb drives can not be performed
by trial and error due to the high microfabrication cost and time
consumption. Designers then make an intensive use of models. Part of the
comb drive modeling works focuse on the development of analytical models
that, beyond taking into account the electrostatic forces between parallel
parts, describe the fringe fields according to different methods and in many
configurations \cite{johnson1995electrophysics}, \cite{yeh2000electrostatic}
, \cite{hammer2010analytical}, \cite{he2016analytical} \cite{li2012improved}
, \cite{he2014calculating}, \cite{li2012improved}, and the analytical models
in the software package Coventor MEMS+ \cite{coventorMEMS+}.
On the other side, the use of direct numerical simulation remains the
reference approach for general configurations. Most often it is carried out
by a finite element method \cite{dong2011analysis}, \cite
{chyuan2008computational}, \cite{ouakad2015numerical}, or a boundary
element method \cite{chyuan2004computational},
\cite{liao2004alternatively}. Despite the impressive increase of
computer power, the time scale required by their use for direct
simulation, optimization or calibration of complex systems is still
incompatible with the time scale of a designer.
Until now, the use of multiscale methods has not been yet explored on this
family of problems despite their periodic structure. However, they can offer
a good compromise between numerical methods adapted to general physics and
geometries, but expensive in simulation time, and analytical methods
developed for particular physics and geometries requiring only a few
computation resources.
In this paper we develop a first comb drive multiscale model based on asymptotic methods. Precisely, we consider a 2-dimensional model for an in-plane comb drive, in a vacuum and in statical longitudinal regime, made by a rotor called $\Omega^a_{\varepsilon,\alpha}$ and a stator called $\Omega^b_{\varepsilon,\alpha}$ (see Figure \ref{Fig1}).
Both of them are composed by a set of $\varepsilon$-periodic fingers, with cross-section of order $\varepsilon$. The goal of this paper is to study the asymptotic behaviour of the longitudinal electrostatic force applied on the rotor with respect to two parameters: the period $\varepsilon$ and the small distance between the rotor and the stator. {\it A priori} estimates show that in this model a discriminating role is played by this distance that we consider of order $\varepsilon^\alpha$. Precisely, we prove that if $\alpha\geq2$ for obtaining asymptotically a force of order $O(1)$, the applied voltage has to be of order $\varepsilon^\alpha V$ and in this case the limit force is given by
\begin{equation}\label{formulaprinciple}-\frac{\epsilon_0}{2}V^2L\left( \hbox{meas}(\omega^a)+\hbox{meas}(\omega^b)\right)\end{equation}
where $\epsilon_0$ is the vacuum permittivity, $V$ is a constant independent of $\varepsilon$, $L$ the comb length, and $\hbox{meas}(\omega^a)$ and $\hbox{meas}(\omega^b)$ the length of the cross section of the reference finger of the rotor and of the stator, respectively (see Figure \ref{Fig1}). This result shows that only the longitudinal forces on the extremities of the rotor's fingers and on the part of the rotor's boundary corresponding to the orthogonal projection of the stator's fingers play a significant role. In particular, this means that the fringe field can be neglected in the asymptotic regime $\alpha\geq2$. We expect that this phenomenon appears when $0\leq\alpha<2$. We also underline that in the limit force there is no contribution of boundary layer effect on the lateral side of the comb, that are expected in other regimes.
The paper is organized in the following way. The geometry of the comb drive is rigorously described in Section \ref{geometryapril2019}. The problem satisfied by the electrical potential in the vacuum between the rotor and the stator is given in Section \ref{Probbb} (see \eqref{J13,2019strong} where the voltage source is normalized by assuming it equal to 1). The main result of this paper, i.e. the proof of formula \eqref{formulaprinciple}, is stated in Theorem \ref{main theoremapril24,2019}. Section \ref{rescscsc} is devoted to rescale the problem given in Section \ref{Probbb} to a problem on a domain where the finger's height is independent of $\varepsilon$ (see Figure \ref{Fig2}). Thus, the problem is split on three subdomains $\Omega^{c,1}_\varepsilon$, $\Omega^{c,2}_\varepsilon$, and $\Omega^{c,3}_\varepsilon$ (see Figure \ref{Fig3}).
Moreover, in Proposition \ref{PropF25,2019} we prove a key result which allows us to transform the longitudinal force applied on the rotor's boundary part $\Gamma^a_{\varepsilon,\alpha}$ (see formula in \eqref{rafbarr} and also p. 225 in \cite{kovetz2000electromagnetic}) into an integral on $\Omega^{c,1}_\varepsilon\cup\Omega^{c,2}_\varepsilon\cup\Omega^{c,3}_\varepsilon$.
{\it A priori} estimates of the rescaled solution of problem \eqref{J13,2019strong} are obtained in Section \ref{apapstst}. They suggest that different regimes depending on $\alpha$ can be expected. Section \ref{casesssalpha=2} is devoted to prove Theorem \ref{main theoremapril24,2019} in the case $\alpha=2$. The proof consists of several steps. In Section \ref{stime dettagliate alpha=2}, further {\it a priori} estimates of the rescaled solution are derived in the case $\alpha=2$. These estimates provide two-scale convergences (the two-scale convergence technique was proposed in \cite{N} and developed in
\cite{A}, see also \cite{Casado}, \cite{CioDaGri}, and \cite{Lenc}). Then, in Section \ref{wweeaakkconv} the two-scale limits are identified on each subdomain $\Omega^{c,1}$, $\Omega^{c,2}$, and $\Omega^{c,3}$ (see Figure \ref{Fig4}). The limit results are improved in Section \ref{corrrresult} by corrector results. Finally in Section \ref{proofmmmmaitttheore}, these correctors allow us to pass to the limit in the formula of the longitudinal force stated in Proposition \ref{PropF25,2019} and to prove Theorem \ref{main theoremapril24,2019} in the case $\alpha=2$. The proof of Theorem \ref{main theoremapril24,2019} in the case $\alpha>2$ is only sketched in Section \ref{sketched}.
Homogenization of oscillating boundaries with fixed amplitude is widely studied and we refer to the following main papers: \cite{AiNaPr}, \cite{ki01ora}, \cite{ki01}, \cite{ki3}, \cite{BaCon}, \cite{ki9}, \cite{Bgg1}, \cite{BgM}, \cite{Blg}, \cite{BoOrRo}, \cite{BrChi}, \cite{BrChTh}, \cite{Chechtar}, \cite{DamPe}, \cite{Pan}, \cite{DeDuMe}, \cite{DeNan}, \cite{Durante1}, \cite{Durantebis}, \cite{EK}, \cite{gagu2}, \cite{gagumu}, \cite{gamel2}, \cite{GSili}, \cite{Lenczner} \cite{LencSmith}, \cite{m1}, \cite{m4}, \cite{NaPrSa}, \cite{Nk}, and \cite{Vinh}.
Also the homogenization of boundaries with oscillations having small amplitude has a wide bibliography, but this argument is beyond the scope of this paper and a reader interested in this subject can see some references quoted in \cite{gagumu}.
\section{The geometry\label{geometryapril2019}}
Let $ \zeta_1,\,\zeta_2,\,\zeta_3,\,\zeta_4\in]0,1[$ be such that
$$\zeta_1<\zeta_2<\zeta_3<\zeta_4,$$
and set $$ \omega^a=]\zeta_1,\zeta_2[,\quad \omega^b=]\zeta_3,\zeta_4[,$$
$$ \hbox{meas}(\omega^a)=\zeta_2-\zeta_1, \quad \hbox{meas}(\omega^b)=\zeta_4-\zeta_3.$$
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{combDriveScheme2.png}
\caption{ The rescaled comb drive \label{Fig2}}
\includegraphics[scale=0.35]{combDriveScheme3.png}
\caption{Decomposition of the rescaled comb drive \label{Fig3}}
\end{figure}
Let $\alpha\in[0,+\infty[$, $L\in ]0,+\infty[$, and $l_1,l_2,l_3\in ]0,+\infty[$ be such that
$$l_1+2<l_2<l_3.$$
For every $\varepsilon\in \left\{\frac{L}{n}:n\in \mathbb{N}\right\}$ set (see Figure \ref{Fig1} for $\alpha>0$ or Figure \ref{Fig2} for $\alpha=0$)
$$\Omega^a_{\varepsilon,\alpha}=\left(]0,L[\times]l_2,l_3[\right)\cup\left(\bigcup_{k=0}^{\frac{L}{\varepsilon}-1}\left(\varepsilon\omega^a+\varepsilon k\right)\times ]l_1+\varepsilon^\alpha,l_2]\right),$$
$$\Omega^b_{\varepsilon,\alpha}=\left(]0,L[\times]0,l_1[\right)\cup\left(\bigcup_{k=0}^{\frac{L}{\varepsilon}-1}\left(\varepsilon\omega^b+\varepsilon k\right)\times [l_1,l_2-\varepsilon^\alpha[\right),$$
$$ \Omega^c_{\varepsilon,\alpha}=\left(]0,L[\times]0,l_3[\right) \setminus\left( \overline{\Omega^a_{\varepsilon,\alpha}}\cup \overline{\Omega^b_{\varepsilon,\alpha}}\right),$$
$$ \Gamma^a_{\varepsilon,\alpha}=\partial \Omega^a_{\varepsilon,\alpha}\cap\partial \Omega^c_{\varepsilon,\alpha},$$
$$ \Gamma^b_{\varepsilon,\alpha}=\partial \Omega^b_{\varepsilon,\alpha}\cap\partial \Omega^c_{\varepsilon,\alpha},$$
$$ \Gamma_{\varepsilon,\alpha}= \Gamma^a_{\varepsilon,\alpha}\cup \Gamma^b_{\varepsilon,\alpha},$$
$$\Gamma=\{0,L\}\times]l_1,l_2[.$$
where $\Omega^a_{\varepsilon,\alpha}$ models the rotor, $\Omega^b_{\varepsilon,\alpha}$ the stator, each one composed of parallel fingers that are
interdigitated, $\Omega^c_{\varepsilon,\alpha}$ the vacuum between the rotor and the stator, and $ \Gamma^a_{\varepsilon,\alpha}$ and $ \Gamma^b_{\varepsilon,\alpha}$ are the parts of the boundary of the rotor and of the stator facing each other.
Moreover, setting (see Figure \ref{Fig3} for $\alpha=0$)
$$ \Omega^{c,1}_{\varepsilon,\alpha}= \Omega^c_{\varepsilon,\alpha}\cap\left(]0,L[\times ]l_1,l_1+\varepsilon^\alpha[\right),$$
$$ \Omega^{c,2}_{\varepsilon,\alpha}= \Omega^c_{\varepsilon,\alpha}\cap\left(]0,L[\times [l_1+\varepsilon^\alpha,l_2-\varepsilon^\alpha]\right),$$
$$ \Omega^{c,3}_{\varepsilon,\alpha}= \Omega^c_{\varepsilon,\alpha}\cap\left(]0,L[\times ]l_2-\varepsilon^\alpha, l_2[\right),$$
the vacuum is split in three parts
$$\Omega^c_{\varepsilon,\alpha}=\Omega^{c,1}_{\varepsilon,\alpha}\cup\Omega^{c,2}_{\varepsilon,\alpha}\cup\Omega^{c,3}_{\varepsilon,\alpha}.$$
Furthermore, set (see Figure \ref{Fig4})
$$\Omega^{c,1} =]0,L[\times]l_1,l_1+1[,\quad \Omega^{c,2} =]0,L[\times]l_1+1,l_2-1[,\quad \Omega^{c,3} =]0,L[\times]l_2-1,l_2[.$$
\begin{Remark} For simplicity we assumed $\varepsilon\in \left\{\frac{L}{n}:n\in \mathbb{N}\right\}$. Of course, with small modifications in the proofs, all results of this paper hold true with $\varepsilon\in ]0,1[$.
\end{Remark}
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{combDriveScheme4.png}
\caption{The limit domains}
\label{Fig4}
\end{figure}
\section{The problem\label{Probbb}}
Let $\alpha\in [0,+\infty[$. Then, for every $\varepsilon$ consider the following normalized problem
\begin{equation}\label{J13,2019strong}\left\{\begin{array}{lll}
-\Delta\phi_{\varepsilon}=0, \hbox{ in } \Omega^c_{\varepsilon,\alpha},\\\\
\phi_{\varepsilon}= 1, \hbox{ on } \Gamma^a_{\varepsilon,\alpha},\\\\
\phi_{\varepsilon}= 0, \hbox{ on } \Gamma^b_{\varepsilon,\alpha},\\\\
\nabla \phi_{\varepsilon}\cdot\nu=0, \hbox{ on } \Gamma,
\end{array}\right.\end{equation}
where $\nu$ denotes the unit normal
to $\Gamma$ exterior to $ \Omega^c_{\varepsilon,\alpha}$. The solution $\phi_{\varepsilon}$ represents the electrical potential in the vacuum $ \Omega^c_{\varepsilon,\alpha}$ when the stator is grounded and the voltage in the rotor is assumed equal to $1$. By setting
$$\mu_{\varepsilon,\alpha}=\left\{\begin{array}{ll}1,\hbox{ on } \Gamma^a_{\varepsilon,\alpha},\\\\
0,\hbox{ on } \Gamma^b_{\varepsilon,\alpha},\end{array}\right.$$
the weak formulation of \eqref{J13,2019strong} is
\begin{equation}\label{J13,2019weak}\left\{\begin{array}{lll}\phi_{\varepsilon}\in H^1_{ \Gamma_{\varepsilon,\alpha}}(\Omega^c_{\varepsilon,\alpha},\mu_{\varepsilon,\alpha}),\\\\\displaystyle{\int_{\Omega^c_{\varepsilon,\alpha}}\nabla \phi_{\varepsilon}\nabla \psi dx=0,\quad \forall\psi \in H^1_{ \Gamma_{\varepsilon,\alpha}}(\Omega^c_{\varepsilon,\alpha},0),}
\end{array}\right.\end{equation}
where for $g\in H^{-\frac{1}{2}} (\Gamma_{\varepsilon,\alpha})$ it is set $$H^1_{ \Gamma_{\varepsilon,\alpha}}(\Omega^c_{\varepsilon,\alpha},g)=\{\psi\in H^1(\Omega^c_{\varepsilon,\alpha}):\psi=g, \hbox{ on }\Gamma_{\varepsilon,\alpha}\}.$$
According to \cite{kovetz2000electromagnetic}, p. 225, the longitudinal electrostatic force on rotor's boundary $\Gamma^a_{\varepsilon,\alpha}$ generated by the electrical potential $\varepsilon^\alpha V \phi_{\varepsilon}$ in the vacuum is given by
\begin{equation}\label{rafbarr}-\frac{\epsilon_0}{2}V^2\int_{\Gamma^a_{\varepsilon,\alpha}} |\varepsilon^\alpha\nabla \phi_{\varepsilon}|^{2}\nu_2 \, ds,\end{equation}
where $\epsilon_0$ is the vacuum permittivity,$V$ is a constant independent of $\varepsilon$, and $\nu_2$ denotes the second component of the unit normal to $\Gamma^a_{\varepsilon,\alpha}$ exterior to $ \Omega^c_{\varepsilon,\alpha}$.
The main result of this paper is the following one.
\begin{Theorem}\label{main theoremapril24,2019} For every $\varepsilon$, let $\phi_\varepsilon$ be the unique solution to \eqref{J13,2019weak} with $\alpha\geq2$ and let $\nu_2$ denote the second component of the unit normal to $\Gamma^a_{\varepsilon,\alpha}$ exterior to $ \Omega^c_{\varepsilon,\alpha}$. Then,
\begin{equation}\label{F25,2019has}\lim_{\varepsilon\rightarrow 0}
\int_{\Gamma^a_{\varepsilon,\alpha}} |\varepsilon^\alpha\nabla \phi_{\varepsilon}|^{2}\nu_2 \, ds =L\left( \hbox{meas}(\omega^a)+\hbox{meas}(\omega^b)\right),\end{equation}
where $L$, $\omega^a$, and $\omega^b$ are defined in Section \ref{geometryapril2019}.
\end{Theorem}
In the sequel, the dependence on $\alpha$ of the domain will be omitted when $\alpha=0$. For instance, $\Omega^a_{\varepsilon,0}$ will be denoted by $\Omega^a_\varepsilon$, and so on.
\section{The rescaling\label{rescscsc}}
By virtue of transformation (see Figure \ref{Fig1} and Figure \ref{Fig2})
\begin{equation}\label{J24,2019trasf} T_{\varepsilon,\alpha}: \Omega^{c}_\varepsilon\rightarrow \Omega^{c}_{\varepsilon,\alpha}
\end{equation}
defined by
\begin{equation}\label{J24,2019trasfpart}\left\{\begin{array}{ll} (x_1,x_2)\in \Omega^{c,1}_\varepsilon\rightarrow\left(x_1,(x_2-l_1)\varepsilon^\alpha+l_1\right)\in \Omega^{c,1}_{\varepsilon,\alpha},\\\\
(x_1,x_2)\in \Omega^{c,2}_\varepsilon\rightarrow \left(x_1,D_\varepsilon(x_2-l_1-1)+l_1+\varepsilon^\alpha\right)\in \Omega^{c,2}_{\varepsilon,\alpha},\\\\
(x_1,x_2)\in \Omega^{c,3}_\varepsilon\rightarrow\left(x_1,(x_2-l_2+1)\varepsilon^\alpha+l_2-\varepsilon^\alpha\right)\in \Omega^{c,3}_{\varepsilon,\alpha},\end{array} \right.\end{equation}with
\begin{equation}\label{J24,2019trasfpartcoff} D_\varepsilon=\frac{l_2-l_1-2\varepsilon^\alpha}{l_2-l_1-2},\end{equation}
problem \eqref{J13,2019weak} is rescaled in the following one
\begin{equation}\label{J13,2019weakrescaled}\left\{\begin{array}{lll} \varphi_\varepsilon\in H^1_{ \Gamma_\varepsilon}(\Omega^c_\varepsilon,\mu_\varepsilon),\\\\
\displaystyle{\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \varepsilon^\alpha \partial_{x_1} \varphi_\varepsilon \partial_{x_1}\psi+\varepsilon^{-\alpha} \partial_{x_2} \varphi_\varepsilon \partial_{x_2}\psi \right)dx}\\\\ \displaystyle{+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon \partial_{x_1} \varphi_\varepsilon \partial_{x_1}\psi+D^{-1}_\varepsilon\partial_{x_2} \varphi_\varepsilon \partial_{x_2}\psi \right)dx=0, \quad\forall\psi \in H^1_{ \Gamma_\varepsilon}(\Omega^c_\varepsilon,0).
}
\end{array}\right.\end{equation}
Remark that
\begin{equation}\label{J15,2019}\lim_{\varepsilon\rightarrow 0} D_\varepsilon= \frac{l_2-l_1}{l_2-l_1-2}.
\end{equation}
Let \begin{equation}\label{F16,2019z}\varphi^\star\in C^\infty(\mathbb{R}\times[l_1,l_2])\end{equation} be such that
\begin{equation}\label{F16,2019z1}\left\{\begin{array}{ll}\varphi^\star(\cdot,x_2)\hbox{ is 1-periodic for every } x_2\in [l_1,l_2],
\\\\ \varphi^\star=1,\hbox{ in }\omega^a\times]l_1+1,l_2[, \quad \varphi^\star=0,\hbox{ in }\omega^b\times]l_1,l_2-1[, \\\\
\varphi^\star=1,\hbox{ on }\mathbb{R}\times\{l_2\}, \quad \varphi^\star=0,\hbox{ on }\mathbb{R}\times\{l_1\},
\end{array}\right.\end{equation}
and for every $\varepsilon\in]0,1[$ set
\begin{equation}\label{F16,2019zq}\varphi^\star_\varepsilon(x_1, x_2)=\varphi^\star\left(\frac{x_1}{\varepsilon}, x_2\right), \hbox{ in }\mathbb{R}\times[l_1,l_2].\end{equation}
The previous rescaling allows us to rewrite formula \eqref{rafbarr}.
\begin{Proposition}\label{PropF25,2019} For every $\varepsilon$, let $\phi_\varepsilon$ be the unique solution to \eqref{J13,2019weak}, $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled}, $ \varphi^\star_\varepsilon$ be defined by \eqref{F16,2019z}-\eqref{F16,2019zq}, $D_\varepsilon$ be defined in \eqref{J24,2019trasfpartcoff}, and let $\nu_2$ denote the second component of the unit normal to $\Gamma^a_{\varepsilon,\alpha}$ exterior to $ \Omega^c_{\varepsilon,\alpha}$. Then, for every $\varepsilon$,
\begin{equation}\label{24Aprile2019serabis}\begin{array}{ll}\displaystyle{
\int_{\Gamma^a_{\varepsilon,\alpha}} |\nabla \phi_{\varepsilon} |^{2}\nu_2 ds=}\\\\
\displaystyle{
\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon} \left(-\partial _{x_{2}}
\varphi^\star_{\varepsilon }\left(\left \vert\partial_{x_1} \varphi_\varepsilon\right\vert^2-\frac{1}{\varepsilon^{2\alpha}}\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2\right)+2\partial_{x_2}
\varphi_{\varepsilon }\partial_{x_1}
\varphi^\star_{\varepsilon }\partial_{x_1}
\varphi_{\varepsilon }
\right)dx}
\\\\
\displaystyle{+
\int_{\Omega^{c,2}_\varepsilon} \left(-\partial _{x_{2}}
\varphi^\star_{\varepsilon }\left(\left \vert\partial_{x_1} \varphi_\varepsilon\right\vert^2-\frac{1}{D_\varepsilon^2}\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2\right)+2\partial_{x_2}
\varphi_{\varepsilon }\partial_{x_1}
\varphi^\star_{\varepsilon }\partial_{x_1}
\varphi_{\varepsilon }
\right)dx.}
\end{array}\end{equation}
\end{Proposition}
\begin{proof}
Let $T_{\varepsilon,\alpha}$ be defined by \eqref{J24,2019trasf}-\eqref{J24,2019trasfpartcoff}. The first step is devoted to proving that
\begin{equation}\label{stylo0}\begin{array}{lll}\displaystyle{
\int_{\Gamma^a_{\varepsilon,\alpha}} |\nabla \phi_{\varepsilon} |^{2}\nu_2 ds}\\\\ \displaystyle{=\int_{\Omega _{\varepsilon ,\alpha }^{c}}\left(-\partial _{x_{2}}\left(
\varphi^\star_{\varepsilon }\circ T_{\varepsilon,\alpha}^{-1}\right)\left\vert\nabla \phi_{\varepsilon}\right\vert^2+2\partial _{x_{2}}\phi_{\varepsilon} \nabla\left( \varphi^\star_{\varepsilon }\circ T_{\varepsilon,\alpha}^{-1}\right)\nabla
\phi_{\varepsilon}\right)dx, \quad\forall\varepsilon,}\end{array}
\end{equation}
from which \eqref{stylo0} follows by changing of variable \eqref{J24,2019trasf} in the second integral.
As we shall show in the following, \begin{equation}\label{7890aprile2019}|\nabla \phi _{\varepsilon }|^{2}\in
W^{1,1}(\Omega _{\varepsilon ,\alpha }^{c}).\end{equation} In particular, also $(\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})|\nabla \phi _{\varepsilon }|^{2}$ belongs to $ W^{1,1}(\Omega _{\varepsilon
,\alpha }^{c})$. Thus, definitions \eqref{J24,2019trasf} and \eqref{F16,2019z}-\eqref{F16,2019zq} allow us to write
\begin{equation}\begin{array}{ll} \label{stylo}\displaystyle{
\int_{\Gamma _{\varepsilon ,\alpha }^{a}}|\nabla \phi _{\varepsilon
}|^{2}\nu _{2}ds =\int_{\Gamma _{\varepsilon ,\alpha
}^{a}}(\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})|\nabla \phi _{\varepsilon }|^{2}\nu _{2}ds }\\\\ \displaystyle{=\int_{\Gamma _{\varepsilon ,\alpha }\cup \Gamma }(\varphi _{\varepsilon
}^{\star }\circ T_{\varepsilon ,\alpha }^{-1})|\nabla \phi _{\varepsilon
}|^{2}\nu _{2}ds, \quad\forall\varepsilon. }
\end{array}\end{equation}
The Green's Formula (for instance, see Th. 6.6-7 in \cite{Ciarlet}) gives
\begin{equation} \label{stylo2}
\int_{\Gamma _{\varepsilon ,\alpha }\cup \Gamma }(\varphi _{\varepsilon
}^{\star }\circ T_{\varepsilon ,\alpha }^{-1})|\nabla \phi _{\varepsilon
}|^{2}\nu _{2}ds=\int_{\Omega _{\varepsilon ,\alpha }^{c}}\partial
_{x_{2}}((\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})|\nabla \phi _{\varepsilon }|^{2})dx , \quad\forall\varepsilon.
\end{equation}
Then, \eqref{stylo} and \eqref{stylo2}) provides
\begin{equation} \label{stylo3}\begin{array}{ll} \displaystyle{
\int_{\Gamma _{\varepsilon ,\alpha }^{a}}|\nabla \phi _{\varepsilon
}|^{2}\nu _{2}ds}\\\\ \displaystyle{=\int_{\Omega _{\varepsilon ,\alpha }^{c}}\partial
_{x_{2}}(\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})|\nabla \phi _{\varepsilon }|^{2}dx +2 \int_{\Omega _{\varepsilon ,\alpha }^{c}}(\varphi _{\varepsilon }^{\star
}\circ T_{\varepsilon ,\alpha }^{-1})\nabla \phi _{\varepsilon }\nabla (\partial
_{x_{2}} \phi _{\varepsilon })dx, \quad\forall\varepsilon.}\end{array}
\end{equation}
On the other side (see below), \begin{equation}\label{MiLenc}\nabla \phi _{\varepsilon } \in W^{1,\frac{3}{2}}(\Omega_{\varepsilon ,\alpha }^{c}).\end{equation} In particular, $(\varphi _{\varepsilon }^{\star }\circ
T_{\varepsilon ,\alpha }^{-1})\nabla \phi _{\varepsilon } $ belongs to $ W^{1,\frac{3}{2}}(\Omega
_{\varepsilon ,\alpha }^{c})$, and $\partial
_{x_{2}}\phi _{\varepsilon }$ belongs to $ W^{1,\frac{3}{2}}(\Omega
_{\varepsilon ,\alpha }^{c})$ which is included in $ W^{1,\frac{6}{5}}(\Omega
_{\varepsilon ,\alpha }^{c})$. Consequently, again applying the Green's Formula as it appears in
Theorem 6.6-7 in \cite{Ciarlet} with exponents $p=\frac{3}{2}$ and $q=\frac{6}{5}$, the last integral in the right-hand side of (\ref{stylo3}) becomes
\begin{equation}\label{stylo4}\begin{array}{ll} \displaystyle{
\int_{\Omega _{\varepsilon
,\alpha }^{c}}(\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})\nabla \phi _{\varepsilon }\nabla (\partial _{x_{2}}\phi_{\varepsilon })dx}\\\\
\displaystyle{
=-\int_{\Omega _{\varepsilon ,\alpha }^{c}}{div}((\varphi
_{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha }^{-1})\nabla \phi
_{\varepsilon })\text{ }\partial _{x_{2}}\phi _{\varepsilon }
dx+\int_{\Gamma _{\varepsilon ,\alpha }\cup \Gamma }(\varphi _{\varepsilon
}^{\star }\circ T_{\varepsilon ,\alpha }^{-1})\partial _{x_{2}}\phi
_{\varepsilon }\nabla \phi _{\varepsilon }\nu ds , \quad\forall\varepsilon,
}
\end{array}\end{equation}
where $\nu $ is the unit normal to $\Gamma _{\varepsilon ,\alpha }\cup
\Gamma $ exterior to $\Omega _{\varepsilon ,\alpha }^{c}$. Since
\[
\int_{\Gamma _{\varepsilon ,\alpha }\cup \Gamma }(\varphi _{\varepsilon
}^{\star }\circ T_{\varepsilon ,\alpha }^{-1})\partial _{x_{2}}\phi
_{\varepsilon }\nabla \phi _{\varepsilon }\nu ds=\int_{\Gamma
_{\varepsilon ,\alpha }^{a}}|\nabla \phi _{\varepsilon }|^{2}\nu _{2}
ds, \quad\forall\varepsilon,
\]
which can be checked by inspectioning on each part of $\Gamma _{\varepsilon
,\alpha }\cup \Gamma $, one can rewrite (\ref{stylo4}) as
\begin{equation}\label{labeslao}\begin{array}{ll} \displaystyle{
\int_{\Omega _{\varepsilon ,\alpha }^{c}}(\varphi _{\varepsilon }^{\star
}\circ T_{\varepsilon ,\alpha }^{-1})\nabla \phi _{\varepsilon }\partial
_{x_{2}}\nabla \phi _{\varepsilon }dx}\\\\ \displaystyle{=-\int_{\Omega _{\varepsilon
,\alpha }^{c}}{div}((\varphi _{\varepsilon }^{\star }\circ
T_{\varepsilon ,\alpha }^{-1})\nabla \phi _{\varepsilon })\text{ }\partial
_{x_{2}}\phi _{\varepsilon }dx+\int_{\Gamma _{\varepsilon ,\alpha
}^{a}}|\nabla \phi _{\varepsilon }|^{2}\nu _{2}ds, \quad\forall\varepsilon.
}
\end{array}\end{equation}
Comparing (\ref{stylo3}) and \eqref{labeslao} gives
\begin{equation}\nonumber\begin{array}{ll} \displaystyle{
\int_{\Gamma _{\varepsilon ,\alpha }^{a}}|\nabla \phi _{\varepsilon
}|^{2}\nu _{2}ds}\\\\ \displaystyle{ =-\int_{\Omega _{\varepsilon ,\alpha
}^{c}}\partial _{x_{2}}(\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon
,\alpha }^{-1})|\nabla \phi _{\varepsilon }|^{2}dx+2\int_{\Omega
_{\varepsilon ,\alpha }^{c}}{div}((\varphi _{\varepsilon }^{\star
}\circ T_{\varepsilon ,\alpha }^{-1})\nabla \phi _{\varepsilon })\partial
_{x_{2}}\phi _{\varepsilon }dx} \\\\ \displaystyle{
=-\int_{\Omega _{\varepsilon ,\alpha }^{c}}\partial _{x_{2}}(\varphi
_{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha }^{-1})|\nabla \phi
_{\varepsilon }|^{2}dx+2\int_{\Omega _{\varepsilon ,\alpha
}^{c}}\nabla (\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})\nabla \phi _{\varepsilon }\partial _{x_{2}}\phi _{\varepsilon
}dx}\\\\
\displaystyle{+\int_{\Omega _{\varepsilon ,\alpha
}^{c}}(\varphi _{\varepsilon }^{\star }\circ T_{\varepsilon ,\alpha
}^{-1})\Delta \phi _{\varepsilon }\partial _{x_{2}}\phi _{\varepsilon }dx, \quad\forall\varepsilon,
}
\end{array}\end{equation}
which provides (\ref{stylo0}) since $\Delta \phi _{\varepsilon }=0$ in $
\Omega _{\varepsilon ,\alpha }^{c}$.
Now, we sketch the proof of \eqref{7890aprile2019}, based on the decomposition of $
\phi _{\varepsilon }$ as a sum of its singular and regular parts $\phi
_{\varepsilon }^{S}\in H^{1}(\Omega _{\varepsilon ,\alpha }^{c})$ and $\phi
_{\varepsilon }^{S}\in H^{2}(\Omega _{\varepsilon ,\alpha }^{c})$. At the
vicinity of any reentering corner with angle $\omega =\frac{3\pi }{2}$, the
expression in polar coordinate of the singular part reads $$\phi
_{\varepsilon }^{S}(r,\theta )=r^{\frac{2}{3}}\sin \left(\frac{2\theta }{3}\right).$$
Thus,
$$|\nabla \phi _{\varepsilon }^{S}|^{2}(r,\theta )=r^{-\frac{2}{3}}\Phi
_{0}(\theta ),$$ with $\Phi _{0}\in C^\infty$.
The expansion of $\nabla |\nabla \phi _{\varepsilon }|^{2}$ in $\phi
_{\varepsilon }^{S}$ and $\phi _{\varepsilon }^{R}$ includes four terms:\begin{equation}\label{nablabnabla}\nabla |\nabla \phi _{\varepsilon }^{S}|^{2}, \quad \nabla \nabla \phi _{\varepsilon }^{S}\nabla
\phi _{\varepsilon }^{R}, \quad \nabla |\nabla \phi _{\varepsilon }^{R}|^{2}, \hbox{ and }\nabla \nabla \phi _{\varepsilon }^{R}\nabla
\phi _{\varepsilon }^{S},\end{equation}
of
which only the first two terms cause regularity problems.
As the first term in \eqref{nablabnabla} is concerned, one has $$\nabla |\nabla \phi _{\varepsilon }^{S}|^{2}(r,\theta
)=r^{-\frac{5}{3}}\Phi _{1}(\theta ),$$ with $\Phi _{1}\in C^\infty$. Then, it is integrable. As the second term in \eqref{nablabnabla} is concerned, one has
$$\nabla \nabla \phi _{\varepsilon }^{S}\nabla
\phi _{\varepsilon }^{R}=(r^{\frac{1}{3}}\nabla \nabla \phi _{\varepsilon
}^{S})(r^{-\frac{1}{3}}\nabla \phi _{\varepsilon }^{R})$$
and
its integrability comes from the
observation that both terms $r^{\frac{1}{3}}\nabla \nabla \phi _{\varepsilon }^{S}$
are $r^{-\frac{1}{3}}\nabla \phi _{\varepsilon }^{R}$ are square
integrable.
The contribution of the corners with mixed conditions, that is
at the ends of $\Gamma $, to the singular part is in $H^{2-\eta }(\Omega
_{\varepsilon ,\alpha }^{c})$ for any positive $\eta $ and does not yield
any regularity issue.
Regularity result \eqref{MiLenc} can be proved with the same arguments.
\end{proof}
\section{{\it A priori} estimates\label{apapstst} }
\begin{Proposition} \label{J18,2019Proposition}For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled}. Then
\begin{equation}\label{J18,2019estimates}\exists c\in]0,+\infty[\quad:\quad\left\{\begin{array}{lll}\displaystyle{ \int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon} \vert \partial_{x_1} \varphi_\varepsilon\vert^2 dx\leq c\left(\varepsilon^{-2-\alpha}+ \varepsilon^{-2\alpha}\right),}\\\\
\displaystyle{ \int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon} \vert\partial_{x_2} \varphi_\varepsilon \vert^2 dx\leq c\left(\varepsilon^{\alpha-2}+ 1\right),
}\\\\ \displaystyle{ \int_{\Omega^{c,2}_\varepsilon} \vert\nabla \varphi_\varepsilon \vert^2 dx\leq c\left(\varepsilon^{-2}+ \varepsilon^{-\alpha}\right),}
\end{array}\right.\quad\forall\varepsilon.\end{equation}
\end{Proposition}
\begin{proof} For every $\varepsilon$, let $\varphi^\star_\varepsilon$ be defined by \eqref{F16,2019z}-\eqref{F16,2019zq}.
Moreover, set
$$Y=]0,1[ \times]l_1,l_2[.$$ Then, one has
\begin{equation} \label{J18,2019}\Vert \varphi^\star_\varepsilon\Vert^2_{L^2(\Omega^c_\varepsilon)}\leq\sum_{k=0}^{\frac{L}{\varepsilon}}\varepsilon\Vert \varphi^\star\Vert^2_{L^2(Y)}= L
\Vert \varphi^\star\Vert^2_{L^2(Y)}, \quad\forall\varepsilon.
\end{equation}
Similarly, one obtains
\begin{equation} \label{derx1}\Vert \partial_{x_1}\varphi^\star_\varepsilon\Vert^2_{L^2(\Omega^c_\varepsilon)}= \frac{L}{\varepsilon^2}
\Vert \partial_{x_1}\varphi^\star\Vert^2_{L^2(Y)}, \quad\forall\varepsilon,
\end{equation}
and
\begin{equation} \label{derx2}\Vert \partial_{x_2}\varphi^\star_\varepsilon\Vert^2_{L^2(\Omega^c_\varepsilon)}= L
\Vert \partial_{x_2}\varphi^\star\Vert^2_{L^2(Y)}, \quad\forall\varepsilon.
\end{equation}
Now choosing $\psi= \varphi_\varepsilon-\varphi^\star_\varepsilon$ as test function in \eqref{J13,2019weakrescaled} and using Young's inequality, \eqref{J15,2019}, and estimates \eqref{derx1} and \eqref{derx2} provide
\begin{equation}\nonumber\begin{array}{lll}\displaystyle{\exists c\in]0,+\infty[\,\,:\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \varepsilon^\alpha\vert\partial_{x_1} \varphi_\varepsilon\vert^2+\varepsilon^{-\alpha} \vert\partial_{x_2} \varphi_\varepsilon \vert^2 \right)dx+
\int_{\Omega^{c,2}_\varepsilon} \vert\nabla \varphi_\varepsilon \vert^2dx
}\\\\ \leq c\left(\varepsilon^{-2}+ \varepsilon^{-\alpha}\right), \quad\forall\varepsilon,
\end{array}\end{equation}
which implies \eqref{J18,2019estimates}.
\end{proof}
\section{The case $\alpha=2$\label{casesssalpha=2}}
This section is devoted to proving Theorem \ref{main theoremapril24,2019} when $\alpha=2$.
\subsection{{\it A priori} estimates \label{stime dettagliate alpha=2}}
Proposition \ref{J18,2019Proposition} immediately implies the following result.
\begin{Corollary} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$. Then,
\begin{equation}\label{J18,2019estimatesalfa=2}\exists c\in]0,+\infty[\quad:\quad\left\{\begin{array}{lll}\displaystyle{ \Vert\varepsilon^2\partial_{x_1} \varphi_\varepsilon\Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\leq c,}\\\\
\displaystyle{ \Vert\partial_{x_2} \varphi_\varepsilon \Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\leq c,
}\\\\ \displaystyle{ \Vert\varepsilon\nabla \varphi_\varepsilon \Vert_{L^2(\Omega^{c,2}_\varepsilon)}\leq c,}
\end{array}\right.\quad\forall\varepsilon.\end{equation}
\end{Corollary}
The next task is devoted to prove the following {\it a priori} estimate.
\begin{Proposition} \label{J21,2019prop} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$. Then,
\begin{equation}\label{J18,2019last}\exists c\in]0,+\infty[\,\,:\Vert \varphi_\varepsilon \Vert_{L^2(\Omega^c_\varepsilon)}\leq c, \quad\forall\varepsilon.
\end{equation}
\end{Proposition}
\begin{proof}
The Dirichlet boundary condition of $\varphi_\varepsilon $ on $\Gamma_\varepsilon$ and the second estimate in \eqref{J18,2019estimatesalfa=2} provide that
\begin{equation}\nonumber\exists c\in]0,+\infty[\,\,:\Vert \varphi_\varepsilon \Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\leq c, \quad\forall\varepsilon.
\end{equation}
The main task is to prove that
\begin{equation}\label{J19,2019ultimo}\exists c\in]0,+\infty[\,\,:\Vert \varphi_\varepsilon \Vert_{L^2(\Omega^{c,2}_\varepsilon)}\leq c, \quad\forall\varepsilon,\end{equation}
which completes the proof.
To this aim, set
$$P=]0,1[\setminus\left(\overline{\omega^a}\cup\overline{\omega^b}\right)=]0,\zeta_1[\cup]\zeta_2,\zeta_3[\cup]\zeta_4,1[.$$
Fix $\varepsilon$. Then, one has
\begin{equation}\label{J19,2019}
\Vert \varphi_\varepsilon \Vert^2_{L^2(\Omega^{c,2}_\varepsilon)}=\sum_{k=0}^{\frac{L}{\varepsilon}-1}\int_{\left(\varepsilon P+\varepsilon k\right)\times ]l_1,l_2[} \vert \varphi_\varepsilon \vert^2 dx.\end{equation}
Now fix $k\in\left\{0,\cdots , \frac{L}{\varepsilon}-1\right\}$. Then, if $x_1\in \varepsilon P+\varepsilon k$, one of the following three cases holds true:
$$x_1\in]\varepsilon k, \varepsilon\zeta_1+\varepsilon k[, \quad x_1\in]\varepsilon\zeta_2+\varepsilon k, \varepsilon\zeta_3+\varepsilon k[,\quad x_1\in]\varepsilon\zeta_4+\varepsilon k,\varepsilon(1+k)[ .$$ In the first case, since
$$\varphi_\varepsilon=1,\hbox{ on }\{\varepsilon\zeta_1+\varepsilon k\}\times]l_1,l_2[,$$
one has
$$\varphi_\varepsilon(x_1,x_2)=1-\int_{x_1}^{\varepsilon\zeta_1+\varepsilon k}\partial_{x_1}\varphi_\varepsilon(t,x_2)dt,\quad\forall x_1\in]\varepsilon k, \varepsilon\zeta_1+\varepsilon k[,\hbox{ for a.e. }x_2\in]l_1,l_2[,$$
which implies
\begin{equation}\label{J19,2019I} \int_{l_1}^{l_2}\int_{\varepsilon k}^{\varepsilon\zeta_1+\varepsilon k}\vert \varphi_\varepsilon(x_1,x_2)\vert^2dx_1dx_2\leq 2(l_2-l_1)\varepsilon+2\varepsilon^2 \int_{l_1}^{l_2}\int_{\varepsilon k}^{\varepsilon\zeta_1+\varepsilon k}\vert \partial_{x_1}\varphi_\varepsilon(x_1,x_2)\vert^2dx_1dx_2.
\end{equation}
Similarly, since
$$\varphi_\varepsilon=0 , \hbox { on }\{\varepsilon\zeta_3+\varepsilon k\}\times]l_1,l_2[ \hbox { and on }\{\varepsilon\zeta_4+\varepsilon k\}\times]l_1,l_2[,$$
in the second and in the third case
one has
\begin{equation}\label{J19,2019II} \int_{l_1}^{l_2}\int_{\varepsilon\zeta_2+\varepsilon k}^{\varepsilon\zeta_3+\varepsilon k}\vert \varphi_\varepsilon(x_1,x_2)\vert^2dx_1dx_2\leq 2\varepsilon^2 \int_{l_1}^{l_2}\int_{\varepsilon\zeta_2+\varepsilon k}^{\varepsilon\zeta_3+\varepsilon k} \vert \partial_{x_1}\varphi_\varepsilon(x_1,x_2)\vert^2dx_1dx_2
\end{equation}
and
\begin{equation}\label{J19,2019III} \int_{l_1}^{l_2}\int_{\varepsilon\zeta_4+\varepsilon k}^{\varepsilon(1+k)}\vert \varphi_\varepsilon(x_1,x_2)\vert^2dx_1dx_2\leq 2\varepsilon^2\int_{l_1}^{l_2}\int_{\varepsilon\zeta_4+\varepsilon k}^{\varepsilon(1+k)} \vert \partial_{x_1}\varphi_\varepsilon(x_1,x_2)\vert^2dx_1dx_2.
\end{equation}
Adding \eqref{J19,2019I}, \eqref{J19,2019II}, and \eqref{J19,2019III} gives
\begin{equation}\nonumber
\int_{\left(\varepsilon P+\varepsilon k\right)\times ]l_1,l_2[} \vert \varphi_\varepsilon \vert^2 dx\leq 2(l_2-l_1)\varepsilon+2\varepsilon^2 \int_{\left(\varepsilon P+\varepsilon k\right)\times ]l_1,l_2[} \vert \partial_{x_1}\varphi_\varepsilon \vert^2 dx,\end{equation}
from which, summing up $k\in\left\{0,\cdots , \frac{L}{\varepsilon}-1\right\}$ and using \eqref{J19,2019} and the third estimate in \eqref{J18,2019estimatesalfa=2}, one obtains \eqref{J19,2019ultimo}.
\end{proof}
\subsection{Weak convergence results\label{wweeaakkconv}}
The next proposition is devoted to studying the limit in $ \Omega^{c,2}$, as $\varepsilon$ tends to zero, of problem \eqref{J13,2019weakrescaled} with $\alpha=2$.
\begin{Proposition} \label{Proposizione Monda3}For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$. Set $$\varphi_{\varepsilon,2}={\varphi_\varepsilon}_{|_{\Omega_\varepsilon^{c,2}}}$$ and
\begin{equation}\label{F13,2019}\overline{\varphi_{\varepsilon,2}}=\left\{\begin{array}{ll}\varphi_{\varepsilon,2},\hbox{ a.e. in }\Omega_\varepsilon^{c,2},\\\\ \displaystyle{1,\hbox{ a.e. in }\bigcup_{k=0}^{\frac{L}{\varepsilon}-1}\left(\varepsilon\omega^a+\varepsilon k\right)\times ]l_1+1,l_2-1[,}\\\\ \displaystyle{0, \hbox{ a.e. in }\bigcup_{k=0}^{\frac{L}{\varepsilon}-1}\left(\varepsilon\omega^b+\varepsilon k\right)\times ]l_1+1,l_2-1[.}\end{array}\right.\end{equation}
Let
\begin{equation}\label{J24,2019all} \varphi_2:y\in[0,1]\longrightarrow \left\{\begin{array}{ll}\dfrac{y+1-\zeta_4}{\zeta_1-\zeta_4+1}, \hbox{ if }y\in[0,\zeta_1],\\\\
1, \hbox{ if }y\in[\zeta_1,\zeta_2],\\\\ \dfrac{y-\zeta_3}{\zeta_2-\zeta_3}, \hbox{ if }y\in[\zeta_2,\zeta_3],\\\\
0, \hbox{ if }y\in[\zeta_3,\zeta_4],\\\\\dfrac{y-\zeta_4}{\zeta_1-\zeta_4+1}, \hbox{ if }y\in[\zeta_4,1].
\end{array}\right.
\end{equation}
Then,
\begin{equation}\label{Monda1bisterforse}\left\{\begin{array}{ll}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to } \varphi_2,\\\\
\varepsilon\partial_{x_1}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }\partial_{y} \varphi_2,\\\\
\varepsilon\partial_{x_2}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }0,\end{array}\right.\end{equation}
as $\varepsilon$ tends to zero.
\end{Proposition}
\begin{proof}
Proposition \ref{J21,2019prop} and the third estimate in \eqref{J18,2019estimatesalfa=2} ensure the existence of a subsequence of $\{\varepsilon\}$, still denoted by $\{\varepsilon\}$, and $u_2 \in L^2\left(\Omega^{c,2}, H^1_{\hbox{per}}(]0,1[)\right)$ (in possible dependence on the subsequence) such that
\begin{equation}\label{Monda1bister}\left\{\begin{array}{ll}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to } u_2,\\\\
\varepsilon\partial_{x_1}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }\partial_{y} u_2,\\\\
\varepsilon\partial_{x_2}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }0,\end{array}\right.\end{equation}
as $\varepsilon$ tends to zero.
The next step is devoted to proving that
\begin{equation}\label{J23,2019fgd} u_2=1, \hbox { a.e. in } \Omega^{c,2}\times\omega^a.\end{equation}
Indeed, the definition of $\overline{ \varphi_{\varepsilon,2}}$ gives
\begin{equation}\label{chiacchiereIIIchie}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,2}}\overline{\varphi_{\varepsilon,2}}(x_1,x_2)\psi \left(x_1,x_2,\frac{x_1}{\varepsilon}\right) dx_1dx_2= \int_{\Omega^{c,2}}\psi \left(x_1,x_2,\frac{x_1}{\varepsilon}\right) dx_1dx_2,} \\\\
\forall\psi\in C_0^\infty(\Omega^{c,2}\times\omega^a),\quad\forall\varepsilon.
\end{array}
\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{chiacchiereIIIchie} and using the first limit in \eqref{Monda1bister} provide
\begin{equation}\nonumber\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,2}\times\omega^a}u_2(x_1,x_2,y)\psi \left(x_1,x_2,y\right) dx_1dx_2dy= \int_{\Omega^{c,2}\times\omega^a}\psi \left(x_1,x_2,y\right) dx_1dx_2dy, }\\\\\forall\psi\in C_0^\infty(\Omega^{c,2}\times\omega^a),
\end{array}
\end{equation}
which implies \eqref{J23,2019fgd}.
\noindent Similarly, one proves that
\begin{equation}\label{J23,2019dfg} u_2=0, \hbox { a.e. in } \Omega^{c,2}\times\omega^b.\end{equation} Finally, choosing $\displaystyle{\psi=\varepsilon^2\chi_1(x_1,x_2)\chi_2\left(\frac{x_1}{\varepsilon}\right)}$ with $\chi_1\in C_0^\infty\left(\Omega^{c,2}\right)$ and $\chi_2\in H^1_{\hbox{per}}\left(]0,1[\right)$ such that $\chi_2=0$ in $\omega^a\cup\omega^b$ as test function in \eqref{J13,2019weakrescaled} with $\alpha=2$ gives
\begin{equation}\label{J23,2019seraIIJ24}\begin{array}{lll} \displaystyle{D_\varepsilon \varepsilon^2\int_{\Omega^{c,2}} \partial_{x_1} \overline{ \varphi_{\varepsilon,2}}\left(\partial_{x_1}\chi_1(x_1,x_2)\chi_2\left(\frac{x_1}{\varepsilon}\right)+\varepsilon^{-1}\chi_1(x_1,x_2)\partial_{y}\chi_2\left(\frac{x_1}{\varepsilon}\right)\right)dx_1dx_2} \\\\
\displaystyle{+D_\varepsilon^{-1}\varepsilon^2\int_{\Omega^{c,2}} \partial_{x_2} \overline{\varphi_{\varepsilon,2} }\partial_{x_2}\chi_1(x_1,x_2)\chi_2\left(\frac{x_1}{\varepsilon}\right)dx_1dx_2=0,}\\\\ \forall \chi_1\in C_0^\infty\left(\Omega^{c,2}\right), \quad\forall \chi_2\in H^1_{\hbox{per}}\left(]0,1[\right)\,\,:\,\, \chi_2=0,\hbox{ in } \omega^a\cup\omega^b,\quad\forall\varepsilon.
\end{array}\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{J23,2019seraIIJ24} and using the second and third limits in \eqref{Monda1bister}, and \eqref{J15,2019}
provide that, for a.e. $(x_1,x_2)$ in $ \Omega^{c,2}$,
\begin{equation}\label{J23,2019seraIJ24}\begin{array}{ll}\displaystyle{ \int_{]0,1[\setminus\left(\omega^a\cup\omega^b\right)}\partial_{y}u_2(x_1,x_2,y)\partial_{y}\chi_2(y)dy=0, }\\\\ \forall \chi_2\in H^1_{\hbox{per}}\left(]0,1[\right)\,\,:\,\, \chi_2=0,\hbox{ in } \omega^a\cup\omega^b.
\end{array}
\end{equation}
Problem \eqref{J23,2019fgd}, \eqref{J23,2019dfg}, and \eqref{J23,2019seraIJ24} is equivalent to the following problem independent of $(x_1,x_2)$
\begin{equation}\left\{\begin{array}{ll}\partial^2_{y^2}u_2=0,\hbox{ in }]0,1[\setminus\left(\omega^a\cup\omega^b\right),\\\\
u_2=1, \hbox { in } \omega^a,\\\\
u_2=0, \hbox { in } \omega^b,\\\\
u_2(0)=u_2(1),\\\\
\partial_y u_2(0)=\partial_yu_2(1),
\end{array}\right.\end{equation}
which admits \eqref{J24,2019all} as unique solution.
Consequently, limits in \eqref{Monda1bister} hold for the whole sequence and \eqref{Monda1bisterforse} is satisfied.
\end{proof}
The next proposition is devoted to studying the limit in $\Omega^{c,3}$ and in $\Omega^{c,1}$, as $\varepsilon$ tends to zero, of problem \eqref{J13,2019weakrescaled} with $\alpha=2$.
\begin{Proposition} \label{Proposizione Eliquis1}For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$. Set $$\varphi_{\varepsilon,3}={\varphi_\varepsilon}_{|_{\Omega_\varepsilon^{c,3}}},\quad \varphi_{\varepsilon,1}={\varphi_\varepsilon}_{|_{\Omega_\varepsilon^{c,1}}},$$
\begin{equation}\label{Eliquis2}\widetilde{\varphi_{\varepsilon,3}}\left\{\begin{array}{ll}\varphi_{\varepsilon,3},\hbox{ a.e. in }\Omega_\varepsilon^{c,3},\\\\ 1,\hbox{ a.e. in }\Omega^{c,3}\setminus \Omega_\varepsilon^{c,3},\end{array}\right.\end{equation}
and
\begin{equation}\label{Eliquis4}\widehat{\varphi_{\varepsilon,1}}=\left\{\begin{array}{ll}\varphi_{\varepsilon,1},\hbox{ a.e. in }\Omega_\varepsilon^{c,1},\\\\ 0,\hbox{ a.e. in }\Omega^{c,1}\setminus \Omega_\varepsilon^{c,1}.\end{array}\right.\end{equation}
Moreover, let
\begin{equation}\label{F12,2019}\varphi_3:(x_1,x_2,y)\in \Omega^{c,3}\times]0,1[\longrightarrow\left\{\begin{array}{ll}x_2+1-l_2, \hbox{ if } y\in \omega^b,\\\\1, \hbox{ if } y\in]0,1[\setminus \omega^b,\end{array}\right.
\end{equation}
and
\begin{equation}\label{F12,2019bis}\varphi_1:(x_1,x_2,y)\in \Omega^{c,1}\times]0,1[\longrightarrow\left\{\begin{array}{ll}x_2-l_1, \hbox{ if } y\in \omega^a,\\\\0, \hbox{ if } y\in]0,1[\setminus \omega^a.\end{array}\right.
\end{equation}
Then
\begin{equation}\label{Monda1bisterforseter}\left\{\begin{array}{ll}\widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to } \varphi_3,\\\\
\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to }\partial_{x_2} \varphi_3,\end{array}\right.\end{equation}
and
\begin{equation}\label{Monda1bisterforsequater}\left\{\begin{array}{ll}\widehat{\varphi_{\varepsilon,1}}\hbox{ two scale converges to } \varphi_1,\\\\
\partial_{x_2}\widehat{\varphi_{\varepsilon,1}}\hbox{ two scale converges to }\partial_{x_2} \varphi_1,\end{array}\right.\end{equation}
as $\varepsilon$ tends to zero.
\end{Proposition}
\begin{proof} The proof will be developed in several steps.
Proposition \ref{J21,2019prop} and the second estimate in \eqref{J18,2019estimatesalfa=2} ensure the existence of a subsequence of $\{\varepsilon\}$, still denoted by $\{\varepsilon\}$, $u_3$, $\xi\in L^2(\Omega^{c,3}\times]0,1[)$, and $w$, $z\in L^2(]0,L[\times]0,1[)$ (in possible dependence on the subsequence) satisfying
\begin{equation} \label{F15,2019a} \widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to } u_3,
\end{equation}
and
\begin{equation}\label{J21,2019I}\left\{ \begin{array}{ll}
\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to }\xi,\\\\ \hbox{ the trace of }
\widetilde{{\varphi_{\varepsilon,3}}} \hbox{ on } ]0,L[\times\{l_2-1\}\hbox{ two scale converges to } w,
\\\\ \hbox{ the trace of }
\widetilde{{\varphi_{\varepsilon,3}}} \hbox{ on } ]0,L[\times\{l_2\}\hbox{ two scale converges to } z,
\end{array}\right.
\end{equation}
as $\varepsilon$ tends to zero.
The first step is devoted to proving that
\begin{equation}\label{J21,2019II}\xi=\partial_{x_2} u_3, \hbox { a.e. in } \Omega^{c,3}\times]0,1[.
\end{equation}
Indeed, integration by parts gives
\begin{equation}\label{chiacchiereI}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}}\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}(x_1,x_2)\psi \left(x_1,x_2,\frac{x_1}{\varepsilon}\right) dx_1dx_2}\\\\
\displaystyle{ =- \int_{\Omega^{c,3}}\widetilde{\varphi_{\varepsilon,3}}(x_1,x_2)\partial_{x_2}\psi \left(x_1,x_2,\frac{x_1}{\varepsilon}\right) dx_1dx_2,\quad\forall\psi\in C_0^\infty(\Omega^{c,3}\times]0,1[),\quad\forall\varepsilon.}
\end{array}
\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{chiacchiereI} and using \eqref{F15,2019a} and the first limit in \eqref{J21,2019I} provide
\begin{equation}\nonumber\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}\times]0,1[}\xi(x_1,x_2,y)\psi (x_1,x_2,y) dx_1dx_2 dy}\\\\\displaystyle{=-
\int_{\Omega^{c,3}\times]0,1[} u_3 (x_1,x_2,y) \partial_{x_2}\psi (x_1,x_2,y) dx_1dx_2 dy, \quad\forall\psi\in C_0^\infty(\Omega^{c,3}\times]0,1[),
}
\end{array}
\end{equation}
which implies \eqref{J21,2019II}. Combining the first limit in \eqref{J21,2019I} with \eqref{J21,2019II}
gives
\begin{equation}\label{Monda2}\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to }\partial_{x_2} u_3,\end{equation}
as $\varepsilon$ tends to zero.
The fact that $u_3$ and $\xi\in L^2(\Omega^{c,3}\times]0,1[)$ combined with \eqref{J21,2019II} provides that for a.e. $y\in ]0,1[$ $u_3(\cdot,\cdot,y)$ has traces on $]0,l[\times\{l_2-1\}$ and on $]0,l[\times\{l_2\}$ belonging to $L^2(]0,l[\times\{l_2-1\})$ and to $L^2(]0,l[\times\{l_2\})$, respectively.
The second step is devoted to proving that
\begin{equation}\label{J22,2019}w(x_1,y)=u_3(x_1, l_2-1,y), \hbox{ a.e in } ]0,L[\times ]0,1[.
\end{equation}
Indeed, integration by parts gives
\begin{equation}\label{chiacchiereII}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}}\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}(x_1,x_2)\psi \left(x_1,\frac{x_1}{\varepsilon}\right) (l_2-x_2)dx_1dx_2}\\\\
\displaystyle{ = \int_{\Omega^{c,3}}\widetilde{\varphi_{\varepsilon,3}}(x_1,x_2)\psi \left(x_1,\frac{x_1}{\varepsilon}\right) dx_1dx_2}
\displaystyle{ - \int_{]0.L[}\widetilde{\varphi_{\varepsilon,3}}(x_1,l_2-1)\psi \left(x_1,\frac{x_1}{\varepsilon}\right) dx_1,}\\\\
\forall\psi\in C_0^\infty(]0,L[\times]0,1[),\quad\forall\varepsilon.
\end{array}
\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{chiacchiereII} and using
\eqref{F15,2019a}, the second limit in \eqref{J21,2019I}, and \eqref{Monda2} provide
\begin{equation}\nonumber\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}\times]0,1[}\partial_{x_2}u_3(x_1,x_2,y)\psi \left(x_1,y\right)(l_2-x_2) dx_1dx_2dy}\\\\
\displaystyle{ = \int_{\Omega^{c,3}\times]0,1[}u_3(x_1,x_2,y)\psi \left(x_1,y\right) dx_1dx_2dy}
\displaystyle{ - \int_{]0.L[\times]0,1[}w(x_1,y)\psi \left(x_1,y\right) dx_1dy,}\\\\
\forall\psi\in C_0^\infty(]0,L[\times]0,1[),
\end{array}
\end{equation}
that is
\begin{equation}\nonumber\begin{array}{ll}\displaystyle{ \int_{]0,L[\times]0,1[}w(x_1,y)\psi \left(x_1,y\right) dx_1dy,= \int_0^1\left(\int_0^L w(x_1,y)\psi(x_1,y)dx_1\right)dy=}\\\\ \displaystyle{ \int_0^1\left(\int_{\Omega^{c,3}} \left( u_3(x_1,x_2,y)\psi(x_1,y) -\partial_{x_2}u_3(x_1,x_2,y)\psi(x_1,y)(l_2-x_2) \right) dx_1dx_2\right)dy}\\\\
\displaystyle{=\int_0^1\left(\int_0^L u_3(x_1,l_2-1,y) \psi(x_1,y)dx_1\right)dy=\int_{]0,L[\times]0,1[} u_3(x_1,l_2-1,y) \psi \left(x_1,y\right) dx_1dy,}\\\\\forall\psi\in C_0^\infty(]0,L[\times]0,1[),
\end{array}
\end{equation}
which implies \eqref{J22,2019}.
Similarly, one proves that
\begin{equation}\label{J22,2019bis}z(x_1,y)=u_3(x_1, l_2,y), \hbox{ a.e in } ]0,L[\times ]0,1[.
\end{equation}
The third step is devoted to proving that
\begin{equation}\label{J23,2019I} u_3(x_1,l_2-1, y)=0,\hbox { a.e. in } ]0,L[\times\omega^b,\end{equation}
Indeed, the boundary condition of $\varphi_\varepsilon$ on $\Gamma^b_\varepsilon$ gives
\begin{equation}\label{chiacchiereIV}\begin{array}{ll}\displaystyle{ \int_{]0,L[}\widetilde{\varphi_{\varepsilon,3}}(x_1,l_2-1)\psi \left(x_1,\frac{x_1}{\varepsilon}\right) dx_1= 0, \quad\forall\psi\in C_0^\infty(]0,L[\times\omega^b),\quad\forall\varepsilon.}
\end{array}
\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{chiacchiereIV} and using the second limit in \eqref{J21,2019I} and \eqref{J22,2019} provide
\begin{equation}\nonumber\begin{array}{ll}\displaystyle{ \int_{]0,L[\times\omega^b}u_3(x_1,l_2-1,y)\psi \left(x_1,y\right) dx_1dy= 0, \quad\forall\psi\in C_0^\infty(]0,L[\times\omega^b),}
\end{array}
\end{equation}
which implies \eqref{J23,2019I}. Similarly, one proves
\begin{equation}\label{J23,2019III} u_3(x_1,l_2, y)=1,\hbox { a.e. in } ]0,L[\times]0,1[.\end{equation}
Arguing as in the proof of \eqref{J23,2019fgd} gives
\begin{equation}\label{J23,2019} u_3=1, \hbox { a.e. in } \Omega^{c,3}\times\omega^a,
\end{equation}
The fourth step is devoted to proving that
\begin{equation}\label{J23,2019seraI}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}\times\left(]0,1[\setminus\omega^a\right)}\partial_{x_2}u_3(x_1,x_2,y)\partial_{x_2}\chi(x_1,x_2,y)dx_1dx_2dy=0,}\\\\ \forall\chi\in C_0^\infty\left(\Omega^{c,3}\times\left(]0,1[\setminus\omega^a\right)\right).\end{array}
\end{equation}
Indeed, choosing $\displaystyle{\psi=\varepsilon^2\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)}$ with $\chi\in C_0^\infty\left(\Omega^{c,3}\times\left(]0,1[\setminus\omega^a\right)\right)$ as test function in \eqref{J13,2019weakrescaled} with $\alpha=2$ gives
\begin{equation}\label{J23,2019seraII}\begin{array}{lll} \displaystyle{\int_{\Omega^{c,3}} \varepsilon^4 \partial_{x_1} \widetilde{ \varphi_{\varepsilon,3}}\left(\partial_{x_1}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)+\varepsilon^{-1}\partial_{y}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)\right)dx_1dx_2} \\\\
\displaystyle{+\int_{\Omega^{c,3}} \partial_{x_2} \widetilde{\varphi_{\varepsilon,3} }\partial_{x_2}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)dx_1dx_2=0,}\quad\forall\chi\in C_0^\infty\left(\Omega^{c,3}\times\left(]0,1[\setminus\omega^a\right)\right)),\quad\forall\varepsilon.
\end{array}\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{J23,2019seraII} and using the first estimate in \eqref{J18,2019estimatesalfa=2}, \eqref{Monda2}, and \eqref{J23,2019} provide \eqref{J23,2019seraI}.
In a similar way, one proves that
there exist a subsequence of $\{\varepsilon\}$, still denoted by $\{\varepsilon\}$ and $u_1 \in L^2(\Omega^{c,1}\times]0,1[)$ (in possible dependence on the subsequence) such that
\begin{equation}\label{Monda1bis}\widehat{\varphi_{\varepsilon,1}}\hbox{ two scale converges to } u_1,\end{equation}
as $\varepsilon$ tends to zero. Moreover, $\partial_{x_2}u_1\in L^2(\Omega^{c,1}\times]0,1[)$ and
\begin{equation}\label{Monda2bis}\partial_{x_2}\widehat{\varphi_{\varepsilon,1}}\hbox{ two scale converges to }\partial_{x_2} u_1,\end{equation}
as $\varepsilon$ tends to zero. Furthermore,
\begin{equation}\label{J23,2019bis} u_1=0, \hbox { a.e. in } \Omega^{c,1}\times\omega^b,
\end{equation}
\begin{equation}\label{J23,2019IIbis} u_1(x_1,l_1+1, y)=1,\hbox { a.e. in } ]0,L[\times\omega^a,\end{equation}
\begin{equation}\label{J23,2019IIIbis} u_1(x_1,l_1, y)=0,\hbox { a.e. in } ]0,L[\times]0,1[,\end{equation}
and
\begin{equation}\label{J23,2019seraIbis}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,1}\times\left(]0,1[\setminus\omega^b\right)}\partial_{x_2}u_1(x_1,x_2,y)\partial_{x_2}\chi(x_1,x_2,y)dx_1dx_2dy=0,}\\\\ \forall\chi\in C_0^\infty\left(\Omega^{c,1}\times\left(]0,1[\setminus\omega^b\right)\right).\end{array}
\end{equation}
The last step is devoted to proving that
\begin{equation}\label{F5,2019}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}\partial_{x_2}u_3(x_1,x_2,y)\partial_{x_2}\chi(x_1,x_2,y)dx_1dx_2dy}\\\\ \displaystyle{ +\int_{\Omega^{c,1}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}\partial_{x_2}u_1(x_1,x_2,y)\partial_{x_2}\chi(x_1,x_2,y)dx_1dx_2dy,} \\\\ \forall\chi\in C_0^\infty\left(\Omega^c\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)\right).\end{array}
\end{equation}
Indeed, choosing $\displaystyle{\psi=\varepsilon^2\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)}$ with $\chi\in C_0^\infty\left(\Omega^c\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)\right)$ as test function in \eqref{J13,2019weakrescaled} with $\alpha=2$ gives
\begin{equation}\label{F2,2019}\begin{array}{lll} \displaystyle{\int_{\Omega^{c,3}} \varepsilon^4 \partial_{x_1} \widetilde{ \varphi_{\varepsilon,3}}\left(\partial_{x_1}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)+\varepsilon^{-1}\partial_{y}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)\right)dx_1dx_2} \\\\
\displaystyle{+\int_{\Omega^{c,3}} \partial_{x_2} \widetilde{\varphi_{\varepsilon,3} }\partial_{x_2}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)dx_1dx_2}\\\\
\displaystyle{+\int_{ \Omega^{c,1}} \varepsilon^4 \partial_{x_1} \widehat{ \varphi_{\varepsilon,3}}\left(\partial_{x_1}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)+\varepsilon^{-1}\partial_{y}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)\right)dx_1dx_2} \\\\
\displaystyle{+\int_{ \Omega^{c,1}} \partial_{x_2} \widehat{\varphi_{\varepsilon,3} }\partial_{x_2}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)dx_1dx_2}\\\\
\displaystyle{+D_\varepsilon \varepsilon^2\int_{\Omega^{c,2}} \partial_{x_1} \overline{ \varphi_{\varepsilon,2}}\left(\partial_{x_1}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)+\varepsilon^{-1}\partial_{y}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)\right)dx_1dx_2} \\\\
\displaystyle{+D_\varepsilon^{-1}\varepsilon^2\int_{\Omega^{c,2}} \partial_{x_2} \overline{\varphi_{\varepsilon,2} }\partial_{x_2}\chi\left(x_1,x_2,\frac{x_1}{\varepsilon}\right)dx_1dx_2=0,}\\\\ \forall\chi\in C_0^\infty\left(\Omega^c\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)\right),\quad\forall\varepsilon.
\end{array}\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{F2,2019} and using the first estimate in \eqref{J18,2019estimatesalfa=2}, \eqref{Monda2}, \eqref{J23,2019}, \eqref{Monda2bis}, \eqref{J23,2019bis}, \eqref{J15,2019}, and the second and third limit in \eqref{Monda1bisterforse} provide
\begin{equation}\label{F5,2019Pasqua2019}\begin{array}{ll}\displaystyle{ \int_{\Omega^{c,3}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}\partial_{x_2}u_3(x_1,x_2,y)\partial_{x_2}\chi(x_1,x_2,y)dx_1dx_2dy}\\\\ \displaystyle{ +\int_{\Omega^{c,1}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}\partial_{x_2}u_1(x_1,x_2,y)\partial_{x_2}\chi(x_1,x_2,y)dx_1dx_2dy+}\\\\ \displaystyle{ \frac{l_2-l_1}{l_2-l_1-2}. \int_{\Omega^{c,2}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}
\partial_{y}\varphi_2(x_1,x_2,y)\partial_{y}\chi(x_1,x_2,y)dx_1dx_2dy,
}
\\\\ \forall\chi\in C_0^\infty\left(\Omega^c\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)\right).\end{array}
\end{equation}
which implies \eqref{F5,2019}, since the last integral in \eqref{F5,2019Pasqua2019} is zero due to \eqref{J24,2019all}.
Finally, \eqref{J23,2019I}, \eqref{J23,2019III}, \eqref{J23,2019}, \eqref{J23,2019seraI}, and \eqref{J23,2019bis}-\eqref{F5,2019} assert that $u_3$ and $u_1$ solve the following problems
\begin{equation}\label{F5bis',2019}\left\{\begin{array}{ll}u_3=1, \hbox { in } \Omega^{c,3}\times\omega^a,\\\\ \left\{\begin{array}{ll}\partial^2_{x^2_2}u_3 (x_1,x_2,y)=0, \hbox{ in }\Omega^{c,3}\times\left(]0,1[\setminus\omega^a\right),\\\\
u_3(x_1,l_2, y)=1,\hbox { in } ]0,L[\times]0,1[,\\\\
u_3(x_1,l_2-1, y)=0,\hbox { in } ]0,L[\times\omega^b,\\\\
\partial_{x_2} u_3(x_1,l_2-1, y)=0,\hbox { in } ]0,L[\times]0,1[\setminus\left(\omega^a\cup\omega^b\right),\end{array}\right.
\end{array}\right.
\end{equation}
and
\begin{equation}\label{F5ter",2019}\left\{\begin{array}{ll}u_1=0, \hbox { in } \Omega^{c,1}\times\omega^b,\\\\ \left\{\begin{array}{ll}
\partial^2_{x^2_2}u_1 (x_1,x_2,y)=0, \hbox{ in }\Omega^{c,1}\times\left(]0,1[\setminus\omega^b\right),\\\\
u_1(x_1,l_1, y)=0,\hbox { in } ]0,L[\times]0,1[,\\\\
u_1(x_1,l_1+1, y)=1,\hbox { in } ]0,L[\times\omega^a,\\\\
\partial_{x_2} u_1(x_1,l_1+1, y)=0,\hbox { in } ]0,L[\times]0,1[\setminus\left(\omega^a\cup\omega^b\right),\end{array}\right.
\end{array}\right.
\end{equation}
respectively, which means that $u_3$ and $u_1$ are given by \eqref{F12,2019} and \eqref{F12,2019bis}, respectively. Consequently, \eqref{F15,2019a}, \eqref{Monda2}, \eqref{Monda1bis}, and \eqref{Monda2bis} hold true for the whole sequence and \eqref{Monda1bisterforseter} and \eqref{Monda1bisterforsequater} are satisfied.
\end{proof}
The following result is an immediate consequence of Proposition \ref{Proposizione Monda3} and Proposition \ref{Proposizione Eliquis1}.
\begin{Corollary}
For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$ and let $\overline{\varphi_{\varepsilon,2}}$,
$\widetilde{\varphi_{\varepsilon,3}}$, and $\widehat{\varphi_{\varepsilon,1}}$ be defined by \eqref{F13,2019}, \eqref{Eliquis2}, and \eqref{Eliquis4}, respectively. Moreover, let $\varphi_2$, $\varphi_3$, and $\varphi_1$ be defined by \eqref{J24,2019all}, \eqref{F12,2019}, and \eqref{F12,2019bis}, respectively. Then
\begin{equation} \nonumber\overline{\varphi_{\varepsilon,2}}\rightharpoonup \frac{1}{2}\left(1+\hbox{meas}(\omega^a)-\hbox{meas}(\omega^b)\right),\quad\varepsilon\partial_{x_1}\overline{\varphi_{\varepsilon,2}}\rightharpoonup0,\quad \varepsilon\partial_{x_2}\overline{\varphi_{\varepsilon,2}}\rightharpoonup0, \hbox{ weakly in }L^2(\Omega^{c,2}),
\end{equation}
\begin{equation} \nonumber\widetilde{\varphi_{\varepsilon,3}}\rightharpoonup (x_2-l_2)\hbox{meas}(\omega^b)+1, \quad\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}\rightharpoonup\hbox{meas}(\omega^b),\hbox{ weakly in }L^2(\Omega^{c,3}),
\end{equation}
and
\begin{equation} \nonumber\widehat{\varphi_{\varepsilon,1}}\rightharpoonup (x_2-l_1)\hbox{meas}(\omega^a),\quad \quad\partial_{x_2}\widehat{\varphi_{\varepsilon,1}}\rightharpoonup\hbox{meas}(\omega^a),\hbox{ weakly in }L^2(\Omega^{c,1}),
\end{equation}
as $\varepsilon$ tends to zero.
\end{Corollary}
\subsection{Corrector results\label{corrrresult}}
Th following proposition is devoted to proving the energies convergence.
\begin{Proposition}\label{encon2019} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$. Moreover, let $\varphi_1$, $\varphi_3$, and $\varphi_2$, be defined by \eqref{F12,2019bis}, \eqref{F12,2019}, and \eqref{J24,2019all}, respectively. Then
\begin{equation}\label{F15,2019energyconv}\begin{array}{lll}
\displaystyle{\lim_{\varepsilon\rightarrow 0}\bigg[\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx} \\\\ \displaystyle{+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon \left\vert\varepsilon\partial_{x_1} \varphi_\varepsilon \right\vert^2+D^{-1}_\varepsilon\left\vert\varepsilon\partial_{x_2} \varphi_\varepsilon \right\vert^2 \right)dx\bigg]
}\\\\
\displaystyle{=
\int_{\Omega^{c,1}\times\omega^a}\left\vert\partial_{x_2} \varphi_1\right\vert^2dxdy+\int_{\Omega^{c,3}\times\omega^b}\left\vert\partial_{x_2} \varphi_3\right\vert^2dxdy}\\\\ \displaystyle{+ \frac{l_2-l_1}{l_2-l_1-2} \int_{\Omega^{c,2}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}\left\vert\partial_{y} \varphi_2\right\vert^2dxdy.}
\end{array}\end{equation}
\end{Proposition}
\begin{proof}
Choosing $\psi= \varepsilon^2\left(\varphi_\varepsilon-\varphi^\star_\varepsilon\right)$ as test function in \eqref{J13,2019weakrescaled}, where $\varphi^\star_\varepsilon$ is defined by \eqref{F16,2019z}-\eqref{F16,2019zq}, gives
\begin{equation}\label{F16,2019tk}\begin{array}{lll}
\displaystyle{\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon \left\vert \varepsilon\partial_{x_1} \varphi_\varepsilon \right\vert^2+D^{-1}_\varepsilon \left\vert \varepsilon\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx
}\\\\
\displaystyle{=\int_{\Omega^{c,1}}\left( \varepsilon^3 \partial_{x_1} \widehat{\varphi_{\varepsilon,1}}\left(\partial_{y}\varphi^\star\right)\left(\frac{x_1}{\varepsilon}, x_2\right)+\partial_{x_2} \widehat{\varphi_{\varepsilon,1}}\partial_{x_2}\varphi^\star\left(\frac{x_1}{\varepsilon}, x_2\right) \right)dx}\\\\
\displaystyle{+\int_{\Omega^{c,3}}\left( \varepsilon^3 \partial_{x_1} \widetilde{\varphi_{\varepsilon,3}}\left(\partial_{y}\varphi^\star\right)\left(\frac{x_1}{\varepsilon}, x_2\right)+\partial_{x_2} \widetilde{\varphi_{\varepsilon,3}}\partial_{x_2}\varphi^\star\left(\frac{x_1}{\varepsilon}, x_2\right)\right)dx}\\\\ \displaystyle{+
\int_{\Omega^{c,2}}\left( D_\varepsilon \varepsilon \partial_{x_1} \overline{\varphi_{\varepsilon,2}}\left(\partial_{y}\varphi^\star\right)\left(\frac{x_1}{\varepsilon}, x_2\right)+D^{-1}_\varepsilon \varepsilon^2\partial_{x_2} \overline{\varphi_{\varepsilon,2}} \partial_{x_2}\varphi^\star\left(\frac{x_1}{\varepsilon}, x_2\right)\right)dx,\quad\forall\varepsilon,}
\end{array}\end{equation}
where
$\widehat{\varphi_{\varepsilon,1}}$, $\widetilde{\varphi_{\varepsilon,3}}$, $\overline{\varphi_{\varepsilon,2}}$ are defined by \eqref{Eliquis4}, \eqref{Eliquis2}, and \eqref{F13,2019}, respectively.
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{F16,2019tk} and using \eqref{J15,2019}, the first estimate in \eqref{J18,2019estimatesalfa=2}, Proposition \ref{Proposizione Monda3}, and Proposition \ref{Proposizione Eliquis1} provide
\begin{equation}\label{F17,2019tkk}\begin{array}{lll}
\displaystyle{\lim_{\varepsilon\rightarrow 0}\bigg[\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx}\\\\ \displaystyle{+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon \left\vert \varepsilon\partial_{x_1} \varphi_\varepsilon \right\vert^2+D^{-1}_\varepsilon \left\vert \varepsilon\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx\bigg]
}\\\\
\displaystyle{=\int_{\Omega^{c,1}\times\omega^a}\partial_{x_2}\varphi^\star dxdy+\int_{\Omega^{c,3}\times\omega^b} \partial_{x_2}\varphi^\star dxdy}
\\\\ \displaystyle{+ \frac{l_2-l_1}{l_2-l_1-2}
\int_{\Omega^{c,2}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)} \partial_{y} \varphi_2\partial_{y}\varphi^\star dxdy.}
\end{array}\end{equation}
As the third integral and fourth integral in \eqref{F17,2019tkk} are concerned, the last two lines in \eqref{F16,2019z1}, \eqref{F12,2019}, and \eqref{F12,2019bis} ensure that
\begin{equation}\label{F17,2019tkkha}\begin{array}{lll}
\displaystyle{\int_{\Omega^{c,1}\times\omega^a}\partial_{x_2}\varphi^\star dxdy+\int_{\Omega^{c,3}\times\omega^b} \partial_{x_2}\varphi^\star dxdy=\int_{\Omega^{c,1}\times\omega^a} 1dxdy+\int_{\Omega^{c,3}\times\omega^b} 1dxdy}\\\\ \displaystyle{=\int_{\Omega^{c,1}\times\omega^a}\left\vert\partial_{x_2} \varphi_1\right\vert^2dxdy+ \int_{\Omega^{c,3}\times\omega^b}\left\vert\partial_{x_2} \varphi_3\right\vert^2dxdy.
}
\end{array}\end{equation}
As the last integral in \eqref{F17,2019tkk} is concerned, the first two lines in \eqref{F16,2019z1}
and \eqref{J24,2019all} ensure that
\begin{equation}\label{F17,2019tkklss}\begin{array}{lll}
\displaystyle{\int_{\Omega^{c,2}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)} \partial_{y} \varphi_2\partial_{y}\varphi^\star dxdy=\left( \frac{1}{\zeta_1-\zeta_4+1}-\frac{1}{\zeta_2-\zeta_3}\right) \int_{\Omega^{c,2}} 1dx}\\\\
\displaystyle{= \int_{\Omega^{c,2}\times\left(]0,1[\setminus\left(\omega^a\cup\omega^b\right)\right)}\left\vert\partial_{y} \varphi_2\right\vert^2dxdy.}
\end{array}\end{equation}
Finally, \eqref{F15,2019energyconv} follows from \eqref{F17,2019tkk}, \eqref{F17,2019tkkha}, and \eqref{F17,2019tkklss}.
\end{proof}
Proposition \ref{Proposizione Monda3}, Proposition \ref{Proposizione Eliquis1}, and Proposition \ref{encon2019} provide the following corrector results.
\begin{Proposition}\label{F19,2019corrres} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha=2$. Moreover, let $\varphi_1$, $\varphi_3$, and $\varphi_2$, be defined by \eqref{F12,2019bis}, \eqref{F12,2019}, and \eqref{J24,2019all}, respectively. Then
\begin{equation}\label{F19,2019cr1}\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}
\int_{\Omega^{c,1}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx=0,
}\end{array}\end{equation}
\begin{equation}\label{F19,2019cr3}\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}
\int_{\Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_3 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx=0,
}\end{array}\end{equation}
and
\begin{equation}\label{F19,2019cr2}\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}
\int_{\Omega^{c,2}_\varepsilon}\left( \left\vert \varepsilon\partial_{x_1} \varphi_\varepsilon -\left(\partial_y\varphi_2\right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2+\left\vert \varepsilon\partial_{x_2} \varphi_\varepsilon (x)\right\vert ^2\right)dx=0.
}\end{array}\end{equation}
\end{Proposition}
\begin{proof} One has
\begin{equation}\nonumber\begin{array}{lll} \displaystyle{
\int_{\Omega^{c,1}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx
}\\\\
\displaystyle{+
\int_{\Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_3 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx
}\\\\
\displaystyle{+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon\left\vert \varepsilon\partial_{x_1} \varphi_\varepsilon -\left(\partial_y\varphi_2\right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2+D_\varepsilon^{-1}\left\vert \varepsilon\partial_{x_2} \varphi_\varepsilon (x)\right\vert ^2\right)dx=}
\end{array}\end{equation}
\begin{equation}\nonumber\begin{array}{lll} \displaystyle{\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon \left\vert \varepsilon\partial_{x_1} \varphi_\varepsilon \right\vert^2+D^{-1}_\varepsilon \left\vert \varepsilon\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)dx}\\\\
\displaystyle{+\int_{\Omega^{c,1}}\left(\left\vert\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2-2\partial_{x_2} \widehat{\varphi_{\varepsilon,1}} (x)\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right)dx}
\\\\
\displaystyle{+\int_{\Omega^{c,3}}\left(\left\vert\left(\partial_{x_2}\varphi_3 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2-2\partial_{x_2} \widetilde{\varphi_{\varepsilon,3}} (x)\left(\partial_{x_2}\varphi_3 \right)\left(\frac{x_1}{\varepsilon}\right)\right)dx}
\\\\
\displaystyle{+D_\varepsilon\int_{\Omega^{c,2}}\left(\left\vert\left(\partial_{y}\varphi_2 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2-2\varepsilon\partial_{x_1} \overline{\varphi_{\varepsilon,2}} (x)\left(\partial_{y}\varphi_2 \right)\left(\frac{x_1}{\varepsilon}\right)\right)dx,\quad\forall\varepsilon.}
\end{array}\end{equation}
where $ \widehat{\varphi_{\varepsilon,1}}$, $\widetilde{\varphi_{\varepsilon,3}}$, and $\overline{\varphi_{\varepsilon,2}} $ are defined by \eqref{Eliquis4}, \eqref{Eliquis2}, and \eqref{F13,2019}, respectively. Passing to the limit, as $\varepsilon\rightarrow 0$, in this equality and using Proposition \ref{Proposizione Monda3}, Proposition \ref{Proposizione Eliquis1}, Proposition \ref{encon2019}, and \eqref{J15,2019} provide
\begin{equation}\nonumber\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}\bigg[
\int_{\Omega^{c,1}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx
}\\\\
\displaystyle{+
\int_{\Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^2\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_3 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx
}\\\\
\displaystyle{+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon\left\vert \varepsilon\partial_{x_1} \varphi_\varepsilon -\left(\partial_y\varphi_2\right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2+D_\varepsilon^{-1}\left\vert \varepsilon\partial_{x_2} \varphi_\varepsilon (x)\right\vert ^2\right)dx\bigg]=0,
}\end{array}\end{equation}
which implies \eqref{F19,2019cr1} thanks to \eqref{J15,2019}.\end{proof}
\subsection{Proof of Theorem \ref{main theoremapril24,2019} with $\alpha=2$\label{proofmmmmaitttheore}}
\begin{proof} Proposition \ref{PropF25,2019} with $\alpha=2$ provides that for every $\varepsilon$
\begin{equation}\label{J27,2019*}\begin{array}{ll}\displaystyle{
\int_{\Gamma^a_{\varepsilon,2}} |\nabla\varepsilon^2 \phi_{\varepsilon} |^{2}\nu_2 ds}\\\\ \displaystyle{=-\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left \vert\varepsilon^2\partial_{x_1} \varphi_\varepsilon\right\vert^2dx+\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2dx+\int_{\Omega^{c,3}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2dx}\\\\
\displaystyle{+2\varepsilon^4
\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\partial_{x_2}
\varphi_{\varepsilon } \partial_{x_1}
\varphi^\star_{\varepsilon }\partial_{x_1}
\varphi_{\varepsilon }dx}
\\\\
\displaystyle{+\varepsilon^2
\int_{\Omega^{c,2}_\varepsilon} \left(-\partial _{x_{2}}
\varphi^\star_{\varepsilon }\left(\left \vert\varepsilon\partial_{x_1} \varphi_\varepsilon\right\vert^2-\frac{1}{D_\varepsilon^2}\left\vert\varepsilon\partial_{x_2} \varphi_\varepsilon\right\vert^2\right)+2\varepsilon\partial_{x_2}
\varphi_{\varepsilon }\partial_{x_1}
\varphi^\star_{\varepsilon }\varepsilon\partial_{x_1}
\varphi_{\varepsilon }\right)dx.}
\end{array}\end{equation}
As the first integral in the right-hand side of \eqref{J27,2019*} is concerned, \eqref{F16,2019z}-\eqref{F16,2019zq}, \eqref{F19,2019cr1}, and \eqref{F19,2019cr3} provide that
\begin{equation}\label{J26,2019I}\begin{array}{ll}\displaystyle{\left\vert \int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left \vert\varepsilon^2\partial_{x_1} \varphi_\varepsilon\right\vert^2 dx\right\vert}\\\\ \displaystyle{\leq\Vert \partial _{x_{2}}
\varphi^\star\Vert_{L^\infty([0,1] \times[l_1,l_2])} \int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left \vert\varepsilon^2\partial_{x_1} \varphi_\varepsilon\right\vert^2 dx\rightarrow 0, }
\end{array}\end{equation}
as $\varepsilon\rightarrow 0$.
As the second integral in the right-hand side of \eqref{J27,2019*} is concerned, one has
\begin{equation}\label{J26,2019II}\begin{array}{ll}\displaystyle{
\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2dx=
\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)+\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx}\\\\
\displaystyle{= \int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx
+\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx}\\\\
\displaystyle{+2\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left(\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right)\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)dx,\quad\forall\varepsilon.
}
\end{array}\end{equation}
where $\varphi_1$ is defined in \eqref{F12,2019bis}. Moreover,
\eqref{F16,2019z}-\eqref{F16,2019zq}, \eqref{F12,2019bis}, \eqref{F19,2019cr1}, and \eqref{F19,2019cr3} provide
\begin{equation}\label{J26,2019IIbis}\left\{\begin{array}{ll}\displaystyle{\left\vert \int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx\right\vert
}\\\\
\displaystyle{\leq\Vert \partial _{x_{2}}
\varphi^\star\Vert_{L^\infty([0,1] \times[l_1,l_2])} \int_{\Omega^{c,1}_\varepsilon} \left\vert\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx\rightarrow 0, }\\\\
\displaystyle{\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx=\int_{\Omega^{c,1}} \partial _{x_{2}}
\varphi^\star\left(\frac{x_1}{\varepsilon},x_2\right)\left\vert\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert^2dx}\\\\
\displaystyle{\rightarrow \int_{\Omega^{c,1}\times\omega^a} \partial _{x_{2}}
\varphi^\star\left(y,x_2\right)dxdy=\hbox{meas}(\omega^a) L,}\\\\
\displaystyle{2\left\vert\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left(\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right)\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)dx\right\vert}
\\\\
\displaystyle{\leq2\Vert \partial _{x_{2}}
\varphi^\star\Vert_{L^\infty([0,1] \times[l_1,l_2])} \Vert \partial _{x_{2}}
\varphi_1\Vert_{L^\infty([0,1] )} \int_{\Omega^{c,1}_\varepsilon} \left\vert\partial_{x_2} \varphi_\varepsilon-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert dx\rightarrow 0, }
\end{array}\right.\end{equation}
as $\varepsilon\rightarrow 0$. Then, combining \eqref{J26,2019II} and \eqref{J26,2019IIbis} gives
\begin{equation}\label{J27,2019+}\begin{array}{ll}\displaystyle{\lim_{\varepsilon\rightarrow 0}
\int_{\Omega^{c,1}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2dx=
\hbox{meas}(\omega^a)L. }
\end{array}\end{equation}
Similarly, one proves that
\begin{equation}\label{J27,2019+-}\begin{array}{ll}\displaystyle{\lim_{\varepsilon\rightarrow 0}
\int_{\Omega^{c,3}_\varepsilon} \partial _{x_{2}}
\varphi^\star_{\varepsilon }\left\vert\partial_{x_2} \varphi_\varepsilon\right\vert^2dx=
\hbox{meas}(\omega^b)L. }
\end{array}\end{equation}
As the fourth integral in the right-hand side of \eqref{J27,2019*} is concerned, \eqref{F16,2019z}-\eqref{F16,2019zq}, and the first two estimates in \eqref{J18,2019estimatesalfa=2} provide
\begin{equation}\label{J27,2019IIbiscx}\begin{array}{ll}
\displaystyle{\left\vert
2\varepsilon^4
\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\partial_{x_2}
\varphi_{\varepsilon } \partial_{x_1}
\varphi^\star_{\varepsilon }\partial_{x_1}
\varphi_{\varepsilon }dx\right\vert
}\\\\
\leq
2\varepsilon\Vert \partial _{x_{1}}
\varphi^\star\Vert_{L^\infty([0,1] \times[l_1,l_2])}
\Vert \varepsilon^2 \partial _{x_{1}}
\varphi_\varepsilon\Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}
\Vert \partial _{x_{2}}
\varphi_\varepsilon\Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\rightarrow0,
\end{array}\end{equation}
as $\varepsilon\rightarrow 0$.
As the last integral in the right-hand side of \eqref{J27,2019*} is concerned, \eqref{J15,2019}, \eqref{F16,2019z}-\eqref{F16,2019zq}, and the last estimate in \eqref{J18,2019estimatesalfa=2} provide
\begin{equation}\label{J27,2019IIbiscxfhf}\begin{array}{ll}
\displaystyle{\left\vert \varepsilon^2
\int_{\Omega^{c,2}_\varepsilon} \left(-\partial _{x_{2}}
\varphi^\star_{\varepsilon }\left(\left \vert\varepsilon\partial_{x_1} \varphi_\varepsilon\right\vert^2-\frac{1}{D_\varepsilon^2}\left\vert\varepsilon\partial_{x_2} \varphi_\varepsilon\right\vert^2\right)+2\varepsilon\partial_{x_2}
\varphi_{\varepsilon }\partial_{x_1}
\varphi^\star_{\varepsilon }\varepsilon\partial_{x_1}
\varphi_{\varepsilon }\right)dx\right\vert}\\\\
\displaystyle{\leq\bigg[\varepsilon^2\Vert \partial _{x_{2}}
\varphi^\star\Vert_{L^\infty([0,1] \times[l_1,l_2])} \int_{\Omega^{c,2}_\varepsilon} \left(\left \vert\varepsilon\partial_{x_1} \varphi_\varepsilon\right\vert^2+\frac{1}{D_\varepsilon^2}\left\vert\varepsilon\partial_{x_2} \varphi_\varepsilon\right\vert^2\right)dx
}\\\\
\displaystyle{+2\varepsilon \Vert \partial _{x_{1}}
\varphi^\star\Vert_{L^\infty([0,1] \times[l_1,l_2])}
\Vert \varepsilon \partial _{x_{1}}
\varphi_\varepsilon\Vert_{L^2(\Omega^{c,2}_\varepsilon)}
\Vert \varepsilon\partial _{x_{2}}
\varphi_\varepsilon\Vert_{L^2(\Omega^{c,2}_\varepsilon)}\bigg]\rightarrow0,
}
\end{array}\end{equation}
as $\varepsilon\rightarrow 0$.
Finally, passing to the limit, as $\varepsilon$ tends to zero, in \eqref{J27,2019*} and using \eqref{J26,2019I}, \eqref{J27,2019+}, \eqref{J27,2019+-}, \eqref{J27,2019IIbiscx}, and \eqref{J27,2019IIbiscxfhf} give \eqref{F25,2019has} when $\alpha=2$.
\end{proof}
\section{The case $\alpha>2$\label{sketched}}
In the case $\alpha>2$, the proof of Theorem \ref{main theoremapril24,2019} will be just sketched.
\subsection{{\it A priori} estimates }
Proposition \ref{J18,2019Proposition} immediately implies the following result.
\begin{Corollary} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$. Then,
\begin{equation}\label{J18,2019estimatesalfa=2Aprile 62019}\exists c\in]0,+\infty[\quad:\quad\left\{\begin{array}{lll}\displaystyle{ \Vert\varepsilon^\alpha\partial_{x_1} \varphi_\varepsilon\Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\leq c,}\\\\
\displaystyle{ \Vert\partial_{x_2} \varphi_\varepsilon \Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\leq c,
}\\\\ \displaystyle{ \Vert\varepsilon^{\frac{\alpha}{2}}\nabla \varphi_\varepsilon \Vert_{L^2(\Omega^{c,2}_\varepsilon)}\leq c,}
\end{array}\right.\quad\forall\varepsilon.\end{equation}
\end{Corollary}
This result provides the following {\it a priori} estimate.
\begin{Proposition} \label{J21,2019prop6Aprile2019} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$. Then,
\begin{equation}\label{J18,2019las6Aprile2019t}\exists c\in]0,+\infty[\quad:\quad\left\{\begin{array}{lll}\displaystyle{ \Vert \varphi_\varepsilon \Vert_{L^2(\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon)}\leq c, }\\\\ \left\Vert \varepsilon^{\frac{\alpha-2}{2}} \varphi_\varepsilon \right\Vert_{L^2(\Omega^{c,2}_\varepsilon)}\leq c,
\end{array}\right.\quad\forall\varepsilon.\end{equation}
\end{Proposition}
\begin{proof} The Dirichlet boundary condition of $\varphi_\varepsilon $ on $\Gamma_\varepsilon$ and the second estimate in \eqref{J18,2019estimatesalfa=2Aprile 62019} provide the first estimate in \eqref{J18,2019las6Aprile2019t}.
Arguing as in the proof of Proposition \ref{J21,2019prop} gives
\begin{equation}\label{J19,2019Aprile62019}
\Vert \varepsilon^{\frac{\alpha-2}{2}} \varphi_\varepsilon \Vert^2_{L^2(\Omega^{c,2}_\varepsilon)}\leq
2(l_2-l_1)\varepsilon^{\alpha-2} +2
\left \Vert \varepsilon^{\frac{\alpha}{2}} \partial_{x_1}\varphi_\varepsilon \right\Vert^2_{L^2(\Omega^{c,2}_\varepsilon)},\quad\forall\varepsilon,
\end{equation}
which implies the second estimate in \eqref{J18,2019las6Aprile2019t}, thanks to the third estimate in \eqref{J18,2019estimatesalfa=2Aprile 62019}.
\end{proof}
\subsection{Weak convergence results}
The next proposition is devoted to studying the limit in $\Omega^{c,2}$, as $\varepsilon$ tends to zero, of problem \eqref{J13,2019weakrescaled} with $\alpha>2$.
\begin{Proposition} \label{Proposizione Monda37Aprile2019}For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$ and let $\overline{\varphi_{\varepsilon,2}}$,
be defined by \eqref{F13,2019}.
Then,
\begin{equation}\label{Monda1bisterforse7Aprile2019}\left\{\begin{array}{ll}\varepsilon^{\frac{\alpha-2}{2}}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to } 0,\\\\
\varepsilon^{\frac{\alpha}{2}}\partial_{x_1}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }0,\\\\
\varepsilon^{\frac{\alpha}{2}}\partial_{x_2}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }0,\end{array}\right.\end{equation}
as $\varepsilon$ tends to zero.
\end{Proposition}
\begin{proof}
The second estimate in \eqref{J18,2019las6Aprile2019t} and the third estimate in \eqref{J18,2019estimatesalfa=2Aprile 62019} ensure the existence of a subsequence of $\{\varepsilon\}$, still denoted by $\{\varepsilon\}$, and $u_2 \in L^2\left(\Omega^{c,2}, H^1_{\hbox{per}}(]0,1[)\right)$ (in possible dependence on the subsequence) such that
\begin{equation}\label{Monda1bister7Aprile2019}\left\{\begin{array}{ll}\varepsilon^{\frac{\alpha-2}{2}}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to } u_2,\\\\
\varepsilon^{\frac{\alpha}{2}}\partial_{x_1}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }\partial_{y} u_2,\\\\
\varepsilon^{\frac{\alpha}{2}}\partial_{x_2}\overline{\varphi_{\varepsilon,2}}\hbox{ two scale converges to }0,\end{array}\right.\end{equation}
as $\varepsilon$ tends to zero.
Arguing as in the proof of Proposition \ref{Proposizione Monda3}, one obtains
\begin{equation}\label{J23,2019fgd7Aprile2019} u_2=0, \hbox { a.e. in } \Omega^{c,2}\times\left(\omega^a\cup\omega^b\right).\end{equation}
Passing to the limit, as $\varepsilon$ tends to zero, in \eqref{J13,2019weakrescaled} with $\alpha>2$ and with test functions
$\displaystyle{\psi=\varepsilon^{\frac{\alpha}{2}+1}\chi_1(x_1,x_2)\chi_2\left(\frac{x_1}{\varepsilon}\right)}$, where $\chi_1\in C_0^\infty\left(\Omega^{c,2}\right)$ and $\chi_2\in H^1_{\hbox{per}}\left(]0,1[\right)$ such that $\chi_2=0$ in $\omega^a\cup\omega^b$, and using \eqref{J15,2019} and the second and third limits in \eqref{Monda1bister7Aprile2019} provide
that, for a.e. $(x_1,x_2)$ in $ \Omega^{c,2}$,
\begin{equation}\label{J23,2019seraIJ247Aprile2019}\begin{array}{ll}\displaystyle{ \int_{]0,1[\setminus\left(\omega^a\cup\omega^b\right)}\partial_{y}u_2(x_1,x_2,y)\partial_{y}\chi_2(y)dy=0, }\\\\ \forall \chi_2\in H^1_{\hbox{per}}\left(]0,1[\right)\,\,:\,\, \chi_2=0,\hbox{ in } \omega^a\cup\omega^b.
\end{array}
\end{equation}
Problem \eqref{J23,2019fgd7Aprile2019} and \eqref{J23,2019seraIJ247Aprile2019} is equivalent to the following problem independent of $(x_1,x_2)$
\begin{equation}\left\{\begin{array}{ll}\partial^2_{y^2}u_2=0,\hbox{ in }]0,1[\setminus\left(\omega^a\cup\omega^b\right),\\\\
u_2=0, \hbox { in } \omega^a\cup\omega^b,\\\\
u_2(0)=u_2(1),\\\\
\partial_y u_2(0)=\partial_yu_2(1),
\end{array}\right.\end{equation}
which admits $u_2=0$ as unique solution.
Consequently, limits in \eqref{Monda1bister7Aprile2019} hold for the whole sequence and \eqref{Monda1bisterforse7Aprile2019} is satisfied.
\end{proof}
The next proposition is devoted to studying the limit in $\Omega^{c,3}$ and in $\Omega^{c,1}$, as $\varepsilon$ tends to zero, of problem \eqref{J13,2019weakrescaled} with $\alpha>2$.
\begin{Proposition} \label{Proposizione Eliquis18Aprile2019}
For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$ and let
$\widetilde{\varphi_{\varepsilon,3}}$ and $\widehat{\varphi_{\varepsilon,1}}$ be defined by \eqref{Eliquis2} and \eqref{Eliquis4}, respectively. Moreover, let $\varphi_3$ and $\varphi_1$ be defined by \eqref{F12,2019},and \eqref{F12,2019bis}, respectively. Then,
\begin{equation}\label{Monda1bisterforseter8Aprile2019}\left\{\begin{array}{ll}\widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to } \varphi_3,\\\\
\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}\hbox{ two scale converges to }\partial_{x_2} \varphi_3,\end{array}\right.\end{equation}
and
\begin{equation}\label{Monda1bisterforsequater8Aprile2019}\left\{\begin{array}{ll}\widehat{\varphi_{\varepsilon,1}}\hbox{ two scale converges to } \varphi_1,\\\\
\partial_{x_2}\widehat{\varphi_{\varepsilon,1}}\hbox{ two scale converges to }\partial_{x_2} \varphi_1,\end{array}\right.\end{equation}
as $\varepsilon$ tends to zero.
\end{Proposition}
\begin{proof} One can repeat the proof of Proposition \ref{Proposizione Eliquis1}, by making attention to use equation \eqref{J13,2019weakrescaled} with $\alpha>2$ instead of $\alpha=2$, and to multiply the test functions by $\varepsilon^\alpha$ instead of $\varepsilon^2$ when it occurs. Really, in this case the proof is simpler than the proof of Proposition \ref{Proposizione Eliquis1} due to the fact that the second limit in \eqref{Monda1bisterforse7Aprile2019} is zero. \end{proof}
The following result is an immediate consequence of Proposition \ref{Proposizione Monda37Aprile2019} and Proposition \ref{Proposizione Eliquis18Aprile2019}.
\begin{Corollary}
For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$ and let $\overline{\varphi_{\varepsilon,2}}$,
$\widetilde{\varphi_{\varepsilon,3}}$, and $\widehat{\varphi_{\varepsilon,1}}$ be defined by \eqref{F13,2019}, \eqref{Eliquis2}, and \eqref{Eliquis4}, respectively. Moreover, let $\varphi_3$ and $\varphi_1$ be defined by \eqref{F12,2019} and \eqref{F12,2019bis}, respectively. Then
\begin{equation} \nonumber\varepsilon^{\frac{\alpha-2}{2}}\overline{\varphi_{\varepsilon,2}}\rightharpoonup 0,\quad\varepsilon^{\frac{\alpha}{2}}\partial_{x_1}\overline{\varphi_{\varepsilon,2}}\rightharpoonup0,\quad \varepsilon^{\frac{\alpha}{2}}\partial_{x_2}\overline{\varphi_{\varepsilon,2}}\rightharpoonup0, \hbox{ weakly in }L^2(\Omega^{c,2}),
\end{equation}
\begin{equation} \nonumber\widetilde{\varphi_{\varepsilon,3}}\rightharpoonup (x_2-l_2)\hbox{meas}(\omega^b)+1, \quad\partial_{x_2}\widetilde{\varphi_{\varepsilon,3}}\rightharpoonup\hbox{meas}(\omega^b),\hbox{ weakly in }L^2(\Omega^{c,3}),
\end{equation}
and
\begin{equation} \nonumber\widehat{\varphi_{\varepsilon,1}}\rightharpoonup (x_2-l_1)\hbox{meas}(\omega^a),\quad \quad\partial_{x_2}\widehat{\varphi_{\varepsilon,1}}\rightharpoonup\hbox{meas}(\omega^a),\hbox{ weakly in }L^2(\Omega^{c,1}),
\end{equation}
as $\varepsilon$ tends to zero.
\end{Corollary}
\subsection{Corrector results}
Arguing as in Proposition \ref{encon2019}, one obtains the following energies convergence.
\begin{Proposition}\label{encon2019Pasqua2019} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$. Moreover, let $\varphi_1$ and $\varphi_3$ be defined by \eqref{F12,2019bis} and \eqref{F12,2019}, respectively. Then
\begin{equation}\nonumber\begin{array}{lll}
\displaystyle{\lim_{\varepsilon\rightarrow 0}\bigg[\int_{\Omega^{c,1}_\varepsilon\cup \Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^\alpha\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert\partial_{x_2} \varphi_\varepsilon \right\vert^2\right)+
\int_{\Omega^{c,2}_\varepsilon}\left( D_\varepsilon \left\vert\varepsilon^{\frac{\alpha}{2}}\partial_{x_1} \varphi_\varepsilon \right\vert^2+D^{-1}_\varepsilon\left\vert\varepsilon^{\frac{\alpha}{2}}\partial_{x_2} \varphi_\varepsilon \right\vert^2 \right)dx\bigg]
}\\\\
\displaystyle{=
\int_{\Omega^{c,1}\times\omega^a}\left\vert\partial_{x_2} \varphi_1\right\vert^2dxdy+\int_{\Omega^{c,3}\times\omega^b}\left\vert\partial_{x_2} \varphi_3\right\vert^2dxdy.}
\end{array}\end{equation}
\end{Proposition}
By arguing as in Proposition \ref{F19,2019corrres}, Proposition \ref{Proposizione Monda37Aprile2019}, Proposition \ref{Proposizione Eliquis18Aprile2019}, and Proposition \ref{encon2019Pasqua2019} provide the following corrector results.
\begin{Proposition}\label{F19,2019corrresPasqua2019} For every $\varepsilon$, let $\varphi_\varepsilon$ be the unique solution to \eqref{J13,2019weakrescaled} with $\alpha>2$. Moreover, let $\varphi_1$ and $\varphi_3$ be defined by \eqref{F12,2019bis} and \eqref{F12,2019}, respectively. Then
\begin{equation}\nonumber\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}
\int_{\Omega^{c,1}_\varepsilon}\left( \left\vert \varepsilon^\alpha\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_1 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx=0,
}\end{array}\end{equation}
\begin{equation}\nonumber\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}
\int_{\Omega^{c,3}_\varepsilon}\left( \left\vert \varepsilon^\alpha\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \partial_{x_2} \varphi_\varepsilon (x)-\left(\partial_{x_2}\varphi_3 \right)\left(\frac{x_1}{\varepsilon}\right)\right\vert ^2\right)dx=0,
}\end{array}\end{equation}
and
\begin{equation}\nonumber\begin{array}{lll} \displaystyle{\lim_{\varepsilon\rightarrow0}
\int_{\Omega^{c,2}_\varepsilon}\left( \left\vert \varepsilon^{\frac{\alpha}{2}}\partial_{x_1} \varphi_\varepsilon \right\vert^2+\left\vert \varepsilon^{\frac{\alpha}{2}}\partial_{x_2} \varphi_\varepsilon (x)\right\vert ^2\right)dx=0.
}\end{array}\end{equation}
\end{Proposition}
Finally, using Proposition \ref{F19,2019corrresPasqua2019}, the proof of Theorem \ref{main theoremapril24,2019}
with $\alpha>2$ follows the same outline of the proof of Theorem \ref{main theoremapril24,2019} with $\alpha=2$.
\section*{Acknowledgments}
The authors thank the
"Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA)"
of the "Istituto Nazionale di Alta Matematica (INdAM)" (Italy), the "Universit\'e de Franche-Comt\'e" (France), and the competitive funding program for interdisciplinary
research of "CNRS" (France) for their financial support. | 118,908 |
Carl Hamstead prepares Brews' Butternut Squash Harvest Salad on Wednesday in the kitchen.
- Filed Under
What turns a salad from ordinary to extraordinary?
For some, it's fresh ingredients. For others, it's all about the dressing. Then there are those who love toppings, from nuts or cheese to fruit or croutons.
Whether you're looking for a classic salad or a unique meal, local restaurants offer something for everyone. ... | 171,883 |
Article content
Derek Boyle plans to jump into the housing market now that Unifor members have ratified a new four-year contract with Fiat Chrysler Automobiles that not only offers each autoworker wage hikes, but bonuses totalling $12,000.
“I’m looking forward to buying a home for my son and me, and this job will play a role in that,” said Boyle, who was hired at the Windsor Assembly Plant about a year ago.
tap here to see other videos from our team.
FCA-Unifor contract expected to boost local economy Back to video
“I voted in favour of it; I think it’s a good deal,” said Boyle. “I’ve had jobs where I received a 42-cent raise after two years. Now, I’m getting raises every year, and a chunk of money for signing, so I’m happy with that.”
Labour peace at the city’s biggest private sector employer could further lift an already buoyant economy that saw unemployment fall last month to 5.7 per cent — its lowest level in 15 years, said Alfie Morgan, professor emeritus of business at the University of Windsor.
The settlement, he added, will likely unleash pent-up demand for big-ticket items — purchases that might have been on hold pending the outcome of contract talks, said Morgan. | 239,028 |
TITLE: Maximum and minimum Expected values when taking colored balls
QUESTION [3 upvotes]: We have a sack with $60$ balls.
From them $15$ balls are red, $15$ green, $15$ blue and $15$ yellow.
We take $30$ balls from the sack.
What's the expected number of balls of the color from which the most balls had been taken? And from the color from which the least balls had been taken?
Expressed in the notation I begun to solve this unsuccesfully:
Let $X_i$ be a random event for the number of balls taken of the color $i$.
I look for: $E[\max(X_1,X_2,X_3,X_4)]$ and $E[\min(X_1,X_2,X_3,X_4)]$
I got that $P(X_i=x)=\frac{\binom{15}{x}\binom{45}{30-x}}{\binom{60}{30}}$
REPLY [6 votes]: The joint distribution of $\boldsymbol X = (X_1, X_2, X_3, X_4)$ that counts the number of balls drawn of each color, is multivariate hypergeometric: $$\Pr[\boldsymbol X = (x_1, x_2, x_3, x_4)] = \binom{60}{30}^{-1} \prod_{k=1}^4 \binom{15}{x_k}, \quad x_1 + x_2 + x_3 + x_4 = 30.$$ Thus the desired expectation is simply $$\operatorname{E}[\max \boldsymbol X] = \sum_{\boldsymbol x \in M} \max(\boldsymbol x) \Pr[\boldsymbol X = \boldsymbol x],$$ where $M$ is the set of all possible outcomes of $\boldsymbol X$. While this is a tedious sum to compute by hand, it is computable using Mathematica:
$$\operatorname{E}[\max \boldsymbol X] = \frac{280571657719508835}{29566145391215356} \approx 9.48963.$$ Similarly, $$\operatorname{E}[\min \boldsymbol X] = \frac{162920523148721505}{29566145391215356} \approx 5.51037.$$
By request, Mathematica code:
Explicit computation of the sum (I make no claims that it's the most elegant or efficient approach):
Total[ Max[#] (Times @@ Binomial[15, #]) & /@ Select[Append[#, 30 - Total[#]] & /@ Tuples[Range[16] - 1, 3], 0 <= Last[#] <= 15 &]] / Binomial[60, 30]
Using the built-in probability distribution:
Expectation[ Max[x1, x2, x3, x4], {x1, x2, x3, x4} \[Distributed] MultivariateHypergeometricDistribution[30, {15, 15, 15, 15}]]
And of course, altering either code to compute the expectation of the minimum is straightforward.
Minor modification and additions can easily generate a plot of the expectation as a function of the number of balls drawn. Depicted below is the expectation of the maximum. It turns out that the explicit calculation (first version) is quite a bit faster than using the built-in distribution for the range of parameters involved. | 76,990 |
Dragon Quest IX Launch Coverage at the New andriasang.com Ticker
See and comment on what's happening around Tokyo!
If you've been visiting the site, you might have noticed a little Twitter feed widget on the right side of blog posts and the main channel.
Previously, the widget was fed with content, usually posted via mobile updates, from Twitter and TwitPic. Today, I made them completely self contained within the site. Click on an image or a link in the feed and you'll be taken to a special page where you can post comments. I also whipped together a quick general feed page.
This part of the site is rough and may even be buggy, so if something doesn't appear to be working right, you can still access the content at Twitter and TwitPic.
I'll be testing this new area of the site out today at the Dragon Quest IX launch event in a couple of hours. Be sure and check back throughout the day and leave some comments in the images and updates I put up!
Loading comments. If comments don't load, make sure Javascript is on in your browser. | 407,002 |
Overview - Surah 2: al-Baqarah (The Cow)). The Surah, revealed in Madinah, deals with a number of issues related to Guidance, Allah’s Governance on Earth, history of previous Muslim Ummah and instructions for the new Muslim Ummah.
The Surah begins with the statement that it is Allah who revealed this book (the Qur'an) for the guidance of those who are conscious of Allah. Only those who seek guidance can benefit from the guidance of this Book. There are three types of human beings:
- Those who believe in the unseen realities, perform prayers, give part of their wealth in charity, believe in what is revealed in this scripture and what was revealed before to other prophets and messengers of Allah. These are the true believers. They shall benefit from this book and they shall be eternally successful.
- Second group consists of those who have decided to reject Allah's message. They are the Kafirs. Since they have made up their minds to reject Islam, no preaching will help them. Allah will punish them on the Day of Judgment because of their rejection.
- Third is the group of people who say that they have believed, but actually they have not believed. They try to be on both sides: sometimes at the side of faith and sometimes at the side of unfaith. They are the hypocrites. They may think that in this way they will gain both sides, but in reality they are also the losers. –peace be upon him- submitted to Allah and this is the message that he and his sons gave to their progeny.
- The change of Qiblah and the response of the hypocrites and fools. Those who have knowledge know that this is the true Qiblah of all the Prophets.
- Follow this direction wherever you are. This is the universal Qiblah for all.
-.
- No fighting during Hajj, rather seek God’s bounty when you return from Hajj.
- Appreciate God’s bounties. All human beings were originally one community. Divisions came later. Be generous and defend your self and your faith.
- Some important questions answered: War in the sacred months, wine and gambling, charity, orphans’ money, divorced women and their situation.
- The laws of divorce
- Continuation of the laws of divorce.
- Rules on the remarriage of the divorced women or the widowers.
- Further rules of divorce
- Fighting in the cause of God: Israelites
- Under the leadership of Prophet David the victory came over the forces of Goliath.
- Emphasis on charity. To Allah belong everything. His Throne extends to heaven and earth. No compulsion in religion. Allah brings out people from darkness unto light.
- Allah’s power over life and death, some examples: Prophet Ibrahim’s dialogue with Namrood, a man in the valley of dead (probably Prophet Ezekiel’s vision of Jerusalem),uary (riba) and its bad effects on individuals and society
- Some rules on loan transactions
- Conclusion and prayer: Everything in the heaven and earth belong to Allah, the prayer of the believers.
The name of the Surah has been mentioned in many authentic hadeeth as ‘al-Baqarah’ as is mentioned by the Prophet, ‘the last two Ayaat from the end of Surah al-Baqarah – whoever reads them at night it will suffice him.’ [Bukhari no. 4753]
Other names used for this Surah include;
- az-Zahra - The Light
- as-Sanaam - The Peak
- al-Fustaat - The Tent/Pavilion
There are 286 Ayat of Surah al-Baqarah.
Overview
- Guidance. The Surah makes it clear what the Straight Path is – who are upon it, who are not, what are their attributes
Allah’s Governance on Earth. The Surah was aptly revealed shortly after the establishment of the Islamic state in Madinah. Indeed, the first story of this Surah is about Adam, the very first Caliph of Allah on Earth. Various facets of the Shariah are explained and expounded for the newly established state, which will in turn only rule by the rulings and commands of Allah. Hence we find rulings/regulations on divorce, Hajj, Zakat, Ramadhan, Jihad, Financial transactions etc
- The Surah revolves around the theme of the methodology in application of the Khilafah of Allah on Earth
As a lesson to the new Muslim Ummah, Surah al-Baqarah deals with the previous ‘Muslim Ummah’ the Children of Israel [and their remnants in Madinah] – the promise of Allah to them, their attributes, how they dealt with the Laws of Allah, and how they were punished. All this providing as a warning to the new Muslim ummah [nation] not to repeat these and the failure to do so will result in similar punishments
- The significance of ayat 143 as the Muslims being the middle nation
- In essence, the themes of the whole Qur'an can be linked back to Surah al-Baqarah.
- Surah al-Baqarah is about the building of a society. Makki Ayat are primary focused on the Individual and Madani Ayat address the Muslims as a community
- Just like the changing of the Qiblah from praying towards Jerusalem to praying towards Makkah, the transformation from the previous Muslim ummah, the Children of Israel to the final Muslim ummah now in Madinah
- ." [Mawdudi, Tafhim]
There a number of names used for this Surah listed by the scholars:
al-Baqarah: This is in reference to the story of the Cow in the incident involving the murder amongst the Children of Israel. The story of the Cow contains the most important lessons for the Believer in relation to the commands of Allah. We learn how we should and how we should not behave with respect to the Shariah and urgency of acting upon the commands and not indulging in excessive questioning. In their implementation of the Law, their excessive questioning and hesitation in implementing the commands of Allah led to their situation only becoming more difficult upon themselves.
Sanaam: Linguistically means the peak or highest point on something or place, for example the sanaam of a camel is in reference to the hump being its highest point. The sanaam of a people are its leaders. Hence, Surah al-Baqarah is the peak with respect to the Qur'an as it contains the most important guidelines in establishing Islam as a system of life. The Prophet [saw] said, ‘Everything has a peak and the peak of the Qur’an is al-Baqarah.’ [Tirmidhee no. 2878].
Fustaat: Ibn Katheer mentions that Khalid bin Ma'dan would refer to this Surah as the fustat of the Qur'an. Fustat can be translated as 'tent' and just as the tent in the battlefield is the head quarters from which all the orders are issued, the Surah is the source/head of the remainder of the Qur'an.
Zahra: Translated as light, this Surah is a light on the path of guidance in this world and the after-life.
- The beginning of the Surah mentions the attributes of Iman [faith] that the Believer has – Ayah (2:3) and (2:4) mention Iman in:
a) al-Ghayb [unseen]
b) Belief in the Revelation sent upon Prophet Muhammad
c) Belief in the Revelations sent upon all the previous Messengers
d) Yaqeen [complete faith] in the Akhirah [afterlife]
- The end of the Surah (2:285) the following aspects of Iman [faith] are mentioned:
a) Belief in Allah
b) Belief in the Angels
c) Belief in the Books [of revelation]
d) Belief in the Messengers – not differientating between any of them [their message was the same]
Combined together they form the first 5 aspects of Iman as mentioned in the Hadith of Jibril [Sahih Muslim – the only aspect of Iman not mentioned in these Ayat but said in the Hadith is Qadr [pre-destination]
Manuscripts / Inscriptions
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713 H 1313 CE
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730 H (1330 CE)
6th Century H (12th century)
391 H (1001 CE)
8th century CE
1005 H (1596 CE)
1130 AH (1717 CE)
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1st Century Hijrah (7th Century CE)
1st Century Hijrah (7th Century CE)
1st Century Hijrah (7th Century CE)
704-705 H (1304-1306 AD)
700 AH (1300–1 AD)
Late 1st century / early 2nd century of Hijra.
Surah al-Baqarah
Surah al-Fatihah and al-Baqarah
- Though it is a Madani Surah, it follows naturally a Makkan Surah, Surah 1: al-Fatihah (The Opening), which ended with the prayer: “Show us the straight way.” It begins with the answer to that prayer, “This is the Book (that) . . . is guidance.”
- In Surah al-Fatihah, one asks to be guided on the way/path of those who have been favoured by Allah and not those who have earnt the anger of Allah nor those who are misguided. Hence, Surah al-Baqarah begins with describing the first category of people and then the second category of people.
- Based on the Hadith of the Prophet, the Maghdub [those whom have the anger of Allah] are the Jews and the Dhal [astray] are the Christians. Hence in the following two Surah’s both communities are addressed in same sequence, with al-Baqarah addressing the Jews and Al-Imran addressing the Christians.’s Companions had a slogan when they fought Musaylimah [the false prophet], ‘O companions of Surah al-Baqarah’ [Ibn Abi Shaybah no. 33572 and Abdur Razzaq in his Musanaf no. 9465].
- The Shayateen [jinn] flee from a house that it is recited].
- Leadership chosen based on relationship to this Surah.
-].
- It contains the Greatest Name of Allah.
- On the authority of Abu Ummamah that the Messenger of Allah said, ‘The Greatest Name of Allah by which if He is supplicated will be answered is in three Surahs, al-Baqarah, Al-Imran and Taha.....i.e. al-Hayyu al-Qayyum.’ [Mustradak al-Hakim no. 1867].
- A Scholar is one who has understood the first seven Surahs of the Qur’an. The Prophet said, ‘Whoever takes from the first seven Surahs of the Qur’an is a Hibr [scholar]’ [Mustradak al-Hakim no. 2070].
- The Prophet is reported to have said, "Whoever recites the last two Ayat of Surah al-Baqarah it would be sufficient for him." [Bukhari]
- It is the longest Surah of the Qur'an with 286 Ayat.
- It has the longest Ayat of the Qur'an (2:282).
- It has the greatest Ayat of the Qur'an (2:255).
- It has the last Ayat of the Qur'an revealed (2:281), according to some narrations.
- Although Nifaq (Hypocrisy) is alluded to, the word Nifaq and Munafiqeen are not mentioned in this Surah by name.
- Taqwa - words derived from و ق ي occur with the frequency of 258 times in 237 ayat. Interestingly, the highest frequency of these words appear in Surah al-Baqarah compared to any other Surah. The Muslim nation's character is built upon Taqwa.
- شطر - Shatr - direction/half etc - is a word only used in Surah al-Baqarah. It appears 5 times.
- The word رَفَثَ [acts leading and including sexual intercouse] appears only in Surah al-Baqarah. Occuring twice in (2:187) and (2:197)
- يَا أَيُّهَا الَّذِينَ آمَنُوا "O you who Believe" - This is first time this phrase appears in the Qur’an, and it does not appear in any Makki Surah. It is mentioned 11 Times in Surah al-Baqarah. The phrase 'O you who Believe' are the the opening words of Surah al-Maidah and it appears more times in Surah al-Maidah than any other Surah, occuring 16 times [confimation needed]
- 'Baqar' - 'adult cow' occurs 5 times in Surah al-Baqarah, which is the most amount compared to all other Surahs. In total, it occurs 9 times in the Qur'an. In Surah Yusuf (12) and Surah al-An'am (6) it occurs twice respectively. 7 out of the 9 times it occurs with respect to Children of Israel.
- ٱلْعِجْلَ - calf - is the word used to describe the child of the Cow, usually from birth to 2 years old. This word occurs the most frequent in this Surah appearing 4 times. It occurs 10 times in the whole Qur'an.
- Words derived from
قتلQ-T-L [to fight] occur 170 times in 122 Ayat in the Qur'an. It occurs 30 times in Surah al-Baqarah, which is the highest frequency compared to any other Surah.
- The word, رمضان Ramadhan only appears once in the Qur'an, appearing in this Surah. (2:185)
- The word الْأَهِلَّةِ - crescent moon - in the (2:189) the only mention of this word in the Qur’an.
- الْحَجِّ The word 'Hajj' appears in the Qur’an 12 times – 8 times in Surah al-Baqarah – the most in the Qur’an.
- رَفَثَ - a word only used in Surah al-Baqarah - used twice (2:187) and (2:197). It is defined as to whatever might lead to sexual intercourse, such as embracing, kissing and talking to women about similar subjects.
- The word يَسْأَلُونَكَ - 'They ask you [O Prophet]' appears in the Qur'an 15 times. It occurs in Surah al-Baqarah 7 times - this highest in any Surah.
- إِلْحَافًا - The triliteral root lām ḥā fā (ل ح ف) occurs only once in the Qur'an (2:273). said, ‘Whoever takes from the first seven Surahs of the Qur’an is a Hibr [scholar]’ [Mustradak al-Hakim no. 2070].].
- Claim of the Qur'an: "This is the Book which contains no doubt."
- Creation of Adam, man's nature, and his destiny.
- The Children of Israel and the People of the Book (Jews and Christians).
- Israelites' sin of worshipping the statue of a calf.
- Punishment of Israelites violation of Sabbath.
- Nature of Jews' belief.
- Allah orders not to prevent the people from coming to Masajid.
- Ibrahim and his sons were neither Jews nor Christians but were Muslims.
- Abraham (Ibrahim), Ishmael (Isma`il), and their building of Ka'bah.
- Change of Qiblah (direction in prayers) towards Ka'bah in Makkah.
- Allah orders not to profess any faith blindly.
- The moon is created to determine the time periods i.e. months and years.
- Hypocrisy vs. True faith.
- Ayat-ul-Kursi (Verse of the Throne of Allah).
- Allah orders the believers to enter into Islam completely.
- Punishment of a murtad (a Muslim who becomes a Non-Muslim).
- It is unlawful to marry a mushrik.
- Victory is not by numbers but by Allah's help.
- Confrontation of Ibrahim and Namrud (the king of his time).
- What makes charity worthless.
- Taking usury is like declaring war against Allah and his Rasool.
- All business dealings relating to deferred payments must be in writing.
- Retaliation against oppression.
- Non compulsion in religion.
- Divine Laws are promulgated about the following categories:
- Food
- Retribution
- Wills
- Fasting
- Bribery
- Jihad
- Self-defense
- Evidence
- Pilgrimage
- Charity
- Drinking
- Bloodwit
- Gambling
- Marriage
- Orphans
- Menstruation
- Oaths
- Divorce
- Alimony
- Nursing
- Widows
- Usury
- Buying on Credit
- Debts
- Loans
- Pledge/Mortgage
- Believers supplication to Allah.
Tafsir Zone
- Surah Al-Baqarah (The Cow) Saad al Ghamidi
- Surah Al-Baqarah- Muhammad Ayub
- Surah Al-Baqarah - Minshawi (tarteel)
- Surah Al-Baqarah - Khalid Jaleel
- Surah Al-Baqarah Mahmoud Khalil Al Hussary
- Surah Al-Baqarah Muhammad Al Luhaydan
- Surah Al-Baqarah Idris Akbar
- Surah Al-Baqarah Muhammad Minshawi
- Surah Al-Baqarah (22-82)-1 Dr Israr Ahmed
- Surah Al-Baqarah (22-82)-2 Dr Israr Ahmed
- Surah Al-Baqarah (83-141)-1 Dr Israr Ahmed
- Surah Al-Baqarah (83-141)-2 Dr Israr Ahmed
- Surah Al-Baqarah (142-188)-1 Dr Israr Ahmed
- Surah Al-Baqarah (142-188)-2 Dr Israr Ahmed
- Surah Al-Baqarah (189-248)-1 Dr Israr Ahmed
- Surah Al-Baqarah (189-248)2 Dr Israr Ahmed
- Surah Al-Baqarah (248-end)1 Dr Israr Ahmed
- Surah Al-Baqarah (248-end)-2 Dr Israr Ahmed | 371,007 |
\begin{document}
\maketitle
\footnote{Mathematical Institute, Graduate School of Science, Tohoku University, Sendai, Japan}
\footnote{MSC2020 Subject Classifications: 60H05, 60G44}
\footnote{Email address: [email protected]}
\renewcommand{\thepage}{\arabic{page}}
\begin{abstract}
Stochastic horizontal lifts and anti-developments of semimartingales with jumps on complete and connected Riemannian manifolds are discussed in this paper. We prove the one-to-one correspondence of some classes of semimartingales on Riemannian manifolds, orthonormal frame bundles and Euclidean spaces by using the stochastic differential geometry with jumps introduced by Cohen (1996). This result is an extension of the research by Pontier and Estrade (1992). We also show that we can construct a martingale with jumps on a sphere from a local martingale on a Euclidean space.
\end{abstract}
\section{Introduction}\label{intro}
Stochastic parallel displacement of a frame along a diffusion was defined in \cite{Mal} and \cite{IkeWata}. This can be understood as the horizontal lift of a diffusion on a manifold to a frame bundle. The horizontal lift of a continuous semimartingale on a manifold to a frame bundle was also considered in \cite{Shi} and \cite{Hsu}. This is an extension of a horizontal lift of a smooth curve on a manifold. Throughout this paper, we assume that we are given a filtered probability space $(\Omega, \mathcal{F},\{ \mathcal{F}_t\}_{0\leq t\leq \infty}, P)$ and the usual hypotheses for $\{ \mathcal{F}_t\}_{0\leq t\leq \infty}$ hold. A stochastic process $X$ valued in a manifold $M$ is called an $M$-valued semimartingale if $f(X)$ is an $\mathbb{R}$-valued semimartingale for all $f\in C^{\infty}(M)$. It is known that for an $M$-valued continuous semimartingale $X$, we can define the Stratonovich integral of 1-form $\phi$ along $X$ and the quadratic variation of 2-tensor $\psi$. They are denoted by $\displaystyle \int \phi(X)\circ dX$ and $\displaystyle \int \psi(X)\, d[X,X]$, respectively. Furthermore, given a torsion-free connection on $M$, we can define the It\^o integral of 1-form $\phi$ denoted by $\displaystyle \int \phi(X)dX$ and the equation
\[
\int \phi(X)\circ dX=\int \phi(X) \, dX+\frac{1}{2}\int \nabla \phi (X)\, d[X,X]
\]
holds. Let $\pi :\mathcal{O}(M) \to M$ be an orthonormal frame bundle. According to \cite{Shi}, given an $M$-valued continuous semimartingale $X$, an $\mathcal{O}(M)$-valued $\mathcal{F}_0$-measurable random variable $u_0$ such that $\pi u_0=X_0$ and a connection $\nabla$ on $M$, the $\mathcal{O}(M)$-valued continuous semimartingale $U$ satisfying $U_0=u_0$, $\pi U=X$ and
\[
\int \theta \circ dU=0
\]
for the connection form $\theta$ corresponding to $\nabla$ is uniquely determined. Furthermore, for an $\mathcal{O}(M)$-valued horizontal semimartingale $U$, the stochastic integral of the solder form $\mathfrak{s}$ along $U$ yields a continuous semimartingale on a Euclidean space. This is called the anti-development of $U$ and $X=\pi U$. Conversely, we can construct an $\mathcal{O}(M)$-valued horizontal semimartingale from a continuous semimartingale $W$ starting at $0$ on a Euclidean space. In fact, there exists an $\mathcal{O}(M)$-valued continuous semimartingale $U$ satisfying
\[
F(U_t)-F(U_0)=\int_0^t L_kF(U_s)\circ dW^k_s,\ F\in C^{\infty}(\mathcal{O}(M))
\]
by the existence and the uniqueness of solutions of stochastic differential equations (SDE's) on manifolds, where $L_k$ is the canonical horizontal vector field on $\mathcal{O}(M)$ and we have used the Einstein summation convention. It immediately follows that the solution is horizontal. Furthermore, by projecting $U$ onto the base space $M$, we obtain a continuous semimartingale on $M$. In \cite{Hsu}, it is shown that this correspondence is one-to-one.\\
The aim of our research is to extend the above result to discontinuous semimartingales on a Riemannian manifold called $\Delta$-semimartingales introduced in \cite{Pic1}. We describe our setting and main theorems.
Let $(M,g)$ be a complete and connected Riemannian manifold. In \cite{Pic1}, to define stochastic integral along $M$-valued discontinuous semimartingales, a $TM\times M$-valued process $(\Delta X,X)$ called a $\Delta$-semimartingale is introduced, which is a pair of an $M$-valued semimartingale $X$ and jump directions $\Delta X_s\in T_{X_{s-}}M$. Since we suppose $M$ is complete and connected, we consider the condition
\begin{align}
\exp_{X_{s-}}\Delta X_s=X_s \label{eq:v}
\end{align}
for a $\Delta$-semimartingale, where $\exp$ is the exponential map with respect to the Levi-Civita connection. Let $\phi$ be a $T^*M$-valued c\`{a}dl\`{a}g process above $X$, that is, $\phi_s\in T_{X_s}^*M$, $s\geq 0$. Given a $\Delta$-semimartingale $(\Delta X,X)$ and a torsion-free connection $\nabla$, the It\^o integral of 1-form $\phi$ along $(\Delta X,X)$ is defined in \cite{Pic1} and denoted by $\displaystyle \int \phi _{s-}\, dX_s$. The quadratic variation of a $T^*M\otimes T^*M$-valued c\`{a}dl\`{a}g process $\psi$ above $X$ is also defined and denoted by $\displaystyle \int \psi_{s-}\, d[X,X]_s$. Furthermore, we define the Stratonovich integral of 1-form $\alpha$ as
\[
\int \alpha (X)\circ dX=\int \alpha (X_-)\, dX+\frac{1}{2}\int \nabla \alpha (X_-)\, d[X,X]^c,
\]
where $\displaystyle \int \nabla \alpha (X_-)\, d[X,X]^c$ is the continuous part of the quadratic variation of $\nabla \alpha$.
The main results of this thesis are as follows. Under suitable conditions, we show the one-to-one correspondence between $\Delta$-semimartingales on manifolds, $\Delta$-horizontal semimartingales on orthonormal frame bundles, and semimartingales on Euclidean spaces starting at $0$. We denote by $\theta$ and $\mathfrak{s}$ the connection form with respect to the Levi-Civita connection and the solder form, respectively. The solder form $\mathfrak{s}$ is a 1-form on $\mathcal{O}(M)$ valued in $\mathbb{R}^d$, where $d=\text{dim}M$. Their precise definition is given in \sref{bundlemetric}. Define the Riemannian metric $\tilde g$ on $\mathcal{O}(M)$ by
\[
\tilde g:=\sum_{\alpha=1}^{\frac{d(d-1)}{2}}\theta^{\alpha}\otimes \theta^{\alpha}+\sum_{k=1}^d\mathfrak{s}^k\otimes \mathfrak{s}^k.
\]
Denote the corresponding Levi-Civita connection by $\tilde \nabla$. Fundamental properties of $\tilde g$ are also given in \sref{bundlemetric}. We can define the It\^o integral of 1-form and the quadratic variation of 2-tensor on $\mathcal{O}(M)$ with respect to $\tilde \nabla$. To state our results more precisely, we introduce two more definitions.
\begin{dfn}
\begin{itemize}
\item [(1)]
Let $(\Delta U,U)$ be an $\mathcal{O}(M)$-valued semimartingale. $(\Delta U,U)$ is said to be horizontal if
\[
\int_0^t \theta (U_s)\circ dU_s=0
\]
for all $t\geq 0$.
\item [(2)]
Let $(\Delta U,U)$ be a horizontal $\Delta$-semimartingale on $\mathcal{O}(M)$. The anti-development of $(\Delta U,U)$ is defined by
\[
W=\int \mathfrak{s}(U)\circ dU.
\]
\end{itemize}
\end{dfn}
In particular, if $(\Delta U,U)$ is horizontal, then
\[
\Delta \int_0^t \theta (U_s)\circ dU_s=\langle \theta (U_{s-}),\Delta U_s\rangle=0.
\]
Thus $\Delta U_s$ is a horizontal tangent vector.
\begin{dfn}
Let $(\Delta U,U)$ be an $\mathcal{O}(M)$-valued $\Delta$-horizontal semimartingale and $(\Delta X,X)$ an $M$-valued $\Delta$-semimartingale. $(\Delta U,U)$ will be called a horizontal lift of $(\Delta X,X)$ if
\[
\pi U=X,\ \pi_*\Delta U=\Delta X.
\]
\end{dfn}
Note that given an $\mathbb{R}^d$-valued semimartingale $W$ with $W_0=0$ and an $\mathcal{O}(M)$-valued $\mathcal{F}_0$-measurable random variable $U_0$, there exists an $\mathcal{O}(M)$-valued semimartingale $U$ such that for $F\in C^{\infty}(\mathcal{O}(M))$,
\begin{align*}
F(U_t)-F(U_0)&=\int_0^t L_kF(U_{s-})\circ dW^k_s\\
&+\sum_{0<s\leq t} \{ F(U_s)-F(U_{s-})-L_kF(U_{s-})\Delta W^k_s\}
\end{align*}
by \cite{Co1} or \eref{exmarcus}. $U$ is called a development of $W$. One of our main results is as follows.
\begin{thm}\label{main}
\begin{itemize}
\item[(1)]Let $W$ an $\mathbb{R}^d$-valued semimartingale with $W_0=0$ and $U$ a development of $W$. Suppose that $U$ does not explode in finite time. Put $\Delta U_s=\Delta W^k_s L_k(U_{s-})$. Then $(\Delta U,U)$ is a $\Delta$-semimartingale satisfying \eqref{eq:v} and it holds that
\[
\int \theta (U_-)\circ dU=\int \theta (U_-)\, dU=0,
\]
\[
\int \mathfrak{s} (U_-)\circ dU=\int \mathfrak{s} (U_-)\, dU=W.
\]
\item[(2)]Let $(\Delta U, U)$ be a horizontal semimartingale satisfying \eqref{eq:v} and $W$ the anti-development of $(\Delta U,U)$. If $(\Delta V, V)$ is the development of $W$ with $V_0=U_0$, then $U=V,\ \Delta U=\Delta V$, $P$-a.s.
\item[(3)]Let $(\Delta U,U)$ be a horizontal $\Delta$-semimartingale satisfying \eqref{eq:v}. Put $X:=\pi U$, $\Delta X:=\pi_*\Delta U$. Then $(\Delta X,X)$ is a $\Delta$-semimartingale satisfying \eqref{eq:v}.
\item[(4)]Let $(\Delta X,X)$ be an $M$-valued $\Delta$-semimartingale and $u_0$ an $\mathcal{O}_{X_0}(M)$-valued $\mathcal{F}_0$-measurable random variable. Then there exists a horizontal lift of $(\Delta X,X)$ with $U_0=u_0$ satisfying \eqref{eq:v} and such a horizontal lift is unique. Furthermore, let $(\varepsilon_1,\dots ,\varepsilon_d)$ be a standard basis of $\mathbb{R} ^d$ and $(\varepsilon^1,\dots , \varepsilon ^d)$ its dual basis. Then it holds that
\[
W_t^i=\int_0^tU_{s-}\varepsilon ^i\, dX_s,
\]
where $U_{t-}:(\mathbb{R}^d)^* \to T_{X_{t-}}^*M$ is defined by
\[
\langle U_{t-}a,v\rangle =\langle a,U_{t-}^{-1}v\rangle,\ a\in (\mathbb{R}^d)^*,\ v\in T_{X_{t-}}^*M,
\]
and $W$ is the anti-development of $U$.
\end{itemize}
\end{thm}
This result is a discontinuous version of the result of \cite{Shi} and a generalization of the result of \cite{PE}. In \cite{PE} this kind of a result was shown only in the case where the jumps of semimartingales can be uniquely connected by a minimal geodesic, but our result includes some cases where this assumption is not satisfied. It is mentioned in \cite{Co2} that one can construct a horizontal lift to the orthonormal frame bundle without the geodesic assumption. Our method is based on \cite{Co2}, but our result includes the uniqueness of the horizontal lift in the framework of $\Delta$-semimartingales.\\
The next theorem says that martingales can be characterized by anti-developments of $\Delta$-semimartingales. To state our second result, we recall the definition of a martingale with jumps on a manifold.
\begin{dfn}[\cite{Pic1}, Definition 4.1]
Let $M$ be a $d$-dimensional manifold with a torsion-free connection $\nabla$, and $(\Delta X, X)$ an $M$-valued $\Delta$-semimartingale. We call $(\Delta X,X)$ a $\nabla$-martingale if for all $T^*M$-valued locally bounded c\`{a}dl\`{a}g processes $\alpha=\{ \alpha _s\}_{s\geq 0}$ above $X$, $\displaystyle \int \alpha _{s-} dX_s$ is a local martingale.
\end{dfn}
Note that the definition of martingales with jumps depends on the direction of jumps $\Delta X$. In \cite{Pic1}, a map $\gamma$ from $M\times M$ to $TM$ called a connection rule is introduced. We review connection rules in \sref{stochasticintegral}. Given a connection rule $\gamma$, we can determine the direction of jumps of a semimartingale $X$ by $\Delta X_s=\gamma (X_{s-},X_s)$. If $(\gamma(X_-,X),X)$ is a $\nabla$-martingale, we call $X$ a $\gamma$-martingale. In \cite{Pic1}, it is mentioned that a connection rule induces a torsion-free connection. Our second result is stated as follows.
\begin{thm}\label{main2}
Let $(\Delta X,X)$ be an $M$-valued $\Delta$-semimartingale satisfying \eqref{eq:v}, $(\Delta U,U)$ a horizontal lift, and $W$ the anti-development. Let $\eta$ be a connection rule which induces the Levi-Civita connection. Then the following are equivalent.
\begin{itemize}
\item[(1)] $X$ is an $\eta$-martingale.
\item[(2)]$\displaystyle Z=W+\sum_{0<s\leq \cdot}(U_{s-}^{-1}\eta (X_{s-},X_s)-\Delta W_s)$ is a local martingale.
\end{itemize}
\end{thm}
By \tref{main2}, we can construct martingales on a sphere with respect to a connection rule from local martingales on Euclidean spaces. See \eref{spmar}.
\indent We give an outline of the paper. First we review connection rules introduced in \cite{Pic1} and define the stochastic integral of 1-form and the quadratic variation of 2-tensor along $\Delta$-semimartingales on manifolds in \sref{stochasticintegral}. Next we review stochastic differential geometry with jumps in \sref{sectionCohen}. Then we review orthonormal frame bundles and Riemannian metrics on them in \sref{bundlemetric}. Finally, we prove our main results in \sref{sectionmain}.
\section{Stochastic integrals on manifolds}\label{stochasticintegral}
To start with, we review connection rules introduced in \cite{Pic1}. A connection rule can determine the direction of jumps on a manifold, which is necessary for the definition of the stochastic integral.
\begin{dfn}
A mapping $\gamma:M\times M\to TM$ is a connection rule if it is measurable, $C^2$ on a neighborhood of the diagonal of $M \times M$, and if it satisfies, for all $x,y\in M$,
\begin{enumerate}
\item[(i)] $\gamma (x,y)\in T_xM;$
\item[(ii)] $\gamma (x,x)=0;$
\item[(iii)] $(d \gamma (x,\cdot))_x=id_{T_xM}.$
\end{enumerate}
\end{dfn}
\begin{ex}\label{conne1}
If $M=\mathbb{R}^d$, the map $\gamma$ defined by
\[
\gamma (x,y)=y-x,\ x,y\in M
\]
is a connection rule.
\end{ex}
\begin{ex}\label{conne2}
Let $M$ be a submanifold of $\mathbb{R}^N$. Let $\Pi _x:\mathbb{R}^N\to T_xM$ be an orthonormal projection for each $x\in M$. Then
\[
\gamma (x,y)=\Pi_x (y-x),\ x,y\in M
\]
is a connection rule.
\end{ex}
\begin{ex}\label{conne3}
Let $M$ be a strongly convex Riemannian manifold. Then
\begin{gather}
\gamma (x,y)=\exp_x^{-1}y,\ x,y\in M \label{conne3def}
\end{gather}
is a connection rule.
\end{ex}
Let $(M,g)$ be a Riemannian manifold. We denote by $C_g$ the set of connection rules $\gamma$ which satisfy the following: for all $x,y\in M$, $\exp_x t\gamma (x,y),\ t\in [0,1]$ is a minimal geodesic connecting $x$ and $y$. If $M$ is a strongly convex Riemannian manifold, $\gamma \in C_g$ can be written as \eqref{conne3def} in \eref{conne3}. In general, as we can observe it in the next theorem, if $(M,g)$ is a complete connected Riemannian manifold, we can take a connection rule $\gamma$ such that $\gamma (x,y)$ is an initial velocity of a minimal geodesic connecting $x$ and $y$ for all $x,y\in M$ even though the cut locus is not empty. We use the notion $\| \cdot\|$ as the norm with respect to the Riemannian metric.
\begin{prop}\label{connection}
Let $(M,g)$ be a complete and connected Riemannian manifold. Then $C_g \neq \varnothing$.
\end{prop}
\begin{proof}
Let $\pi :UM \to M$ be a unit tangent bundle. Define $t:UM\to [0,\infty]$ by
\[
t(u):=\sup \{ t\geq 0 \mid d(\pi u, \exp (tu))=t \},
\]
where $d$ is the Riemannian distance. Put
\[
D_x:=\{tu \in TM \mid u\in U_xM,\ t\in [0,t(u)]\},
\]
\[
D:=\bigsqcup _{x\in M}D_x.
\]
Then $D$ is a closed subset of $TM$. Define $F:TM\to M\times M$ by
\[
F(u):=(\pi u, \exp(u)).
\]
Then $F$ is a continuous map because the solution of the geodesic equations depends continuously on the initial value. Put
\[
\Phi(x,y):=\{ u\in T_xM \mid \exp _xu=y,\ \| u\| =d(x,y)\}.
\]
Then for a compact subset $C \subset TM$, we have
\begin{align*}
\Phi^{-1}(C)&:=\{(x,y)\mid \Phi(x,y) \cap C \neq \varnothing \}\\
&=F(C\cap D).
\end{align*}
Since $F$ is continuous, this is a compact subset in $M\times M$. In particular, $\Phi ^{-1}(C)$ is a Borel subset. Therefore by measurable section theorem (\cite{Par}, Theorem 5.2), there exists a map $\gamma:M\times M \to TM$ such that $\gamma (x,y)\in \Phi (x,y)$ and $\gamma^{-1}(C)$ is a Borel set for any compact subset $C\subset TM$. Then $\gamma$ is Borel measurable and this is a connection rule we want.
\end{proof}
As mentioned in \cite{Pic1}, a connection rule induces a torsion-free connection. The connection induced by a connection rule given in \esref{conne1} through \ref{conne3} is the Levi-Civita connection.\\
\indent Next we review the definition of $\Delta$-semimartingale introduced in \cite{Pic1}, which is a pair of a c\`{a}dl\`{a}g semimartingale and directions of jumps.
\begin{dfn}\label{Pic3.1}
Let $Y=(\Delta X, X)$ be an adapted $TM\times M$-valued process. $Y$ is called a $\Delta$-semimartingale if it satisfies the following:
\begin{itemize}
\item[(i)] $X$ is an $M$-valued semimartingale;
\item[(ii)] $\Delta X_s\in T_{X_{s-}}M$ for all $s>0$;
\item[(iii)] $\Delta X_0\in T_{X_0}M,\ \Delta X_0=0$;
\item[(iv)] for all connection rules $\gamma$ and $T^*M$-valued c\`{a}dl\`{a}g processes $\phi$,
\begin{gather}
\sum_{0<s\leq t} \langle \phi _{s-}, \Delta X_s-\gamma(X_{s-},X_s)\rangle<\infty,\ \text{for all }t>0.\label{iv}
\end{gather}
\end{itemize}
\end{dfn}
\begin{rem}
According to \cite{Pic1}, it is sufficient that condition (iv) is satisfied for some connection rule.
\end{rem}
\begin{rem}\label{v}
We will consider the case where $(M,g)$ is a complete, connected Riemannian manifold in later sections. In this case we consider condition \eqref{eq:v} for $\Delta$-semimartingale.
\end{rem}
The following \psref{Pic3.2}, \ref{Pic3.5} and \ref{Pic3.6} shown in \cite{Pic1} determine the It\^o integral of 1-form and the quadratic variation of 2-tensor along $\Delta$-semimartingales.
\begin{prop}[\cite{Pic1}, Proposition 3.2]\label{Pic3.2}
Let $\gamma$ be a connection rule, $X$ an $M$-valued semimartingale, and $\phi$ a $T^*M$-valued process above $X$. Let $\{ U_i\}_{i=1}^{\infty}$ be an atlas such that $\gamma$ is differentiable on each $U_i\times U_i$ and each $U_i$ appears an infinite number of times in the sequence. Put $\theta_0=0$ and
\[
\theta_i=\inf \{ s\geq \theta _{i-1}\mid X_t \notin U_i \}.
\]
Let
\[
\Delta^n :0=T_0^n<T_1^n<\dots <T_{k_n}^n
\]
be a random partition tending to infinity, i.e.
\[
\lim _{n\to \infty} |\Delta ^n | = 0,
\]
where $|\Delta ^n | =\max \{ |T_i^n-T_{i-1}^n| ;i=1,\dots ,k_n\}$ and
\[
\lim_{n\to \infty} T_{k_n}^n=\infty.
\]
Moreover, suppose that $\Delta ^n$ contains $\{ \theta _i \land T_{k_n}^n \}_{i=1}^{\infty}$ for all $n$.
Put
\[
J_t^n:=\sum _{i=1}^{k_n} \langle \phi _{T_{i-1}^n\land t}, \gamma (X_{T_{i-1}^n\land t},X_{T_i^n\land t})\rangle.
\]
Then $J_t^n$ converges in probability as $n\to \infty$ for every $t\geq 0$ and the limit $J_t$ is independent of the partition. Furthermore, the process $\{ J_t \}_{t\geq 0}$ has a modification which is a c\`{a}dl\`{a}g semimartingale.
\end{prop}
We denote $\{ J_t\}_{t\geq 0}$ by $\displaystyle \int \phi_-\, \gamma dX$. This can be considered as the stochastic integral along $(\gamma (X_-,X),X)$. The It\^o integral along the general $\Delta$-semimartingale is defined in the following proposition.
\begin{prop}[\cite{Pic1}, Proposition 3.5]\label{Pic3.5}
Let $\nabla$ be a connection without torsion and $(\Delta X,X)$ a $\Delta$-semimartingale. Then there exists a unique linear map from the space of $T^*M$-valued locally bounded predictable processes above $X_-$ to the space of $\mathbb{R}$-valued semimartingales denoted by
\begin{gather}
\alpha \mapsto \int \alpha \, dX\label{integralmap}
\end{gather}
which satisfies the following conditions.
\begin{enumerate}
\item Let $\phi=\{ \phi _s\}_{s\geq 0}$ be a $T^*M$-valued c\`{a}dl\`{a}g process above $X$. Then
\[
\int _0^t\phi_{s-} \, dX_s=\int_0^t \phi_{s-} \, \gamma dX_s+\sum_{0<s\leq t} \langle \phi_s, \Delta X_s-\gamma(X_{s-},X_s) \rangle,
\]
where $\gamma$ is a connection rule which induces $\nabla$. (The right-hand side does not depend on the choice of $\gamma$.)
\item Let $K_t$ be an $\mathbb{R}$-valued locally bounded predictable process. Then
\[
\int K_s\alpha_s \, dX_s=\int K_s\, d\left(\int_0^s \alpha _u dX_u \right).
\]
\end{enumerate}
Furthermore, $\{ \Delta \int_0^t \alpha_s \, dX_s \}_{t\geq 0}$ is indistinguishable from $\displaystyle \{ \sum_{0<s\leq t}\langle \alpha_s, \Delta X_s \rangle \}_{t\geq 0}$.
\end{prop}
The map \eqref{integralmap} determined in \pref{Pic3.5} is called the It\^o integral.\\
\indent Next, we review the quadratic variation of 2-tensor along a $\Delta$-semimartingale.
\begin{prop}[\cite{Pic1}, Proposition 3.6]\label{Pic3.6}
Let $(\Delta X,X)$ be a $\Delta$-semimartingale. Then there exists a unique linear map from the space of $T^*M\otimes T^*M$-valued locally bounded predictable processes above $X$ to the space of $\mathbb{R}$-valued locally finite variation processes denoted by
\begin{gather}
b \mapsto \int b\, d[X,X] \label{quadmap}
\end{gather}
which satisfies the following conditions.
\begin{itemize}
\item[(i)]
Let $b=\{b_s\}_{s\geq 0}$ be a $T^*M\otimes T^*M$-valued c\`{a}dl\`{a}g adapted process above $X$ and
\[
\Delta^n :0=T_0^n<T_1^n<\dots <T_{k_n}^n,\ n=1,2,\dots
\]
a sequence of random partitions tending to infinity. Then for any connection rule $\gamma$,
\begin{align*}
&\sum_{k=0}^{k_n}b_{T_k^{k_n}}(\gamma(X_{T^n_k},X_{T^n_{k+1}}),\gamma(X_{T^n_k},X_{T^n_{k+1}}))\\
+&\sum_{0<s\leq t}b_{s-}(\Delta X_s-\gamma(X_{s-},X_s),\Delta X_s-\gamma(X_{s-},X_s))
\end{align*}
converges to $\int_0^t b_{s-}\, d[X,X]_s$ in probability as $n\to \infty$.
\item[(ii)] Let $K_s$ be an $\mathbb{R}$-valued locally bounded predictable process. Then
\[
\int_0^t K_sb_s\, d[X,X]_s=\int_0^tK_s\, d\left( \int_0^sb_u\ d[X,X]_u\right).
\]
\end{itemize}
Furthermore, put
\[
\int_0^tb_s\, d[X,X]^d_s:=\sum_{0<s\leq t}b_s(\Delta X_s,\Delta X_s),
\]
\[
\int_0^tb_s\, d[X,X]^c_s:=\int_0^tb_s\, d[X,X]_s-\int_0^tb_s\, d[X,X]^d_s.
\]
Then $\displaystyle \int b\ d[X,X]^c$ is continuous and does not depend on $\Delta X$.
\end{prop}
The map \eqref{quadmap} determined in \pref{Pic3.6} is called the quadratic variation of $(\Delta X,X)$ with respect to 2-tensor.
\begin{rem}\label{quad}
In \cite{Pic1}, it is mentioned that for a $T^*M$-valued c\`{a}dl\`{a}g process $\phi$ above $X$, the quadratic variation of $\displaystyle \int \phi_-\ dX$ is
\[
[\int \phi_- \, dX,\int \phi_-\, dX]=\int \phi_- \otimes \phi_- \, d[X,X].
\]
Let $\psi$ be another $T^*M$-valued c\`{a}dl\`{a}g process above $X$. Then
\[
\int \phi_-\otimes \psi_- \, d[X,X]=\int \psi_- \otimes \phi_-\, d[X,X],
\]
since for all $s\geq 0$ and $v\in T_{X_{s-}}M$,
\[
\phi_{s-}\otimes \psi_{s-}(v,v)=\psi_{s-}\otimes \phi_{s-}(v,v).
\]
Therefore we obtain
\begin{align*}
[\int \phi_- \, dX,\int \psi_-\, dX]=\int \phi_- \otimes \psi_- \, d[X,X].
\end{align*}
\end{rem}
We introduce the Stratonovich integral of 1-form along the $\Delta$-semimartingale in \dref{Stratonovich}. This definition is an extension of that of \cite{PE}.
\begin{dfn}\label{Stratonovich}
Let $\nabla$ be a torsion-free connection and $(\Delta X,X)$ an $M$-valued $\Delta$-semimartingale. For $\alpha \in \Omega^1(M)$, we define
\[
\int_0^t \alpha \circ dX:=\int_0^t \alpha (X_{s-})\, dX_s+\frac{1}{2}\int_0^t (\nabla \alpha)(X_{s-})\, d[X,X]^c_s.
\]
This is called the Stratonovich integral of 1-form along $(\Delta X,X)$. We also denote the integral by $\displaystyle \int_0^t \alpha(X_-)\circ dX.$
\end{dfn}
\begin{prop}\label{falpha}
For $\alpha \in \Omega^1(M)$ and $f\in C^{\infty}(M)$,
\begin{gather}
\int f \alpha \circ dX=\int f(X_-)\circ d\left(\int \alpha \circ dX\right).\label{intfa}
\end{gather}
\end{prop}
In \eqref{intfa}, the right-hand side is the Stratonovich integral of $\mathbb{R}$-valued semimartingale $f(X)$ along $\mathbb{R}$-valued semimartingale $\displaystyle \int \alpha \circ dX$; namely, for $\mathbb{R}$-valued semimartingales $Y$ and $Z$,
\[
\int Y_-\circ dZ=\int Y_-\, dZ+\frac{1}{2}[Y,Z]^c.
\]
\begin{proof}
We begin with the left-hand side of \eqref{intfa}:
\begin{align*}
\int _0^t (f\alpha) (X_s) \circ dX_s=&\int _0^tf\alpha (X_{s-})\ dX_s+\frac{1}{2}\int_0^t (\nabla f\alpha)(X_{s-})\ d[X,X]^c_s\\
=&\int_0^t f(X_{s-})\ d\left( \int_0^{\cdot} \alpha \, dX \right)+\frac{1}{2}\int_0^tf(X_{s-})\ d\left(\int_0^{\cdot} (\nabla \alpha )\ d[X,X]^c\right)\\
&+\frac{1}{2}\int_0^t \alpha \otimes df \ d[X,X]^c_s.
\end{align*}
On the other hand,
\begin{align*}
&\int_0^t f(X_{s-})\circ d\left(\int_0^{\cdot}\alpha \circ dX \right)\\
=&\int_0^t f(X_{s-})\ d\left( \int_0^{\cdot}\alpha \circ dX\right)+\frac{1}{2}\int_0^td[f(X),\int_0^{\cdot}\alpha \ dX]^c_s\\
=&\int_0^t f(X_{s-})\ d\left( \int_0^{\cdot}\alpha \ dX\right)+\frac{1}{2}\int_0^tf(X_{s-})\ d\left(\int_0^{\cdot}\nabla \alpha (X_-)\ d[X,X]^c \right)\\
&+\frac{1}{2}[f(X),\int_0^{\cdot}\alpha \ dX]^c_t.
\end{align*}
Furthermore,
\begin{align*}
[f(X),\int_0^{\cdot}\alpha \ dX]^c_t&=[\int_0^{\cdot}df(X_-)\ dX,\int_0^{\cdot}\alpha \ dX]^c_t\\
&=\int_0^tdf\otimes \alpha (X_-)\ d[X,X]^c.
\end{align*}
Therefore we obtain
\[
\int f \alpha \circ dX=\int f(X_-)\circ d\left(\int \alpha \circ dX\right),
\]
and this is precisely the assertion of the proposition.
\end{proof}
\begin{prop}
Let $(\Delta X, X)$ be a $\Delta$-semimaritngale. Then for $\alpha$, $\beta \in \Omega^1(M)$,
\[
[\int \alpha (X_-)\circ dX, \int \beta (X_-)\circ dX]=\int \alpha \otimes \beta (X_-)\, d[X,X].
\]
\end{prop}
\begin{proof}
By \dref{Stratonovich} and \rref{quad}, we have
\begin{align*}
[\int \alpha (X_-)\circ dX, \int \beta (X_-)\circ dX]=&[\int \alpha(X_-)\, dX+\frac{1}{2}\nabla \alpha (X_-)\, d[X,X]^c,\\ &\int \beta (X_-)\, dX+\frac{1}{2}\int \nabla \beta (X_-)\, d[X,X]^c]\\
=&[\int \alpha (X_-)\, dX,\int \beta(X_-)\, dX]\\
=&\int \alpha \otimes \beta (X_-)\, d[X,X].
\end{align*}
This is our claim.
\end{proof}
Since the stochastic integral along a $\Delta$-semimartingale has a c\`{a}dl\`{a}g modification by \pref{Pic3.2}, we can consider the stochastic integral on a random interval.
\begin{dfn}
Let $S$,$T$ be stopping times with $S<T$ and $(\Delta X,X)$ a $\Delta$-semimartingale. For a $T^*M$-valued c\`{a}dl\`{a}g process $\phi$ above $X$,
\[
\int_{(S,T]}\phi_{s-}\, dX_s:=\int_0^T\phi_{s-}\, dX_s-\int_0^S\phi_{s-}\, dX_s,
\]
\[
\int_{\{ T\}}\phi_{s-}\, dX_s:=\langle \phi _{T-}, \Delta X_T\rangle,
\]
\[
\int_{(S,T)}\phi_{s-}\, dX_s:=\int_{(S,T]}\phi_{s-}\, dX_s-\int_{\{ T\}}\phi_{s-}\, dX_s.
\]
We define the quadratic variation and the Stratonovich integral on $(S,T]$, $\{ T\}$, $(S,T)$ in the same way.
\end{dfn}
\begin{prop}\label{coordinate}
Let $(\Delta X, X)$ be a $\Delta$-semimartingale and $(U;x^1,\dots ,x^d)$ a coordinate neighborhood. Let $\alpha \in \Omega^1(M)$, $b$ be a 2-tensor field with
\[
\alpha=\alpha _idx^i,\ b=b_{ij}dx^i\otimes dx^j\ \text{on}\ U.
\]
Let $S$, $T$ be stopping times such that $S<T$ and $X_s\in U$ for $s\in (S,T)$. Then
\begin{align}
\int_{(S,T)}\alpha (X_{s-}) \circ dX_s=&\int_{(S,T)}\alpha_i(X_{s-}) \circ dX^i_s+\sum_{S<s<T}\langle \alpha(X_{s-}),\Delta X_s-\Delta X^i_s \frac{\partial}{\partial x^i}\rangle, \label{eq:1}\\
\int_{(S,T)}b(X_{s-})\, d[X,X]_s=&\int_{(S,T)}b_{ij}\, d[X^i,X^j]_s\notag \\
+\sum_{S<s<T}\{b(X_{s-})&(\Delta X_s,\Delta X_s)-b(X_{s-})(\Delta X^i_s \frac{\partial}{\partial x^i},\Delta X^i_s \frac{\partial}{\partial x^i})\}, \label{eq:2}
\end{align}
where $X^i=x^i(X)$ on $U$.
\end{prop}
\begin{proof}
We begin with the left-hand side of \eqref{eq:1}:
\begin{align*}
\int_{(S,T)}\alpha (X_-)\circ dX=&\int_{(S,T)}\alpha_i(X_-)\ d\left(\int dx^idX \right)\\
&+\frac{1}{2}\int_{(S,T)}\left( \frac{\partial \alpha_i}{\partial x^j}-\Gamma ^k_{ij}\alpha _k \right)(X_-)d\left( \int dx^i\otimes dx^j \ d[X,X]^c\right).
\end{align*}
On the other hand,
\begin{align*}
&\int_{(S,T)}\alpha_i(X_-)\circ d\left( \int dx^i\circ dX \right)\\
=&\int_{(S,T)}\alpha_i(X_-)\circ d\left(\int dx^i dX \right)+\frac{1}{2}\int _{(S,T)}\alpha_i(X_-)\circ d\left(\int \nabla dx^i d[X,X]^c \right)\\
=&\int_{(S,T)}\alpha_i(X_-)\ d\left(\int dx^i\ dX \right)+\frac{1}{2}\int_{(S,T)}\left(\frac{\partial \alpha_k}{\partial x^j}-\Gamma ^i_{jk}\alpha_i \right)(X_-)\ dx^j\otimes dx^k\ d[X,X]^c.
\end{align*}
Therefore
\[
\int_{(S,T)}\alpha (X_-)\circ dX=\int_{(S,T)}\alpha_i(X_-)\circ d\left( \int dx^i\circ dX\right).
\]
Put $\displaystyle \Delta X_s=a^i_s\left( \frac{\partial}{\partial x^i} \right)_{X_{s-}}$, then
\begin{align*}
\int_{(S,T)}\alpha_i(X_-)& \circ d\left( \int dx^i\circ dX \right)\\
&=\int_{(S,T)}\alpha_i(X_-)\circ d\left(X^i-X^i_0-\sum_{0<s\leq \cdot}(X^i_s-X^i_{s-}-a^i_s) \right)\\
&=\int_{(S,T)}\alpha_i(X_-)\circ dX^i+\sum_{S<s<T}\langle \alpha (X_{s-}),\Delta X_s-\Delta X^i_s\frac{\partial}{\partial x^i}\rangle.
\end{align*}
\eqref{eq:2} follows in the same way.
\end{proof}
\section{Second-order stochastic differential geometry with jumps}\label{sectionCohen}
In \cites{Co1, Co2}, Cohen formulated the stochastic integral of order 2 along a c\`{a}dl\`{a}g semimartingale valued in a manifold and the stochastic defferential equation. In this section we summarize results about them. See \cites{Co1, Co2}, \cites{AVMU}, \cites{Vecc} for details.
Let $M$ be a $d$-dimensional $C^{\infty}(M)$ manifold. A linear map $L:C^2(M)\to \mathbb{R}$ is called a second-order differential operator without constant at $x\in M$ if for a local coordinate $(x^i)$ including $x$, there exist $a^{ij}\in \mathbb{R}$, $b^k\in \mathbb{R}$ ($i,j,k=1,\dots,d$) such that $L$ is denoted by
\[
Lf(x)=\sum_{i,j=1}^da^{ij}\frac{\partial^2f}{\partial x^i \partial x^j}(x)+\sum_{k=1}^db^k\frac{\partial f}{\partial x^k}(x),\ f\in C^2(M).
\]
This definition does not depend on local coordinates.
Denote the vector space of all second-order differential operators at $x\in M$ by $\mathbb{T}_xM$. The space
\[
\mathbb{T}M=\bigsqcup_{x\in M}\mathbb{T}_xM
\]
is called the second-order tangent bundle on $M$. Let $\mathbb{T}^*_xM$ be the dual space of $\mathbb{T}_xM$ for each $x\in M$. The space
\[
\mathbb{T}^*M=\bigsqcup_{x\in M}\mathbb{T}^*_xM
\]
is called the second-order cotangent bundle on $M$.
\begin{ex}
Suppose $f:M\to \mathbb{R}$ is $C^2$ at $x\in M$. Define $d^2f(x):\mathbb{T}^*M\to \mathbb{R}$ by
\[
d^2f(x)(L)=Lf(x).
\]
Then $d^2f(x) \in \mathbb{T}_x^*M$.
\end{ex}
\begin{dfn}
Let $f:M\to \mathbb{R}$ be a Borel measurable function. $f$ is called a form of order 2 specified in $x$ if $f$ is twice differentiable at $x$ and $f(x)=0$. We call $x$ the base point of $f$ and denote it by $\pi (f)$. Define
\begin{gather*}
\overset{\triangle}{\mathbb{T}^*}_xM=\{f:M\to \mathbb{R}\mid f\text{ is a form of order 2 specified in }x \},\\
\overset{\triangle}{\mathbb{T}^*}M=\bigsqcup_{x\in M} \overset{\triangle}{\mathbb{T}^*}_xM,
\end{gather*}
and
\[
\mathcal{G}_x:\overset{\triangle}{\mathbb{T}^*}_xM\to \mathbb{T}^*_xM,\ \mathcal{G}_xf=d^2f(x).
\]
\end{dfn}
In this section, we will use the notion of $\overset{\triangle}{\mathbb{T}^*}M$-valued predictable locally bounded processes. We refer the reader to \cite{Co1} for the definition, but throughout this paper, we only use processes such as \eqref{process1} and \eqref{process2}.
\begin{ex}
For an $M$-valued semimartingale $X$ and $f\in C^2(M)$, put
\begin{gather}
\theta_t(x)=f(x)-f(X_{t-}).\label{process1}
\end{gather}
Then $\theta$ is a $\overset{\triangle}{\mathbb{T}^*}M$-valued locally bounded predictable process.
\end{ex}
\begin{thm}[\cite{Co1}, Theorem 1]\label{Coint}
Let $X$ be an $M$-valued semimartingale and $\theta$ a $\overset{\triangle}{\mathbb{T}^*}M$-valued predictable locally bounded process above $X$. Then there exists a unique linear mapping $\displaystyle \theta \mapsto \int \theta \ \overset{\triangle}{d}X$ such that $\displaystyle \int \theta \ \overset{\triangle}{d}X$ is an $\mathbb{R}$-valued semimartingale which is null at $0$ and satisfies for all $f\in C^2(M)$ and for all $\mathbb{R}$-valued locally bounded predictable processes $K$,
\begin{item}
\item[(i)] $\displaystyle \theta _s(z)=f(z)-f(X_{s-}),\ s\geq 0,\ z\in M\Rightarrow \int_0^t\theta_s \ \overset{\triangle}{d}X_s=f(X_t)-f(X_0),$
\item[(ii)] $\displaystyle \int K_s \theta _s \ \overset{\triangle}{d}X_s=\int K_s\, d\left(\int_0^s \theta _u \ \overset{\triangle}{d}X_u \right)$,
\item[(iii)]$\displaystyle \mathcal{G}_{X_{s-}}\theta _s=0,\ s\geq 0\Rightarrow \int_0^t \theta_s \ \overset{\triangle}{d}X_s=\sum_{0<s\leq t}\theta _s(X_s)$.
\end{item}
\end{thm}
The next theorem means that Cohen's stochastic integral defined in \tref{Coint} can recover Picard's stochastic integral defined in \pref{Pic3.5}.
\begin{thm}[\cite{Co2}, Proposition 7]\label{CoPic}
Let $\gamma$ be a connection rule on $M$, $X$ an $M$-valued semimartingale and $\phi$ a $T^*M$-valued c\`{a}dl\`{a}g process. Put
\begin{gather}
\left( \overset{\triangle}{\gamma}_{X_{s-}}\phi_{s-}\right)(y)=\langle \phi_{s-},\gamma(X_{s-},y)\rangle,\ y\in M.\label{process2}
\end{gather}
Then $\overset{\triangle}{\gamma}_{X_{s-}}\phi_{s-}\in \overset{\triangle}{\mathbb{T}^*}_{X_{s-}}M$ and
\[
\int \overset{\triangle}{\gamma}_{X_{s-}}\phi_{s-}\, \overset{\triangle}{d}X_s=\int \phi_{s-}\, \gamma dX_s,
\]
where the right-hand side is defined in \pref{Pic3.5}.
\end{thm}
Next we review the theory of SDE's on manifolds with jumps discussed in \cites{Co1, Co2}.
\begin{dfn}
Let $M$ and $N$ be manifolds. Suppose that $C$ is a closed submanifold of $M\times N$, such that the projection $p_1$ from $C$ to $M$ is onto and a submersion. A measurable map $\phi:C\times M \to N$ will be called a constraint coefficient from $C\times M$ to $N$ if
\begin{itemize}
\item[(i)]for each $(x,y)\in C$, $\phi (x,y,x)=y$,
\item[(ii)]$\phi$ is $C^3$ in a neighborhood of $\{ (z,\ p_1(z))| z\in C\}$,
\item[(iii)]for all $x\in M$ and $z\in C$, $(x,\ \phi (z, x))\in C$.
\end{itemize}
\end{dfn}
\begin{prop}[\cite{Co1}, Proposition 3]\label{Copro3}
Let $M$ and $N$ be manifolds, $X$ an $M$-valued semimartingale, $Y$ an $N$-valued semimartingale, and $\phi$ a constraint coefficient from $C\times M$ to $N$. For a $\overset{\triangle}{\mathbb{T}^*}N$-valued predictable locally bounded process $\alpha$ above $Y_-$, put
\[
\beta_t(z)=\alpha_t (\phi (X_{t-}, Y_{t-},z)).
\]
Then $\beta$ is a $\overset{\triangle}{\mathbb{T}^*}M$-valued predictable locally bounded process above $X_-$. We denote $\beta$ by $\phi^*\alpha$.
\end{prop}
\begin{dfn}\label{solution}
Let $M$ and $N$ be manifolds, $C$ a closed submanifold of $N$, and $\phi:C\times M \to N$ a constraint coefficient from $C\times M$ to $N$. Fix an $M$-valued semimartingale $X$ and an $N$-valued $\mathcal{F}_0$-measurable random variable $y_0$ with $(X_0,y_0)\in C$. A pair of a positive predictable stopping time $\eta$ and an $N$-valued semimartingale $Y$ on $[0,\eta)$ is called a solution of the SDE
\begin{align}
\left\{ \begin{array}{ll}
\overset{\triangle}{d}Y=\phi(Y,\overset{\triangle}{d}X),\\
Y_0=y_0,
\end{array} \right.\label{SDE}
\end{align}
if $Y_0=y_0$, $(X,Y)\in C$ and for all $\overset{\triangle}{\mathbb{T}^*}N$-valued locally bounded predictable processes $\alpha$ with $\alpha_t\in \overset{\triangle}{\mathbb{T}^*}_{Y_t}N$ on $[0,\eta)$,
\[
\int \alpha \overset{\triangle}{d}Y=\int \phi^*\alpha \overset{\triangle}{d}X.
\]
\end{dfn}
\begin{thm}[Theorem 2 of \cite{Co1} and Theorem 1 of \cite{Co3}]\label{solution2}
Equation \eqref{SDE} admits a unique solution.
\end{thm}
\begin{ex}\label{exmarcus}
Let $A_i$ ($i=1,\dots,r$) be a complete vector field on $M$. Define $\phi:\mathbb{R}^r \times M\times \mathbb{R}^r\to M$ by
\[
\phi(a,x,b):=\text{Exp}\sum_{k=1}^r(b^k-a^k)A_k(x),
\]
where Exp is the exponential map determined by the integral curve. Then $\phi$ is a constraint coefficient from $\mathbb{R}^r\times M \times \mathbb{R}^r$ to $M$. Let $W$ be a $d$-dimensional semimartingale with $W_0=0$. Then the SDE
\begin{gather}
\overset{\triangle}{d}X=\phi(X,\overset{\triangle}{d}W) \label{SDE2}
\end{gather}
admits a unique solution $(X,\eta)$. Furthermore, $X$ satisfies that for all $f\in C^{\infty}(M)$,
\begin{align*}
f(X_t)=&f(X_0)+\int_0^t A_kf(X_{s-})\, dW^k_s+\frac{1}{2}\int_0^tA_iA_jf(X_{s-})\, d[W^i,W^j]^c_s\\
&+\sum_{0<s\leq t}\{ f(X_s)-f(X_{s-})-A_kf(X_{s-})\Delta W^k_s \}.
\end{align*}
\end{ex}
In \sref{stochasticintegral}, we have reviewed the quadratic variation in \pref{Pic3.6} and defined Stratonovich integral in \dref{Stratonovich}. \pref{intvec} says that we can write the integral along the solution of \eqref{SDE2} as the integral along the semimartingale on a Euclidean space.
\begin{prop}\label{intvec}
Put $\Delta X:=A_k(X_-)\Delta W^k$ under the conditions stated in \eref{exmarcus}. Suppose $(\Delta X,X)$ is a $\Delta$-semimartingale. Then for $\alpha \in \Omega^1(M)$ and $\beta \in \Gamma(T^*M\otimes T^*M)$,
\begin{align*}
&\int \alpha (X)\circ dX=\int \langle \alpha,A_k\rangle (X)\circ dW,\\
&\int \beta (X)\, d[X,X]=\int \beta (X_-)(A_k(X_-),A_l(X_-))\, d[W^k,W^l].
\end{align*}
\end{prop}
\begin{proof}
Let $(U;x^1,\dots ,x^d)$ be a coordinate neighborhood and $S$, $T$ stopping times such that $S<T$ and $X_s\in U$, $s\in (S,T)$. Put $x^i(X)=X^i$, $A_kx^i=A^i_k$, $\alpha =\alpha_idx^i$, $\beta=\beta_{ij}dx^i\otimes dx^j$. Then for $t\in (S,T)$, it holds that
\begin{gather}
X^i_t-X^i_S=\int_S^tA^i_k(X)\circ dW^k+\sum_{S<s\leq t}\{ \Delta X^i_s-A_k^i(X_s-)\Delta W^k_s\},\label{vectorfield1}\\
\Delta X=A_K(X_-)\Delta W^k\label{vectorfield2}.
\end{gather}
Therefore by \pref{coordinate},
\begin{align*}
\int_{(S,t]}\alpha (X)\circ dX=&\int_{(S,t]}\alpha_i(X_{s-})A^i_k(X_{s-})\circ dW^k_s\\
&+\sum_{S<s\leq t}\langle \alpha (X_{s-}),\Delta X^i_s\frac{\partial}{\partial x^i}-A_k(X_{s-})\Delta W^k_s\rangle \\
&+\sum_{S<s\leq t}\langle \alpha (X_{s-}),\Delta X_s-\Delta X^i_s\frac{\partial}{\partial x^i}\rangle \\
=&\int_{(S,t]}\alpha_i(X_{s-})A^i_k(X_{s-})\circ dW^k_s.
\end{align*}
Similarly, by \pref{coordinate}, it holds that
\begin{align*}
\int_{(S,T)} \beta(X_-)\ d[X,X]=&\int_{(S,T)}\beta_{ij}\ d[X^i,X^j]-\sum_{S<s<T}\beta_{ij}(X_{s-})(\Delta X_s^i\Delta X_s^j\\
&-\langle dx^i,\Delta X_s\rangle \langle dx^j,\Delta X_s\rangle).
\end{align*}
By \eqref{vectorfield1} and \eqref{vectorfield2}, we obtain
\begin{align*}
\int_{(S,T)}\beta \, d[X,X]
=&\int_{(S,T)}\beta(A_k(X_{s-}),A_l(X_{s-}))\, d[W^k,W^l]_s.
\end{align*}
This is our claim.
\end{proof}
\section{The Riemannian metric and the Levi-Civita connection on $\mathcal{O}(M)$}\label{bundlemetric}
To define the stochastic integrals on $\mathcal{O}(M)$, we introduce the Riemannian metric and the Levi-Civita connection on $\mathcal{O}(M)$. First we recall the notion of orthonormal frame bundles. Fundamental properties of orthonormal frame bundles mentioned in this section are based on \cite{KN} and \cite{IkeWata}. Let $(M,g)$ be a Riemannian manifold. Set
\begin{gather*}
\mathcal{O}_x(M):=\{ u:\mathbb{R}^d\to T_xM\mid u\text{ is a linear isometric}\},\\
\mathcal{O}(M):=\bigsqcup_{x\in M}\mathcal{O}_x(M),\\
\pi: \mathcal{O}(M)\to M,\ \pi(u)=x,\ u\in \mathcal{O}_x(M).
\end{gather*}
Let $O(d)$ be an orthogonal group and $\mathfrak{o}(d)$ its Lie algebra. $O(d)$ acts on $\mathcal{O}(M)$ and the action is defined by
\[
ua:=u\circ a\in \mathcal{O}_x(M),\ x\in M,\ u\in \mathcal{O}_x(M),\ a\in O(d).
\]
We define a map $R_a:\mathcal{O}(M)\to \mathcal{O}(M)$ by
\[
R_au:=ua,\ u\in \mathcal{O}(M)
\]
for $a\in O(d)$. It is well known that $\mathcal{O}(M)$ is a differentiable manifold and $\pi :\mathcal{O}(M)\to M$ is a principal $O(d)$-bundle. For each $u\in \mathcal{O}(M)$, $\ker \pi_{*u}$ is called the vertical subspace of $T_u\mathcal{O}(M)$ and each vector in $\ker \pi_{*u}$ is said to be vertical. For $X\in \mathfrak{o}(d)$, the matrix exponential $\exp tX$ determines a one parameter subgroup of $O(d)$. The vertical vector field $X^{\sharp}$ is defined by
\[
X^{\sharp}(u)=\left( \frac{d}{dt}\right)_{t=0}u\exp tX,\ u\in \mathcal{O}(M).
\]
Fix an orthonormal basis $\{X_{\alpha} \}_{\alpha =1,\dots, \frac{d(d-1)}{2}}$ of $\mathfrak{o}(d)$ with respect to the standard inner product. Then $\{X_{\alpha}^{\sharp}(u) \}_{\alpha =1,\dots, \frac{d(d-1)}{2}}$ is the basis of $\ker \pi_{*u}$ for each $u\in \mathcal{O}(M)$.\\
\indent By pulling back to $\mathcal{O}(M)$, a $(p,q)$-tensor $T$ can be seen as an $(\mathbb{R}^d)^p\otimes ((\mathbb{R}^{d})^*)^q$-valued function. The pullback of $T$ is defined by
\[
\pi^*T(u)=u^{-1}T_{\pi u}.
\]
This is called the scalarization of $T$. The tensor transformation rule can be denoted by
\[
\pi^*T(ua)=a^{-1}\pi^*T(u),\ u\in \mathcal{O}(M),\ a\in O(d).
\]
\begin{ex}
Let $X$ be a $C^{\infty}$-vector field on $M$. Then the scalarization of $X$ is
\[
\pi^*X(u)=u^{-1}X(\pi u).
\]
\end{ex}
\begin{ex}
The scalarization of a 1-form $\alpha \in \Omega^1(M)$ is
\[
\pi^*\alpha(u)=u^{-1}\alpha(\pi u),
\]
and it holds that
\[
\langle \pi^*\alpha (u),w\rangle_{\mathbb{R}^d} =\langle \alpha (\pi u), uw\rangle,\ u\in \mathcal{O}(M),\ w\in \mathbb{R}^d.
\]
\end{ex}
Let $\theta$ be a connection form on $\mathcal{O}(M)$, that is, $\theta$ is an $\mathfrak{o}(d)$-valued 1-form satisfying the following:
\begin{itemize}
\item[(1)] for all $X\in \mathfrak{o}(d)$, $\langle \theta,X^{\sharp}\rangle=X$,
\item[(2)] for all $g\in O(d)$, $R_g^*\theta=Ad(g^{-1})\circ \theta$,
\end{itemize}
where $R_g^*\theta$ is the pull-back of $\theta$ by $R_g$ and $Ad:O(d)\to \text{End}(\mathfrak{o}(d))$ is the adjoint representation. For a connection form $\theta$, put
\[
H_u=\{ A\in T_u\mathcal{O}(M)\mid \langle \theta,A\rangle=0\},\ u\in \mathcal{O}(M).
\]
This is called the horizontal subspace of $T_u\mathcal{O}(M)$. The restriction of $\pi_{*u}$ to $H_u$ denoted by $\pi_{*\mid H_u}:H_u\to T_{\pi u}M$ is a linear isomorphism for each $u\in \mathcal{O}(M)$ and for $X\in T_{\pi u}M$,
\[
\tilde X=(\pi_{*\mid H_u})^{-1}(X)
\]
is called the horizontal lift of $X$. It is well known that the horizontal lift of a $C^{\infty}$-vector field on $M$ is a $C^{\infty}$-vector field on $\mathcal{O}(M)$. A connection form $\theta$ induces a connection on $M$. In fact,
\[
\nabla_X Y(\pi u)=u(\tilde X\pi^*Y(u)),\ u\in \mathcal{O}(M),
\]
determines a connection on $M$. We can observe that a covariant derivative of a vector field on $M$ can be seen as a derivative of a vector-valued function on $\mathcal{O}(M)$ by scalarization.
\begin{dfn}
The solder form $\mathfrak{s}\in \Omega^1 (\mathcal{O}(M);\mathbb{R}^d)$ is defined by
\[
\mathfrak{s}_u(A)=u^{-1}\pi_*A,\ u\in \mathcal{O}(M),\ A\in T_u\mathcal{O}(M).
\]
\end{dfn}
\begin{dfn}
The horizontal vector fields on $\mathcal{O}(M)$ defined by
\[
L_k(u)=(\pi_{*\mid H_u})^{-1}(u\varepsilon_k)\ (k=1,\dots,d)
\]
are called canonical horizontal vector fields.
\end{dfn}
$\{ L_k, X_{\alpha}^{\sharp} \}^{k=1,\dots,d}_{\alpha =1,\dots, \frac{d(d-1)}{2}}$ is a basis of each tangent space on $\mathcal{O}(M)$ and $\{ \mathfrak{s}^k,\theta_{\alpha}\}$ is its dual basis.
\begin{dfn}
Let $\nabla$ be a connection on $M$, and $\theta$ a connection form which corresponds to $\nabla$. Let $\mathfrak{s}$ be a solder form determined by $\nabla$. Fix orthonormal bases of $\mathbb{R}^d$ and $\mathfrak{o}(d)$ and denote them by $\{ \varepsilon_i\}$ and $\{ X_{\alpha} \}$, respectively. Then $\theta$ and $\mathfrak{s}$ can be written as
\begin{gather*}
\theta=\theta^{\alpha}X_{\alpha},\ \mathfrak{s}=\mathfrak{s}^k\varepsilon_k.
\end{gather*}
Define the Riemannian metric $\tilde g$ on $\mathcal{O}(M)$ by
\[
\tilde g=\sum_{\alpha}\theta^{\alpha}\otimes \theta^{\alpha}+\sum_{k}\mathfrak{s}^k\otimes \mathfrak{s}^k.
\]
Denote the Levi-Civita connection corresponding to $\tilde g$ by $\tilde \nabla$.
\end{dfn}
The Riemannian metric $\tilde g$, the Levi-Civita connection $\tilde \nabla$ and geodesics on $\mathcal{O}(M)$ are considered in \cite{PE}. Covariant derivatives with respect to connections on soldered principal fiber bundles are computed in \cite{BB} under a more general situation. Set
\begin{gather*}
\tilde \nabla L_j=\sum_{\beta}\omega_j^{\beta}\ X^{\sharp}_{\beta}+\sum_{i}\omega_j^i\ L_i,\\
\tilde \nabla X^{\sharp}_{\alpha}=\sum_{\beta} \omega_{\alpha}^{\beta}\ X^{\sharp}_{\beta}+\sum_i\omega_{\alpha}^i\ L_i.
\end{gather*}
Since $\tilde \nabla$ is torsion-free, it holds that
\[
\omega_j^{\alpha}=-\omega_{\alpha}^j.
\]
We can write
\begin{gather*}
\omega_{\beta}^{\alpha}=\sum_{\gamma}\omega_{\beta \gamma}^{\alpha}\ \theta^{\gamma}+\sum_k\omega_{\beta k}^{\alpha}\ \mathfrak{s}^k,\\
\omega_j^{\alpha}=\sum_{\gamma}\omega_{j \gamma}^{\alpha}\ \theta^{\gamma}+\sum_k\omega_{j k}^{\alpha}\ \mathfrak{s}^k,\\
\omega_j^i=\sum_{\gamma}\omega_{j \gamma}^i\ \theta^{\gamma}+\sum_k\omega_{jk}^i\ \mathfrak{s}^k.
\end{gather*}
In view of calculations in \cite[p.\ 897]{BB}, it holds that
\begin{gather}
\omega_{\beta}^{\alpha}=\frac{1}{2}\sum_{\gamma}c_{\beta \gamma}^{\alpha}\ \theta^{\gamma},\nonumber \\
\omega_j^{\alpha}=-\frac{1}{2}\sum_k\Omega_{j k}^{\alpha}\ \mathfrak{s}^k,\nonumber \\
\omega_j^i=\sum_{\gamma}\{ (X_{\gamma}^{\sharp})^{ij}-\frac{1}{2}\Omega_{ij}^{\gamma} \} \theta^{\gamma}\label{co},
\end{gather}
where
\[
c_{\alpha \beta}^{\gamma},\ \alpha,\beta,\gamma=1,\dots, \frac{d(d-1)}{2}
\]
are the structure constants with respect to an orthonormal basis $\{X_{\alpha} \}_{\alpha =1,\dots, \frac{d(d-1)}{2}}$ of $\mathfrak{o}(d)$ defined through
\[
[X_{\alpha}^{\sharp},X_{\beta}^{\sharp}]_{\mathfrak{o}(d)}=c_{\alpha \beta}^{\gamma}\ X_{\gamma}^{\sharp},
\]
and
\[
\Omega_{ij}^{\alpha},\ i,j=1,\dots,d,\ \alpha=1,\dots,\frac{d(d-1)}{2}
\]
are the components of the curvature form $\Omega^{\theta}$ defined through
\begin{align*}
\Omega^{\theta}=&d\theta +\frac{1}{2}[\theta,\theta]\\
=&\frac{1}{2}\sum_{i,j,\alpha}\Omega^{\alpha}_{i j}\ X_{\alpha}^{\sharp}\ \mathfrak{s}^i\wedge \mathfrak{s}^j\ (\Omega^{\alpha}_{ij}=-\Omega^{\alpha}_{ji}).
\end{align*}
Since the standard inner product of $\mathfrak{o}(d)$ is $O(d)$-invariant, $\{c_{\alpha \beta}^{\gamma} \}$ is totally anti-symmetric in $\alpha, \beta, \gamma$. The following \psref{covariant} and \ref{integralcurve} can be easily obtained by \eqref{co}.
\begin{prop}\label{covariant}
For any $u\in \mathcal{O}(M)$ and $A\in T_u\mathcal{O}(M)$, it holds that
\begin{gather*}
\tilde \nabla \theta (A,A)=0.
\end{gather*}
Furthermore, if $A$ is horizontal, then
\begin{gather*}
\tilde \nabla \mathfrak{s}(A,A)=0.
\end{gather*}
\end{prop}
\begin{proof}
Any tangent vector $A$ can be denoted by
\[
A=a^iL_i(u)+b^{\alpha}X_{\alpha}^{\sharp},\ a^i,b^{\alpha}\in \mathbb{R},\ i=1,\dots,d,\ \alpha=1,\dots,\frac{d(d-1)}{2}.
\]
Therefore we can write
\begin{align*}
\tilde \nabla \theta^{\alpha} (A,A)=&a^ka^l(\tilde \nabla \theta^{\alpha})(L_k,L_l)+a^kb^{\gamma}\tilde \nabla \theta^{\alpha}(L_k,X_{\gamma}^{\sharp})\\
&+b^{\beta}a^l\tilde \nabla \theta^{\alpha}(X_{\beta}^{\sharp},L_l)+b^{\beta}b^{\gamma}(\tilde \nabla \theta^{\alpha})(X_{\beta}^{\sharp},X_{\gamma}^{\sharp}).
\end{align*}
By using \eqref{co}, it holds that
\begin{align*}
&a^ka^l(\tilde \nabla \theta^{\alpha})(L_k,L_l)=a^ka^l\langle \omega^{\alpha}_l,L_k \rangle
= \frac{a^ka^l}{2} \Omega_{l k}^{\alpha},\\
&\tilde \nabla \theta^{\alpha}(L_k,X_{\gamma}^{\sharp})=-\langle \omega_{\gamma}^{\alpha},L_k\rangle=0,\\
&\tilde \nabla \theta^{\alpha}(X_{\beta}^{\sharp},L_l)=-\langle \omega_l^{\alpha},X_{\beta}^{\sharp}\rangle=0,\\
&b^{\beta}b^{\gamma}(\tilde \nabla \theta^{\alpha})(X_{\beta}^{\sharp},X_{\gamma}^{\sharp})=b^{\beta}b^{\gamma}\langle \omega_{\gamma}^{\alpha},X_{\beta}^{\sharp} \rangle
=-b^{\beta}b^{\gamma}c^{\alpha}_{\beta \gamma}.
\end{align*}
Since $\Omega _{lk}^{\alpha}=-\Omega_{kl}^{\alpha}$, we obtain
\[
a^ka^l(\tilde \nabla \theta^{\alpha})(L_k,L_l)=0.
\]
Similarly, we obtain
\[
b^{\beta}b^{\gamma}(\tilde \nabla \theta^{\alpha})(X_{\beta}^{\sharp},X_{\gamma}^{\sharp})=0.
\]
Therefore we deduce $\tilde \nabla \theta (A,A)=0.$ Next suppose $A$ is horizontal. Then we obtain
\begin{align*}
\tilde \nabla \mathfrak{s}^j(A,A)=&a^ka^l \tilde \nabla \mathfrak{s}^j (L_k,L_l)
=-a^ka^l\langle \omega_l^j\ L_k \rangle
=0.
\end{align*}
This proves the proposition.
\end{proof}
\begin{prop}\label{integralcurve}
\begin{itemize}
\item[(1)]Integral curves of the horizontal vector field $a^kL_k$, $a_k\in \mathbb{R}^k,\ k=1,\dots,d$, are geodesics with respect to $\tilde \nabla$.
\item[(2)]Integral curves of the vertical vector field $b^{\alpha}X_{\alpha}^{\sharp}$, $b^{\alpha}\in \mathbb{R}^d,\ \alpha=1,\dots,\frac{d(d-1)}{2}$, are geodesics with respect to $\tilde \nabla$.
\end{itemize}
\end{prop}
\begin{proof}
Let $u(t)$ be a curve on $\mathcal{O}(M)$ satisfying
\[
\frac{du}{dt}(t)=a^kL_k(u(t)).
\]
Then it holds that
\begin{align*}
\tilde \nabla_{\frac{d}{dt}}\frac{du}{dt}={}& a^ka^l(\tilde \nabla_{L_l}L_k)(u(t))\\
={}&a^ka^l(\omega_{kl}^{\alpha}X_{\alpha}^{\sharp})\\
={}&-\frac{1}{2}(a^ka^l\Omega_{k l}^{\alpha})X_{\alpha}^{\sharp}\\
={}&0.
\end{align*}
Next let us denote the integral curve of $b^{\alpha}X_{\alpha}^{\sharp}$ by $v(t)$. Then
\begin{align*}
\tilde \nabla_{\frac{d}{dt}}\frac{dv}{dt}={}&b^{\beta}b^{\gamma}(\tilde \nabla_{X_{\beta}^{\sharp}}X_{\gamma}^{\sharp})(v(t))\\
={}&\frac{b^{\beta}b^{\gamma}}{2}c_{\beta \gamma}^{\lambda}\ X_{\lambda}^{\sharp}(v(t))\\
={}&0.
\end{align*}
This completes the proof.
\end{proof}
The next proposition proved in \cite{PE} expresses the relation between geodesics on $\mathcal{O}(M)$ and those on $M$.
\begin{prop}[\cite{PE}, Proposition 1.9]\label{geodesic}
\begin{itemize}
\item[(1)]Let $x$ and $y$ be two points in $M$ and $c$ a minimal geodesic from $x$ to $y$. Let us denote the parallel transport along $c$ by $P_c:T_xM\to T_yM$. Let $u\in \mathcal{O}_x(M)$, $v\in \mathcal{O}_y(M)$ with $v=P_c\circ u$. Then minimal geodesics from $u$ to $v$ with respect to $\tilde g$ are horizontal and the horizontal lift of $c$ starting at $u$ is one of minimal geodesics from $u$ to $v$. Furthermore, if minimal geodesics from $x$ to $y$ on $M$ are unique, then minimal geodesics from $u$ to $v$ are also unique.
\item[(2)] Let $\tau$ be a geodesic on $\mathcal{O}(M)$ with respect to $\tilde g$. Suppose that $\tau'(0)$ is horizontal. Then $\tau$ is a horizontal curve and $\pi \circ \tau$ is a geodesic on $M$.
\end{itemize}
\end{prop}
We prepare a lemma and propositions for later use in \sref{sectionmain}.
\begin{lem}\label{innerprod}
Let $u\in \mathcal{O}(M)$, $A,B\in T_u\mathcal{O}(M)$ and $a\in O(d)$. Then
\[
\tilde{g}(R_{a*}A,R_{a*}B)=\tilde{g}(A,B).
\]
\end{lem}
\begin{proof}
By definition, it holds that
\begin{align*}
\tilde{g}(R_{a*}A,R_{a*}B)=&\langle \langle \theta (ua),R_{a*}A\rangle, \langle \theta (ua),R_{a*}B\rangle \rangle_{\mathfrak{o}(d)}\\
&+\langle \langle \mathfrak{s}(ua),R_{a*}A\rangle,\langle \mathfrak{s}(ua),R_{a*}B\rangle \rangle_{\mathbb{R}^d}.
\end{align*}
Since $\langle \theta (ua),R_{a*}A\rangle=Ad(a^{-1})\langle \theta (u),A\rangle$ and the metric on $\mathfrak{o}(d)$ is $Ad$-invariant,
\[
\langle \langle \theta (ua),R_{a*}A\rangle, \langle \theta (ua),R_{a*}B\rangle \rangle_{\mathfrak{o}(d)}=\langle \langle\theta (u),A\rangle,\langle \theta (u),B\rangle \rangle_{\mathfrak{o}(d)}.
\]
Furthermore,
\begin{align*}
\langle \mathfrak{s}(ua),R_{a*}A\rangle&=(ua)^{-1}\pi_*R_{a*}A\\
&=a^{-1}\circ u^{-1}(\pi \circ R_a)_*A\\
&=a^{-1}\circ u^{-1}(\pi _*A)\\
&=a^{-1}\langle \mathfrak{s}(u),A\rangle.
\end{align*}
Since $a$ is isometric,
\[
\langle \langle \mathfrak{s}(ua),R_{a*}A\rangle ,\langle \mathfrak{s}(ua),R_{a*}B\rangle \rangle_{\mathbb{R}^d}=\langle \langle \mathfrak{s}(u),A\rangle,\langle \mathfrak{s}(u),B\rangle \rangle_{\mathbb{R}^d}.
\]
Therefore $\tilde{g}(R_{a*}A,R_{a*}B)=\tilde{g}(A,B).$
\end{proof}
\begin{prop}\label{minimal}
Let $u,v\in \mathcal{O}(M)$. Then for all $a\in O(d)$,
\[
d_{\mathcal{O}(M)}(ua,va)= d_{\mathcal{O}(M)}(u,v),
\]
where $d_{\mathcal{O}(M)}$ is the Riemannian distance with respect to $\tilde g$.
\end{prop}
\begin{proof}
For any $\varepsilon>0$, there exists a curve $\tau_{\varepsilon}:[0,1]\to \mathcal{O}(M)$ with $\tau_{\epsilon}(0)=u$, $\tau_{\epsilon}(1)=v$ satisfying $\displaystyle |\frac{d\tau_{\varepsilon}}{dt}|\leq d_{\mathcal{O}(M)}(u,v)+\varepsilon$. Then by \lref{innerprod},
\begin{align*}
\int_0^1|\frac{d}{dt}R_a\tau_{\varepsilon}|dt&=\int_0^1 |R_{a*}\frac{d\tau_{\varepsilon}}{dt}|dt\\
&=\int_0^1|\frac{d\tau_{\varepsilon}}{dt}|dt\\
&\le d_{\mathcal{O}(M)}(u,v)+\varepsilon.
\end{align*}
Thus
\[
d_{\mathcal{O}(M)}(ua,va)\leq d_{\mathcal{O}(M)}(u,v)+\varepsilon.
\]
Since $\varepsilon$ is arbitrary,
\[
d_{\mathcal{O}(M)}(ua,va)\leq d_{\mathcal{O}(M)}(u,v).
\]
This inequality holds for all $u,v\in \mathcal{O}$ and $a\in O(d)$. Therefore
\[
d_{\mathcal{O}(M)}(ua,va)= d_{\mathcal{O}(M)}(u,v),
\]
and the proposition follows.
\end{proof}
\begin{prop}\label{geo}
Let $u,v\in \mathcal{O}(M)$ and $\tau (t)\ (t\in [0,1])$ a minimal geodesic from $u$ to $v$. Then for all $a\in O(d)$, $R_a\tau$ is a minimal geodesic from $ua$ to $va$.
\end{prop}
\begin{proof}
By \lref{innerprod} and \pref{minimal}, we obtain
\begin{align*}
\int_0^1|\frac{d}{dt}R_a\tau|dt&=\int |R_{a*}\frac{d\tau}{dt}|dt\\
&=\int_0^1|\frac{d\tau}{dt}|dt\\
&=d_{\mathcal{O}(M)}(u,v)\\
&=d_{\mathcal{O}(M)}(ua,va).
\end{align*}
Therefore $R_a\tau$ is a minimal geodesic.
\end{proof}
\section{Proofs of \tsref{main} and \ref{main2}}\label{sectionmain}
Let $(M,g)$ be a complete and connected Riemannian manifold and $\mathcal{O}(M)$ an orthonormal frame bundle on $M$. We use the notation defined in the previous section. In this section, we prove \tsref{main} and \ref{main2}.
\subsection{Proofs of \tref{main} (1)--(3)}
\begin{proof}[Proof of \tref{main} (2)]
By the definition of the stochastic development given just above \tref{main}, it holds that for all $F\in C^{\infty}(\mathcal{O}(M))$,
\begin{align*}
F(V_t)-F(V_0)=&\int_0^t L_kF(V_s)\circ dW^k_s\\
&+\sum_{0<s\leq t} \{ F(V_s)-F(V_{s-})-L_kF(V_{s-})\Delta W^k_s\}.
\end{align*}
On the other hand, by It\^o's formula,
\begin{align}
F(U_t)-F(U_0)=&\int_0^tdF(U_s)\circ dU_s \nonumber \\
&+\sum_{0<s\leq t} \{ F(U_s)-F(U_{s-})-\langle dF(U_{s-}),\Delta U_s\rangle \}.\label{UIto}
\end{align}
Since $(\Delta U,U)$ is horizontal and satisfies $W^k=\langle \mathfrak{s}^k,\Delta U\rangle$, it holds that
\[
\Delta U_s=W^k_s L_k(U_{s-})=\Delta V_s,\ s\geq 0.
\]
Note that $dF=L_kF\mathfrak{s}^k+X_{\alpha}^{\sharp} F\theta ^{\alpha}$. Then by \pref{falpha}, \eqref{UIto} can be written as
\begin{align*}
F(U_t)-F(U_0)=&\int_0^tL_kF(U_s)\circ dW^k_s\\
&+\sum_{0<s\leq t} \{ F(U_s)-F(U_{s-})-L_kF(U_{s-})\Delta W^k_s\}.
\end{align*}
This implies that $U$ is also the stochastic development of $W$. Therefore by uniqueness of the solution of the SDE (see \cite{Co1} or \tref{solution2}), $U=V$, $P$-a.s. Thus we deduce $(\Delta U,U)=(\Delta V,V).$
\end{proof}
\begin{lem}\label{finite}
Let $(\Delta X,X)$ a $TM\times M$-valued process satisfying (i), (ii), (iii) in \dref{Pic3.1} and \eqref{eq:v}. Suppose for all $\omega \in \Omega$ and $t>0$,
\[
\sum_{0<s\leq t}|\Delta X_s(\omega)|^2<\infty.
\]
Then for all $\gamma \in C_g$, $\omega \in \Omega$ and $t>0$, the number of $s\in [0,t]$ with
\[
\Delta X_s(\omega) \neq \gamma (X_{s-}(\omega),X_s(\omega))
\]
is finite. Furthermore, $(\Delta X,X)$ is a $\Delta$-semimartingale.
\end{lem}
\begin{proof}
Fix $t\geq 0$, $\omega \in \Omega$, and a connection rule $\gamma \in C_g$. Let $r:M\to [0,\infty]$ be an injective radius. Then $r$ is positive and continuous on $M$. Since $\overline{X(\omega, [0,t])}$ is compact, $r$ admits the minimum value $r_0$ on the set. Since $\displaystyle \sum_{0<s\leq t}|\Delta X_s(\omega)|^2<\infty$, the number of $s\in [0,t]$ with $|\Delta X_s(\omega)|\geq r_0$ is finite. For $s$ with $|\Delta X_s|<r_0$, we have $\Delta X_s(\omega)=\gamma (X_{s-}(\omega),X_s(\omega))$. Therefore the number of $s\in [0,t]$ with $\Delta X_s(\omega)\neq \gamma(X_{s-}(\omega),X_s(\omega))$ is finite. Furthermore, for all $T^*M$-valued c\`{a}dl\`{a}g processes $\phi$ above $X$, \eqref{iv} is satisfied and hence $(\Delta X,X)$ is a $\Delta$-semimartingale.
\end{proof}
\begin{proof}[Proof of \tref{main} (1)(3)]Since $\text{Exp}_{U_{s-}}\Delta W^k_s L_k=U_s$, $\Delta U$ is the initial velocity of the geodesic from $U_{s-}$ to $U_s$ with regard to $\tilde g$ and $\Delta X_s$ is the initial velocity of the geodesic from $X_{s-}$ to $X_s$. Since $[W,W]_t(\omega)<\infty$, $\displaystyle \sum_{0<s\leq t}|\Delta W_s(\omega)|^2<\infty$ for any fixed $t\geq 0$. Hence $\displaystyle \sum_{0<s\leq t} |\Delta X_s(\omega)|^2<\infty$, $\displaystyle \sum_{0<s\leq t} |\Delta U_s(\omega)|^2<\infty$ because
\[
|\Delta W|=|\Delta U|=|\Delta X|.
\]
Therefore by the previous lemma, $(\Delta X,X)$ and $(\Delta U,U)$ are $\Delta$-semimartingales. This completes the proof of \tref{main} (3) because any $\Delta$-semimartingale on $\mathcal{O}(M)$ satisfying \eqref{eq:v} can be seen as the development of an $\mathbb{R}^d$-valued semimartingale by (2). By \psref{intvec} and \ref{covariant},
\begin{gather*}
\int \theta (U_-)\circ dU=\int \langle \theta, L_k\rangle \circ dW^k=0,\\
\int \tilde \nabla \theta (U_-)\, d[U,U]^c=\int \tilde \nabla \theta (L_k(U_-),L_l(U_-))\, d[W^k,W^l]^c=0.
\end{gather*}
Then we obtain
\[
\int \theta (U_-)\, dU=0.
\]
Similarly, it holds that
\begin{gather*}
\int \mathfrak{s}^k (U_-)\circ dU=\int \langle \mathfrak{s}^k,L_l\rangle \circ dW^l=\int \delta^k_l \circ dW^l=W^k,\\
\int \tilde \nabla \mathfrak{s}(U_-)\, d[U,U]^c=\int \tilde \nabla \mathfrak{s}(L_k(U_-),L_l(U_-))\, d[W^k,W^l]^c=0.
\end{gather*}
Hence we obtain
\[
\int \mathfrak{s}(U_-)\, dU=W.
\]
This completes the proof of (1).
\end{proof}
\subsection{Proof of \tref{main} (4)}
In this subsection, we prove (4). We divide \tref{main} (4) into \tsref{4-1}, \ref{4-2}, and \ref{4-3}. Let $\gamma$ be a connection rule on $M$ such that $\gamma \in C_g$. (We can take such a connection rule as shown in \pref{connection}.)
Put
\begin{gather*}
C=\{ (x,u)\in M\times \mathcal{O}(M)\mid \pi u=x \},\\
\phi:C\times M\to \mathcal{O}(M),\ \phi (x,u,y):=\tilde \eta_{x,y} (1),\ x,y\in M,\ u\in \mathcal{O}_x(M),
\end{gather*}
where $\eta_{x,y} (t)$ is the geodesic with $\eta_{x,y} (0)=x,\ \eta_{x,y}'(0)=\gamma(x,y)$, and $\tilde \eta_{x.y}$ is a horizontal lift of $\eta$ whose initial value is $u$. Since $\gamma$ is measurable on $M\times M$ and differentiable on the diagonal set of $M\times M$, $\phi$ is a constraint coefficient of SDE's with jumps. Suppose that an $M$-valued semimartingale $X$ is defined on $[0,\infty )$, that is, $X$ does not explode in finite time. Define the connection rule $\tilde \gamma$ on $(\mathcal{O}(M),\tilde g)$ as follows:
\begin{align*}
&\tilde \gamma(u,v)\\
&=\left\{ \begin{array}{ll}
\text{The initial velocity of minimal geodesic from}\ u\ \text{to}\ v,\ &(u,v)\in D_{\mathcal{O}(M)},\\
\left( \pi _*|_{H_u}\right)^{-1}\gamma(\pi u, \pi v),\ &(u,v)\in C_{\mathcal{O}(M)},
\end{array}\right.
\end{align*}
where
\begin{align*}
D_{\mathcal{O}(M)}&=\{ (u,v)\in \mathcal{O}(M)\times \mathcal{O}(M)\mid u\ \text{and}\ v\ \text{can be connected}\\
&\hspace{50mm} \text{by a unique geodesic with respect to}\ \tilde g\},\\
C_{\mathcal{O}(M)}&=\mathcal{O}(M)\times \mathcal{O}(M)\backslash D_{\mathcal{O}(M)}.
\end{align*}
For later use, we start with three lemmas.
\begin{lem}\label{tildegamma}
For $u,v\in \mathcal{O}(M)$ and $a\in O(d)$,
\[
R_{a*}\tilde{\gamma}(u,v)=\tilde{\gamma}(ua,va).
\]
\end{lem}
\begin{proof}
It holds that $(u,v)\in D_{\mathcal{O}(M)}\Leftrightarrow (ua,va)\in D_{\mathcal{O}(M)}$ for $u,v\in \mathcal{O}(M)$ and $a\in O(d)$ by \pref{geo}. First suppose $(u,v)\in D_{\mathcal{O}(M)}$. Put $\tau (t)=\exp_ut\tilde{\gamma}(u,v)$. Then $R_{a}\tau (t)$ is a unique minimal geodesic from $ua$ to $va$ by \pref{geo}. Therefore
\[
R_{a*} \tilde \gamma(u,v)=R_{a*}\frac{d\tau}{dt}=\frac{d}{dt}R_a\tau (t)=\tilde \gamma (ua,va).
\]
Next we suppose that $(u,v)\in C_{\mathcal{O}(M)}$, then $\tilde \gamma(u,v)$ is the horizontal lift of $\gamma (\pi u,\pi v)$ at $u$. Thus $R_{a*}\tilde \gamma(u,v)$ is the horizontal lift of $\gamma(\pi u,\pi v)$ at $ua$ and equals $\tilde \gamma (ua,va)$.
\end{proof}
\begin{lem}\label{horizon2}
Let $(\Delta U,U)$ be an $\mathcal{O}(M)$-valued $\Delta$-semimartingale satisfying \eqref{eq:v}. Then $(\Delta U,U)$ is horizontal if and only if it holds that
\begin{gather}
\int \theta(U_-)\, dU=0.\label{itohor}
\end{gather}
\end{lem}
\begin{proof}
By (1) and (3) of \tref{main}, if $(\Delta U,U)$ is horizontal, then \eqref{itohor} holds. Conversely, suppose that \eqref{itohor} holds. For any stopping times $S,T$ with $S\leq T$, it holds that
\begin{gather}
\tilde \nabla \theta (U_S)(\tilde \gamma(U_S,U_T),\tilde \gamma (U_S,U_T))=0 \label{zero1}
\end{gather}
by \pref{covariant}. Similarly, it holds that for $s\geq 0$,
\begin{equation}\label{zero2}
\begin{split}
\tilde \nabla \theta (U_{s-})(\tilde \gamma(U_{s-},U_s),\tilde \gamma(U_{s-},U_s))=0,\\
\tilde \nabla \theta (U_{s-})(\Delta U_s,\Delta U_s)=0.
\end{split}
\end{equation}
\eqref{zero1} and \eqref{zero2} imply
\[
\int \tilde \nabla \theta (U_-)\, d[U,U]=0.
\]
Therefore $(\Delta U,U)$ is horizontal.
\end{proof}
\begin{lem}\label{connectionhorizon}
Let $X$ be an $M$-valued semimartingale. Fix an $\mathcal{F}_0$-measurable $\mathcal{O}(M)$-valued random variable $u_0$ such that $u_0\in \mathcal{O}_{X_0}(M)$. A semimartingale $U$ valued in $\mathcal{O}(M)$ is defined by the solution of the following SDE\\
\begin{equation}\label{connectionSDE}
\left\{ \begin{array}{ll}
\overset{\triangle}{d}U=\phi(U,\overset{\triangle}{d}X),\\
U_0=u_0.
\end{array} \right.
\end{equation}
Then $U$ does not explode in finite time with probability one and
\begin{gather}
\int \theta \circ \tilde \gamma dU=\int \theta \ \tilde \gamma dU=0\label{gamma1}.
\end{gather}
In particular, $(\tilde{\gamma} (U_-,U),U)$ is a horizontal lift of $(\gamma (X_-,X),X)$. Furthermore, it holds that
\begin{gather}
\int \frak{s} \circ \tilde \gamma dU=\int \frak{s} \ \tilde \gamma dU,\label{gamma2}\\
\int \frak{s}^k(U_-) \, \tilde \gamma dU=\int U_-\varepsilon ^k\ \gamma dX,\ k=1,\dots,d.\label{gamma3}
\end{gather}
\end{lem}
\begin{proof}
Since $U$ is the solution of \eqref{connectionSDE}, it holds that
\[
\phi(X_{t-},U_{t_-},X_t)=U_t.
\]
This implies that $U_{t-}$ and $U_t$ can be connected by a horizontal minimal geodesic with respect to the metric $\tilde g$ and one of the minimal geodesics is the horizontal lift of $\exp_{X_{t-}} t\gamma (X_{t-},X_t)$ by \pref{geodesic}. Therefore $\tilde \gamma (U_{t-},U_t)$ is horizontal by the definition of $\tilde \gamma$. Let $\zeta$ be an explosion time of $U$ and assume
\[
P(\zeta <\infty)>0.
\]
Then for $\omega \in \{ \zeta <\infty \}$, $\{ U_t(\omega) \}_{0\leq t<\zeta (\omega)}$ is not relatively compact. On the other hand, since $X$ does not explode in finite time,
\[
A(\omega):=\{ X_t(\omega) \mid 0\leq t \leq \zeta(\omega) \}
\]
is relatively compact in $M$. Now it holds that $\{ U_t(\omega) \}_{0\leq t<\zeta (\omega)}\subset \pi ^{-1}(\overline{A(\omega)})$ and the right-hand side is compact because $O(d)$ is compact. This is a contradiction. Therefore $\zeta =\infty$, $P$-a.s. Next we will show the second claim. Since $U$ is a solution of the SDE \eqref{connectionSDE}, for any $\overset{\triangle}{\mathbb{T}}\mathcal{O}(M)$-valued c\`{a}dl\`{a}g process $\Theta$, we have
\[
\int \Theta \ \overset{\triangle}{d}U=\int \Theta (\phi (X_-,U_-,\cdot)) \ \overset{\triangle}{d}X.
\]
(See \cites{Co1} or \dref{solution}.) Therefore by \tref{CoPic} or \cite[Proposition 7]{Co2}, it holds that
\begin{align*}
\int \theta (U_-)\ \tilde \gamma dU=&\int \theta_{U_-}(\tilde \gamma (U_-,\cdot))\ \overset{\triangle}{d}U\\
=&\int \theta_{U_-}(\tilde \gamma (U_-,\phi(X_-,U_-,\cdot)))\ \overset{\triangle}{d}X\\
=&0.
\end{align*}
Therefore by \lref{horizon2}, \eqref{gamma1} holds and consequently, $(\tilde \gamma (U_-,U),U)$ is a horizontal lift of $(\Delta X,X)$.
Therefore \eqref{gamma2} can be obtained by \tref{main} (1). Finally we will show \eqref{gamma3}.
We begin with the left-hand side of the claimed equation:
\begin{align*}
\int \mathfrak{s}^k(U_-)\ \tilde \gamma dU=&\int \pi^*(U_-\varepsilon^k) \ \tilde \gamma dU\\
=&\int \pi^*(U_-\varepsilon^k)(\tilde \gamma(U_-,\cdot))\ \overset{\triangle}{d}U\\
=&\int \pi^*(U_-\varepsilon^k)(\tilde \gamma (U_-,\phi (X_-,U_-,\cdot)))\ \overset{\triangle}{d}X\\
=&\int U_-\varepsilon^k(\gamma (X_-,\cdot ))\ \overset{\triangle}{d}X\\
=&\int (U_-\varepsilon^k)\ \gamma dX.
\end{align*}
Therefore we obtain \eqref{gamma3} and this completes the proof.
\end{proof}
First we show the existence of the horizontal lift of $(\Delta X,X)$.
\begin{thm}\label{4-1}
Let $(\Delta X,X)$ be an $M$-valued $\Delta$-semimartingale and $u_0$ an $\mathcal{O}_{X_0}(M)$-valued $\mathcal{F}_0$-measurable random variable. Then there exists a horizontal lift of $(\Delta X,X)$ with $U_0=u_0$ satisfying \eqref{eq:v}.
\end{thm}
\begin{proof}
Fix a connection rule $\gamma \in C_g$, where $C_g$ is the set of geodesic connection rules defined in \sref{stochasticintegral}. Let $U^{\gamma}$ be the horizontal lift of $(\gamma (X_-,X),X)$. If an $\mathcal{O}(M)$-valued semimartingale $U$ satisfies $\pi U=X$, there exists an $O(d)$-valued process $a_s$ such that $U=U^{\gamma}a$. We will specify the process $a_s$. For each $s\geq 0$, put
\begin{gather*}
c_s^1(t):=\exp t\gamma (X_{s-},X_s),\ t\in[0,1],\\
c_s^2(t):=\exp t\Delta X_s,\ t\in [0,1],\\
(c_s^1)^{-1}(t):=c_s^1(1-t),\ t\in [0,1],
\end{gather*}
and
\begin{align*}
c_s^2\cdot \left( c_s^1\right)^{-1}=\left\{ \begin{array}{ll}
(c_s^1)^{-1}(2t),\ t\in [0,\frac{1}{2}],\\
c_s^2(2t-1),\ t\in [\frac{1}{2},1].
\end{array}\right.
\end{align*}
Denote by $\widetilde{c_s^2\cdot \left( c_s^1\right)^{-1}}$ the horizontal lift of $c_s^2\cdot \left( c_s^1\right)^{-1}$ starting at $U_s^{\gamma}$. Then there exists a unique element $b_s\in O(d)$ satisfying
\begin{gather}
\widetilde{c_s^2\cdot \left( c_s^1\right)^{-1}}(1)=U_s^{\gamma}b_s.\label{bt}
\end{gather}
Since $b_s$ equals the unit element $e$ in $O(d)$ for $s\geq 0$ with $\gamma (X_{s-},X_s)=\Delta X_s$, $b_s$ equals the unit element except for finite number of $s\in [0,t]$ for fixed $t\geq 0$ by \lref{finite}. Let $0<T_1<T_2<\cdots$ be a sequence of stopping times which exhausts the time $s$ with $\gamma (X_{s-},X_s)\neq \Delta X_s$. We define $O(d)$-valued processes $\delta_s$, $a_s$ as follows:\\
\begin{equation}\label{at}
\begin{split}
a_s=\delta _s=e,\ s\in [0,T_1),\\
\delta_s=\left( a_{T_{i-1}}\right)^{-1}b_{T_i}a_{T_{i-1}},\ s\in [T_i,T_{i+1}),\\
a_s=a_{T_{i-1}}\delta_{T_i},\ s\in [T_i,T_{i+1}).
\end{split}
\end{equation}
Then $a_t$ satisfies
\begin{gather}
a_t=\prod_{0<s\leq t} b_{t-s}\label{atbt}
\end{gather}
for each $t$. We put
\[
U_s:=U^{\gamma}_sa_s,
\]
and
\[
\Delta U_s:=(\text{the horizontal lift of }\Delta X_s\ \text{at }U_{s-}).
\]
Obviously we have $\pi U=X$ and $\pi_*\Delta U=\Delta X$. Let us prove that $(\Delta U,U)$ is a $\Delta$-semimartingale satisfying \eqref{eq:v}. Denote the horizontal lift of $\Delta X_s$ at $U_{s-}^{\gamma}$ as $\Delta U_s'$. Then
\[
\Delta U_s=R_{a_{s-}*}\Delta U_s',
\]
where
\[
R_a:\mathcal{O}(M)\to \mathcal{O}(M),\ R_au=ua,\ a\in O(d).
\]
By the definition of $b_s$, it holds that
\[
\exp_{U_{s-}^{\gamma}}\Delta U'_s=U_{s-}^{\gamma}b_s.
\]
Therefore we obtain
\begin{align*}
\exp_{U_{s-}}\Delta U_s=&\exp_{U_{s-}a_{s-}}\left( R_{a_{s-}*}\Delta U_s'\right) \\
=&\left( \exp_{U_{s-}^{\gamma}}\Delta U'_s\right) a_{s-}\\
=&U_s^{\gamma}b_s a_{s-}\\
=&U_s^{\gamma}a_s\\
=&U_s.
\end{align*}
At the second equality, we use \pref{geodesic} (2). Thus $(\Delta U,U)$ satisfies \eqref{eq:v} and consequently it is a $\Delta$-semimartingale by \lref{finite}. Next we prove that $(\Delta U, U)$ is horizontal. It suffices to show that
\begin{align}
\int \theta(U_-) dU=0,\label{ti1}
\end{align}
by \lref{horizon2}. For each $i$, it is obvious that
\begin{gather}
\langle \theta (U_{T_i-}),\Delta U_{T_i}\rangle=0\label{ti2}
\end{gather}
by the definition of $\Delta U$.
For $r,s\in (T_i,T_{i+1})$ with $r<s$, by \lref{tildegamma},
\begin{align*}
\langle \theta (U_{r}),\tilde \gamma (U_r,U_s)\rangle=&\langle \theta (U_r ^{\gamma}a_{T_i}),\tilde \gamma (U_r^{\gamma}a_{T_i}, U_s^{\gamma}a_{T_i})\rangle \\
=& \langle \theta (R_{a_{T_i}}U_r^{\gamma}),R_{a_{T_i}*}\tilde \gamma (U_r^{\gamma}, U_s^{\gamma})\rangle \\
=&Ad(a_{T_i}) \langle \theta (U_r^{\gamma}),\tilde \gamma (U_r^{\gamma}, U_s^{\gamma})\rangle.
\end{align*}
Therefore it holds that
\begin{align}
\int _{(T_i,T_{i+1})} \theta (U_{s-})dU_s=Ad(a_{T_i})\int _{(T_i,T_{i+1})} \theta (U_s)\tilde \gamma dU_s=0 \label{ti3}
\end{align}
for each $i$. Combining \eqref{ti2} and \eqref{ti3}, we obtain \eqref{ti1}. Therefore we can deduce that $(\Delta U,U)$ is a horizontal lift of $(\Delta X,X)$.
\end{proof}
Next we show the uniqueness of the horizontal lift. Let $(\Delta U,U)$ be a horizontal lift of $(\Delta X,X)$ satisfying $U_0=u_0$ and $\exp_{U_{s-}}\Delta U_s=U_s$. Then there exists an $O(d)$-valued adapted process $a_t$ satisfying $U=U^{\gamma}a$ and such $a_t$ is unique since the action of $O(d)$ to each fiber of $\mathcal{O}(M)$ is free. Note that it holds that
\[
\Delta U_s=(\text{the horizontal lift of }\Delta X_s\ \text{at }U_{s-})
\]
by the definition of horizontal lifts. We will show that the process $a_t$ equals the process which is constructed in the proof of the existence of the horizontal lift. We denote $U^{\gamma}$ by $V$ to simplify the notation.
\begin{lem}\label{lemat}
It holds that
\[
a_{s-}(\omega)\neq a_s(\omega)\Rightarrow \gamma (X_{s-}(\omega),X_s(\omega))\neq \Delta X_s (\omega)
\]
for $s\geq 0$, $\omega \in \Omega$.
\end{lem}
\begin{proof}
If $s$ and $\omega$ satisfy
\[
\gamma (X_{s-}(\omega),X_s(\omega))=\Delta X_s (\omega),
\]
then each of $\Delta U_s(\omega)$ and $\Delta V_s(\omega)$ is the horizontal lift of $\gamma(X_{s-}(\omega),X_s(\omega))$ at $U_{s-}$ and $V_{s-}$, respectively. Therefore we have
\[
\exp t\Delta U_s(\omega)=\left( \exp t\Delta V_s(\omega)\right)a_{s-}(\omega),\ t\in [0,1].
\]
In particular, $U_s(\omega)=V_s(\omega)a_{s-}(\omega)$. On the other hand, the process $a_s$ satisfies $U_s(\omega)=V_s(\omega)a_s(\omega)$. Thus we can deduce $a_s(\omega)=a_{s-}(\omega)$.
\end{proof}
Applyng \lref{finite}, for any fixed $t\geq 0$ and $\omega \in \Omega$, the number of $s\in [0,t]$ with $a_s(\omega)\neq a_{s-}(\omega)$ is finite. Let $T_1<T_2<\cdots$ be a sequence of stopping times which exhausts the jumps of $a_t$. We can also observe that $\Delta U_s=\tilde \gamma (U_{s-},U_s)$, $s\in (T_i,T_{i+1})$. Next we will show that $a_s$ is constant on each $[T_i,T_{i+1})$. This can be shown in the same way as Theorem 3.2 in \cite{PE} by Pontier and Estrade.
\begin{lem}\label{lemj}
Suppose $j$ is the canonical 1-form, which is a 1-form on $O(d)$ valued in $\mathfrak{o}(d)$ defined by
\[
j(fA)=A,\ f\in O(d),\ A\in \mathfrak{o}(d).
\]
Then it holds that
\begin{gather}
\int_{(T_i,T_{i+1})}j\circ da=0\label{canonicalform}
\end{gather}
and consequently, $a_t$ is constant on $(T_i,T_{i+1})$ for each $i$.
\end{lem}
\begin{proof}
Put
\begin{gather*}
\phi:\mathcal{O}(M)\times O(d)\to \mathcal{O}(M),\ \phi(u,g)=u\cdot g,\\
\phi_u:O(d)\to \mathcal{O}(M),\ \phi_u(g)=u\cdot g.
\end{gather*}
Then
\[
U_t=\phi (V_t,a_t).
\]
Let $(u^{\alpha})$ be a local coordinate of $\mathcal{O}(M)$. Suppose that $V$ lives in the coordinate neighborhood on $[\sigma, \tau)\subset [T_i,T_{i+1})$, where $\sigma$ and $\tau$ are stopping times. We denote $V^{\alpha}=u^{\alpha}(V)$, and $U^{\alpha}=u^{\alpha}(U)$ on $[\sigma,\tau)$. Then by It\^o's formula,
\begin{align*}
U_t^{\alpha}-U_{\sigma}^{\alpha}=&\int_{(\sigma,t]}\{ \frac{\partial \phi^{\alpha}}{\partial u^{\beta}}(V_{s-},a_s)\circ dV^{\beta}_s+\frac{\partial \phi^{\alpha}}{\partial a^k}(V_{s-},a_s)\circ da^k_s\}\\
+&\sum_{\sigma<s\leq t}\{ \phi^{\alpha}(V_s,a_s)-\phi^{\alpha}(V_{s-},a_s)-\frac{\partial \phi^{\alpha}}{\partial u^{\beta}}(V_{s-},a_s)\Delta V_s^{\beta}\}
\end{align*}
for $t\in (\sigma, \tau)$. Therefore by \pref{coordinate},
\begin{align}
\int_{(\sigma,t]}\theta \circ dU=&\int_{(\sigma,t]}\theta_{\alpha}\circ dU^{\alpha}+\sum_{\sigma<s\leq t}\langle \theta, \Delta U-\Delta U^{\alpha}\frac{\partial}{\partial u^{\alpha}}\rangle \nonumber \\
=&\int_{(\sigma,t]}\theta_{\alpha}\{ \frac{\partial \phi^{\alpha}}{\partial u^{\beta}}(V_{s-},a_s)\circ dV^{\beta}_s+\frac{\partial \phi^{\alpha}}{\partial a^k}(V_{s-},a_s)\circ da^k_s \} \nonumber \\
+&\sum_{\sigma<s\leq t} \theta_{\alpha}\{ \phi^{\alpha}(V_s,a_s)-\phi^{\alpha}(V_{s-},a_s)-\frac{\partial \phi^{\alpha}}{\partial u^{\beta}}(V_{s-},a_s)\Delta V_s^{\beta} \} \nonumber \\
+&\sum_{\sigma<s\leq t}(\langle \theta,\Delta U\rangle -\theta_{\alpha}\Delta U^{\alpha}),\label{hor1}
\end{align}
where $\displaystyle \theta=\theta_{\alpha}du^{\alpha}$ and $\theta_{\alpha}\in C^{\infty}(\mathcal{O}(M);\mathfrak{o}(d))$, which is an $\mathfrak{o}(d)$-valued $C^{\infty}$-function. Note that it holds that
\begin{gather*}
\theta_{\alpha}\frac{\partial \phi^{\alpha}}{\partial u^{\beta}}(u, a)=\left(Ad(a^{-1})\theta \right)_{\beta}(u),\\
\theta_{\alpha}\frac{\partial \phi^{\alpha}}{\partial a^k}(u,a)=\langle d\left(\phi_u^*\theta \right),\frac{\partial}{\partial a^k}\rangle=\langle j,\frac{\partial}{\partial a^k}\rangle.
\end{gather*}
Therefore \eqref{hor1} can be rewritten as
\begin{align}
\int_{(\sigma,t]}\theta \circ dU=&\int_{(\sigma,t]}\{ \left( Ad(a^{-1})\theta \right)_{\beta}(V)\circ dV^{\beta}+\langle j,\frac{\partial}{\partial a^k}\rangle (a)\circ da^k\} \nonumber \\
&+\sum_{\sigma<s\leq t}\left( \langle \theta(U_{s-}), \Delta U_s\rangle -\langle Ad(a_s^{-1})\theta,\frac{\partial}{\partial u^{\beta}}\rangle \Delta V_s^{\beta} \right).\label{hor2}
\end{align}
Moreover, it holds that
\begin{align}
\int_{(\sigma, t]}Ad(a^{-1})(\theta)\circ dV=&\int_{(\sigma,t]}\left( Ad(a^{-1})\theta \right)_{\beta}\circ dV^{\beta} \nonumber \\
&+\sum_{\sigma<s\leq t}\{ \langle Ad(a_s^{-1})\theta,\Delta V_s\rangle-\langle Ad(a_s^{-1})\theta,\frac{\partial}{\partial u^{\beta}}\rangle \Delta V_s^{\beta} \} \label{hor3}
\end{align}
by \pref{coordinate} and
\begin{align}
\langle \theta (U_{s-}),\Delta U_s\rangle=&\langle \theta (V_{s-}a_s),R_{a_s*}\Delta V_s\rangle \nonumber \\
=&\langle R_{a_s}^*\theta (V_{s-}),\Delta V_s\rangle \nonumber \\
=& \langle Ad(a_s^{-1}) \theta (V_{s-}),\Delta V_s\rangle. \label{hor4}
\end{align}
Substituting \eqref{hor3} and \eqref{hor4} into \eqref{hor2}, we can deduce that
\begin{align*}
0=&\int_{(\sigma,t]}\theta \circ dU\\
=&\int_{(\sigma,t]}Ad(a^{-1})\theta \circ dV+\int_{(\sigma,t]}j\circ da\\
=&\int_{(\sigma,t]}j\circ da.
\end{align*}
Thus \eqref{canonicalform} follows and this implies that $a_s$ is constant on $(T_i,T_{i+1})$ for each $i$.
\end{proof}
By \lref{lemj}, we obtain the following uniqueness result.
\begin{thm}\label{4-2}
$(\Delta U,U)$ is uniquely determined.
\end{thm}
\begin{proof}
It suffice to show that $a_t$ equals the process defined in \eqref{at}. Let $b_t$ be an $O(d)$-valued process defined through \eqref{bt}. Denote the horizontal lift of $\Delta X_s$ at $V_s$ by $\Delta U'_s$. Then for $s\geq 0$,
\begin{align*}
V_s a_s=&U_s\\
=&\exp_{V_{s-}a_{s-}} \left(R_{a_{s-}*}\Delta U_s'\right) \\
=&\left( \exp_{V_{s-}}\Delta U'_s\right) a_{s-}\\
=&V_sb_sa_{s-}.
\end{align*}
Since $a_t$ is constant on $(T_i,T_{i+1})$ for each $i$, it holds that
\[
a_t=\prod_{0\leq s<t}b_{t-s},\ t\geq 0.
\]
This completes the proof by \eqref{atbt}.
\end{proof}
We end the proof of \tref{main} (4) with the following theorem.
\begin{thm}\label{4-3}
It holds that
\[
W_t^i=\int_0^tU_{s-}\varepsilon ^i\, dX_s,
\]
where $W$ is the anti-development of $(\Delta U,U)$.
\end{thm}
\begin{proof}
Since
\[
\int_{(0,t]}U_{s-}\varepsilon^i\ dX=\sum_{m=1}^{\infty}\int_{(T_m\wedge t,T_{m+1}\wedge t]}U_{s-}\varepsilon^i\ dX,
\]
it suffices to show
\begin{gather}
\int_{(T_m\wedge t,T_{m+1}\wedge t)}U_{s-}\varepsilon^i\ dX=\int _{(T_m\wedge t,T_{m+1}\wedge t)}\mathfrak{s}^i(U_{s-})\ dU_s, \label{anti1}\\
\langle U_{T_m\wedge t-}\varepsilon^i, \Delta X_{T_m\wedge t}\rangle= \langle \mathfrak{s}^i(U_{T_m\wedge t-}),\Delta U_{T_m\wedge t}\rangle \label{anti2}
\end{gather}
for each $m$. \eqref{anti2} can be easily obtained by the definition of the solder form. Thus we will show \eqref{anti1}. Put $c_j^i(s)=a_i^j(s)$.
Since $a_t$ is constant and $\Delta X=\gamma (X_-,X)$ on $(T_m,T_{m+1})$ for each $m$, we have
\begin{align}
\int_{(T_m\wedge t,T_{m+1}\wedge t)}U_{s-}\varepsilon^i\ dX_s=&\int_{(T_m\wedge t,T_{m+1}\wedge t)}U_{s-}\varepsilon^i \ \gamma dX_s \nonumber\\
=&\int_{(T_m\wedge t,T_{m+1}\wedge t)}\left(U_{s-}^{\gamma}a\right) \varepsilon^i \ \gamma dX_s\nonumber\\
=&c^i_j\int_{(T_m\wedge t,T_{m+1}\wedge t)}U_{s-}^{\gamma} \varepsilon^j\ \gamma dX_s\nonumber\\
=&c^i_j\int_{(T_m\wedge t,T_{m+1}\wedge t)}\mathfrak{s}^j(U_{s-}^{\gamma})\ \tilde{\gamma}dU^{\gamma}_s,\label{anti3}
\end{align}
by \eqref{gamma3}, where we write $c_j^i=c_j^i(T_m\wedge t)$, $a=a(T_m\wedge t)$ to simplify the notation.
Furthermore, for stopping times $S,T$ with $T_i<S\leq T<T_{i+1}$, it holds that
\begin{align*}
c^i_j\langle \mathfrak{s}^j(U_S^{\gamma}), \tilde \gamma (U_S^{\gamma},U_T^{\gamma})\rangle=& c_j^i\langle \mathfrak{s}^j(U_{S}^{\gamma}),R_{a^{-1}*}R_{a*}\tilde \gamma (U_{S}^{\gamma},U_{T}^{\gamma})\rangle \\
=&\langle \mathfrak{s}^i (U_{S}),\tilde \gamma(U_{S},U_{T})\rangle
\end{align*}
by \lref{tildegamma}. Therefore it holds that
\begin{align}
c^i_j \int_{(T_m\wedge t,T_{m+1}\wedge t)}\mathfrak{s}^j(U_{s-}^{\gamma})\ \tilde \gamma dU_s^{\gamma}= \int _{(T_m\wedge t,T_{m+1}\wedge t)}\mathfrak{s}^i(U_{s-})\ dU_s.\label{anti4}
\end{align}
Combining \eqref{anti3} and \eqref{anti4}, we obtain \eqref{anti1} and the claim follows.
\end{proof}
Combining \tsref{4-1}, \ref{4-2} and \ref{4-3}, we complete the proof of \tref{main} (4).
\subsection{Proof of \tref{main2}}
We end this section with the proof of \tref{main2} and its example.
\begin{proof}[Proof of \tref{main2}]
(1) $\Rightarrow$(2): It holds that for each $i=1,\dots,d$,
\begin{align}
Z_t^i&=W_t^i+\sum_{0<s\leq t}\langle \varepsilon^i,U_{s-}^{-1}\eta (X_{s-},X_s)-\Delta W_s\rangle \notag \\
&=\int_0^t U_{s-}\varepsilon^i \, dX_s+\sum_{0<s\leq t}\langle U_{s-}\varepsilon^i, \eta (X_{s-},X_s)-\Delta W_s\rangle \notag \\
&=\int _0^t U_{s-}\varepsilon^i\, \eta dX_s \label{eq:a}.
\end{align}
Therefore $Z$ is a local martingale.\\
(2) $\Rightarrow$(1): Let $\phi$ be a $T^*M$-valued c\`adl\`ag process above $X$. Then it holds that
\begin{align*}
\int_0^t \phi_{s-}\ \eta dX_s &=\int_0^t\langle U_{s-}^{-1}\phi_s,\varepsilon_i\rangle \, d\left(\int_0^s U_{r-}\varepsilon^i\ \eta dX_r \right).
\end{align*}
Therefore by \eqref{eq:a}, $X$ is an $\eta$-martingale.
\end{proof}
\begin{ex}\label{spmar}
We consider the case
\[
M=S^d:=\{ (x^1,\dots,x^{d+1})\in \mathbb{R}^{d+1} \mid (x^1)^2+\cdots +(x^{d+1})^2=1\}.
\]
Let $\eta$ be a connection rule on $S^d$ given by \eref{conne2}. Then it holds that
\[
\eta(X_{s-},X_s)=\frac{\sin |\Delta W_s|}{|\Delta W_s|}\Delta X_s.
\]
Define $f:\mathbb{R}\to \mathbb{R}$ by
\begin{align*}
f(\theta)=\left\{
\begin{array}{ll} \frac{\sin \theta-\theta}{\theta},\ &\theta \neq 0,\\
0,\ &\theta = 0.
\end{array}
\right.
\end{align*}
Then $X$ is an $\eta$-martingale if and only if
\[
Z=W+\sum_{0<s\leq \cdot}f(|\Delta W_s|)\Delta W_s
\]
is a local martingale. In particular, for an $\mathbb{R}$-valued local martingale $Z$ satisfying $|\Delta Z_s|\leq 1$, $s\geq 0$, put
\[
W=Z+\sum_{0<s\leq \cdot}g(|\Delta Z_s|)\Delta Z_s,
\]
where $g$ is a function on $[0,1]$ defined by
\begin{align*}
g(\theta)=\left\{
\begin{array}{ll} \frac{\arcsin \theta-\theta}{\theta},\ &\theta \neq 0,\\
0,\ &\theta = 0.
\end{array}
\right.
\end{align*}
Then the development of $W$ is an $S^d$-valued $\eta$-martingale.
\end{ex}
\end{document} | 140,706 |
"Batman Begins" and "King Kong" topped the winners at the 32nd annual Saturn Awards, taking home 3 statuettes each in ceremonies in Universal City,
Talking about "Batman Begins" and "King Kong", both besides scoring great in box office, also are making sensation at the 32nd annual Saturn Awards held May 2nd, 2006 in Universal City, Calif.
Bringing home three awards each, as for "Batman Begins" the flick nabs honor for the categories of Best Fantasy Film, Best Actor for Christian Bale, and Best Writing. Meanwhile, "King Kong" wins the categories of Best Actress for Naomi Watts, Best Director for Peter Jackson, and Best Special Effects.
Also scores winning at the event are "Star Wars: Episode III - Revenge of the Sith", "Sin City", "The Exorcism of Emily Rose", and "Corpse Bride" among others. Take a look at the complete list of the winners at. | 82,099 |
PLAQUEMINE, LA. -- Toxic floodwaters are draining into Lake Pontchartrain, enabling rescue workers to recover New Orleans' dead. Millions of Americans watch with horror, and wonder how could this have happened in the richest country on earth.
At the same time, an unneeded $231 million bridge to a sparsely inhabited part of Alaska is to be built.
Yes, there is a connection.
The bridge stands as a monument to a corrupt Washington culture - a culture that has mismanaged and plundered our nation's treasury. The exact culture that allowed Hurricane Katrina to wreak more destruction than it ever should have. A corrupt culture that has cost this nation hundreds of billions of dollars in rebuilding costs. It has cost us a city, a flourishing cultural center, and it has cost us lives.
The bridge is aptly named "Don Young's Way," after Alaska's Republican Rep. Don Young, the House Transportation Committee chairman who gleefully described his having stuffed the transportation bill "like a turkey." | 60,018 |
I’ll be pre-ordering the iPhone 4 June fifteenth, expect one other article in 30 days. In it, A�my resolution will be made on whether or to not preserve this cellphone after making an attempt out the brand new iPhone 4 for about 15 days, midway via my 30 days of having the EVO 4G.
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Third-party libraries implementation.
Smartphones are mainly targeted as a result of customers don’t update their working systems when new versions are launched. That is one challenge users have confronted. In addition, some carriers and manufacturers of android devices are releasing older OS platforms available in the market, giving hackers a chance to invade the units. Customers ought to make sure that they get the most recent secure versions of working methods instantly they’re released available in the market.
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I started a second app. Funds is a problem! Operating System.
The brand new Desire HD is no different from the old Desire the place the interface is worried; the same seven page home display system lets you view all screens at once by pinching the screen. You also get FriendStream which enables customers to view all of there communications in one utility; dwell feeds from Facebook, Twitter and different networks can all be filtered into the one stream of information.
The digicam itself operates at eight megapixels, (3264x 2448 decision), that means that the ensuing still images are of very top quality. To make it simpler to take top quality image, numerous picture enhancing features are included. Upon getting opened the digicam application, every little thing is properly laid out and really straightforward to use. There’s a settings icon which presents choices similar to flash, capturing mode, scene mode, publicity worth and focus modes. Customers can easily change these to swimsuit their very own preferences. There may be also an icon to instantly flip the LED flash on or off, so you may get the best outcomes for the actual lighting circumstances of the shot. If you’ll primarily be taking pictures of individuals, one can find it very useful to have the face detection mode enabled. This mechanically detects faces and focuses on them robotically, which is one much less factor to worry about when trying to take the perfect picture.
Conclusion
The tremendous smart interface integrates your pal’s social community exercise with there relevant contacts within the phonebook, this implies you can take a look at the most recent doings and happenings direct from your contacts listing. From the get go you might be introduced with up to twenty dwelling screens which may each house there personal widget. | 135,932 |
TITLE: Characteristic polynomial modulo 12
QUESTION [4 upvotes]: Consider the vector space $V =\left\{a_0+a_1x+\cdots+a_{11}x^{11},\;a_i\in\mathbb{R}\right\}$. Define a linear operator $A$ on $V$ by $A(x^i) = x^{i+4}$ where $i + 4$ is taken modulo $12$.
Find $(a)$ the minimal polynomial of $A$ and $(b)$ the characteristic polynomial of $A$.
My try:
I coulnot find another way so I tried the brute force method. $\:$I found the matrix representation of the operator is $$A=\begin{bmatrix} 0&0&0&0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&0&0&0&1&0&0\\0&0&0&0&0&0&0&0&0&0&1&0\\0&0&0&0&0&0&0&0&0&0&0&1\\1&0&0&0&0&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0&0&0&0&0\\0&0&1&0&0&0&0&0&0&0&0&0\\0&0&0&1&0&0&0&0&0&0&0&0\\0&0&0&0&1&0&0&0&0&0&0&0\\0&0&0&0&0&1&0&0&0&0&0&0\\0&0&0&0&0&0&1&0&0&0&0&0\\0&0&0&0&0&0&0&1&0&0&0&0\\\end{bmatrix}.$$
The characteristic polynomial I found to be $\lambda^{12}-4\lambda^9+6\lambda^6-4\lambda^3+1$.$\:$(It took me almost 40 minutes. $\:$Is there another way to do this problem? Provide hints or suggestions please.
$\rule{17cm}{1pt}$
Taking forward the answer provided by $\textbf{Servaes}$ "The minimal polynomial of $A|_{U_i}$ is still $X^3−1$." Taking $U_1=span\{x_1,x_5,x_9\}$ we see $A(x)=x^5,\:A^2(x)=x^9,\:A^3(x)=x\implies (A^3-I)=0$ and since it factors into linear irredeucible factors, we have the minimal polynomial of $A|_{U_i}$ is $X^3−1$.
Completing the proof: We show that characteristic polynomial of $A$ is the product of characteristic polynomials of $A|_{U_i}$ where $V=\oplus U_i$. We have seen that minimal polynomial of $A|_{U_i}$ is $X^3−1$ which is precisely the characteristic polynomial. So let $p_i(\lambda)$ is characteristic polynomial corresponding to eigenvalue $\lambda_i$ and invariant subspace $U_i$ and $p(\lambda)$ is the characteristic polynomial of A. Then $\displaystyle p(\lambda_i)=0 \:\forall\;i \implies p_i(\lambda)|p(\lambda) \:\forall\;i \implies p(\lambda)=\prod_ip_i(\lambda)$.
$\Big($$p(\lambda)$ is atleast $\displaystyle\prod_ip_i(\lambda)$. If $\exists\lambda\neq\lambda_i \forall\: i$ such that $p(\lambda)=0$ then $V$ is not $\oplus U_i$ $\Big)$
$\rule{17cm}{1pt}$
Minimal polynomial : $X^3-1\qquad$ Characteristic polynomial : $(X^3-1)^4$.
REPLY [5 votes]: It usually helps to find some nontrivial relation that the given operator satisfies. Clearly
$$A^3(x^i)=x^{i+12}=x^i,$$
for all $i$, so $A$ is a zero of $X^3-I$. This factors as
$$X^3-I=(X-I)(X^2+X+I),$$
which has no repeated factors, so this is the minimal polynomial of $A$. The characteristic polynomial has the same irreducible factors and has degree $12$, so the minimal polynomial equals
$$(X-I)^a(X^2+X+I)^b,$$
for some positive integers $a$ and $b$ satisfying $a+2b=12$.
Note that $a$ is the algebraic multiplicity of the eigenvalue $1$, which is at least $4$ because
$$1+x^4+x^8,\qquad x+x^5+x^9,\qquad x^2+x^6+x^{10},\qquad x^3+x^7+x^{11},$$
are four linearly independent eigenvectors with eigenvalue $1$. So $(a,b)$ is either $(4,4)$, $(6,3)$, $(8,2)$ or $(10,1)$.
The following is a bit contrived and implicitly assumes some slightly advanced ideas, but it is an (almost) computation-free way of determining the characteristic polynomial:
For $i\in\{1,2,3,4\}$ let $U_i:=\operatorname{span}(x^i,A(x^i),A^2(x^i))$. Then $U_i\cap U_j=0$ whenever $i\neq j$, and the $U_i$ together span $V$ and are invariant under $A$. This yields a decomposition
$$V=U_1\oplus U_2\oplus U_3\oplus U_4,$$
of $A$-invariant subspaces. The minimal polynomial of $A\vert_{U_i}$ is still $X^3-1$ (verify this!), hence it is the characteristic polynomial of the restriction. The characteristic polynomial of $A$ is the product of the characteristic polynomials of the $A\vert_{U_i}$, so it is $(X^3-1)^4$. | 12,343 |
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